Optimal. Leaf size=107 \[ -\frac {b \sinh (x) \cosh (x)}{2 a^2}+\frac {b x \left (a^2-2 b^2\right )}{2 a^4}-\frac {2 b^4 \tanh ^{-1}\left (\frac {a-b \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{a^4 \sqrt {a^2+b^2}}-\frac {\left (2 a^2-3 b^2\right ) \cosh (x)}{3 a^3}+\frac {\sinh ^2(x) \cosh (x)}{3 a} \]
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Rubi [A] time = 0.46, antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {3853, 4104, 3919, 3831, 2660, 618, 206} \[ \frac {b x \left (a^2-2 b^2\right )}{2 a^4}-\frac {\left (2 a^2-3 b^2\right ) \cosh (x)}{3 a^3}-\frac {2 b^4 \tanh ^{-1}\left (\frac {a-b \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{a^4 \sqrt {a^2+b^2}}-\frac {b \sinh (x) \cosh (x)}{2 a^2}+\frac {\sinh ^2(x) \cosh (x)}{3 a} \]
Antiderivative was successfully verified.
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Rule 206
Rule 618
Rule 2660
Rule 3831
Rule 3853
Rule 3919
Rule 4104
Rubi steps
\begin {align*} \int \frac {\sinh ^3(x)}{a+b \text {csch}(x)} \, dx &=\frac {\cosh (x) \sinh ^2(x)}{3 a}-\frac {i \int \frac {\left (-3 i b-2 i a \text {csch}(x)-2 i b \text {csch}^2(x)\right ) \sinh ^2(x)}{a+b \text {csch}(x)} \, dx}{3 a}\\ &=-\frac {b \cosh (x) \sinh (x)}{2 a^2}+\frac {\cosh (x) \sinh ^2(x)}{3 a}+\frac {\int \frac {\left (-2 \left (2 a^2-3 b^2\right )-a b \text {csch}(x)+3 b^2 \text {csch}^2(x)\right ) \sinh (x)}{a+b \text {csch}(x)} \, dx}{6 a^2}\\ &=-\frac {\left (2 a^2-3 b^2\right ) \cosh (x)}{3 a^3}-\frac {b \cosh (x) \sinh (x)}{2 a^2}+\frac {\cosh (x) \sinh ^2(x)}{3 a}+\frac {i \int \frac {-3 i b \left (a^2-2 b^2\right )-3 i a b^2 \text {csch}(x)}{a+b \text {csch}(x)} \, dx}{6 a^3}\\ &=\frac {b \left (a^2-2 b^2\right ) x}{2 a^4}-\frac {\left (2 a^2-3 b^2\right ) \cosh (x)}{3 a^3}-\frac {b \cosh (x) \sinh (x)}{2 a^2}+\frac {\cosh (x) \sinh ^2(x)}{3 a}+\frac {b^4 \int \frac {\text {csch}(x)}{a+b \text {csch}(x)} \, dx}{a^4}\\ &=\frac {b \left (a^2-2 b^2\right ) x}{2 a^4}-\frac {\left (2 a^2-3 b^2\right ) \cosh (x)}{3 a^3}-\frac {b \cosh (x) \sinh (x)}{2 a^2}+\frac {\cosh (x) \sinh ^2(x)}{3 a}+\frac {b^3 \int \frac {1}{1+\frac {a \sinh (x)}{b}} \, dx}{a^4}\\ &=\frac {b \left (a^2-2 b^2\right ) x}{2 a^4}-\frac {\left (2 a^2-3 b^2\right ) \cosh (x)}{3 a^3}-\frac {b \cosh (x) \sinh (x)}{2 a^2}+\frac {\cosh (x) \sinh ^2(x)}{3 a}+\frac {\left (2 b^3\right ) \operatorname {Subst}\left (\int \frac {1}{1+\frac {2 a x}{b}-x^2} \, dx,x,\tanh \left (\frac {x}{2}\right )\right )}{a^4}\\ &=\frac {b \left (a^2-2 b^2\right ) x}{2 a^4}-\frac {\left (2 a^2-3 b^2\right ) \cosh (x)}{3 a^3}-\frac {b \cosh (x) \sinh (x)}{2 a^2}+\frac {\cosh (x) \sinh ^2(x)}{3 a}-\frac {\left (4 b^3\right ) \operatorname {Subst}\left (\int \frac {1}{4 \left (1+\frac {a^2}{b^2}\right )-x^2} \, dx,x,\frac {2 a}{b}-2 \tanh \left (\frac {x}{2}\right )\right )}{a^4}\\ &=\frac {b \left (a^2-2 b^2\right ) x}{2 a^4}-\frac {2 b^4 \tanh ^{-1}\left (\frac {b \left (\frac {a}{b}-\tanh \left (\frac {x}{2}\right )\right )}{\sqrt {a^2+b^2}}\right )}{a^4 \sqrt {a^2+b^2}}-\frac {\left (2 a^2-3 b^2\right ) \cosh (x)}{3 a^3}-\frac {b \cosh (x) \sinh (x)}{2 a^2}+\frac {\cosh (x) \sinh ^2(x)}{3 a}\\ \end {align*}
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Mathematica [A] time = 0.43, size = 104, normalized size = 0.97 \[ \frac {\left (12 a b^2-9 a^3\right ) \cosh (x)+a^3 \cosh (3 x)+3 b \left (\frac {8 b^3 \tan ^{-1}\left (\frac {a-b \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-a^2-b^2}}+2 a^2 x-a^2 \sinh (2 x)-4 b^2 x\right )}{12 a^4} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.43, size = 807, normalized size = 7.54 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 155, normalized size = 1.45 \[ \frac {b^{4} \log \left (\frac {{\left | 2 \, a e^{x} + 2 \, b - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, a e^{x} + 2 \, b + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{\sqrt {a^{2} + b^{2}} a^{4}} + \frac {a^{2} e^{\left (3 \, x\right )} - 3 \, a b e^{\left (2 \, x\right )} - 9 \, a^{2} e^{x} + 12 \, b^{2} e^{x}}{24 \, a^{3}} + \frac {{\left (a^{2} b - 2 \, b^{3}\right )} x}{2 \, a^{4}} + \frac {{\left (3 \, a^{2} b e^{x} + a^{3} - 3 \, {\left (3 \, a^{3} - 4 \, a b^{2}\right )} e^{\left (2 \, x\right )}\right )} e^{\left (-3 \, x\right )}}{24 \, a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.16, size = 262, normalized size = 2.45 \[ -\frac {1}{3 a \left (\tanh \left (\frac {x}{2}\right )-1\right )^{3}}-\frac {1}{2 a \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}-\frac {b}{2 a^{2} \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}+\frac {1}{2 a \left (\tanh \left (\frac {x}{2}\right )-1\right )}-\frac {b}{2 a^{2} \left (\tanh \left (\frac {x}{2}\right )-1\right )}-\frac {b^{2}}{a^{3} \left (\tanh \left (\frac {x}{2}\right )-1\right )}-\frac {b \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{2 a^{2}}+\frac {b^{3} \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{a^{4}}+\frac {1}{3 a \left (\tanh \left (\frac {x}{2}\right )+1\right )^{3}}-\frac {1}{2 a \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {b}{2 a^{2} \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {1}{2 a \left (\tanh \left (\frac {x}{2}\right )+1\right )}-\frac {b}{2 a^{2} \left (\tanh \left (\frac {x}{2}\right )+1\right )}+\frac {b^{2}}{a^{3} \left (\tanh \left (\frac {x}{2}\right )+1\right )}+\frac {b \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{2 a^{2}}-\frac {b^{3} \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{a^{4}}+\frac {2 b^{4} \arctanh \left (\frac {2 \tanh \left (\frac {x}{2}\right ) b -2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{a^{4} \sqrt {a^{2}+b^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 157, normalized size = 1.47 \[ \frac {b^{4} \log \left (\frac {a e^{\left (-x\right )} - b - \sqrt {a^{2} + b^{2}}}{a e^{\left (-x\right )} - b + \sqrt {a^{2} + b^{2}}}\right )}{\sqrt {a^{2} + b^{2}} a^{4}} - \frac {{\left (3 \, a b e^{\left (-x\right )} - a^{2} + 3 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )} e^{\left (-2 \, x\right )}\right )} e^{\left (3 \, x\right )}}{24 \, a^{3}} + \frac {3 \, a b e^{\left (-2 \, x\right )} + a^{2} e^{\left (-3 \, x\right )} - 3 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )} e^{\left (-x\right )}}{24 \, a^{3}} + \frac {{\left (a^{2} b - 2 \, b^{3}\right )} x}{2 \, a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.83, size = 199, normalized size = 1.86 \[ \frac {{\mathrm {e}}^{-3\,x}}{24\,a}+\frac {{\mathrm {e}}^{3\,x}}{24\,a}+\frac {x\,\left (a^2\,b-2\,b^3\right )}{2\,a^4}-\frac {{\mathrm {e}}^x\,\left (3\,a^2-4\,b^2\right )}{8\,a^3}+\frac {b\,{\mathrm {e}}^{-2\,x}}{8\,a^2}-\frac {b\,{\mathrm {e}}^{2\,x}}{8\,a^2}-\frac {{\mathrm {e}}^{-x}\,\left (3\,a^2-4\,b^2\right )}{8\,a^3}-\frac {b^4\,\ln \left (-\frac {2\,b^4\,{\mathrm {e}}^x}{a^5}-\frac {2\,b^4\,\left (a-b\,{\mathrm {e}}^x\right )}{a^5\,\sqrt {a^2+b^2}}\right )}{a^4\,\sqrt {a^2+b^2}}+\frac {b^4\,\ln \left (\frac {2\,b^4\,\left (a-b\,{\mathrm {e}}^x\right )}{a^5\,\sqrt {a^2+b^2}}-\frac {2\,b^4\,{\mathrm {e}}^x}{a^5}\right )}{a^4\,\sqrt {a^2+b^2}} \]
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 \[ \int \frac {\sinh ^{3}{\relax (x )}}{a + b \operatorname {csch}{\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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