Optimal. Leaf size=186 \[ -\frac {5 a^4 (A-2 C) \sin (c+d x)}{2 d}+\frac {a^4 (13 A+2 C) \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac {(9 A-4 C) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{3 d}+2 a^4 x (2 A+3 C)-\frac {(15 A-2 C) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{6 d}+\frac {2 a A \tan (c+d x) (a \cos (c+d x)+a)^3}{d}+\frac {A \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^4}{2 d} \]
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Rubi [A] time = 0.61, antiderivative size = 186, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.212, Rules used = {3044, 2975, 2976, 2968, 3023, 2735, 3770} \[ -\frac {5 a^4 (A-2 C) \sin (c+d x)}{2 d}+\frac {a^4 (13 A+2 C) \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac {(15 A-2 C) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{6 d}-\frac {(9 A-4 C) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{3 d}+2 a^4 x (2 A+3 C)+\frac {2 a A \tan (c+d x) (a \cos (c+d x)+a)^3}{d}+\frac {A \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^4}{2 d} \]
Antiderivative was successfully verified.
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Rule 2735
Rule 2968
Rule 2975
Rule 2976
Rule 3023
Rule 3044
Rule 3770
Rubi steps
\begin {align*} \int (a+a \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx &=\frac {A (a+a \cos (c+d x))^4 \sec (c+d x) \tan (c+d x)}{2 d}+\frac {\int (a+a \cos (c+d x))^4 (4 a A-a (3 A-2 C) \cos (c+d x)) \sec ^2(c+d x) \, dx}{2 a}\\ &=\frac {2 a A (a+a \cos (c+d x))^3 \tan (c+d x)}{d}+\frac {A (a+a \cos (c+d x))^4 \sec (c+d x) \tan (c+d x)}{2 d}+\frac {\int (a+a \cos (c+d x))^3 \left (a^2 (13 A+2 C)-a^2 (15 A-2 C) \cos (c+d x)\right ) \sec (c+d x) \, dx}{2 a}\\ &=-\frac {(15 A-2 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{6 d}+\frac {2 a A (a+a \cos (c+d x))^3 \tan (c+d x)}{d}+\frac {A (a+a \cos (c+d x))^4 \sec (c+d x) \tan (c+d x)}{2 d}+\frac {\int (a+a \cos (c+d x))^2 \left (3 a^3 (13 A+2 C)-4 a^3 (9 A-4 C) \cos (c+d x)\right ) \sec (c+d x) \, dx}{6 a}\\ &=-\frac {(15 A-2 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{6 d}-\frac {(9 A-4 C) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{3 d}+\frac {2 a A (a+a \cos (c+d x))^3 \tan (c+d x)}{d}+\frac {A (a+a \cos (c+d x))^4 \sec (c+d x) \tan (c+d x)}{2 d}+\frac {\int (a+a \cos (c+d x)) \left (6 a^4 (13 A+2 C)-30 a^4 (A-2 C) \cos (c+d x)\right ) \sec (c+d x) \, dx}{12 a}\\ &=-\frac {(15 A-2 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{6 d}-\frac {(9 A-4 C) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{3 d}+\frac {2 a A (a+a \cos (c+d x))^3 \tan (c+d x)}{d}+\frac {A (a+a \cos (c+d x))^4 \sec (c+d x) \tan (c+d x)}{2 d}+\frac {\int \left (6 a^5 (13 A+2 C)+\left (-30 a^5 (A-2 C)+6 a^5 (13 A+2 C)\right ) \cos (c+d x)-30 a^5 (A-2 C) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx}{12 a}\\ &=-\frac {5 a^4 (A-2 C) \sin (c+d x)}{2 d}-\frac {(15 A-2 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{6 d}-\frac {(9 A-4 C) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{3 d}+\frac {2 a A (a+a \cos (c+d x))^3 \tan (c+d x)}{d}+\frac {A (a+a \cos (c+d x))^4 \sec (c+d x) \tan (c+d x)}{2 d}+\frac {\int \left (6 a^5 (13 A+2 C)+24 a^5 (2 A+3 C) \cos (c+d x)\right ) \sec (c+d x) \, dx}{12 a}\\ &=2 a^4 (2 A+3 C) x-\frac {5 a^4 (A-2 C) \sin (c+d x)}{2 d}-\frac {(15 A-2 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{6 d}-\frac {(9 A-4 C) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{3 d}+\frac {2 a A (a+a \cos (c+d x))^3 \tan (c+d x)}{d}+\frac {A (a+a \cos (c+d x))^4 \sec (c+d x) \tan (c+d x)}{2 d}+\frac {1}{2} \left (a^4 (13 A+2 C)\right ) \int \sec (c+d x) \, dx\\ &=2 a^4 (2 A+3 C) x+\frac {a^4 (13 A+2 C) \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac {5 a^4 (A-2 C) \sin (c+d x)}{2 d}-\frac {(15 A-2 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{6 d}-\frac {(9 A-4 C) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{3 d}+\frac {2 a A (a+a \cos (c+d x))^3 \tan (c+d x)}{d}+\frac {A (a+a \cos (c+d x))^4 \sec (c+d x) \tan (c+d x)}{2 d}\\ \end {align*}
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Mathematica [B] time = 6.24, size = 756, normalized size = 4.06 \[ \frac {1}{8} x (2 A+3 C) \sec ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) (a \cos (c+d x)+a)^4+\frac {(4 A+27 C) \sin (c) \cos (d x) \sec ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) (a \cos (c+d x)+a)^4}{64 d}+\frac {(4 A+27 C) \cos (c) \sin (d x) \sec ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) (a \cos (c+d x)+a)^4}{64 d}+\frac {(-13 A-2 C) \sec ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) (a \cos (c+d x)+a)^4 \log \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}{32 d}+\frac {(13 A+2 C) \sec ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) (a \cos (c+d x)+a)^4 \log \left (\sin \left (\frac {c}{2}+\frac {d x}{2}\right )+\cos \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}{32 d}+\frac {A \sin \left (\frac {d x}{2}\right ) \sec ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) (a \cos (c+d x)+a)^4}{4 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 {A \sin \left (\frac {d x}{2}\right ) \sec ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) (a \cos (c+d x)+a)^4}{4 d \left (\sin \left (\frac {c}{2}\right )+\cos \left (\frac {c}{2}\right )\right ) \left (\sin \left (\frac {c}{2}+\frac {d x}{2}\right )+\cos \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}+\frac {A \sec ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) (a \cos (c+d x)+a)^4}{64 d \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 \sec ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) (a \cos (c+d x)+a)^4}{64 d \left (\sin \left (\frac {c}{2}+\frac {d x}{2}\right )+\cos \left (\frac {c}{2}+\frac {d x}{2}\right )\right )^2}+\frac {C \sin (2 c) \cos (2 d x) \sec ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) (a \cos (c+d x)+a)^4}{16 d}+\frac {C \sin (3 c) \cos (3 d x) \sec ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) (a \cos (c+d x)+a)^4}{192 d}+\frac {C \cos (2 c) \sin (2 d x) \sec ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) (a \cos (c+d x)+a)^4}{16 d}+\frac {C \cos (3 c) \sin (3 d x) \sec ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) (a \cos (c+d x)+a)^4}{192 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 171, normalized size = 0.92 \[ \frac {24 \, {\left (2 \, A + 3 \, C\right )} a^{4} d x \cos \left (d x + c\right )^{2} + 3 \, {\left (13 \, A + 2 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (13 \, A + 2 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (2 \, C a^{4} \cos \left (d x + c\right )^{4} + 12 \, C a^{4} \cos \left (d x + c\right )^{3} + 2 \, {\left (3 \, A + 20 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} + 24 \, A a^{4} \cos \left (d x + c\right ) + 3 \, A a^{4}\right )} \sin \left (d x + c\right )}{12 \, d \cos \left (d x + c\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.64, size = 248, normalized size = 1.33 \[ \frac {12 \, {\left (2 \, A a^{4} + 3 \, C a^{4}\right )} {\left (d x + c\right )} + 3 \, {\left (13 \, A a^{4} + 2 \, C a^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 3 \, {\left (13 \, A a^{4} + 2 \, C a^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {6 \, {\left (7 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 9 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{2}} + \frac {4 \, {\left (3 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 15 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 38 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 27 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.40, size = 190, normalized size = 1.02 \[ \frac {A \,a^{4} \sin \left (d x +c \right )}{d}+\frac {a^{4} C \sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right )}{3 d}+\frac {20 a^{4} C \sin \left (d x +c \right )}{3 d}+4 A \,a^{4} x +\frac {4 A \,a^{4} c}{d}+\frac {2 a^{4} C \cos \left (d x +c \right ) \sin \left (d x +c \right )}{d}+6 a^{4} C x +\frac {6 a^{4} C c}{d}+\frac {13 A \,a^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2 d}+\frac {4 A \,a^{4} \tan \left (d x +c \right )}{d}+\frac {A \,a^{4} \sec \left (d x +c \right ) \tan \left (d x +c \right )}{2 d}+\frac {a^{4} C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 211, normalized size = 1.13 \[ \frac {48 \, {\left (d x + c\right )} A a^{4} - 4 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{4} + 12 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{4} + 48 \, {\left (d x + c\right )} C a^{4} - 3 \, A a^{4} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 36 \, A a^{4} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 6 \, C a^{4} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 12 \, A a^{4} \sin \left (d x + c\right ) + 72 \, C a^{4} \sin \left (d x + c\right ) + 48 \, A a^{4} \tan \left (d x + c\right )}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.14, size = 244, normalized size = 1.31 \[ \frac {A\,a^4\,\sin \left (c+d\,x\right )}{d}+\frac {20\,C\,a^4\,\sin \left (c+d\,x\right )}{3\,d}+\frac {8\,A\,a^4\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {13\,A\,a^4\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {12\,C\,a^4\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {2\,C\,a^4\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {4\,A\,a^4\,\sin \left (c+d\,x\right )}{d\,\cos \left (c+d\,x\right )}+\frac {A\,a^4\,\sin \left (c+d\,x\right )}{2\,d\,{\cos \left (c+d\,x\right )}^2}+\frac {C\,a^4\,{\cos \left (c+d\,x\right )}^2\,\sin \left (c+d\,x\right )}{3\,d}+\frac {2\,C\,a^4\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )}{d} \]
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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