Integrand size = 35, antiderivative size = 249 \[ \int \frac {A+C \sec ^2(c+d x)}{\sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^3} \, dx=-\frac {(119 A+9 C) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{10 a^3 d}+\frac {(11 A+C) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{2 a^3 d}+\frac {(11 A+C) \sin (c+d x)}{2 a^3 d \sqrt {\sec (c+d x)}}-\frac {(A+C) \sin (c+d x)}{5 d \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^3}-\frac {2 A \sin (c+d x)}{3 a d \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^2}-\frac {(119 A+9 C) \sin (c+d x)}{30 d \sqrt {\sec (c+d x)} \left (a^3+a^3 \sec (c+d x)\right )} \]
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Time = 0.59 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4170, 4105, 3872, 3854, 3856, 2720, 2719} \[ \int \frac {A+C \sec ^2(c+d x)}{\sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^3} \, dx=\frac {(11 A+C) \sin (c+d x)}{2 a^3 d \sqrt {\sec (c+d x)}}-\frac {(119 A+9 C) \sin (c+d x)}{30 d \sqrt {\sec (c+d x)} \left (a^3 \sec (c+d x)+a^3\right )}+\frac {(11 A+C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{2 a^3 d}-\frac {(119 A+9 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{10 a^3 d}-\frac {(A+C) \sin (c+d x)}{5 d \sqrt {\sec (c+d x)} (a \sec (c+d x)+a)^3}-\frac {2 A \sin (c+d x)}{3 a d \sqrt {\sec (c+d x)} (a \sec (c+d x)+a)^2} \]
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Rule 2719
Rule 2720
Rule 3854
Rule 3856
Rule 3872
Rule 4105
Rule 4170
Rubi steps \begin{align*} \text {integral}& = -\frac {(A+C) \sin (c+d x)}{5 d \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^3}-\frac {\int \frac {-\frac {1}{2} a (13 A+3 C)+\frac {1}{2} a (7 A-3 C) \sec (c+d x)}{\sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^2} \, dx}{5 a^2} \\ & = -\frac {(A+C) \sin (c+d x)}{5 d \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^3}-\frac {2 A \sin (c+d x)}{3 a d \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^2}-\frac {\int \frac {-\frac {3}{2} a^2 (23 A+3 C)+25 a^2 A \sec (c+d x)}{\sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))} \, dx}{15 a^4} \\ & = -\frac {(A+C) \sin (c+d x)}{5 d \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^3}-\frac {2 A \sin (c+d x)}{3 a d \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^2}-\frac {(119 A+9 C) \sin (c+d x)}{30 d \sqrt {\sec (c+d x)} \left (a^3+a^3 \sec (c+d x)\right )}-\frac {\int \frac {-\frac {45}{4} a^3 (11 A+C)+\frac {3}{4} a^3 (119 A+9 C) \sec (c+d x)}{\sec ^{\frac {3}{2}}(c+d x)} \, dx}{15 a^6} \\ & = -\frac {(A+C) \sin (c+d x)}{5 d \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^3}-\frac {2 A \sin (c+d x)}{3 a d \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^2}-\frac {(119 A+9 C) \sin (c+d x)}{30 d \sqrt {\sec (c+d x)} \left (a^3+a^3 \sec (c+d x)\right )}+\frac {(3 (11 A+C)) \int \frac {1}{\sec ^{\frac {3}{2}}(c+d x)} \, dx}{4 a^3}-\frac {(119 A+9 C) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx}{20 a^3} \\ & = \frac {(11 A+C) \sin (c+d x)}{2 a^3 d \sqrt {\sec (c+d x)}}-\frac {(A+C) \sin (c+d x)}{5 d \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^3}-\frac {2 A \sin (c+d x)}{3 a d \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^2}-\frac {(119 A+9 C) \sin (c+d x)}{30 d \sqrt {\sec (c+d x)} \left (a^3+a^3 \sec (c+d x)\right )}+\frac {(11 A+C) \int \sqrt {\sec (c+d x)} \, dx}{4 a^3}-\frac {\left ((119 A+9 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{20 a^3} \\ & = -\frac {(119 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)}}{10 a^3 d}+\frac {(11 A+C) \sin (c+d x)}{2 a^3 d \sqrt {\sec (c+d x)}}-\frac {(A+C) \sin (c+d x)}{5 d \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^3}-\frac {2 A \sin (c+d x)}{3 a d \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^2}-\frac {(119 A+9 C) \sin (c+d x)}{30 d \sqrt {\sec (c+d x)} \left (a^3+a^3 \sec (c+d x)\right )}+\frac {\left ((11 A+C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{4 a^3} \\ & = -\frac {(119 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)}}{10 a^3 d}+\frac {(11 A+C) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{2 a^3 d}+\frac {(11 A+C) \sin (c+d x)}{2 a^3 d \sqrt {\sec (c+d x)}}-\frac {(A+C) \sin (c+d x)}{5 d \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^3}-\frac {2 A \sin (c+d x)}{3 a d \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^2}-\frac {(119 A+9 C) \sin (c+d x)}{30 d \sqrt {\sec (c+d x)} \left (a^3+a^3 \sec (c+d x)\right )} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.02 (sec) , antiderivative size = 1008, normalized size of antiderivative = 4.05 \[ \int \frac {A+C \sec ^2(c+d x)}{\sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^3} \, dx=\frac {238 \sqrt {2} A e^{-i d x} \sqrt {\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt {1+e^{2 i (c+d x)}} \cos ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) \csc \left (\frac {c}{2}\right ) \left (-3 \sqrt {1+e^{2 i (c+d x)}}+e^{2 i d x} \left (-1+e^{2 i c}\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-e^{2 i (c+d x)}\right )\right ) \sec \left (\frac {c}{2}\right ) \sec (c+d x) \left (A+C \sec ^2(c+d x)\right )}{15 d (A+2 C+A \cos (2 c+2 d x)) (a+a \sec (c+d x))^3}+\frac {6 \sqrt {2} C e^{-i d x} \sqrt {\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt {1+e^{2 i (c+d x)}} \cos ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) \csc \left (\frac {c}{2}\right ) \left (-3 \sqrt {1+e^{2 i (c+d x)}}+e^{2 i d x} \left (-1+e^{2 i c}\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-e^{2 i (c+d x)}\right )\right ) \sec \left (\frac {c}{2}\right ) \sec (c+d x) \left (A+C \sec ^2(c+d x)\right )}{5 d (A+2 C+A \cos (2 c+2 d x)) (a+a \sec (c+d x))^3}+\frac {44 A \cos ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {\cos (c+d x)} \csc \left (\frac {c}{2}\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sec \left (\frac {c}{2}\right ) \sec ^{\frac {3}{2}}(c+d x) \left (A+C \sec ^2(c+d x)\right ) \sin (c)}{d (A+2 C+A \cos (2 c+2 d x)) (a+a \sec (c+d x))^3}+\frac {4 C \cos ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {\cos (c+d x)} \csc \left (\frac {c}{2}\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sec \left (\frac {c}{2}\right ) \sec ^{\frac {3}{2}}(c+d x) \left (A+C \sec ^2(c+d x)\right ) \sin (c)}{d (A+2 C+A \cos (2 c+2 d x)) (a+a \sec (c+d x))^3}+\frac {\cos ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec ^{\frac {3}{2}}(c+d x) \left (A+C \sec ^2(c+d x)\right ) \left (\frac {4 (89 A+9 C+30 A \cos (2 c)) \cos (d x) \csc \left (\frac {c}{2}\right ) \sec \left (\frac {c}{2}\right )}{5 d}+\frac {16 A \cos (2 d x) \sin (2 c)}{3 d}-\frac {4 \sec \left (\frac {c}{2}\right ) \sec ^5\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (A \sin \left (\frac {d x}{2}\right )+C \sin \left (\frac {d x}{2}\right )\right )}{5 d}+\frac {16 \sec \left (\frac {c}{2}\right ) \sec ^3\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (11 A \sin \left (\frac {d x}{2}\right )+6 C \sin \left (\frac {d x}{2}\right )\right )}{15 d}-\frac {8 \sec \left (\frac {c}{2}\right ) \sec \left (\frac {c}{2}+\frac {d x}{2}\right ) \left (43 A \sin \left (\frac {d x}{2}\right )+9 C \sin \left (\frac {d x}{2}\right )\right )}{3 d}-\frac {96 A \cos (c) \sin (d x)}{d}+\frac {16 A \cos (2 c) \sin (2 d x)}{3 d}-\frac {8 (43 A+9 C) \tan \left (\frac {c}{2}\right )}{3 d}+\frac {16 (11 A+6 C) \sec ^2\left (\frac {c}{2}+\frac {d x}{2}\right ) \tan \left (\frac {c}{2}\right )}{15 d}-\frac {4 (A+C) \sec ^4\left (\frac {c}{2}+\frac {d x}{2}\right ) \tan \left (\frac {c}{2}\right )}{5 d}\right )}{(A+2 C+A \cos (2 c+2 d x)) (a+a \sec (c+d x))^3} \]
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Time = 3.69 (sec) , antiderivative size = 465, normalized size of antiderivative = 1.87
method | result | size |
default | \(-\frac {\sqrt {\left (2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \left (160 A \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+468 A \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+330 A \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+714 A \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+108 C \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+30 C \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+54 C \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-1058 A \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}-198 C \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+474 A \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+114 C \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-47 A \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-27 C \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+3 A +3 C \right )}{60 a^{3} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} \sqrt {-2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, d}\) | \(465\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.14 (sec) , antiderivative size = 487, normalized size of antiderivative = 1.96 \[ \int \frac {A+C \sec ^2(c+d x)}{\sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^3} \, dx=-\frac {15 \, {\left (\sqrt {2} {\left (11 i \, A + i \, C\right )} \cos \left (d x + c\right )^{3} + 3 \, \sqrt {2} {\left (11 i \, A + i \, C\right )} \cos \left (d x + c\right )^{2} + 3 \, \sqrt {2} {\left (11 i \, A + i \, C\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (11 i \, A + i \, C\right )}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 15 \, {\left (\sqrt {2} {\left (-11 i \, A - i \, C\right )} \cos \left (d x + c\right )^{3} + 3 \, \sqrt {2} {\left (-11 i \, A - i \, C\right )} \cos \left (d x + c\right )^{2} + 3 \, \sqrt {2} {\left (-11 i \, A - i \, C\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (-11 i \, A - i \, C\right )}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 3 \, {\left (\sqrt {2} {\left (119 i \, A + 9 i \, C\right )} \cos \left (d x + c\right )^{3} + 3 \, \sqrt {2} {\left (119 i \, A + 9 i \, C\right )} \cos \left (d x + c\right )^{2} + 3 \, \sqrt {2} {\left (119 i \, A + 9 i \, C\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (119 i \, A + 9 i \, C\right )}\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 \, {\left (\sqrt {2} {\left (-119 i \, A - 9 i \, C\right )} \cos \left (d x + c\right )^{3} + 3 \, \sqrt {2} {\left (-119 i \, A - 9 i \, C\right )} \cos \left (d x + c\right )^{2} + 3 \, \sqrt {2} {\left (-119 i \, A - 9 i \, C\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (-119 i \, A - 9 i \, C\right )}\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 {2 \, {\left (20 \, A \cos \left (d x + c\right )^{4} + 3 \, {\left (79 \, A + 9 \, C\right )} \cos \left (d x + c\right )^{3} + 4 \, {\left (94 \, A + 9 \, C\right )} \cos \left (d x + c\right )^{2} + 15 \, {\left (11 \, A + C\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{60 \, {\left (a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} + 3 \, a^{3} d \cos \left (d x + c\right ) + a^{3} d\right )}} \]
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Timed out. \[ \int \frac {A+C \sec ^2(c+d x)}{\sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^3} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {A+C \sec ^2(c+d x)}{\sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^3} \, dx=\text {Exception raised: RuntimeError} \]
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\[ \int \frac {A+C \sec ^2(c+d x)}{\sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^3} \, dx=\int { \frac {C \sec \left (d x + c\right )^{2} + A}{{\left (a \sec \left (d x + c\right ) + a\right )}^{3} \sec \left (d x + c\right )^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {A+C \sec ^2(c+d x)}{\sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^3} \, dx=\int \frac {A+\frac {C}{{\cos \left (c+d\,x\right )}^2}}{{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^3\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{3/2}} \,d x \]
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