Optimal. Leaf size=181 \[ -\frac {119 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{10 a^3 d}+\frac {11 F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{2 a^3 d}+\frac {11 \sqrt {\cos (c+d x)} \sin (c+d x)}{2 a^3 d}-\frac {\sqrt {\cos (c+d x)} \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac {2 \sqrt {\cos (c+d x)} \sin (c+d x)}{3 a d (a+a \sec (c+d x))^2}-\frac {119 \sqrt {\cos (c+d x)} \sin (c+d x)}{30 d \left (a^3+a^3 \sec (c+d x)\right )} \]
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Rubi [A]
time = 0.28, antiderivative size = 181, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 8, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {4349, 3902,
4105, 3872, 3854, 3856, 2720, 2719} \begin {gather*} \frac {11 F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{2 a^3 d}-\frac {119 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{10 a^3 d}+\frac {11 \sin (c+d x) \sqrt {\cos (c+d x)}}{2 a^3 d}-\frac {119 \sin (c+d x) \sqrt {\cos (c+d x)}}{30 d \left (a^3 \sec (c+d x)+a^3\right )}-\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 a d (a \sec (c+d x)+a)^2}-\frac {\sin (c+d x) \sqrt {\cos (c+d x)}}{5 d (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 3902
Rule 4105
Rule 4349
Rubi steps
\begin {align*} \int \frac {\cos ^{\frac {3}{2}}(c+d x)}{(a+a \sec (c+d x))^3} \, dx &=\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^3} \, dx\\ &=-\frac {\sqrt {\cos (c+d x)} \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {-\frac {13 a}{2}+\frac {7}{2} a \sec (c+d x)}{\sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^2} \, dx}{5 a^2}\\ &=-\frac {\sqrt {\cos (c+d x)} \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac {2 \sqrt {\cos (c+d x)} \sin (c+d x)}{3 a d (a+a \sec (c+d x))^2}-\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {-\frac {69 a^2}{2}+25 a^2 \sec (c+d x)}{\sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))} \, dx}{15 a^4}\\ &=-\frac {\sqrt {\cos (c+d x)} \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac {2 \sqrt {\cos (c+d x)} \sin (c+d x)}{3 a d (a+a \sec (c+d x))^2}-\frac {119 \sqrt {\cos (c+d x)} \sin (c+d x)}{30 d \left (a^3+a^3 \sec (c+d x)\right )}-\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {-\frac {495 a^3}{4}+\frac {357}{4} a^3 \sec (c+d x)}{\sec ^{\frac {3}{2}}(c+d x)} \, dx}{15 a^6}\\ &=-\frac {\sqrt {\cos (c+d x)} \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac {2 \sqrt {\cos (c+d x)} \sin (c+d x)}{3 a d (a+a \sec (c+d x))^2}-\frac {119 \sqrt {\cos (c+d x)} \sin (c+d x)}{30 d \left (a^3+a^3 \sec (c+d x)\right )}-\frac {\left (119 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx}{20 a^3}+\frac {\left (33 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sec ^{\frac {3}{2}}(c+d x)} \, dx}{4 a^3}\\ &=\frac {11 \sqrt {\cos (c+d x)} \sin (c+d x)}{2 a^3 d}-\frac {\sqrt {\cos (c+d x)} \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac {2 \sqrt {\cos (c+d x)} \sin (c+d x)}{3 a d (a+a \sec (c+d x))^2}-\frac {119 \sqrt {\cos (c+d x)} \sin (c+d x)}{30 d \left (a^3+a^3 \sec (c+d x)\right )}-\frac {119 \int \sqrt {\cos (c+d x)} \, dx}{20 a^3}+\frac {\left (11 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\sec (c+d x)} \, dx}{4 a^3}\\ &=-\frac {119 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{10 a^3 d}+\frac {11 \sqrt {\cos (c+d x)} \sin (c+d x)}{2 a^3 d}-\frac {\sqrt {\cos (c+d x)} \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac {2 \sqrt {\cos (c+d x)} \sin (c+d x)}{3 a d (a+a \sec (c+d x))^2}-\frac {119 \sqrt {\cos (c+d x)} \sin (c+d x)}{30 d \left (a^3+a^3 \sec (c+d x)\right )}+\frac {11 \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{4 a^3}\\ &=-\frac {119 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{10 a^3 d}+\frac {11 F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{2 a^3 d}+\frac {11 \sqrt {\cos (c+d x)} \sin (c+d x)}{2 a^3 d}-\frac {\sqrt {\cos (c+d x)} \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac {2 \sqrt {\cos (c+d x)} \sin (c+d x)}{3 a d (a+a \sec (c+d x))^2}-\frac {119 \sqrt {\cos (c+d x)} \sin (c+d x)}{30 d \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 = 2.18, size = 375, normalized size = 2.07 \begin {gather*} \frac {\cos ^6\left (\frac {1}{2} (c+d x)\right ) \left (-\frac {4 i \sqrt {2} e^{-i (c+d x)} \left (119 \left (1+e^{2 i (c+d x)}\right )+119 \left (-1+e^{2 i c}\right ) \sqrt {1+e^{2 i (c+d x)}} \, _2F_1\left (-\frac {1}{4},\frac {1}{2};\frac {3}{4};-e^{2 i (c+d x)}\right )+55 e^{i (c+d x)} \left (-1+e^{2 i c}\right ) \sqrt {1+e^{2 i (c+d x)}} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};-e^{2 i (c+d x)}\right )\right ) \sec ^3(c+d x)}{d \left (-1+e^{2 i c}\right ) \sqrt {e^{-i (c+d x)} \left (1+e^{2 i (c+d x)}\right )}}+\frac {720 \cot (c)+708 \csc (c)+\frac {1}{4} \sec \left (\frac {c}{2}\right ) \sec ^5\left (\frac {1}{2} (c+d x)\right ) \left (1061 \sin \left (\frac {d x}{2}\right )-709 \sin \left (c+\frac {d x}{2}\right )+715 \sin \left (c+\frac {3 d x}{2}\right )-170 \sin \left (2 c+\frac {3 d x}{2}\right )+202 \sin \left (2 c+\frac {5 d x}{2}\right )+25 \sin \left (3 c+\frac {5 d x}{2}\right )+5 \sin \left (3 c+\frac {7 d x}{2}\right )+5 \sin \left (4 c+\frac {7 d x}{2}\right )\right )}{3 d \cos ^{\frac {5}{2}}(c+d x)}\right )}{5 a^3 (1+\sec (c+d x))^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 283, normalized size = 1.56
method | result | size |
default | \(-\frac {\sqrt {\left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (160 \left (\cos ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+468 \left (\cos ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+330 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \left (\cos ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+714 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1}\, \left (\cos ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-1058 \left (\cos ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+474 \left (\cos ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-47 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3\right )}{60 a^{3} \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}\, \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, d}\) | \(283\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \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 = 0.70, size = 354, normalized size = 1.96 \begin {gather*} \frac {2 \, {\left (20 \, \cos \left (d x + c\right )^{3} + 237 \, \cos \left (d x + c\right )^{2} + 376 \, \cos \left (d x + c\right ) + 165\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - 165 \, {\left (i \, \sqrt {2} \cos \left (d x + c\right )^{3} + 3 i \, \sqrt {2} \cos \left (d x + c\right )^{2} + 3 i \, \sqrt {2} \cos \left (d x + c\right ) + i \, \sqrt {2}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 165 \, {\left (-i \, \sqrt {2} \cos \left (d x + c\right )^{3} - 3 i \, \sqrt {2} \cos \left (d x + c\right )^{2} - 3 i \, \sqrt {2} \cos \left (d x + c\right ) - i \, \sqrt {2}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 357 \, {\left (i \, \sqrt {2} \cos \left (d x + c\right )^{3} + 3 i \, \sqrt {2} \cos \left (d x + c\right )^{2} + 3 i \, \sqrt {2} \cos \left (d x + c\right ) + i \, \sqrt {2}\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 ) - 357 \, {\left (-i \, \sqrt {2} \cos \left (d x + c\right )^{3} - 3 i \, \sqrt {2} \cos \left (d x + c\right )^{2} - 3 i \, \sqrt {2} \cos \left (d x + c\right ) - i \, \sqrt {2}\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 )}{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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {\cos ^{\frac {3}{2}}{\left (c + d x \right )}}{\sec ^{3}{\left (c + d x \right )} + 3 \sec ^{2}{\left (c + d x \right )} + 3 \sec {\left (c + d x \right )} + 1}\, dx}{a^{3}} \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 = -0.01 \begin {gather*} \int \frac {{\cos \left (c+d\,x\right )}^{3/2}}{{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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