Optimal. Leaf size=57 \[ \frac {a \tanh ^{-1}(\cosh (x))}{b^2}-\frac {\sqrt {a^2-b^2} \tanh ^{-1}\left (\frac {(b+a \coth (x)) \sinh (x)}{\sqrt {a^2-b^2}}\right )}{b^2}-\frac {\text {csch}(x)}{b} \]
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Rubi [A]
time = 0.08, 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 = {3591, 3567,
3855, 3590, 212} \begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 3567
Rule 3590
Rule 3591
Rule 3855
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 ) \text {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 = 0.08, size = 65, normalized size = 1.14 \begin {gather*} -\frac {2 \sqrt {-a+b} \sqrt {a+b} \text {ArcTan}\left (\frac {a+b \tanh \left (\frac {x}{2}\right )}{\sqrt {-a+b} \sqrt {a+b}}\right )+b \text {csch}(x)+a \log \left (\tanh \left (\frac {x}{2}\right )\right )}{b^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.77, size = 85, normalized size = 1.49
method | result | size |
default | \(\frac {\tanh \left (\frac {x}{2}\right )}{2 b}-\frac {1}{2 b \tanh \left (\frac {x}{2}\right )}-\frac {a \ln \left (\tanh \left (\frac {x}{2}\right )\right )}{b^{2}}+\frac {\left (4 a^{2}-4 b^{2}\right ) \arctan \left (\frac {2 \tanh \left (\frac {x}{2}\right ) b +2 a}{2 \sqrt {-a^{2}+b^{2}}}\right )}{2 b^{2} \sqrt {-a^{2}+b^{2}}}\) | \(85\) |
risch | \(-\frac {2 \,{\mathrm e}^{x}}{b \left ({\mathrm e}^{2 x}-1\right )}+\frac {a \ln \left ({\mathrm e}^{x}+1\right )}{b^{2}}-\frac {a \ln \left ({\mathrm e}^{x}-1\right )}{b^{2}}+\frac {\sqrt {a^{2}-b^{2}}\, \ln \left ({\mathrm e}^{x}-\frac {\sqrt {a^{2}-b^{2}}}{a +b}\right )}{b^{2}}-\frac {\sqrt {a^{2}-b^{2}}\, \ln \left ({\mathrm e}^{x}+\frac {\sqrt {a^{2}-b^{2}}}{a +b}\right )}{b^{2}}\) | \(112\) |
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 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 165 vs.
\(2 (53) = 106\).
time = 0.41, size = 384, normalized size = 6.74 \begin {gather*} \left [\frac {\sqrt {a^{2} - b^{2}} {\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 1\right )} \log \left (\frac {{\left (a + b\right )} \cosh \left (x\right )^{2} + 2 \, {\left (a + b\right )} \cosh \left (x\right ) \sinh \left (x\right ) + {\left (a + b\right )} \sinh \left (x\right )^{2} - 2 \, \sqrt {a^{2} - b^{2}} {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} + a - b}{{\left (a + b\right )} \cosh \left (x\right )^{2} + 2 \, {\left (a + b\right )} \cosh \left (x\right ) \sinh \left (x\right ) + {\left (a + b\right )} \sinh \left (x\right )^{2} - a + b}\right ) - 2 \, b \cosh \left (x\right ) + {\left (a \cosh \left (x\right )^{2} + 2 \, a \cosh \left (x\right ) \sinh \left (x\right ) + a \sinh \left (x\right )^{2} - a\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) - {\left (a \cosh \left (x\right )^{2} + 2 \, a \cosh \left (x\right ) \sinh \left (x\right ) + a \sinh \left (x\right )^{2} - a\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\right ) - 2 \, b \sinh \left (x\right )}{b^{2} \cosh \left (x\right )^{2} + 2 \, b^{2} \cosh \left (x\right ) \sinh \left (x\right ) + b^{2} \sinh \left (x\right )^{2} - b^{2}}, \frac {2 \, \sqrt {-a^{2} + b^{2}} {\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 1\right )} \arctan \left (\frac {\sqrt {-a^{2} + b^{2}}}{{\left (a + b\right )} \cosh \left (x\right ) + {\left (a + b\right )} \sinh \left (x\right )}\right ) - 2 \, b \cosh \left (x\right ) + {\left (a \cosh \left (x\right )^{2} + 2 \, a \cosh \left (x\right ) \sinh \left (x\right ) + a \sinh \left (x\right )^{2} - a\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) - {\left (a \cosh \left (x\right )^{2} + 2 \, a \cosh \left (x\right ) \sinh \left (x\right ) + a \sinh \left (x\right )^{2} - a\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\right ) - 2 \, b \sinh \left (x\right )}{b^{2} \cosh \left (x\right )^{2} + 2 \, b^{2} \cosh \left (x\right ) \sinh \left (x\right ) + b^{2} \sinh \left (x\right )^{2} - b^{2}}\right ] \end {gather*}
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 \begin {gather*} \int \frac {\operatorname {csch}^{3}{\left (x \right )}}{a + b \coth {\left (x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 85, normalized size = 1.49 \begin {gather*} \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 )}} \end {gather*}
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 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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