Integrand size = 43, antiderivative size = 271 \[ \int \frac {(a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {9}{2}}(c+d x)} \, dx=\frac {4 a^3 (17 A+21 B+27 C) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{15 d}+\frac {4 a^3 (11 A+13 B+21 C) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{21 d}+\frac {4 a^3 (32 A+41 B+42 C) \sin (c+d x)}{105 d \sqrt {\sec (c+d x)}}+\frac {2 A (a+a \sec (c+d x))^3 \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 (2 A+3 B) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{21 a d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 (73 A+99 B+63 C) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{315 d \sec ^{\frac {3}{2}}(c+d x)} \]
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Time = 0.71 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.163, Rules used = {4171, 4102, 4081, 3872, 3856, 2719, 2720} \[ \int \frac {(a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {9}{2}}(c+d x)} \, dx=\frac {2 (73 A+99 B+63 C) \sin (c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{315 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {4 a^3 (32 A+41 B+42 C) \sin (c+d x)}{105 d \sqrt {\sec (c+d x)}}+\frac {4 a^3 (11 A+13 B+21 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d}+\frac {4 a^3 (17 A+21 B+27 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 (2 A+3 B) \sin (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{21 a d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 A \sin (c+d x) (a \sec (c+d x)+a)^3}{9 d \sec ^{\frac {7}{2}}(c+d x)} \]
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Rule 2719
Rule 2720
Rule 3856
Rule 3872
Rule 4081
Rule 4102
Rule 4171
Rubi steps \begin{align*} \text {integral}& = \frac {2 A (a+a \sec (c+d x))^3 \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 \int \frac {(a+a \sec (c+d x))^3 \left (\frac {3}{2} a (2 A+3 B)+\frac {1}{2} a (A+9 C) \sec (c+d x)\right )}{\sec ^{\frac {7}{2}}(c+d x)} \, dx}{9 a} \\ & = \frac {2 A (a+a \sec (c+d x))^3 \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 (2 A+3 B) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{21 a d \sec ^{\frac {5}{2}}(c+d x)}+\frac {4 \int \frac {(a+a \sec (c+d x))^2 \left (\frac {1}{4} a^2 (73 A+99 B+63 C)+\frac {1}{4} a^2 (13 A+9 B+63 C) \sec (c+d x)\right )}{\sec ^{\frac {5}{2}}(c+d x)} \, dx}{63 a} \\ & = \frac {2 A (a+a \sec (c+d x))^3 \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 (2 A+3 B) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{21 a d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 (73 A+99 B+63 C) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{315 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {8 \int \frac {(a+a \sec (c+d x)) \left (\frac {9}{4} a^3 (32 A+41 B+42 C)+\frac {3}{4} a^3 (23 A+24 B+63 C) \sec (c+d x)\right )}{\sec ^{\frac {3}{2}}(c+d x)} \, dx}{315 a} \\ & = \frac {4 a^3 (32 A+41 B+42 C) \sin (c+d x)}{105 d \sqrt {\sec (c+d x)}}+\frac {2 A (a+a \sec (c+d x))^3 \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 (2 A+3 B) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{21 a d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 (73 A+99 B+63 C) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{315 d \sec ^{\frac {3}{2}}(c+d x)}-\frac {16 \int \frac {-\frac {63}{8} a^4 (17 A+21 B+27 C)-\frac {45}{8} a^4 (11 A+13 B+21 C) \sec (c+d x)}{\sqrt {\sec (c+d x)}} \, dx}{945 a} \\ & = \frac {4 a^3 (32 A+41 B+42 C) \sin (c+d x)}{105 d \sqrt {\sec (c+d x)}}+\frac {2 A (a+a \sec (c+d x))^3 \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 (2 A+3 B) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{21 a d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 (73 A+99 B+63 C) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{315 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {1}{21} \left (2 a^3 (11 A+13 B+21 C)\right ) \int \sqrt {\sec (c+d x)} \, dx+\frac {1}{15} \left (2 a^3 (17 A+21 B+27 C)\right ) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx \\ & = \frac {4 a^3 (32 A+41 B+42 C) \sin (c+d x)}{105 d \sqrt {\sec (c+d x)}}+\frac {2 A (a+a \sec (c+d x))^3 \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 (2 A+3 B) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{21 a d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 (73 A+99 B+63 C) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{315 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {1}{21} \left (2 a^3 (11 A+13 B+21 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx+\frac {1}{15} \left (2 a^3 (17 A+21 B+27 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx \\ & = \frac {4 a^3 (17 A+21 B+27 C) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{15 d}+\frac {4 a^3 (11 A+13 B+21 C) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{21 d}+\frac {4 a^3 (32 A+41 B+42 C) \sin (c+d x)}{105 d \sqrt {\sec (c+d x)}}+\frac {2 A (a+a \sec (c+d x))^3 \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 (2 A+3 B) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{21 a d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 (73 A+99 B+63 C) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{315 d \sec ^{\frac {3}{2}}(c+d x)} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 6.31 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.79 \[ \int \frac {(a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {9}{2}}(c+d x)} \, dx=\frac {a^3 \sqrt {\sec (c+d x)} \left (480 (11 A+13 B+21 C) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )-224 i (17 A+21 B+27 C) e^{i (c+d x)} \sqrt {1+e^{2 i (c+d x)}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-e^{2 i (c+d x)}\right )+2 \cos (c+d x) (5712 i A+7056 i B+9072 i C+30 (97 A+107 B+84 C) \sin (c+d x)+14 (73 A+54 B+18 C) \sin (2 (c+d x))+270 A \sin (3 (c+d x))+90 B \sin (3 (c+d x))+35 A \sin (4 (c+d x)))\right )}{2520 d} \]
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Time = 8.76 (sec) , antiderivative size = 514, normalized size of antiderivative = 1.90
method | result | size |
default | \(-\frac {4 \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}}\, a^{3} \left (-560 A \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+\left (2200 A +360 B \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{8} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-3412 A -1296 B -252 C \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (2702 A +1806 B +882 C \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-738 A -624 B -378 C \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+165 A \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \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 )-357 A \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \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 )+195 B \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \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 )-441 B \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \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 )+315 C \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \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 )-567 C \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \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 )\right )}{315 \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}\) | \(514\) |
parts | \(\text {Expression too large to display}\) | \(1122\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.12 (sec) , antiderivative size = 249, normalized size of antiderivative = 0.92 \[ \int \frac {(a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {9}{2}}(c+d x)} \, dx=-\frac {2 \, {\left (15 i \, \sqrt {2} {\left (11 \, A + 13 \, B + 21 \, C\right )} a^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 15 i \, \sqrt {2} {\left (11 \, A + 13 \, B + 21 \, C\right )} a^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 21 i \, \sqrt {2} {\left (17 \, A + 21 \, B + 27 \, C\right )} a^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 21 i \, \sqrt {2} {\left (17 \, A + 21 \, B + 27 \, C\right )} a^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - \frac {{\left (35 \, A a^{3} \cos \left (d x + c\right )^{4} + 45 \, {\left (3 \, A + B\right )} a^{3} \cos \left (d x + c\right )^{3} + 7 \, {\left (34 \, A + 27 \, B + 9 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} + 15 \, {\left (22 \, A + 26 \, B + 21 \, C\right )} a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}\right )}}{315 \, d} \]
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Timed out. \[ \int \frac {(a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {9}{2}}(c+d x)} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {(a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {9}{2}}(c+d x)} \, dx=\text {Timed out} \]
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\[ \int \frac {(a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {9}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{3}}{\sec \left (d x + c\right )^{\frac {9}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {9}{2}}(c+d x)} \, dx=\int \frac {{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^3\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{9/2}} \,d x \]
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