Optimal. Leaf size=160 \[ \frac {56 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 a^2 d}-\frac {5 F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{a^2 d}-\frac {5 \sqrt {\cos (c+d x)} \sin (c+d x)}{a^2 d}+\frac {56 \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{15 a^2 d}-\frac {3 \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{a^2 d (1+\sec (c+d x))}-\frac {\cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{3 d (a+a \sec (c+d x))^2} \]
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Rubi [A]
time = 0.20, antiderivative size = 160, 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, 2719, 2720} \begin {gather*} -\frac {5 F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{a^2 d}+\frac {56 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 a^2 d}+\frac {56 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{15 a^2 d}-\frac {5 \sin (c+d x) \sqrt {\cos (c+d x)}}{a^2 d}-\frac {3 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{a^2 d (\sec (c+d x)+1)}-\frac {\sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{3 d (a \sec (c+d x)+a)^2} \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 {5}{2}}(c+d x)}{(a+a \sec (c+d x))^2} \, dx &=\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^2} \, dx\\ &=-\frac {\cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{3 d (a+a \sec (c+d x))^2}-\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {-\frac {11 a}{2}+\frac {7}{2} a \sec (c+d x)}{\sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))} \, dx}{3 a^2}\\ &=-\frac {3 \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{a^2 d (1+\sec (c+d x))}-\frac {\cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{3 d (a+a \sec (c+d x))^2}-\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {-28 a^2+\frac {45}{2} a^2 \sec (c+d x)}{\sec ^{\frac {5}{2}}(c+d x)} \, dx}{3 a^4}\\ &=-\frac {3 \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{a^2 d (1+\sec (c+d x))}-\frac {\cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{3 d (a+a \sec (c+d x))^2}-\frac {\left (15 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sec ^{\frac {3}{2}}(c+d x)} \, dx}{2 a^2}+\frac {\left (28 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sec ^{\frac {5}{2}}(c+d x)} \, dx}{3 a^2}\\ &=-\frac {5 \sqrt {\cos (c+d x)} \sin (c+d x)}{a^2 d}+\frac {56 \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{15 a^2 d}-\frac {3 \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{a^2 d (1+\sec (c+d x))}-\frac {\cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{3 d (a+a \sec (c+d x))^2}-\frac {\left (5 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\sec (c+d x)} \, dx}{2 a^2}+\frac {\left (28 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx}{5 a^2}\\ &=-\frac {5 \sqrt {\cos (c+d x)} \sin (c+d x)}{a^2 d}+\frac {56 \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{15 a^2 d}-\frac {3 \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{a^2 d (1+\sec (c+d x))}-\frac {\cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{3 d (a+a \sec (c+d x))^2}-\frac {5 \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{2 a^2}+\frac {28 \int \sqrt {\cos (c+d x)} \, dx}{5 a^2}\\ &=\frac {56 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 a^2 d}-\frac {5 F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{a^2 d}-\frac {5 \sqrt {\cos (c+d x)} \sin (c+d x)}{a^2 d}+\frac {56 \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{15 a^2 d}-\frac {3 \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{a^2 d (1+\sec (c+d x))}-\frac {\cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{3 d (a+a \sec (c+d x))^2}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 2.39, size = 366, normalized size = 2.29 \begin {gather*} \frac {\cos ^4\left (\frac {1}{2} (c+d x)\right ) \left (\frac {4 i \sqrt {2} e^{-i (c+d x)} \left (56 \left (1+e^{2 i (c+d x)}\right )+56 \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 )+25 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 ^2(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 {2 \left (-216 \cot (c)-120 \csc (c)-40 \cos (d x) \sin (c)+6 \cos (2 d x) \sin (2 c)-120 \sec \left (\frac {c}{2}\right ) \sec \left (\frac {1}{2} (c+d x)\right ) \sin \left (\frac {d x}{2}\right )+5 \sec \left (\frac {c}{2}\right ) \sec ^3\left (\frac {1}{2} (c+d x)\right ) \sin \left (\frac {d x}{2}\right )-40 \cos (c) \sin (d x)+6 \cos (2 c) \sin (2 d x)+5 \sec ^2\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {c}{2}\right )\right )}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )}{5 a^2 (1+\sec (c+d x))^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 283, normalized size = 1.77
method | result | size |
default | \(-\frac {\sqrt {\left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (96 \left (\cos ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-352 \left (\cos ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+120 \left (\cos ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-150 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \left (\cos ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-336 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1}\, \left (\cos ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+266 \left (\cos ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-135 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+5\right )}{30 a^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, d}\) | \(283\) |
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 = 0.75, size = 288, normalized size = 1.80 \begin {gather*} \frac {2 \, {\left (6 \, \cos \left (d x + c\right )^{3} - 8 \, \cos \left (d x + c\right )^{2} - 94 \, \cos \left (d x + c\right ) - 75\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - 75 \, {\left (-i \, \sqrt {2} \cos \left (d x + c\right )^{2} - 2 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 ) - 75 \, {\left (i \, \sqrt {2} \cos \left (d x + c\right )^{2} + 2 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 ) - 168 \, {\left (-i \, \sqrt {2} \cos \left (d x + c\right )^{2} - 2 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 ) - 168 \, {\left (i \, \sqrt {2} \cos \left (d x + c\right )^{2} + 2 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 )}{30 \, {\left (a^{2} d \cos \left (d x + c\right )^{2} + 2 \, a^{2} d \cos \left (d x + c\right ) + a^{2} d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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 )}^{5/2}}{{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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