Optimal. Leaf size=131 \[ -\frac {7 x}{16 a^3}-\frac {7 \cos ^5(c+d x)}{30 a^3 d}-\frac {7 \cos (c+d x) \sin (c+d x)}{16 a^3 d}-\frac {7 \cos ^3(c+d x) \sin (c+d x)}{24 a^3 d}-\frac {\cos ^9(c+d x)}{3 d (a+a \sin (c+d x))^3}-\frac {\cos ^7(c+d x)}{6 d \left (a^3+a^3 \sin (c+d x)\right )} \]
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Rubi [A]
time = 0.12, antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {2938, 2758,
2761, 2715, 8} \begin {gather*} -\frac {7 \cos ^5(c+d x)}{30 a^3 d}-\frac {\cos ^7(c+d x)}{6 d \left (a^3 \sin (c+d x)+a^3\right )}-\frac {7 \sin (c+d x) \cos ^3(c+d x)}{24 a^3 d}-\frac {7 \sin (c+d x) \cos (c+d x)}{16 a^3 d}-\frac {7 x}{16 a^3}-\frac {\cos ^9(c+d x)}{3 d (a \sin (c+d x)+a)^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2715
Rule 2758
Rule 2761
Rule 2938
Rubi steps
\begin {align*} \int \frac {\cos ^8(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^3} \, dx &=-\frac {\cos ^9(c+d x)}{3 d (a+a \sin (c+d x))^3}-\frac {\int \frac {\cos ^8(c+d x)}{(a+a \sin (c+d x))^2} \, dx}{a}\\ &=-\frac {\cos ^9(c+d x)}{3 d (a+a \sin (c+d x))^3}-\frac {\cos ^7(c+d x)}{6 d \left (a^3+a^3 \sin (c+d x)\right )}-\frac {7 \int \frac {\cos ^6(c+d x)}{a+a \sin (c+d x)} \, dx}{6 a^2}\\ &=-\frac {7 \cos ^5(c+d x)}{30 a^3 d}-\frac {\cos ^9(c+d x)}{3 d (a+a \sin (c+d x))^3}-\frac {\cos ^7(c+d x)}{6 d \left (a^3+a^3 \sin (c+d x)\right )}-\frac {7 \int \cos ^4(c+d x) \, dx}{6 a^3}\\ &=-\frac {7 \cos ^5(c+d x)}{30 a^3 d}-\frac {7 \cos ^3(c+d x) \sin (c+d x)}{24 a^3 d}-\frac {\cos ^9(c+d x)}{3 d (a+a \sin (c+d x))^3}-\frac {\cos ^7(c+d x)}{6 d \left (a^3+a^3 \sin (c+d x)\right )}-\frac {7 \int \cos ^2(c+d x) \, dx}{8 a^3}\\ &=-\frac {7 \cos ^5(c+d x)}{30 a^3 d}-\frac {7 \cos (c+d x) \sin (c+d x)}{16 a^3 d}-\frac {7 \cos ^3(c+d x) \sin (c+d x)}{24 a^3 d}-\frac {\cos ^9(c+d x)}{3 d (a+a \sin (c+d x))^3}-\frac {\cos ^7(c+d x)}{6 d \left (a^3+a^3 \sin (c+d x)\right )}-\frac {7 \int 1 \, dx}{16 a^3}\\ &=-\frac {7 x}{16 a^3}-\frac {7 \cos ^5(c+d x)}{30 a^3 d}-\frac {7 \cos (c+d x) \sin (c+d x)}{16 a^3 d}-\frac {7 \cos ^3(c+d x) \sin (c+d x)}{24 a^3 d}-\frac {\cos ^9(c+d x)}{3 d (a+a \sin (c+d x))^3}-\frac {\cos ^7(c+d x)}{6 d \left (a^3+a^3 \sin (c+d x)\right )}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(366\) vs. \(2(131)=262\).
time = 1.13, size = 366, normalized size = 2.79 \begin {gather*} \frac {-21 (1+40 d x) \cos \left (\frac {c}{2}\right )-600 \cos \left (\frac {c}{2}+d x\right )-600 \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 )-140 \cos \left (\frac {5 c}{2}+3 d x\right )-140 \cos \left (\frac {7 c}{2}+3 d x\right )+105 \cos \left (\frac {7 c}{2}+4 d x\right )-105 \cos \left (\frac {9 c}{2}+4 d x\right )+36 \cos \left (\frac {9 c}{2}+5 d x\right )+36 \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 )+21 \sin \left (\frac {c}{2}\right )-840 d x \sin \left (\frac {c}{2}\right )+600 \sin \left (\frac {c}{2}+d x\right )-600 \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 )+140 \sin \left (\frac {5 c}{2}+3 d x\right )-140 \sin \left (\frac {7 c}{2}+3 d x\right )+105 \sin \left (\frac {7 c}{2}+4 d x\right )+105 \sin \left (\frac {9 c}{2}+4 d x\right )-36 \sin \left (\frac {9 c}{2}+5 d x\right )+36 \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^3 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 = 181, normalized size = 1.38
method | result | size |
risch | \(-\frac {7 x}{16 a^{3}}-\frac {5 \cos \left (d x +c \right )}{8 a^{3} d}-\frac {\sin \left (6 d x +6 c \right )}{192 d \,a^{3}}+\frac {3 \cos \left (5 d x +5 c \right )}{80 d \,a^{3}}+\frac {7 \sin \left (4 d x +4 c \right )}{64 d \,a^{3}}-\frac {7 \cos \left (3 d x +3 c \right )}{48 d \,a^{3}}+\frac {\sin \left (2 d x +2 c \right )}{64 d \,a^{3}}\) | \(107\) |
derivativedivides | \(\frac {\frac {4 \left (-\frac {7 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32}-\frac {\left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {73 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{96}-\frac {9 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {37 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}-\frac {11 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-\frac {37 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}-\left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {73 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{96}-\frac {17 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{10}+\frac {7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{32}-\frac {11}{30}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}-\frac {7 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}}{d \,a^{3}}\) | \(181\) |
default | \(\frac {\frac {4 \left (-\frac {7 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32}-\frac {\left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {73 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{96}-\frac {9 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {37 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}-\frac {11 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-\frac {37 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}-\left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {73 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{96}-\frac {17 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{10}+\frac {7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{32}-\frac {11}{30}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}-\frac {7 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}}{d \,a^{3}}\) | \(181\) |
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. 393 vs.
\(2 (119) = 238\).
time = 0.52, size = 393, normalized size = 3.00 \begin {gather*} \frac {\frac {\frac {105 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {816 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {365 \, \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 {1110 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {1760 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {1110 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {2160 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {365 \, \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 {105 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} - 176}{a^{3} + \frac {6 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {15 \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {20 \, a^{3} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {15 \, a^{3} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {6 \, a^{3} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac {a^{3} \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}}} - \frac {105 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}}{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 = 0.53 \begin {gather*} \frac {144 \, \cos \left (d x + c\right )^{5} - 320 \, \cos \left (d x + c\right )^{3} - 105 \, d x - 5 \, {\left (8 \, \cos \left (d x + c\right )^{5} - 50 \, \cos \left (d x + c\right )^{3} + 21 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, a^{3} 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. 2535 vs.
\(2 (121) = 242\).
time = 216.16, size = 2535, normalized size = 19.35 \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 = 0.48, size = 179, normalized size = 1.37 \begin {gather*} -\frac {\frac {105 \, {\left (d x + c\right )}}{a^{3}} + \frac {2 \, {\left (105 \, \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} - 365 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 2160 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 1110 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 1760 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 1110 \, \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} + 365 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 816 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 105 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 176\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{6} a^{3}}}{240 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 12.49, size = 173, normalized size = 1.32 \begin {gather*} -\frac {7\,x}{16\,a^3}-\frac {\frac {7\,{\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 {73\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{24}+18\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-\frac {37\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{4}+\frac {44\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{3}+\frac {37\,{\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 {73\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24}+\frac {34\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{5}-\frac {7\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8}+\frac {22}{15}}{a^3\,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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