Optimal. Leaf size=54 \[ -\frac {a x}{b \left (a^2-b^2\right )}+\frac {x}{b (a+b \coth (x))}+\frac {\log (b \cosh (x)+a \sinh (x))}{a^2-b^2} \]
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Rubi [A]
time = 0.06, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {5575, 3565,
3611} \begin {gather*} -\frac {a x}{b \left (a^2-b^2\right )}+\frac {\log (a \sinh (x)+b \cosh (x))}{a^2-b^2}+\frac {x}{b (a+b \coth (x))} \end {gather*}
Antiderivative was successfully verified.
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Rule 3565
Rule 3611
Rule 5575
Rubi steps
\begin {align*} \int \frac {x \text {csch}^2(x)}{(a+b \coth (x))^2} \, dx &=\frac {x}{b (a+b \coth (x))}-\frac {\int \frac {1}{a+b \coth (x)} \, dx}{b}\\ &=-\frac {a x}{b \left (a^2-b^2\right )}+\frac {x}{b (a+b \coth (x))}+\frac {i \int \frac {-i b-i a \coth (x)}{a+b \coth (x)} \, dx}{a^2-b^2}\\ &=-\frac {a x}{b \left (a^2-b^2\right )}+\frac {x}{b (a+b \coth (x))}+\frac {\log (b \cosh (x)+a \sinh (x))}{a^2-b^2}\\ \end {align*}
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Mathematica [A]
time = 0.14, size = 49, normalized size = 0.91 \begin {gather*} \frac {a x-b \log (b \cosh (x)+a \sinh (x))}{-a^2 b+b^3}+\frac {x \sinh (x)}{b^2 \cosh (x)+a b \sinh (x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.30, size = 73, normalized size = 1.35
method | result | size |
risch | \(-\frac {2 x}{a^{2}-b^{2}}-\frac {2 x}{\left ({\mathrm e}^{2 x} a +b \,{\mathrm e}^{2 x}-a +b \right ) \left (a +b \right )}+\frac {\ln \left ({\mathrm e}^{2 x}-\frac {a -b}{a +b}\right )}{a^{2}-b^{2}}\) | \(73\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.41, size = 68, normalized size = 1.26 \begin {gather*} \frac {2 \, x e^{\left (2 \, x\right )}}{a^{2} - 2 \, a b + b^{2} - {\left (a^{2} - b^{2}\right )} e^{\left (2 \, x\right )}} + \frac {\log \left (\frac {{\left (a + b\right )} e^{\left (2 \, x\right )} - a + b}{a + b}\right )}{a^{2} - b^{2}} \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. 184 vs.
\(2 (54) = 108\).
time = 0.42, size = 184, normalized size = 3.41 \begin {gather*} \frac {2 \, {\left (a + b\right )} x \cosh \left (x\right )^{2} + 4 \, {\left (a + b\right )} x \cosh \left (x\right ) \sinh \left (x\right ) + 2 \, {\left (a + b\right )} x \sinh \left (x\right )^{2} - {\left ({\left (a + b\right )} \cosh \left (x\right )^{2} + 2 \, {\left (a + b\right )} \cosh \left (x\right ) \sinh \left (x\right ) + {\left (a + b\right )} \sinh \left (x\right )^{2} - a + b\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 )}{a^{3} - a^{2} b - a b^{2} + b^{3} - {\left (a^{3} + a^{2} b - a b^{2} - b^{3}\right )} \cosh \left (x\right )^{2} - 2 \, {\left (a^{3} + a^{2} b - a b^{2} - b^{3}\right )} \cosh \left (x\right ) \sinh \left (x\right ) - {\left (a^{3} + a^{2} b - a b^{2} - b^{3}\right )} \sinh \left (x\right )^{2}} \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 {x \operatorname {csch}^{2}{\left (x \right )}}{\left (a + b \coth {\left (x \right )}\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 169 vs.
\(2 (54) = 108\).
time = 0.42, size = 169, normalized size = 3.13 \begin {gather*} -\frac {2 \, a x e^{\left (2 \, x\right )} + 2 \, b x e^{\left (2 \, x\right )} - a e^{\left (2 \, x\right )} \log \left (a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} - a + b\right ) - b e^{\left (2 \, x\right )} \log \left (a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} - a + b\right ) + a \log \left (a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} - a + b\right ) - b \log \left (a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} - a + b\right )}{a^{3} e^{\left (2 \, x\right )} + a^{2} b e^{\left (2 \, x\right )} - a b^{2} e^{\left (2 \, x\right )} - b^{3} e^{\left (2 \, x\right )} - a^{3} + a^{2} b + a b^{2} - b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.28, size = 68, normalized size = 1.26 \begin {gather*} \frac {\ln \left (b-a+a\,{\mathrm {e}}^{2\,x}+b\,{\mathrm {e}}^{2\,x}\right )}{a^2-b^2}-\frac {2\,x}{a^2-b^2}-\frac {2\,x}{\left (a+b\right )\,\left (b-a+{\mathrm {e}}^{2\,x}\,\left (a+b\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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