Optimal. Leaf size=87 \[ \frac {\cos ^3(c+d x)}{3 a d}-\frac {\cos (c+d x)}{a d}+\frac {\sin ^3(c+d x) \cos (c+d x)}{4 a d}+\frac {3 \sin (c+d x) \cos (c+d x)}{8 a d}-\frac {3 x}{8 a} \]
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Rubi [A] time = 0.13, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {2839, 2633, 2635, 8} \[ \frac {\cos ^3(c+d x)}{3 a d}-\frac {\cos (c+d x)}{a d}+\frac {\sin ^3(c+d x) \cos (c+d x)}{4 a d}+\frac {3 \sin (c+d x) \cos (c+d x)}{8 a d}-\frac {3 x}{8 a} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2633
Rule 2635
Rule 2839
Rubi steps
\begin {align*} \int \frac {\cos ^2(c+d x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac {\int \sin ^3(c+d x) \, dx}{a}-\frac {\int \sin ^4(c+d x) \, dx}{a}\\ &=\frac {\cos (c+d x) \sin ^3(c+d x)}{4 a d}-\frac {3 \int \sin ^2(c+d x) \, dx}{4 a}-\frac {\operatorname {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{a d}\\ &=-\frac {\cos (c+d x)}{a d}+\frac {\cos ^3(c+d x)}{3 a d}+\frac {3 \cos (c+d x) \sin (c+d x)}{8 a d}+\frac {\cos (c+d x) \sin ^3(c+d x)}{4 a d}-\frac {3 \int 1 \, dx}{8 a}\\ &=-\frac {3 x}{8 a}-\frac {\cos (c+d x)}{a d}+\frac {\cos ^3(c+d x)}{3 a d}+\frac {3 \cos (c+d x) \sin (c+d x)}{8 a d}+\frac {\cos (c+d x) \sin ^3(c+d x)}{4 a d}\\ \end {align*}
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Mathematica [B] time = 1.72, size = 271, normalized size = 3.11 \[ \frac {-72 d x \sin \left (\frac {c}{2}\right )+72 \sin \left (\frac {c}{2}+d x\right )-72 \sin \left (\frac {3 c}{2}+d x\right )+24 \sin \left (\frac {3 c}{2}+2 d x\right )+24 \sin \left (\frac {5 c}{2}+2 d x\right )-8 \sin \left (\frac {5 c}{2}+3 d x\right )+8 \sin \left (\frac {7 c}{2}+3 d x\right )-3 \sin \left (\frac {7 c}{2}+4 d x\right )-3 \sin \left (\frac {9 c}{2}+4 d x\right )+24 \cos \left (\frac {c}{2}\right ) (c-3 d x)-72 \cos \left (\frac {c}{2}+d x\right )-72 \cos \left (\frac {3 c}{2}+d x\right )+24 \cos \left (\frac {3 c}{2}+2 d x\right )-24 \cos \left (\frac {5 c}{2}+2 d x\right )+8 \cos \left (\frac {5 c}{2}+3 d x\right )+8 \cos \left (\frac {7 c}{2}+3 d x\right )-3 \cos \left (\frac {7 c}{2}+4 d x\right )+3 \cos \left (\frac {9 c}{2}+4 d x\right )+24 c \sin \left (\frac {c}{2}\right )-48 \sin \left (\frac {c}{2}\right )}{192 a d \left (\sin \left (\frac {c}{2}\right )+\cos \left (\frac {c}{2}\right )\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 58, normalized size = 0.67 \[ \frac {8 \, \cos \left (d x + c\right )^{3} - 9 \, d x - 3 \, {\left (2 \, \cos \left (d x + c\right )^{3} - 5 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) - 24 \, \cos \left (d x + c\right )}{24 \, a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 114, normalized size = 1.31 \[ -\frac {\frac {9 \, {\left (d x + c\right )}}{a} + \frac {2 \, {\left (9 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 33 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 48 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 33 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 64 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 9 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 16\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{4} a}}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.29, size = 245, normalized size = 2.82 \[ -\frac {3 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}-\frac {11 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}-\frac {4 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {11 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}-\frac {16 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}-\frac {4}{3 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}-\frac {3 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.43, size = 237, normalized size = 2.72 \[ \frac {\frac {\frac {9 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {64 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {33 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {48 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {33 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {9 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - 16}{a + \frac {4 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {6 \, a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {4 \, a \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {a \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}} - \frac {9 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a}}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.61, size = 79, normalized size = 0.91 \[ \frac {{\cos \left (c+d\,x\right )}^3}{3\,a\,d}-\frac {\cos \left (c+d\,x\right )}{a\,d}-\frac {3\,x}{8\,a}-\frac {{\cos \left (c+d\,x\right )}^3\,\sin \left (c+d\,x\right )}{4\,a\,d}+\frac {5\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )}{8\,a\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 19.76, size = 1049, normalized size = 12.06 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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