Integrand size = 23, antiderivative size = 277 \[ \int \frac {\sec ^{\frac {3}{2}}(c+d x)}{(a+b \cos (c+d x))^2} \, dx=-\frac {\left (2 a^2-3 b^2\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{a^2 \left (a^2-b^2\right ) d}+\frac {b \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{a \left (a^2-b^2\right ) d}-\frac {b \left (5 a^2-3 b^2\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{a^2 (a-b) (a+b)^2 d}+\frac {\left (2 a^2-3 b^2\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{a^2 \left (a^2-b^2\right ) d}+\frac {b^2 \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{a \left (a^2-b^2\right ) d (b+a \sec (c+d x))} \]
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Time = 0.75 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {3317, 3930, 4187, 4191, 3934, 2884, 3872, 3856, 2719, 2720} \[ \int \frac {\sec ^{\frac {3}{2}}(c+d x)}{(a+b \cos (c+d x))^2} \, dx=\frac {b^2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)}+\frac {\left (2 a^2-3 b^2\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{a^2 d \left (a^2-b^2\right )}+\frac {b \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{a d \left (a^2-b^2\right )}-\frac {\left (2 a^2-3 b^2\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{a^2 d \left (a^2-b^2\right )}-\frac {b \left (5 a^2-3 b^2\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right )}{a^2 d (a-b) (a+b)^2} \]
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Rule 2719
Rule 2720
Rule 2884
Rule 3317
Rule 3856
Rule 3872
Rule 3930
Rule 3934
Rule 4187
Rule 4191
Rubi steps \begin{align*} \text {integral}& = \int \frac {\sec ^{\frac {7}{2}}(c+d x)}{(b+a \sec (c+d x))^2} \, dx \\ & = \frac {b^2 \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{a \left (a^2-b^2\right ) d (b+a \sec (c+d x))}+\frac {\int \frac {\sqrt {\sec (c+d x)} \left (\frac {b^2}{2}-a b \sec (c+d x)+\frac {1}{2} \left (2 a^2-3 b^2\right ) \sec ^2(c+d x)\right )}{b+a \sec (c+d x)} \, dx}{a \left (a^2-b^2\right )} \\ & = \frac {\left (2 a^2-3 b^2\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{a^2 \left (a^2-b^2\right ) d}+\frac {b^2 \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{a \left (a^2-b^2\right ) d (b+a \sec (c+d x))}+\frac {2 \int \frac {-\frac {1}{4} b \left (2 a^2-3 b^2\right )-\frac {1}{2} a \left (a^2-2 b^2\right ) \sec (c+d x)-\frac {1}{4} b \left (4 a^2-3 b^2\right ) \sec ^2(c+d x)}{\sqrt {\sec (c+d x)} (b+a \sec (c+d x))} \, dx}{a^2 \left (a^2-b^2\right )} \\ & = \frac {\left (2 a^2-3 b^2\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{a^2 \left (a^2-b^2\right ) d}+\frac {b^2 \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{a \left (a^2-b^2\right ) d (b+a \sec (c+d x))}+\frac {2 \int \frac {-\frac {1}{4} b^2 \left (2 a^2-3 b^2\right )-\left (-\frac {1}{4} a b \left (2 a^2-3 b^2\right )+\frac {1}{2} a b \left (a^2-2 b^2\right )\right ) \sec (c+d x)}{\sqrt {\sec (c+d x)}} \, dx}{a^2 b^2 \left (a^2-b^2\right )}-\frac {\left (b \left (5 a^2-3 b^2\right )\right ) \int \frac {\sec ^{\frac {3}{2}}(c+d x)}{b+a \sec (c+d x)} \, dx}{2 a^2 \left (a^2-b^2\right )} \\ & = \frac {\left (2 a^2-3 b^2\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{a^2 \left (a^2-b^2\right ) d}+\frac {b^2 \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{a \left (a^2-b^2\right ) d (b+a \sec (c+d x))}+\frac {b \int \sqrt {\sec (c+d x)} \, dx}{2 a \left (a^2-b^2\right )}-\frac {\left (2 a^2-3 b^2\right ) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx}{2 a^2 \left (a^2-b^2\right )}-\frac {\left (b \left (5 a^2-3 b^2\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))} \, dx}{2 a^2 \left (a^2-b^2\right )} \\ & = -\frac {b \left (5 a^2-3 b^2\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{a^2 (a-b) (a+b)^2 d}+\frac {\left (2 a^2-3 b^2\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{a^2 \left (a^2-b^2\right ) d}+\frac {b^2 \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{a \left (a^2-b^2\right ) d (b+a \sec (c+d x))}+\frac {\left (b \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{2 a \left (a^2-b^2\right )}-\frac {\left (\left (2 a^2-3 b^2\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{2 a^2 \left (a^2-b^2\right )} \\ & = -\frac {\left (2 a^2-3 b^2\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{a^2 \left (a^2-b^2\right ) d}+\frac {b \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{a \left (a^2-b^2\right ) d}-\frac {b \left (5 a^2-3 b^2\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{a^2 (a-b) (a+b)^2 d}+\frac {\left (2 a^2-3 b^2\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{a^2 \left (a^2-b^2\right ) d}+\frac {b^2 \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{a \left (a^2-b^2\right ) d (b+a \sec (c+d x))} \\ \end{align*}
Time = 2.78 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.27 \[ \int \frac {\sec ^{\frac {3}{2}}(c+d x)}{(a+b \cos (c+d x))^2} \, dx=\frac {\frac {2 a \left (2 a^2 b-3 b^3+2 a \left (a^2-b^2\right ) \sec (c+d x)\right ) \sin (c+d x)}{\left (a^2-b^2\right ) (a+b \cos (c+d x)) \sqrt {\sec (c+d x)}}+\frac {\cot (c+d x) \left (-2 a^3 \sec ^{\frac {3}{2}}(c+d x)+3 a b^2 \sec ^{\frac {3}{2}}(c+d x)+2 a^3 \cos (2 (c+d x)) \sec ^{\frac {3}{2}}(c+d x)-3 a b^2 \cos (2 (c+d x)) \sec ^{\frac {3}{2}}(c+d x)+2 a \left (2 a^2-3 b^2\right ) E\left (\left .\arcsin \left (\sqrt {\sec (c+d x)}\right )\right |-1\right ) \sqrt {-\tan ^2(c+d x)}-2 \left (2 a^3+4 a^2 b-3 a b^2-3 b^3\right ) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right ) \sqrt {-\tan ^2(c+d x)}+10 a^2 b \operatorname {EllipticPi}\left (-\frac {a}{b},\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right ) \sqrt {-\tan ^2(c+d x)}-6 b^3 \operatorname {EllipticPi}\left (-\frac {a}{b},\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right ) \sqrt {-\tan ^2(c+d x)}\right )}{(a-b) (a+b)}}{2 a^3 d} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(846\) vs. \(2(339)=678\).
Time = 5.98 (sec) , antiderivative size = 847, normalized size of antiderivative = 3.06
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Timed out. \[ \int \frac {\sec ^{\frac {3}{2}}(c+d x)}{(a+b \cos (c+d x))^2} \, dx=\text {Timed out} \]
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\[ \int \frac {\sec ^{\frac {3}{2}}(c+d x)}{(a+b \cos (c+d x))^2} \, dx=\int \frac {\sec ^{\frac {3}{2}}{\left (c + d x \right )}}{\left (a + b \cos {\left (c + d x \right )}\right )^{2}}\, dx \]
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\[ \int \frac {\sec ^{\frac {3}{2}}(c+d x)}{(a+b \cos (c+d x))^2} \, dx=\int { \frac {\sec \left (d x + c\right )^{\frac {3}{2}}}{{\left (b \cos \left (d x + c\right ) + a\right )}^{2}} \,d x } \]
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\[ \int \frac {\sec ^{\frac {3}{2}}(c+d x)}{(a+b \cos (c+d x))^2} \, dx=\int { \frac {\sec \left (d x + c\right )^{\frac {3}{2}}}{{\left (b \cos \left (d x + c\right ) + a\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {\sec ^{\frac {3}{2}}(c+d x)}{(a+b \cos (c+d x))^2} \, dx=\int \frac {{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{3/2}}{{\left (a+b\,\cos \left (c+d\,x\right )\right )}^2} \,d x \]
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