Optimal. Leaf size=100 \[ \frac{b^2 x}{a \left (a^2+b^2\right )}+\frac{a x}{a^2+b^2}+\frac{2 b^3 \tanh ^{-1}\left (\frac{a-b \tanh \left (\frac{x}{2}\right )}{\sqrt{a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2}}-\frac{a \tanh (x)}{a^2+b^2}+\frac{b \text{sech}(x)}{a^2+b^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.220667, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.692, Rules used = {3898, 2902, 2606, 8, 3473, 2735, 2660, 618, 204} \[ \frac{b^2 x}{a \left (a^2+b^2\right )}+\frac{a x}{a^2+b^2}+\frac{2 b^3 \tanh ^{-1}\left (\frac{a-b \tanh \left (\frac{x}{2}\right )}{\sqrt{a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2}}-\frac{a \tanh (x)}{a^2+b^2}+\frac{b \text{sech}(x)}{a^2+b^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3898
Rule 2902
Rule 2606
Rule 8
Rule 3473
Rule 2735
Rule 2660
Rule 618
Rule 204
Rubi steps
\begin{align*} \int \frac{\tanh ^2(x)}{a+b \text{csch}(x)} \, dx &=i \int \frac{\sinh (x) \tanh ^2(x)}{i b+i a \sinh (x)} \, dx\\ &=\frac{a \int \tanh ^2(x) \, dx}{a^2+b^2}-\frac{b \int \text{sech}(x) \tanh (x) \, dx}{a^2+b^2}+\frac{\left (i b^2\right ) \int \frac{\sinh (x)}{i b+i a \sinh (x)} \, dx}{a^2+b^2}\\ &=\frac{b^2 x}{a \left (a^2+b^2\right )}-\frac{a \tanh (x)}{a^2+b^2}+\frac{a \int 1 \, dx}{a^2+b^2}+\frac{b \operatorname{Subst}(\int 1 \, dx,x,\text{sech}(x))}{a^2+b^2}-\frac{\left (i b^3\right ) \int \frac{1}{i b+i a \sinh (x)} \, dx}{a \left (a^2+b^2\right )}\\ &=\frac{a x}{a^2+b^2}+\frac{b^2 x}{a \left (a^2+b^2\right )}+\frac{b \text{sech}(x)}{a^2+b^2}-\frac{a \tanh (x)}{a^2+b^2}-\frac{\left (2 i b^3\right ) \operatorname{Subst}\left (\int \frac{1}{i b+2 i a x-i b x^2} \, dx,x,\tanh \left (\frac{x}{2}\right )\right )}{a \left (a^2+b^2\right )}\\ &=\frac{a x}{a^2+b^2}+\frac{b^2 x}{a \left (a^2+b^2\right )}+\frac{b \text{sech}(x)}{a^2+b^2}-\frac{a \tanh (x)}{a^2+b^2}+\frac{\left (4 i b^3\right ) \operatorname{Subst}\left (\int \frac{1}{-4 \left (a^2+b^2\right )-x^2} \, dx,x,2 i a-2 i b \tanh \left (\frac{x}{2}\right )\right )}{a \left (a^2+b^2\right )}\\ &=\frac{a x}{a^2+b^2}+\frac{b^2 x}{a \left (a^2+b^2\right )}+\frac{2 b^3 \tanh ^{-1}\left (\frac{a-b \tanh \left (\frac{x}{2}\right )}{\sqrt{a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2}}+\frac{b \text{sech}(x)}{a^2+b^2}-\frac{a \tanh (x)}{a^2+b^2}\\ \end{align*}
Mathematica [A] time = 0.318356, size = 82, normalized size = 0.82 \[ -\frac{a \tanh (x)}{a^2+b^2}+\frac{b \text{sech}(x)}{a^2+b^2}+\frac{\frac{2 b^3 \tan ^{-1}\left (\frac{a-b \tanh \left (\frac{x}{2}\right )}{\sqrt{-a^2-b^2}}\right )}{\left (-a^2-b^2\right )^{3/2}}+x}{a} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.04, size = 95, normalized size = 1. \begin{align*}{\frac{1}{a}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) }-2\,{\frac{{b}^{3}}{a \left ({a}^{2}+{b}^{2} \right ) ^{3/2}}{\it Artanh} \left ( 1/2\,{\frac{2\,\tanh \left ( x/2 \right ) b-2\,a}{\sqrt{{a}^{2}+{b}^{2}}}} \right ) }-{\frac{1}{a}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) }+2\,{\frac{-a\tanh \left ( x/2 \right ) +b}{ \left ({a}^{2}+{b}^{2} \right ) \left ( \left ( \tanh \left ( x/2 \right ) \right ) ^{2}+1 \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.65838, size = 899, normalized size = 8.99 \begin{align*} \frac{2 \, a^{4} + 2 \, a^{2} b^{2} +{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} x \cosh \left (x\right )^{2} +{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} x \sinh \left (x\right )^{2} +{\left (b^{3} \cosh \left (x\right )^{2} + 2 \, b^{3} \cosh \left (x\right ) \sinh \left (x\right ) + b^{3} \sinh \left (x\right )^{2} + b^{3}\right )} \sqrt{a^{2} + b^{2}} \log \left (\frac{a^{2} \cosh \left (x\right )^{2} + a^{2} \sinh \left (x\right )^{2} + 2 \, a b \cosh \left (x\right ) + a^{2} + 2 \, b^{2} + 2 \,{\left (a^{2} \cosh \left (x\right ) + a b\right )} \sinh \left (x\right ) + 2 \, \sqrt{a^{2} + b^{2}}{\left (a \cosh \left (x\right ) + a \sinh \left (x\right ) + b\right )}}{a \cosh \left (x\right )^{2} + a \sinh \left (x\right )^{2} + 2 \, b \cosh \left (x\right ) + 2 \,{\left (a \cosh \left (x\right ) + b\right )} \sinh \left (x\right ) - a}\right ) +{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} x + 2 \,{\left (a^{3} b + a b^{3}\right )} \cosh \left (x\right ) + 2 \,{\left (a^{3} b + a b^{3} +{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} x \cosh \left (x\right )\right )} \sinh \left (x\right )}{a^{5} + 2 \, a^{3} b^{2} + a b^{4} +{\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} \cosh \left (x\right )^{2} + 2 \,{\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} \cosh \left (x\right ) \sinh \left (x\right ) +{\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} \sinh \left (x\right )^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\tanh ^{2}{\left (x \right )}}{a + b \operatorname{csch}{\left (x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.21591, size = 138, normalized size = 1.38 \begin{align*} -\frac{b^{3} \log \left (\frac{{\left | 2 \, a e^{x} + 2 \, b - 2 \, \sqrt{a^{2} + b^{2}} \right |}}{{\left | 2 \, a e^{x} + 2 \, b + 2 \, \sqrt{a^{2} + b^{2}} \right |}}\right )}{{\left (a^{3} + a b^{2}\right )} \sqrt{a^{2} + b^{2}}} + \frac{x}{a} + \frac{2 \,{\left (b e^{x} + a\right )}}{{\left (a^{2} + b^{2}\right )}{\left (e^{\left (2 \, x\right )} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]