Indefinite Integrals with hyperbolic tangent (th x)
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| $ \int th ax dx=\dfrac{\ln ch ax}{a} +C $ |
| $ \int th^{2} ax dx=x-\dfrac{th ax}{a} +C $ |
| $ \int th^{3} ax dx=\dfrac{1}{a}\dfrac{\ln ch ax}{a}-\dfrac{th^{2} ax}{2a} +C $ |
| $ \int\dfrac{th^{n} ax}{ch^{2} ax} dx=\dfrac{th^{n+1} ax}{(n+1)a} +C $ |
| $ \int\dfrac{dx}{th ax ch^{2} ax} dx=\dfrac{1}{a}\ln th ax +C $ |
| $ \int\dfrac{dx}{th ax} dx=\dfrac{1}{a}\ln sh ax +C $ |
| $ \int x th ax dx=\dfrac{1}{a^{2}}\biggl\{\dfrac{(ax)^{3}}{3}-\dfrac{(ax)^{5}}{15}+\dfrac{2(ax)^{7}}{105}\cdots+\dfrac{(-1)^{n-1}2^{2n}(2^{2n}-1)B_{n}(ax)^{2n+1}}{(2n+1)|}\biggl\} +C $ |
| $ \int x th^{2} ax dx=\dfrac{x^{2}}{2}-\dfrac{x th ax}{a}+\dfrac{1}{a^{2}}\ln ch ax +C $ |
| $ \int\dfrac{th ax}{x} dx=\biggl\{ ax-\dfrac{(ax)^{3}}{9}+\dfrac{2(ax)^{5}}{75}-\cdots+\dfrac{(-1)^{n-1}2^{2n}(2^{2n}-1)B_{n}(ax)^{2n-1}}{(2n-1)(2n)!}\biggl\} +C $ |
| $ \int\dfrac{dx}{p+q th ax}=\dfrac{px}{p^{2}-q^{2}}-\dfrac{q}{a(p^{2}-q^{2})}\ln(q sh ax+p ch ax) +C $ |
| $ \int th^{n} ax dx=-\dfrac{th^{n+1} ax}{a(n-1)}+ \int th^{n-2} ax dx $ |
