Optimal. Leaf size=76 \[ -\frac{b x}{a^2-b^2}+\frac{\left (a^2+b^2\right ) \log (\cosh (x))}{a^3}+\frac{b^4 \log (a \sinh (x)+b \cosh (x))}{a^3 \left (a^2-b^2\right )}+\frac{b \tanh (x)}{a^2}-\frac{\tanh ^2(x)}{2 a} \]
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Rubi [A] time = 0.325755, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385, Rules used = {3569, 3649, 3652, 3530, 3475} \[ -\frac{b x}{a^2-b^2}+\frac{\left (a^2+b^2\right ) \log (\cosh (x))}{a^3}+\frac{b^4 \log (a \sinh (x)+b \cosh (x))}{a^3 \left (a^2-b^2\right )}+\frac{b \tanh (x)}{a^2}-\frac{\tanh ^2(x)}{2 a} \]
Antiderivative was successfully verified.
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Rule 3569
Rule 3649
Rule 3652
Rule 3530
Rule 3475
Rubi steps
\begin{align*} \int \frac{\tanh ^3(x)}{a+b \coth (x)} \, dx &=-\frac{\tanh ^2(x)}{2 a}-\frac{i \int \frac{\left (-2 i b+2 i a \coth (x)+2 i b \coth ^2(x)\right ) \tanh ^2(x)}{a+b \coth (x)} \, dx}{2 a}\\ &=\frac{b \tanh (x)}{a^2}-\frac{\tanh ^2(x)}{2 a}-\frac{\int \frac{\left (-2 \left (a^2+b^2\right )+2 b^2 \coth ^2(x)\right ) \tanh (x)}{a+b \coth (x)} \, dx}{2 a^2}\\ &=-\frac{b x}{a^2-b^2}+\frac{b \tanh (x)}{a^2}-\frac{\tanh ^2(x)}{2 a}+\frac{\left (i b^4\right ) \int \frac{-i b-i a \coth (x)}{a+b \coth (x)} \, dx}{a^3 \left (a^2-b^2\right )}+\frac{\left (a^2+b^2\right ) \int \tanh (x) \, dx}{a^3}\\ &=-\frac{b x}{a^2-b^2}+\frac{\left (a^2+b^2\right ) \log (\cosh (x))}{a^3}+\frac{b^4 \log (b \cosh (x)+a \sinh (x))}{a^3 \left (a^2-b^2\right )}+\frac{b \tanh (x)}{a^2}-\frac{\tanh ^2(x)}{2 a}\\ \end{align*}
Mathematica [A] time = 0.303121, size = 88, normalized size = 1.16 \[ \frac{a^2 \left (a^2-b^2\right ) \text{sech}^2(x)+2 \left (a b \left (a^2-b^2\right ) \tanh (x)+\left (a^4-b^4\right ) \log (\cosh (x))-a^3 b x+b^4 \log (a \sinh (x)+b \cosh (x))\right )}{2 a^3 (a-b) (a+b)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.053, size = 167, normalized size = 2.2 \begin{align*} -32\,{\frac{\ln \left ( \tanh \left ( x/2 \right ) +1 \right ) }{32\,a-32\,b}}-32\,{\frac{\ln \left ( \tanh \left ( x/2 \right ) -1 \right ) }{32\,a+32\,b}}+2\,{\frac{ \left ( \tanh \left ( x/2 \right ) \right ) ^{3}b}{{a}^{2} \left ( \left ( \tanh \left ( x/2 \right ) \right ) ^{2}+1 \right ) ^{2}}}-2\,{\frac{ \left ( \tanh \left ( x/2 \right ) \right ) ^{2}}{a \left ( \left ( \tanh \left ( x/2 \right ) \right ) ^{2}+1 \right ) ^{2}}}+2\,{\frac{\tanh \left ( x/2 \right ) b}{{a}^{2} \left ( \left ( \tanh \left ( x/2 \right ) \right ) ^{2}+1 \right ) ^{2}}}+{\frac{1}{a}\ln \left ( \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}+1 \right ) }+{\frac{{b}^{2}}{{a}^{3}}\ln \left ( \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}+1 \right ) }+{\frac{{b}^{4}}{ \left ( a+b \right ) \left ( a-b \right ){a}^{3}}\ln \left ( \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}b+2\,a\tanh \left ( x/2 \right ) +b \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.73305, size = 127, normalized size = 1.67 \begin{align*} \frac{b^{4} \log \left (-{\left (a - b\right )} e^{\left (-2 \, x\right )} + a + b\right )}{a^{5} - a^{3} b^{2}} + \frac{2 \,{\left ({\left (a + b\right )} e^{\left (-2 \, x\right )} + b\right )}}{2 \, a^{2} e^{\left (-2 \, x\right )} + a^{2} e^{\left (-4 \, x\right )} + a^{2}} + \frac{x}{a + b} + \frac{{\left (a^{2} + b^{2}\right )} \log \left (e^{\left (-2 \, x\right )} + 1\right )}{a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 3.24519, size = 1539, normalized size = 20.25 \begin{align*} \text{result too large to display} \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 \coth{\left (x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17562, size = 131, normalized size = 1.72 \begin{align*} \frac{b^{4} \log \left ({\left | a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} - a + b \right |}\right )}{a^{5} - a^{3} b^{2}} - \frac{x}{a - b} + \frac{{\left (a^{2} + b^{2}\right )} \log \left (e^{\left (2 \, x\right )} + 1\right )}{a^{3}} - \frac{2 \,{\left (a b -{\left (a^{2} - a b\right )} e^{\left (2 \, x\right )}\right )}}{a^{3}{\left (e^{\left (2 \, x\right )} + 1\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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