Integrand size = 23, antiderivative size = 251 \[ \int \frac {\sec ^2(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=\frac {2 b \sec (c+d x)}{\left (a^2-b^2\right ) d \sqrt {a+b \sin (c+d x)}}-\frac {\left (a^2+3 b^2\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{\left (a^2-b^2\right )^2 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {a \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{\left (a^2-b^2\right ) d \sqrt {a+b \sin (c+d x)}}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a b-\left (a^2+3 b^2\right ) \sin (c+d x)\right )}{\left (a^2-b^2\right )^2 d} \]
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Time = 0.26 (sec) , antiderivative size = 251, 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 = {2773, 2945, 2831, 2742, 2740, 2734, 2732} \[ \int \frac {\sec ^2(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a b-\left (a^2+3 b^2\right ) \sin (c+d x)\right )}{d \left (a^2-b^2\right )^2}+\frac {2 b \sec (c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \sin (c+d x)}}+\frac {a \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \left (a^2-b^2\right ) \sqrt {a+b \sin (c+d x)}}-\frac {\left (a^2+3 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \left (a^2-b^2\right )^2 \sqrt {\frac {a+b \sin (c+d x)}{a+b}}} \]
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Rule 2732
Rule 2734
Rule 2740
Rule 2742
Rule 2773
Rule 2831
Rule 2945
Rubi steps \begin{align*} \text {integral}& = \frac {2 b \sec (c+d x)}{\left (a^2-b^2\right ) d \sqrt {a+b \sin (c+d x)}}-\frac {2 \int \frac {\sec ^2(c+d x) \left (-\frac {a}{2}+\frac {3}{2} b \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{a^2-b^2} \\ & = \frac {2 b \sec (c+d x)}{\left (a^2-b^2\right ) d \sqrt {a+b \sin (c+d x)}}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a b-\left (a^2+3 b^2\right ) \sin (c+d x)\right )}{\left (a^2-b^2\right )^2 d}+\frac {2 \int \frac {-a b^2-\frac {1}{4} b \left (a^2+3 b^2\right ) \sin (c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx}{\left (a^2-b^2\right )^2} \\ & = \frac {2 b \sec (c+d x)}{\left (a^2-b^2\right ) d \sqrt {a+b \sin (c+d x)}}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a b-\left (a^2+3 b^2\right ) \sin (c+d x)\right )}{\left (a^2-b^2\right )^2 d}+\frac {a \int \frac {1}{\sqrt {a+b \sin (c+d x)}} \, dx}{2 \left (a^2-b^2\right )}-\frac {\left (a^2+3 b^2\right ) \int \sqrt {a+b \sin (c+d x)} \, dx}{2 \left (a^2-b^2\right )^2} \\ & = \frac {2 b \sec (c+d x)}{\left (a^2-b^2\right ) d \sqrt {a+b \sin (c+d x)}}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a b-\left (a^2+3 b^2\right ) \sin (c+d x)\right )}{\left (a^2-b^2\right )^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 \left (a^2-b^2\right )^2 \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\left (a \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 \left (a^2-b^2\right ) \sqrt {a+b \sin (c+d x)}} \\ & = \frac {2 b \sec (c+d x)}{\left (a^2-b^2\right ) d \sqrt {a+b \sin (c+d x)}}-\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)}}{\left (a^2-b^2\right )^2 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {a \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}}}{\left (a^2-b^2\right ) d \sqrt {a+b \sin (c+d x)}}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a b-\left (a^2+3 b^2\right ) \sin (c+d x)\right )}{\left (a^2-b^2\right )^2 d} \\ \end{align*}
Time = 1.22 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.82 \[ \int \frac {\sec ^2(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=\frac {\left (a^3+a^2 b+3 a b^2+3 b^3\right ) E\left (\frac {1}{4} (-2 c+\pi -2 d x)|\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}-a \left (a^2-b^2\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}-\frac {1}{2} \sec (c+d x) \left (3 a^2 b+b^3+b \left (a^2+3 b^2\right ) \cos (2 (c+d x))-2 a \left (a^2-b^2\right ) \sin (c+d x)\right )}{(a-b)^2 (a+b)^2 d \sqrt {a+b \sin (c+d x)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(1063\) vs. \(2(305)=610\).
Time = 1.56 (sec) , antiderivative size = 1064, normalized size of antiderivative = 4.24
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.12 (sec) , antiderivative size = 685, normalized size of antiderivative = 2.73 \[ \int \frac {\sec ^2(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=\frac {2 \, {\left (\sqrt {2} {\left (a^{3} b - 3 \, a b^{3}\right )} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + \sqrt {2} {\left (a^{4} - 3 \, a^{2} b^{2}\right )} \cos \left (d x + c\right )\right )} \sqrt {i \, b} {\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 \, {\left (\sqrt {2} {\left (a^{3} b - 3 \, a b^{3}\right )} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + \sqrt {2} {\left (a^{4} - 3 \, a^{2} b^{2}\right )} \cos \left (d x + c\right )\right )} \sqrt {-i \, b} {\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 \, {\left (\sqrt {2} {\left (-i \, a^{2} b^{2} - 3 i \, b^{4}\right )} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + \sqrt {2} {\left (-i \, a^{3} b - 3 i \, a b^{3}\right )} \cos \left (d x + c\right )\right )} \sqrt {i \, b} {\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 \, {\left (\sqrt {2} {\left (i \, a^{2} b^{2} + 3 i \, b^{4}\right )} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + \sqrt {2} {\left (i \, a^{3} b + 3 i \, a b^{3}\right )} \cos \left (d x + c\right )\right )} \sqrt {-i \, b} {\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 (a^{2} b^{2} - b^{4} + {\left (a^{2} b^{2} + 3 \, b^{4}\right )} \cos \left (d x + c\right )^{2} - {\left (a^{3} b - a b^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt {b \sin \left (d x + c\right ) + a}}{6 \, {\left ({\left (a^{4} b^{2} - 2 \, a^{2} b^{4} + b^{6}\right )} d \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{5} b - 2 \, a^{3} b^{3} + a b^{5}\right )} d \cos \left (d x + c\right )\right )}} \]
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\[ \int \frac {\sec ^2(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=\int \frac {\sec ^{2}{\left (c + d x \right )}}{\left (a + b \sin {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {\sec ^2(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=\int { \frac {\sec \left (d x + c\right )^{2}}{{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {\sec ^2(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=\int { \frac {\sec \left (d x + c\right )^{2}}{{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\sec ^2(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=\int \frac {1}{{\cos \left (c+d\,x\right )}^2\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^{3/2}} \,d x \]
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