Optimal. Leaf size=57 \[ -\frac {2 a^2 \tanh ^{-1}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{b^2 \sqrt {a^2+b^2}}-\frac {a x}{b^2}+\frac {\cosh (x)}{b} \]
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Rubi [A] time = 0.12, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {2746, 12, 2735, 2660, 618, 206} \[ -\frac {2 a^2 \tanh ^{-1}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{b^2 \sqrt {a^2+b^2}}-\frac {a x}{b^2}+\frac {\cosh (x)}{b} \]
Antiderivative was successfully verified.
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Rule 12
Rule 206
Rule 618
Rule 2660
Rule 2735
Rule 2746
Rubi steps
\begin {align*} \int \frac {\sinh ^2(x)}{a+b \sinh (x)} \, dx &=\frac {\cosh (x)}{b}-\frac {\int \frac {a \sinh (x)}{a+b \sinh (x)} \, dx}{b}\\ &=\frac {\cosh (x)}{b}-\frac {a \int \frac {\sinh (x)}{a+b \sinh (x)} \, dx}{b}\\ &=-\frac {a x}{b^2}+\frac {\cosh (x)}{b}+\frac {a^2 \int \frac {1}{a+b \sinh (x)} \, dx}{b^2}\\ &=-\frac {a x}{b^2}+\frac {\cosh (x)}{b}+\frac {\left (2 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{a+2 b x-a x^2} \, dx,x,\tanh \left (\frac {x}{2}\right )\right )}{b^2}\\ &=-\frac {a x}{b^2}+\frac {\cosh (x)}{b}-\frac {\left (4 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{4 \left (a^2+b^2\right )-x^2} \, dx,x,2 b-2 a \tanh \left (\frac {x}{2}\right )\right )}{b^2}\\ &=-\frac {a x}{b^2}-\frac {2 a^2 \tanh ^{-1}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{b^2 \sqrt {a^2+b^2}}+\frac {\cosh (x)}{b}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 61, normalized size = 1.07 \[ \frac {a \left (\frac {2 a \tan ^{-1}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-a^2-b^2}}-x\right )+b \cosh (x)}{b^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.58, size = 238, normalized size = 4.18 \[ \frac {a^{2} b + b^{3} - 2 \, {\left (a^{3} + a b^{2}\right )} x \cosh \relax (x) + {\left (a^{2} b + b^{3}\right )} \cosh \relax (x)^{2} + {\left (a^{2} b + b^{3}\right )} \sinh \relax (x)^{2} + 2 \, {\left (a^{2} \cosh \relax (x) + a^{2} \sinh \relax (x)\right )} \sqrt {a^{2} + b^{2}} \log \left (\frac {b^{2} \cosh \relax (x)^{2} + b^{2} \sinh \relax (x)^{2} + 2 \, a b \cosh \relax (x) + 2 \, a^{2} + b^{2} + 2 \, {\left (b^{2} \cosh \relax (x) + a b\right )} \sinh \relax (x) - 2 \, \sqrt {a^{2} + b^{2}} {\left (b \cosh \relax (x) + b \sinh \relax (x) + a\right )}}{b \cosh \relax (x)^{2} + b \sinh \relax (x)^{2} + 2 \, a \cosh \relax (x) + 2 \, {\left (b \cosh \relax (x) + a\right )} \sinh \relax (x) - b}\right ) - 2 \, {\left ({\left (a^{3} + a b^{2}\right )} x - {\left (a^{2} b + b^{3}\right )} \cosh \relax (x)\right )} \sinh \relax (x)}{2 \, {\left ({\left (a^{2} b^{2} + b^{4}\right )} \cosh \relax (x) + {\left (a^{2} b^{2} + b^{4}\right )} \sinh \relax (x)\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 86, normalized size = 1.51 \[ \frac {a^{2} \log \left (\frac {{\left | 2 \, b e^{x} + 2 \, a - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, b e^{x} + 2 \, a + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{\sqrt {a^{2} + b^{2}} b^{2}} - \frac {a x}{b^{2}} + \frac {e^{\left (-x\right )}}{2 \, b} + \frac {e^{x}}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 92, normalized size = 1.61 \[ \frac {2 a^{2} \arctanh \left (\frac {2 a \tanh \left (\frac {x}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{b^{2} \sqrt {a^{2}+b^{2}}}-\frac {1}{b \left (\tanh \left (\frac {x}{2}\right )-1\right )}+\frac {a \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{b^{2}}+\frac {1}{b \left (\tanh \left (\frac {x}{2}\right )+1\right )}-\frac {a \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.50, size = 84, normalized size = 1.47 \[ \frac {a^{2} \log \left (\frac {b e^{\left (-x\right )} - a - \sqrt {a^{2} + b^{2}}}{b e^{\left (-x\right )} - a + \sqrt {a^{2} + b^{2}}}\right )}{\sqrt {a^{2} + b^{2}} b^{2}} - \frac {a x}{b^{2}} + \frac {e^{\left (-x\right )}}{2 \, b} + \frac {e^{x}}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.54, size = 129, normalized size = 2.26 \[ \frac {{\mathrm {e}}^{-x}}{2\,b}+\frac {{\mathrm {e}}^x}{2\,b}-\frac {a\,x}{b^2}-\frac {a^2\,\ln \left (-\frac {2\,a^2\,{\mathrm {e}}^x}{b^3}-\frac {2\,a^2\,\left (b-a\,{\mathrm {e}}^x\right )}{b^3\,\sqrt {a^2+b^2}}\right )}{b^2\,\sqrt {a^2+b^2}}+\frac {a^2\,\ln \left (\frac {2\,a^2\,\left (b-a\,{\mathrm {e}}^x\right )}{b^3\,\sqrt {a^2+b^2}}-\frac {2\,a^2\,{\mathrm {e}}^x}{b^3}\right )}{b^2\,\sqrt {a^2+b^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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