Optimal. Leaf size=32 \[ \left (\frac {a}{b^2}+\frac {1}{a}\right ) \log (a+b \text {csch}(x))+\frac {\log (\sinh (x))}{a}-\frac {\text {csch}(x)}{b} \]
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Rubi [A] time = 0.07, antiderivative size = 32, 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 = {3885, 894} \[ \left (\frac {a}{b^2}+\frac {1}{a}\right ) \log (a+b \text {csch}(x))+\frac {\log (\sinh (x))}{a}-\frac {\text {csch}(x)}{b} \]
Antiderivative was successfully verified.
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Rule 894
Rule 3885
Rubi steps
\begin {align*} \int \frac {\coth ^3(x)}{a+b \text {csch}(x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {-b^2-x^2}{x (a+x)} \, dx,x,b \text {csch}(x)\right )}{b^2}\\ &=\frac {\operatorname {Subst}\left (\int \left (-1-\frac {b^2}{a x}+\frac {a^2+b^2}{a (a+x)}\right ) \, dx,x,b \text {csch}(x)\right )}{b^2}\\ &=-\frac {\text {csch}(x)}{b}+\left (\frac {1}{a}+\frac {a}{b^2}\right ) \log (a+b \text {csch}(x))+\frac {\log (\sinh (x))}{a}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 37, normalized size = 1.16 \[ \frac {\left (a^2+b^2\right ) \log (a \sinh (x)+b)+a^2 (-\log (\sinh (x)))-a b \text {csch}(x)}{a b^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.61, size = 199, normalized size = 6.22 \[ -\frac {b^{2} x \cosh \relax (x)^{2} + b^{2} x \sinh \relax (x)^{2} - b^{2} x + 2 \, a b \cosh \relax (x) - {\left ({\left (a^{2} + b^{2}\right )} \cosh \relax (x)^{2} + 2 \, {\left (a^{2} + b^{2}\right )} \cosh \relax (x) \sinh \relax (x) + {\left (a^{2} + b^{2}\right )} \sinh \relax (x)^{2} - a^{2} - b^{2}\right )} \log \left (\frac {2 \, {\left (a \sinh \relax (x) + b\right )}}{\cosh \relax (x) - \sinh \relax (x)}\right ) + {\left (a^{2} \cosh \relax (x)^{2} + 2 \, a^{2} \cosh \relax (x) \sinh \relax (x) + a^{2} \sinh \relax (x)^{2} - a^{2}\right )} \log \left (\frac {2 \, \sinh \relax (x)}{\cosh \relax (x) - \sinh \relax (x)}\right ) + 2 \, {\left (b^{2} x \cosh \relax (x) + a b\right )} \sinh \relax (x)}{a b^{2} \cosh \relax (x)^{2} + 2 \, a b^{2} \cosh \relax (x) \sinh \relax (x) + a b^{2} \sinh \relax (x)^{2} - a b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.13, size = 80, normalized size = 2.50 \[ -\frac {a \log \left ({\left | -e^{\left (-x\right )} + e^{x} \right |}\right )}{b^{2}} + \frac {{\left (a^{2} + b^{2}\right )} \log \left ({\left | -a {\left (e^{\left (-x\right )} - e^{x}\right )} + 2 \, b \right |}\right )}{a b^{2}} + \frac {a {\left (e^{\left (-x\right )} - e^{x}\right )} + 2 \, b}{b^{2} {\left (e^{\left (-x\right )} - e^{x}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.14, size = 106, normalized size = 3.31 \[ \frac {\tanh \left (\frac {x}{2}\right )}{2 b}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{a}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{a}+\frac {a \ln \left (\left (\tanh ^{2}\left (\frac {x}{2}\right )\right ) b -2 a \tanh \left (\frac {x}{2}\right )-b \right )}{b^{2}}+\frac {\ln \left (\left (\tanh ^{2}\left (\frac {x}{2}\right )\right ) b -2 a \tanh \left (\frac {x}{2}\right )-b \right )}{a}-\frac {1}{2 b \tanh \left (\frac {x}{2}\right )}-\frac {a \ln \left (\tanh \left (\frac {x}{2}\right )\right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.32, size = 82, normalized size = 2.56 \[ \frac {x}{a} + \frac {2 \, e^{\left (-x\right )}}{b e^{\left (-2 \, x\right )} - b} - \frac {a \log \left (e^{\left (-x\right )} + 1\right )}{b^{2}} - \frac {a \log \left (e^{\left (-x\right )} - 1\right )}{b^{2}} + \frac {{\left (a^{2} + b^{2}\right )} \log \left (-2 \, b e^{\left (-x\right )} + a e^{\left (-2 \, x\right )} - a\right )}{a b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.84, size = 261, normalized size = 8.16 \[ \frac {2\,{\mathrm {e}}^x}{b-b\,{\mathrm {e}}^{2\,x}}-\frac {x}{a}+\frac {\ln \left (16\,a^5\,{\mathrm {e}}^{2\,x}-4\,a\,b^4-16\,a^5-16\,a^3\,b^2+8\,b^5\,{\mathrm {e}}^x+16\,a^3\,b^2\,{\mathrm {e}}^{2\,x}+32\,a^4\,b\,{\mathrm {e}}^x+4\,a\,b^4\,{\mathrm {e}}^{2\,x}+32\,a^2\,b^3\,{\mathrm {e}}^x\right )}{a}+\frac {a\,\ln \left (16\,a^5\,{\mathrm {e}}^{2\,x}-4\,a\,b^4-16\,a^5-16\,a^3\,b^2+8\,b^5\,{\mathrm {e}}^x+16\,a^3\,b^2\,{\mathrm {e}}^{2\,x}+32\,a^4\,b\,{\mathrm {e}}^x+4\,a\,b^4\,{\mathrm {e}}^{2\,x}+32\,a^2\,b^3\,{\mathrm {e}}^x\right )}{b^2}-\frac {a\,\ln \left (16\,a^6\,{\mathrm {e}}^{2\,x}+4\,b^6\,{\mathrm {e}}^{2\,x}-16\,a^6-4\,b^6-20\,a^2\,b^4-32\,a^4\,b^2+20\,a^2\,b^4\,{\mathrm {e}}^{2\,x}+32\,a^4\,b^2\,{\mathrm {e}}^{2\,x}\right )}{b^2} \]
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 \[ \int \frac {\coth ^{3}{\relax (x )}}{a + b \operatorname {csch}{\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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