Integrand size = 23, antiderivative size = 279 \[ \int \frac {\sec ^{\frac {7}{2}}(c+d x)}{(a+b \sec (c+d x))^2} \, dx=-\frac {\left (3 a^2-2 b^2\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{b^2 \left (a^2-b^2\right ) d}-\frac {a \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{b \left (a^2-b^2\right ) d}-\frac {a \left (3 a^2-5 b^2\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 a}{a+b},\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{(a-b) b^2 (a+b)^2 d}+\frac {\left (3 a^2-2 b^2\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{b^2 \left (a^2-b^2\right ) d}-\frac {a^2 \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{b \left (a^2-b^2\right ) d (a+b \sec (c+d x))} \]
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Time = 0.71 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {3930, 4187, 4191, 3934, 2884, 3872, 3856, 2719, 2720} \[ \int \frac {\sec ^{\frac {7}{2}}(c+d x)}{(a+b \sec (c+d x))^2} \, dx=-\frac {a^2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{b d \left (a^2-b^2\right ) (a+b \sec (c+d x))}+\frac {\left (3 a^2-2 b^2\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{b^2 d \left (a^2-b^2\right )}-\frac {a \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{b d \left (a^2-b^2\right )}-\frac {\left (3 a^2-2 b^2\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{b^2 d \left (a^2-b^2\right )}-\frac {a \left (3 a^2-5 b^2\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticPi}\left (\frac {2 a}{a+b},\frac {1}{2} (c+d x),2\right )}{b^2 d (a-b) (a+b)^2} \]
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Rule 2719
Rule 2720
Rule 2884
Rule 3856
Rule 3872
Rule 3930
Rule 3934
Rule 4187
Rule 4191
Rubi steps \begin{align*} \text {integral}& = -\frac {a^2 \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{b \left (a^2-b^2\right ) d (a+b \sec (c+d x))}-\frac {\int \frac {\sqrt {\sec (c+d x)} \left (\frac {a^2}{2}-a b \sec (c+d x)-\frac {1}{2} \left (3 a^2-2 b^2\right ) \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx}{b \left (a^2-b^2\right )} \\ & = \frac {\left (3 a^2-2 b^2\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{b^2 \left (a^2-b^2\right ) d}-\frac {a^2 \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{b \left (a^2-b^2\right ) d (a+b \sec (c+d x))}-\frac {2 \int \frac {\frac {1}{4} a \left (3 a^2-2 b^2\right )+\frac {1}{2} b \left (2 a^2-b^2\right ) \sec (c+d x)+\frac {1}{4} a \left (3 a^2-4 b^2\right ) \sec ^2(c+d x)}{\sqrt {\sec (c+d x)} (a+b \sec (c+d x))} \, dx}{b^2 \left (a^2-b^2\right )} \\ & = \frac {\left (3 a^2-2 b^2\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{b^2 \left (a^2-b^2\right ) d}-\frac {a^2 \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{b \left (a^2-b^2\right ) d (a+b \sec (c+d x))}-\frac {2 \int \frac {\frac {1}{4} a^2 \left (3 a^2-2 b^2\right )-\left (\frac {1}{4} a b \left (3 a^2-2 b^2\right )-\frac {1}{2} a b \left (2 a^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 (a \left (3 a^2-5 b^2\right )\right ) \int \frac {\sec ^{\frac {3}{2}}(c+d x)}{a+b \sec (c+d x)} \, dx}{2 b^2 \left (a^2-b^2\right )} \\ & = \frac {\left (3 a^2-2 b^2\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{b^2 \left (a^2-b^2\right ) d}-\frac {a^2 \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{b \left (a^2-b^2\right ) d (a+b \sec (c+d x))}-\frac {a \int \sqrt {\sec (c+d x)} \, dx}{2 b \left (a^2-b^2\right )}-\frac {\left (3 a^2-2 b^2\right ) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx}{2 b^2 \left (a^2-b^2\right )}-\frac {\left (a \left (3 a^2-5 b^2\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)} (b+a \cos (c+d x))} \, dx}{2 b^2 \left (a^2-b^2\right )} \\ & = -\frac {a \left (3 a^2-5 b^2\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 a}{a+b},\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{(a-b) b^2 (a+b)^2 d}+\frac {\left (3 a^2-2 b^2\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{b^2 \left (a^2-b^2\right ) d}-\frac {a^2 \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{b \left (a^2-b^2\right ) d (a+b \sec (c+d x))}-\frac {\left (a \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{2 b \left (a^2-b^2\right )}-\frac {\left (\left (3 a^2-2 b^2\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{2 b^2 \left (a^2-b^2\right )} \\ & = -\frac {\left (3 a^2-2 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)}}{b^2 \left (a^2-b^2\right ) d}-\frac {a \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{b \left (a^2-b^2\right ) d}-\frac {a \left (3 a^2-5 b^2\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 a}{a+b},\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{(a-b) b^2 (a+b)^2 d}+\frac {\left (3 a^2-2 b^2\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{b^2 \left (a^2-b^2\right ) d}-\frac {a^2 \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{b \left (a^2-b^2\right ) d (a+b \sec (c+d x))} \\ \end{align*}
Time = 3.96 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.26 \[ \int \frac {\sec ^{\frac {7}{2}}(c+d x)}{(a+b \sec (c+d x))^2} \, dx=\frac {\frac {2 b \left (-3 a^3+2 a b^2+2 b \left (-a^2+b^2\right ) \sec (c+d x)\right ) \sin (c+d x)}{\left (-a^2+b^2\right ) (b+a \cos (c+d x)) \sqrt {\sec (c+d x)}}+\frac {\cot (c+d x) \left (-3 a^2 b \sec ^{\frac {3}{2}}(c+d x)+2 b^3 \sec ^{\frac {3}{2}}(c+d x)+3 a^2 b \cos (2 (c+d x)) \sec ^{\frac {3}{2}}(c+d x)-2 b^3 \cos (2 (c+d x)) \sec ^{\frac {3}{2}}(c+d x)+2 b \left (3 a^2-2 b^2\right ) E\left (\left .\arcsin \left (\sqrt {\sec (c+d x)}\right )\right |-1\right ) \sqrt {-\tan ^2(c+d x)}-2 \left (3 a^3+3 a^2 b-4 a b^2-2 b^3\right ) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right ) \sqrt {-\tan ^2(c+d x)}+6 a^3 \operatorname {EllipticPi}\left (-\frac {b}{a},\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right ) \sqrt {-\tan ^2(c+d x)}-10 a b^2 \operatorname {EllipticPi}\left (-\frac {b}{a},\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right ) \sqrt {-\tan ^2(c+d x)}\right )}{(a-b) (a+b)}}{2 b^3 d} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(840\) vs. \(2(341)=682\).
Time = 111.12 (sec) , antiderivative size = 841, normalized size of antiderivative = 3.01
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Timed out. \[ \int \frac {\sec ^{\frac {7}{2}}(c+d x)}{(a+b \sec (c+d x))^2} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {\sec ^{\frac {7}{2}}(c+d x)}{(a+b \sec (c+d x))^2} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {\sec ^{\frac {7}{2}}(c+d x)}{(a+b \sec (c+d x))^2} \, dx=\text {Timed out} \]
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\[ \int \frac {\sec ^{\frac {7}{2}}(c+d x)}{(a+b \sec (c+d x))^2} \, dx=\int { \frac {\sec \left (d x + c\right )^{\frac {7}{2}}}{{\left (b \sec \left (d x + c\right ) + a\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {\sec ^{\frac {7}{2}}(c+d x)}{(a+b \sec (c+d x))^2} \, dx=\int \frac {{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{7/2}}{{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^2} \,d x \]
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