Integrand size = 25, antiderivative size = 441 \[ \int \frac {\cos ^3(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx=-\frac {2 b (4 a+3 b) \cos ^2(e+f x) \sin (e+f x)}{3 a^2 (a+b)^2 f \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )}}-\frac {b \cos ^4(e+f x) \sin (e+f x)}{3 a (a+b) f \left (a+b-a \sin ^2(e+f x)\right ) \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )}}+\frac {\left (a^2+11 a b+8 b^2\right ) \sin (e+f x) \left (a+b-a \sin ^2(e+f x)\right )}{3 a^3 (a+b)^2 f \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )}}+\frac {2 (a+2 b) \left (a^2-4 a b-4 b^2\right ) E\left (\arcsin (\sin (e+f x))\left |\frac {a}{a+b}\right .\right ) \left (a+b-a \sin ^2(e+f x)\right )}{3 a^4 (a+b)^2 f \sqrt {\cos ^2(e+f x)} \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )} \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}}}-\frac {b \left (a^2-16 a b-16 b^2\right ) \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),\frac {a}{a+b}\right ) \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}}}{3 a^4 (a+b) f \sqrt {\cos ^2(e+f x)} \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )}} \]
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Time = 0.71 (sec) , antiderivative size = 441, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {4233, 1985, 1986, 424, 540, 542, 538, 437, 435, 432, 430} \[ \int \frac {\cos ^3(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx=-\frac {2 b (4 a+3 b) \sin (e+f x) \cos ^2(e+f x)}{3 a^2 f (a+b)^2 \sqrt {\sec ^2(e+f x) \left (-a \sin ^2(e+f x)+a+b\right )}}-\frac {b \left (a^2-16 a b-16 b^2\right ) \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),\frac {a}{a+b}\right )}{3 a^4 f (a+b) \sqrt {\cos ^2(e+f x)} \sqrt {\sec ^2(e+f x) \left (-a \sin ^2(e+f x)+a+b\right )}}+\frac {2 (a+2 b) \left (a^2-4 a b-4 b^2\right ) \left (-a \sin ^2(e+f x)+a+b\right ) E\left (\arcsin (\sin (e+f x))\left |\frac {a}{a+b}\right .\right )}{3 a^4 f (a+b)^2 \sqrt {\cos ^2(e+f x)} \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}} \sqrt {\sec ^2(e+f x) \left (-a \sin ^2(e+f x)+a+b\right )}}+\frac {\left (a^2+11 a b+8 b^2\right ) \sin (e+f x) \left (-a \sin ^2(e+f x)+a+b\right )}{3 a^3 f (a+b)^2 \sqrt {\sec ^2(e+f x) \left (-a \sin ^2(e+f x)+a+b\right )}}-\frac {b \sin (e+f x) \cos ^4(e+f x)}{3 a f (a+b) \left (-a \sin ^2(e+f x)+a+b\right ) \sqrt {\sec ^2(e+f x) \left (-a \sin ^2(e+f x)+a+b\right )}} \]
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Rule 424
Rule 430
Rule 432
Rule 435
Rule 437
Rule 538
Rule 540
Rule 542
Rule 1985
Rule 1986
Rule 4233
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1-x^2}{\left (a+\frac {b}{1-x^2}\right )^{5/2}} \, dx,x,\sin (e+f x)\right )}{f} \\ & = \frac {\text {Subst}\left (\int \frac {1-x^2}{\left (\frac {a+b-a x^2}{1-x^2}\right )^{5/2}} \, dx,x,\sin (e+f x)\right )}{f} \\ & = \frac {\sqrt {a+b-a \sin ^2(e+f x)} \text {Subst}\left (\int \frac {\left (1-x^2\right )^{7/2}}{\left (a+b-a x^2\right )^{5/2}} \, dx,x,\sin (e+f x)\right )}{f \sqrt {\cos ^2(e+f x)} \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )}} \\ & = -\frac {b \cos ^4(e+f x) \sin (e+f x)}{3 a (a+b) f \left (a+b-a \sin ^2(e+f x)\right ) \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )}}-\frac {\sqrt {a+b-a \sin ^2(e+f x)} \text {Subst}\left (\int \frac {\left (1-x^2\right )^{3/2} \left (-3 a-b+3 (a+2 b) x^2\right )}{\left (a+b-a x^2\right )^{3/2}} \, dx,x,\sin (e+f x)\right )}{3 a (a+b) f \sqrt {\cos ^2(e+f x)} \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )}} \\ & = -\frac {2 b (4 a+3 b) \cos ^2(e+f x) \sin (e+f x)}{3 a^2 (a+b)^2 f \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )}}-\frac {b \cos ^4(e+f x) \sin (e+f x)}{3 a (a+b) f \left (a+b-a \sin ^2(e+f x)\right ) \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )}}-\frac {\sqrt {a+b-a \sin ^2(e+f x)} \text {Subst}\left (\int \frac {\sqrt {1-x^2} \left (-3 (a+b) (a+2 b)+3 \left (a^2+11 a b+8 b^2\right ) x^2\right )}{\sqrt {a+b-a x^2}} \, dx,x,\sin (e+f x)\right )}{3 a^2 (a+b)^2 f \sqrt {\cos ^2(e+f x)} \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )}} \\ & = -\frac {2 b (4 a+3 b) \cos ^2(e+f x) \sin (e+f x)}{3 a^2 (a+b)^2 f \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )}}-\frac {b \cos ^4(e+f x) \sin (e+f x)}{3 a (a+b) f \left (a+b-a \sin ^2(e+f x)\right ) \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )}}+\frac {\left (a^2+11 a b+8 b^2\right ) \sin (e+f x) \left (a+b-a \sin ^2(e+f x)\right )}{3 a^3 (a+b)^2 f \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )}}+\frac {\sqrt {a+b-a \sin ^2(e+f x)} \text {Subst}\left (\int \frac {3 (a+b) \left (2 a^2-5 a b-8 b^2\right )-6 (a+2 b) \left (a^2-4 a b-4 b^2\right ) x^2}{\sqrt {1-x^2} \sqrt {a+b-a x^2}} \, dx,x,\sin (e+f x)\right )}{9 a^3 (a+b)^2 f \sqrt {\cos ^2(e+f x)} \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )}} \\ & = -\frac {2 b (4 a+3 b) \cos ^2(e+f x) \sin (e+f x)}{3 a^2 (a+b)^2 f \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )}}-\frac {b \cos ^4(e+f x) \sin (e+f x)}{3 a (a+b) f \left (a+b-a \sin ^2(e+f x)\right ) \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )}}+\frac {\left (a^2+11 a b+8 b^2\right ) \sin (e+f x) \left (a+b-a \sin ^2(e+f x)\right )}{3 a^3 (a+b)^2 f \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )}}-\frac {\left (b \left (a^2-16 a b-16 b^2\right ) \sqrt {a+b-a \sin ^2(e+f x)}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {a+b-a x^2}} \, dx,x,\sin (e+f x)\right )}{3 a^4 (a+b) f \sqrt {\cos ^2(e+f x)} \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )}}+\frac {\left (2 (a+2 b) \left (a^2-4 a b-4 b^2\right ) \sqrt {a+b-a \sin ^2(e+f x)}\right ) \text {Subst}\left (\int \frac {\sqrt {a+b-a x^2}}{\sqrt {1-x^2}} \, dx,x,\sin (e+f x)\right )}{3 a^4 (a+b)^2 f \sqrt {\cos ^2(e+f x)} \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )}} \\ & = -\frac {2 b (4 a+3 b) \cos ^2(e+f x) \sin (e+f x)}{3 a^2 (a+b)^2 f \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )}}-\frac {b \cos ^4(e+f x) \sin (e+f x)}{3 a (a+b) f \left (a+b-a \sin ^2(e+f x)\right ) \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )}}+\frac {\left (a^2+11 a b+8 b^2\right ) \sin (e+f x) \left (a+b-a \sin ^2(e+f x)\right )}{3 a^3 (a+b)^2 f \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )}}+\frac {\left (2 (a+2 b) \left (a^2-4 a b-4 b^2\right ) \left (a+b-a \sin ^2(e+f x)\right )\right ) \text {Subst}\left (\int \frac {\sqrt {1-\frac {a x^2}{a+b}}}{\sqrt {1-x^2}} \, dx,x,\sin (e+f x)\right )}{3 a^4 (a+b)^2 f \sqrt {\cos ^2(e+f x)} \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )} \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}}}-\frac {\left (b \left (a^2-16 a b-16 b^2\right ) \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1-\frac {a x^2}{a+b}}} \, dx,x,\sin (e+f x)\right )}{3 a^4 (a+b) f \sqrt {\cos ^2(e+f x)} \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )}} \\ & = -\frac {2 b (4 a+3 b) \cos ^2(e+f x) \sin (e+f x)}{3 a^2 (a+b)^2 f \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )}}-\frac {b \cos ^4(e+f x) \sin (e+f x)}{3 a (a+b) f \left (a+b-a \sin ^2(e+f x)\right ) \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )}}+\frac {\left (a^2+11 a b+8 b^2\right ) \sin (e+f x) \left (a+b-a \sin ^2(e+f x)\right )}{3 a^3 (a+b)^2 f \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )}}+\frac {2 (a+2 b) \left (a^2-4 a b-4 b^2\right ) E\left (\arcsin (\sin (e+f x))\left |\frac {a}{a+b}\right .\right ) \left (a+b-a \sin ^2(e+f x)\right )}{3 a^4 (a+b)^2 f \sqrt {\cos ^2(e+f x)} \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )} \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}}}-\frac {b \left (a^2-16 a b-16 b^2\right ) \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),\frac {a}{a+b}\right ) \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}}}{3 a^4 (a+b) f \sqrt {\cos ^2(e+f x)} \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )}} \\ \end{align*}
\[ \int \frac {\cos ^3(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx=\int \frac {\cos ^3(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx \]
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Result contains complex when optimal does not.
Time = 9.93 (sec) , antiderivative size = 24449, normalized size of antiderivative = 55.44
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\[ \int \frac {\cos ^3(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx=\int { \frac {\cos \left (f x + e\right )^{3}}{{\left (b \sec \left (f x + e\right )^{2} + a\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\cos ^3(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {\cos ^3(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx=\int { \frac {\cos \left (f x + e\right )^{3}}{{\left (b \sec \left (f x + e\right )^{2} + a\right )}^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {\cos ^3(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx=\int { \frac {\cos \left (f x + e\right )^{3}}{{\left (b \sec \left (f x + e\right )^{2} + a\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\cos ^3(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx=\int \frac {{\cos \left (e+f\,x\right )}^3}{{\left (a+\frac {b}{{\cos \left (e+f\,x\right )}^2}\right )}^{5/2}} \,d x \]
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