Optimal. Leaf size=73 \[ -\frac {\cos ^3(c+d x)}{3 a d}+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 a d}-\frac {\sin (c+d x) \cos (c+d x)}{8 a d}-\frac {x}{8 a} \]
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Rubi [A] time = 0.11, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2839, 2565, 30, 2568, 2635, 8} \[ -\frac {\cos ^3(c+d x)}{3 a d}+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 a d}-\frac {\sin (c+d x) \cos (c+d x)}{8 a d}-\frac {x}{8 a} \]
Antiderivative was successfully verified.
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Rule 8
Rule 30
Rule 2565
Rule 2568
Rule 2635
Rule 2839
Rubi steps
\begin {align*} \int \frac {\cos ^4(c+d x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx &=\frac {\int \cos ^2(c+d x) \sin (c+d x) \, dx}{a}-\frac {\int \cos ^2(c+d x) \sin ^2(c+d x) \, dx}{a}\\ &=\frac {\cos ^3(c+d x) \sin (c+d x)}{4 a d}-\frac {\int \cos ^2(c+d x) \, dx}{4 a}-\frac {\operatorname {Subst}\left (\int x^2 \, dx,x,\cos (c+d x)\right )}{a d}\\ &=-\frac {\cos ^3(c+d x)}{3 a d}-\frac {\cos (c+d x) \sin (c+d x)}{8 a d}+\frac {\cos ^3(c+d x) \sin (c+d x)}{4 a d}-\frac {\int 1 \, dx}{8 a}\\ &=-\frac {x}{8 a}-\frac {\cos ^3(c+d x)}{3 a d}-\frac {\cos (c+d x) \sin (c+d x)}{8 a d}+\frac {\cos ^3(c+d x) \sin (c+d x)}{4 a d}\\ \end {align*}
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Mathematica [B] time = 1.68, size = 219, normalized size = 3.00 \[ -\frac {24 d x \sin \left (\frac {c}{2}\right )-24 \sin \left (\frac {c}{2}+d x\right )+24 \sin \left (\frac {3 c}{2}+d x\right )-8 \sin \left (\frac {5 c}{2}+3 d x\right )+8 \sin \left (\frac {7 c}{2}+3 d x\right )-3 \sin \left (\frac {7 c}{2}+4 d x\right )-3 \sin \left (\frac {9 c}{2}+4 d x\right )-24 \cos \left (\frac {c}{2}\right ) (c-d x)+24 \cos \left (\frac {c}{2}+d x\right )+24 \cos \left (\frac {3 c}{2}+d x\right )+8 \cos \left (\frac {5 c}{2}+3 d x\right )+8 \cos \left (\frac {7 c}{2}+3 d x\right )-3 \cos \left (\frac {7 c}{2}+4 d x\right )+3 \cos \left (\frac {9 c}{2}+4 d x\right )-24 c \sin \left (\frac {c}{2}\right )+48 \sin \left (\frac {c}{2}\right )}{192 a d \left (\sin \left (\frac {c}{2}\right )+\cos \left (\frac {c}{2}\right )\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 50, normalized size = 0.68 \[ -\frac {8 \, \cos \left (d x + c\right )^{3} + 3 \, d x - 3 \, {\left (2 \, \cos \left (d x + c\right )^{3} - \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{24 \, a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 127, normalized size = 1.74 \[ -\frac {\frac {3 \, {\left (d x + c\right )}}{a} + \frac {2 \, {\left (3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 24 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 21 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 24 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 21 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 8 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 8\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{4} a}}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.17, size = 279, normalized size = 3.82 \[ -\frac {\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}-\frac {2 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {7 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}-\frac {2 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}-\frac {7 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}-\frac {2 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}-\frac {2}{3 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}-\frac {\arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.42, size = 257, normalized size = 3.52 \[ \frac {\frac {\frac {3 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {8 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {21 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {24 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {21 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {24 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {3 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - 8}{a + \frac {4 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {6 \, a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {4 \, a \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {a \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}} - \frac {3 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a}}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.80, size = 43, normalized size = 0.59 \[ -\frac {6\,\cos \left (c+d\,x\right )+2\,\cos \left (3\,c+3\,d\,x\right )-\frac {3\,\sin \left (4\,c+4\,d\,x\right )}{4}+3\,d\,x}{24\,a\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 24.78, size = 1134, normalized size = 15.53 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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