Optimal. Leaf size=35 \[ \frac{\left (1-\frac{a^2}{b^2}\right ) \log (a+b \text{sech}(x))}{a}+\frac{\log (\cosh (x))}{a}+\frac{\text{sech}(x)}{b} \]
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Rubi [A] time = 0.0750012, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {3885, 894} \[ \frac{\left (1-\frac{a^2}{b^2}\right ) \log (a+b \text{sech}(x))}{a}+\frac{\log (\cosh (x))}{a}+\frac{\text{sech}(x)}{b} \]
Antiderivative was successfully verified.
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Rule 3885
Rule 894
Rubi steps
\begin{align*} \int \frac{\tanh ^3(x)}{a+b \text{sech}(x)} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{b^2-x^2}{x (a+x)} \, dx,x,b \text{sech}(x)\right )}{b^2}\\ &=-\frac{\operatorname{Subst}\left (\int \left (-1+\frac{b^2}{a x}+\frac{a^2-b^2}{a (a+x)}\right ) \, dx,x,b \text{sech}(x)\right )}{b^2}\\ &=\frac{\log (\cosh (x))}{a}+\frac{\left (1-\frac{a^2}{b^2}\right ) \log (a+b \text{sech}(x))}{a}+\frac{\text{sech}(x)}{b}\\ \end{align*}
Mathematica [A] time = 0.0840565, size = 37, normalized size = 1.06 \[ \frac{\left (b^2-a^2\right ) \log (a \cosh (x)+b)+a^2 \log (\cosh (x))+a b \text{sech}(x)}{a b^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.03, size = 107, normalized size = 3.1 \begin{align*} -{\frac{1}{a}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) }-{\frac{1}{a}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) }+{\frac{a}{{b}^{2}}\ln \left ( \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}+1 \right ) }+2\,{\frac{1}{b \left ( \left ( \tanh \left ( x/2 \right ) \right ) ^{2}+1 \right ) }}-{\frac{a}{{b}^{2}}\ln \left ( a \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}- \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}b+a+b \right ) }+{\frac{1}{a}\ln \left ( a \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}- \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}b+a+b \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.56517, size = 90, normalized size = 2.57 \begin{align*} \frac{x}{a} + \frac{2 \, e^{\left (-x\right )}}{b e^{\left (-2 \, x\right )} + b} + \frac{a \log \left (e^{\left (-2 \, x\right )} + 1\right )}{b^{2}} - \frac{{\left (a^{2} - b^{2}\right )} \log \left (2 \, b e^{\left (-x\right )} + a e^{\left (-2 \, x\right )} + a\right )}{a b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.5721, size = 543, normalized size = 15.51 \begin{align*} -\frac{b^{2} x \cosh \left (x\right )^{2} + b^{2} x \sinh \left (x\right )^{2} + b^{2} x - 2 \, a b \cosh \left (x\right ) +{\left ({\left (a^{2} - b^{2}\right )} \cosh \left (x\right )^{2} + 2 \,{\left (a^{2} - b^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right ) +{\left (a^{2} - b^{2}\right )} \sinh \left (x\right )^{2} + a^{2} - b^{2}\right )} \log \left (\frac{2 \,{\left (a \cosh \left (x\right ) + b\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) -{\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 )} \log \left (\frac{2 \, \cosh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) + 2 \,{\left (b^{2} x \cosh \left (x\right ) - a b\right )} \sinh \left (x\right )}{a b^{2} \cosh \left (x\right )^{2} + 2 \, a b^{2} \cosh \left (x\right ) \sinh \left (x\right ) + a b^{2} \sinh \left (x\right )^{2} + a b^{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 ^{3}{\left (x \right )}}{a + b \operatorname{sech}{\left (x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.13766, size = 99, normalized size = 2.83 \begin{align*} \frac{a \log \left (e^{\left (-x\right )} + e^{x}\right )}{b^{2}} - \frac{{\left (a^{2} - b^{2}\right )} \log \left ({\left | a{\left (e^{\left (-x\right )} + e^{x}\right )} + 2 \, b \right |}\right )}{a b^{2}} - \frac{a{\left (e^{\left (-x\right )} + e^{x}\right )} - 2 \, b}{b^{2}{\left (e^{\left (-x\right )} + e^{x}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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