Integrand size = 12, antiderivative size = 118 \[ \int \frac {1}{(b \sinh (c+d x))^{7/2}} \, dx=-\frac {2 \cosh (c+d x)}{5 b d (b \sinh (c+d x))^{5/2}}+\frac {6 \cosh (c+d x)}{5 b^3 d \sqrt {b \sinh (c+d x)}}+\frac {6 i E\left (\left .\frac {1}{2} \left (i c-\frac {\pi }{2}+i d x\right )\right |2\right ) \sqrt {b \sinh (c+d x)}}{5 b^4 d \sqrt {i \sinh (c+d x)}} \]
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Time = 0.05 (sec) , antiderivative size = 118, 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 = {2716, 2721, 2719} \[ \int \frac {1}{(b \sinh (c+d x))^{7/2}} \, dx=\frac {6 i E\left (\left .\frac {1}{2} \left (i c+i d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {b \sinh (c+d x)}}{5 b^4 d \sqrt {i \sinh (c+d x)}}+\frac {6 \cosh (c+d x)}{5 b^3 d \sqrt {b \sinh (c+d x)}}-\frac {2 \cosh (c+d x)}{5 b d (b \sinh (c+d x))^{5/2}} \]
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Rule 2716
Rule 2719
Rule 2721
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \cosh (c+d x)}{5 b d (b \sinh (c+d x))^{5/2}}-\frac {3 \int \frac {1}{(b \sinh (c+d x))^{3/2}} \, dx}{5 b^2} \\ & = -\frac {2 \cosh (c+d x)}{5 b d (b \sinh (c+d x))^{5/2}}+\frac {6 \cosh (c+d x)}{5 b^3 d \sqrt {b \sinh (c+d x)}}-\frac {3 \int \sqrt {b \sinh (c+d x)} \, dx}{5 b^4} \\ & = -\frac {2 \cosh (c+d x)}{5 b d (b \sinh (c+d x))^{5/2}}+\frac {6 \cosh (c+d x)}{5 b^3 d \sqrt {b \sinh (c+d x)}}-\frac {\left (3 \sqrt {b \sinh (c+d x)}\right ) \int \sqrt {i \sinh (c+d x)} \, dx}{5 b^4 \sqrt {i \sinh (c+d x)}} \\ & = -\frac {2 \cosh (c+d x)}{5 b d (b \sinh (c+d x))^{5/2}}+\frac {6 \cosh (c+d x)}{5 b^3 d \sqrt {b \sinh (c+d x)}}+\frac {6 i E\left (\left .\frac {1}{2} \left (i c-\frac {\pi }{2}+i d x\right )\right |2\right ) \sqrt {b \sinh (c+d x)}}{5 b^4 d \sqrt {i \sinh (c+d x)}} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.67 \[ \int \frac {1}{(b \sinh (c+d x))^{7/2}} \, dx=-\frac {2 \left (-3 \cosh (c+d x)+\coth (c+d x) \text {csch}(c+d x)+3 E\left (\left .\frac {1}{4} (-2 i c+\pi -2 i d x)\right |2\right ) \sqrt {i \sinh (c+d x)}\right )}{5 b^3 d \sqrt {b \sinh (c+d x)}} \]
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Time = 0.93 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.74
method | result | size |
default | \(-\frac {6 \sqrt {-i \left (\sinh \left (d x +c \right )+i\right )}\, \sqrt {2}\, \sqrt {-i \left (i-\sinh \left (d x +c \right )\right )}\, \sqrt {i \sinh \left (d x +c \right )}\, \sinh \left (d x +c \right )^{2} \operatorname {EllipticE}\left (\sqrt {-i \left (\sinh \left (d x +c \right )+i\right )}, \frac {\sqrt {2}}{2}\right )-3 \sqrt {-i \left (\sinh \left (d x +c \right )+i\right )}\, \sqrt {2}\, \sqrt {-i \left (i-\sinh \left (d x +c \right )\right )}\, \sqrt {i \sinh \left (d x +c \right )}\, \sinh \left (d x +c \right )^{2} \operatorname {EllipticF}\left (\sqrt {-i \left (\sinh \left (d x +c \right )+i\right )}, \frac {\sqrt {2}}{2}\right )-6 \sinh \left (d x +c \right )^{4}-4 \sinh \left (d x +c \right )^{2}+2}{5 b^{3} \sinh \left (d x +c \right )^{2} \cosh \left (d x +c \right ) \sqrt {b \sinh \left (d x +c \right )}\, d}\) | \(205\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 675, normalized size of antiderivative = 5.72 \[ \int \frac {1}{(b \sinh (c+d x))^{7/2}} \, dx=\frac {2 \, {\left (3 \, {\left (\sqrt {2} \cosh \left (d x + c\right )^{6} + 6 \, \sqrt {2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{5} + \sqrt {2} \sinh \left (d x + c\right )^{6} + 3 \, {\left (5 \, \sqrt {2} \cosh \left (d x + c\right )^{2} - \sqrt {2}\right )} \sinh \left (d x + c\right )^{4} - 3 \, \sqrt {2} \cosh \left (d x + c\right )^{4} + 4 \, {\left (5 \, \sqrt {2} \cosh \left (d x + c\right )^{3} - 3 \, \sqrt {2} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + 3 \, {\left (5 \, \sqrt {2} \cosh \left (d x + c\right )^{4} - 6 \, \sqrt {2} \cosh \left (d x + c\right )^{2} + \sqrt {2}\right )} \sinh \left (d x + c\right )^{2} + 3 \, \sqrt {2} \cosh \left (d x + c\right )^{2} + 6 \, {\left (\sqrt {2} \cosh \left (d x + c\right )^{5} - 2 \, \sqrt {2} \cosh \left (d x + c\right )^{3} + \sqrt {2} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) - \sqrt {2}\right )} \sqrt {b} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right )\right ) + 2 \, {\left (3 \, \cosh \left (d x + c\right )^{6} + 18 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{5} + 3 \, \sinh \left (d x + c\right )^{6} + {\left (45 \, \cosh \left (d x + c\right )^{2} - 8\right )} \sinh \left (d x + c\right )^{4} - 8 \, \cosh \left (d x + c\right )^{4} + 4 \, {\left (15 \, \cosh \left (d x + c\right )^{3} - 8 \, \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + {\left (45 \, \cosh \left (d x + c\right )^{4} - 48 \, \cosh \left (d x + c\right )^{2} + 1\right )} \sinh \left (d x + c\right )^{2} + \cosh \left (d x + c\right )^{2} + 2 \, {\left (9 \, \cosh \left (d x + c\right )^{5} - 16 \, \cosh \left (d x + c\right )^{3} + \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )\right )} \sqrt {b \sinh \left (d x + c\right )}\right )}}{5 \, {\left (b^{4} d \cosh \left (d x + c\right )^{6} + 6 \, b^{4} d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{5} + b^{4} d \sinh \left (d x + c\right )^{6} - 3 \, b^{4} d \cosh \left (d x + c\right )^{4} + 3 \, b^{4} d \cosh \left (d x + c\right )^{2} - b^{4} d + 3 \, {\left (5 \, b^{4} d \cosh \left (d x + c\right )^{2} - b^{4} d\right )} \sinh \left (d x + c\right )^{4} + 4 \, {\left (5 \, b^{4} d \cosh \left (d x + c\right )^{3} - 3 \, b^{4} d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + 3 \, {\left (5 \, b^{4} d \cosh \left (d x + c\right )^{4} - 6 \, b^{4} d \cosh \left (d x + c\right )^{2} + b^{4} d\right )} \sinh \left (d x + c\right )^{2} + 6 \, {\left (b^{4} d \cosh \left (d x + c\right )^{5} - 2 \, b^{4} d \cosh \left (d x + c\right )^{3} + b^{4} d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )\right )}} \]
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\[ \int \frac {1}{(b \sinh (c+d x))^{7/2}} \, dx=\int \frac {1}{\left (b \sinh {\left (c + d x \right )}\right )^{\frac {7}{2}}}\, dx \]
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\[ \int \frac {1}{(b \sinh (c+d x))^{7/2}} \, dx=\int { \frac {1}{\left (b \sinh \left (d x + c\right )\right )^{\frac {7}{2}}} \,d x } \]
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\[ \int \frac {1}{(b \sinh (c+d x))^{7/2}} \, dx=\int { \frac {1}{\left (b \sinh \left (d x + c\right )\right )^{\frac {7}{2}}} \,d x } \]
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Timed out. \[ \int \frac {1}{(b \sinh (c+d x))^{7/2}} \, dx=\int \frac {1}{{\left (b\,\mathrm {sinh}\left (c+d\,x\right )\right )}^{7/2}} \,d x \]
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