Optimal. Leaf size=129 \[ \frac {23 a^3 x}{16}+\frac {4 a^3 \sin (c+d x)}{d}+\frac {23 a^3 \cos (c+d x) \sin (c+d x)}{16 d}+\frac {23 a^3 \cos ^3(c+d x) \sin (c+d x)}{24 d}+\frac {a^3 \cos ^5(c+d x) \sin (c+d x)}{6 d}-\frac {7 a^3 \sin ^3(c+d x)}{3 d}+\frac {3 a^3 \sin ^5(c+d x)}{5 d} \]
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Rubi [A]
time = 0.10, antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3876, 2713,
2715, 8} \begin {gather*} \frac {3 a^3 \sin ^5(c+d x)}{5 d}-\frac {7 a^3 \sin ^3(c+d x)}{3 d}+\frac {4 a^3 \sin (c+d x)}{d}+\frac {a^3 \sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac {23 a^3 \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac {23 a^3 \sin (c+d x) \cos (c+d x)}{16 d}+\frac {23 a^3 x}{16} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2713
Rule 2715
Rule 3876
Rubi steps
\begin {align*} \int \cos ^6(c+d x) (a+a \sec (c+d x))^3 \, dx &=\int \left (a^3 \cos ^3(c+d x)+3 a^3 \cos ^4(c+d x)+3 a^3 \cos ^5(c+d x)+a^3 \cos ^6(c+d x)\right ) \, dx\\ &=a^3 \int \cos ^3(c+d x) \, dx+a^3 \int \cos ^6(c+d x) \, dx+\left (3 a^3\right ) \int \cos ^4(c+d x) \, dx+\left (3 a^3\right ) \int \cos ^5(c+d x) \, dx\\ &=\frac {3 a^3 \cos ^3(c+d x) \sin (c+d x)}{4 d}+\frac {a^3 \cos ^5(c+d x) \sin (c+d x)}{6 d}+\frac {1}{6} \left (5 a^3\right ) \int \cos ^4(c+d x) \, dx+\frac {1}{4} \left (9 a^3\right ) \int \cos ^2(c+d x) \, dx-\frac {a^3 \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{d}-\frac {\left (3 a^3\right ) \text {Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,-\sin (c+d x)\right )}{d}\\ &=\frac {4 a^3 \sin (c+d x)}{d}+\frac {9 a^3 \cos (c+d x) \sin (c+d x)}{8 d}+\frac {23 a^3 \cos ^3(c+d x) \sin (c+d x)}{24 d}+\frac {a^3 \cos ^5(c+d x) \sin (c+d x)}{6 d}-\frac {7 a^3 \sin ^3(c+d x)}{3 d}+\frac {3 a^3 \sin ^5(c+d x)}{5 d}+\frac {1}{8} \left (5 a^3\right ) \int \cos ^2(c+d x) \, dx+\frac {1}{8} \left (9 a^3\right ) \int 1 \, dx\\ &=\frac {9 a^3 x}{8}+\frac {4 a^3 \sin (c+d x)}{d}+\frac {23 a^3 \cos (c+d x) \sin (c+d x)}{16 d}+\frac {23 a^3 \cos ^3(c+d x) \sin (c+d x)}{24 d}+\frac {a^3 \cos ^5(c+d x) \sin (c+d x)}{6 d}-\frac {7 a^3 \sin ^3(c+d x)}{3 d}+\frac {3 a^3 \sin ^5(c+d x)}{5 d}+\frac {1}{16} \left (5 a^3\right ) \int 1 \, dx\\ &=\frac {23 a^3 x}{16}+\frac {4 a^3 \sin (c+d x)}{d}+\frac {23 a^3 \cos (c+d x) \sin (c+d x)}{16 d}+\frac {23 a^3 \cos ^3(c+d x) \sin (c+d x)}{24 d}+\frac {a^3 \cos ^5(c+d x) \sin (c+d x)}{6 d}-\frac {7 a^3 \sin ^3(c+d x)}{3 d}+\frac {3 a^3 \sin ^5(c+d x)}{5 d}\\ \end {align*}
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Mathematica [A]
time = 0.21, size = 73, normalized size = 0.57 \begin {gather*} \frac {a^3 (1380 d x+2520 \sin (c+d x)+945 \sin (2 (c+d x))+380 \sin (3 (c+d x))+135 \sin (4 (c+d x))+36 \sin (5 (c+d x))+5 \sin (6 (c+d x)))}{960 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 143, normalized size = 1.11
method | result | size |
risch | \(\frac {23 a^{3} x}{16}+\frac {21 a^{3} \sin \left (d x +c \right )}{8 d}+\frac {a^{3} \sin \left (6 d x +6 c \right )}{192 d}+\frac {3 a^{3} \sin \left (5 d x +5 c \right )}{80 d}+\frac {9 a^{3} \sin \left (4 d x +4 c \right )}{64 d}+\frac {19 a^{3} \sin \left (3 d x +3 c \right )}{48 d}+\frac {63 a^{3} \sin \left (2 d x +2 c \right )}{64 d}\) | \(107\) |
derivativedivides | \(\frac {a^{3} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+\frac {3 a^{3} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+3 a^{3} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {a^{3} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}}{d}\) | \(143\) |
default | \(\frac {a^{3} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+\frac {3 a^{3} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+3 a^{3} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {a^{3} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}}{d}\) | \(143\) |
norman | \(\frac {\frac {23 a^{3} x}{16}+\frac {105 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}+\frac {a^{3} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}+\frac {353 a^{3} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{40 d}-\frac {1303 a^{3} \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{40 d}-\frac {1339 a^{3} \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{120 d}+\frac {989 a^{3} \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{120 d}+\frac {253 a^{3} \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}+\frac {23 a^{3} \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}+\frac {23 a^{3} x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}+\frac {23 a^{3} x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}-\frac {23 a^{3} x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}-\frac {115 a^{3} x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}-\frac {23 a^{3} x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}+\frac {23 a^{3} x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}+\frac {23 a^{3} x \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}+\frac {23 a^{3} x \left (\tan ^{16}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}\) | \(325\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 143, normalized size = 1.11 \begin {gather*} \frac {192 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} a^{3} - 5 \, {\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} - 320 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} a^{3} + 90 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{3}}{960 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.64, size = 89, normalized size = 0.69 \begin {gather*} \frac {345 \, a^{3} d x + {\left (40 \, a^{3} \cos \left (d x + c\right )^{5} + 144 \, a^{3} \cos \left (d x + c\right )^{4} + 230 \, a^{3} \cos \left (d x + c\right )^{3} + 272 \, a^{3} \cos \left (d x + c\right )^{2} + 345 \, a^{3} \cos \left (d x + c\right ) + 544 \, a^{3}\right )} \sin \left (d x + c\right )}{240 \, 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 = 0.53, size = 128, normalized size = 0.99 \begin {gather*} \frac {345 \, {\left (d x + c\right )} a^{3} + \frac {2 \, {\left (345 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 1955 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 4554 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 5814 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3165 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1575 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{6}}}{240 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.49, size = 121, normalized size = 0.94 \begin {gather*} \frac {23\,a^3\,x}{16}+\frac {\frac {23\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{8}+\frac {391\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{24}+\frac {759\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{20}+\frac {969\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{20}+\frac {211\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{8}+\frac {105\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8}}{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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