Optimal. Leaf size=156 \[ \frac {a^3 (5 A+6 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {(5 A+3 C) \tan (c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{3 d}-\frac {5 a^3 A \sin (c+d x)}{2 d}+3 a^3 C x+\frac {A \tan (c+d x) \sec (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{2 a d}+\frac {A \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^3}{3 d} \]
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Rubi [A] time = 0.50, antiderivative size = 156, 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, 2975, 2968, 3023, 2735, 3770} \[ \frac {a^3 (5 A+6 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {(5 A+3 C) \tan (c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{3 d}-\frac {5 a^3 A \sin (c+d x)}{2 d}+\frac {A \tan (c+d x) \sec (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{2 a d}+3 a^3 C x+\frac {A \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^3}{3 d} \]
Antiderivative was successfully verified.
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Rule 2735
Rule 2968
Rule 2975
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 ^4(c+d x) \, dx &=\frac {A (a+a \cos (c+d x))^3 \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac {\int (a+a \cos (c+d x))^3 (3 a A-a (A-3 C) \cos (c+d x)) \sec ^3(c+d x) \, dx}{3 a}\\ &=\frac {A \left (a^2+a^2 \cos (c+d x)\right )^2 \sec (c+d x) \tan (c+d x)}{2 a d}+\frac {A (a+a \cos (c+d x))^3 \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac {\int (a+a \cos (c+d x))^2 \left (2 a^2 (5 A+3 C)-a^2 (5 A-6 C) \cos (c+d x)\right ) \sec ^2(c+d x) \, dx}{6 a}\\ &=\frac {(5 A+3 C) \left (a^3+a^3 \cos (c+d x)\right ) \tan (c+d x)}{3 d}+\frac {A \left (a^2+a^2 \cos (c+d x)\right )^2 \sec (c+d x) \tan (c+d x)}{2 a d}+\frac {A (a+a \cos (c+d x))^3 \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac {\int (a+a \cos (c+d x)) \left (3 a^3 (5 A+6 C)-15 a^3 A \cos (c+d x)\right ) \sec (c+d x) \, dx}{6 a}\\ &=\frac {(5 A+3 C) \left (a^3+a^3 \cos (c+d x)\right ) \tan (c+d x)}{3 d}+\frac {A \left (a^2+a^2 \cos (c+d x)\right )^2 \sec (c+d x) \tan (c+d x)}{2 a d}+\frac {A (a+a \cos (c+d x))^3 \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac {\int \left (3 a^4 (5 A+6 C)+\left (-15 a^4 A+3 a^4 (5 A+6 C)\right ) \cos (c+d x)-15 a^4 A \cos ^2(c+d x)\right ) \sec (c+d x) \, dx}{6 a}\\ &=-\frac {5 a^3 A \sin (c+d x)}{2 d}+\frac {(5 A+3 C) \left (a^3+a^3 \cos (c+d x)\right ) \tan (c+d x)}{3 d}+\frac {A \left (a^2+a^2 \cos (c+d x)\right )^2 \sec (c+d x) \tan (c+d x)}{2 a d}+\frac {A (a+a \cos (c+d x))^3 \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac {\int \left (3 a^4 (5 A+6 C)+18 a^4 C \cos (c+d x)\right ) \sec (c+d x) \, dx}{6 a}\\ &=3 a^3 C x-\frac {5 a^3 A \sin (c+d x)}{2 d}+\frac {(5 A+3 C) \left (a^3+a^3 \cos (c+d x)\right ) \tan (c+d x)}{3 d}+\frac {A \left (a^2+a^2 \cos (c+d x)\right )^2 \sec (c+d x) \tan (c+d x)}{2 a d}+\frac {A (a+a \cos (c+d x))^3 \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac {1}{2} \left (a^3 (5 A+6 C)\right ) \int \sec (c+d x) \, dx\\ &=3 a^3 C x+\frac {a^3 (5 A+6 C) \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac {5 a^3 A \sin (c+d x)}{2 d}+\frac {(5 A+3 C) \left (a^3+a^3 \cos (c+d x)\right ) \tan (c+d x)}{3 d}+\frac {A \left (a^2+a^2 \cos (c+d x)\right )^2 \sec (c+d x) \tan (c+d x)}{2 a d}+\frac {A (a+a \cos (c+d x))^3 \sec ^2(c+d x) \tan (c+d x)}{3 d}\\ \end {align*}
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Mathematica [B] time = 6.37, size = 832, normalized size = 5.33 \[ \frac {3}{8} C x (\cos (c+d x) a+a)^3 \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right )+\frac {(-5 A-6 C) (\cos (c+d x) a+a)^3 \log \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right ) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right )}{16 d}+\frac {(5 A+6 C) (\cos (c+d x) a+a)^3 \log \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right ) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right )}{16 d}+\frac {C \cos (d x) (\cos (c+d x) a+a)^3 \sin (c) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right )}{8 d}+\frac {C \cos (c) (\cos (c+d x) a+a)^3 \sin (d x) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right )}{8 d}+\frac {(\cos (c+d x) a+a)^3 \left (11 A \sin \left (\frac {d x}{2}\right )+3 C \sin \left (\frac {d x}{2}\right )\right ) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right )}{24 d \left (\cos \left (\frac {c}{2}\right )-\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}+\frac {(\cos (c+d x) a+a)^3 \left (11 A \sin \left (\frac {d x}{2}\right )+3 C \sin \left (\frac {d x}{2}\right )\right ) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right )}{24 d \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}+\frac {(\cos (c+d x) a+a)^3 \left (5 A \cos \left (\frac {c}{2}\right )-4 A \sin \left (\frac {c}{2}\right )\right ) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right )}{48 d \left (\cos \left (\frac {c}{2}\right )-\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )^2}+\frac {(\cos (c+d x) a+a)^3 \left (-5 A \cos \left (\frac {c}{2}\right )-4 A \sin \left (\frac {c}{2}\right )\right ) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right )}{48 d \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )^2}+\frac {A (\cos (c+d x) a+a)^3 \sin \left (\frac {d x}{2}\right ) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right )}{48 d \left (\cos \left (\frac {c}{2}\right )-\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )^3}+\frac {A (\cos (c+d x) a+a)^3 \sin \left (\frac {d x}{2}\right ) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right )}{48 d \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 151, normalized size = 0.97 \[ \frac {36 \, C a^{3} d x \cos \left (d x + c\right )^{3} + 3 \, {\left (5 \, A + 6 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (5 \, A + 6 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (6 \, C a^{3} \cos \left (d x + c\right )^{3} + 2 \, {\left (11 \, A + 3 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} + 9 \, A a^{3} \cos \left (d x + c\right ) + 2 \, A a^{3}\right )} \sin \left (d x + c\right )}{12 \, d \cos \left (d x + c\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.51, size = 219, normalized size = 1.40 \[ \frac {18 \, {\left (d x + c\right )} C a^{3} + \frac {12 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1} + 3 \, {\left (5 \, A a^{3} + 6 \, C a^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 3 \, {\left (5 \, A a^{3} + 6 \, C a^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (15 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 40 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 12 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 33 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{3}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.37, size = 152, normalized size = 0.97 \[ \frac {5 A \,a^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2 d}+\frac {a^{3} C \sin \left (d x +c \right )}{d}+\frac {11 A \,a^{3} \tan \left (d x +c \right )}{3 d}+3 a^{3} C x +\frac {3 C \,a^{3} c}{d}+\frac {3 A \,a^{3} \sec \left (d x +c \right ) \tan \left (d x +c \right )}{2 d}+\frac {3 C \,a^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {A \,a^{3} \tan \left (d x +c \right ) \left (\sec ^{2}\left (d x +c \right )\right )}{3 d}+\frac {C \,a^{3} \tan \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.69, size = 177, normalized size = 1.13 \[ \frac {4 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a^{3} + 36 \, {\left (d x + c\right )} C a^{3} - 9 \, A a^{3} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 6 \, A a^{3} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 18 \, C a^{3} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 12 \, C a^{3} \sin \left (d x + c\right ) + 36 \, A a^{3} \tan \left (d x + c\right ) + 12 \, C a^{3} \tan \left (d x + c\right )}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.04, size = 199, normalized size = 1.28 \[ \frac {C\,a^3\,\sin \left (c+d\,x\right )}{d}+\frac {5\,A\,a^3\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {6\,C\,a^3\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {6\,C\,a^3\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {11\,A\,a^3\,\sin \left (c+d\,x\right )}{3\,d\,\cos \left (c+d\,x\right )}+\frac {3\,A\,a^3\,\sin \left (c+d\,x\right )}{2\,d\,{\cos \left (c+d\,x\right )}^2}+\frac {A\,a^3\,\sin \left (c+d\,x\right )}{3\,d\,{\cos \left (c+d\,x\right )}^3}+\frac {C\,a^3\,\sin \left (c+d\,x\right )}{d\,\cos \left (c+d\,x\right )} \]
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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