Optimal. Leaf size=70 \[ -\frac {2 \cos ^3(c+d x)}{3 a^2 d}-\frac {\sin (c+d x) \cos (c+d x)}{a^2 d}-\frac {x}{a^2}-\frac {\cos ^5(c+d x)}{d (a \sin (c+d x)+a)^2} \]
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Rubi [A] time = 0.11, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {2859, 2682, 2635, 8} \[ -\frac {2 \cos ^3(c+d x)}{3 a^2 d}-\frac {\sin (c+d x) \cos (c+d x)}{a^2 d}-\frac {x}{a^2}-\frac {\cos ^5(c+d x)}{d (a \sin (c+d x)+a)^2} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2635
Rule 2682
Rule 2859
Rubi steps
\begin {align*} \int \frac {\cos ^4(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^2} \, dx &=-\frac {\cos ^5(c+d x)}{d (a+a \sin (c+d x))^2}-\frac {2 \int \frac {\cos ^4(c+d x)}{a+a \sin (c+d x)} \, dx}{a}\\ &=-\frac {2 \cos ^3(c+d x)}{3 a^2 d}-\frac {\cos ^5(c+d x)}{d (a+a \sin (c+d x))^2}-\frac {2 \int \cos ^2(c+d x) \, dx}{a^2}\\ &=-\frac {2 \cos ^3(c+d x)}{3 a^2 d}-\frac {\cos (c+d x) \sin (c+d x)}{a^2 d}-\frac {\cos ^5(c+d x)}{d (a+a \sin (c+d x))^2}-\frac {\int 1 \, dx}{a^2}\\ &=-\frac {x}{a^2}-\frac {2 \cos ^3(c+d x)}{3 a^2 d}-\frac {\cos (c+d x) \sin (c+d x)}{a^2 d}-\frac {\cos ^5(c+d x)}{d (a+a \sin (c+d x))^2}\\ \end {align*}
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Mathematica [B] time = 0.81, size = 204, normalized size = 2.91 \[ \frac {-24 d x \sin \left (\frac {c}{2}\right )+21 \sin \left (\frac {c}{2}+d x\right )-21 \sin \left (\frac {3 c}{2}+d x\right )+6 \sin \left (\frac {3 c}{2}+2 d x\right )+6 \sin \left (\frac {5 c}{2}+2 d x\right )-\sin \left (\frac {5 c}{2}+3 d x\right )+\sin \left (\frac {7 c}{2}+3 d x\right )-2 \cos \left (\frac {c}{2}\right ) (12 d x+1)-21 \cos \left (\frac {c}{2}+d x\right )-21 \cos \left (\frac {3 c}{2}+d x\right )+6 \cos \left (\frac {3 c}{2}+2 d x\right )-6 \cos \left (\frac {5 c}{2}+2 d x\right )+\cos \left (\frac {5 c}{2}+3 d x\right )+\cos \left (\frac {7 c}{2}+3 d x\right )+2 \sin \left (\frac {c}{2}\right )}{24 a^2 d \left (\sin \left (\frac {c}{2}\right )+\cos \left (\frac {c}{2}\right )\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 43, normalized size = 0.61 \[ \frac {\cos \left (d x + c\right )^{3} - 3 \, d x + 3 \, \cos \left (d x + c\right ) \sin \left (d x + c\right ) - 6 \, \cos \left (d x + c\right )}{3 \, a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 88, normalized size = 1.26 \[ -\frac {\frac {3 \, {\left (d x + c\right )}}{a^{2}} + \frac {2 \, {\left (3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 12 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 5\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3} a^{2}}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.35, size = 177, normalized size = 2.53 \[ -\frac {2 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}-\frac {2 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}-\frac {8 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}-\frac {10}{3 d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}-\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.44, size = 184, normalized size = 2.63 \[ \frac {2 \, {\left (\frac {\frac {3 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {12 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {3 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {3 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - 5}{a^{2} + \frac {3 \, a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {3 \, a^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {a^{2} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}} - \frac {3 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}}\right )}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.64, size = 55, normalized size = 0.79 \[ \frac {\cos \left (3\,c+3\,d\,x\right )}{12\,a^2\,d}-\frac {7\,\cos \left (c+d\,x\right )}{4\,a^2\,d}-\frac {x}{a^2}+\frac {\sin \left (2\,c+2\,d\,x\right )}{2\,a^2\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 36.06, size = 694, normalized size = 9.91 \[ \begin {cases} - \frac {3 d x \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{3 a^{2} d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{2} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{2} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 3 a^{2} d} - \frac {9 d x \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{3 a^{2} d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{2} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{2} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 3 a^{2} d} - \frac {9 d x \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{3 a^{2} d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{2} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{2} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 3 a^{2} d} - \frac {3 d x}{3 a^{2} d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{2} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{2} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 3 a^{2} d} - \frac {6 \tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{3 a^{2} d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{2} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{2} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 3 a^{2} d} - \frac {6 \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{3 a^{2} d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{2} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{2} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 3 a^{2} d} - \frac {24 \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{3 a^{2} d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{2} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{2} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 3 a^{2} d} + \frac {6 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{3 a^{2} d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{2} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{2} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 3 a^{2} d} - \frac {10}{3 a^{2} d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{2} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{2} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 3 a^{2} d} & \text {for}\: d \neq 0 \\\frac {x \sin {\relax (c )} \cos ^{4}{\relax (c )}}{\left (a \sin {\relax (c )} + a\right )^{2}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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