Optimal. Leaf size=102 \[ -\frac {\cos ^5(c+d x)}{5 a^2 d}+\frac {\cos ^3(c+d x)}{a^2 d}-\frac {2 \cos (c+d x)}{a^2 d}+\frac {\sin ^3(c+d x) \cos (c+d x)}{2 a^2 d}+\frac {3 \sin (c+d x) \cos (c+d x)}{4 a^2 d}-\frac {3 x}{4 a^2} \]
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Rubi [A] time = 0.20, antiderivative size = 102, 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, 2633, 2635, 8} \[ -\frac {\cos ^5(c+d x)}{5 a^2 d}+\frac {\cos ^3(c+d x)}{a^2 d}-\frac {2 \cos (c+d x)}{a^2 d}+\frac {\sin ^3(c+d x) \cos (c+d x)}{2 a^2 d}+\frac {3 \sin (c+d x) \cos (c+d x)}{4 a^2 d}-\frac {3 x}{4 a^2} \]
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 ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx &=\frac {\int \sin ^3(c+d x) (a-a \sin (c+d x))^2 \, dx}{a^4}\\ &=\frac {\int \left (a^2 \sin ^3(c+d x)-2 a^2 \sin ^4(c+d x)+a^2 \sin ^5(c+d x)\right ) \, dx}{a^4}\\ &=\frac {\int \sin ^3(c+d x) \, dx}{a^2}+\frac {\int \sin ^5(c+d x) \, dx}{a^2}-\frac {2 \int \sin ^4(c+d x) \, dx}{a^2}\\ &=\frac {\cos (c+d x) \sin ^3(c+d x)}{2 a^2 d}-\frac {3 \int \sin ^2(c+d x) \, dx}{2 a^2}-\frac {\operatorname {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{a^2 d}-\frac {\operatorname {Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,\cos (c+d x)\right )}{a^2 d}\\ &=-\frac {2 \cos (c+d x)}{a^2 d}+\frac {\cos ^3(c+d x)}{a^2 d}-\frac {\cos ^5(c+d x)}{5 a^2 d}+\frac {3 \cos (c+d x) \sin (c+d x)}{4 a^2 d}+\frac {\cos (c+d x) \sin ^3(c+d x)}{2 a^2 d}-\frac {3 \int 1 \, dx}{4 a^2}\\ &=-\frac {3 x}{4 a^2}-\frac {2 \cos (c+d x)}{a^2 d}+\frac {\cos ^3(c+d x)}{a^2 d}-\frac {\cos ^5(c+d x)}{5 a^2 d}+\frac {3 \cos (c+d x) \sin (c+d x)}{4 a^2 d}+\frac {\cos (c+d x) \sin ^3(c+d x)}{2 a^2 d}\\ \end {align*}
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Mathematica [B] time = 1.32, size = 308, normalized size = 3.02 \[ -\frac {120 d x \sin \left (\frac {c}{2}\right )-110 \sin \left (\frac {c}{2}+d x\right )+110 \sin \left (\frac {3 c}{2}+d x\right )-40 \sin \left (\frac {3 c}{2}+2 d x\right )-40 \sin \left (\frac {5 c}{2}+2 d x\right )+15 \sin \left (\frac {5 c}{2}+3 d x\right )-15 \sin \left (\frac {7 c}{2}+3 d x\right )+5 \sin \left (\frac {7 c}{2}+4 d x\right )+5 \sin \left (\frac {9 c}{2}+4 d x\right )-\sin \left (\frac {9 c}{2}+5 d x\right )+\sin \left (\frac {11 c}{2}+5 d x\right )+5 \cos \left (\frac {c}{2}\right ) (24 d x+1)+110 \cos \left (\frac {c}{2}+d x\right )+110 \cos \left (\frac {3 c}{2}+d x\right )-40 \cos \left (\frac {3 c}{2}+2 d x\right )+40 \cos \left (\frac {5 c}{2}+2 d x\right )-15 \cos \left (\frac {5 c}{2}+3 d x\right )-15 \cos \left (\frac {7 c}{2}+3 d x\right )+5 \cos \left (\frac {7 c}{2}+4 d x\right )-5 \cos \left (\frac {9 c}{2}+4 d x\right )+\cos \left (\frac {9 c}{2}+5 d x\right )+\cos \left (\frac {11 c}{2}+5 d x\right )-5 \sin \left (\frac {c}{2}\right )}{160 a^2 d \left (\sin \left (\frac {c}{2}\right )+\cos \left (\frac {c}{2}\right )\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 68, normalized size = 0.67 \[ -\frac {4 \, \cos \left (d x + c\right )^{5} - 20 \, \cos \left (d x + c\right )^{3} + 15 \, d x + 5 \, {\left (2 \, \cos \left (d x + c\right )^{3} - 5 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) + 40 \, \cos \left (d x + c\right )}{20 \, a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 127, normalized size = 1.25 \[ -\frac {\frac {15 \, {\left (d x + c\right )}}{a^{2}} + \frac {2 \, {\left (15 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 70 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 40 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 200 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 70 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 120 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 15 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 24\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{5} a^{2}}}{20 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.40, size = 279, normalized size = 2.74 \[ -\frac {3 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a^{2} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}-\frac {7 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{2} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}-\frac {4 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{2} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}-\frac {20 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{2} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}+\frac {7 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{2} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}-\frac {12 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{2} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}+\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a^{2} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}-\frac {12}{5 a^{2} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}-\frac {3 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d \,a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.49, size = 290, normalized size = 2.84 \[ \frac {\frac {\frac {15 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {120 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {70 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {200 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {40 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {70 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {15 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - 24}{a^{2} + \frac {5 \, a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {10 \, a^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {10 \, a^{2} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {5 \, a^{2} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {a^{2} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}}} - \frac {15 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}}}{10 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.72, size = 89, normalized size = 0.87 \[ \frac {3\,\cos \left (3\,c+3\,d\,x\right )}{16\,a^2\,d}-\frac {11\,\cos \left (c+d\,x\right )}{8\,a^2\,d}-\frac {3\,x}{4\,a^2}-\frac {\cos \left (5\,c+5\,d\,x\right )}{80\,a^2\,d}+\frac {\sin \left (2\,c+2\,d\,x\right )}{2\,a^2\,d}-\frac {\sin \left (4\,c+4\,d\,x\right )}{16\,a^2\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 94.09, size = 1608, normalized size = 15.76 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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