Optimal. Leaf size=82 \[ \frac {2 b \sqrt {a-b} \sqrt {a+b} \tan ^{-1}\left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{a^3}-\frac {\sinh (x) (2 b-a \cosh (x))}{2 a^2}-\frac {x \left (a^2-2 b^2\right )}{2 a^3} \]
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Rubi [A] time = 0.21, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {3872, 2865, 2735, 2659, 205} \[ -\frac {x \left (a^2-2 b^2\right )}{2 a^3}+\frac {2 b \sqrt {a-b} \sqrt {a+b} \tan ^{-1}\left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{a^3}-\frac {\sinh (x) (2 b-a \cosh (x))}{2 a^2} \]
Antiderivative was successfully verified.
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Rule 205
Rule 2659
Rule 2735
Rule 2865
Rule 3872
Rubi steps
\begin {align*} \int \frac {\sinh ^2(x)}{a+b \text {sech}(x)} \, dx &=-\int \frac {\cosh (x) \sinh ^2(x)}{-b-a \cosh (x)} \, dx\\ &=-\frac {(2 b-a \cosh (x)) \sinh (x)}{2 a^2}+\frac {\int \frac {-a b+\left (a^2-2 b^2\right ) \cosh (x)}{-b-a \cosh (x)} \, dx}{2 a^2}\\ &=-\frac {\left (a^2-2 b^2\right ) x}{2 a^3}-\frac {(2 b-a \cosh (x)) \sinh (x)}{2 a^2}-\frac {\left (b \left (a^2-b^2\right )\right ) \int \frac {1}{-b-a \cosh (x)} \, dx}{a^3}\\ &=-\frac {\left (a^2-2 b^2\right ) x}{2 a^3}-\frac {(2 b-a \cosh (x)) \sinh (x)}{2 a^2}-\frac {\left (2 b \left (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 )}{a^3}\\ &=-\frac {\left (a^2-2 b^2\right ) x}{2 a^3}+\frac {2 \sqrt {a-b} b \sqrt {a+b} \tan ^{-1}\left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{a^3}-\frac {(2 b-a \cosh (x)) \sinh (x)}{2 a^2}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 76, normalized size = 0.93 \[ \frac {-8 b \sqrt {a^2-b^2} \tan ^{-1}\left (\frac {(b-a) \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2}}\right )-2 a^2 x+a^2 \sinh (2 x)-4 a b \sinh (x)+4 b^2 x}{4 a^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.43, size = 536, normalized size = 6.54 \[ \left [\frac {a^{2} \cosh \relax (x)^{4} + a^{2} \sinh \relax (x)^{4} - 4 \, a b \cosh \relax (x)^{3} - 4 \, {\left (a^{2} - 2 \, b^{2}\right )} x \cosh \relax (x)^{2} + 4 \, {\left (a^{2} \cosh \relax (x) - a b\right )} \sinh \relax (x)^{3} + 4 \, a b \cosh \relax (x) + 2 \, {\left (3 \, a^{2} \cosh \relax (x)^{2} - 6 \, a b \cosh \relax (x) - 2 \, {\left (a^{2} - 2 \, b^{2}\right )} x\right )} \sinh \relax (x)^{2} + 8 \, {\left (b \cosh \relax (x)^{2} + 2 \, b \cosh \relax (x) \sinh \relax (x) + b \sinh \relax (x)^{2}\right )} \sqrt {-a^{2} + b^{2}} \log \left (\frac {a^{2} \cosh \relax (x)^{2} + a^{2} \sinh \relax (x)^{2} + 2 \, a b \cosh \relax (x) - a^{2} + 2 \, b^{2} + 2 \, {\left (a^{2} \cosh \relax (x) + a b\right )} \sinh \relax (x) + 2 \, \sqrt {-a^{2} + b^{2}} {\left (a \cosh \relax (x) + a \sinh \relax (x) + b\right )}}{a \cosh \relax (x)^{2} + a \sinh \relax (x)^{2} + 2 \, b \cosh \relax (x) + 2 \, {\left (a \cosh \relax (x) + b\right )} \sinh \relax (x) + a}\right ) - a^{2} + 4 \, {\left (a^{2} \cosh \relax (x)^{3} - 3 \, a b \cosh \relax (x)^{2} - 2 \, {\left (a^{2} - 2 \, b^{2}\right )} x \cosh \relax (x) + a b\right )} \sinh \relax (x)}{8 \, {\left (a^{3} \cosh \relax (x)^{2} + 2 \, a^{3} \cosh \relax (x) \sinh \relax (x) + a^{3} \sinh \relax (x)^{2}\right )}}, \frac {a^{2} \cosh \relax (x)^{4} + a^{2} \sinh \relax (x)^{4} - 4 \, a b \cosh \relax (x)^{3} - 4 \, {\left (a^{2} - 2 \, b^{2}\right )} x \cosh \relax (x)^{2} + 4 \, {\left (a^{2} \cosh \relax (x) - a b\right )} \sinh \relax (x)^{3} + 4 \, a b \cosh \relax (x) + 2 \, {\left (3 \, a^{2} \cosh \relax (x)^{2} - 6 \, a b \cosh \relax (x) - 2 \, {\left (a^{2} - 2 \, b^{2}\right )} x\right )} \sinh \relax (x)^{2} - 16 \, {\left (b \cosh \relax (x)^{2} + 2 \, b \cosh \relax (x) \sinh \relax (x) + b \sinh \relax (x)^{2}\right )} \sqrt {a^{2} - b^{2}} \arctan \left (-\frac {a \cosh \relax (x) + a \sinh \relax (x) + b}{\sqrt {a^{2} - b^{2}}}\right ) - a^{2} + 4 \, {\left (a^{2} \cosh \relax (x)^{3} - 3 \, a b \cosh \relax (x)^{2} - 2 \, {\left (a^{2} - 2 \, b^{2}\right )} x \cosh \relax (x) + a b\right )} \sinh \relax (x)}{8 \, {\left (a^{3} \cosh \relax (x)^{2} + 2 \, a^{3} \cosh \relax (x) \sinh \relax (x) + a^{3} \sinh \relax (x)^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 100, normalized size = 1.22 \[ \frac {a e^{\left (2 \, x\right )} - 4 \, b e^{x}}{8 \, a^{2}} - \frac {{\left (a^{2} - 2 \, b^{2}\right )} x}{2 \, a^{3}} + \frac {{\left (4 \, a b e^{x} - a^{2}\right )} e^{\left (-2 \, x\right )}}{8 \, a^{3}} + \frac {2 \, {\left (a^{2} b - b^{3}\right )} \arctan \left (\frac {a e^{x} + b}{\sqrt {a^{2} - b^{2}}}\right )}{\sqrt {a^{2} - b^{2}} a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.12, size = 213, normalized size = 2.60 \[ \frac {1}{2 a \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}+\frac {1}{2 a \left (\tanh \left (\frac {x}{2}\right )-1\right )}+\frac {b}{a^{2} \left (\tanh \left (\frac {x}{2}\right )-1\right )}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{2 a}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right ) b^{2}}{a^{3}}-\frac {1}{2 a \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {1}{2 a \left (\tanh \left (\frac {x}{2}\right )+1\right )}+\frac {b}{a^{2} \left (\tanh \left (\frac {x}{2}\right )+1\right )}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{2 a}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right ) b^{2}}{a^{3}}+\frac {2 b \arctan \left (\frac {\left (a -b \right ) \tanh \left (\frac {x}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{a \sqrt {\left (a +b \right ) \left (a -b \right )}}-\frac {2 b^{3} \arctan \left (\frac {\left (a -b \right ) \tanh \left (\frac {x}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{a^{3} \sqrt {\left (a +b \right ) \left (a -b \right )}} \]
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 = 1.67, size = 173, normalized size = 2.11 \[ \frac {{\mathrm {e}}^{2\,x}}{8\,a}-\frac {{\mathrm {e}}^{-2\,x}}{8\,a}-\frac {b\,{\mathrm {e}}^x}{2\,a^2}+\frac {b\,{\mathrm {e}}^{-x}}{2\,a^2}-\frac {x\,\left (a^2-2\,b^2\right )}{2\,a^3}+\frac {b\,\ln \left (-\frac {2\,b\,{\mathrm {e}}^x\,\left (a^2-b^2\right )}{a^4}-\frac {2\,b\,\sqrt {a+b}\,\left (a+b\,{\mathrm {e}}^x\right )\,\sqrt {b-a}}{a^4}\right )\,\sqrt {a+b}\,\sqrt {b-a}}{a^3}-\frac {b\,\ln \left (\frac {2\,b\,\sqrt {a+b}\,\left (a+b\,{\mathrm {e}}^x\right )\,\sqrt {b-a}}{a^4}-\frac {2\,b\,{\mathrm {e}}^x\,\left (a^2-b^2\right )}{a^4}\right )\,\sqrt {a+b}\,\sqrt {b-a}}{a^3} \]
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 \[ \int \frac {\sinh ^{2}{\relax (x )}}{a + b \operatorname {sech}{\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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