Optimal. Leaf size=135 \[ -\frac {\cos ^7(c+d x)}{7 a^2 d}+\frac {3 \cos ^5(c+d x)}{5 a^2 d}-\frac {2 \cos ^3(c+d x)}{3 a^2 d}+\frac {\sin ^3(c+d x) \cos ^3(c+d x)}{3 a^2 d}+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 a^2 d}-\frac {\sin (c+d x) \cos (c+d x)}{8 a^2 d}-\frac {x}{8 a^2} \]
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Rubi [A] time = 0.35, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {2875, 2873, 2565, 14, 2568, 2635, 8, 270} \[ -\frac {\cos ^7(c+d x)}{7 a^2 d}+\frac {3 \cos ^5(c+d x)}{5 a^2 d}-\frac {2 \cos ^3(c+d x)}{3 a^2 d}+\frac {\sin ^3(c+d x) \cos ^3(c+d x)}{3 a^2 d}+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 a^2 d}-\frac {\sin (c+d x) \cos (c+d x)}{8 a^2 d}-\frac {x}{8 a^2} \]
Antiderivative was successfully verified.
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Rule 8
Rule 14
Rule 270
Rule 2565
Rule 2568
Rule 2635
Rule 2873
Rule 2875
Rubi steps
\begin {align*} \int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx &=\frac {\int \cos ^2(c+d x) \sin ^3(c+d x) (a-a \sin (c+d x))^2 \, dx}{a^4}\\ &=\frac {\int \left (a^2 \cos ^2(c+d x) \sin ^3(c+d x)-2 a^2 \cos ^2(c+d x) \sin ^4(c+d x)+a^2 \cos ^2(c+d x) \sin ^5(c+d x)\right ) \, dx}{a^4}\\ &=\frac {\int \cos ^2(c+d x) \sin ^3(c+d x) \, dx}{a^2}+\frac {\int \cos ^2(c+d x) \sin ^5(c+d x) \, dx}{a^2}-\frac {2 \int \cos ^2(c+d x) \sin ^4(c+d x) \, dx}{a^2}\\ &=\frac {\cos ^3(c+d x) \sin ^3(c+d x)}{3 a^2 d}-\frac {\int \cos ^2(c+d x) \sin ^2(c+d x) \, dx}{a^2}-\frac {\operatorname {Subst}\left (\int x^2 \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{a^2 d}-\frac {\operatorname {Subst}\left (\int x^2 \left (1-x^2\right )^2 \, dx,x,\cos (c+d x)\right )}{a^2 d}\\ &=\frac {\cos ^3(c+d x) \sin (c+d x)}{4 a^2 d}+\frac {\cos ^3(c+d x) \sin ^3(c+d x)}{3 a^2 d}-\frac {\int \cos ^2(c+d x) \, dx}{4 a^2}-\frac {\operatorname {Subst}\left (\int \left (x^2-x^4\right ) \, dx,x,\cos (c+d x)\right )}{a^2 d}-\frac {\operatorname {Subst}\left (\int \left (x^2-2 x^4+x^6\right ) \, dx,x,\cos (c+d x)\right )}{a^2 d}\\ &=-\frac {2 \cos ^3(c+d x)}{3 a^2 d}+\frac {3 \cos ^5(c+d x)}{5 a^2 d}-\frac {\cos ^7(c+d x)}{7 a^2 d}-\frac {\cos (c+d x) \sin (c+d x)}{8 a^2 d}+\frac {\cos ^3(c+d x) \sin (c+d x)}{4 a^2 d}+\frac {\cos ^3(c+d x) \sin ^3(c+d x)}{3 a^2 d}-\frac {\int 1 \, dx}{8 a^2}\\ &=-\frac {x}{8 a^2}-\frac {2 \cos ^3(c+d x)}{3 a^2 d}+\frac {3 \cos ^5(c+d x)}{5 a^2 d}-\frac {\cos ^7(c+d x)}{7 a^2 d}-\frac {\cos (c+d x) \sin (c+d x)}{8 a^2 d}+\frac {\cos ^3(c+d x) \sin (c+d x)}{4 a^2 d}+\frac {\cos ^3(c+d x) \sin ^3(c+d x)}{3 a^2 d}\\ \end {align*}
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Mathematica [B] time = 3.15, size = 418, normalized size = 3.10 \[ -\frac {1680 d x \sin \left (\frac {c}{2}\right )-1365 \sin \left (\frac {c}{2}+d x\right )+1365 \sin \left (\frac {3 c}{2}+d x\right )-210 \sin \left (\frac {3 c}{2}+2 d x\right )-210 \sin \left (\frac {5 c}{2}+2 d x\right )-175 \sin \left (\frac {5 c}{2}+3 d x\right )+175 \sin \left (\frac {7 c}{2}+3 d x\right )-210 \sin \left (\frac {7 c}{2}+4 d x\right )-210 \sin \left (\frac {9 c}{2}+4 d x\right )+147 \sin \left (\frac {9 c}{2}+5 d x\right )-147 \sin \left (\frac {11 c}{2}+5 d x\right )+70 \sin \left (\frac {11 c}{2}+6 d x\right )+70 \sin \left (\frac {13 c}{2}+6 d x\right )-15 \sin \left (\frac {13 c}{2}+7 d x\right )+15 \sin \left (\frac {15 c}{2}+7 d x\right )+210 \cos \left (\frac {c}{2}\right ) (8 d x+1)+1365 \cos \left (\frac {c}{2}+d x\right )+1365 \cos \left (\frac {3 c}{2}+d x\right )-210 \cos \left (\frac {3 c}{2}+2 d x\right )+210 \cos \left (\frac {5 c}{2}+2 d x\right )+175 \cos \left (\frac {5 c}{2}+3 d x\right )+175 \cos \left (\frac {7 c}{2}+3 d x\right )-210 \cos \left (\frac {7 c}{2}+4 d x\right )+210 \cos \left (\frac {9 c}{2}+4 d x\right )-147 \cos \left (\frac {9 c}{2}+5 d x\right )-147 \cos \left (\frac {11 c}{2}+5 d x\right )+70 \cos \left (\frac {11 c}{2}+6 d x\right )-70 \cos \left (\frac {13 c}{2}+6 d x\right )+15 \cos \left (\frac {13 c}{2}+7 d x\right )+15 \cos \left (\frac {15 c}{2}+7 d x\right )-210 \sin \left (\frac {c}{2}\right )}{13440 a^2 d \left (\sin \left (\frac {c}{2}\right )+\cos \left (\frac {c}{2}\right )\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.71, size = 80, normalized size = 0.59 \[ -\frac {120 \, \cos \left (d x + c\right )^{7} - 504 \, \cos \left (d x + c\right )^{5} + 560 \, \cos \left (d x + c\right )^{3} + 105 \, d x + 35 \, {\left (8 \, \cos \left (d x + c\right )^{5} - 14 \, \cos \left (d x + c\right )^{3} + 3 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{840 \, a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 179, normalized size = 1.33 \[ -\frac {\frac {105 \, {\left (d x + c\right )}}{a^{2}} + \frac {2 \, {\left (105 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 700 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 1680 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} - 3395 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 7280 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 1120 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 3395 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 2016 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 700 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1232 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 105 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 176\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{7} a^{2}}}{840 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.34, size = 415, normalized size = 3.07 \[ -\frac {\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}-\frac {5 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}-\frac {4 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}+\frac {97 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 a^{2} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}-\frac {52 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a^{2} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}+\frac {8 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}-\frac {97 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}-\frac {24 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}+\frac {5 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a^{2} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}-\frac {44 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}-\frac {44}{105 d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}-\frac {\arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d \,a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.43, size = 416, normalized size = 3.08 \[ \frac {\frac {\frac {105 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {1232 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {700 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {2016 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {3395 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {1120 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {7280 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {3395 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {1680 \, \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} - \frac {700 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} - \frac {105 \, \sin \left (d x + c\right )^{13}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{13}} - 176}{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 {105 \, \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 = 12.66, size = 173, normalized size = 1.28 \[ -\frac {x}{8\,a^2}-\frac {\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{4}+\frac {5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{3}+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-\frac {97\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{12}+\frac {52\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{3}-\frac {8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{3}+\frac {97\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{12}+\frac {24\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{5}-\frac {5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3}+\frac {44\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{15}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}+\frac {44}{105}}{a^2\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^7} \]
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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