Optimal. Leaf size=69 \[ -\frac{2 a^2 \tanh ^{-1}\left (\frac{b-a \tanh \left (\frac{x}{2}\right )}{\sqrt{a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2}}-\frac{a \tanh (x)}{a^2+b^2}-\frac{b \text{sech}(x)}{a^2+b^2} \]
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Rubi [A] time = 0.0914194, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.538, Rules used = {2727, 3767, 8, 2606, 2660, 618, 206} \[ -\frac{2 a^2 \tanh ^{-1}\left (\frac{b-a \tanh \left (\frac{x}{2}\right )}{\sqrt{a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2}}-\frac{a \tanh (x)}{a^2+b^2}-\frac{b \text{sech}(x)}{a^2+b^2} \]
Antiderivative was successfully verified.
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Rule 2727
Rule 3767
Rule 8
Rule 2606
Rule 2660
Rule 618
Rule 206
Rubi steps
\begin{align*} \int \frac{\tanh ^2(x)}{a+b \sinh (x)} \, dx &=-\frac{a \int \text{sech}^2(x) \, dx}{a^2+b^2}+\frac{a^2 \int \frac{1}{a+b \sinh (x)} \, dx}{a^2+b^2}+\frac{b \int \text{sech}(x) \tanh (x) \, dx}{a^2+b^2}\\ &=-\frac{(i a) \operatorname{Subst}(\int 1 \, dx,x,-i \tanh (x))}{a^2+b^2}+\frac{\left (2 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{a+2 b x-a x^2} \, dx,x,\tanh \left (\frac{x}{2}\right )\right )}{a^2+b^2}-\frac{b \operatorname{Subst}(\int 1 \, dx,x,\text{sech}(x))}{a^2+b^2}\\ &=-\frac{b \text{sech}(x)}{a^2+b^2}-\frac{a \tanh (x)}{a^2+b^2}-\frac{\left (4 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{4 \left (a^2+b^2\right )-x^2} \, dx,x,2 b-2 a \tanh \left (\frac{x}{2}\right )\right )}{a^2+b^2}\\ &=-\frac{2 a^2 \tanh ^{-1}\left (\frac{b-a \tanh \left (\frac{x}{2}\right )}{\sqrt{a^2+b^2}}\right )}{\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 = 0.188303, size = 69, normalized size = 1. \[ \frac{a \left (\frac{2 a \tan ^{-1}\left (\frac{b-a \tanh \left (\frac{x}{2}\right )}{\sqrt{-a^2-b^2}}\right )}{\sqrt{-a^2-b^2}}-\tanh (x)\right )-b \text{sech}(x)}{a^2+b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.029, size = 84, normalized size = 1.2 \begin{align*} 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 ) }}+8\,{\frac{{a}^{2}}{ \left ( 4\,{a}^{2}+4\,{b}^{2} \right ) \sqrt{{a}^{2}+{b}^{2}}}{\it Artanh} \left ( 1/2\,{\frac{2\,a\tanh \left ( x/2 \right ) -2\,b}{\sqrt{{a}^{2}+{b}^{2}}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Fricas [B] time = 2.05387, size = 691, normalized size = 10.01 \begin{align*} \frac{2 \, a^{3} + 2 \, a b^{2} +{\left (a^{2} \cosh \left (x\right )^{2} + 2 \, a^{2} \cosh \left (x\right ) \sinh \left (x\right ) + a^{2} \sinh \left (x\right )^{2} + a^{2}\right )} \sqrt{a^{2} + b^{2}} \log \left (\frac{b^{2} \cosh \left (x\right )^{2} + b^{2} \sinh \left (x\right )^{2} + 2 \, a b \cosh \left (x\right ) + 2 \, a^{2} + b^{2} + 2 \,{\left (b^{2} \cosh \left (x\right ) + a b\right )} \sinh \left (x\right ) - 2 \, \sqrt{a^{2} + b^{2}}{\left (b \cosh \left (x\right ) + b \sinh \left (x\right ) + a\right )}}{b \cosh \left (x\right )^{2} + b \sinh \left (x\right )^{2} + 2 \, a \cosh \left (x\right ) + 2 \,{\left (b \cosh \left (x\right ) + a\right )} \sinh \left (x\right ) - b}\right ) - 2 \,{\left (a^{2} b + b^{3}\right )} \cosh \left (x\right ) - 2 \,{\left (a^{2} b + b^{3}\right )} \sinh \left (x\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4} +{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \cosh \left (x\right )^{2} + 2 \,{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \cosh \left (x\right ) \sinh \left (x\right ) +{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \sinh \left (x\right )^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\tanh ^{2}{\left (x \right )}}{a + b \sinh{\left (x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19777, size = 117, normalized size = 1.7 \begin{align*} \frac{a^{2} \log \left (\frac{{\left | 2 \, b e^{x} + 2 \, a - 2 \, \sqrt{a^{2} + b^{2}} \right |}}{{\left | 2 \, b e^{x} + 2 \, a + 2 \, \sqrt{a^{2} + b^{2}} \right |}}\right )}{{\left (a^{2} + b^{2}\right )}^{\frac{3}{2}}} - \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.
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