Optimal. Leaf size=114 \[ \frac {a^3 (7 B+6 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {(2 B+C) \tan (c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{d}-\frac {5 a^3 B \sin (c+d x)}{2 d}+a^3 x (B+3 C)+\frac {a B \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^2}{2 d} \]
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Rubi [A] time = 0.43, antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {3029, 2975, 2968, 3023, 2735, 3770} \[ \frac {a^3 (7 B+6 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {(2 B+C) \tan (c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{d}-\frac {5 a^3 B \sin (c+d x)}{2 d}+a^3 x (B+3 C)+\frac {a B \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^2}{2 d} \]
Antiderivative was successfully verified.
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Rule 2735
Rule 2968
Rule 2975
Rule 3023
Rule 3029
Rule 3770
Rubi steps
\begin {align*} \int (a+a \cos (c+d x))^3 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx &=\int (a+a \cos (c+d x))^3 (B+C \cos (c+d x)) \sec ^3(c+d x) \, dx\\ &=\frac {a B (a+a \cos (c+d x))^2 \sec (c+d x) \tan (c+d x)}{2 d}+\frac {1}{2} \int (a+a \cos (c+d x))^2 (2 a (2 B+C)-a (B-2 C) \cos (c+d x)) \sec ^2(c+d x) \, dx\\ &=\frac {(2 B+C) \left (a^3+a^3 \cos (c+d x)\right ) \tan (c+d x)}{d}+\frac {a B (a+a \cos (c+d x))^2 \sec (c+d x) \tan (c+d x)}{2 d}+\frac {1}{2} \int (a+a \cos (c+d x)) \left (a^2 (7 B+6 C)-5 a^2 B \cos (c+d x)\right ) \sec (c+d x) \, dx\\ &=\frac {(2 B+C) \left (a^3+a^3 \cos (c+d x)\right ) \tan (c+d x)}{d}+\frac {a B (a+a \cos (c+d x))^2 \sec (c+d x) \tan (c+d x)}{2 d}+\frac {1}{2} \int \left (a^3 (7 B+6 C)+\left (-5 a^3 B+a^3 (7 B+6 C)\right ) \cos (c+d x)-5 a^3 B \cos ^2(c+d x)\right ) \sec (c+d x) \, dx\\ &=-\frac {5 a^3 B \sin (c+d x)}{2 d}+\frac {(2 B+C) \left (a^3+a^3 \cos (c+d x)\right ) \tan (c+d x)}{d}+\frac {a B (a+a \cos (c+d x))^2 \sec (c+d x) \tan (c+d x)}{2 d}+\frac {1}{2} \int \left (a^3 (7 B+6 C)+2 a^3 (B+3 C) \cos (c+d x)\right ) \sec (c+d x) \, dx\\ &=a^3 (B+3 C) x-\frac {5 a^3 B \sin (c+d x)}{2 d}+\frac {(2 B+C) \left (a^3+a^3 \cos (c+d x)\right ) \tan (c+d x)}{d}+\frac {a B (a+a \cos (c+d x))^2 \sec (c+d x) \tan (c+d x)}{2 d}+\frac {1}{2} \left (a^3 (7 B+6 C)\right ) \int \sec (c+d x) \, dx\\ &=a^3 (B+3 C) x+\frac {a^3 (7 B+6 C) \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac {5 a^3 B \sin (c+d x)}{2 d}+\frac {(2 B+C) \left (a^3+a^3 \cos (c+d x)\right ) \tan (c+d x)}{d}+\frac {a B (a+a \cos (c+d x))^2 \sec (c+d x) \tan (c+d x)}{2 d}\\ \end {align*}
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Mathematica [A] time = 1.84, size = 208, normalized size = 1.82 \[ \frac {a^3 \left (4 (3 B+C) \tan (c+d x)+\frac {B}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}-\frac {B}{\left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^2}-14 B \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+14 B \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )+4 B c+4 B d x+4 C \sin (c+d x)-12 C \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+12 C \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )+12 c C+12 C d x\right )}{4 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 137, normalized size = 1.20 \[ \frac {4 \, {\left (B + 3 \, C\right )} a^{3} d x \cos \left (d x + c\right )^{2} + {\left (7 \, B + 6 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left (7 \, B + 6 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (2 \, C a^{3} \cos \left (d x + c\right )^{2} + 2 \, {\left (3 \, B + C\right )} a^{3} \cos \left (d x + c\right ) + B a^{3}\right )} \sin \left (d x + c\right )}{4 \, d \cos \left (d x + c\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.07, size = 192, normalized size = 1.68 \[ \frac {\frac {4 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1} + 2 \, {\left (B a^{3} + 3 \, C a^{3}\right )} {\left (d x + c\right )} + {\left (7 \, B a^{3} + 6 \, C a^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - {\left (7 \, B a^{3} + 6 \, C a^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (5 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 7 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{2}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.31, size = 144, normalized size = 1.26 \[ \frac {a^{3} C \sin \left (d x +c \right )}{d}+a^{3} B x +\frac {a^{3} B c}{d}+3 a^{3} C x +\frac {3 C \,a^{3} c}{d}+\frac {7 a^{3} B \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2 d}+\frac {3 C \,a^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {3 a^{3} B \tan \left (d x +c \right )}{d}+\frac {C \,a^{3} \tan \left (d x +c \right )}{d}+\frac {a^{3} B \sec \left (d x +c \right ) \tan \left (d x +c \right )}{2 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.84, size = 165, normalized size = 1.45 \[ \frac {4 \, {\left (d x + c\right )} B a^{3} + 12 \, {\left (d x + c\right )} C a^{3} - B a^{3} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 6 \, B a^{3} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 6 \, C a^{3} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 4 \, C a^{3} \sin \left (d x + c\right ) + 12 \, B a^{3} \tan \left (d x + c\right ) + 4 \, C a^{3} \tan \left (d x + c\right )}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.25, size = 207, normalized size = 1.82 \[ \frac {C\,a^3\,\sin \left (c+d\,x\right )}{d}+\frac {2\,B\,a^3\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {7\,B\,a^3\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {6\,C\,a^3\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {6\,C\,a^3\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {3\,B\,a^3\,\sin \left (c+d\,x\right )}{d\,\cos \left (c+d\,x\right )}+\frac {B\,a^3\,\sin \left (c+d\,x\right )}{2\,d\,{\cos \left (c+d\,x\right )}^2}+\frac {C\,a^3\,\sin \left (c+d\,x\right )}{d\,\cos \left (c+d\,x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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