Optimal. Leaf size=20 \[ \frac {\log (\sinh (x))}{a}-\frac {\log (a+b \sinh (x))}{a} \]
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Rubi [A]
time = 0.03, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {2800, 36, 29,
31} \begin {gather*} \frac {\log (\sinh (x))}{a}-\frac {\log (a+b \sinh (x))}{a} \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 2800
Rubi steps
\begin {align*} \int \frac {\coth (x)}{a+b \sinh (x)} \, dx &=\text {Subst}\left (\int \frac {1}{x (a+x)} \, dx,x,b \sinh (x)\right )\\ &=\frac {\text {Subst}\left (\int \frac {1}{x} \, dx,x,b \sinh (x)\right )}{a}-\frac {\text {Subst}\left (\int \frac {1}{a+x} \, dx,x,b \sinh (x)\right )}{a}\\ &=\frac {\log (\sinh (x))}{a}-\frac {\log (a+b \sinh (x))}{a}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 20, normalized size = 1.00 \begin {gather*} \frac {\log (\sinh (x))}{a}-\frac {\log (a+b \sinh (x))}{a} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.46, size = 21, normalized size = 1.05
method | result | size |
derivativedivides | \(\frac {\ln \left (\sinh \left (x \right )\right )}{a}-\frac {\ln \left (a +b \sinh \left (x \right )\right )}{a}\) | \(21\) |
default | \(\frac {\ln \left (\sinh \left (x \right )\right )}{a}-\frac {\ln \left (a +b \sinh \left (x \right )\right )}{a}\) | \(21\) |
risch | \(\frac {\ln \left ({\mathrm e}^{2 x}-1\right )}{a}-\frac {\ln \left ({\mathrm e}^{2 x}+\frac {2 a \,{\mathrm e}^{x}}{b}-1\right )}{a}\) | \(33\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 46 vs.
\(2 (20) = 40\).
time = 0.28, size = 46, normalized size = 2.30 \begin {gather*} -\frac {\log \left (-2 \, a e^{\left (-x\right )} + b e^{\left (-2 \, x\right )} - b\right )}{a} + \frac {\log \left (e^{\left (-x\right )} + 1\right )}{a} + \frac {\log \left (e^{\left (-x\right )} - 1\right )}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 40, normalized size = 2.00 \begin {gather*} -\frac {\log \left (\frac {2 \, {\left (b \sinh \left (x\right ) + a\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) - \log \left (\frac {2 \, \sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right )}{a} \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 {\coth {\left (x \right )}}{a + b \sinh {\left (x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 39, normalized size = 1.95 \begin {gather*} -\frac {\log \left ({\left | -b {\left (e^{\left (-x\right )} - e^{x}\right )} + 2 \, a \right |}\right )}{a} + \frac {\log \left ({\left | -e^{\left (-x\right )} + e^{x} \right |}\right )}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.89, size = 195, normalized size = 9.75 \begin {gather*} \frac {2\,\mathrm {atan}\left (\frac {a\,\sqrt {-a^2}+b\,{\mathrm {e}}^x\,\sqrt {-a^2}-2\,a\,{\mathrm {e}}^{2\,x}\,\sqrt {-a^2}-b\,{\mathrm {e}}^{3\,x}\,\sqrt {-a^2}}{a^2}\right )}{\sqrt {-a^2}}-\frac {2\,\mathrm {atan}\left (\left (4\,a^4\,b\,\sqrt {-a^2}+4\,a^2\,b^3\,\sqrt {-a^2}\right )\,\left (\frac {1}{8\,a\,b\,{\left (a^2+b^2\right )}^2}-{\mathrm {e}}^x\,\left (\frac {1}{16\,b^2\,{\left (a^2+b^2\right )}^2}-\frac {{\left (a^2+2\,b^2\right )}^2}{16\,a^4\,b^2\,{\left (a^2+b^2\right )}^2}\right )+\frac {a^2+2\,b^2}{8\,a^3\,b\,{\left (a^2+b^2\right )}^2}\right )\right )}{\sqrt {-a^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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