Optimal. Leaf size=87 \[ -\frac {2 \cos ^3(c+d x)}{3 a^2 d}+\frac {2 \cos (c+d x)}{a^2 d}-\frac {\sin ^3(c+d x) \cos (c+d x)}{4 a^2 d}-\frac {7 \sin (c+d x) \cos (c+d x)}{8 a^2 d}+\frac {7 x}{8 a^2} \]
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Rubi [A] time = 0.20, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 5, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {2869, 2757, 2635, 8, 2633} \[ -\frac {2 \cos ^3(c+d x)}{3 a^2 d}+\frac {2 \cos (c+d x)}{a^2 d}-\frac {\sin ^3(c+d x) \cos (c+d x)}{4 a^2 d}-\frac {7 \sin (c+d x) \cos (c+d x)}{8 a^2 d}+\frac {7 x}{8 a^2} \]
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 ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx &=\frac {\int \sin ^2(c+d x) (a-a \sin (c+d x))^2 \, dx}{a^4}\\ &=\frac {\int \left (a^2 \sin ^2(c+d x)-2 a^2 \sin ^3(c+d x)+a^2 \sin ^4(c+d x)\right ) \, dx}{a^4}\\ &=\frac {\int \sin ^2(c+d x) \, dx}{a^2}+\frac {\int \sin ^4(c+d x) \, dx}{a^2}-\frac {2 \int \sin ^3(c+d x) \, dx}{a^2}\\ &=-\frac {\cos (c+d x) \sin (c+d x)}{2 a^2 d}-\frac {\cos (c+d x) \sin ^3(c+d x)}{4 a^2 d}+\frac {\int 1 \, dx}{2 a^2}+\frac {3 \int \sin ^2(c+d x) \, dx}{4 a^2}+\frac {2 \operatorname {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{a^2 d}\\ &=\frac {x}{2 a^2}+\frac {2 \cos (c+d x)}{a^2 d}-\frac {2 \cos ^3(c+d x)}{3 a^2 d}-\frac {7 \cos (c+d x) \sin (c+d x)}{8 a^2 d}-\frac {\cos (c+d x) \sin ^3(c+d x)}{4 a^2 d}+\frac {3 \int 1 \, dx}{8 a^2}\\ &=\frac {7 x}{8 a^2}+\frac {2 \cos (c+d x)}{a^2 d}-\frac {2 \cos ^3(c+d x)}{3 a^2 d}-\frac {7 \cos (c+d x) \sin (c+d x)}{8 a^2 d}-\frac {\cos (c+d x) \sin ^3(c+d x)}{4 a^2 d}\\ \end {align*}
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Mathematica [B] time = 1.35, size = 258, normalized size = 2.97 \[ \frac {168 d x \sin \left (\frac {c}{2}\right )-144 \sin \left (\frac {c}{2}+d x\right )+144 \sin \left (\frac {3 c}{2}+d x\right )-48 \sin \left (\frac {3 c}{2}+2 d x\right )-48 \sin \left (\frac {5 c}{2}+2 d x\right )+16 \sin \left (\frac {5 c}{2}+3 d x\right )-16 \sin \left (\frac {7 c}{2}+3 d x\right )+3 \sin \left (\frac {7 c}{2}+4 d x\right )+3 \sin \left (\frac {9 c}{2}+4 d x\right )+168 d x \cos \left (\frac {c}{2}\right )+144 \cos \left (\frac {c}{2}+d x\right )+144 \cos \left (\frac {3 c}{2}+d x\right )-48 \cos \left (\frac {3 c}{2}+2 d x\right )+48 \cos \left (\frac {5 c}{2}+2 d x\right )-16 \cos \left (\frac {5 c}{2}+3 d x\right )-16 \cos \left (\frac {7 c}{2}+3 d x\right )+3 \cos \left (\frac {7 c}{2}+4 d x\right )-3 \cos \left (\frac {9 c}{2}+4 d x\right )+8 \sin \left (\frac {c}{2}\right )}{192 a^2 d \left (\sin \left (\frac {c}{2}\right )+\cos \left (\frac {c}{2}\right )\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 58, normalized size = 0.67 \[ -\frac {16 \, \cos \left (d x + c\right )^{3} - 21 \, d x - 3 \, {\left (2 \, \cos \left (d x + c\right )^{3} - 9 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) - 48 \, \cos \left (d x + c\right )}{24 \, a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 114, normalized size = 1.31 \[ \frac {\frac {21 \, {\left (d x + c\right )}}{a^{2}} + \frac {2 \, {\left (21 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 45 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 96 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 45 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 128 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 21 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 32\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{4} a^{2}}}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.36, size = 245, normalized size = 2.82 \[ \frac {7 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{2} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {15 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{2} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {8 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}-\frac {15 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{2} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {32 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a^{2} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}-\frac {7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a^{2} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {8}{3 a^{2} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {7 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d \,a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.47, size = 247, normalized size = 2.84 \[ -\frac {\frac {\frac {21 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {128 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {45 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {96 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {45 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {21 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - 32}{a^{2} + \frac {4 \, a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {6 \, a^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {4 \, a^{2} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {a^{2} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}} - \frac {21 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}}}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.66, size = 79, normalized size = 0.91 \[ \frac {7\,x}{8\,a^2}+\frac {2\,\cos \left (c+d\,x\right )}{a^2\,d}-\frac {2\,{\cos \left (c+d\,x\right )}^3}{3\,a^2\,d}+\frac {{\cos \left (c+d\,x\right )}^3\,\sin \left (c+d\,x\right )}{4\,a^2\,d}-\frac {9\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )}{8\,a^2\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 59.86, size = 1153, normalized size = 13.25 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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