Optimal. Leaf size=56 \[ -\frac {2 \sqrt {a^2+b^2} \tanh ^{-1}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{a^2}+\frac {b \tanh ^{-1}(\cosh (x))}{a^2}-\frac {\coth (x)}{a} \]
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Rubi [A] time = 0.23, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {2723, 3056, 3001, 3770, 2660, 618, 206} \[ -\frac {2 \sqrt {a^2+b^2} \tanh ^{-1}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{a^2}+\frac {b \tanh ^{-1}(\cosh (x))}{a^2}-\frac {\coth (x)}{a} \]
Antiderivative was successfully verified.
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Rule 206
Rule 618
Rule 2660
Rule 2723
Rule 3001
Rule 3056
Rule 3770
Rubi steps
\begin {align*} \int \frac {\coth ^2(x)}{a+b \sinh (x)} \, dx &=\int \frac {\text {csch}^2(x) \left (1+\sinh ^2(x)\right )}{a+b \sinh (x)} \, dx\\ &=-\frac {\coth (x)}{a}+\frac {i \int \frac {\text {csch}(x) (i b-i a \sinh (x))}{a+b \sinh (x)} \, dx}{a}\\ &=-\frac {\coth (x)}{a}-\frac {b \int \text {csch}(x) \, dx}{a^2}+\frac {\left (a^2+b^2\right ) \int \frac {1}{a+b \sinh (x)} \, dx}{a^2}\\ &=\frac {b \tanh ^{-1}(\cosh (x))}{a^2}-\frac {\coth (x)}{a}+\frac {\left (2 \left (a^2+b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a+2 b x-a x^2} \, dx,x,\tanh \left (\frac {x}{2}\right )\right )}{a^2}\\ &=\frac {b \tanh ^{-1}(\cosh (x))}{a^2}-\frac {\coth (x)}{a}-\frac {\left (4 \left (a^2+b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{4 \left (a^2+b^2\right )-x^2} \, dx,x,2 b-2 a \tanh \left (\frac {x}{2}\right )\right )}{a^2}\\ &=\frac {b \tanh ^{-1}(\cosh (x))}{a^2}-\frac {2 \sqrt {a^2+b^2} \tanh ^{-1}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{a^2}-\frac {\coth (x)}{a}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 82, normalized size = 1.46 \[ -\frac {\text {csch}\left (\frac {x}{2}\right ) \text {sech}\left (\frac {x}{2}\right ) \left (\sinh (x) \left (2 \sqrt {-a^2-b^2} \tan ^{-1}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2-b^2}}\right )+b \log \left (\tanh \left (\frac {x}{2}\right )\right )\right )+a \cosh (x)\right )}{2 a^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.62, size = 228, normalized size = 4.07 \[ \frac {\sqrt {a^{2} + b^{2}} {\left (\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2} - 1\right )} \log \left (\frac {b^{2} \cosh \relax (x)^{2} + b^{2} \sinh \relax (x)^{2} + 2 \, a b \cosh \relax (x) + 2 \, a^{2} + b^{2} + 2 \, {\left (b^{2} \cosh \relax (x) + a b\right )} \sinh \relax (x) - 2 \, \sqrt {a^{2} + b^{2}} {\left (b \cosh \relax (x) + b \sinh \relax (x) + a\right )}}{b \cosh \relax (x)^{2} + b \sinh \relax (x)^{2} + 2 \, a \cosh \relax (x) + 2 \, {\left (b \cosh \relax (x) + a\right )} \sinh \relax (x) - b}\right ) + {\left (b \cosh \relax (x)^{2} + 2 \, b \cosh \relax (x) \sinh \relax (x) + b \sinh \relax (x)^{2} - b\right )} \log \left (\cosh \relax (x) + \sinh \relax (x) + 1\right ) - {\left (b \cosh \relax (x)^{2} + 2 \, b \cosh \relax (x) \sinh \relax (x) + b \sinh \relax (x)^{2} - b\right )} \log \left (\cosh \relax (x) + \sinh \relax (x) - 1\right ) - 2 \, a}{a^{2} \cosh \relax (x)^{2} + 2 \, a^{2} \cosh \relax (x) \sinh \relax (x) + a^{2} \sinh \relax (x)^{2} - a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.50, size = 95, normalized size = 1.70 \[ \frac {b \log \left (e^{x} + 1\right )}{a^{2}} - \frac {b \log \left ({\left | e^{x} - 1 \right |}\right )}{a^{2}} + \frac {\sqrt {a^{2} + b^{2}} \log \left (\frac {{\left | 2 \, b e^{x} + 2 \, a - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, b e^{x} + 2 \, a + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{a^{2}} - \frac {2}{a {\left (e^{\left (2 \, x\right )} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 107, normalized size = 1.91 \[ -\frac {\tanh \left (\frac {x}{2}\right )}{2 a}+\frac {2 \arctanh \left (\frac {2 a \tanh \left (\frac {x}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{\sqrt {a^{2}+b^{2}}}+\frac {2 b^{2} \arctanh \left (\frac {2 a \tanh \left (\frac {x}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{a^{2} \sqrt {a^{2}+b^{2}}}-\frac {1}{2 a \tanh \left (\frac {x}{2}\right )}-\frac {b \ln \left (\tanh \left (\frac {x}{2}\right )\right )}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 97, normalized size = 1.73 \[ \frac {b \log \left (e^{\left (-x\right )} + 1\right )}{a^{2}} - \frac {b \log \left (e^{\left (-x\right )} - 1\right )}{a^{2}} + \frac {\sqrt {a^{2} + b^{2}} \log \left (\frac {b e^{\left (-x\right )} - a - \sqrt {a^{2} + b^{2}}}{b e^{\left (-x\right )} - a + \sqrt {a^{2} + b^{2}}}\right )}{a^{2}} + \frac {2}{a e^{\left (-2 \, x\right )} - a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.84, size = 304, normalized size = 5.43 \[ \frac {2}{a-a\,{\mathrm {e}}^{2\,x}}-\frac {b\,\ln \left (32\,a^2+32\,b^2-32\,a^2\,{\mathrm {e}}^x-32\,b^2\,{\mathrm {e}}^x\right )}{a^2}+\frac {b\,\ln \left (32\,a^2+32\,b^2+32\,a^2\,{\mathrm {e}}^x+32\,b^2\,{\mathrm {e}}^x\right )}{a^2}+\frac {\ln \left (128\,a^4\,{\mathrm {e}}^x-64\,a\,b^3-64\,a^3\,b-32\,b^3\,\sqrt {a^2+b^2}+32\,b^4\,{\mathrm {e}}^x+128\,a^3\,{\mathrm {e}}^x\,\sqrt {a^2+b^2}+160\,a^2\,b^2\,{\mathrm {e}}^x-64\,a^2\,b\,\sqrt {a^2+b^2}+96\,a\,b^2\,{\mathrm {e}}^x\,\sqrt {a^2+b^2}\right )\,\sqrt {a^2+b^2}}{a^2}-\frac {\ln \left (32\,b^3\,\sqrt {a^2+b^2}-64\,a\,b^3-64\,a^3\,b+128\,a^4\,{\mathrm {e}}^x+32\,b^4\,{\mathrm {e}}^x-128\,a^3\,{\mathrm {e}}^x\,\sqrt {a^2+b^2}+160\,a^2\,b^2\,{\mathrm {e}}^x+64\,a^2\,b\,\sqrt {a^2+b^2}-96\,a\,b^2\,{\mathrm {e}}^x\,\sqrt {a^2+b^2}\right )\,\sqrt {a^2+b^2}}{a^2} \]
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 \[ \int \frac {\coth ^{2}{\relax (x )}}{a + b \sinh {\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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