Integrand size = 12, antiderivative size = 116 \[ \int (b \sinh (c+d x))^{7/2} \, dx=-\frac {10 i b^4 \operatorname {EllipticF}\left (\frac {1}{2} \left (i c-\frac {\pi }{2}+i d x\right ),2\right ) \sqrt {i \sinh (c+d x)}}{21 d \sqrt {b \sinh (c+d x)}}-\frac {10 b^3 \cosh (c+d x) \sqrt {b \sinh (c+d x)}}{21 d}+\frac {2 b \cosh (c+d x) (b \sinh (c+d x))^{5/2}}{7 d} \]
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Time = 0.04 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2715, 2721, 2720} \[ \int (b \sinh (c+d x))^{7/2} \, dx=-\frac {10 i b^4 \sqrt {i \sinh (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (i c+i d x-\frac {\pi }{2}\right ),2\right )}{21 d \sqrt {b \sinh (c+d x)}}-\frac {10 b^3 \cosh (c+d x) \sqrt {b \sinh (c+d x)}}{21 d}+\frac {2 b \cosh (c+d x) (b \sinh (c+d x))^{5/2}}{7 d} \]
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Rule 2715
Rule 2720
Rule 2721
Rubi steps \begin{align*} \text {integral}& = \frac {2 b \cosh (c+d x) (b \sinh (c+d x))^{5/2}}{7 d}-\frac {1}{7} \left (5 b^2\right ) \int (b \sinh (c+d x))^{3/2} \, dx \\ & = -\frac {10 b^3 \cosh (c+d x) \sqrt {b \sinh (c+d x)}}{21 d}+\frac {2 b \cosh (c+d x) (b \sinh (c+d x))^{5/2}}{7 d}+\frac {1}{21} \left (5 b^4\right ) \int \frac {1}{\sqrt {b \sinh (c+d x)}} \, dx \\ & = -\frac {10 b^3 \cosh (c+d x) \sqrt {b \sinh (c+d x)}}{21 d}+\frac {2 b \cosh (c+d x) (b \sinh (c+d x))^{5/2}}{7 d}+\frac {\left (5 b^4 \sqrt {i \sinh (c+d x)}\right ) \int \frac {1}{\sqrt {i \sinh (c+d x)}} \, dx}{21 \sqrt {b \sinh (c+d x)}} \\ & = -\frac {10 i b^4 \operatorname {EllipticF}\left (\frac {1}{2} \left (i c-\frac {\pi }{2}+i d x\right ),2\right ) \sqrt {i \sinh (c+d x)}}{21 d \sqrt {b \sinh (c+d x)}}-\frac {10 b^3 \cosh (c+d x) \sqrt {b \sinh (c+d x)}}{21 d}+\frac {2 b \cosh (c+d x) (b \sinh (c+d x))^{5/2}}{7 d} \\ \end{align*}
Time = 0.22 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.66 \[ \int (b \sinh (c+d x))^{7/2} \, dx=\frac {b^3 \left (-23 \cosh (c+d x)+3 \cosh (3 (c+d x))-\frac {20 \operatorname {EllipticF}\left (\frac {1}{4} (-2 i c+\pi -2 i d x),2\right )}{\sqrt {i \sinh (c+d x)}}\right ) \sqrt {b \sinh (c+d x)}}{42 d} \]
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Time = 1.06 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.05
method | result | size |
default | \(\frac {b^{4} \left (5 i \sqrt {1-i \sinh \left (d x +c \right )}\, \sqrt {2}\, \sqrt {1+i \sinh \left (d x +c \right )}\, \sqrt {i \sinh \left (d x +c \right )}\, \operatorname {EllipticF}\left (\sqrt {1-i \sinh \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right )+6 \cosh \left (d x +c \right )^{4} \sinh \left (d x +c \right )-16 \cosh \left (d x +c \right )^{2} \sinh \left (d x +c \right )\right )}{21 \cosh \left (d x +c \right ) \sqrt {b \sinh \left (d x +c \right )}\, d}\) | \(122\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 394, normalized size of antiderivative = 3.40 \[ \int (b \sinh (c+d x))^{7/2} \, dx=\frac {40 \, {\left (\sqrt {2} b^{3} \cosh \left (d x + c\right )^{3} + 3 \, \sqrt {2} b^{3} \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right ) + 3 \, \sqrt {2} b^{3} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + \sqrt {2} b^{3} \sinh \left (d x + c\right )^{3}\right )} \sqrt {b} {\rm weierstrassPInverse}\left (4, 0, \cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right ) + {\left (3 \, b^{3} \cosh \left (d x + c\right )^{6} + 18 \, b^{3} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{5} + 3 \, b^{3} \sinh \left (d x + c\right )^{6} - 23 \, b^{3} \cosh \left (d x + c\right )^{4} - 23 \, b^{3} \cosh \left (d x + c\right )^{2} + {\left (45 \, b^{3} \cosh \left (d x + c\right )^{2} - 23 \, b^{3}\right )} \sinh \left (d x + c\right )^{4} + 4 \, {\left (15 \, b^{3} \cosh \left (d x + c\right )^{3} - 23 \, b^{3} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + 3 \, b^{3} + {\left (45 \, b^{3} \cosh \left (d x + c\right )^{4} - 138 \, b^{3} \cosh \left (d x + c\right )^{2} - 23 \, b^{3}\right )} \sinh \left (d x + c\right )^{2} + 2 \, {\left (9 \, b^{3} \cosh \left (d x + c\right )^{5} - 46 \, b^{3} \cosh \left (d x + c\right )^{3} - 23 \, b^{3} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )\right )} \sqrt {b \sinh \left (d x + c\right )}}{84 \, {\left (d \cosh \left (d x + c\right )^{3} + 3 \, d \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right ) + 3 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + d \sinh \left (d x + c\right )^{3}\right )}} \]
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Timed out. \[ \int (b \sinh (c+d x))^{7/2} \, dx=\text {Timed out} \]
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\[ \int (b \sinh (c+d x))^{7/2} \, dx=\int { \left (b \sinh \left (d x + c\right )\right )^{\frac {7}{2}} \,d x } \]
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\[ \int (b \sinh (c+d x))^{7/2} \, dx=\int { \left (b \sinh \left (d x + c\right )\right )^{\frac {7}{2}} \,d x } \]
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Timed out. \[ \int (b \sinh (c+d x))^{7/2} \, dx=\int {\left (b\,\mathrm {sinh}\left (c+d\,x\right )\right )}^{7/2} \,d x \]
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