Optimal. Leaf size=160 \[ \frac {1}{8} a^2 (6 B+7 C) x+\frac {a^2 (9 B+10 C) \sin (c+d x)}{5 d}+\frac {a^2 (6 B+7 C) \cos (c+d x) \sin (c+d x)}{8 d}+\frac {a^2 (6 B+5 C) \cos ^3(c+d x) \sin (c+d x)}{20 d}+\frac {B \cos ^4(c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{5 d}-\frac {a^2 (9 B+10 C) \sin ^3(c+d x)}{15 d} \]
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Rubi [A]
time = 0.23, antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.175, Rules used = {4157, 4102,
4081, 3872, 2713, 2715, 8} \begin {gather*} -\frac {a^2 (9 B+10 C) \sin ^3(c+d x)}{15 d}+\frac {a^2 (9 B+10 C) \sin (c+d x)}{5 d}+\frac {a^2 (6 B+5 C) \sin (c+d x) \cos ^3(c+d x)}{20 d}+\frac {a^2 (6 B+7 C) \sin (c+d x) \cos (c+d x)}{8 d}+\frac {B \sin (c+d x) \cos ^4(c+d x) \left (a^2 \sec (c+d x)+a^2\right )}{5 d}+\frac {1}{8} a^2 x (6 B+7 C) \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2713
Rule 2715
Rule 3872
Rule 4081
Rule 4102
Rule 4157
Rubi steps
\begin {align*} \int \cos ^6(c+d x) (a+a \sec (c+d x))^2 \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\int \cos ^5(c+d x) (a+a \sec (c+d x))^2 (B+C \sec (c+d x)) \, dx\\ &=\frac {B \cos ^4(c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{5 d}+\frac {1}{5} \int \cos ^4(c+d x) (a+a \sec (c+d x)) (a (6 B+5 C)+a (3 B+5 C) \sec (c+d x)) \, dx\\ &=\frac {a^2 (6 B+5 C) \cos ^3(c+d x) \sin (c+d x)}{20 d}+\frac {B \cos ^4(c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{5 d}-\frac {1}{20} \int \cos ^3(c+d x) \left (-4 a^2 (9 B+10 C)-5 a^2 (6 B+7 C) \sec (c+d x)\right ) \, dx\\ &=\frac {a^2 (6 B+5 C) \cos ^3(c+d x) \sin (c+d x)}{20 d}+\frac {B \cos ^4(c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{5 d}+\frac {1}{4} \left (a^2 (6 B+7 C)\right ) \int \cos ^2(c+d x) \, dx+\frac {1}{5} \left (a^2 (9 B+10 C)\right ) \int \cos ^3(c+d x) \, dx\\ &=\frac {a^2 (6 B+7 C) \cos (c+d x) \sin (c+d x)}{8 d}+\frac {a^2 (6 B+5 C) \cos ^3(c+d x) \sin (c+d x)}{20 d}+\frac {B \cos ^4(c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{5 d}+\frac {1}{8} \left (a^2 (6 B+7 C)\right ) \int 1 \, dx-\frac {\left (a^2 (9 B+10 C)\right ) \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{5 d}\\ &=\frac {1}{8} a^2 (6 B+7 C) x+\frac {a^2 (9 B+10 C) \sin (c+d x)}{5 d}+\frac {a^2 (6 B+7 C) \cos (c+d x) \sin (c+d x)}{8 d}+\frac {a^2 (6 B+5 C) \cos ^3(c+d x) \sin (c+d x)}{20 d}+\frac {B \cos ^4(c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{5 d}-\frac {a^2 (9 B+10 C) \sin ^3(c+d x)}{15 d}\\ \end {align*}
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Mathematica [A]
time = 0.41, size = 108, normalized size = 0.68 \begin {gather*} \frac {a^2 (360 B c+360 B d x+420 C d x+60 (11 B+12 C) \sin (c+d x)+240 (B+C) \sin (2 (c+d x))+90 B \sin (3 (c+d x))+80 C \sin (3 (c+d x))+30 B \sin (4 (c+d x))+15 C \sin (4 (c+d x))+6 B \sin (5 (c+d x)))}{480 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.41, size = 186, normalized size = 1.16
method | result | size |
risch | \(\frac {3 a^{2} B x}{4}+\frac {7 a^{2} x C}{8}+\frac {11 a^{2} B \sin \left (d x +c \right )}{8 d}+\frac {3 \sin \left (d x +c \right ) a^{2} C}{2 d}+\frac {a^{2} B \sin \left (5 d x +5 c \right )}{80 d}+\frac {a^{2} B \sin \left (4 d x +4 c \right )}{16 d}+\frac {\sin \left (4 d x +4 c \right ) a^{2} C}{32 d}+\frac {3 a^{2} B \sin \left (3 d x +3 c \right )}{16 d}+\frac {\sin \left (3 d x +3 c \right ) a^{2} C}{6 d}+\frac {a^{2} B \sin \left (2 d x +2 c \right )}{2 d}+\frac {\sin \left (2 d x +2 c \right ) a^{2} C}{2 d}\) | \(172\) |
derivativedivides | \(\frac {\frac {a^{2} B \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+a^{2} C \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+2 a^{2} B \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {2 a^{2} C \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+\frac {a^{2} B \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+a^{2} C \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\) | \(186\) |
default | \(\frac {\frac {a^{2} B \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+a^{2} C \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+2 a^{2} B \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {2 a^{2} C \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+\frac {a^{2} B \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+a^{2} C \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\) | \(186\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 178, normalized size = 1.11 \begin {gather*} \frac {32 \, {\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 )} B a^{2} - 160 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{2} + 30 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{2} - 320 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{2} + 15 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{2} + 120 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{2}}{480 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.39, size = 110, normalized size = 0.69 \begin {gather*} \frac {15 \, {\left (6 \, B + 7 \, C\right )} a^{2} d x + {\left (24 \, B a^{2} \cos \left (d x + c\right )^{4} + 30 \, {\left (2 \, B + C\right )} a^{2} \cos \left (d x + c\right )^{3} + 8 \, {\left (9 \, B + 10 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 15 \, {\left (6 \, B + 7 \, C\right )} a^{2} \cos \left (d x + c\right ) + 16 \, {\left (9 \, B + 10 \, C\right )} a^{2}\right )} \sin \left (d x + c\right )}{120 \, 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 = 0.52, size = 210, normalized size = 1.31 \begin {gather*} \frac {15 \, {\left (6 \, B a^{2} + 7 \, C a^{2}\right )} {\left (d x + c\right )} + \frac {2 \, {\left (90 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 105 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 420 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 490 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 864 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 800 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 540 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 790 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 390 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 375 \, C a^{2} \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 )}^{5}}}{120 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.67, size = 247, normalized size = 1.54 \begin {gather*} \frac {\left (\frac {3\,B\,a^2}{2}+\frac {7\,C\,a^2}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (7\,B\,a^2+\frac {49\,C\,a^2}{6}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {72\,B\,a^2}{5}+\frac {40\,C\,a^2}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (9\,B\,a^2+\frac {79\,C\,a^2}{6}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {13\,B\,a^2}{2}+\frac {25\,C\,a^2}{4}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {a^2\,\mathrm {atan}\left (\frac {a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (6\,B+7\,C\right )}{4\,\left (\frac {3\,B\,a^2}{2}+\frac {7\,C\,a^2}{4}\right )}\right )\,\left (6\,B+7\,C\right )}{4\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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