Optimal. Leaf size=84 \[ \frac {\cos ^3(c+d x)}{a^3 d}-\frac {4 \cos (c+d x)}{a^3 d}+\frac {\sin ^3(c+d x) \cos (c+d x)}{4 a^3 d}+\frac {15 \sin (c+d x) \cos (c+d x)}{8 a^3 d}-\frac {15 x}{8 a^3} \]
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Rubi [A] time = 0.17, antiderivative size = 105, normalized size of antiderivative = 1.25, number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {2859, 2679, 2682, 2635, 8} \[ -\frac {5 \cos ^3(c+d x)}{4 a^3 d}-\frac {3 \cos ^5(c+d x)}{4 d \left (a^3 \sin (c+d x)+a^3\right )}-\frac {15 \sin (c+d x) \cos (c+d x)}{8 a^3 d}-\frac {15 x}{8 a^3}-\frac {\cos ^7(c+d x)}{d (a \sin (c+d x)+a)^3} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2635
Rule 2679
Rule 2682
Rule 2859
Rubi steps
\begin {align*} \int \frac {\cos ^6(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^3} \, dx &=-\frac {\cos ^7(c+d x)}{d (a+a \sin (c+d x))^3}-\frac {3 \int \frac {\cos ^6(c+d x)}{(a+a \sin (c+d x))^2} \, dx}{a}\\ &=-\frac {\cos ^7(c+d x)}{d (a+a \sin (c+d x))^3}-\frac {3 \cos ^5(c+d x)}{4 d \left (a^3+a^3 \sin (c+d x)\right )}-\frac {15 \int \frac {\cos ^4(c+d x)}{a+a \sin (c+d x)} \, dx}{4 a^2}\\ &=-\frac {5 \cos ^3(c+d x)}{4 a^3 d}-\frac {\cos ^7(c+d x)}{d (a+a \sin (c+d x))^3}-\frac {3 \cos ^5(c+d x)}{4 d \left (a^3+a^3 \sin (c+d x)\right )}-\frac {15 \int \cos ^2(c+d x) \, dx}{4 a^3}\\ &=-\frac {5 \cos ^3(c+d x)}{4 a^3 d}-\frac {15 \cos (c+d x) \sin (c+d x)}{8 a^3 d}-\frac {\cos ^7(c+d x)}{d (a+a \sin (c+d x))^3}-\frac {3 \cos ^5(c+d x)}{4 d \left (a^3+a^3 \sin (c+d x)\right )}-\frac {15 \int 1 \, dx}{8 a^3}\\ &=-\frac {15 x}{8 a^3}-\frac {5 \cos ^3(c+d x)}{4 a^3 d}-\frac {15 \cos (c+d x) \sin (c+d x)}{8 a^3 d}-\frac {\cos ^7(c+d x)}{d (a+a \sin (c+d x))^3}-\frac {3 \cos ^5(c+d x)}{4 d \left (a^3+a^3 \sin (c+d x)\right )}\\ \end {align*}
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Mathematica [B] time = 1.37, size = 255, normalized size = 3.04 \[ -\frac {120 d x \sin \left (\frac {c}{2}\right )-104 \sin \left (\frac {c}{2}+d x\right )+104 \sin \left (\frac {3 c}{2}+d x\right )-32 \sin \left (\frac {3 c}{2}+2 d x\right )-32 \sin \left (\frac {5 c}{2}+2 d x\right )+8 \sin \left (\frac {5 c}{2}+3 d x\right )-8 \sin \left (\frac {7 c}{2}+3 d x\right )+\sin \left (\frac {7 c}{2}+4 d x\right )+\sin \left (\frac {9 c}{2}+4 d x\right )+\cos \left (\frac {c}{2}\right ) (120 d x+1)+104 \cos \left (\frac {c}{2}+d x\right )+104 \cos \left (\frac {3 c}{2}+d x\right )-32 \cos \left (\frac {3 c}{2}+2 d x\right )+32 \cos \left (\frac {5 c}{2}+2 d x\right )-8 \cos \left (\frac {5 c}{2}+3 d x\right )-8 \cos \left (\frac {7 c}{2}+3 d x\right )+\cos \left (\frac {7 c}{2}+4 d x\right )-\cos \left (\frac {9 c}{2}+4 d x\right )-\sin \left (\frac {c}{2}\right )}{64 a^3 d \left (\sin \left (\frac {c}{2}\right )+\cos \left (\frac {c}{2}\right )\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.72, size = 58, normalized size = 0.69 \[ \frac {8 \, \cos \left (d x + c\right )^{3} - 15 \, d x - {\left (2 \, \cos \left (d x + c\right )^{3} - 17 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) - 32 \, \cos \left (d x + c\right )}{8 \, a^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 127, normalized size = 1.51 \[ -\frac {\frac {15 \, {\left (d x + c\right )}}{a^{3}} + \frac {2 \, {\left (15 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 8 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 23 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 72 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 23 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 88 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 15 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 24\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{4} a^{3}}}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.43, size = 279, normalized size = 3.32 \[ -\frac {15 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{3} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}-\frac {2 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{3} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}-\frac {23 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{3} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}-\frac {18 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{3} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {23 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{3} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}-\frac {22 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{3} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {15 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a^{3} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}-\frac {6}{a^{3} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}-\frac {15 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.44, size = 267, normalized size = 3.18 \[ \frac {\frac {\frac {15 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {88 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {23 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {72 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {23 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {8 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {15 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - 24}{a^{3} + \frac {4 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {6 \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {4 \, a^{3} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {a^{3} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}} - \frac {15 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.00, size = 78, normalized size = 0.93 \[ \frac {{\cos \left (c+d\,x\right )}^3}{a^3\,d}-\frac {4\,\cos \left (c+d\,x\right )}{a^3\,d}-\frac {15\,x}{8\,a^3}-\frac {{\cos \left (c+d\,x\right )}^3\,\sin \left (c+d\,x\right )}{4\,a^3\,d}+\frac {17\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )}{8\,a^3\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 166.36, size = 1246, normalized size = 14.83 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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