Integrand size = 35, antiderivative size = 251 \[ \int (a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {5}{2}}(c+d x) \, dx=-\frac {4 a^3 (5 A-9 C) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{5 d}+\frac {4 a^3 (5 A+3 C) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{3 d}-\frac {8 a^3 (10 A-3 C) \sin (c+d x)}{15 d \sqrt {\sec (c+d x)}}-\frac {2 (35 A-3 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{15 d \sqrt {\sec (c+d x)}}+\frac {4 A \left (a^2+a^2 \cos (c+d x)\right )^2 \sqrt {\sec (c+d x)} \sin (c+d x)}{a d}+\frac {2 A (a+a \cos (c+d x))^3 \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{3 d} \]
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Time = 0.74 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.257, Rules used = {4306, 3123, 3054, 3055, 3047, 3102, 2827, 2720, 2719} \[ \int (a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {5}{2}}(c+d x) \, dx=-\frac {8 a^3 (10 A-3 C) \sin (c+d x)}{15 d \sqrt {\sec (c+d x)}}-\frac {2 (35 A-3 C) \sin (c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{15 d \sqrt {\sec (c+d x)}}+\frac {4 a^3 (5 A+3 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}-\frac {4 a^3 (5 A-9 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {4 A \sin (c+d x) \sqrt {\sec (c+d x)} \left (a^2 \cos (c+d x)+a^2\right )^2}{a d}+\frac {2 A \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^3}{3 d} \]
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Rule 2719
Rule 2720
Rule 2827
Rule 3047
Rule 3054
Rule 3055
Rule 3102
Rule 3123
Rule 4306
Rubi steps \begin{align*} \text {integral}& = \left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {(a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {5}{2}}(c+d x)} \, dx \\ & = \frac {2 A (a+a \cos (c+d x))^3 \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{3 d}+\frac {\left (2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {(a+a \cos (c+d x))^3 \left (3 a A-\frac {1}{2} a (5 A-3 C) \cos (c+d x)\right )}{\cos ^{\frac {3}{2}}(c+d x)} \, dx}{3 a} \\ & = \frac {4 A \left (a^2+a^2 \cos (c+d x)\right )^2 \sqrt {\sec (c+d x)} \sin (c+d x)}{a d}+\frac {2 A (a+a \cos (c+d x))^3 \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{3 d}+\frac {\left (4 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {(a+a \cos (c+d x))^2 \left (\frac {1}{4} a^2 (25 A+3 C)-\frac {1}{4} a^2 (35 A-3 C) \cos (c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx}{3 a} \\ & = -\frac {2 (35 A-3 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{15 d \sqrt {\sec (c+d x)}}+\frac {4 A \left (a^2+a^2 \cos (c+d x)\right )^2 \sqrt {\sec (c+d x)} \sin (c+d x)}{a d}+\frac {2 A (a+a \cos (c+d x))^3 \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{3 d}+\frac {\left (8 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {(a+a \cos (c+d x)) \left (\frac {9}{4} a^3 (5 A+C)-\frac {3}{2} a^3 (10 A-3 C) \cos (c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx}{15 a} \\ & = -\frac {2 (35 A-3 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{15 d \sqrt {\sec (c+d x)}}+\frac {4 A \left (a^2+a^2 \cos (c+d x)\right )^2 \sqrt {\sec (c+d x)} \sin (c+d x)}{a d}+\frac {2 A (a+a \cos (c+d x))^3 \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{3 d}+\frac {\left (8 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\frac {9}{4} a^4 (5 A+C)+\left (-\frac {3}{2} a^4 (10 A-3 C)+\frac {9}{4} a^4 (5 A+C)\right ) \cos (c+d x)-\frac {3}{2} a^4 (10 A-3 C) \cos ^2(c+d x)}{\sqrt {\cos (c+d x)}} \, dx}{15 a} \\ & = -\frac {8 a^3 (10 A-3 C) \sin (c+d x)}{15 d \sqrt {\sec (c+d x)}}-\frac {2 (35 A-3 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{15 d \sqrt {\sec (c+d x)}}+\frac {4 A \left (a^2+a^2 \cos (c+d x)\right )^2 \sqrt {\sec (c+d x)} \sin (c+d x)}{a d}+\frac {2 A (a+a \cos (c+d x))^3 \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{3 d}+\frac {\left (16 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\frac {15}{8} a^4 (5 A+3 C)-\frac {9}{8} a^4 (5 A-9 C) \cos (c+d x)}{\sqrt {\cos (c+d x)}} \, dx}{45 a} \\ & = -\frac {8 a^3 (10 A-3 C) \sin (c+d x)}{15 d \sqrt {\sec (c+d x)}}-\frac {2 (35 A-3 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{15 d \sqrt {\sec (c+d x)}}+\frac {4 A \left (a^2+a^2 \cos (c+d x)\right )^2 \sqrt {\sec (c+d x)} \sin (c+d x)}{a d}+\frac {2 A (a+a \cos (c+d x))^3 \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{3 d}-\frac {1}{5} \left (2 a^3 (5 A-9 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx+\frac {1}{3} \left (2 a^3 (5 A+3 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx \\ & = -\frac {4 a^3 (5 A-9 C) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{5 d}+\frac {4 a^3 (5 A+3 C) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{3 d}-\frac {8 a^3 (10 A-3 C) \sin (c+d x)}{15 d \sqrt {\sec (c+d x)}}-\frac {2 (35 A-3 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{15 d \sqrt {\sec (c+d x)}}+\frac {4 A \left (a^2+a^2 \cos (c+d x)\right )^2 \sqrt {\sec (c+d x)} \sin (c+d x)}{a d}+\frac {2 A (a+a \cos (c+d x))^3 \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{3 d} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 4.17 (sec) , antiderivative size = 221, normalized size of antiderivative = 0.88 \[ \int (a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {5}{2}}(c+d x) \, dx=\frac {a^3 e^{-i d x} \sec ^{\frac {3}{2}}(c+d x) (\cos (d x)+i \sin (d x)) \left (-120 i A+216 i C-120 i A \cos (2 (c+d x))+216 i C \cos (2 (c+d x))+80 (5 A+3 C) \cos ^{\frac {3}{2}}(c+d x) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+8 i (5 A-9 C) \left (1+e^{2 i (c+d x)}\right )^{3/2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-e^{2 i (c+d x)}\right )+40 A \sin (c+d x)+30 C \sin (c+d x)+180 A \sin (2 (c+d x))+6 C \sin (2 (c+d x))+30 C \sin (3 (c+d x))+3 C \sin (4 (c+d x))\right )}{60 d} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(703\) vs. \(2(277)=554\).
Time = 494.78 (sec) , antiderivative size = 704, normalized size of antiderivative = 2.80
method | result | size |
default | \(\text {Expression too large to display}\) | \(704\) |
parts | \(\text {Expression too large to display}\) | \(1078\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.11 (sec) , antiderivative size = 239, normalized size of antiderivative = 0.95 \[ \int (a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {5}{2}}(c+d x) \, dx=-\frac {2 \, {\left (5 i \, \sqrt {2} {\left (5 \, A + 3 \, C\right )} a^{3} \cos \left (d x + c\right ) {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 5 i \, \sqrt {2} {\left (5 \, A + 3 \, C\right )} a^{3} \cos \left (d x + c\right ) {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 3 i \, \sqrt {2} {\left (5 \, A - 9 \, C\right )} a^{3} \cos \left (d x + c\right ) {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 3 i \, \sqrt {2} {\left (5 \, A - 9 \, C\right )} a^{3} \cos \left (d x + c\right ) {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - \frac {{\left (3 \, C a^{3} \cos \left (d x + c\right )^{3} + 15 \, C a^{3} \cos \left (d x + c\right )^{2} + 45 \, A a^{3} \cos \left (d x + c\right ) + 5 \, A a^{3}\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}\right )}}{15 \, d \cos \left (d x + c\right )} \]
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Timed out. \[ \int (a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {5}{2}}(c+d x) \, dx=\text {Timed out} \]
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\[ \int (a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {5}{2}}(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (a \cos \left (d x + c\right ) + a\right )}^{3} \sec \left (d x + c\right )^{\frac {5}{2}} \,d x } \]
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\[ \int (a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {5}{2}}(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (a \cos \left (d x + c\right ) + a\right )}^{3} \sec \left (d x + c\right )^{\frac {5}{2}} \,d x } \]
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Timed out. \[ \int (a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {5}{2}}(c+d x) \, dx=\int \left (C\,{\cos \left (c+d\,x\right )}^2+A\right )\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{5/2}\,{\left (a+a\,\cos \left (c+d\,x\right )\right )}^3 \,d x \]
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