Optimal. Leaf size=85 \[ -\frac {a \log (1-\coth (x))}{4 (a+b)^2}+\frac {a \log (1+\coth (x))}{4 (a-b)^2}-\frac {a^2 b \log (a+b \coth (x))}{\left (a^2-b^2\right )^2}-\frac {(b-a \coth (x)) \sinh ^2(x)}{2 \left (a^2-b^2\right )} \]
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Rubi [A]
time = 0.11, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {3597, 1661,
815} \begin {gather*} -\frac {a^2 b \log (a+b \coth (x))}{\left (a^2-b^2\right )^2}-\frac {\sinh ^2(x) (b-a \coth (x))}{2 \left (a^2-b^2\right )}-\frac {a \log (1-\coth (x))}{4 (a+b)^2}+\frac {a \log (\coth (x)+1)}{4 (a-b)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 815
Rule 1661
Rule 3597
Rubi steps
\begin {align*} \int \frac {\cosh ^2(x)}{a+b \coth (x)} \, dx &=-\left (b \text {Subst}\left (\int \frac {x^2}{(a+x) \left (-b^2+x^2\right )^2} \, dx,x,b \coth (x)\right )\right )\\ &=-\frac {(b-a \coth (x)) \sinh ^2(x)}{2 \left (a^2-b^2\right )}-\frac {\text {Subst}\left (\int \frac {\frac {a^2 b^2}{a^2-b^2}-\frac {a b^2 x}{a^2-b^2}}{(a+x) \left (-b^2+x^2\right )} \, dx,x,b \coth (x)\right )}{2 b}\\ &=-\frac {(b-a \coth (x)) \sinh ^2(x)}{2 \left (a^2-b^2\right )}-\frac {\text {Subst}\left (\int \left (-\frac {a b}{2 (a+b)^2 (b-x)}+\frac {2 a^2 b^2}{(a-b)^2 (a+b)^2 (a+x)}-\frac {a b}{2 (a-b)^2 (b+x)}\right ) \, dx,x,b \coth (x)\right )}{2 b}\\ &=-\frac {a \log (1-\coth (x))}{4 (a+b)^2}+\frac {a \log (1+\coth (x))}{4 (a-b)^2}-\frac {a^2 b \log (a+b \coth (x))}{\left (a^2-b^2\right )^2}-\frac {(b-a \coth (x)) \sinh ^2(x)}{2 \left (a^2-b^2\right )}\\ \end {align*}
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Mathematica [A]
time = 0.17, size = 73, normalized size = 0.86 \begin {gather*} \frac {\left (-a^2 b+b^3\right ) \cosh (2 x)+a \left (2 \left (a^2+b^2\right ) x-4 a b \log (b \cosh (x)+a \sinh (x))+\left (a^2-b^2\right ) \sinh (2 x)\right )}{4 (a-b)^2 (a+b)^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.73, size = 146, normalized size = 1.72
method | result | size |
risch | \(\frac {a x}{2 \left (a +b \right )^{2}}+\frac {{\mathrm e}^{2 x}}{8 a +8 b}-\frac {{\mathrm e}^{-2 x}}{8 \left (a -b \right )}+\frac {2 a^{2} b x}{a^{4}-2 a^{2} b^{2}+b^{4}}-\frac {a^{2} b \ln \left ({\mathrm e}^{2 x}-\frac {a -b}{a +b}\right )}{a^{4}-2 a^{2} b^{2}+b^{4}}\) | \(100\) |
default | \(\frac {2}{\left (4 a +4 b \right ) \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}+\frac {4}{\left (8 a +8 b \right ) \left (\tanh \left (\frac {x}{2}\right )-1\right )}-\frac {a \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{2 \left (a +b \right )^{2}}-\frac {a^{2} b \ln \left (b \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )+2 a \tanh \left (\frac {x}{2}\right )+b \right )}{\left (a -b \right )^{2} \left (a +b \right )^{2}}-\frac {2}{\left (4 a -4 b \right ) \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {4}{\left (8 a -8 b \right ) \left (\tanh \left (\frac {x}{2}\right )+1\right )}+\frac {a \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{2 \left (a -b \right )^{2}}\) | \(146\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 80, normalized size = 0.94 \begin {gather*} -\frac {a^{2} b \log \left (-{\left (a - b\right )} e^{\left (-2 \, x\right )} + a + b\right )}{a^{4} - 2 \, a^{2} b^{2} + b^{4}} + \frac {a x}{2 \, {\left (a^{2} + 2 \, a b + b^{2}\right )}} + \frac {e^{\left (2 \, x\right )}}{8 \, {\left (a + b\right )}} - \frac {e^{\left (-2 \, x\right )}}{8 \, {\left (a - b\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 334 vs.
\(2 (80) = 160\).
time = 0.36, size = 334, normalized size = 3.93 \begin {gather*} \frac {{\left (a^{3} - a^{2} b - a b^{2} + b^{3}\right )} \cosh \left (x\right )^{4} + 4 \, {\left (a^{3} - a^{2} b - a b^{2} + b^{3}\right )} \cosh \left (x\right ) \sinh \left (x\right )^{3} + {\left (a^{3} - a^{2} b - a b^{2} + b^{3}\right )} \sinh \left (x\right )^{4} + 4 \, {\left (a^{3} + 2 \, a^{2} b + a b^{2}\right )} x \cosh \left (x\right )^{2} - a^{3} - a^{2} b + a b^{2} + b^{3} + 2 \, {\left (3 \, {\left (a^{3} - a^{2} b - a b^{2} + b^{3}\right )} \cosh \left (x\right )^{2} + 2 \, {\left (a^{3} + 2 \, a^{2} b + a b^{2}\right )} x\right )} \sinh \left (x\right )^{2} - 8 \, {\left (a^{2} b \cosh \left (x\right )^{2} + 2 \, a^{2} b \cosh \left (x\right ) \sinh \left (x\right ) + a^{2} b \sinh \left (x\right )^{2}\right )} \log \left (\frac {2 \, {\left (b \cosh \left (x\right ) + a \sinh \left (x\right )\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) + 4 \, {\left ({\left (a^{3} - a^{2} b - a b^{2} + b^{3}\right )} \cosh \left (x\right )^{3} + 2 \, {\left (a^{3} + 2 \, a^{2} b + a b^{2}\right )} x \cosh \left (x\right )\right )} \sinh \left (x\right )}{8 \, {\left ({\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \cosh \left (x\right )^{2} + 2 \, {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \cosh \left (x\right ) \sinh \left (x\right ) + {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \sinh \left (x\right )^{2}\right )}} \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 {\cosh ^{2}{\left (x \right )}}{a + b \coth {\left (x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 104, normalized size = 1.22 \begin {gather*} -\frac {a^{2} b \log \left ({\left | -a e^{\left (2 \, x\right )} - b e^{\left (2 \, x\right )} + a - b \right |}\right )}{a^{4} - 2 \, a^{2} b^{2} + b^{4}} + \frac {a x}{2 \, {\left (a^{2} - 2 \, a b + b^{2}\right )}} - \frac {{\left (2 \, a e^{\left (2 \, x\right )} + a - b\right )} e^{\left (-2 \, x\right )}}{8 \, {\left (a^{2} - 2 \, a b + b^{2}\right )}} + \frac {e^{\left (2 \, x\right )}}{8 \, {\left (a + b\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.40, size = 82, normalized size = 0.96 \begin {gather*} \frac {{\mathrm {e}}^{2\,x}}{8\,a+8\,b}-\frac {{\mathrm {e}}^{-2\,x}}{8\,a-8\,b}+\frac {a\,x}{2\,{\left (a-b\right )}^2}-\frac {a^2\,b\,\ln \left (b-a+a\,{\mathrm {e}}^{2\,x}+b\,{\mathrm {e}}^{2\,x}\right )}{a^4-2\,a^2\,b^2+b^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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