Optimal. Leaf size=32 \[ -\frac {\log (a+b \sinh (x))}{a^2}+\frac {\log (\sinh (x))}{a^2}+\frac {1}{a (a+b \sinh (x))} \]
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Rubi [A] time = 0.05, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2721, 44} \[ -\frac {\log (a+b \sinh (x))}{a^2}+\frac {\log (\sinh (x))}{a^2}+\frac {1}{a (a+b \sinh (x))} \]
Antiderivative was successfully verified.
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Rule 44
Rule 2721
Rubi steps
\begin {align*} \int \frac {\coth (x)}{(a+b \sinh (x))^2} \, dx &=\operatorname {Subst}\left (\int \frac {1}{x (a+x)^2} \, dx,x,b \sinh (x)\right )\\ &=\operatorname {Subst}\left (\int \left (\frac {1}{a^2 x}-\frac {1}{a (a+x)^2}-\frac {1}{a^2 (a+x)}\right ) \, dx,x,b \sinh (x)\right )\\ &=\frac {\log (\sinh (x))}{a^2}-\frac {\log (a+b \sinh (x))}{a^2}+\frac {1}{a (a+b \sinh (x))}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 27, normalized size = 0.84 \[ \frac {\frac {a}{a+b \sinh (x)}-\log (a+b \sinh (x))+\log (\sinh (x))}{a^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.16, size = 158, normalized size = 4.94 \[ \frac {2 \, a \cosh \relax (x) - {\left (b \cosh \relax (x)^{2} + b \sinh \relax (x)^{2} + 2 \, a \cosh \relax (x) + 2 \, {\left (b \cosh \relax (x) + a\right )} \sinh \relax (x) - b\right )} \log \left (\frac {2 \, {\left (b \sinh \relax (x) + a\right )}}{\cosh \relax (x) - \sinh \relax (x)}\right ) + {\left (b \cosh \relax (x)^{2} + b \sinh \relax (x)^{2} + 2 \, a \cosh \relax (x) + 2 \, {\left (b \cosh \relax (x) + a\right )} \sinh \relax (x) - b\right )} \log \left (\frac {2 \, \sinh \relax (x)}{\cosh \relax (x) - \sinh \relax (x)}\right ) + 2 \, a \sinh \relax (x)}{a^{2} b \cosh \relax (x)^{2} + a^{2} b \sinh \relax (x)^{2} + 2 \, a^{3} \cosh \relax (x) - a^{2} b + 2 \, {\left (a^{2} b \cosh \relax (x) + a^{3}\right )} \sinh \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.15, size = 75, normalized size = 2.34 \[ -\frac {\log \left ({\left | -b {\left (e^{\left (-x\right )} - e^{x}\right )} + 2 \, a \right |}\right )}{a^{2}} + \frac {\log \left ({\left | -e^{\left (-x\right )} + e^{x} \right |}\right )}{a^{2}} + \frac {b {\left (e^{\left (-x\right )} - e^{x}\right )} - 4 \, a}{{\left (b {\left (e^{\left (-x\right )} - e^{x}\right )} - 2 \, a\right )} a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 33, normalized size = 1.03 \[ \frac {\ln \left (\sinh \relax (x )\right )}{a^{2}}-\frac {\ln \left (a +b \sinh \relax (x )\right )}{a^{2}}+\frac {1}{a \left (a +b \sinh \relax (x )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.33, size = 75, normalized size = 2.34 \[ \frac {2 \, e^{\left (-x\right )}}{2 \, a^{2} e^{\left (-x\right )} - a b e^{\left (-2 \, x\right )} + a b} - \frac {\log \left (-2 \, a e^{\left (-x\right )} + b e^{\left (-2 \, x\right )} - b\right )}{a^{2}} + \frac {\log \left (e^{\left (-x\right )} + 1\right )}{a^{2}} + \frac {\log \left (e^{\left (-x\right )} - 1\right )}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.97, size = 240, normalized size = 7.50 \[ \frac {2\,\mathrm {atan}\left (\frac {a\,\sqrt {-a^4}+b\,{\mathrm {e}}^x\,\sqrt {-a^4}-2\,a\,{\mathrm {e}}^{2\,x}\,\sqrt {-a^4}-b\,{\mathrm {e}}^{3\,x}\,\sqrt {-a^4}}{a^3}\right )-2\,\mathrm {atan}\left (\left (4\,a^5\,b\,\sqrt {-a^4}+4\,a^3\,b^3\,\sqrt {-a^4}\right )\,\left (\frac {1}{8\,a^3\,b\,{\left (a^2+b^2\right )}^2}-{\mathrm {e}}^x\,\left (\frac {1}{16\,a^2\,b^2\,{\left (a^2+b^2\right )}^2}-\frac {{\left (a^2+2\,b^2\right )}^2}{16\,a^6\,b^2\,{\left (a^2+b^2\right )}^2}\right )+\frac {a^2+2\,b^2}{8\,a^5\,b\,{\left (a^2+b^2\right )}^2}\right )\right )}{\sqrt {-a^4}}+\frac {2\,b^3\,{\mathrm {e}}^x\,\left (a^2+b^2\right )}{a\,\left (a^2\,b^3+b^5\right )\,\left (2\,a\,{\mathrm {e}}^x-b+b\,{\mathrm {e}}^{2\,x}\right )} \]
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 \[ \int \frac {\coth {\relax (x )}}{\left (a + b \sinh {\relax (x )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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