Optimal. Leaf size=185 \[ \frac {125 a^3 x}{128}-\frac {a^3 \tanh ^{-1}(\cos (c+d x))}{d}+\frac {a^3 \cos (c+d x)}{d}+\frac {a^3 \cos ^3(c+d x)}{3 d}+\frac {a^3 \cos ^5(c+d x)}{5 d}-\frac {3 a^3 \cos ^7(c+d x)}{7 d}+\frac {125 a^3 \cos (c+d x) \sin (c+d x)}{128 d}+\frac {125 a^3 \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac {25 a^3 \cos ^5(c+d x) \sin (c+d x)}{48 d}-\frac {a^3 \cos ^7(c+d x) \sin (c+d x)}{8 d} \]
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Rubi [A]
time = 0.17, antiderivative size = 185, normalized size of antiderivative = 1.00, number of steps
used = 17, number of rules used = 9, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2952, 2715, 8,
2672, 308, 212, 2645, 30, 2648} \begin {gather*} -\frac {3 a^3 \cos ^7(c+d x)}{7 d}+\frac {a^3 \cos ^5(c+d x)}{5 d}+\frac {a^3 \cos ^3(c+d x)}{3 d}+\frac {a^3 \cos (c+d x)}{d}-\frac {a^3 \sin (c+d x) \cos ^7(c+d x)}{8 d}+\frac {25 a^3 \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac {125 a^3 \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac {125 a^3 \sin (c+d x) \cos (c+d x)}{128 d}-\frac {a^3 \tanh ^{-1}(\cos (c+d x))}{d}+\frac {125 a^3 x}{128} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 30
Rule 212
Rule 308
Rule 2645
Rule 2648
Rule 2672
Rule 2715
Rule 2952
Rubi steps
\begin {align*} \int \cos ^5(c+d x) \cot (c+d x) (a+a \sin (c+d x))^3 \, dx &=\int \left (3 a^3 \cos ^6(c+d x)+a^3 \cos ^5(c+d x) \cot (c+d x)+3 a^3 \cos ^6(c+d x) \sin (c+d x)+a^3 \cos ^6(c+d x) \sin ^2(c+d x)\right ) \, dx\\ &=a^3 \int \cos ^5(c+d x) \cot (c+d x) \, dx+a^3 \int \cos ^6(c+d x) \sin ^2(c+d x) \, dx+\left (3 a^3\right ) \int \cos ^6(c+d x) \, dx+\left (3 a^3\right ) \int \cos ^6(c+d x) \sin (c+d x) \, dx\\ &=\frac {a^3 \cos ^5(c+d x) \sin (c+d x)}{2 d}-\frac {a^3 \cos ^7(c+d x) \sin (c+d x)}{8 d}+\frac {1}{8} a^3 \int \cos ^6(c+d x) \, dx+\frac {1}{2} \left (5 a^3\right ) \int \cos ^4(c+d x) \, dx-\frac {a^3 \text {Subst}\left (\int \frac {x^6}{1-x^2} \, dx,x,\cos (c+d x)\right )}{d}-\frac {\left (3 a^3\right ) \text {Subst}\left (\int x^6 \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac {3 a^3 \cos ^7(c+d x)}{7 d}+\frac {5 a^3 \cos ^3(c+d x) \sin (c+d x)}{8 d}+\frac {25 a^3 \cos ^5(c+d x) \sin (c+d x)}{48 d}-\frac {a^3 \cos ^7(c+d x) \sin (c+d x)}{8 d}+\frac {1}{48} \left (5 a^3\right ) \int \cos ^4(c+d x) \, dx+\frac {1}{8} \left (15 a^3\right ) \int \cos ^2(c+d x) \, dx-\frac {a^3 \text {Subst}\left (\int \left (-1-x^2-x^4+\frac {1}{1-x^2}\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=\frac {a^3 \cos (c+d x)}{d}+\frac {a^3 \cos ^3(c+d x)}{3 d}+\frac {a^3 \cos ^5(c+d x)}{5 d}-\frac {3 a^3 \cos ^7(c+d x)}{7 d}+\frac {15 a^3 \cos (c+d x) \sin (c+d x)}{16 d}+\frac {125 a^3 \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac {25 a^3 \cos ^5(c+d x) \sin (c+d x)}{48 d}-\frac {a^3 \cos ^7(c+d x) \sin (c+d x)}{8 d}+\frac {1}{64} \left (5 a^3\right ) \int \cos ^2(c+d x) \, dx+\frac {1}{16} \left (15 a^3\right ) \int 1 \, dx-\frac {a^3 \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\cos (c+d x)\right )}{d}\\ &=\frac {15 a^3 x}{16}-\frac {a^3 \tanh ^{-1}(\cos (c+d x))}{d}+\frac {a^3 \cos (c+d x)}{d}+\frac {a^3 \cos ^3(c+d x)}{3 d}+\frac {a^3 \cos ^5(c+d x)}{5 d}-\frac {3 a^3 \cos ^7(c+d x)}{7 d}+\frac {125 a^3 \cos (c+d x) \sin (c+d x)}{128 d}+\frac {125 a^3 \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac {25 a^3 \cos ^5(c+d x) \sin (c+d x)}{48 d}-\frac {a^3 \cos ^7(c+d x) \sin (c+d x)}{8 d}+\frac {1}{128} \left (5 a^3\right ) \int 1 \, dx\\ &=\frac {125 a^3 x}{128}-\frac {a^3 \tanh ^{-1}(\cos (c+d x))}{d}+\frac {a^3 \cos (c+d x)}{d}+\frac {a^3 \cos ^3(c+d x)}{3 d}+\frac {a^3 \cos ^5(c+d x)}{5 d}-\frac {3 a^3 \cos ^7(c+d x)}{7 d}+\frac {125 a^3 \cos (c+d x) \sin (c+d x)}{128 d}+\frac {125 a^3 \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac {25 a^3 \cos ^5(c+d x) \sin (c+d x)}{48 d}-\frac {a^3 \cos ^7(c+d x) \sin (c+d x)}{8 d}\\ \end {align*}
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Mathematica [A]
time = 0.45, size = 122, normalized size = 0.66 \begin {gather*} \frac {a^3 \left (105000 c+105000 d x+122640 \cos (c+d x)+560 \cos (3 (c+d x))-3696 \cos (5 (c+d x))-720 \cos (7 (c+d x))-107520 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+107520 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+77280 \sin (2 (c+d x))+14280 \sin (4 (c+d x))+1120 \sin (6 (c+d x))-105 \sin (8 (c+d x))\right )}{107520 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.24, size = 177, normalized size = 0.96
method | result | size |
derivativedivides | \(\frac {a^{3} \left (\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{3}+\cos \left (d x +c \right )+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )+3 a^{3} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )-\frac {3 a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{7}+a^{3} \left (-\frac {\left (\cos ^{7}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{8}+\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{48}+\frac {5 d x}{128}+\frac {5 c}{128}\right )}{d}\) | \(177\) |
default | \(\frac {a^{3} \left (\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{3}+\cos \left (d x +c \right )+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )+3 a^{3} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )-\frac {3 a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{7}+a^{3} \left (-\frac {\left (\cos ^{7}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{8}+\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{48}+\frac {5 d x}{128}+\frac {5 c}{128}\right )}{d}\) | \(177\) |
risch | \(\frac {125 a^{3} x}{128}+\frac {73 a^{3} {\mathrm e}^{i \left (d x +c \right )}}{128 d}+\frac {73 a^{3} {\mathrm e}^{-i \left (d x +c \right )}}{128 d}-\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{d}+\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d}-\frac {a^{3} \sin \left (8 d x +8 c \right )}{1024 d}-\frac {3 a^{3} \cos \left (7 d x +7 c \right )}{448 d}+\frac {a^{3} \sin \left (6 d x +6 c \right )}{96 d}-\frac {11 a^{3} \cos \left (5 d x +5 c \right )}{320 d}+\frac {17 a^{3} \sin \left (4 d x +4 c \right )}{128 d}+\frac {a^{3} \cos \left (3 d x +3 c \right )}{192 d}+\frac {23 a^{3} \sin \left (2 d x +2 c \right )}{32 d}\) | \(200\) |
norman | \(\frac {\frac {125 a^{3} x}{128}+\frac {232 a^{3}}{105 d}-\frac {259 a^{3} \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}-\frac {1861 a^{3} \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d}-\frac {1817 a^{3} \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d}+\frac {24 a^{3} \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {1817 a^{3} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d}+\frac {3805 a^{3} \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d}-\frac {3805 a^{3} \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d}+\frac {568 a^{3} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 d}+\frac {128 a^{3} \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {1216 a^{3} \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 d}+\frac {232 a^{3} \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {259 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{64 d}+\frac {1861 a^{3} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d}+\frac {1856 a^{3} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{105 d}+\frac {125 a^{3} x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {875 a^{3} x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32}+\frac {875 a^{3} x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {4375 a^{3} x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64}+\frac {875 a^{3} x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {875 a^{3} x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32}+\frac {125 a^{3} x \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {125 a^{3} x \left (\tan ^{16}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}+\frac {a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}\) | \(450\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 171, normalized size = 0.92 \begin {gather*} -\frac {46080 \, a^{3} \cos \left (d x + c\right )^{7} - 3584 \, {\left (6 \, \cos \left (d x + c\right )^{5} + 10 \, \cos \left (d x + c\right )^{3} + 30 \, \cos \left (d x + c\right ) - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} a^{3} - 35 \, {\left (64 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 120 \, d x + 120 \, c - 3 \, \sin \left (8 \, d x + 8 \, c\right ) - 24 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{3} + 1680 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{3}}{107520 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.42, size = 154, normalized size = 0.83 \begin {gather*} -\frac {5760 \, a^{3} \cos \left (d x + c\right )^{7} - 2688 \, a^{3} \cos \left (d x + c\right )^{5} - 4480 \, a^{3} \cos \left (d x + c\right )^{3} - 13125 \, a^{3} d x - 13440 \, a^{3} \cos \left (d x + c\right ) + 6720 \, a^{3} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 6720 \, a^{3} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 35 \, {\left (48 \, a^{3} \cos \left (d x + c\right )^{7} - 200 \, a^{3} \cos \left (d x + c\right )^{5} - 250 \, a^{3} \cos \left (d x + c\right )^{3} - 375 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{13440 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.54, size = 277, normalized size = 1.50 \begin {gather*} \frac {13125 \, {\left (d x + c\right )} a^{3} + 13440 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - \frac {2 \, {\left (27195 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{15} + 65135 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} - 161280 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{12} + 63595 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 286720 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 133175 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 519680 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 133175 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 544768 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 63595 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 254464 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 65135 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 118784 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 27195 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 14848 \, a^{3}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{8}}}{13440 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 10.85, size = 429, normalized size = 2.32 \begin {gather*} \frac {a^3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}+\frac {125\,a^3\,\mathrm {atan}\left (\frac {15625\,a^6}{4096\,\left (\frac {125\,a^6}{32}-\frac {15625\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4096}\right )}+\frac {125\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{32\,\left (\frac {125\,a^6}{32}-\frac {15625\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4096}\right )}\right )}{64\,d}+\frac {-\frac {259\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{64}-\frac {1861\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{192}+24\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-\frac {1817\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{192}+\frac {128\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{3}-\frac {3805\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{192}+\frac {232\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{3}+\frac {3805\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{192}+\frac {1216\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{15}+\frac {1817\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{192}+\frac {568\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{15}+\frac {1861\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{192}+\frac {1856\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{105}+\frac {259\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64}+\frac {232\,a^3}{105}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+28\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+56\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+70\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+56\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+28\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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