Optimal. Leaf size=145 \[ -\frac {(6 A-5 C) \sin (c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{6 d}+\frac {3 a^3 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac {1}{2} a^3 x (6 A+5 C)+\frac {5 a^3 C \sin (c+d x)}{2 d}-\frac {(3 A-C) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{3 a d}+\frac {A \tan (c+d x) (a \cos (c+d x)+a)^3}{d} \]
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Rubi [A] time = 0.45, antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3044, 2976, 2968, 3023, 2735, 3770} \[ -\frac {(6 A-5 C) \sin (c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{6 d}-\frac {(3 A-C) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{3 a d}+\frac {3 a^3 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac {1}{2} a^3 x (6 A+5 C)+\frac {5 a^3 C \sin (c+d x)}{2 d}+\frac {A \tan (c+d x) (a \cos (c+d x)+a)^3}{d} \]
Antiderivative was successfully verified.
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Rule 2735
Rule 2968
Rule 2976
Rule 3023
Rule 3044
Rule 3770
Rubi steps
\begin {align*} \int (a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx &=\frac {A (a+a \cos (c+d x))^3 \tan (c+d x)}{d}+\frac {\int (a+a \cos (c+d x))^3 (3 a A-a (3 A-C) \cos (c+d x)) \sec (c+d x) \, dx}{a}\\ &=-\frac {(3 A-C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{3 a d}+\frac {A (a+a \cos (c+d x))^3 \tan (c+d x)}{d}+\frac {\int (a+a \cos (c+d x))^2 \left (9 a^2 A-a^2 (6 A-5 C) \cos (c+d x)\right ) \sec (c+d x) \, dx}{3 a}\\ &=-\frac {(3 A-C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{3 a d}-\frac {(6 A-5 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{6 d}+\frac {A (a+a \cos (c+d x))^3 \tan (c+d x)}{d}+\frac {\int (a+a \cos (c+d x)) \left (18 a^3 A+15 a^3 C \cos (c+d x)\right ) \sec (c+d x) \, dx}{6 a}\\ &=-\frac {(3 A-C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{3 a d}-\frac {(6 A-5 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{6 d}+\frac {A (a+a \cos (c+d x))^3 \tan (c+d x)}{d}+\frac {\int \left (18 a^4 A+\left (18 a^4 A+15 a^4 C\right ) \cos (c+d x)+15 a^4 C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx}{6 a}\\ &=\frac {5 a^3 C \sin (c+d x)}{2 d}-\frac {(3 A-C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{3 a d}-\frac {(6 A-5 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{6 d}+\frac {A (a+a \cos (c+d x))^3 \tan (c+d x)}{d}+\frac {\int \left (18 a^4 A+3 a^4 (6 A+5 C) \cos (c+d x)\right ) \sec (c+d x) \, dx}{6 a}\\ &=\frac {1}{2} a^3 (6 A+5 C) x+\frac {5 a^3 C \sin (c+d x)}{2 d}-\frac {(3 A-C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{3 a d}-\frac {(6 A-5 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{6 d}+\frac {A (a+a \cos (c+d x))^3 \tan (c+d x)}{d}+\left (3 a^3 A\right ) \int \sec (c+d x) \, dx\\ &=\frac {1}{2} a^3 (6 A+5 C) x+\frac {3 a^3 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac {5 a^3 C \sin (c+d x)}{2 d}-\frac {(3 A-C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{3 a d}-\frac {(6 A-5 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{6 d}+\frac {A (a+a \cos (c+d x))^3 \tan (c+d x)}{d}\\ \end {align*}
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Mathematica [B] time = 1.97, size = 298, normalized size = 2.06 \[ \frac {1}{96} a^3 (\cos (c+d x)+1)^3 \sec ^6\left (\frac {1}{2} (c+d x)\right ) \left (\frac {3 (4 A+15 C) \sin (c) \cos (d x)}{d}+\frac {3 (4 A+15 C) \cos (c) \sin (d x)}{d}+\frac {12 A \sin \left (\frac {d x}{2}\right )}{d \left (\cos \left (\frac {c}{2}\right )-\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {12 A \sin \left (\frac {d x}{2}\right )}{d \left (\sin \left (\frac {c}{2}\right )+\cos \left (\frac {c}{2}\right )\right ) \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )}-\frac {36 A \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{d}+\frac {36 A \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )}{d}+6 x (6 A+5 C)+\frac {9 C \sin (2 c) \cos (2 d x)}{d}+\frac {C \sin (3 c) \cos (3 d x)}{d}+\frac {9 C \cos (2 c) \sin (2 d x)}{d}+\frac {C \cos (3 c) \sin (3 d x)}{d}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.53, size = 138, normalized size = 0.95 \[ \frac {3 \, {\left (6 \, A + 5 \, C\right )} a^{3} d x \cos \left (d x + c\right ) + 9 \, A a^{3} \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) - 9 \, A a^{3} \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) + {\left (2 \, C a^{3} \cos \left (d x + c\right )^{3} + 9 \, C a^{3} \cos \left (d x + c\right )^{2} + 2 \, {\left (3 \, A + 11 \, C\right )} a^{3} \cos \left (d x + c\right ) + 6 \, A a^{3}\right )} \sin \left (d x + c\right )}{6 \, d \cos \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.49, size = 210, normalized size = 1.45 \[ \frac {18 \, A a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 18 \, A a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {12 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1} + 3 \, {\left (6 \, A a^{3} + 5 \, C a^{3}\right )} {\left (d x + c\right )} + \frac {2 \, {\left (6 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 15 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 12 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 40 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 6 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 33 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.32, size = 146, normalized size = 1.01 \[ \frac {a^{3} A \sin \left (d x +c \right )}{d}+\frac {C \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right ) a^{3}}{3 d}+\frac {11 a^{3} C \sin \left (d x +c \right )}{3 d}+3 A x \,a^{3}+\frac {3 A \,a^{3} c}{d}+\frac {3 C \,a^{3} \cos \left (d x +c \right ) \sin \left (d x +c \right )}{2 d}+\frac {5 a^{3} C x}{2}+\frac {5 C \,a^{3} c}{2 d}+\frac {3 A \,a^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {A \,a^{3} \tan \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.50, size = 137, normalized size = 0.94 \[ \frac {36 \, {\left (d x + c\right )} A a^{3} - 4 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{3} + 9 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{3} + 12 \, {\left (d x + c\right )} C a^{3} + 18 \, A a^{3} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 12 \, A a^{3} \sin \left (d x + c\right ) + 36 \, C a^{3} \sin \left (d x + c\right ) + 12 \, A a^{3} \tan \left (d x + c\right )}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.00, size = 189, normalized size = 1.30 \[ \frac {A\,a^3\,\sin \left (c+d\,x\right )}{d}+\frac {11\,C\,a^3\,\sin \left (c+d\,x\right )}{3\,d}+\frac {6\,A\,a^3\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {6\,A\,a^3\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {5\,C\,a^3\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {A\,a^3\,\sin \left (c+d\,x\right )}{d\,\cos \left (c+d\,x\right )}+\frac {C\,a^3\,{\cos \left (c+d\,x\right )}^2\,\sin \left (c+d\,x\right )}{3\,d}+\frac {3\,C\,a^3\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )}{2\,d} \]
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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