Optimal. Leaf size=225 \[ -\frac {7664 \sin (c+d x)}{315 a^5 d}-\frac {3832 \sin (c+d x) \cos ^2(c+d x)}{315 d \left (a^5 \cos (c+d x)+a^5\right )}+\frac {31 \sin (c+d x) \cos (c+d x)}{2 a^5 d}+\frac {31 x}{2 a^5}-\frac {577 \sin (c+d x) \cos ^3(c+d x)}{315 a^3 d (a \cos (c+d x)+a)^2}-\frac {28 \sin (c+d x) \cos ^4(c+d x)}{45 a^2 d (a \cos (c+d x)+a)^3}-\frac {\sin (c+d x) \cos ^6(c+d x)}{9 d (a \cos (c+d x)+a)^5}-\frac {17 \sin (c+d x) \cos ^5(c+d x)}{63 a d (a \cos (c+d x)+a)^4} \]
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Rubi [A] time = 0.52, antiderivative size = 225, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2765, 2977, 2734} \[ -\frac {7664 \sin (c+d x)}{315 a^5 d}-\frac {28 \sin (c+d x) \cos ^4(c+d x)}{45 a^2 d (a \cos (c+d x)+a)^3}-\frac {577 \sin (c+d x) \cos ^3(c+d x)}{315 a^3 d (a \cos (c+d x)+a)^2}-\frac {3832 \sin (c+d x) \cos ^2(c+d x)}{315 d \left (a^5 \cos (c+d x)+a^5\right )}+\frac {31 \sin (c+d x) \cos (c+d x)}{2 a^5 d}+\frac {31 x}{2 a^5}-\frac {\sin (c+d x) \cos ^6(c+d x)}{9 d (a \cos (c+d x)+a)^5}-\frac {17 \sin (c+d x) \cos ^5(c+d x)}{63 a d (a \cos (c+d x)+a)^4} \]
Antiderivative was successfully verified.
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Rule 2734
Rule 2765
Rule 2977
Rubi steps
\begin {align*} \int \frac {\cos ^7(c+d x)}{(a+a \cos (c+d x))^5} \, dx &=-\frac {\cos ^6(c+d x) \sin (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac {\int \frac {\cos ^5(c+d x) (6 a-11 a \cos (c+d x))}{(a+a \cos (c+d x))^4} \, dx}{9 a^2}\\ &=-\frac {\cos ^6(c+d x) \sin (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac {17 \cos ^5(c+d x) \sin (c+d x)}{63 a d (a+a \cos (c+d x))^4}-\frac {\int \frac {\cos ^4(c+d x) \left (85 a^2-111 a^2 \cos (c+d x)\right )}{(a+a \cos (c+d x))^3} \, dx}{63 a^4}\\ &=-\frac {\cos ^6(c+d x) \sin (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac {17 \cos ^5(c+d x) \sin (c+d x)}{63 a d (a+a \cos (c+d x))^4}-\frac {28 \cos ^4(c+d x) \sin (c+d x)}{45 a^2 d (a+a \cos (c+d x))^3}-\frac {\int \frac {\cos ^3(c+d x) \left (784 a^3-947 a^3 \cos (c+d x)\right )}{(a+a \cos (c+d x))^2} \, dx}{315 a^6}\\ &=-\frac {\cos ^6(c+d x) \sin (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac {17 \cos ^5(c+d x) \sin (c+d x)}{63 a d (a+a \cos (c+d x))^4}-\frac {28 \cos ^4(c+d x) \sin (c+d x)}{45 a^2 d (a+a \cos (c+d x))^3}-\frac {577 \cos ^3(c+d x) \sin (c+d x)}{315 a^3 d (a+a \cos (c+d x))^2}-\frac {\int \frac {\cos ^2(c+d x) \left (5193 a^4-6303 a^4 \cos (c+d x)\right )}{a+a \cos (c+d x)} \, dx}{945 a^8}\\ &=-\frac {\cos ^6(c+d x) \sin (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac {17 \cos ^5(c+d x) \sin (c+d x)}{63 a d (a+a \cos (c+d x))^4}-\frac {28 \cos ^4(c+d x) \sin (c+d x)}{45 a^2 d (a+a \cos (c+d x))^3}-\frac {577 \cos ^3(c+d x) \sin (c+d x)}{315 a^3 d (a+a \cos (c+d x))^2}-\frac {3832 \cos ^2(c+d x) \sin (c+d x)}{315 d \left (a^5+a^5 \cos (c+d x)\right )}-\frac {\int \cos (c+d x) \left (22992 a^5-29295 a^5 \cos (c+d x)\right ) \, dx}{945 a^{10}}\\ &=\frac {31 x}{2 a^5}-\frac {7664 \sin (c+d x)}{315 a^5 d}+\frac {31 \cos (c+d x) \sin (c+d x)}{2 a^5 d}-\frac {\cos ^6(c+d x) \sin (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac {17 \cos ^5(c+d x) \sin (c+d x)}{63 a d (a+a \cos (c+d x))^4}-\frac {28 \cos ^4(c+d x) \sin (c+d x)}{45 a^2 d (a+a \cos (c+d x))^3}-\frac {577 \cos ^3(c+d x) \sin (c+d x)}{315 a^3 d (a+a \cos (c+d x))^2}-\frac {3832 \cos ^2(c+d x) \sin (c+d x)}{315 d \left (a^5+a^5 \cos (c+d x)\right )}\\ \end {align*}
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Mathematica [A] time = 0.78, size = 345, normalized size = 1.53 \[ \frac {\sec \left (\frac {c}{2}\right ) \sec ^9\left (\frac {1}{2} (c+d x)\right ) \left (7194600 \sin \left (c+\frac {d x}{2}\right )-7472241 \sin \left (c+\frac {3 d x}{2}\right )+3432975 \sin \left (2 c+\frac {3 d x}{2}\right )-3871989 \sin \left (2 c+\frac {5 d x}{2}\right )+801675 \sin \left (3 c+\frac {5 d x}{2}\right )-1186056 \sin \left (3 c+\frac {7 d x}{2}\right )-17640 \sin \left (4 c+\frac {7 d x}{2}\right )-175184 \sin \left (4 c+\frac {9 d x}{2}\right )-45360 \sin \left (5 c+\frac {9 d x}{2}\right )-3465 \sin \left (5 c+\frac {11 d x}{2}\right )-3465 \sin \left (6 c+\frac {11 d x}{2}\right )+315 \sin \left (6 c+\frac {13 d x}{2}\right )+315 \sin \left (7 c+\frac {13 d x}{2}\right )+4921560 d x \cos \left (c+\frac {d x}{2}\right )+3281040 d x \cos \left (c+\frac {3 d x}{2}\right )+3281040 d x \cos \left (2 c+\frac {3 d x}{2}\right )+1406160 d x \cos \left (2 c+\frac {5 d x}{2}\right )+1406160 d x \cos \left (3 c+\frac {5 d x}{2}\right )+351540 d x \cos \left (3 c+\frac {7 d x}{2}\right )+351540 d x \cos \left (4 c+\frac {7 d x}{2}\right )+39060 d x \cos \left (4 c+\frac {9 d x}{2}\right )+39060 d x \cos \left (5 c+\frac {9 d x}{2}\right )-9163224 \sin \left (\frac {d x}{2}\right )+4921560 d x \cos \left (\frac {d x}{2}\right )\right )}{1290240 a^5 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.88, size = 207, normalized size = 0.92 \[ \frac {9765 \, d x \cos \left (d x + c\right )^{5} + 48825 \, d x \cos \left (d x + c\right )^{4} + 97650 \, d x \cos \left (d x + c\right )^{3} + 97650 \, d x \cos \left (d x + c\right )^{2} + 48825 \, d x \cos \left (d x + c\right ) + 9765 \, d x + {\left (315 \, \cos \left (d x + c\right )^{6} - 1575 \, \cos \left (d x + c\right )^{5} - 28828 \, \cos \left (d x + c\right )^{4} - 87440 \, \cos \left (d x + c\right )^{3} - 112119 \, \cos \left (d x + c\right )^{2} - 66875 \, \cos \left (d x + c\right ) - 15328\right )} \sin \left (d x + c\right )}{630 \, {\left (a^{5} d \cos \left (d x + c\right )^{5} + 5 \, a^{5} d \cos \left (d x + c\right )^{4} + 10 \, a^{5} d \cos \left (d x + c\right )^{3} + 10 \, a^{5} d \cos \left (d x + c\right )^{2} + 5 \, a^{5} d \cos \left (d x + c\right ) + a^{5} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.47, size = 145, normalized size = 0.64 \[ \frac {\frac {78120 \, {\left (d x + c\right )}}{a^{5}} - \frac {5040 \, {\left (11 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 9 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2} a^{5}} - \frac {35 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 450 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 3024 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 15750 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 110565 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{45}}}{5040 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 179, normalized size = 0.80 \[ -\frac {\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )}{144 d \,a^{5}}+\frac {5 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{56 d \,a^{5}}-\frac {3 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d \,a^{5}}+\frac {25 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \,a^{5}}-\frac {351 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{16 d \,a^{5}}-\frac {11 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{5} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\frac {9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d \,a^{5} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {31 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.43, size = 224, normalized size = 1.00 \[ -\frac {\frac {5040 \, {\left (\frac {9 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {11 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{a^{5} + \frac {2 \, a^{5} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {a^{5} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} + \frac {\frac {110565 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {15750 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {3024 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {450 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {35 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}}}{a^{5}} - \frac {156240 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{5}}}{5040 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.58, size = 181, normalized size = 0.80 \[ -\frac {35\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-590\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+4584\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-23288\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+129824\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+55440\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-10080\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-78120\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\left (c+d\,x\right )}{5040\,a^5\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 64.31, size = 588, normalized size = 2.61 \[ \begin {cases} \frac {78120 d x \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{5040 a^{5} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 10080 a^{5} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 5040 a^{5} d} + \frac {156240 d x \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{5040 a^{5} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 10080 a^{5} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 5040 a^{5} d} + \frac {78120 d x}{5040 a^{5} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 10080 a^{5} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 5040 a^{5} d} - \frac {35 \tan ^{13}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{5040 a^{5} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 10080 a^{5} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 5040 a^{5} d} + \frac {380 \tan ^{11}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{5040 a^{5} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 10080 a^{5} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 5040 a^{5} d} - \frac {2159 \tan ^{9}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{5040 a^{5} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 10080 a^{5} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 5040 a^{5} d} + \frac {10152 \tan ^{7}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{5040 a^{5} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 10080 a^{5} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 5040 a^{5} d} - \frac {82089 \tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{5040 a^{5} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 10080 a^{5} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 5040 a^{5} d} - \frac {260820 \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{5040 a^{5} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 10080 a^{5} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 5040 a^{5} d} - \frac {155925 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{5040 a^{5} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 10080 a^{5} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 5040 a^{5} d} & \text {for}\: d \neq 0 \\\frac {x \cos ^{7}{\relax (c )}}{\left (a \cos {\relax (c )} + a\right )^{5}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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