Optimal. Leaf size=12 \[ -\frac {\log (a+b \coth (x))}{b} \]
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Rubi [A] time = 0.04, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3506, 31} \[ -\frac {\log (a+b \coth (x))}{b} \]
Antiderivative was successfully verified.
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Rule 31
Rule 3506
Rubi steps
\begin {align*} \int \frac {\text {csch}^2(x)}{a+b \coth (x)} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {1}{a+x} \, dx,x,b \coth (x)\right )}{b}\\ &=-\frac {\log (a+b \coth (x))}{b}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 20, normalized size = 1.67 \[ \frac {\log (\sinh (x))-\log (a \sinh (x)+b \cosh (x))}{b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.41, size = 43, normalized size = 3.58 \[ -\frac {\log \left (\frac {2 \, {\left (b \cosh \relax (x) + a \sinh \relax (x)\right )}}{\cosh \relax (x) - \sinh \relax (x)}\right ) - \log \left (\frac {2 \, \sinh \relax (x)}{\cosh \relax (x) - \sinh \relax (x)}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.11, size = 46, normalized size = 3.83 \[ -\frac {{\left (a + b\right )} \log \left ({\left | a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} - a + b \right |}\right )}{a b + b^{2}} + \frac {\log \left ({\left | e^{\left (2 \, x\right )} - 1 \right |}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 13, normalized size = 1.08 \[ -\frac {\ln \left (a +b \coth \relax (x )\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 12, normalized size = 1.00 \[ -\frac {\log \left (b \coth \relax (x) + a\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.16, size = 51, normalized size = 4.25 \[ -\frac {2\,\mathrm {atan}\left (\frac {a\,{\mathrm {e}}^{2\,x}\,\sqrt {-b^2}-a\,\sqrt {-b^2}+b\,{\mathrm {e}}^{2\,x}\,\sqrt {-b^2}}{b^2}\right )}{\sqrt {-b^2}} \]
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 \[ \int \frac {\operatorname {csch}^{2}{\relax (x )}}{a + b \coth {\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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