Integrand size = 13, antiderivative size = 107 \[ \int \frac {\sinh ^3(x)}{a+b \text {csch}(x)} \, dx=\frac {b \left (a^2-2 b^2\right ) x}{2 a^4}-\frac {2 b^4 \text {arctanh}\left (\frac {a-b \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{a^4 \sqrt {a^2+b^2}}-\frac {\left (2 a^2-3 b^2\right ) \cosh (x)}{3 a^3}-\frac {b \cosh (x) \sinh (x)}{2 a^2}+\frac {\cosh (x) \sinh ^2(x)}{3 a} \]
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Time = 0.32 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {3938, 4189, 4004, 3916, 2739, 632, 212} \[ \int \frac {\sinh ^3(x)}{a+b \text {csch}(x)} \, dx=-\frac {b \sinh (x) \cosh (x)}{2 a^2}-\frac {2 b^4 \text {arctanh}\left (\frac {a-b \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{a^4 \sqrt {a^2+b^2}}+\frac {b x \left (a^2-2 b^2\right )}{2 a^4}-\frac {\left (2 a^2-3 b^2\right ) \cosh (x)}{3 a^3}+\frac {\sinh ^2(x) \cosh (x)}{3 a} \]
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Rule 212
Rule 632
Rule 2739
Rule 3916
Rule 3938
Rule 4004
Rule 4189
Rubi steps \begin{align*} \text {integral}& = \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 ) \text {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 ) \text {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 \text {arctanh}\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*}
Time = 1.52 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.97 \[ \int \frac {\sinh ^3(x)}{a+b \text {csch}(x)} \, dx=\frac {\left (-9 a^3+12 a b^2\right ) \cosh (x)+a^3 \cosh (3 x)+3 b \left (2 a^2 x-4 b^2 x+\frac {8 b^3 \arctan \left (\frac {a-b \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-a^2-b^2}}-a^2 \sinh (2 x)\right )}{12 a^4} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(197\) vs. \(2(93)=186\).
Time = 0.42 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.85
method | result | size |
default | \(-\frac {2 b^{4} \operatorname {arctanh}\left (\frac {-2 b \tanh \left (\frac {x}{2}\right )+2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{a^{4} \sqrt {a^{2}+b^{2}}}+\frac {1}{3 a \left (\tanh \left (\frac {x}{2}\right )+1\right )^{3}}-\frac {a -b}{2 a^{2} \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {a^{2}+a b -2 b^{2}}{2 a^{3} \left (\tanh \left (\frac {x}{2}\right )+1\right )}+\frac {b \left (a^{2}-2 b^{2}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{2 a^{4}}-\frac {1}{3 a \left (\tanh \left (\frac {x}{2}\right )-1\right )^{3}}-\frac {a +b}{2 a^{2} \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}-\frac {-a^{2}+a b +2 b^{2}}{2 a^{3} \left (\tanh \left (\frac {x}{2}\right )-1\right )}-\frac {b \left (a^{2}-2 b^{2}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{2 a^{4}}\) | \(198\) |
risch | \(\frac {b x}{2 a^{2}}-\frac {b^{3} x}{a^{4}}+\frac {{\mathrm e}^{3 x}}{24 a}-\frac {b \,{\mathrm e}^{2 x}}{8 a^{2}}-\frac {3 \,{\mathrm e}^{x}}{8 a}+\frac {{\mathrm e}^{x} b^{2}}{2 a^{3}}-\frac {3 \,{\mathrm e}^{-x}}{8 a}+\frac {{\mathrm e}^{-x} b^{2}}{2 a^{3}}+\frac {b \,{\mathrm e}^{-2 x}}{8 a^{2}}+\frac {{\mathrm e}^{-3 x}}{24 a}+\frac {b^{4} \ln \left ({\mathrm e}^{x}+\frac {b \sqrt {a^{2}+b^{2}}-a^{2}-b^{2}}{\sqrt {a^{2}+b^{2}}\, a}\right )}{\sqrt {a^{2}+b^{2}}\, a^{4}}-\frac {b^{4} \ln \left ({\mathrm e}^{x}+\frac {b \sqrt {a^{2}+b^{2}}+a^{2}+b^{2}}{\sqrt {a^{2}+b^{2}}\, a}\right )}{\sqrt {a^{2}+b^{2}}\, a^{4}}\) | \(201\) |
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Leaf count of result is larger than twice the leaf count of optimal. 807 vs. \(2 (95) = 190\).
Time = 0.29 (sec) , antiderivative size = 807, normalized size of antiderivative = 7.54 \[ \int \frac {\sinh ^3(x)}{a+b \text {csch}(x)} \, dx=\text {Too large to display} \]
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\[ \int \frac {\sinh ^3(x)}{a+b \text {csch}(x)} \, dx=\int \frac {\sinh ^{3}{\left (x \right )}}{a + b \operatorname {csch}{\left (x \right )}}\, dx \]
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Time = 0.27 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.47 \[ \int \frac {\sinh ^3(x)}{a+b \text {csch}(x)} \, dx=\frac {b^{4} \log \left (\frac {a e^{\left (-x\right )} - b - \sqrt {a^{2} + b^{2}}}{a e^{\left (-x\right )} - b + \sqrt {a^{2} + b^{2}}}\right )}{\sqrt {a^{2} + b^{2}} a^{4}} - \frac {{\left (3 \, a b e^{\left (-x\right )} - a^{2} + 3 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )} e^{\left (-2 \, x\right )}\right )} e^{\left (3 \, x\right )}}{24 \, a^{3}} + \frac {3 \, a b e^{\left (-2 \, x\right )} + a^{2} e^{\left (-3 \, x\right )} - 3 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )} e^{\left (-x\right )}}{24 \, a^{3}} + \frac {{\left (a^{2} b - 2 \, b^{3}\right )} x}{2 \, a^{4}} \]
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Time = 0.28 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.45 \[ \int \frac {\sinh ^3(x)}{a+b \text {csch}(x)} \, dx=\frac {b^{4} \log \left (\frac {{\left | 2 \, a e^{x} + 2 \, b - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, a e^{x} + 2 \, b + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{\sqrt {a^{2} + b^{2}} a^{4}} + \frac {a^{2} e^{\left (3 \, x\right )} - 3 \, a b e^{\left (2 \, x\right )} - 9 \, a^{2} e^{x} + 12 \, b^{2} e^{x}}{24 \, a^{3}} + \frac {{\left (a^{2} b - 2 \, b^{3}\right )} x}{2 \, a^{4}} + \frac {{\left (3 \, a^{2} b e^{x} + a^{3} - 3 \, {\left (3 \, a^{3} - 4 \, a b^{2}\right )} e^{\left (2 \, x\right )}\right )} e^{\left (-3 \, x\right )}}{24 \, a^{4}} \]
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Time = 2.53 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.86 \[ \int \frac {\sinh ^3(x)}{a+b \text {csch}(x)} \, dx=\frac {{\mathrm {e}}^{-3\,x}}{24\,a}+\frac {{\mathrm {e}}^{3\,x}}{24\,a}+\frac {x\,\left (a^2\,b-2\,b^3\right )}{2\,a^4}-\frac {{\mathrm {e}}^x\,\left (3\,a^2-4\,b^2\right )}{8\,a^3}+\frac {b\,{\mathrm {e}}^{-2\,x}}{8\,a^2}-\frac {b\,{\mathrm {e}}^{2\,x}}{8\,a^2}-\frac {{\mathrm {e}}^{-x}\,\left (3\,a^2-4\,b^2\right )}{8\,a^3}-\frac {b^4\,\ln \left (-\frac {2\,b^4\,{\mathrm {e}}^x}{a^5}-\frac {2\,b^4\,\left (a-b\,{\mathrm {e}}^x\right )}{a^5\,\sqrt {a^2+b^2}}\right )}{a^4\,\sqrt {a^2+b^2}}+\frac {b^4\,\ln \left (\frac {2\,b^4\,\left (a-b\,{\mathrm {e}}^x\right )}{a^5\,\sqrt {a^2+b^2}}-\frac {2\,b^4\,{\mathrm {e}}^x}{a^5}\right )}{a^4\,\sqrt {a^2+b^2}} \]
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