Integrand size = 13, antiderivative size = 132 \[ \int \frac {\sinh ^4(x)}{a+b \text {sech}(x)} \, dx=\frac {\left (3 a^4-12 a^2 b^2+8 b^4\right ) x}{8 a^5}-\frac {2 (a-b)^{3/2} b (a+b)^{3/2} \arctan \left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{a^5}+\frac {\left (8 b \left (a^2-b^2\right )-a \left (3 a^2-4 b^2\right ) \cosh (x)\right ) \sinh (x)}{8 a^4}-\frac {(4 b-3 a \cosh (x)) \sinh ^3(x)}{12 a^2} \]
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Time = 0.25 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {3957, 2944, 2814, 2738, 211} \[ \int \frac {\sinh ^4(x)}{a+b \text {sech}(x)} \, dx=-\frac {2 b (a-b)^{3/2} (a+b)^{3/2} \arctan \left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{a^5}-\frac {\sinh ^3(x) (4 b-3 a \cosh (x))}{12 a^2}+\frac {\sinh (x) \left (8 b \left (a^2-b^2\right )-a \left (3 a^2-4 b^2\right ) \cosh (x)\right )}{8 a^4}+\frac {x \left (3 a^4-12 a^2 b^2+8 b^4\right )}{8 a^5} \]
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Rule 211
Rule 2738
Rule 2814
Rule 2944
Rule 3957
Rubi steps \begin{align*} \text {integral}& = -\int \frac {\cosh (x) \sinh ^4(x)}{-b-a \cosh (x)} \, dx \\ & = -\frac {(4 b-3 a \cosh (x)) \sinh ^3(x)}{12 a^2}+\frac {\int \frac {\left (-a b+\left (3 a^2-4 b^2\right ) \cosh (x)\right ) \sinh ^2(x)}{-b-a \cosh (x)} \, dx}{4 a^2} \\ & = \frac {\left (8 b \left (a^2-b^2\right )-a \left (3 a^2-4 b^2\right ) \cosh (x)\right ) \sinh (x)}{8 a^4}-\frac {(4 b-3 a \cosh (x)) \sinh ^3(x)}{12 a^2}-\frac {\int \frac {-a b \left (5 a^2-4 b^2\right )+\left (3 a^4-12 a^2 b^2+8 b^4\right ) \cosh (x)}{-b-a \cosh (x)} \, dx}{8 a^4} \\ & = \frac {\left (3 a^4-12 a^2 b^2+8 b^4\right ) x}{8 a^5}+\frac {\left (8 b \left (a^2-b^2\right )-a \left (3 a^2-4 b^2\right ) \cosh (x)\right ) \sinh (x)}{8 a^4}-\frac {(4 b-3 a \cosh (x)) \sinh ^3(x)}{12 a^2}+\frac {\left (b \left (a^2-b^2\right )^2\right ) \int \frac {1}{-b-a \cosh (x)} \, dx}{a^5} \\ & = \frac {\left (3 a^4-12 a^2 b^2+8 b^4\right ) x}{8 a^5}+\frac {\left (8 b \left (a^2-b^2\right )-a \left (3 a^2-4 b^2\right ) \cosh (x)\right ) \sinh (x)}{8 a^4}-\frac {(4 b-3 a \cosh (x)) \sinh ^3(x)}{12 a^2}+\frac {\left (2 b \left (a^2-b^2\right )^2\right ) \text {Subst}\left (\int \frac {1}{-a-b-(a-b) x^2} \, dx,x,\tanh \left (\frac {x}{2}\right )\right )}{a^5} \\ & = \frac {\left (3 a^4-12 a^2 b^2+8 b^4\right ) x}{8 a^5}-\frac {2 (a-b)^{3/2} b (a+b)^{3/2} \arctan \left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{a^5}+\frac {\left (8 b \left (a^2-b^2\right )-a \left (3 a^2-4 b^2\right ) \cosh (x)\right ) \sinh (x)}{8 a^4}-\frac {(4 b-3 a \cosh (x)) \sinh ^3(x)}{12 a^2} \\ \end{align*}
Time = 0.79 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.66 \[ \int \frac {\sinh ^4(x)}{a+b \text {sech}(x)} \, dx=\frac {36 a^4 x-144 a^2 b^2 x+96 b^4 x+\frac {192 a^4 b \arctan \left (\frac {(-a+b) \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}-\frac {384 a^2 b^3 \arctan \left (\frac {(-a+b) \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}+\frac {192 b^5 \arctan \left (\frac {(-a+b) \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}+24 a b \left (5 a^2-4 b^2\right ) \sinh (x)-24 a^2 \left (a^2-b^2\right ) \sinh (2 x)-8 a^3 b \sinh (3 x)+3 a^4 \sinh (4 x)}{96 a^5} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(304\) vs. \(2(116)=232\).
Time = 48.29 (sec) , antiderivative size = 305, normalized size of antiderivative = 2.31
method | result | size |
risch | \(\frac {3 x}{8 a}-\frac {3 x \,b^{2}}{2 a^{3}}+\frac {x \,b^{4}}{a^{5}}+\frac {{\mathrm e}^{4 x}}{64 a}-\frac {b \,{\mathrm e}^{3 x}}{24 a^{2}}-\frac {{\mathrm e}^{2 x}}{8 a}+\frac {{\mathrm e}^{2 x} b^{2}}{8 a^{3}}+\frac {5 b \,{\mathrm e}^{x}}{8 a^{2}}-\frac {b^{3} {\mathrm e}^{x}}{2 a^{4}}-\frac {5 b \,{\mathrm e}^{-x}}{8 a^{2}}+\frac {b^{3} {\mathrm e}^{-x}}{2 a^{4}}+\frac {{\mathrm e}^{-2 x}}{8 a}-\frac {{\mathrm e}^{-2 x} b^{2}}{8 a^{3}}+\frac {b \,{\mathrm e}^{-3 x}}{24 a^{2}}-\frac {{\mathrm e}^{-4 x}}{64 a}+\frac {\sqrt {-a^{2}+b^{2}}\, b \ln \left ({\mathrm e}^{x}-\frac {\sqrt {-a^{2}+b^{2}}-b}{a}\right )}{a^{3}}-\frac {\sqrt {-a^{2}+b^{2}}\, b^{3} \ln \left ({\mathrm e}^{x}-\frac {\sqrt {-a^{2}+b^{2}}-b}{a}\right )}{a^{5}}-\frac {\sqrt {-a^{2}+b^{2}}\, b \ln \left ({\mathrm e}^{x}+\frac {b +\sqrt {-a^{2}+b^{2}}}{a}\right )}{a^{3}}+\frac {\sqrt {-a^{2}+b^{2}}\, b^{3} \ln \left ({\mathrm e}^{x}+\frac {b +\sqrt {-a^{2}+b^{2}}}{a}\right )}{a^{5}}\) | \(305\) |
default | \(\frac {1}{4 a \left (\tanh \left (\frac {x}{2}\right )-1\right )^{4}}-\frac {-3 a -2 b}{6 a^{2} \left (\tanh \left (\frac {x}{2}\right )-1\right )^{3}}-\frac {a^{2}-4 a b -4 b^{2}}{8 a^{3} \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}+\frac {\left (-3 a^{4}+12 a^{2} b^{2}-8 b^{4}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{8 a^{5}}-\frac {3 a^{3}+8 a^{2} b -4 a \,b^{2}-8 b^{3}}{8 a^{4} \left (\tanh \left (\frac {x}{2}\right )-1\right )}-\frac {2 b \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \arctan \left (\frac {\left (a -b \right ) \tanh \left (\frac {x}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{a^{5} \sqrt {\left (a +b \right ) \left (a -b \right )}}-\frac {1}{4 a \left (\tanh \left (\frac {x}{2}\right )+1\right )^{4}}+\frac {\left (3 a^{4}-12 a^{2} b^{2}+8 b^{4}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{8 a^{5}}-\frac {-3 a -2 b}{6 a^{2} \left (\tanh \left (\frac {x}{2}\right )+1\right )^{3}}-\frac {-a^{2}+4 a b +4 b^{2}}{8 a^{3} \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {3 a^{3}+8 a^{2} b -4 a \,b^{2}-8 b^{3}}{8 a^{4} \left (\tanh \left (\frac {x}{2}\right )+1\right )}\) | \(310\) |
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Leaf count of result is larger than twice the leaf count of optimal. 866 vs. \(2 (115) = 230\).
Time = 0.27 (sec) , antiderivative size = 1812, normalized size of antiderivative = 13.73 \[ \int \frac {\sinh ^4(x)}{a+b \text {sech}(x)} \, dx=\text {Too large to display} \]
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\[ \int \frac {\sinh ^4(x)}{a+b \text {sech}(x)} \, dx=\int \frac {\sinh ^{4}{\left (x \right )}}{a + b \operatorname {sech}{\left (x \right )}}\, dx \]
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Exception generated. \[ \int \frac {\sinh ^4(x)}{a+b \text {sech}(x)} \, dx=\text {Exception raised: ValueError} \]
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none
Time = 0.27 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.49 \[ \int \frac {\sinh ^4(x)}{a+b \text {sech}(x)} \, dx=\frac {3 \, a^{3} e^{\left (4 \, x\right )} - 8 \, a^{2} b e^{\left (3 \, x\right )} - 24 \, a^{3} e^{\left (2 \, x\right )} + 24 \, a b^{2} e^{\left (2 \, x\right )} + 120 \, a^{2} b e^{x} - 96 \, b^{3} e^{x}}{192 \, a^{4}} + \frac {{\left (3 \, a^{4} - 12 \, a^{2} b^{2} + 8 \, b^{4}\right )} x}{8 \, a^{5}} + \frac {{\left (8 \, a^{3} b e^{x} - 3 \, a^{4} - 24 \, {\left (5 \, a^{3} b - 4 \, a b^{3}\right )} e^{\left (3 \, x\right )} + 24 \, {\left (a^{4} - a^{2} b^{2}\right )} e^{\left (2 \, x\right )}\right )} e^{\left (-4 \, x\right )}}{192 \, a^{5}} - \frac {2 \, {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} \arctan \left (\frac {a e^{x} + b}{\sqrt {a^{2} - b^{2}}}\right )}{\sqrt {a^{2} - b^{2}} a^{5}} \]
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Time = 2.60 (sec) , antiderivative size = 275, normalized size of antiderivative = 2.08 \[ \int \frac {\sinh ^4(x)}{a+b \text {sech}(x)} \, dx=\frac {{\mathrm {e}}^{4\,x}}{64\,a}-\frac {{\mathrm {e}}^{-4\,x}}{64\,a}+\frac {x\,\left (3\,a^4-12\,a^2\,b^2+8\,b^4\right )}{8\,a^5}-\frac {{\mathrm {e}}^{-x}\,\left (5\,a^2\,b-4\,b^3\right )}{8\,a^4}+\frac {{\mathrm {e}}^{-2\,x}\,\left (a^2-b^2\right )}{8\,a^3}-\frac {{\mathrm {e}}^{2\,x}\,\left (a^2-b^2\right )}{8\,a^3}+\frac {b\,{\mathrm {e}}^{-3\,x}}{24\,a^2}-\frac {b\,{\mathrm {e}}^{3\,x}}{24\,a^2}+\frac {{\mathrm {e}}^x\,\left (5\,a^2\,b-4\,b^3\right )}{8\,a^4}+\frac {b\,\ln \left (\frac {2\,{\mathrm {e}}^x\,\left (a^4\,b-2\,a^2\,b^3+b^5\right )}{a^6}-\frac {2\,b\,{\left (a+b\right )}^{3/2}\,\left (a+b\,{\mathrm {e}}^x\right )\,{\left (b-a\right )}^{3/2}}{a^6}\right )\,{\left (a+b\right )}^{3/2}\,{\left (b-a\right )}^{3/2}}{a^5}-\frac {b\,\ln \left (\frac {2\,{\mathrm {e}}^x\,\left (a^4\,b-2\,a^2\,b^3+b^5\right )}{a^6}+\frac {2\,b\,{\left (a+b\right )}^{3/2}\,\left (a+b\,{\mathrm {e}}^x\right )\,{\left (b-a\right )}^{3/2}}{a^6}\right )\,{\left (a+b\right )}^{3/2}\,{\left (b-a\right )}^{3/2}}{a^5} \]
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