Optimal. Leaf size=104 \[ \frac {3 x}{16 a^2}+\frac {\cos ^5(c+d x)}{10 a^2 d}+\frac {3 \cos (c+d x) \sin (c+d x)}{16 a^2 d}+\frac {\cos ^3(c+d x) \sin (c+d x)}{8 a^2 d}+\frac {\cos ^3(c+d x) (a-a \sin (c+d x))^3}{6 a^5 d} \]
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Rubi [A]
time = 0.18, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {2954, 2949,
2748, 2715, 8} \begin {gather*} \frac {\cos ^3(c+d x) (a-a \sin (c+d x))^3}{6 a^5 d}+\frac {\cos ^5(c+d x)}{10 a^2 d}+\frac {\sin (c+d x) \cos ^3(c+d x)}{8 a^2 d}+\frac {3 \sin (c+d x) \cos (c+d x)}{16 a^2 d}+\frac {3 x}{16 a^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2715
Rule 2748
Rule 2949
Rule 2954
Rubi steps
\begin {align*} \int \frac {\cos ^6(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx &=\frac {\int \cos ^2(c+d x) \sin ^2(c+d x) (a-a \sin (c+d x))^2 \, dx}{a^4}\\ &=\frac {\cos ^3(c+d x) (a-a \sin (c+d x))^3}{6 a^5 d}+\frac {\int \cos ^4(c+d x) (a-a \sin (c+d x)) \, dx}{2 a^3}\\ &=\frac {\cos ^5(c+d x)}{10 a^2 d}+\frac {\cos ^3(c+d x) (a-a \sin (c+d x))^3}{6 a^5 d}+\frac {\int \cos ^4(c+d x) \, dx}{2 a^2}\\ &=\frac {\cos ^5(c+d x)}{10 a^2 d}+\frac {\cos ^3(c+d x) \sin (c+d x)}{8 a^2 d}+\frac {\cos ^3(c+d x) (a-a \sin (c+d x))^3}{6 a^5 d}+\frac {3 \int \cos ^2(c+d x) \, dx}{8 a^2}\\ &=\frac {\cos ^5(c+d x)}{10 a^2 d}+\frac {3 \cos (c+d x) \sin (c+d x)}{16 a^2 d}+\frac {\cos ^3(c+d x) \sin (c+d x)}{8 a^2 d}+\frac {\cos ^3(c+d x) (a-a \sin (c+d x))^3}{6 a^5 d}+\frac {3 \int 1 \, dx}{16 a^2}\\ &=\frac {3 x}{16 a^2}+\frac {\cos ^5(c+d x)}{10 a^2 d}+\frac {3 \cos (c+d x) \sin (c+d x)}{16 a^2 d}+\frac {\cos ^3(c+d x) \sin (c+d x)}{8 a^2 d}+\frac {\cos ^3(c+d x) (a-a \sin (c+d x))^3}{6 a^5 d}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(362\) vs. \(2(104)=208\).
time = 1.39, size = 362, normalized size = 3.48 \begin {gather*} \frac {360 d x \cos \left (\frac {c}{2}\right )+240 \cos \left (\frac {c}{2}+d x\right )+240 \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 )+40 \cos \left (\frac {5 c}{2}+3 d x\right )+40 \cos \left (\frac {7 c}{2}+3 d x\right )-45 \cos \left (\frac {7 c}{2}+4 d x\right )+45 \cos \left (\frac {9 c}{2}+4 d x\right )-24 \cos \left (\frac {9 c}{2}+5 d x\right )-24 \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 )+50 \sin \left (\frac {c}{2}\right )+360 d x \sin \left (\frac {c}{2}\right )-240 \sin \left (\frac {c}{2}+d x\right )+240 \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 )-40 \sin \left (\frac {5 c}{2}+3 d x\right )+40 \sin \left (\frac {7 c}{2}+3 d x\right )-45 \sin \left (\frac {7 c}{2}+4 d x\right )-45 \sin \left (\frac {9 c}{2}+4 d x\right )+24 \sin \left (\frac {9 c}{2}+5 d x\right )-24 \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 )}{1920 a^2 d \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.15, size = 153, normalized size = 1.47
method | result | size |
risch | \(\frac {3 x}{16 a^{2}}+\frac {\cos \left (d x +c \right )}{4 a^{2} d}+\frac {\sin \left (6 d x +6 c \right )}{192 a^{2} d}-\frac {\cos \left (5 d x +5 c \right )}{40 d \,a^{2}}-\frac {3 \sin \left (4 d x +4 c \right )}{64 a^{2} d}+\frac {\cos \left (3 d x +3 c \right )}{24 d \,a^{2}}-\frac {\sin \left (2 d x +2 c \right )}{64 a^{2} d}\) | \(107\) |
derivativedivides | \(\frac {\frac {8 \left (\frac {3 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64}-\frac {13 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192}+\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {25 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32}+\frac {2 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+\frac {25 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32}+\frac {13 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192}+\frac {2 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{64}+\frac {1}{15}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}+\frac {3 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}}{a^{2} d}\) | \(153\) |
default | \(\frac {\frac {8 \left (\frac {3 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64}-\frac {13 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192}+\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {25 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32}+\frac {2 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+\frac {25 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32}+\frac {13 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192}+\frac {2 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{64}+\frac {1}{15}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}+\frac {3 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}}{a^{2} d}\) | \(153\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 353 vs.
\(2 (95) = 190\).
time = 0.52, size = 353, normalized size = 3.39 \begin {gather*} -\frac {\frac {\frac {45 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {384 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {65 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {750 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {640 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {750 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {960 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {65 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {45 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} - 64}{a^{2} + \frac {6 \, a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {15 \, a^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {20 \, a^{2} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {15 \, a^{2} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {6 \, a^{2} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac {a^{2} \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}}} - \frac {45 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}}}{120 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 70, normalized size = 0.67 \begin {gather*} -\frac {96 \, \cos \left (d x + c\right )^{5} - 160 \, \cos \left (d x + c\right )^{3} - 45 \, d x - 5 \, {\left (8 \, \cos \left (d x + c\right )^{5} - 26 \, \cos \left (d x + c\right )^{3} + 9 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, a^{2} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 2271 vs.
\(2 (94) = 188\).
time = 77.86, size = 2271, normalized size = 21.84 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.44, size = 153, normalized size = 1.47 \begin {gather*} \frac {\frac {45 \, {\left (d x + c\right )}}{a^{2}} + \frac {2 \, {\left (45 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 65 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 960 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 750 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 640 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 750 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 65 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 384 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 45 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 64\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{6} a^{2}}}{240 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 11.70, size = 146, normalized size = 1.40 \begin {gather*} \frac {3\,x}{16\,a^2}+\frac {\frac {3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{8}-\frac {13\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{24}+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-\frac {25\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{4}+\frac {16\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{3}+\frac {25\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{4}+\frac {13\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24}+\frac {16\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{5}-\frac {3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8}+\frac {8}{15}}{a^2\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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