Integrand size = 10, antiderivative size = 80 \[ \int \sinh ^{\frac {5}{2}}(a+b x) \, dx=\frac {6 i E\left (\left .\frac {1}{2} \left (i a-\frac {\pi }{2}+i b x\right )\right |2\right ) \sqrt {\sinh (a+b x)}}{5 b \sqrt {i \sinh (a+b x)}}+\frac {2 \cosh (a+b x) \sinh ^{\frac {3}{2}}(a+b x)}{5 b} \]
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Time = 0.03 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {2715, 2721, 2719} \[ \int \sinh ^{\frac {5}{2}}(a+b x) \, dx=\frac {2 \sinh ^{\frac {3}{2}}(a+b x) \cosh (a+b x)}{5 b}+\frac {6 i \sqrt {\sinh (a+b x)} E\left (\left .\frac {1}{2} \left (i a+i b x-\frac {\pi }{2}\right )\right |2\right )}{5 b \sqrt {i \sinh (a+b x)}} \]
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Rule 2715
Rule 2719
Rule 2721
Rubi steps \begin{align*} \text {integral}& = \frac {2 \cosh (a+b x) \sinh ^{\frac {3}{2}}(a+b x)}{5 b}-\frac {3}{5} \int \sqrt {\sinh (a+b x)} \, dx \\ & = \frac {2 \cosh (a+b x) \sinh ^{\frac {3}{2}}(a+b x)}{5 b}-\frac {\left (3 \sqrt {\sinh (a+b x)}\right ) \int \sqrt {i \sinh (a+b x)} \, dx}{5 \sqrt {i \sinh (a+b x)}} \\ & = \frac {6 i E\left (\left .\frac {1}{2} \left (i a-\frac {\pi }{2}+i b x\right )\right |2\right ) \sqrt {\sinh (a+b x)}}{5 b \sqrt {i \sinh (a+b x)}}+\frac {2 \cosh (a+b x) \sinh ^{\frac {3}{2}}(a+b x)}{5 b} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.85 \[ \int \sinh ^{\frac {5}{2}}(a+b x) \, dx=\frac {-6 E\left (\left .\frac {1}{4} (-2 i a+\pi -2 i b x)\right |2\right ) \sqrt {i \sinh (a+b x)}+\sinh (a+b x) \sinh (2 (a+b x))}{5 b \sqrt {\sinh (a+b x)}} \]
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Time = 0.75 (sec) , antiderivative size = 164, normalized size of antiderivative = 2.05
method | result | size |
default | \(\frac {-\frac {6 \sqrt {1-i \sinh \left (b x +a \right )}\, \sqrt {2}\, \sqrt {1+i \sinh \left (b x +a \right )}\, \sqrt {i \sinh \left (b x +a \right )}\, \operatorname {EllipticE}\left (\sqrt {1-i \sinh \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right )}{5}+\frac {3 \sqrt {1-i \sinh \left (b x +a \right )}\, \sqrt {2}\, \sqrt {1+i \sinh \left (b x +a \right )}\, \sqrt {i \sinh \left (b x +a \right )}\, \operatorname {EllipticF}\left (\sqrt {1-i \sinh \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right )}{5}+\frac {2 \cosh \left (b x +a \right )^{4}}{5}-\frac {2 \cosh \left (b x +a \right )^{2}}{5}}{\cosh \left (b x +a \right ) \sqrt {\sinh \left (b x +a \right )}\, b}\) | \(164\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 202, normalized size of antiderivative = 2.52 \[ \int \sinh ^{\frac {5}{2}}(a+b x) \, dx=\frac {12 \, {\left (\sqrt {2} \cosh \left (b x + a\right )^{2} + 2 \, \sqrt {2} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sqrt {2} \sinh \left (b x + a\right )^{2}\right )} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )\right ) + {\left (\cosh \left (b x + a\right )^{4} + 4 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + \sinh \left (b x + a\right )^{4} + 6 \, {\left (\cosh \left (b x + a\right )^{2} + 2\right )} \sinh \left (b x + a\right )^{2} + 12 \, \cosh \left (b x + a\right )^{2} + 4 \, {\left (\cosh \left (b x + a\right )^{3} + 6 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) - 1\right )} \sqrt {\sinh \left (b x + a\right )}}{10 \, {\left (b \cosh \left (b x + a\right )^{2} + 2 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b \sinh \left (b x + a\right )^{2}\right )}} \]
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\[ \int \sinh ^{\frac {5}{2}}(a+b x) \, dx=\int \sinh ^{\frac {5}{2}}{\left (a + b x \right )}\, dx \]
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\[ \int \sinh ^{\frac {5}{2}}(a+b x) \, dx=\int { \sinh \left (b x + a\right )^{\frac {5}{2}} \,d x } \]
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\[ \int \sinh ^{\frac {5}{2}}(a+b x) \, dx=\int { \sinh \left (b x + a\right )^{\frac {5}{2}} \,d x } \]
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Timed out. \[ \int \sinh ^{\frac {5}{2}}(a+b x) \, dx=\int {\mathrm {sinh}\left (a+b\,x\right )}^{5/2} \,d x \]
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