Optimal. Leaf size=207 \[ -\frac {21 F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{2 a^3 d}+\frac {231 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{10 a^3 d}-\frac {63 \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{10 d \left (a^3 \cos (c+d x)+a^3\right )}+\frac {77 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{10 a^3 d}-\frac {21 \sin (c+d x) \sqrt {\cos (c+d x)}}{2 a^3 d}-\frac {\sin (c+d x) \cos ^{\frac {9}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}-\frac {4 \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{5 a d (a \cos (c+d x)+a)^2} \]
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Rubi [A] time = 0.34, antiderivative size = 207, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {2765, 2977, 2748, 2635, 2641, 2639} \[ -\frac {21 F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{2 a^3 d}+\frac {231 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{10 a^3 d}-\frac {63 \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{10 d \left (a^3 \cos (c+d x)+a^3\right )}+\frac {77 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{10 a^3 d}-\frac {21 \sin (c+d x) \sqrt {\cos (c+d x)}}{2 a^3 d}-\frac {\sin (c+d x) \cos ^{\frac {9}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}-\frac {4 \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{5 a d (a \cos (c+d x)+a)^2} \]
Antiderivative was successfully verified.
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Rule 2635
Rule 2639
Rule 2641
Rule 2748
Rule 2765
Rule 2977
Rubi steps
\begin {align*} \int \frac {\cos ^{\frac {11}{2}}(c+d x)}{(a+a \cos (c+d x))^3} \, dx &=-\frac {\cos ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac {\int \frac {\cos ^{\frac {7}{2}}(c+d x) \left (\frac {9 a}{2}-\frac {15}{2} a \cos (c+d x)\right )}{(a+a \cos (c+d x))^2} \, dx}{5 a^2}\\ &=-\frac {\cos ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac {4 \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{5 a d (a+a \cos (c+d x))^2}-\frac {\int \frac {\cos ^{\frac {5}{2}}(c+d x) \left (42 a^2-\frac {105}{2} a^2 \cos (c+d x)\right )}{a+a \cos (c+d x)} \, dx}{15 a^4}\\ &=-\frac {\cos ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac {4 \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{5 a d (a+a \cos (c+d x))^2}-\frac {63 \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{10 d \left (a^3+a^3 \cos (c+d x)\right )}-\frac {\int \cos ^{\frac {3}{2}}(c+d x) \left (\frac {945 a^3}{4}-\frac {1155}{4} a^3 \cos (c+d x)\right ) \, dx}{15 a^6}\\ &=-\frac {\cos ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac {4 \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{5 a d (a+a \cos (c+d x))^2}-\frac {63 \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{10 d \left (a^3+a^3 \cos (c+d x)\right )}-\frac {63 \int \cos ^{\frac {3}{2}}(c+d x) \, dx}{4 a^3}+\frac {77 \int \cos ^{\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 {9}{2}}(c+d x) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac {4 \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{5 a d (a+a \cos (c+d x))^2}-\frac {63 \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{10 d \left (a^3+a^3 \cos (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 {9}{2}}(c+d x) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac {4 \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{5 a d (a+a \cos (c+d x))^2}-\frac {63 \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{10 d \left (a^3+a^3 \cos (c+d x)\right )}\\ \end {align*}
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Mathematica [C] time = 2.75, size = 388, normalized size = 1.87 \[ \frac {2 \cos ^6\left (\frac {1}{2} (c+d x)\right ) \left (-\sqrt {\cos (c+d x)} \left (\frac {1}{16} \sec \left (\frac {c}{2}\right ) \left (-770 \sin \left (c+\frac {d x}{2}\right )+840 \sin \left (c+\frac {3 d x}{2}\right )-150 \sin \left (2 c+\frac {3 d x}{2}\right )+238 \sin \left (2 c+\frac {5 d x}{2}\right )+40 \sin \left (3 c+\frac {5 d x}{2}\right )+5 \sin \left (3 c+\frac {7 d x}{2}\right )+5 \sin \left (4 c+\frac {7 d x}{2}\right )-\sin \left (4 c+\frac {9 d x}{2}\right )-\sin \left (5 c+\frac {9 d x}{2}\right )+1210 \sin \left (\frac {d x}{2}\right )\right ) \sec ^5\left (\frac {1}{2} (c+d x)\right )+264 \cot (c)+198 \csc (c)\right )+\frac {42 i \sqrt {2} e^{-i (c+d x)} \left (11 \left (-1+e^{2 i c}\right ) \sqrt {1+e^{2 i (c+d x)}} \, _2F_1\left (-\frac {1}{4},\frac {1}{2};\frac {3}{4};-e^{2 i (c+d x)}\right )+5 \left (-1+e^{2 i c}\right ) e^{i (c+d x)} \sqrt {1+e^{2 i (c+d x)}} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};-e^{2 i (c+d x)}\right )+11 \left (1+e^{2 i (c+d x)}\right )\right )}{\left (-1+e^{2 i c}\right ) \sqrt {e^{-i (c+d x)} \left (1+e^{2 i (c+d x)}\right )}}\right )}{5 a^3 d (\cos (c+d x)+1)^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.11, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\cos \left (d x + c\right )^{\frac {11}{2}}}{a^{3} \cos \left (d x + c\right )^{3} + 3 \, a^{3} \cos \left (d x + c\right )^{2} + 3 \, a^{3} \cos \left (d x + c\right ) + a^{3}}, x\right ) \]
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 \[ \int \frac {\cos \left (d x + c\right )^{\frac {11}{2}}}{{\left (a \cos \left (d x + c\right ) + a\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.64, 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 \[ \int \frac {\cos \left (d x + c\right )^{\frac {11}{2}}}{{\left (a \cos \left (d x + c\right ) + a\right )}^{3}}\,{d x} \]
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 \[ \int \frac {{\cos \left (c+d\,x\right )}^{11/2}}{{\left (a+a\,\cos \left (c+d\,x\right )\right )}^3} \,d x \]
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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