Integrand size = 13, antiderivative size = 57 \[ \int \frac {\cosh ^3(x)}{a+b \text {csch}(x)} \, dx=-\frac {b \left (a^2+b^2\right ) \log (b+a \sinh (x))}{a^4}+\frac {\left (a^2+b^2\right ) \sinh (x)}{a^3}-\frac {b \sinh ^2(x)}{2 a^2}+\frac {\sinh ^3(x)}{3 a} \]
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Time = 0.11 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {3957, 2916, 12, 786} \[ \int \frac {\cosh ^3(x)}{a+b \text {csch}(x)} \, dx=-\frac {b \sinh ^2(x)}{2 a^2}-\frac {b \left (a^2+b^2\right ) \log (a \sinh (x)+b)}{a^4}+\frac {\left (a^2+b^2\right ) \sinh (x)}{a^3}+\frac {\sinh ^3(x)}{3 a} \]
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Rule 12
Rule 786
Rule 2916
Rule 3957
Rubi steps \begin{align*} \text {integral}& = i \int \frac {\cosh ^3(x) \sinh (x)}{i b+i a \sinh (x)} \, dx \\ & = -\frac {i \text {Subst}\left (\int \frac {x \left (a^2-x^2\right )}{a (i b+x)} \, dx,x,i a \sinh (x)\right )}{a^3} \\ & = -\frac {i \text {Subst}\left (\int \frac {x \left (a^2-x^2\right )}{i b+x} \, dx,x,i a \sinh (x)\right )}{a^4} \\ & = -\frac {i \text {Subst}\left (\int \left (a^2 \left (1+\frac {b^2}{a^2}\right )-\frac {b \left (a^2+b^2\right )}{b-i x}+i b x-x^2\right ) \, dx,x,i a \sinh (x)\right )}{a^4} \\ & = -\frac {b \left (a^2+b^2\right ) \log (b+a \sinh (x))}{a^4}+\frac {\left (a^2+b^2\right ) \sinh (x)}{a^3}-\frac {b \sinh ^2(x)}{2 a^2}+\frac {\sinh ^3(x)}{3 a} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.98 \[ \int \frac {\cosh ^3(x)}{a+b \text {csch}(x)} \, dx=\frac {-6 b \left (a^2+b^2\right ) \log (b+a \sinh (x))+6 a \left (a^2+b^2\right ) \sinh (x)-3 a^2 b \sinh ^2(x)+2 a^3 \sinh ^3(x)}{6 a^4} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(135\) vs. \(2(53)=106\).
Time = 14.75 (sec) , antiderivative size = 136, normalized size of antiderivative = 2.39
method | result | size |
risch | \(\frac {b x}{a^{2}}+\frac {b^{3} x}{a^{4}}+\frac {{\mathrm e}^{3 x}}{24 a}-\frac {b \,{\mathrm e}^{2 x}}{8 a^{2}}+\frac {3 \,{\mathrm e}^{x}}{8 a}+\frac {{\mathrm e}^{x} b^{2}}{2 a^{3}}-\frac {3 \,{\mathrm e}^{-x}}{8 a}-\frac {{\mathrm e}^{-x} b^{2}}{2 a^{3}}-\frac {b \,{\mathrm e}^{-2 x}}{8 a^{2}}-\frac {{\mathrm e}^{-3 x}}{24 a}-\frac {b \ln \left ({\mathrm e}^{2 x}+\frac {2 b \,{\mathrm e}^{x}}{a}-1\right )}{a^{2}}-\frac {b^{3} \ln \left ({\mathrm e}^{2 x}+\frac {2 b \,{\mathrm e}^{x}}{a}-1\right )}{a^{4}}\) | \(136\) |
default | \(\frac {2 b \left (-\frac {a^{2}}{2}-\frac {b^{2}}{2}\right ) \ln \left (-\tanh \left (\frac {x}{2}\right )^{2} b +2 a \tanh \left (\frac {x}{2}\right )+b \right )}{a^{4}}-\frac {1}{3 a \left (\tanh \left (\frac {x}{2}\right )+1\right )^{3}}-\frac {-a +b}{2 a^{2} \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {2 a^{2}-a b +2 b^{2}}{2 a^{3} \left (\tanh \left (\frac {x}{2}\right )+1\right )}+\frac {\left (a^{2}+b^{2}\right ) b \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{a^{4}}-\frac {1}{3 a \left (\tanh \left (\frac {x}{2}\right )-1\right )^{3}}-\frac {a +b}{2 a^{2} \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}-\frac {2 a^{2}+a b +2 b^{2}}{2 a^{3} \left (\tanh \left (\frac {x}{2}\right )-1\right )}+\frac {\left (a^{2}+b^{2}\right ) b \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{a^{4}}\) | \(191\) |
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Leaf count of result is larger than twice the leaf count of optimal. 476 vs. \(2 (53) = 106\).
Time = 0.26 (sec) , antiderivative size = 476, normalized size of antiderivative = 8.35 \[ \int \frac {\cosh ^3(x)}{a+b \text {csch}(x)} \, dx=\frac {a^{3} \cosh \left (x\right )^{6} + a^{3} \sinh \left (x\right )^{6} - 3 \, a^{2} b \cosh \left (x\right )^{5} + 3 \, {\left (2 \, a^{3} \cosh \left (x\right ) - a^{2} b\right )} \sinh \left (x\right )^{5} + 24 \, {\left (a^{2} b + b^{3}\right )} x \cosh \left (x\right )^{3} + 3 \, {\left (3 \, a^{3} + 4 \, a b^{2}\right )} \cosh \left (x\right )^{4} + 3 \, {\left (5 \, a^{3} \cosh \left (x\right )^{2} - 5 \, a^{2} b \cosh \left (x\right ) + 3 \, a^{3} + 4 \, a b^{2}\right )} \sinh \left (x\right )^{4} - 3 \, a^{2} b \cosh \left (x\right ) + 2 \, {\left (10 \, a^{3} \cosh \left (x\right )^{3} - 15 \, a^{2} b \cosh \left (x\right )^{2} + 12 \, {\left (a^{2} b + b^{3}\right )} x + 6 \, {\left (3 \, a^{3} + 4 \, a b^{2}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} - a^{3} - 3 \, {\left (3 \, a^{3} + 4 \, a b^{2}\right )} \cosh \left (x\right )^{2} + 3 \, {\left (5 \, a^{3} \cosh \left (x\right )^{4} - 10 \, a^{2} b \cosh \left (x\right )^{3} - 3 \, a^{3} - 4 \, a b^{2} + 24 \, {\left (a^{2} b + b^{3}\right )} x \cosh \left (x\right ) + 6 \, {\left (3 \, a^{3} + 4 \, a b^{2}\right )} \cosh \left (x\right )^{2}\right )} \sinh \left (x\right )^{2} - 24 \, {\left ({\left (a^{2} b + b^{3}\right )} \cosh \left (x\right )^{3} + 3 \, {\left (a^{2} b + b^{3}\right )} \cosh \left (x\right )^{2} \sinh \left (x\right ) + 3 \, {\left (a^{2} b + b^{3}\right )} \cosh \left (x\right ) \sinh \left (x\right )^{2} + {\left (a^{2} b + b^{3}\right )} \sinh \left (x\right )^{3}\right )} \log \left (\frac {2 \, {\left (a \sinh \left (x\right ) + b\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) + 3 \, {\left (2 \, a^{3} \cosh \left (x\right )^{5} - 5 \, a^{2} b \cosh \left (x\right )^{4} + 24 \, {\left (a^{2} b + b^{3}\right )} x \cosh \left (x\right )^{2} + 4 \, {\left (3 \, a^{3} + 4 \, a b^{2}\right )} \cosh \left (x\right )^{3} - a^{2} b - 2 \, {\left (3 \, a^{3} + 4 \, a b^{2}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )}{24 \, {\left (a^{4} \cosh \left (x\right )^{3} + 3 \, a^{4} \cosh \left (x\right )^{2} \sinh \left (x\right ) + 3 \, a^{4} \cosh \left (x\right ) \sinh \left (x\right )^{2} + a^{4} \sinh \left (x\right )^{3}\right )}} \]
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\[ \int \frac {\cosh ^3(x)}{a+b \text {csch}(x)} \, dx=\int \frac {\cosh ^{3}{\left (x \right )}}{a + b \operatorname {csch}{\left (x \right )}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 127 vs. \(2 (53) = 106\).
Time = 0.18 (sec) , antiderivative size = 127, normalized size of antiderivative = 2.23 \[ \int \frac {\cosh ^3(x)}{a+b \text {csch}(x)} \, dx=-\frac {{\left (3 \, a b e^{\left (-x\right )} - a^{2} - 3 \, {\left (3 \, a^{2} + 4 \, b^{2}\right )} e^{\left (-2 \, x\right )}\right )} e^{\left (3 \, x\right )}}{24 \, a^{3}} - \frac {3 \, a b e^{\left (-2 \, x\right )} + a^{2} e^{\left (-3 \, x\right )} + 3 \, {\left (3 \, a^{2} + 4 \, b^{2}\right )} e^{\left (-x\right )}}{24 \, a^{3}} - \frac {{\left (a^{2} b + b^{3}\right )} x}{a^{4}} - \frac {{\left (a^{2} b + b^{3}\right )} \log \left (-2 \, b e^{\left (-x\right )} + a e^{\left (-2 \, x\right )} - a\right )}{a^{4}} \]
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Time = 0.27 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.70 \[ \int \frac {\cosh ^3(x)}{a+b \text {csch}(x)} \, dx=-\frac {a^{2} {\left (e^{\left (-x\right )} - e^{x}\right )}^{3} + 3 \, a b {\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 12 \, a^{2} {\left (e^{\left (-x\right )} - e^{x}\right )} + 12 \, b^{2} {\left (e^{\left (-x\right )} - e^{x}\right )}}{24 \, a^{3}} - \frac {{\left (a^{2} b + b^{3}\right )} \log \left ({\left | -a {\left (e^{\left (-x\right )} - e^{x}\right )} + 2 \, b \right |}\right )}{a^{4}} \]
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Time = 2.41 (sec) , antiderivative size = 121, normalized size of antiderivative = 2.12 \[ \int \frac {\cosh ^3(x)}{a+b \text {csch}(x)} \, dx=\frac {{\mathrm {e}}^{3\,x}}{24\,a}-\frac {{\mathrm {e}}^{-3\,x}}{24\,a}+\frac {x\,\left (a^2\,b+b^3\right )}{a^4}+\frac {{\mathrm {e}}^x\,\left (3\,a^2+4\,b^2\right )}{8\,a^3}-\frac {b\,{\mathrm {e}}^{-2\,x}}{8\,a^2}-\frac {b\,{\mathrm {e}}^{2\,x}}{8\,a^2}-\frac {\ln \left (2\,b\,{\mathrm {e}}^x-a+a\,{\mathrm {e}}^{2\,x}\right )\,\left (a^2\,b+b^3\right )}{a^4}-\frac {{\mathrm {e}}^{-x}\,\left (3\,a^2+4\,b^2\right )}{8\,a^3} \]
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