Integrand size = 13, antiderivative size = 61 \[ \int \frac {\sinh ^3(x)}{a+b \text {sech}(x)} \, dx=-\frac {\left (a^2-b^2\right ) \cosh (x)}{a^3}-\frac {b \cosh ^2(x)}{2 a^2}+\frac {\cosh ^3(x)}{3 a}+\frac {b \left (a^2-b^2\right ) \log (b+a \cosh (x))}{a^4} \]
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Time = 0.12 (sec) , antiderivative size = 61, 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} \[ \int \frac {\sinh ^3(x)}{a+b \text {sech}(x)} \, dx=-\frac {b \cosh ^2(x)}{2 a^2}+\frac {b \left (a^2-b^2\right ) \log (a \cosh (x)+b)}{a^4}-\frac {\left (a^2-b^2\right ) \cosh (x)}{a^3}+\frac {\cosh ^3(x)}{3 a} \]
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Rule 12
Rule 786
Rule 2916
Rule 3957
Rubi steps \begin{align*} \text {integral}& = -\int \frac {\cosh (x) \sinh ^3(x)}{-b-a \cosh (x)} \, dx \\ & = \frac {\text {Subst}\left (\int \frac {x \left (a^2-x^2\right )}{a (-b+x)} \, dx,x,-a \cosh (x)\right )}{a^3} \\ & = \frac {\text {Subst}\left (\int \frac {x \left (a^2-x^2\right )}{-b+x} \, dx,x,-a \cosh (x)\right )}{a^4} \\ & = \frac {\text {Subst}\left (\int \left (a^2 \left (1-\frac {b^2}{a^2}\right )+\frac {-a^2 b+b^3}{b-x}-b x-x^2\right ) \, dx,x,-a \cosh (x)\right )}{a^4} \\ & = -\frac {\left (a^2-b^2\right ) \cosh (x)}{a^3}-\frac {b \cosh ^2(x)}{2 a^2}+\frac {\cosh ^3(x)}{3 a}+\frac {b \left (a^2-b^2\right ) \log (b+a \cosh (x))}{a^4} \\ \end{align*}
Time = 0.22 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.08 \[ \int \frac {\sinh ^3(x)}{a+b \text {sech}(x)} \, dx=\frac {\left (-9 a^3+12 a b^2\right ) \cosh (x)-3 a^2 b \cosh (2 x)+a^3 \cosh (3 x)+12 a^2 b \log (b+a \cosh (x))-12 b^3 \log (b+a \cosh (x))}{12 a^4} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(135\) vs. \(2(57)=114\).
Time = 10.41 (sec) , antiderivative size = 136, normalized size of antiderivative = 2.23
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 | \(-\frac {a^{2}-a b -2 b^{2}}{2 a^{3} \left (\tanh \left (\frac {x}{2}\right )+1\right )}-\frac {b \left (a^{2}-b^{2}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{a^{4}}-\frac {a +b}{2 a^{2} \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {1}{3 a \left (\tanh \left (\frac {x}{2}\right )+1\right )^{3}}+\frac {b \left (a^{3}-a^{2} b -a \,b^{2}+b^{3}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )^{2} a -\tanh \left (\frac {x}{2}\right )^{2} b +a +b \right )}{a^{4} \left (a -b \right )}-\frac {-a^{2}+a b +2 b^{2}}{2 a^{3} \left (\tanh \left (\frac {x}{2}\right )-1\right )}-\frac {1}{3 a \left (\tanh \left (\frac {x}{2}\right )-1\right )^{3}}-\frac {a +b}{2 a^{2} \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}-\frac {b \left (a^{2}-b^{2}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{a^{4}}\) | \(209\) |
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Leaf count of result is larger than twice the leaf count of optimal. 490 vs. \(2 (57) = 114\).
Time = 0.26 (sec) , antiderivative size = 490, normalized size of antiderivative = 8.03 \[ \int \frac {\sinh ^3(x)}{a+b \text {sech}(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 \cosh \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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\[ \int \frac {\sinh ^3(x)}{a+b \text {sech}(x)} \, dx=\int \frac {\sinh ^{3}{\left (x \right )}}{a + b \operatorname {sech}{\left (x \right )}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 128 vs. \(2 (57) = 114\).
Time = 0.20 (sec) , antiderivative size = 128, normalized size of antiderivative = 2.10 \[ \int \frac {\sinh ^3(x)}{a+b \text {sech}(x)} \, dx=-\frac {{\left (3 \, a b e^{\left (-x\right )} - a^{2} + 3 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )} e^{\left (-2 \, x\right )}\right )} e^{\left (3 \, x\right )}}{24 \, a^{3}} - \frac {3 \, a b e^{\left (-2 \, x\right )} - a^{2} e^{\left (-3 \, x\right )} + 3 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )} e^{\left (-x\right )}}{24 \, a^{3}} + \frac {{\left (a^{2} b - b^{3}\right )} x}{a^{4}} + \frac {{\left (a^{2} b - b^{3}\right )} \log \left (2 \, b e^{\left (-x\right )} + a e^{\left (-2 \, x\right )} + a\right )}{a^{4}} \]
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Time = 0.28 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.43 \[ \int \frac {\sinh ^3(x)}{a+b \text {sech}(x)} \, dx=\frac {a^{2} {\left (e^{\left (-x\right )} + e^{x}\right )}^{3} - 3 \, a b {\left (e^{\left (-x\right )} + e^{x}\right )}^{2} - 12 \, a^{2} {\left (e^{\left (-x\right )} + e^{x}\right )} + 12 \, b^{2} {\left (e^{\left (-x\right )} + e^{x}\right )}}{24 \, a^{3}} + \frac {{\left (a^{2} b - b^{3}\right )} \log \left ({\left | a {\left (e^{\left (-x\right )} + e^{x}\right )} + 2 \, b \right |}\right )}{a^{4}} \]
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Time = 2.22 (sec) , antiderivative size = 123, normalized size of antiderivative = 2.02 \[ \int \frac {\sinh ^3(x)}{a+b \text {sech}(x)} \, dx=\frac {{\mathrm {e}}^{-3\,x}}{24\,a}+\frac {{\mathrm {e}}^{3\,x}}{24\,a}-\frac {x\,\left (a^2\,b-b^3\right )}{a^4}-\frac {{\mathrm {e}}^x\,\left (3\,a^2-4\,b^2\right )}{8\,a^3}-\frac {b\,{\mathrm {e}}^{-2\,x}}{8\,a^2}-\frac {b\,{\mathrm {e}}^{2\,x}}{8\,a^2}+\frac {\ln \left (a+2\,b\,{\mathrm {e}}^x+a\,{\mathrm {e}}^{2\,x}\right )\,\left (a^2\,b-b^3\right )}{a^4}-\frac {{\mathrm {e}}^{-x}\,\left (3\,a^2-4\,b^2\right )}{8\,a^3} \]
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