Integrand size = 10, antiderivative size = 80 \[ \int \sinh ^{\frac {3}{2}}(a+b x) \, dx=\frac {2 i \operatorname {EllipticF}\left (\frac {1}{2} \left (i a-\frac {\pi }{2}+i b x\right ),2\right ) \sqrt {i \sinh (a+b x)}}{3 b \sqrt {\sinh (a+b x)}}+\frac {2 \cosh (a+b x) \sqrt {\sinh (a+b x)}}{3 b} \]
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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, 2720} \[ \int \sinh ^{\frac {3}{2}}(a+b x) \, dx=\frac {2 \sqrt {\sinh (a+b x)} \cosh (a+b x)}{3 b}+\frac {2 i \sqrt {i \sinh (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (i a+i b x-\frac {\pi }{2}\right ),2\right )}{3 b \sqrt {\sinh (a+b x)}} \]
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Rule 2715
Rule 2720
Rule 2721
Rubi steps \begin{align*} \text {integral}& = \frac {2 \cosh (a+b x) \sqrt {\sinh (a+b x)}}{3 b}-\frac {1}{3} \int \frac {1}{\sqrt {\sinh (a+b x)}} \, dx \\ & = \frac {2 \cosh (a+b x) \sqrt {\sinh (a+b x)}}{3 b}-\frac {\sqrt {i \sinh (a+b x)} \int \frac {1}{\sqrt {i \sinh (a+b x)}} \, dx}{3 \sqrt {\sinh (a+b x)}} \\ & = \frac {2 i \operatorname {EllipticF}\left (\frac {1}{2} \left (i a-\frac {\pi }{2}+i b x\right ),2\right ) \sqrt {i \sinh (a+b x)}}{3 b \sqrt {\sinh (a+b x)}}+\frac {2 \cosh (a+b x) \sqrt {\sinh (a+b x)}}{3 b} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.07 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.04 \[ \int \sinh ^{\frac {3}{2}}(a+b x) \, dx=\frac {\sinh (2 (a+b x))-2 \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},\cosh (2 (a+b x))+\sinh (2 (a+b x))\right ) \sqrt {1-\cosh (2 a+2 b x)-\sinh (2 a+2 b x)}}{3 b \sqrt {\sinh (a+b x)}} \]
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Time = 0.78 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.25
method | result | size |
default | \(\frac {-\frac {i \sqrt {1-i \sinh \left (b x +a \right )}\, \sqrt {2}\, \sqrt {1+i \sinh \left (b x +a \right )}\, \sqrt {i \sinh \left (b x +a \right )}\, \operatorname {EllipticF}\left (\sqrt {1-i \sinh \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right )}{3}+\frac {2 \cosh \left (b x +a \right )^{2} \sinh \left (b x +a \right )}{3}}{\cosh \left (b x +a \right ) \sqrt {\sinh \left (b x +a \right )}\, b}\) | \(100\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.29 \[ \int \sinh ^{\frac {3}{2}}(a+b x) \, dx=-\frac {2 \, {\left (\sqrt {2} \cosh \left (b x + a\right ) + \sqrt {2} \sinh \left (b x + a\right )\right )} {\rm weierstrassPInverse}\left (4, 0, \cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) - {\left (\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} + 1\right )} \sqrt {\sinh \left (b x + a\right )}}{3 \, {\left (b \cosh \left (b x + a\right ) + b \sinh \left (b x + a\right )\right )}} \]
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\[ \int \sinh ^{\frac {3}{2}}(a+b x) \, dx=\int \sinh ^{\frac {3}{2}}{\left (a + b x \right )}\, dx \]
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\[ \int \sinh ^{\frac {3}{2}}(a+b x) \, dx=\int { \sinh \left (b x + a\right )^{\frac {3}{2}} \,d x } \]
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\[ \int \sinh ^{\frac {3}{2}}(a+b x) \, dx=\int { \sinh \left (b x + a\right )^{\frac {3}{2}} \,d x } \]
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Timed out. \[ \int \sinh ^{\frac {3}{2}}(a+b x) \, dx=\int {\mathrm {sinh}\left (a+b\,x\right )}^{3/2} \,d x \]
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