Optimal. Leaf size=110 \[ \frac {a^3 (3 B+C) \tanh ^{-1}(\sin (c+d x))}{d}-\frac {(2 B-C) \sin (c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{2 d}+\frac {1}{2} a^3 x (6 B+7 C)+\frac {5 a^3 C \sin (c+d x)}{2 d}+\frac {a B \tan (c+d x) (a \cos (c+d x)+a)^2}{d} \]
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Rubi [A] time = 0.40, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.175, Rules used = {3029, 2975, 2976, 2968, 3023, 2735, 3770} \[ \frac {a^3 (3 B+C) \tanh ^{-1}(\sin (c+d x))}{d}-\frac {(2 B-C) \sin (c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{2 d}+\frac {1}{2} a^3 x (6 B+7 C)+\frac {5 a^3 C \sin (c+d x)}{2 d}+\frac {a B \tan (c+d x) (a \cos (c+d x)+a)^2}{d} \]
Antiderivative was successfully verified.
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Rule 2735
Rule 2968
Rule 2975
Rule 2976
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 ^3(c+d x) \, dx &=\int (a+a \cos (c+d x))^3 (B+C \cos (c+d x)) \sec ^2(c+d x) \, dx\\ &=\frac {a B (a+a \cos (c+d x))^2 \tan (c+d x)}{d}+\int (a+a \cos (c+d x))^2 (a (3 B+C)-a (2 B-C) \cos (c+d x)) \sec (c+d x) \, dx\\ &=-\frac {(2 B-C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{2 d}+\frac {a B (a+a \cos (c+d x))^2 \tan (c+d x)}{d}+\frac {1}{2} \int (a+a \cos (c+d x)) \left (2 a^2 (3 B+C)+5 a^2 C \cos (c+d x)\right ) \sec (c+d x) \, dx\\ &=-\frac {(2 B-C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{2 d}+\frac {a B (a+a \cos (c+d x))^2 \tan (c+d x)}{d}+\frac {1}{2} \int \left (2 a^3 (3 B+C)+\left (5 a^3 C+2 a^3 (3 B+C)\right ) \cos (c+d x)+5 a^3 C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx\\ &=\frac {5 a^3 C \sin (c+d x)}{2 d}-\frac {(2 B-C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{2 d}+\frac {a B (a+a \cos (c+d x))^2 \tan (c+d x)}{d}+\frac {1}{2} \int \left (2 a^3 (3 B+C)+a^3 (6 B+7 C) \cos (c+d x)\right ) \sec (c+d x) \, dx\\ &=\frac {1}{2} a^3 (6 B+7 C) x+\frac {5 a^3 C \sin (c+d x)}{2 d}-\frac {(2 B-C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{2 d}+\frac {a B (a+a \cos (c+d x))^2 \tan (c+d x)}{d}+\left (a^3 (3 B+C)\right ) \int \sec (c+d x) \, dx\\ &=\frac {1}{2} a^3 (6 B+7 C) x+\frac {a^3 (3 B+C) \tanh ^{-1}(\sin (c+d x))}{d}+\frac {5 a^3 C \sin (c+d x)}{2 d}-\frac {(2 B-C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{2 d}+\frac {a B (a+a \cos (c+d x))^2 \tan (c+d x)}{d}\\ \end {align*}
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Mathematica [B] time = 1.73, size = 272, normalized size = 2.47 \[ \frac {1}{32} a^3 (\cos (c+d x)+1)^3 \sec ^6\left (\frac {1}{2} (c+d x)\right ) \left (\frac {4 (B+3 C) \sin (c) \cos (d x)}{d}+\frac {4 (B+3 C) \cos (c) \sin (d x)}{d}-\frac {4 (3 B+C) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{d}+\frac {4 (3 B+C) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )}{d}+\frac {4 B \sin \left (\frac {d x}{2}\right )}{d \left (\cos \left (\frac {c}{2}\right )-\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {4 B \sin \left (\frac {d x}{2}\right )}{d \left (\sin \left (\frac {c}{2}\right )+\cos \left (\frac {c}{2}\right )\right ) \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )}+2 x (6 B+7 C)+\frac {C \sin (2 c) \cos (2 d x)}{d}+\frac {C \cos (2 c) \sin (2 d x)}{d}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 127, normalized size = 1.15 \[ \frac {{\left (6 \, B + 7 \, C\right )} a^{3} d x \cos \left (d x + c\right ) + {\left (3 \, B + C\right )} a^{3} \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left (3 \, B + C\right )} a^{3} \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) + {\left (C a^{3} \cos \left (d x + c\right )^{2} + 2 \, {\left (B + 3 \, C\right )} a^{3} \cos \left (d x + c\right ) + 2 \, B a^{3}\right )} \sin \left (d x + c\right )}{2 \, d \cos \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.56, size = 192, normalized size = 1.75 \[ -\frac {\frac {4 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1} - {\left (6 \, B a^{3} + 7 \, C a^{3}\right )} {\left (d x + c\right )} - 2 \, {\left (3 \, B a^{3} + C a^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) + 2 \, {\left (3 \, B a^{3} + C a^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (2 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 5 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 7 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.24, size = 145, normalized size = 1.32 \[ \frac {C \,a^{3} \cos \left (d x +c \right ) \sin \left (d x +c \right )}{2 d}+\frac {7 a^{3} C x}{2}+\frac {7 C \,a^{3} c}{2 d}+\frac {a^{3} B \sin \left (d x +c \right )}{d}+\frac {3 a^{3} C \sin \left (d x +c \right )}{d}+3 a^{3} B x +\frac {3 a^{3} B c}{d}+\frac {3 a^{3} B \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {C \,a^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {a^{3} B \tan \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.56, size = 140, normalized size = 1.27 \[ \frac {12 \, {\left (d x + c\right )} B a^{3} + {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{3} + 12 \, {\left (d x + c\right )} C a^{3} + 6 \, B a^{3} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 2 \, C a^{3} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 4 \, B a^{3} \sin \left (d x + c\right ) + 12 \, C a^{3} \sin \left (d x + c\right ) + 4 \, B a^{3} \tan \left (d x + c\right )}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.25, size = 197, normalized size = 1.79 \[ \frac {B\,a^3\,\sin \left (c+d\,x\right )}{d}+\frac {3\,C\,a^3\,\sin \left (c+d\,x\right )}{d}+\frac {6\,B\,a^3\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {6\,B\,a^3\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {7\,C\,a^3\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {2\,C\,a^3\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {B\,a^3\,\sin \left (c+d\,x\right )}{d\,\cos \left (c+d\,x\right )}+\frac {C\,a^3\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )}{2\,d} \]
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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