Optimal. Leaf size=127 \[ \frac {2 \cos (c+d x)}{d \left (a^8 \sin (c+d x)+a^8\right )}+\frac {x}{a^8}+\frac {2 \cos ^5(c+d x)}{5 a^3 d (a \sin (c+d x)+a)^5}-\frac {2 \cos ^3(c+d x)}{3 a^2 d \left (a^2 \sin (c+d x)+a^2\right )^3}-\frac {2 \cos ^7(c+d x)}{7 a d (a \sin (c+d x)+a)^7} \]
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Rubi [A] time = 0.18, antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2680, 8} \[ \frac {2 \cos ^5(c+d x)}{5 a^3 d (a \sin (c+d x)+a)^5}-\frac {2 \cos ^3(c+d x)}{3 a^2 d \left (a^2 \sin (c+d x)+a^2\right )^3}+\frac {2 \cos (c+d x)}{d \left (a^8 \sin (c+d x)+a^8\right )}+\frac {x}{a^8}-\frac {2 \cos ^7(c+d x)}{7 a d (a \sin (c+d x)+a)^7} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2680
Rubi steps
\begin {align*} \int \frac {\cos ^8(c+d x)}{(a+a \sin (c+d x))^8} \, dx &=-\frac {2 \cos ^7(c+d x)}{7 a d (a+a \sin (c+d x))^7}-\frac {\int \frac {\cos ^6(c+d x)}{(a+a \sin (c+d x))^6} \, dx}{a^2}\\ &=-\frac {2 \cos ^7(c+d x)}{7 a d (a+a \sin (c+d x))^7}+\frac {2 \cos ^5(c+d x)}{5 a^3 d (a+a \sin (c+d x))^5}+\frac {\int \frac {\cos ^4(c+d x)}{(a+a \sin (c+d x))^4} \, dx}{a^4}\\ &=-\frac {2 \cos ^7(c+d x)}{7 a d (a+a \sin (c+d x))^7}+\frac {2 \cos ^5(c+d x)}{5 a^3 d (a+a \sin (c+d x))^5}-\frac {2 \cos ^3(c+d x)}{3 a^5 d (a+a \sin (c+d x))^3}-\frac {\int \frac {\cos ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx}{a^6}\\ &=-\frac {2 \cos ^7(c+d x)}{7 a d (a+a \sin (c+d x))^7}+\frac {2 \cos ^5(c+d x)}{5 a^3 d (a+a \sin (c+d x))^5}-\frac {2 \cos ^3(c+d x)}{3 a^5 d (a+a \sin (c+d x))^3}+\frac {2 \cos (c+d x)}{d \left (a^8+a^8 \sin (c+d x)\right )}+\frac {\int 1 \, dx}{a^8}\\ &=\frac {x}{a^8}-\frac {2 \cos ^7(c+d x)}{7 a d (a+a \sin (c+d x))^7}+\frac {2 \cos ^5(c+d x)}{5 a^3 d (a+a \sin (c+d x))^5}-\frac {2 \cos ^3(c+d x)}{3 a^5 d (a+a \sin (c+d x))^3}+\frac {2 \cos (c+d x)}{d \left (a^8+a^8 \sin (c+d x)\right )}\\ \end {align*}
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Mathematica [B] time = 6.08, size = 275, normalized size = 2.17 \[ -\frac {2 \sqrt {2} \left (\frac {1}{2} (1-\sin (c+d x))-1\right )^4 \left (\frac {\sin ^{-1}\left (\frac {\sqrt {1-\sin (c+d x)}}{\sqrt {2}}\right ) \sqrt {1-\sin (c+d x)}}{\sqrt {2} \sqrt {\frac {1}{2} (\sin (c+d x)-1)+1}}+\frac {(1-\sin (c+d x))^4}{112 \left (\frac {1}{2} (1-\sin (c+d x))-1\right )^4}+\frac {(1-\sin (c+d x))^3}{40 \left (\frac {1}{2} (1-\sin (c+d x))-1\right )^3}+\frac {(1-\sin (c+d x))^2}{12 \left (\frac {1}{2} (1-\sin (c+d x))-1\right )^2}+\frac {1-\sin (c+d x)}{2 \left (\frac {1}{2} (1-\sin (c+d x))-1\right )}\right ) \cos ^9(c+d x)}{a^8 d \left (\frac {1}{2} (\sin (c+d x)-1)+1\right )^{7/2} (1-\sin (c+d x))^5 (\sin (c+d x)+1)^{9/2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.76, size = 244, normalized size = 1.92 \[ \frac {{\left (105 \, d x - 352\right )} \cos \left (d x + c\right )^{4} - {\left (315 \, d x + 568\right )} \cos \left (d x + c\right )^{3} - 24 \, {\left (35 \, d x - 31\right )} \cos \left (d x + c\right )^{2} + 840 \, d x + 60 \, {\left (7 \, d x + 12\right )} \cos \left (d x + c\right ) - {\left ({\left (105 \, d x + 352\right )} \cos \left (d x + c\right )^{3} + 12 \, {\left (35 \, d x - 18\right )} \cos \left (d x + c\right )^{2} - 840 \, d x - 60 \, {\left (7 \, d x + 16\right )} \cos \left (d x + c\right ) - 240\right )} \sin \left (d x + c\right ) - 240}{105 \, {\left (a^{8} d \cos \left (d x + c\right )^{4} - 3 \, a^{8} d \cos \left (d x + c\right )^{3} - 8 \, a^{8} d \cos \left (d x + c\right )^{2} + 4 \, a^{8} d \cos \left (d x + c\right ) + 8 \, a^{8} d - {\left (a^{8} d \cos \left (d x + c\right )^{3} + 4 \, a^{8} d \cos \left (d x + c\right )^{2} - 4 \, a^{8} d \cos \left (d x + c\right ) - 8 \, a^{8} d\right )} \sin \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.18, size = 99, normalized size = 0.78 \[ \frac {\frac {105 \, {\left (d x + c\right )}}{a^{8}} + \frac {16 \, {\left (105 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 175 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 490 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 294 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 133 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 19\right )}}{a^{8} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{7}}}{105 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.28, size = 146, normalized size = 1.15 \[ \frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{8} d}-\frac {256}{7 a^{8} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7}}+\frac {128}{a^{8} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{6}}-\frac {896}{5 a^{8} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}+\frac {128}{a^{8} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}-\frac {160}{3 a^{8} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {16}{a^{8} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.64, size = 295, normalized size = 2.32 \[ \frac {2 \, {\left (\frac {8 \, {\left (\frac {133 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {294 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {490 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {175 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {105 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + 19\right )}}{a^{8} + \frac {7 \, a^{8} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {21 \, a^{8} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {35 \, a^{8} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {35 \, a^{8} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {21 \, a^{8} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {7 \, a^{8} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {a^{8} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}} + \frac {105 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{8}}\right )}}{105 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 7.77, size = 91, normalized size = 0.72 \[ \frac {x}{a^8}+\frac {16\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\frac {80\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{3}+\frac {224\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3}+\frac {224\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{5}+\frac {304\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{15}+\frac {304}{105}}{a^8\,d\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}^7} \]
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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