Integrand size = 23, antiderivative size = 30 \[ \int \sec ^4(c+d x) (a+a \sin (c+d x))^{5/2} \, dx=\frac {2 a \sec ^3(c+d x) (a+a \sin (c+d x))^{3/2}}{3 d} \]
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Time = 0.04 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {2752} \[ \int \sec ^4(c+d x) (a+a \sin (c+d x))^{5/2} \, dx=\frac {2 a \sec ^3(c+d x) (a \sin (c+d x)+a)^{3/2}}{3 d} \]
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Rule 2752
Rubi steps \begin{align*} \text {integral}& = \frac {2 a \sec ^3(c+d x) (a+a \sin (c+d x))^{3/2}}{3 d} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \sec ^4(c+d x) (a+a \sin (c+d x))^{5/2} \, dx=\frac {2 a \sec ^3(c+d x) (a (1+\sin (c+d x)))^{3/2}}{3 d} \]
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Time = 109.12 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.57
method | result | size |
default | \(-\frac {2 a^{3} \left (1+\sin \left (d x +c \right )\right )}{3 \left (\sin \left (d x +c \right )-1\right ) \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\) | \(47\) |
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Time = 0.28 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.43 \[ \int \sec ^4(c+d x) (a+a \sin (c+d x))^{5/2} \, dx=-\frac {2 \, \sqrt {a \sin \left (d x + c\right ) + a} a^{2}}{3 \, {\left (d \cos \left (d x + c\right ) \sin \left (d x + c\right ) - d \cos \left (d x + c\right )\right )}} \]
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Timed out. \[ \int \sec ^4(c+d x) (a+a \sin (c+d x))^{5/2} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 184 vs. \(2 (26) = 52\).
Time = 0.42 (sec) , antiderivative size = 184, normalized size of antiderivative = 6.13 \[ \int \sec ^4(c+d x) (a+a \sin (c+d x))^{5/2} \, dx=-\frac {2 \, {\left (a^{\frac {5}{2}} + \frac {4 \, a^{\frac {5}{2}} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {6 \, a^{\frac {5}{2}} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {4 \, a^{\frac {5}{2}} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {a^{\frac {5}{2}} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}\right )}}{3 \, d {\left (\frac {3 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {3 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - 1\right )} {\left (\frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}^{\frac {5}{2}}} \]
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Time = 0.31 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.27 \[ \int \sec ^4(c+d x) (a+a \sin (c+d x))^{5/2} \, dx=-\frac {\sqrt {2} a^{\frac {5}{2}} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{6 \, d \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}} \]
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Time = 6.91 (sec) , antiderivative size = 225, normalized size of antiderivative = 7.50 \[ \int \sec ^4(c+d x) (a+a \sin (c+d x))^{5/2} \, dx=\frac {4\,a^2\,\sqrt {a\,\left (\sin \left (c+d\,x\right )+1\right )}\,\left ({\sin \left (c+d\,x\right )}^2\,4{}\mathrm {i}+\sin \left (c+d\,x\right )\,1{}\mathrm {i}-2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+2\,{\sin \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )}^2-2\,\sin \left (2\,c+2\,d\,x\right )+\sin \left (3\,c+3\,d\,x\right )\,1{}\mathrm {i}-4{}\mathrm {i}\right )}{3\,d\,\left (8\,{\sin \left (c+d\,x\right )}^2+4\,\sin \left (c+d\,x\right )-2\,{\sin \left (2\,c+2\,d\,x\right )}^2+4\,\sin \left (3\,c+3\,d\,x\right )-8\right )}+\frac {4\,a^2\,\sqrt {a\,\left (\sin \left (c+d\,x\right )+1\right )}\,\left (\sin \left (2\,c+2\,d\,x\right )+4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-{\sin \left (c+d\,x\right )}^2\,2{}\mathrm {i}-2+2{}\mathrm {i}\right )}{3\,d\,\left (4\,{\sin \left (c+d\,x\right )}^2+\sin \left (c+d\,x\right )+\sin \left (3\,c+3\,d\,x\right )-4\right )} \]
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