Optimal. Leaf size=97 \[ -\frac {\cos ^5(c+d x)}{5 a d}+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 a d}-\frac {\sin (c+d x) \cos ^3(c+d x)}{24 a d}-\frac {\sin (c+d x) \cos (c+d x)}{16 a d}-\frac {x}{16 a} \]
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Rubi [A] time = 0.13, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 7, 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} \[ -\frac {\cos ^5(c+d x)}{5 a d}+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 a d}-\frac {\sin (c+d x) \cos ^3(c+d x)}{24 a d}-\frac {\sin (c+d x) \cos (c+d x)}{16 a d}-\frac {x}{16 a} \]
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 ^6(c+d x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx &=\frac {\int \cos ^4(c+d x) \sin (c+d x) \, dx}{a}-\frac {\int \cos ^4(c+d x) \sin ^2(c+d x) \, dx}{a}\\ &=\frac {\cos ^5(c+d x) \sin (c+d x)}{6 a d}-\frac {\int \cos ^4(c+d x) \, dx}{6 a}-\frac {\operatorname {Subst}\left (\int x^4 \, dx,x,\cos (c+d x)\right )}{a d}\\ &=-\frac {\cos ^5(c+d x)}{5 a d}-\frac {\cos ^3(c+d x) \sin (c+d x)}{24 a d}+\frac {\cos ^5(c+d x) \sin (c+d x)}{6 a d}-\frac {\int \cos ^2(c+d x) \, dx}{8 a}\\ &=-\frac {\cos ^5(c+d x)}{5 a d}-\frac {\cos (c+d x) \sin (c+d x)}{16 a d}-\frac {\cos ^3(c+d x) \sin (c+d x)}{24 a d}+\frac {\cos ^5(c+d x) \sin (c+d x)}{6 a d}-\frac {\int 1 \, dx}{16 a}\\ &=-\frac {x}{16 a}-\frac {\cos ^5(c+d x)}{5 a d}-\frac {\cos (c+d x) \sin (c+d x)}{16 a d}-\frac {\cos ^3(c+d x) \sin (c+d x)}{24 a d}+\frac {\cos ^5(c+d x) \sin (c+d x)}{6 a d}\\ \end {align*}
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Mathematica [B] time = 5.08, size = 377, normalized size = 3.89 \[ -\frac {120 d x \sin \left (\frac {c}{2}\right )-120 \sin \left (\frac {c}{2}+d x\right )+120 \sin \left (\frac {3 c}{2}+d x\right )+15 \sin \left (\frac {3 c}{2}+2 d x\right )+15 \sin \left (\frac {5 c}{2}+2 d x\right )-60 \sin \left (\frac {5 c}{2}+3 d x\right )+60 \sin \left (\frac {7 c}{2}+3 d x\right )-15 \sin \left (\frac {7 c}{2}+4 d x\right )-15 \sin \left (\frac {9 c}{2}+4 d x\right )-12 \sin \left (\frac {9 c}{2}+5 d x\right )+12 \sin \left (\frac {11 c}{2}+5 d x\right )-5 \sin \left (\frac {11 c}{2}+6 d x\right )-5 \sin \left (\frac {13 c}{2}+6 d x\right )-30 \cos \left (\frac {c}{2}\right ) (5 c-4 d x)+120 \cos \left (\frac {c}{2}+d x\right )+120 \cos \left (\frac {3 c}{2}+d x\right )+15 \cos \left (\frac {3 c}{2}+2 d x\right )-15 \cos \left (\frac {5 c}{2}+2 d x\right )+60 \cos \left (\frac {5 c}{2}+3 d x\right )+60 \cos \left (\frac {7 c}{2}+3 d x\right )-15 \cos \left (\frac {7 c}{2}+4 d x\right )+15 \cos \left (\frac {9 c}{2}+4 d x\right )+12 \cos \left (\frac {9 c}{2}+5 d x\right )+12 \cos \left (\frac {11 c}{2}+5 d x\right )-5 \cos \left (\frac {11 c}{2}+6 d x\right )+5 \cos \left (\frac {13 c}{2}+6 d x\right )-150 c \sin \left (\frac {c}{2}\right )+300 \sin \left (\frac {c}{2}\right )}{1920 a d \left (\sin \left (\frac {c}{2}\right )+\cos \left (\frac {c}{2}\right )\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.85, size = 60, normalized size = 0.62 \[ -\frac {48 \, \cos \left (d x + c\right )^{5} + 15 \, d x - 5 \, {\left (8 \, \cos \left (d x + c\right )^{5} - 2 \, \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.15, size = 179, normalized size = 1.85 \[ -\frac {\frac {15 \, {\left (d x + c\right )}}{a} + \frac {2 \, {\left (15 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 240 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} - 235 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 240 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 390 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 480 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 390 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 480 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 235 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 48 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 15 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 48\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{6} a}}{240 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.21, size = 415, normalized size = 4.28 \[ -\frac {\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}-\frac {2 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}+\frac {47 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}-\frac {2 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}-\frac {13 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}-\frac {4 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}+\frac {13 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}-\frac {4 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}-\frac {47 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}-\frac {2 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}-\frac {2}{5 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}-\frac {\arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.52, size = 379, normalized size = 3.91 \[ \frac {\frac {\frac {15 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {48 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {235 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {480 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {390 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {480 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {390 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {240 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {235 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {240 \, \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} - \frac {15 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} - 48}{a + \frac {6 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {15 \, a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {20 \, a \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {15 \, a \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {6 \, a \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac {a \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}}} - \frac {15 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a}}{120 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 12.49, size = 173, normalized size = 1.78 \[ -\frac {x}{16\,a}-\frac {\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{8}+2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-\frac {47\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{24}+2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+\frac {13\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{4}+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-\frac {13\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{4}+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\frac {47\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24}+\frac {2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{5}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8}+\frac {2}{5}}{a\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^6} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 49.69, size = 2307, normalized size = 23.78 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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