Optimal. Leaf size=104 \[ \frac {7 \cos ^5(c+d x)}{30 a^2 d}+\frac {\cos ^7(c+d x)}{6 d \left (a^2 \sin (c+d x)+a^2\right )}+\frac {7 \sin (c+d x) \cos ^3(c+d x)}{24 a^2 d}+\frac {7 \sin (c+d x) \cos (c+d x)}{16 a^2 d}+\frac {7 x}{16 a^2} \]
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Rubi [A] time = 0.11, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2679, 2682, 2635, 8} \[ \frac {7 \cos ^5(c+d x)}{30 a^2 d}+\frac {\cos ^7(c+d x)}{6 d \left (a^2 \sin (c+d x)+a^2\right )}+\frac {7 \sin (c+d x) \cos ^3(c+d x)}{24 a^2 d}+\frac {7 \sin (c+d x) \cos (c+d x)}{16 a^2 d}+\frac {7 x}{16 a^2} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2635
Rule 2679
Rule 2682
Rubi steps
\begin {align*} \int \frac {\cos ^8(c+d x)}{(a+a \sin (c+d x))^2} \, dx &=\frac {\cos ^7(c+d x)}{6 d \left (a^2+a^2 \sin (c+d x)\right )}+\frac {7 \int \frac {\cos ^6(c+d x)}{a+a \sin (c+d x)} \, dx}{6 a}\\ &=\frac {7 \cos ^5(c+d x)}{30 a^2 d}+\frac {\cos ^7(c+d x)}{6 d \left (a^2+a^2 \sin (c+d x)\right )}+\frac {7 \int \cos ^4(c+d x) \, dx}{6 a^2}\\ &=\frac {7 \cos ^5(c+d x)}{30 a^2 d}+\frac {7 \cos ^3(c+d x) \sin (c+d x)}{24 a^2 d}+\frac {\cos ^7(c+d x)}{6 d \left (a^2+a^2 \sin (c+d x)\right )}+\frac {7 \int \cos ^2(c+d x) \, dx}{8 a^2}\\ &=\frac {7 \cos ^5(c+d x)}{30 a^2 d}+\frac {7 \cos (c+d x) \sin (c+d x)}{16 a^2 d}+\frac {7 \cos ^3(c+d x) \sin (c+d x)}{24 a^2 d}+\frac {\cos ^7(c+d x)}{6 d \left (a^2+a^2 \sin (c+d x)\right )}+\frac {7 \int 1 \, dx}{16 a^2}\\ &=\frac {7 x}{16 a^2}+\frac {7 \cos ^5(c+d x)}{30 a^2 d}+\frac {7 \cos (c+d x) \sin (c+d x)}{16 a^2 d}+\frac {7 \cos ^3(c+d x) \sin (c+d x)}{24 a^2 d}+\frac {\cos ^7(c+d x)}{6 d \left (a^2+a^2 \sin (c+d x)\right )}\\ \end {align*}
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Mathematica [A] time = 1.12, size = 151, normalized size = 1.45 \[ -\frac {\left (\sqrt {\sin (c+d x)+1} \left (40 \sin ^6(c+d x)-136 \sin ^5(c+d x)+86 \sin ^4(c+d x)+202 \sin ^3(c+d x)-327 \sin ^2(c+d x)+39 \sin (c+d x)+96\right )-210 \sin ^{-1}\left (\frac {\sqrt {1-\sin (c+d x)}}{\sqrt {2}}\right ) \sqrt {1-\sin (c+d x)}\right ) \cos ^9(c+d x)}{240 a^2 d (\sin (c+d x)-1)^5 (\sin (c+d x)+1)^{9/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.81, size = 60, normalized size = 0.58 \[ \frac {96 \, \cos \left (d x + c\right )^{5} + 105 \, d x - 5 \, {\left (8 \, \cos \left (d x + c\right )^{5} - 14 \, \cos \left (d x + c\right )^{3} - 21 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.80, size = 179, normalized size = 1.72 \[ \frac {\frac {105 \, {\left (d x + c\right )}}{a^{2}} - \frac {2 \, {\left (135 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 480 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 445 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 480 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 330 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 960 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 330 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 960 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 445 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 96 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 135 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 96\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{6} a^{2}}}{240 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.22, size = 415, normalized size = 3.99 \[ -\frac {9 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a^{2} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}+\frac {4 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{2} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}-\frac {89 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 a^{2} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}+\frac {4 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{2} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}+\frac {11 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{2} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}+\frac {8 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{2} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}-\frac {11 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{2} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}+\frac {8 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{2} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}+\frac {89 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 a^{2} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}+\frac {4 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 a^{2} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}+\frac {9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a^{2} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}+\frac {4}{5 a^{2} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}+\frac {7 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.63, size = 393, normalized size = 3.78 \[ \frac {\frac {\frac {135 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {96 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {445 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {960 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {330 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {960 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {330 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {480 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {445 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} + \frac {480 \, \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} - \frac {135 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} + 96}{a^{2} + \frac {6 \, a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {15 \, a^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {20 \, a^{2} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {15 \, a^{2} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {6 \, a^{2} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac {a^{2} \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}}} + \frac {105 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}}}{120 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.22, size = 172, normalized size = 1.65 \[ \frac {7\,x}{16\,a^2}+\frac {-\frac {9\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{8}+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-\frac {89\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{24}+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+\frac {11\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{4}+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-\frac {11\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{4}+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\frac {89\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24}+\frac {4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{5}+\frac {9\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8}+\frac {4}{5}}{a^2\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^6} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 148.51, size = 2531, normalized size = 24.34 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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