Optimal. Leaf size=57 \[ -\frac {\sqrt {a^2-b^2} \tanh ^{-1}\left (\frac {\sinh (x) (a \coth (x)+b)}{\sqrt {a^2-b^2}}\right )}{b^2}+\frac {a \tanh ^{-1}(\cosh (x))}{b^2}-\frac {\text {csch}(x)}{b} \]
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Rubi [A] time = 0.11, antiderivative size = 57, normalized size of antiderivative = 1.00, 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 = {3510, 3486, 3770, 3509, 206} \[ -\frac {\sqrt {a^2-b^2} \tanh ^{-1}\left (\frac {\sinh (x) (a \coth (x)+b)}{\sqrt {a^2-b^2}}\right )}{b^2}+\frac {a \tanh ^{-1}(\cosh (x))}{b^2}-\frac {\text {csch}(x)}{b} \]
Antiderivative was successfully verified.
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Rule 206
Rule 3486
Rule 3509
Rule 3510
Rule 3770
Rubi steps
\begin {align*} \int \frac {\text {csch}^3(x)}{a+b \coth (x)} \, dx &=-\frac {\int (a-b \coth (x)) \text {csch}(x) \, dx}{b^2}+\frac {\left (a^2-b^2\right ) \int \frac {\text {csch}(x)}{a+b \coth (x)} \, dx}{b^2}\\ &=-\frac {\text {csch}(x)}{b}-\frac {a \int \text {csch}(x) \, dx}{b^2}-\frac {\left (a^2-b^2\right ) \operatorname {Subst}\left (\int \frac {1}{a^2-b^2-x^2} \, dx,x,i (-i b-i a \coth (x)) \sinh (x)\right )}{b^2}\\ &=\frac {a \tanh ^{-1}(\cosh (x))}{b^2}-\frac {\sqrt {a^2-b^2} \tanh ^{-1}\left (\frac {(b+a \coth (x)) \sinh (x)}{\sqrt {a^2-b^2}}\right )}{b^2}-\frac {\text {csch}(x)}{b}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 65, normalized size = 1.14 \[ -\frac {2 \sqrt {b-a} \sqrt {a+b} \tan ^{-1}\left (\frac {a+b \tanh \left (\frac {x}{2}\right )}{\sqrt {b-a} \sqrt {a+b}}\right )+a \log \left (\tanh \left (\frac {x}{2}\right )\right )+b \text {csch}(x)}{b^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 384, normalized size = 6.74 \[ \left [\frac {\sqrt {a^{2} - b^{2}} {\left (\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2} - 1\right )} \log \left (\frac {{\left (a + b\right )} \cosh \relax (x)^{2} + 2 \, {\left (a + b\right )} \cosh \relax (x) \sinh \relax (x) + {\left (a + b\right )} \sinh \relax (x)^{2} - 2 \, \sqrt {a^{2} - b^{2}} {\left (\cosh \relax (x) + \sinh \relax (x)\right )} + a - b}{{\left (a + b\right )} \cosh \relax (x)^{2} + 2 \, {\left (a + b\right )} \cosh \relax (x) \sinh \relax (x) + {\left (a + b\right )} \sinh \relax (x)^{2} - a + b}\right ) - 2 \, b \cosh \relax (x) + {\left (a \cosh \relax (x)^{2} + 2 \, a \cosh \relax (x) \sinh \relax (x) + a \sinh \relax (x)^{2} - a\right )} \log \left (\cosh \relax (x) + \sinh \relax (x) + 1\right ) - {\left (a \cosh \relax (x)^{2} + 2 \, a \cosh \relax (x) \sinh \relax (x) + a \sinh \relax (x)^{2} - a\right )} \log \left (\cosh \relax (x) + \sinh \relax (x) - 1\right ) - 2 \, b \sinh \relax (x)}{b^{2} \cosh \relax (x)^{2} + 2 \, b^{2} \cosh \relax (x) \sinh \relax (x) + b^{2} \sinh \relax (x)^{2} - b^{2}}, \frac {2 \, \sqrt {-a^{2} + b^{2}} {\left (\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2} - 1\right )} \arctan \left (\frac {\sqrt {-a^{2} + b^{2}}}{{\left (a + b\right )} \cosh \relax (x) + {\left (a + b\right )} \sinh \relax (x)}\right ) - 2 \, b \cosh \relax (x) + {\left (a \cosh \relax (x)^{2} + 2 \, a \cosh \relax (x) \sinh \relax (x) + a \sinh \relax (x)^{2} - a\right )} \log \left (\cosh \relax (x) + \sinh \relax (x) + 1\right ) - {\left (a \cosh \relax (x)^{2} + 2 \, a \cosh \relax (x) \sinh \relax (x) + a \sinh \relax (x)^{2} - a\right )} \log \left (\cosh \relax (x) + \sinh \relax (x) - 1\right ) - 2 \, b \sinh \relax (x)}{b^{2} \cosh \relax (x)^{2} + 2 \, b^{2} \cosh \relax (x) \sinh \relax (x) + b^{2} \sinh \relax (x)^{2} - b^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 85, normalized size = 1.49 \[ \frac {a \log \left (e^{x} + 1\right )}{b^{2}} - \frac {a \log \left ({\left | e^{x} - 1 \right |}\right )}{b^{2}} + \frac {2 \, {\left (a^{2} - b^{2}\right )} \arctan \left (\frac {a e^{x} + b e^{x}}{\sqrt {-a^{2} + b^{2}}}\right )}{\sqrt {-a^{2} + b^{2}} b^{2}} - \frac {2 \, e^{x}}{b {\left (e^{\left (2 \, x\right )} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.11, size = 115, normalized size = 2.02 \[ \frac {\tanh \left (\frac {x}{2}\right )}{2 b}+\frac {2 \arctan \left (\frac {2 \tanh \left (\frac {x}{2}\right ) b +2 a}{2 \sqrt {-a^{2}+b^{2}}}\right ) a^{2}}{b^{2} \sqrt {-a^{2}+b^{2}}}-\frac {2 \arctan \left (\frac {2 \tanh \left (\frac {x}{2}\right ) b +2 a}{2 \sqrt {-a^{2}+b^{2}}}\right )}{\sqrt {-a^{2}+b^{2}}}-\frac {1}{2 b \tanh \left (\frac {x}{2}\right )}-\frac {a \ln \left (\tanh \left (\frac {x}{2}\right )\right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.46, size = 230, normalized size = 4.04 \[ \frac {2\,{\mathrm {e}}^x}{b-b\,{\mathrm {e}}^{2\,x}}-\frac {a\,\ln \left (32\,a\,b^2-64\,a^2\,b+32\,a^3-32\,a^3\,{\mathrm {e}}^x-32\,a\,b^2\,{\mathrm {e}}^x+64\,a^2\,b\,{\mathrm {e}}^x\right )}{b^2}+\frac {a\,\ln \left (32\,a\,b^2-64\,a^2\,b+32\,a^3+32\,a^3\,{\mathrm {e}}^x+32\,a\,b^2\,{\mathrm {e}}^x-64\,a^2\,b\,{\mathrm {e}}^x\right )}{b^2}+\frac {\ln \left (32\,a\,\sqrt {a^2-b^2}-32\,b\,\sqrt {a^2-b^2}-32\,a^2\,{\mathrm {e}}^x+32\,b^2\,{\mathrm {e}}^x\right )\,\sqrt {a^2-b^2}}{b^2}-\frac {\ln \left (32\,a\,\sqrt {a^2-b^2}-32\,b\,\sqrt {a^2-b^2}+32\,a^2\,{\mathrm {e}}^x-32\,b^2\,{\mathrm {e}}^x\right )\,\sqrt {a^2-b^2}}{b^2} \]
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 \[ \int \frac {\operatorname {csch}^{3}{\relax (x )}}{a + b \coth {\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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