Optimal. Leaf size=59 \[ -\frac {a x}{b^2}+\frac {2 \sqrt {a-b} \sqrt {a+b} \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{b^2}+\frac {\sinh (x)}{b} \]
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Rubi [A] time = 0.11, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {2695, 2735, 2659, 208} \[ -\frac {a x}{b^2}+\frac {2 \sqrt {a-b} \sqrt {a+b} \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{b^2}+\frac {\sinh (x)}{b} \]
Antiderivative was successfully verified.
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Rule 208
Rule 2659
Rule 2695
Rule 2735
Rubi steps
\begin {align*} \int \frac {\sinh ^2(x)}{a+b \cosh (x)} \, dx &=\frac {\sinh (x)}{b}+\frac {\int \frac {-b-a \cosh (x)}{a+b \cosh (x)} \, dx}{b}\\ &=-\frac {a x}{b^2}+\frac {\sinh (x)}{b}-\left (1-\frac {a^2}{b^2}\right ) \int \frac {1}{a+b \cosh (x)} \, dx\\ &=-\frac {a x}{b^2}+\frac {\sinh (x)}{b}-\left (2 \left (1-\frac {a^2}{b^2}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a+b-(a-b) x^2} \, dx,x,\tanh \left (\frac {x}{2}\right )\right )\\ &=-\frac {a x}{b^2}+\frac {2 \sqrt {a-b} \sqrt {a+b} \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{b^2}+\frac {\sinh (x)}{b}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 54, normalized size = 0.92 \[ \frac {2 \sqrt {b^2-a^2} \tan ^{-1}\left (\frac {(a-b) \tanh \left (\frac {x}{2}\right )}{\sqrt {b^2-a^2}}\right )-a x+b \sinh (x)}{b^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.29, size = 279, normalized size = 4.73 \[ \left [-\frac {2 \, a x \cosh \relax (x) - b \cosh \relax (x)^{2} - b \sinh \relax (x)^{2} - 2 \, \sqrt {a^{2} - b^{2}} {\left (\cosh \relax (x) + \sinh \relax (x)\right )} \log \left (\frac {b^{2} \cosh \relax (x)^{2} + b^{2} \sinh \relax (x)^{2} + 2 \, a b \cosh \relax (x) + 2 \, a^{2} - b^{2} + 2 \, {\left (b^{2} \cosh \relax (x) + a b\right )} \sinh \relax (x) - 2 \, \sqrt {a^{2} - b^{2}} {\left (b \cosh \relax (x) + b \sinh \relax (x) + a\right )}}{b \cosh \relax (x)^{2} + b \sinh \relax (x)^{2} + 2 \, a \cosh \relax (x) + 2 \, {\left (b \cosh \relax (x) + a\right )} \sinh \relax (x) + b}\right ) + 2 \, {\left (a x - b \cosh \relax (x)\right )} \sinh \relax (x) + b}{2 \, {\left (b^{2} \cosh \relax (x) + b^{2} \sinh \relax (x)\right )}}, -\frac {2 \, a x \cosh \relax (x) - b \cosh \relax (x)^{2} - b \sinh \relax (x)^{2} + 4 \, \sqrt {-a^{2} + b^{2}} {\left (\cosh \relax (x) + \sinh \relax (x)\right )} \arctan \left (-\frac {\sqrt {-a^{2} + b^{2}} {\left (b \cosh \relax (x) + b \sinh \relax (x) + a\right )}}{a^{2} - b^{2}}\right ) + 2 \, {\left (a x - b \cosh \relax (x)\right )} \sinh \relax (x) + b}{2 \, {\left (b^{2} \cosh \relax (x) + b^{2} \sinh \relax (x)\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 68, normalized size = 1.15 \[ -\frac {a x}{b^{2}} - \frac {e^{\left (-x\right )}}{2 \, b} + \frac {e^{x}}{2 \, b} + \frac {2 \, {\left (a^{2} - b^{2}\right )} \arctan \left (\frac {b e^{x} + a}{\sqrt {-a^{2} + b^{2}}}\right )}{\sqrt {-a^{2} + b^{2}} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 129, normalized size = 2.19 \[ -\frac {1}{b \left (\tanh \left (\frac {x}{2}\right )-1\right )}+\frac {a \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{b^{2}}-\frac {1}{b \left (\tanh \left (\frac {x}{2}\right )+1\right )}-\frac {a \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{b^{2}}+\frac {2 \arctanh \left (\frac {\left (a -b \right ) \tanh \left (\frac {x}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right ) a^{2}}{b^{2} \sqrt {\left (a +b \right ) \left (a -b \right )}}-\frac {2 \arctanh \left (\frac {\left (a -b \right ) \tanh \left (\frac {x}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.04, size = 139, normalized size = 2.36 \[ \frac {{\mathrm {e}}^x}{2\,b}-\frac {{\mathrm {e}}^{-x}}{2\,b}-\frac {a\,x}{b^2}+\frac {\ln \left (-\frac {2\,{\mathrm {e}}^x\,\left (a^2-b^2\right )}{b^3}-\frac {2\,\sqrt {a+b}\,\left (b+a\,{\mathrm {e}}^x\right )\,\sqrt {a-b}}{b^3}\right )\,\sqrt {a+b}\,\sqrt {a-b}}{b^2}-\frac {\ln \left (\frac {2\,\sqrt {a+b}\,\left (b+a\,{\mathrm {e}}^x\right )\,\sqrt {a-b}}{b^3}-\frac {2\,{\mathrm {e}}^x\,\left (a^2-b^2\right )}{b^3}\right )\,\sqrt {a+b}\,\sqrt {a-b}}{b^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 93.14, size = 892, normalized size = 15.12 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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