Optimal. Leaf size=147 \[ \frac {\cos ^7(c+d x)}{7 a^2 d}-\frac {4 \cos ^5(c+d x)}{5 a^2 d}+\frac {5 \cos ^3(c+d x)}{3 a^2 d}-\frac {2 \cos (c+d x)}{a^2 d}+\frac {\sin ^5(c+d x) \cos (c+d x)}{3 a^2 d}+\frac {5 \sin ^3(c+d x) \cos (c+d x)}{12 a^2 d}+\frac {5 \sin (c+d x) \cos (c+d x)}{8 a^2 d}-\frac {5 x}{8 a^2} \]
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Rubi [A] time = 0.22, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 5, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {2869, 2757, 2633, 2635, 8} \[ \frac {\cos ^7(c+d x)}{7 a^2 d}-\frac {4 \cos ^5(c+d x)}{5 a^2 d}+\frac {5 \cos ^3(c+d x)}{3 a^2 d}-\frac {2 \cos (c+d x)}{a^2 d}+\frac {\sin ^5(c+d x) \cos (c+d x)}{3 a^2 d}+\frac {5 \sin ^3(c+d x) \cos (c+d x)}{12 a^2 d}+\frac {5 \sin (c+d x) \cos (c+d x)}{8 a^2 d}-\frac {5 x}{8 a^2} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2633
Rule 2635
Rule 2757
Rule 2869
Rubi steps
\begin {align*} \int \frac {\cos ^4(c+d x) \sin ^5(c+d x)}{(a+a \sin (c+d x))^2} \, dx &=\frac {\int \sin ^5(c+d x) (a-a \sin (c+d x))^2 \, dx}{a^4}\\ &=\frac {\int \left (a^2 \sin ^5(c+d x)-2 a^2 \sin ^6(c+d x)+a^2 \sin ^7(c+d x)\right ) \, dx}{a^4}\\ &=\frac {\int \sin ^5(c+d x) \, dx}{a^2}+\frac {\int \sin ^7(c+d x) \, dx}{a^2}-\frac {2 \int \sin ^6(c+d x) \, dx}{a^2}\\ &=\frac {\cos (c+d x) \sin ^5(c+d x)}{3 a^2 d}-\frac {5 \int \sin ^4(c+d x) \, dx}{3 a^2}-\frac {\operatorname {Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,\cos (c+d x)\right )}{a^2 d}-\frac {\operatorname {Subst}\left (\int \left (1-3 x^2+3 x^4-x^6\right ) \, dx,x,\cos (c+d x)\right )}{a^2 d}\\ &=-\frac {2 \cos (c+d x)}{a^2 d}+\frac {5 \cos ^3(c+d x)}{3 a^2 d}-\frac {4 \cos ^5(c+d x)}{5 a^2 d}+\frac {\cos ^7(c+d x)}{7 a^2 d}+\frac {5 \cos (c+d x) \sin ^3(c+d x)}{12 a^2 d}+\frac {\cos (c+d x) \sin ^5(c+d x)}{3 a^2 d}-\frac {5 \int \sin ^2(c+d x) \, dx}{4 a^2}\\ &=-\frac {2 \cos (c+d x)}{a^2 d}+\frac {5 \cos ^3(c+d x)}{3 a^2 d}-\frac {4 \cos ^5(c+d x)}{5 a^2 d}+\frac {\cos ^7(c+d x)}{7 a^2 d}+\frac {5 \cos (c+d x) \sin (c+d x)}{8 a^2 d}+\frac {5 \cos (c+d x) \sin ^3(c+d x)}{12 a^2 d}+\frac {\cos (c+d x) \sin ^5(c+d x)}{3 a^2 d}-\frac {5 \int 1 \, dx}{8 a^2}\\ &=-\frac {5 x}{8 a^2}-\frac {2 \cos (c+d x)}{a^2 d}+\frac {5 \cos ^3(c+d x)}{3 a^2 d}-\frac {4 \cos ^5(c+d x)}{5 a^2 d}+\frac {\cos ^7(c+d x)}{7 a^2 d}+\frac {5 \cos (c+d x) \sin (c+d x)}{8 a^2 d}+\frac {5 \cos (c+d x) \sin ^3(c+d x)}{12 a^2 d}+\frac {\cos (c+d x) \sin ^5(c+d x)}{3 a^2 d}\\ \end {align*}
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Mathematica [B] time = 4.83, size = 418, normalized size = 2.84 \[ \frac {-8400 d x \sin \left (\frac {c}{2}\right )+7875 \sin \left (\frac {c}{2}+d x\right )-7875 \sin \left (\frac {3 c}{2}+d x\right )+3150 \sin \left (\frac {3 c}{2}+2 d x\right )+3150 \sin \left (\frac {5 c}{2}+2 d x\right )-1435 \sin \left (\frac {5 c}{2}+3 d x\right )+1435 \sin \left (\frac {7 c}{2}+3 d x\right )-630 \sin \left (\frac {7 c}{2}+4 d x\right )-630 \sin \left (\frac {9 c}{2}+4 d x\right )+231 \sin \left (\frac {9 c}{2}+5 d x\right )-231 \sin \left (\frac {11 c}{2}+5 d x\right )+70 \sin \left (\frac {11 c}{2}+6 d x\right )+70 \sin \left (\frac {13 c}{2}+6 d x\right )-15 \sin \left (\frac {13 c}{2}+7 d x\right )+15 \sin \left (\frac {15 c}{2}+7 d x\right )-210 \cos \left (\frac {c}{2}\right ) (40 d x+1)-7875 \cos \left (\frac {c}{2}+d x\right )-7875 \cos \left (\frac {3 c}{2}+d x\right )+3150 \cos \left (\frac {3 c}{2}+2 d x\right )-3150 \cos \left (\frac {5 c}{2}+2 d x\right )+1435 \cos \left (\frac {5 c}{2}+3 d x\right )+1435 \cos \left (\frac {7 c}{2}+3 d x\right )-630 \cos \left (\frac {7 c}{2}+4 d x\right )+630 \cos \left (\frac {9 c}{2}+4 d x\right )-231 \cos \left (\frac {9 c}{2}+5 d x\right )-231 \cos \left (\frac {11 c}{2}+5 d x\right )+70 \cos \left (\frac {11 c}{2}+6 d x\right )-70 \cos \left (\frac {13 c}{2}+6 d x\right )+15 \cos \left (\frac {13 c}{2}+7 d x\right )+15 \cos \left (\frac {15 c}{2}+7 d x\right )+210 \sin \left (\frac {c}{2}\right )}{13440 a^2 d \left (\sin \left (\frac {c}{2}\right )+\cos \left (\frac {c}{2}\right )\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 88, normalized size = 0.60 \[ \frac {120 \, \cos \left (d x + c\right )^{7} - 672 \, \cos \left (d x + c\right )^{5} + 1400 \, \cos \left (d x + c\right )^{3} - 525 \, d x + 35 \, {\left (8 \, \cos \left (d x + c\right )^{5} - 26 \, \cos \left (d x + c\right )^{3} + 33 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) - 1680 \, \cos \left (d x + c\right )}{840 \, a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 166, normalized size = 1.13 \[ -\frac {\frac {525 \, {\left (d x + c\right )}}{a^{2}} + \frac {2 \, {\left (525 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 3500 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 9905 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 4480 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 24640 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 9905 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 17472 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 3500 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 5824 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 525 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 832\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{7} a^{2}}}{840 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.44, size = 381, normalized size = 2.59 \[ -\frac {5 \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{2} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}-\frac {25 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a^{2} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}-\frac {283 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 a^{2} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}-\frac {32 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a^{2} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}-\frac {176 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a^{2} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}+\frac {283 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 a^{2} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}-\frac {208 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 a^{2} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}+\frac {25 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a^{2} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}-\frac {208 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 a^{2} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}+\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a^{2} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}-\frac {208}{105 a^{2} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}-\frac {5 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d \,a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.49, size = 396, normalized size = 2.69 \[ \frac {\frac {\frac {525 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {5824 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {3500 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {17472 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {9905 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {24640 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {4480 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {9905 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {3500 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} - \frac {525 \, \sin \left (d x + c\right )^{13}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{13}} - 832}{a^{2} + \frac {7 \, a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {21 \, a^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {35 \, a^{2} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {35 \, a^{2} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {21 \, a^{2} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac {7 \, a^{2} \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}} + \frac {a^{2} \sin \left (d x + c\right )^{14}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{14}}} - \frac {525 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}}}{420 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 12.22, size = 160, normalized size = 1.09 \[ -\frac {5\,x}{8\,a^2}-\frac {\frac {5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{4}+\frac {25\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{3}+\frac {283\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{12}+\frac {32\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{3}+\frac {176\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{3}-\frac {283\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{12}+\frac {208\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{5}-\frac {25\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3}+\frac {208\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{15}-\frac {5\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}+\frac {208}{105}}{a^2\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^7} \]
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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