Optimal. Leaf size=68 \[ \frac {\left (a^2+b^2\right ) x}{\left (a^2-b^2\right )^2}-\frac {a}{\left (a^2-b^2\right ) (b+a \coth (x))}-\frac {2 a b \log (a \cosh (x)+b \sinh (x))}{\left (a^2-b^2\right )^2} \]
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Rubi [A]
time = 0.09, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3164, 3564,
3612, 3611} \begin {gather*} \frac {x \left (a^2+b^2\right )}{\left (a^2-b^2\right )^2}-\frac {a}{\left (a^2-b^2\right ) (a \coth (x)+b)}-\frac {2 a b \log (a \cosh (x)+b \sinh (x))}{\left (a^2-b^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 3164
Rule 3564
Rule 3611
Rule 3612
Rubi steps
\begin {align*} \int \frac {\sinh ^2(x)}{(a \cosh (x)+b \sinh (x))^2} \, dx &=-\int \frac {1}{(-i b-i a \coth (x))^2} \, dx\\ &=-\frac {a}{\left (a^2-b^2\right ) (b+a \coth (x))}-\frac {\int \frac {-i b+i a \coth (x)}{-i b-i a \coth (x)} \, dx}{a^2-b^2}\\ &=\frac {\left (a^2+b^2\right ) x}{\left (a^2-b^2\right )^2}-\frac {a}{\left (a^2-b^2\right ) (b+a \coth (x))}-\frac {(2 i a b) \int \frac {-a-b \coth (x)}{-i b-i a \coth (x)} \, dx}{\left (a^2-b^2\right )^2}\\ &=\frac {\left (a^2+b^2\right ) x}{\left (a^2-b^2\right )^2}-\frac {a}{\left (a^2-b^2\right ) (b+a \coth (x))}-\frac {2 a b \log (a \cosh (x)+b \sinh (x))}{\left (a^2-b^2\right )^2}\\ \end {align*}
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Mathematica [A]
time = 0.19, size = 61, normalized size = 0.90 \begin {gather*} \frac {\left (a^2+b^2\right ) x-2 a b \log (a \cosh (x)+b \sinh (x))-\frac {a (a-b) (a+b) \sinh (x)}{a \cosh (x)+b \sinh (x)}}{(a-b)^2 (a+b)^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.65, size = 101, normalized size = 1.49
method | result | size |
default | \(-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{\left (a +b \right )^{2}}+\frac {2 a \left (\frac {\left (-a^{2}+b^{2}\right ) \tanh \left (\frac {x}{2}\right )}{a +2 b \tanh \left (\frac {x}{2}\right )+a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )}-b \ln \left (a +2 b \tanh \left (\frac {x}{2}\right )+a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )\right )\right )}{\left (a -b \right )^{2} \left (a +b \right )^{2}}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{\left (a -b \right )^{2}}\) | \(101\) |
risch | \(\frac {x}{a^{2}+2 a b +b^{2}}+\frac {4 a b x}{a^{4}-2 a^{2} b^{2}+b^{4}}+\frac {2 a^{2}}{\left (a -b \right ) \left (a^{2}+2 a b +b^{2}\right ) \left ({\mathrm e}^{2 x} a +b \,{\mathrm e}^{2 x}+a -b \right )}-\frac {2 a b \ln \left ({\mathrm e}^{2 x}+\frac {a -b}{a +b}\right )}{a^{4}-2 a^{2} b^{2}+b^{4}}\) | \(121\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 104, normalized size = 1.53 \begin {gather*} -\frac {2 \, a 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 {2 \, a^{2}}{a^{4} - 2 \, a^{2} b^{2} + b^{4} + {\left (a^{4} - 2 \, a^{3} b + 2 \, a b^{3} - b^{4}\right )} e^{\left (-2 \, x\right )}} + \frac {x}{a^{2} + 2 \, a b + 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. 348 vs.
\(2 (68) = 136\).
time = 0.42, size = 348, normalized size = 5.12 \begin {gather*} \frac {{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} x \cosh \left (x\right )^{2} + 2 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} x \cosh \left (x\right ) \sinh \left (x\right ) + {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} x \sinh \left (x\right )^{2} + 2 \, a^{3} - 2 \, a^{2} b + {\left (a^{3} + a^{2} b - a b^{2} - b^{3}\right )} x - 2 \, {\left (a^{2} b - a b^{2} + {\left (a^{2} b + a b^{2}\right )} \cosh \left (x\right )^{2} + 2 \, {\left (a^{2} b + a b^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right ) + {\left (a^{2} b + a b^{2}\right )} \sinh \left (x\right )^{2}\right )} \log \left (\frac {2 \, {\left (a \cosh \left (x\right ) + b \sinh \left (x\right )\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right )}{a^{5} - a^{4} b - 2 \, a^{3} b^{2} + 2 \, a^{2} b^{3} + a b^{4} - b^{5} + {\left (a^{5} + a^{4} b - 2 \, a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4} + b^{5}\right )} \cosh \left (x\right )^{2} + 2 \, {\left (a^{5} + a^{4} b - 2 \, a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4} + b^{5}\right )} \cosh \left (x\right ) \sinh \left (x\right ) + {\left (a^{5} + a^{4} b - 2 \, a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4} + b^{5}\right )} \sinh \left (x\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 983 vs.
\(2 (56) = 112\).
time = 0.68, size = 983, normalized size = 14.46 \begin {gather*} \begin {cases} \tilde {\infty } x & \text {for}\: a = 0 \wedge b = 0 \\\frac {x}{b^{2}} & \text {for}\: a = 0 \\\frac {2 x \sinh ^{2}{\left (x \right )}}{8 b^{2} \sinh ^{2}{\left (x \right )} - 16 b^{2} \sinh {\left (x \right )} \cosh {\left (x \right )} + 8 b^{2} \cosh ^{2}{\left (x \right )}} - \frac {4 x \sinh {\left (x \right )} \cosh {\left (x \right )}}{8 b^{2} \sinh ^{2}{\left (x \right )} - 16 b^{2} \sinh {\left (x \right )} \cosh {\left (x \right )} + 8 b^{2} \cosh ^{2}{\left (x \right )}} + \frac {2 x \cosh ^{2}{\left (x \right )}}{8 b^{2} \sinh ^{2}{\left (x \right )} - 16 b^{2} \sinh {\left (x \right )} \cosh {\left (x \right )} + 8 b^{2} \cosh ^{2}{\left (x \right )}} + \frac {3 \sinh ^{2}{\left (x \right )}}{8 b^{2} \sinh ^{2}{\left (x \right )} - 16 b^{2} \sinh {\left (x \right )} \cosh {\left (x \right )} + 8 b^{2} \cosh ^{2}{\left (x \right )}} - \frac {\cosh ^{2}{\left (x \right )}}{8 b^{2} \sinh ^{2}{\left (x \right )} - 16 b^{2} \sinh {\left (x \right )} \cosh {\left (x \right )} + 8 b^{2} \cosh ^{2}{\left (x \right )}} & \text {for}\: a = - b \\\frac {2 x \sinh ^{2}{\left (x \right )}}{8 b^{2} \sinh ^{2}{\left (x \right )} + 16 b^{2} \sinh {\left (x \right )} \cosh {\left (x \right )} + 8 b^{2} \cosh ^{2}{\left (x \right )}} + \frac {4 x \sinh {\left (x \right )} \cosh {\left (x \right )}}{8 b^{2} \sinh ^{2}{\left (x \right )} + 16 b^{2} \sinh {\left (x \right )} \cosh {\left (x \right )} + 8 b^{2} \cosh ^{2}{\left (x \right )}} + \frac {2 x \cosh ^{2}{\left (x \right )}}{8 b^{2} \sinh ^{2}{\left (x \right )} + 16 b^{2} \sinh {\left (x \right )} \cosh {\left (x \right )} + 8 b^{2} \cosh ^{2}{\left (x \right )}} - \frac {3 \sinh ^{2}{\left (x \right )}}{8 b^{2} \sinh ^{2}{\left (x \right )} + 16 b^{2} \sinh {\left (x \right )} \cosh {\left (x \right )} + 8 b^{2} \cosh ^{2}{\left (x \right )}} + \frac {\cosh ^{2}{\left (x \right )}}{8 b^{2} \sinh ^{2}{\left (x \right )} + 16 b^{2} \sinh {\left (x \right )} \cosh {\left (x \right )} + 8 b^{2} \cosh ^{2}{\left (x \right )}} & \text {for}\: a = b \\\frac {x - \frac {\sinh {\left (x \right )}}{\cosh {\left (x \right )}}}{a^{2}} & \text {for}\: b = 0 \\\frac {a^{4} \cosh {\left (x \right )}}{a^{5} b \cosh {\left (x \right )} + a^{4} b^{2} \sinh {\left (x \right )} - 2 a^{3} b^{3} \cosh {\left (x \right )} - 2 a^{2} b^{4} \sinh {\left (x \right )} + a b^{5} \cosh {\left (x \right )} + b^{6} \sinh {\left (x \right )}} + \frac {a^{3} b x \cosh {\left (x \right )}}{a^{5} b \cosh {\left (x \right )} + a^{4} b^{2} \sinh {\left (x \right )} - 2 a^{3} b^{3} \cosh {\left (x \right )} - 2 a^{2} b^{4} \sinh {\left (x \right )} + a b^{5} \cosh {\left (x \right )} + b^{6} \sinh {\left (x \right )}} + \frac {a^{2} b^{2} x \sinh {\left (x \right )}}{a^{5} b \cosh {\left (x \right )} + a^{4} b^{2} \sinh {\left (x \right )} - 2 a^{3} b^{3} \cosh {\left (x \right )} - 2 a^{2} b^{4} \sinh {\left (x \right )} + a b^{5} \cosh {\left (x \right )} + b^{6} \sinh {\left (x \right )}} - \frac {2 a^{2} b^{2} \log {\left (\cosh {\left (x \right )} + \frac {b \sinh {\left (x \right )}}{a} \right )} \cosh {\left (x \right )}}{a^{5} b \cosh {\left (x \right )} + a^{4} b^{2} \sinh {\left (x \right )} - 2 a^{3} b^{3} \cosh {\left (x \right )} - 2 a^{2} b^{4} \sinh {\left (x \right )} + a b^{5} \cosh {\left (x \right )} + b^{6} \sinh {\left (x \right )}} - \frac {a^{2} b^{2} \cosh {\left (x \right )}}{a^{5} b \cosh {\left (x \right )} + a^{4} b^{2} \sinh {\left (x \right )} - 2 a^{3} b^{3} \cosh {\left (x \right )} - 2 a^{2} b^{4} \sinh {\left (x \right )} + a b^{5} \cosh {\left (x \right )} + b^{6} \sinh {\left (x \right )}} + \frac {a b^{3} x \cosh {\left (x \right )}}{a^{5} b \cosh {\left (x \right )} + a^{4} b^{2} \sinh {\left (x \right )} - 2 a^{3} b^{3} \cosh {\left (x \right )} - 2 a^{2} b^{4} \sinh {\left (x \right )} + a b^{5} \cosh {\left (x \right )} + b^{6} \sinh {\left (x \right )}} - \frac {2 a b^{3} \log {\left (\cosh {\left (x \right )} + \frac {b \sinh {\left (x \right )}}{a} \right )} \sinh {\left (x \right )}}{a^{5} b \cosh {\left (x \right )} + a^{4} b^{2} \sinh {\left (x \right )} - 2 a^{3} b^{3} \cosh {\left (x \right )} - 2 a^{2} b^{4} \sinh {\left (x \right )} + a b^{5} \cosh {\left (x \right )} + b^{6} \sinh {\left (x \right )}} + \frac {b^{4} x \sinh {\left (x \right )}}{a^{5} b \cosh {\left (x \right )} + a^{4} b^{2} \sinh {\left (x \right )} - 2 a^{3} b^{3} \cosh {\left (x \right )} - 2 a^{2} b^{4} \sinh {\left (x \right )} + a b^{5} \cosh {\left (x \right )} + b^{6} \sinh {\left (x \right )}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 113, normalized size = 1.66 \begin {gather*} -\frac {2 \, a 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 {x}{a^{2} - 2 \, a b + b^{2}} + \frac {2 \, {\left (a b e^{\left (2 \, x\right )} + a^{2} - a b\right )}}{{\left (a^{3} - a^{2} b - a b^{2} + b^{3}\right )} {\left (a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} + a - b\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.47, size = 108, normalized size = 1.59 \begin {gather*} \frac {\frac {a^2\,\mathrm {cosh}\left (x\right )}{b\,\left (a^2-b^2\right )}+\frac {a\,x\,\mathrm {cosh}\left (x\right )\,\left (a^2+b^2\right )}{{\left (a^2-b^2\right )}^2}+\frac {b\,x\,\mathrm {sinh}\left (x\right )\,\left (a^2+b^2\right )}{{\left (a^2-b^2\right )}^2}}{a\,\mathrm {cosh}\left (x\right )+b\,\mathrm {sinh}\left (x\right )}+\ln \left (a\,\mathrm {cosh}\left (x\right )+b\,\mathrm {sinh}\left (x\right )\right )\,\left (\frac {1}{2\,{\left (a+b\right )}^2}-\frac {1}{2\,{\left (a-b\right )}^2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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