Optimal. Leaf size=60 \[ \frac {a x}{a^2-b^2}-\frac {b^3 \log (a \sinh (x)+b \cosh (x))}{a^2 \left (a^2-b^2\right )}-\frac {b \log (\cosh (x))}{a^2}-\frac {\tanh (x)}{a} \]
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Rubi [A] time = 0.19, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {3569, 3651, 3530, 3475} \[ \frac {a x}{a^2-b^2}-\frac {b^3 \log (a \sinh (x)+b \cosh (x))}{a^2 \left (a^2-b^2\right )}-\frac {b \log (\cosh (x))}{a^2}-\frac {\tanh (x)}{a} \]
Antiderivative was successfully verified.
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Rule 3475
Rule 3530
Rule 3569
Rule 3651
Rubi steps
\begin {align*} \int \frac {\tanh ^2(x)}{a+b \coth (x)} \, dx &=-\frac {\tanh (x)}{a}-\frac {i \int \frac {\left (-i b+i a \coth (x)+i b \coth ^2(x)\right ) \tanh (x)}{a+b \coth (x)} \, dx}{a}\\ &=\frac {a x}{a^2-b^2}-\frac {\tanh (x)}{a}-\frac {b \int \tanh (x) \, dx}{a^2}-\frac {\left (i b^3\right ) \int \frac {-i b-i a \coth (x)}{a+b \coth (x)} \, dx}{a^2 \left (a^2-b^2\right )}\\ &=\frac {a x}{a^2-b^2}-\frac {b \log (\cosh (x))}{a^2}-\frac {b^3 \log (b \cosh (x)+a \sinh (x))}{a^2 \left (a^2-b^2\right )}-\frac {\tanh (x)}{a}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 64, normalized size = 1.07 \[ \frac {\left (a b^2-a^3\right ) \tanh (x)+a^3 x+\left (b^3-a^2 b\right ) \log (\cosh (x))-b^3 \log (a \sinh (x)+b \cosh (x))}{a^4-a^2 b^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.44, size = 264, normalized size = 4.40 \[ \frac {{\left (a^{3} + a^{2} b\right )} x \cosh \relax (x)^{2} + 2 \, {\left (a^{3} + a^{2} b\right )} x \cosh \relax (x) \sinh \relax (x) + {\left (a^{3} + a^{2} b\right )} x \sinh \relax (x)^{2} + 2 \, a^{3} - 2 \, a b^{2} + {\left (a^{3} + a^{2} b\right )} x - {\left (b^{3} \cosh \relax (x)^{2} + 2 \, b^{3} \cosh \relax (x) \sinh \relax (x) + b^{3} \sinh \relax (x)^{2} + b^{3}\right )} \log \left (\frac {2 \, {\left (b \cosh \relax (x) + a \sinh \relax (x)\right )}}{\cosh \relax (x) - \sinh \relax (x)}\right ) - {\left (a^{2} b - b^{3} + {\left (a^{2} b - b^{3}\right )} \cosh \relax (x)^{2} + 2 \, {\left (a^{2} b - b^{3}\right )} \cosh \relax (x) \sinh \relax (x) + {\left (a^{2} b - b^{3}\right )} \sinh \relax (x)^{2}\right )} \log \left (\frac {2 \, \cosh \relax (x)}{\cosh \relax (x) - \sinh \relax (x)}\right )}{a^{4} - a^{2} b^{2} + {\left (a^{4} - a^{2} b^{2}\right )} \cosh \relax (x)^{2} + 2 \, {\left (a^{4} - a^{2} b^{2}\right )} \cosh \relax (x) \sinh \relax (x) + {\left (a^{4} - a^{2} b^{2}\right )} \sinh \relax (x)^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 74, normalized size = 1.23 \[ -\frac {b^{3} \log \left ({\left | a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} - a + b \right |}\right )}{a^{4} - a^{2} b^{2}} + \frac {x}{a - b} - \frac {b \log \left (e^{\left (2 \, x\right )} + 1\right )}{a^{2}} + \frac {2}{a {\left (e^{\left (2 \, x\right )} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 110, normalized size = 1.83 \[ -\frac {b^{3} \ln \left (\left (\tanh ^{2}\left (\frac {x}{2}\right )\right ) b +2 a \tanh \left (\frac {x}{2}\right )+b \right )}{\left (a +b \right ) \left (a -b \right ) a^{2}}-\frac {16 \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{16 a +16 b}+\frac {16 \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{16 a -16 b}-\frac {2 \tanh \left (\frac {x}{2}\right )}{a \left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right )}-\frac {b \ln \left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right )}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 67, normalized size = 1.12 \[ -\frac {b^{3} \log \left (-{\left (a - b\right )} e^{\left (-2 \, x\right )} + a + b\right )}{a^{4} - a^{2} b^{2}} + \frac {x}{a + b} - \frac {b \log \left (e^{\left (-2 \, x\right )} + 1\right )}{a^{2}} - \frac {2}{a e^{\left (-2 \, x\right )} + a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.45, size = 73, normalized size = 1.22 \[ \frac {2}{a\,\left ({\mathrm {e}}^{2\,x}+1\right )}+\frac {x}{a-b}-\frac {b^3\,\ln \left (b-a+a\,{\mathrm {e}}^{2\,x}+b\,{\mathrm {e}}^{2\,x}\right )}{a^4-a^2\,b^2}-\frac {b\,\ln \left ({\mathrm {e}}^{2\,x}+1\right )}{a^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\tanh ^{2}{\relax (x )}}{a + b \coth {\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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