Optimal. Leaf size=57 \[ \frac {2 \sqrt {a^2+b^2} \tanh ^{-1}\left (\frac {a-b \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{a b}+\frac {x}{a}-\frac {\tanh ^{-1}(\cosh (x))}{b} \]
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Rubi [A] time = 0.18, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.615, Rules used = {3894, 4051, 3770, 3919, 3831, 2660, 618, 206} \[ \frac {2 \sqrt {a^2+b^2} \tanh ^{-1}\left (\frac {a-b \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{a b}+\frac {x}{a}-\frac {\tanh ^{-1}(\cosh (x))}{b} \]
Antiderivative was successfully verified.
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Rule 206
Rule 618
Rule 2660
Rule 3770
Rule 3831
Rule 3894
Rule 3919
Rule 4051
Rubi steps
\begin {align*} \int \frac {\coth ^2(x)}{a+b \text {csch}(x)} \, dx &=-\int \frac {-1-\text {csch}^2(x)}{a+b \text {csch}(x)} \, dx\\ &=\frac {i \int \frac {-i b+i a \text {csch}(x)}{a+b \text {csch}(x)} \, dx}{b}+\frac {\int \text {csch}(x) \, dx}{b}\\ &=\frac {x}{a}-\frac {\tanh ^{-1}(\cosh (x))}{b}-\frac {\left (a^2+b^2\right ) \int \frac {\text {csch}(x)}{a+b \text {csch}(x)} \, dx}{a b}\\ &=\frac {x}{a}-\frac {\tanh ^{-1}(\cosh (x))}{b}-\left (\frac {1}{a}+\frac {a}{b^2}\right ) \int \frac {1}{1+\frac {a \sinh (x)}{b}} \, dx\\ &=\frac {x}{a}-\frac {\tanh ^{-1}(\cosh (x))}{b}-\left (2 \left (\frac {1}{a}+\frac {a}{b^2}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1+\frac {2 a x}{b}-x^2} \, dx,x,\tanh \left (\frac {x}{2}\right )\right )\\ &=\frac {x}{a}-\frac {\tanh ^{-1}(\cosh (x))}{b}+\left (4 \left (\frac {1}{a}+\frac {a}{b^2}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{4 \left (1+\frac {a^2}{b^2}\right )-x^2} \, dx,x,\frac {2 a}{b}-2 \tanh \left (\frac {x}{2}\right )\right )\\ &=\frac {x}{a}-\frac {\tanh ^{-1}(\cosh (x))}{b}+\frac {2 \sqrt {a^2+b^2} \tanh ^{-1}\left (\frac {b \left (\frac {a}{b}-\tanh \left (\frac {x}{2}\right )\right )}{\sqrt {a^2+b^2}}\right )}{a b}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 65, normalized size = 1.14 \[ \frac {2 \sqrt {-a^2-b^2} \tan ^{-1}\left (\frac {a-b \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2-b^2}}\right )+a \log \left (\tanh \left (\frac {x}{2}\right )\right )+b x}{a b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.63, size = 141, normalized size = 2.47 \[ \frac {b x - a \log \left (\cosh \relax (x) + \sinh \relax (x) + 1\right ) + a \log \left (\cosh \relax (x) + \sinh \relax (x) - 1\right ) + \sqrt {a^{2} + b^{2}} \log \left (\frac {a^{2} \cosh \relax (x)^{2} + a^{2} \sinh \relax (x)^{2} + 2 \, a b \cosh \relax (x) + a^{2} + 2 \, b^{2} + 2 \, {\left (a^{2} \cosh \relax (x) + a b\right )} \sinh \relax (x) + 2 \, \sqrt {a^{2} + b^{2}} {\left (a \cosh \relax (x) + a \sinh \relax (x) + b\right )}}{a \cosh \relax (x)^{2} + a \sinh \relax (x)^{2} + 2 \, b \cosh \relax (x) + 2 \, {\left (a \cosh \relax (x) + b\right )} \sinh \relax (x) - a}\right )}{a b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 89, normalized size = 1.56 \[ \frac {x}{a} - \frac {\log \left (e^{x} + 1\right )}{b} + \frac {\log \left ({\left | e^{x} - 1 \right |}\right )}{b} - \frac {\sqrt {a^{2} + b^{2}} \log \left (\frac {{\left | 2 \, a e^{x} + 2 \, b - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, a e^{x} + 2 \, b + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{a b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.13, size = 110, normalized size = 1.93 \[ -\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{a}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{a}-\frac {2 a \arctanh \left (\frac {2 \tanh \left (\frac {x}{2}\right ) b -2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{b \sqrt {a^{2}+b^{2}}}-\frac {2 b \arctanh \left (\frac {2 \tanh \left (\frac {x}{2}\right ) b -2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{a \sqrt {a^{2}+b^{2}}}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 90, normalized size = 1.58 \[ \frac {x}{a} - \frac {\log \left (e^{\left (-x\right )} + 1\right )}{b} + \frac {\log \left (e^{\left (-x\right )} - 1\right )}{b} - \frac {\sqrt {a^{2} + b^{2}} \log \left (\frac {a e^{\left (-x\right )} - b - \sqrt {a^{2} + b^{2}}}{a e^{\left (-x\right )} - b + \sqrt {a^{2} + b^{2}}}\right )}{a b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.34, size = 316, normalized size = 5.54 \[ \frac {x}{a}+\frac {\ln \left (32\,a^2\,b+32\,b^3-32\,b^3\,{\mathrm {e}}^x-32\,a^2\,b\,{\mathrm {e}}^x\right )}{b}-\frac {\ln \left (32\,a^2\,b+32\,b^3+32\,b^3\,{\mathrm {e}}^x+32\,a^2\,b\,{\mathrm {e}}^x\right )}{b}+\frac {\ln \left (128\,b^5\,{\mathrm {e}}^x-64\,a^3\,b^2-64\,a\,b^4-128\,b^4\,{\mathrm {e}}^x\,\sqrt {a^2+b^2}+32\,a^4\,b\,{\mathrm {e}}^x+160\,a^2\,b^3\,{\mathrm {e}}^x+64\,a\,b^3\,\sqrt {a^2+b^2}+32\,a^3\,b\,\sqrt {a^2+b^2}-96\,a^2\,b^2\,{\mathrm {e}}^x\,\sqrt {a^2+b^2}\right )\,\sqrt {a^2+b^2}}{a\,b}-\frac {\ln \left (128\,b^5\,{\mathrm {e}}^x-64\,a^3\,b^2-64\,a\,b^4+128\,b^4\,{\mathrm {e}}^x\,\sqrt {a^2+b^2}+32\,a^4\,b\,{\mathrm {e}}^x+160\,a^2\,b^3\,{\mathrm {e}}^x-64\,a\,b^3\,\sqrt {a^2+b^2}-32\,a^3\,b\,\sqrt {a^2+b^2}+96\,a^2\,b^2\,{\mathrm {e}}^x\,\sqrt {a^2+b^2}\right )\,\sqrt {a^2+b^2}}{a\,b} \]
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 \[ \int \frac {\coth ^{2}{\relax (x )}}{a + b \operatorname {csch}{\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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