Integrand size = 23, antiderivative size = 255 \[ \int \frac {1}{\cos ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))^3} \, dx=\frac {3 a \left (a^2-3 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{4 b^2 \left (a^2-b^2\right )^2 d}+\frac {\left (a^2-7 b^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{4 b \left (a^2-b^2\right )^2 d}+\frac {3 \left (a^4-2 a^2 b^2+5 b^4\right ) \operatorname {EllipticPi}\left (\frac {2 a}{a+b},\frac {1}{2} (c+d x),2\right )}{4 (a-b)^2 b^2 (a+b)^3 d}-\frac {a^2 \sin (c+d x)}{2 b \left (a^2-b^2\right ) d \cos ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2}-\frac {3 a^2 \left (a^2-3 b^2\right ) \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d \sqrt {\cos (c+d x)} (a+b \sec (c+d x))} \]
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Time = 0.90 (sec) , antiderivative size = 255, 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 = {4349, 3930, 4183, 4191, 3934, 2884, 3872, 3856, 2719, 2720} \[ \int \frac {1}{\cos ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))^3} \, dx=\frac {\left (a^2-7 b^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{4 b d \left (a^2-b^2\right )^2}+\frac {3 a \left (a^2-3 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{4 b^2 d \left (a^2-b^2\right )^2}-\frac {a^2 \sin (c+d x)}{2 b d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2}-\frac {3 a^2 \left (a^2-3 b^2\right ) \sin (c+d x)}{4 b^2 d \left (a^2-b^2\right )^2 \sqrt {\cos (c+d x)} (a+b \sec (c+d x))}+\frac {3 \left (a^4-2 a^2 b^2+5 b^4\right ) \operatorname {EllipticPi}\left (\frac {2 a}{a+b},\frac {1}{2} (c+d x),2\right )}{4 b^2 d (a-b)^2 (a+b)^3} \]
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Rule 2719
Rule 2720
Rule 2884
Rule 3856
Rule 3872
Rule 3930
Rule 3934
Rule 4183
Rule 4191
Rule 4349
Rubi steps \begin{align*} \text {integral}& = \left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sec ^{\frac {7}{2}}(c+d x)}{(a+b \sec (c+d x))^3} \, dx \\ & = -\frac {a^2 \sin (c+d x)}{2 b \left (a^2-b^2\right ) d \cos ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2}-\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sqrt {\sec (c+d x)} \left (\frac {a^2}{2}-2 a b \sec (c+d x)-\frac {1}{2} \left (3 a^2-4 b^2\right ) \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^2} \, dx}{2 b \left (a^2-b^2\right )} \\ & = -\frac {a^2 \sin (c+d x)}{2 b \left (a^2-b^2\right ) d \cos ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2}-\frac {3 a^2 \left (a^2-3 b^2\right ) \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d \sqrt {\cos (c+d x)} (a+b \sec (c+d x))}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\frac {3}{4} a^2 \left (a^2-3 b^2\right )+a b \left (a^2-4 b^2\right ) \sec (c+d x)+\frac {1}{4} \left (3 a^4-5 a^2 b^2+8 b^4\right ) \sec ^2(c+d x)}{\sqrt {\sec (c+d x)} (a+b \sec (c+d x))} \, dx}{2 b^2 \left (a^2-b^2\right )^2} \\ & = -\frac {a^2 \sin (c+d x)}{2 b \left (a^2-b^2\right ) d \cos ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2}-\frac {3 a^2 \left (a^2-3 b^2\right ) \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d \sqrt {\cos (c+d x)} (a+b \sec (c+d x))}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\frac {3}{4} a^3 \left (a^2-3 b^2\right )-\left (-a^2 b \left (a^2-4 b^2\right )+\frac {3}{4} a^2 b \left (a^2-3 b^2\right )\right ) \sec (c+d x)}{\sqrt {\sec (c+d x)}} \, dx}{2 a^2 b^2 \left (a^2-b^2\right )^2}+\frac {\left (3 \left (a^4-2 a^2 b^2+5 b^4\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sec ^{\frac {3}{2}}(c+d x)}{a+b \sec (c+d x)} \, dx}{8 b^2 \left (a^2-b^2\right )^2} \\ & = -\frac {a^2 \sin (c+d x)}{2 b \left (a^2-b^2\right ) d \cos ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2}-\frac {3 a^2 \left (a^2-3 b^2\right ) \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d \sqrt {\cos (c+d x)} (a+b \sec (c+d x))}+\frac {\left (3 \left (a^4-2 a^2 b^2+5 b^4\right )\right ) \int \frac {1}{\sqrt {\cos (c+d x)} (b+a \cos (c+d x))} \, dx}{8 b^2 \left (a^2-b^2\right )^2}+\frac {\left (\left (a^2-7 b^2\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\sec (c+d x)} \, dx}{8 b \left (a^2-b^2\right )^2}+\frac {\left (3 a \left (a^2-3 b^2\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx}{8 b^2 \left (a^2-b^2\right )^2} \\ & = \frac {3 \left (a^4-2 a^2 b^2+5 b^4\right ) \operatorname {EllipticPi}\left (\frac {2 a}{a+b},\frac {1}{2} (c+d x),2\right )}{4 (a-b)^2 b^2 (a+b)^3 d}-\frac {a^2 \sin (c+d x)}{2 b \left (a^2-b^2\right ) d \cos ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2}-\frac {3 a^2 \left (a^2-3 b^2\right ) \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d \sqrt {\cos (c+d x)} (a+b \sec (c+d x))}+\frac {\left (a^2-7 b^2\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{8 b \left (a^2-b^2\right )^2}+\frac {\left (3 a \left (a^2-3 b^2\right )\right ) \int \sqrt {\cos (c+d x)} \, dx}{8 b^2 \left (a^2-b^2\right )^2} \\ & = \frac {3 a \left (a^2-3 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{4 b^2 \left (a^2-b^2\right )^2 d}+\frac {\left (a^2-7 b^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{4 b \left (a^2-b^2\right )^2 d}+\frac {3 \left (a^4-2 a^2 b^2+5 b^4\right ) \operatorname {EllipticPi}\left (\frac {2 a}{a+b},\frac {1}{2} (c+d x),2\right )}{4 (a-b)^2 b^2 (a+b)^3 d}-\frac {a^2 \sin (c+d x)}{2 b \left (a^2-b^2\right ) d \cos ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2}-\frac {3 a^2 \left (a^2-3 b^2\right ) \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d \sqrt {\cos (c+d x)} (a+b \sec (c+d x))} \\ \end{align*}
Time = 3.15 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.16 \[ \int \frac {1}{\cos ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))^3} \, dx=\frac {-\frac {4 a^2 \sqrt {\cos (c+d x)} \left (5 a^2 b-11 b^3+3 a \left (a^2-3 b^2\right ) \cos (c+d x)\right ) \sin (c+d x)}{\left (a^2-b^2\right )^2 (b+a \cos (c+d x))^2}+\frac {\frac {2 \left (9 a^4-19 a^2 b^2+16 b^4\right ) \operatorname {EllipticPi}\left (\frac {2 a}{a+b},\frac {1}{2} (c+d x),2\right )}{a+b}+\frac {16 b \left (a^2-4 b^2\right ) \left ((a+b) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )-b \operatorname {EllipticPi}\left (\frac {2 a}{a+b},\frac {1}{2} (c+d x),2\right )\right )}{a+b}+\frac {6 \left (a^2-3 b^2\right ) \left (-2 a b E\left (\left .\arcsin \left (\sqrt {\cos (c+d x)}\right )\right |-1\right )+2 b (a+b) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\cos (c+d x)}\right ),-1\right )+\left (a^2-2 b^2\right ) \operatorname {EllipticPi}\left (-\frac {a}{b},\arcsin \left (\sqrt {\cos (c+d x)}\right ),-1\right )\right ) \sin (c+d x)}{b \sqrt {\sin ^2(c+d x)}}}{(a-b)^2 (a+b)^2}}{16 b^2 d} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(1202\) vs. \(2(319)=638\).
Time = 15.59 (sec) , antiderivative size = 1203, normalized size of antiderivative = 4.72
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Timed out. \[ \int \frac {1}{\cos ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))^3} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {1}{\cos ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))^3} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {1}{\cos ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))^3} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{\cos ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))^3} \, dx=\int { \frac {1}{{\left (b \sec \left (d x + c\right ) + a\right )}^{3} \cos \left (d x + c\right )^{\frac {7}{2}}} \,d x } \]
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Timed out. \[ \int \frac {1}{\cos ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))^3} \, dx=\int \frac {1}{{\cos \left (c+d\,x\right )}^{7/2}\,{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^3} \,d x \]
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