Optimal. Leaf size=207 \[ -\frac {21 F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{2 a^3 d}+\frac {231 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{10 a^3 d}+\frac {77 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{10 a^3 d}-\frac {21 \sin (c+d x) \sqrt {\cos (c+d x)}}{2 a^3 d}-\frac {63 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{10 d \left (a^3 \sec (c+d x)+a^3\right )}-\frac {4 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 a d (a \sec (c+d x)+a)^2}-\frac {\sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d (a \sec (c+d x)+a)^3} \]
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Rubi [A] time = 0.43, antiderivative size = 207, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {4264, 3817, 4020, 3787, 3769, 3771, 2639, 2641} \[ -\frac {21 F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{2 a^3 d}+\frac {231 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{10 a^3 d}+\frac {77 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{10 a^3 d}-\frac {21 \sin (c+d x) \sqrt {\cos (c+d x)}}{2 a^3 d}-\frac {63 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{10 d \left (a^3 \sec (c+d x)+a^3\right )}-\frac {4 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 a d (a \sec (c+d x)+a)^2}-\frac {\sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d (a \sec (c+d x)+a)^3} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 2641
Rule 3769
Rule 3771
Rule 3787
Rule 3817
Rule 4020
Rule 4264
Rubi steps
\begin {align*} \int \frac {\cos ^{\frac {5}{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 {5}{2}}(c+d x) (a+a \sec (c+d x))^3} \, dx\\ &=-\frac {\cos ^{\frac {3}{2}}(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 {15 a}{2}+\frac {9}{2} a \sec (c+d x)}{\sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^2} \, dx}{5 a^2}\\ &=-\frac {\cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac {4 \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{5 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 {105 a^2}{2}+42 a^2 \sec (c+d x)}{\sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))} \, dx}{15 a^4}\\ &=-\frac {\cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac {4 \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{5 a d (a+a \sec (c+d x))^2}-\frac {63 \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{10 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 {1155 a^3}{4}+\frac {945}{4} a^3 \sec (c+d x)}{\sec ^{\frac {5}{2}}(c+d x)} \, dx}{15 a^6}\\ &=-\frac {\cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac {4 \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{5 a d (a+a \sec (c+d x))^2}-\frac {63 \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{10 d \left (a^3+a^3 \sec (c+d x)\right )}-\frac {\left (63 \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 {\left (77 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sec ^{\frac {5}{2}}(c+d x)} \, dx}{4 a^3}\\ &=-\frac {21 \sqrt {\cos (c+d x)} \sin (c+d x)}{2 a^3 d}+\frac {77 \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{10 a^3 d}-\frac {\cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac {4 \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{5 a d (a+a \sec (c+d x))^2}-\frac {63 \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{10 d \left (a^3+a^3 \sec (c+d x)\right )}-\frac {\left (21 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\sec (c+d x)} \, dx}{4 a^3}+\frac {\left (231 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx}{20 a^3}\\ &=-\frac {21 \sqrt {\cos (c+d x)} \sin (c+d x)}{2 a^3 d}+\frac {77 \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{10 a^3 d}-\frac {\cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac {4 \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{5 a d (a+a \sec (c+d x))^2}-\frac {63 \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{10 d \left (a^3+a^3 \sec (c+d x)\right )}-\frac {21 \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{4 a^3}+\frac {231 \int \sqrt {\cos (c+d x)} \, dx}{20 a^3}\\ &=\frac {231 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{10 a^3 d}-\frac {21 F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{2 a^3 d}-\frac {21 \sqrt {\cos (c+d x)} \sin (c+d x)}{2 a^3 d}+\frac {77 \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{10 a^3 d}-\frac {\cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac {4 \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{5 a d (a+a \sec (c+d x))^2}-\frac {63 \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{10 d \left (a^3+a^3 \sec (c+d x)\right )}\\ \end {align*}
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Mathematica [C] time = 2.57, size = 391, normalized size = 1.89 \[ \frac {2 \cos ^6\left (\frac {1}{2} (c+d x)\right ) \left (\frac {\frac {1}{16} \sec \left (\frac {c}{2}\right ) \left (770 \sin \left (c+\frac {d x}{2}\right )-840 \sin \left (c+\frac {3 d x}{2}\right )+150 \sin \left (2 c+\frac {3 d x}{2}\right )-238 \sin \left (2 c+\frac {5 d x}{2}\right )-40 \sin \left (3 c+\frac {5 d x}{2}\right )-5 \sin \left (3 c+\frac {7 d x}{2}\right )-5 \sin \left (4 c+\frac {7 d x}{2}\right )+\sin \left (4 c+\frac {9 d x}{2}\right )+\sin \left (5 c+\frac {9 d x}{2}\right )-1210 \sin \left (\frac {d x}{2}\right )\right ) \sec ^5\left (\frac {1}{2} (c+d x)\right )-264 \cot (c)-198 \csc (c)}{\cos ^{\frac {5}{2}}(c+d x)}+\frac {42 i \sqrt {2} e^{-i (c+d x)} \left (11 \left (-1+e^{2 i c}\right ) \sqrt {1+e^{2 i (c+d x)}} \, _2F_1\left (-\frac {1}{4},\frac {1}{2};\frac {3}{4};-e^{2 i (c+d x)}\right )+5 \left (-1+e^{2 i c}\right ) e^{i (c+d x)} \sqrt {1+e^{2 i (c+d x)}} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};-e^{2 i (c+d x)}\right )+11 \left (1+e^{2 i (c+d x)}\right )\right ) \sec ^3(c+d x)}{\left (-1+e^{2 i c}\right ) \sqrt {e^{-i (c+d x)} \left (1+e^{2 i (c+d x)}\right )}}\right )}{5 a^3 d (\sec (c+d x)+1)^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.54, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\cos \left (d x + c\right )^{\frac {5}{2}}}{a^{3} \sec \left (d x + c\right )^{3} + 3 \, a^{3} \sec \left (d x + c\right )^{2} + 3 \, a^{3} \sec \left (d x + c\right ) + a^{3}}, x\right ) \]
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 \[ \int \frac {\cos \left (d x + c\right )^{\frac {5}{2}}}{{\left (a \sec \left (d x + c\right ) + a\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 3.72, size = 296, normalized size = 1.43 \[ -\frac {\sqrt {\left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (64 \left (\cos ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-288 \left (\cos ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-76 \left (\cos ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-210 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \left (\cos ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-462 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1}\, \left (\cos ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+530 \left (\cos ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-248 \left (\cos ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+19 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right )}{20 a^{3} \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}\, \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, d} \]
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 \[ \int \frac {\cos \left (d x + c\right )^{\frac {5}{2}}}{{\left (a \sec \left (d x + c\right ) + a\right )}^{3}}\,{d x} \]
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 \[ \int \frac {{\cos \left (c+d\,x\right )}^{5/2}}{{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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