Optimal. Leaf size=67 \[ \frac {x}{b^2}-\frac {2 a \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{\sqrt {a-b} b^2 \sqrt {a+b}}-\frac {\sinh (x)}{b (a+b \cosh (x))} \]
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Rubi [A]
time = 0.08, antiderivative size = 67, 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 = {2772, 2814,
2738, 214} \begin {gather*} -\frac {2 a \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{b^2 \sqrt {a-b} \sqrt {a+b}}-\frac {\sinh (x)}{b (a+b \cosh (x))}+\frac {x}{b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 214
Rule 2738
Rule 2772
Rule 2814
Rubi steps
\begin {align*} \int \frac {\sinh ^2(x)}{(a+b \cosh (x))^2} \, dx &=-\frac {\sinh (x)}{b (a+b \cosh (x))}+\frac {\int \frac {\cosh (x)}{a+b \cosh (x)} \, dx}{b}\\ &=\frac {x}{b^2}-\frac {\sinh (x)}{b (a+b \cosh (x))}-\frac {a \int \frac {1}{a+b \cosh (x)} \, dx}{b^2}\\ &=\frac {x}{b^2}-\frac {\sinh (x)}{b (a+b \cosh (x))}-\frac {(2 a) \text {Subst}\left (\int \frac {1}{a+b-(a-b) x^2} \, dx,x,\tanh \left (\frac {x}{2}\right )\right )}{b^2}\\ &=\frac {x}{b^2}-\frac {2 a \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{\sqrt {a-b} b^2 \sqrt {a+b}}-\frac {\sinh (x)}{b (a+b \cosh (x))}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 61, normalized size = 0.91 \begin {gather*} \frac {x+\frac {2 a \text {ArcTan}\left (\frac {(a-b) \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2+b^2}}\right )}{\sqrt {-a^2+b^2}}-\frac {b \sinh (x)}{a+b \cosh (x)}}{b^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.46, size = 99, normalized size = 1.48
method | result | size |
default | \(\frac {\frac {2 b \tanh \left (\frac {x}{2}\right )}{a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-b \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-a -b}-\frac {2 a \arctanh \left (\frac {\left (a -b \right ) \tanh \left (\frac {x}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}}{b^{2}}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{b^{2}}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{b^{2}}\) | \(99\) |
risch | \(\frac {x}{b^{2}}+\frac {2 a \,{\mathrm e}^{x}+2 b}{b^{2} \left (b \,{\mathrm e}^{2 x}+2 a \,{\mathrm e}^{x}+b \right )}+\frac {a \ln \left ({\mathrm e}^{x}+\frac {a \sqrt {a^{2}-b^{2}}+a^{2}-b^{2}}{b \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, b^{2}}-\frac {a \ln \left ({\mathrm e}^{x}+\frac {a \sqrt {a^{2}-b^{2}}-a^{2}+b^{2}}{b \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, b^{2}}\) | \(148\) |
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 \begin {gather*} \text {Exception raised: ValueError} \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. 316 vs.
\(2 (57) = 114\).
time = 0.45, size = 700, normalized size = 10.45 \begin {gather*} \left [\frac {{\left (a^{2} b - b^{3}\right )} x \cosh \left (x\right )^{2} + {\left (a^{2} b - b^{3}\right )} x \sinh \left (x\right )^{2} + 2 \, a^{2} b - 2 \, b^{3} + {\left (a b \cosh \left (x\right )^{2} + a b \sinh \left (x\right )^{2} + 2 \, a^{2} \cosh \left (x\right ) + a b + 2 \, {\left (a b \cosh \left (x\right ) + a^{2}\right )} \sinh \left (x\right )\right )} \sqrt {a^{2} - b^{2}} \log \left (\frac {b^{2} \cosh \left (x\right )^{2} + b^{2} \sinh \left (x\right )^{2} + 2 \, a b \cosh \left (x\right ) + 2 \, a^{2} - b^{2} + 2 \, {\left (b^{2} \cosh \left (x\right ) + a b\right )} \sinh \left (x\right ) + 2 \, \sqrt {a^{2} - b^{2}} {\left (b \cosh \left (x\right ) + b \sinh \left (x\right ) + a\right )}}{b \cosh \left (x\right )^{2} + b \sinh \left (x\right )^{2} + 2 \, a \cosh \left (x\right ) + 2 \, {\left (b \cosh \left (x\right ) + a\right )} \sinh \left (x\right ) + b}\right ) + {\left (a^{2} b - b^{3}\right )} x + 2 \, {\left (a^{3} - a b^{2} + {\left (a^{3} - a b^{2}\right )} x\right )} \cosh \left (x\right ) + 2 \, {\left (a^{3} - a b^{2} + {\left (a^{2} b - b^{3}\right )} x \cosh \left (x\right ) + {\left (a^{3} - a b^{2}\right )} x\right )} \sinh \left (x\right )}{a^{2} b^{3} - b^{5} + {\left (a^{2} b^{3} - b^{5}\right )} \cosh \left (x\right )^{2} + {\left (a^{2} b^{3} - b^{5}\right )} \sinh \left (x\right )^{2} + 2 \, {\left (a^{3} b^{2} - a b^{4}\right )} \cosh \left (x\right ) + 2 \, {\left (a^{3} b^{2} - a b^{4} + {\left (a^{2} b^{3} - b^{5}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )}, \frac {{\left (a^{2} b - b^{3}\right )} x \cosh \left (x\right )^{2} + {\left (a^{2} b - b^{3}\right )} x \sinh \left (x\right )^{2} + 2 \, a^{2} b - 2 \, b^{3} + 2 \, {\left (a b \cosh \left (x\right )^{2} + a b \sinh \left (x\right )^{2} + 2 \, a^{2} \cosh \left (x\right ) + a b + 2 \, {\left (a b \cosh \left (x\right ) + a^{2}\right )} \sinh \left (x\right )\right )} \sqrt {-a^{2} + b^{2}} \arctan \left (-\frac {\sqrt {-a^{2} + b^{2}} {\left (b \cosh \left (x\right ) + b \sinh \left (x\right ) + a\right )}}{a^{2} - b^{2}}\right ) + {\left (a^{2} b - b^{3}\right )} x + 2 \, {\left (a^{3} - a b^{2} + {\left (a^{3} - a b^{2}\right )} x\right )} \cosh \left (x\right ) + 2 \, {\left (a^{3} - a b^{2} + {\left (a^{2} b - b^{3}\right )} x \cosh \left (x\right ) + {\left (a^{3} - a b^{2}\right )} x\right )} \sinh \left (x\right )}{a^{2} b^{3} - b^{5} + {\left (a^{2} b^{3} - b^{5}\right )} \cosh \left (x\right )^{2} + {\left (a^{2} b^{3} - b^{5}\right )} \sinh \left (x\right )^{2} + 2 \, {\left (a^{3} b^{2} - a b^{4}\right )} \cosh \left (x\right ) + 2 \, {\left (a^{3} b^{2} - a b^{4} + {\left (a^{2} b^{3} - b^{5}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 68, normalized size = 1.01 \begin {gather*} -\frac {2 \, a \arctan \left (\frac {b e^{x} + a}{\sqrt {-a^{2} + b^{2}}}\right )}{\sqrt {-a^{2} + b^{2}} b^{2}} + \frac {x}{b^{2}} + \frac {2 \, {\left (a e^{x} + b\right )}}{{\left (b e^{\left (2 \, x\right )} + 2 \, a e^{x} + b\right )} b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.12, size = 139, normalized size = 2.07 \begin {gather*} \frac {x}{b^2}+\frac {\frac {2}{b}+\frac {2\,a\,{\mathrm {e}}^x}{b^2}}{b+2\,a\,{\mathrm {e}}^x+b\,{\mathrm {e}}^{2\,x}}+\frac {a\,\ln \left (\frac {2\,a\,{\mathrm {e}}^x}{b^3}-\frac {2\,a\,\left (b+a\,{\mathrm {e}}^x\right )}{b^3\,\sqrt {a+b}\,\sqrt {a-b}}\right )}{b^2\,\sqrt {a+b}\,\sqrt {a-b}}-\frac {a\,\ln \left (\frac {2\,a\,{\mathrm {e}}^x}{b^3}+\frac {2\,a\,\left (b+a\,{\mathrm {e}}^x\right )}{b^3\,\sqrt {a+b}\,\sqrt {a-b}}\right )}{b^2\,\sqrt {a+b}\,\sqrt {a-b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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