Optimal. Leaf size=286 \[ \frac {4 a^3 (5 A+7 C) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{5 d}+\frac {4 a^3 (105 A+143 C) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{231 d}+\frac {8 a^3 (35 A+44 C) \sin (c+d x)}{385 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {4 a^3 (105 A+143 C) \sin (c+d x)}{231 d \sqrt {\sec (c+d x)}}+\frac {2 A (a+a \sec (c+d x))^3 \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)}+\frac {4 A \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{33 a d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 (35 A+33 C) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{231 d \sec ^{\frac {5}{2}}(c+d x)} \]
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Rubi [A]
time = 0.41, antiderivative size = 286, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 8, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.229, Rules used = {4172, 4102,
4081, 3872, 3854, 3856, 2720, 2719} \begin {gather*} \frac {8 a^3 (35 A+44 C) \sin (c+d x)}{385 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 (35 A+33 C) \sin (c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{231 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {4 a^3 (105 A+143 C) \sin (c+d x)}{231 d \sqrt {\sec (c+d x)}}+\frac {4 a^3 (105 A+143 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{231 d}+\frac {4 a^3 (5 A+7 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) \left (a^2 \sec (c+d x)+a^2\right )^2}{33 a d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 A \sin (c+d x) (a \sec (c+d x)+a)^3}{11 d \sec ^{\frac {9}{2}}(c+d x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2719
Rule 2720
Rule 3854
Rule 3856
Rule 3872
Rule 4081
Rule 4102
Rule 4172
Rubi steps
\begin {align*} \int \frac {(a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right )}{\sec ^{\frac {11}{2}}(c+d x)} \, dx &=\frac {2 A (a+a \sec (c+d x))^3 \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)}+\frac {2 \int \frac {(a+a \sec (c+d x))^3 \left (3 a A+\frac {1}{2} a (3 A+11 C) \sec (c+d x)\right )}{\sec ^{\frac {9}{2}}(c+d x)} \, dx}{11 a}\\ &=\frac {2 A (a+a \sec (c+d x))^3 \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)}+\frac {4 A \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{33 a d \sec ^{\frac {7}{2}}(c+d x)}+\frac {4 \int \frac {(a+a \sec (c+d x))^2 \left (\frac {3}{4} a^2 (35 A+33 C)+\frac {9}{4} a^2 (5 A+11 C) \sec (c+d x)\right )}{\sec ^{\frac {7}{2}}(c+d x)} \, dx}{99 a}\\ &=\frac {2 A (a+a \sec (c+d x))^3 \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)}+\frac {4 A \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{33 a d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 (35 A+33 C) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{231 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {8 \int \frac {(a+a \sec (c+d x)) \left (\frac {9}{2} a^3 (35 A+44 C)+\frac {45}{4} a^3 (7 A+11 C) \sec (c+d x)\right )}{\sec ^{\frac {5}{2}}(c+d x)} \, dx}{693 a}\\ &=\frac {8 a^3 (35 A+44 C) \sin (c+d x)}{385 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 A (a+a \sec (c+d x))^3 \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)}+\frac {4 A \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{33 a d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 (35 A+33 C) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{231 d \sec ^{\frac {5}{2}}(c+d x)}-\frac {16 \int \frac {-\frac {45}{8} a^4 (105 A+143 C)-\frac {693}{8} a^4 (5 A+7 C) \sec (c+d x)}{\sec ^{\frac {3}{2}}(c+d x)} \, dx}{3465 a}\\ &=\frac {8 a^3 (35 A+44 C) \sin (c+d x)}{385 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 A (a+a \sec (c+d x))^3 \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)}+\frac {4 A \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{33 a d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 (35 A+33 C) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{231 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {1}{5} \left (2 a^3 (5 A+7 C)\right ) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx+\frac {1}{77} \left (2 a^3 (105 A+143 C)\right ) \int \frac {1}{\sec ^{\frac {3}{2}}(c+d x)} \, dx\\ &=\frac {8 a^3 (35 A+44 C) \sin (c+d x)}{385 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {4 a^3 (105 A+143 C) \sin (c+d x)}{231 d \sqrt {\sec (c+d x)}}+\frac {2 A (a+a \sec (c+d x))^3 \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)}+\frac {4 A \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{33 a d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 (35 A+33 C) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{231 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {1}{231} \left (2 a^3 (105 A+143 C)\right ) \int \sqrt {\sec (c+d x)} \, dx+\frac {1}{5} \left (2 a^3 (5 A+7 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx\\ &=\frac {4 a^3 (5 A+7 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 {8 a^3 (35 A+44 C) \sin (c+d x)}{385 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {4 a^3 (105 A+143 C) \sin (c+d x)}{231 d \sqrt {\sec (c+d x)}}+\frac {2 A (a+a \sec (c+d x))^3 \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)}+\frac {4 A \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{33 a d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 (35 A+33 C) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{231 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {1}{231} \left (2 a^3 (105 A+143 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+7 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 (105 A+143 C) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{231 d}+\frac {8 a^3 (35 A+44 C) \sin (c+d x)}{385 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {4 a^3 (105 A+143 C) \sin (c+d x)}{231 d \sqrt {\sec (c+d x)}}+\frac {2 A (a+a \sec (c+d x))^3 \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)}+\frac {4 A \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{33 a d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 (35 A+33 C) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{231 d \sec ^{\frac {5}{2}}(c+d x)}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 3.47, size = 228, normalized size = 0.80 \begin {gather*} \frac {a^3 e^{-i d x} \sqrt {\sec (c+d x)} (\cos (d x)+i \sin (d x)) \left (160 (105 A+143 C) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )-2464 i (5 A+7 C) e^{i (c+d x)} \sqrt {1+e^{2 i (c+d x)}} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-e^{2 i (c+d x)}\right )+\cos (c+d x) (36960 i A+51744 i C+10 (1953 A+2354 C) \sin (c+d x)+308 (25 A+18 C) \sin (2 (c+d x))+2835 A \sin (3 (c+d x))+660 C \sin (3 (c+d x))+770 A \sin (4 (c+d x))+105 A \sin (5 (c+d x)))\right )}{9240 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 3.26, size = 436, normalized size = 1.52
method | result | size |
default | \(-\frac {4 \sqrt {\left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, a^{3} \left (3360 A \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-14560 A \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (25760 A +1320 C \right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-24080 A -4752 C \right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (13090 A +6622 C \right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-2940 A -2288 C \right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+525 A \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-1155 A \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+715 C \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-1617 C \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )\right )}{1155 \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, d}\) | \(436\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.94, size = 247, normalized size = 0.86 \begin {gather*} -\frac {2 \, {\left (5 i \, \sqrt {2} {\left (105 \, A + 143 \, C\right )} a^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 5 i \, \sqrt {2} {\left (105 \, A + 143 \, C\right )} a^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 231 i \, \sqrt {2} {\left (5 \, A + 7 \, 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 ) + 231 i \, \sqrt {2} {\left (5 \, A + 7 \, 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 (105 \, A a^{3} \cos \left (d x + c\right )^{5} + 385 \, A a^{3} \cos \left (d x + c\right )^{4} + 15 \, {\left (42 \, A + 11 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} + 77 \, {\left (10 \, A + 9 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} + 10 \, {\left (105 \, A + 143 \, 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 )}}{1155 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\left (A+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^3}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{11/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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