Optimal. Leaf size=40 \[ -\frac {\left (a^2-b^2\right ) \log (a+b \coth (x))}{b^3}+\frac {a \coth (x)}{b^2}-\frac {\coth ^2(x)}{2 b} \]
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Rubi [A] time = 0.07, antiderivative size = 40, normalized size of antiderivative = 1.00, 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 = {3506, 697} \[ -\frac {\left (a^2-b^2\right ) \log (a+b \coth (x))}{b^3}+\frac {a \coth (x)}{b^2}-\frac {\coth ^2(x)}{2 b} \]
Antiderivative was successfully verified.
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Rule 697
Rule 3506
Rubi steps
\begin {align*} \int \frac {\text {csch}^4(x)}{a+b \coth (x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1-\frac {x^2}{b^2}}{a+x} \, dx,x,b \coth (x)\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {a}{b^2}-\frac {x}{b^2}+\frac {-a^2+b^2}{b^2 (a+x)}\right ) \, dx,x,b \coth (x)\right )}{b}\\ &=\frac {a \coth (x)}{b^2}-\frac {\coth ^2(x)}{2 b}-\frac {\left (a^2-b^2\right ) \log (a+b \coth (x))}{b^3}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 50, normalized size = 1.25 \[ \frac {2 \left (a^2-b^2\right ) (\log (\sinh (x))-\log (a \sinh (x)+b \cosh (x)))+2 a b \coth (x)-b^2 \text {csch}^2(x)}{2 b^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.42, size = 434, normalized size = 10.85 \[ \frac {2 \, {\left (a b - b^{2}\right )} \cosh \relax (x)^{2} + 4 \, {\left (a b - b^{2}\right )} \cosh \relax (x) \sinh \relax (x) + 2 \, {\left (a b - b^{2}\right )} \sinh \relax (x)^{2} - 2 \, a b - {\left ({\left (a^{2} - b^{2}\right )} \cosh \relax (x)^{4} + 4 \, {\left (a^{2} - b^{2}\right )} \cosh \relax (x) \sinh \relax (x)^{3} + {\left (a^{2} - b^{2}\right )} \sinh \relax (x)^{4} - 2 \, {\left (a^{2} - b^{2}\right )} \cosh \relax (x)^{2} + 2 \, {\left (3 \, {\left (a^{2} - b^{2}\right )} \cosh \relax (x)^{2} - a^{2} + b^{2}\right )} \sinh \relax (x)^{2} + a^{2} - b^{2} + 4 \, {\left ({\left (a^{2} - b^{2}\right )} \cosh \relax (x)^{3} - {\left (a^{2} - b^{2}\right )} \cosh \relax (x)\right )} \sinh \relax (x)\right )} \log \left (\frac {2 \, {\left (b \cosh \relax (x) + a \sinh \relax (x)\right )}}{\cosh \relax (x) - \sinh \relax (x)}\right ) + {\left ({\left (a^{2} - b^{2}\right )} \cosh \relax (x)^{4} + 4 \, {\left (a^{2} - b^{2}\right )} \cosh \relax (x) \sinh \relax (x)^{3} + {\left (a^{2} - b^{2}\right )} \sinh \relax (x)^{4} - 2 \, {\left (a^{2} - b^{2}\right )} \cosh \relax (x)^{2} + 2 \, {\left (3 \, {\left (a^{2} - b^{2}\right )} \cosh \relax (x)^{2} - a^{2} + b^{2}\right )} \sinh \relax (x)^{2} + a^{2} - b^{2} + 4 \, {\left ({\left (a^{2} - b^{2}\right )} \cosh \relax (x)^{3} - {\left (a^{2} - b^{2}\right )} \cosh \relax (x)\right )} \sinh \relax (x)\right )} \log \left (\frac {2 \, \sinh \relax (x)}{\cosh \relax (x) - \sinh \relax (x)}\right )}{b^{3} \cosh \relax (x)^{4} + 4 \, b^{3} \cosh \relax (x) \sinh \relax (x)^{3} + b^{3} \sinh \relax (x)^{4} - 2 \, b^{3} \cosh \relax (x)^{2} + b^{3} + 2 \, {\left (3 \, b^{3} \cosh \relax (x)^{2} - b^{3}\right )} \sinh \relax (x)^{2} + 4 \, {\left (b^{3} \cosh \relax (x)^{3} - b^{3} \cosh \relax (x)\right )} \sinh \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.12, size = 106, normalized size = 2.65 \[ -\frac {{\left (a^{3} + a^{2} b - a b^{2} - b^{3}\right )} \log \left ({\left | a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} - a + b \right |}\right )}{a b^{3} + b^{4}} + \frac {{\left (a^{2} - b^{2}\right )} \log \left ({\left | e^{\left (2 \, x\right )} - 1 \right |}\right )}{b^{3}} - \frac {2 \, {\left (a b - {\left (a b - b^{2}\right )} e^{\left (2 \, x\right )}\right )}}{b^{3} {\left (e^{\left (2 \, x\right )} - 1\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.12, size = 116, normalized size = 2.90 \[ -\frac {\tanh ^{2}\left (\frac {x}{2}\right )}{8 b}+\frac {a \tanh \left (\frac {x}{2}\right )}{2 b^{2}}-\frac {\ln \left (\left (\tanh ^{2}\left (\frac {x}{2}\right )\right ) b +2 a \tanh \left (\frac {x}{2}\right )+b \right ) a^{2}}{b^{3}}+\frac {\ln \left (\left (\tanh ^{2}\left (\frac {x}{2}\right )\right ) b +2 a \tanh \left (\frac {x}{2}\right )+b \right )}{b}-\frac {1}{8 b \tanh \left (\frac {x}{2}\right )^{2}}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )\right ) a^{2}}{b^{3}}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )\right )}{b}+\frac {a}{2 b^{2} \tanh \left (\frac {x}{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.33, size = 110, normalized size = 2.75 \[ \frac {2 \, {\left ({\left (a + b\right )} e^{\left (-2 \, x\right )} - a\right )}}{2 \, b^{2} e^{\left (-2 \, x\right )} - b^{2} e^{\left (-4 \, x\right )} - b^{2}} - \frac {{\left (a^{2} - b^{2}\right )} \log \left (-{\left (a - b\right )} e^{\left (-2 \, x\right )} + a + b\right )}{b^{3}} + \frac {{\left (a^{2} - b^{2}\right )} \log \left (e^{\left (-x\right )} + 1\right )}{b^{3}} + \frac {{\left (a^{2} - b^{2}\right )} \log \left (e^{\left (-x\right )} - 1\right )}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.44, size = 88, normalized size = 2.20 \[ \frac {2\,\left (a-b\right )}{b^2\,\left ({\mathrm {e}}^{2\,x}-1\right )}-\frac {2}{b\,\left ({\mathrm {e}}^{4\,x}-2\,{\mathrm {e}}^{2\,x}+1\right )}-\frac {\ln \left (b-a+a\,{\mathrm {e}}^{2\,x}+b\,{\mathrm {e}}^{2\,x}\right )\,\left (a+b\right )\,\left (a-b\right )}{b^3}+\frac {\ln \left ({\mathrm {e}}^{2\,x}-1\right )\,\left (a+b\right )\,\left (a-b\right )}{b^3} \]
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 \[ \int \frac {\operatorname {csch}^{4}{\relax (x )}}{a + b \coth {\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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