Optimal. Leaf size=290 \[ \frac {7 (33 A+7 C) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{10 a^3 d}-\frac {(63 A+13 C) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{6 a^3 d}+\frac {7 (33 A+7 C) \sin (c+d x)}{30 a^3 d \sec ^{\frac {3}{2}}(c+d x)}-\frac {(63 A+13 C) \sin (c+d x)}{6 a^3 d \sqrt {\sec (c+d x)}}-\frac {(A+C) \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^3}-\frac {2 (6 A+C) \sin (c+d x)}{15 a d \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^2}-\frac {(63 A+13 C) \sin (c+d x)}{10 d \sec ^{\frac {3}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right )} \]
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Rubi [A]
time = 0.38, antiderivative size = 290, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 7, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4170, 4105,
3872, 3854, 3856, 2719, 2720} \begin {gather*} -\frac {(63 A+13 C) \sin (c+d x)}{10 d \sec ^{\frac {3}{2}}(c+d x) \left (a^3 \sec (c+d x)+a^3\right )}+\frac {7 (33 A+7 C) \sin (c+d x)}{30 a^3 d \sec ^{\frac {3}{2}}(c+d x)}-\frac {(63 A+13 C) \sin (c+d x)}{6 a^3 d \sqrt {\sec (c+d x)}}-\frac {(63 A+13 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{6 a^3 d}+\frac {7 (33 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 )}{10 a^3 d}-\frac {2 (6 A+C) \sin (c+d x)}{15 a d \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^2}-\frac {(A+C) \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 2719
Rule 2720
Rule 3854
Rule 3856
Rule 3872
Rule 4105
Rule 4170
Rubi steps
\begin {align*} \int \frac {A+C \sec ^2(c+d x)}{\sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^3} \, dx &=-\frac {(A+C) \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^3}-\frac {\int \frac {-\frac {5}{2} a (3 A+C)+\frac {1}{2} a (9 A-C) \sec (c+d x)}{\sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^2} \, dx}{5 a^2}\\ &=-\frac {(A+C) \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^3}-\frac {2 (6 A+C) \sin (c+d x)}{15 a d \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^2}-\frac {\int \frac {-\frac {5}{2} a^2 (21 A+5 C)+7 a^2 (6 A+C) \sec (c+d x)}{\sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))} \, dx}{15 a^4}\\ &=-\frac {(A+C) \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^3}-\frac {2 (6 A+C) \sin (c+d x)}{15 a d \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^2}-\frac {(63 A+13 C) \sin (c+d x)}{10 d \sec ^{\frac {3}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right )}-\frac {\int \frac {-\frac {35}{4} a^3 (33 A+7 C)+\frac {15}{4} a^3 (63 A+13 C) \sec (c+d x)}{\sec ^{\frac {5}{2}}(c+d x)} \, dx}{15 a^6}\\ &=-\frac {(A+C) \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^3}-\frac {2 (6 A+C) \sin (c+d x)}{15 a d \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^2}-\frac {(63 A+13 C) \sin (c+d x)}{10 d \sec ^{\frac {3}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right )}+\frac {(7 (33 A+7 C)) \int \frac {1}{\sec ^{\frac {5}{2}}(c+d x)} \, dx}{12 a^3}-\frac {(63 A+13 C) \int \frac {1}{\sec ^{\frac {3}{2}}(c+d x)} \, dx}{4 a^3}\\ &=\frac {7 (33 A+7 C) \sin (c+d x)}{30 a^3 d \sec ^{\frac {3}{2}}(c+d x)}-\frac {(63 A+13 C) \sin (c+d x)}{6 a^3 d \sqrt {\sec (c+d x)}}-\frac {(A+C) \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^3}-\frac {2 (6 A+C) \sin (c+d x)}{15 a d \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^2}-\frac {(63 A+13 C) \sin (c+d x)}{10 d \sec ^{\frac {3}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right )}+\frac {(7 (33 A+7 C)) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx}{20 a^3}-\frac {(63 A+13 C) \int \sqrt {\sec (c+d x)} \, dx}{12 a^3}\\ &=\frac {7 (33 A+7 C) \sin (c+d x)}{30 a^3 d \sec ^{\frac {3}{2}}(c+d x)}-\frac {(63 A+13 C) \sin (c+d x)}{6 a^3 d \sqrt {\sec (c+d x)}}-\frac {(A+C) \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^3}-\frac {2 (6 A+C) \sin (c+d x)}{15 a d \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^2}-\frac {(63 A+13 C) \sin (c+d x)}{10 d \sec ^{\frac {3}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right )}+\frac {\left (7 (33 A+7 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{20 a^3}-\frac {\left ((63 A+13 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{12 a^3}\\ &=\frac {7 (33 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)}}{10 a^3 d}-\frac {(63 A+13 C) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{6 a^3 d}+\frac {7 (33 A+7 C) \sin (c+d x)}{30 a^3 d \sec ^{\frac {3}{2}}(c+d x)}-\frac {(63 A+13 C) \sin (c+d x)}{6 a^3 d \sqrt {\sec (c+d x)}}-\frac {(A+C) \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^3}-\frac {2 (6 A+C) \sin (c+d x)}{15 a d \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^2}-\frac {(63 A+13 C) \sin (c+d x)}{10 d \sec ^{\frac {3}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right )}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 7.43, size = 1052, normalized size = 3.63 \begin {gather*} -\frac {154 \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 ) \, _2F_1\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 {98 \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 ) \, _2F_1\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 {84 A \cos ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {\cos (c+d x)} \csc \left (\frac {c}{2}\right ) F\left (\left .\frac {1}{2} (c+d x)\right |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 {52 C \cos ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {\cos (c+d x)} \csc \left (\frac {c}{2}\right ) F\left (\left .\frac {1}{2} (c+d x)\right |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)}{3 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 {2 (329 A+78 C+133 A \cos (2 c)+20 C \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)}{d}+\frac {8 A \cos (3 d x) \sin (3 c)}{5 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 {184 \sec \left (\frac {c}{2}\right ) \sec \left (\frac {c}{2}+\frac {d x}{2}\right ) \left (3 A \sin \left (\frac {d x}{2}\right )+C \sin \left (\frac {d x}{2}\right )\right )}{3 d}-\frac {8 \sec \left (\frac {c}{2}\right ) \sec ^3\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (27 A \sin \left (\frac {d x}{2}\right )+17 C \sin \left (\frac {d x}{2}\right )\right )}{15 d}+\frac {8 (133 A+20 C) \cos (c) \sin (d x)}{5 d}-\frac {16 A \cos (2 c) \sin (2 d x)}{d}+\frac {8 A \cos (3 c) \sin (3 d x)}{5 d}+\frac {184 (3 A+C) \tan \left (\frac {c}{2}\right )}{3 d}-\frac {8 (27 A+17 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} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 4.62, size = 479, normalized size = 1.65
method | result | size |
default | \(-\frac {\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 )}\, \left (192 A \left (\cos ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-864 A \left (\cos ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-228 A \left (\cos ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-630 A \left (\cos ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{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 )-1386 A \left (\cos ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{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 )-348 C \left (\cos ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-130 C \left (\cos ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{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 )-294 C \left (\cos ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{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 )+1590 A \left (\cos ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+578 C \left (\cos ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-744 A \left (\cos ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-264 C \left (\cos ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+57 A \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+37 C \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-3 A -3 C \right )}{60 a^{3} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} \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}\) | \(479\) |
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 = 0.67, size = 500, normalized size = 1.72 \begin {gather*} -\frac {5 \, {\left (\sqrt {2} {\left (-63 i \, A - 13 i \, C\right )} \cos \left (d x + c\right )^{3} + 3 \, \sqrt {2} {\left (-63 i \, A - 13 i \, C\right )} \cos \left (d x + c\right )^{2} + 3 \, \sqrt {2} {\left (-63 i \, A - 13 i \, C\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (-63 i \, A - 13 i \, C\right )}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 5 \, {\left (\sqrt {2} {\left (63 i \, A + 13 i \, C\right )} \cos \left (d x + c\right )^{3} + 3 \, \sqrt {2} {\left (63 i \, A + 13 i \, C\right )} \cos \left (d x + c\right )^{2} + 3 \, \sqrt {2} {\left (63 i \, A + 13 i \, C\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (63 i \, A + 13 i \, C\right )}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 21 \, {\left (\sqrt {2} {\left (-33 i \, A - 7 i \, C\right )} \cos \left (d x + c\right )^{3} + 3 \, \sqrt {2} {\left (-33 i \, A - 7 i \, C\right )} \cos \left (d x + c\right )^{2} + 3 \, \sqrt {2} {\left (-33 i \, A - 7 i \, C\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (-33 i \, A - 7 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 ) + 21 \, {\left (\sqrt {2} {\left (33 i \, A + 7 i \, C\right )} \cos \left (d x + c\right )^{3} + 3 \, \sqrt {2} {\left (33 i \, A + 7 i \, C\right )} \cos \left (d x + c\right )^{2} + 3 \, \sqrt {2} {\left (33 i \, A + 7 i \, C\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (33 i \, A + 7 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 (12 \, A \cos \left (d x + c\right )^{5} - 24 \, A \cos \left (d x + c\right )^{4} - 3 \, {\left (147 \, A + 29 \, C\right )} \cos \left (d x + c\right )^{3} - 2 \, {\left (357 \, A + 73 \, C\right )} \cos \left (d x + c\right )^{2} - 5 \, {\left (63 \, A + 13 \, 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 )}} \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 {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 )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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