Integrand size = 23, antiderivative size = 203 \[ \int \sec ^2(c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\frac {a b \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{d}+\frac {\sec (c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^{3/2}}{d}-\frac {\left (a^2+3 b^2\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {a \left (a^2-b^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{d \sqrt {a+b \sin (c+d x)}} \]
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Time = 0.19 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {2770, 2832, 2831, 2742, 2740, 2734, 2732} \[ \int \sec ^2(c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\frac {a \left (a^2-b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}-\frac {\left (a^2+3 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {a b \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{d}+\frac {\sec (c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^{3/2}}{d} \]
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Rule 2732
Rule 2734
Rule 2740
Rule 2742
Rule 2770
Rule 2831
Rule 2832
Rubi steps \begin{align*} \text {integral}& = \frac {\sec (c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^{3/2}}{d}-\int \sqrt {a+b \sin (c+d x)} \left (\frac {3 b^2}{2}+\frac {3}{2} a b \sin (c+d x)\right ) \, dx \\ & = \frac {a b \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{d}+\frac {\sec (c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^{3/2}}{d}-\frac {2}{3} \int \frac {3 a b^2+\frac {3}{4} b \left (a^2+3 b^2\right ) \sin (c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx \\ & = \frac {a b \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{d}+\frac {\sec (c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^{3/2}}{d}+\frac {1}{2} \left (a \left (a^2-b^2\right )\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}} \, dx-\frac {1}{2} \left (a^2+3 b^2\right ) \int \sqrt {a+b \sin (c+d x)} \, dx \\ & = \frac {a b \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{d}+\frac {\sec (c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^{3/2}}{d}-\frac {\left (\left (a^2+3 b^2\right ) \sqrt {a+b \sin (c+d x)}\right ) \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}} \, dx}{2 \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\left (a \left (a^2-b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}\right ) \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}} \, dx}{2 \sqrt {a+b \sin (c+d x)}} \\ & = \frac {a b \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{d}+\frac {\sec (c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^{3/2}}{d}-\frac {\left (a^2+3 b^2\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {a \left (a^2-b^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{d \sqrt {a+b \sin (c+d x)}} \\ \end{align*}
Time = 0.66 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.00 \[ \int \sec ^2(c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\frac {2 a^2 b \sec (c+d x)+\left (a^3+a^2 b+3 a b^2+3 b^3\right ) E\left (\frac {1}{4} (-2 c+\pi -2 d x)|\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}-a \left (a^2-b^2\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}+a^3 \tan (c+d x)+3 a b^2 \tan (c+d x)+a^2 b \sin (c+d x) \tan (c+d x)+b^3 \sin (c+d x) \tan (c+d x)}{d \sqrt {a+b \sin (c+d x)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(1038\) vs. \(2(257)=514\).
Time = 6.27 (sec) , antiderivative size = 1039, normalized size of antiderivative = 5.12
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.14 (sec) , antiderivative size = 474, normalized size of antiderivative = 2.33 \[ \int \sec ^2(c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\frac {2 \, \sqrt {2} {\left (a^{3} - 3 \, a b^{2}\right )} \sqrt {i \, b} \cos \left (d x + c\right ) {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 i \, a^{3} - 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) - 3 i \, b \sin \left (d x + c\right ) - 2 i \, a}{3 \, b}\right ) + 2 \, \sqrt {2} {\left (a^{3} - 3 \, a b^{2}\right )} \sqrt {-i \, b} \cos \left (d x + c\right ) {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (-8 i \, a^{3} + 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) + 3 i \, b \sin \left (d x + c\right ) + 2 i \, a}{3 \, b}\right ) - 3 \, \sqrt {2} {\left (-i \, a^{2} b - 3 i \, b^{3}\right )} \sqrt {i \, b} \cos \left (d x + c\right ) {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 i \, a^{3} - 9 i \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 i \, a^{3} - 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) - 3 i \, b \sin \left (d x + c\right ) - 2 i \, a}{3 \, b}\right )\right ) - 3 \, \sqrt {2} {\left (i \, a^{2} b + 3 i \, b^{3}\right )} \sqrt {-i \, b} \cos \left (d x + c\right ) {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (-8 i \, a^{3} + 9 i \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (-8 i \, a^{3} + 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) + 3 i \, b \sin \left (d x + c\right ) + 2 i \, a}{3 \, b}\right )\right ) + 6 \, {\left (2 \, a b^{2} + {\left (a^{2} b + b^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt {b \sin \left (d x + c\right ) + a}}{6 \, b d \cos \left (d x + c\right )} \]
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Timed out. \[ \int \sec ^2(c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\text {Timed out} \]
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\[ \int \sec ^2(c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\int { {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sec \left (d x + c\right )^{2} \,d x } \]
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Timed out. \[ \int \sec ^2(c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\text {Timed out} \]
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Timed out. \[ \int \sec ^2(c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\int \frac {{\left (a+b\,\sin \left (c+d\,x\right )\right )}^{5/2}}{{\cos \left (c+d\,x\right )}^2} \,d x \]
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