Optimal. Leaf size=162 \[ -\frac {5 a^4 (A-B) \sin (c+d x)}{2 d}+\frac {a^4 (13 A+8 B) \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac {(6 A+B) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{2 d}+\frac {1}{2} a^4 x (8 A+13 B)+\frac {(5 A+2 B) \tan (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{2 d}+\frac {a A \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^3}{2 d} \]
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Rubi [A] time = 0.48, antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {2975, 2976, 2968, 3023, 2735, 3770} \[ -\frac {5 a^4 (A-B) \sin (c+d x)}{2 d}+\frac {a^4 (13 A+8 B) \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac {(6 A+B) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{2 d}+\frac {(5 A+2 B) \tan (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{2 d}+\frac {1}{2} a^4 x (8 A+13 B)+\frac {a A \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^3}{2 d} \]
Antiderivative was successfully verified.
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Rule 2735
Rule 2968
Rule 2975
Rule 2976
Rule 3023
Rule 3770
Rubi steps
\begin {align*} \int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \sec ^3(c+d x) \, dx &=\frac {a A (a+a \cos (c+d x))^3 \sec (c+d x) \tan (c+d x)}{2 d}+\frac {1}{2} \int (a+a \cos (c+d x))^3 (a (5 A+2 B)-2 a (A-B) \cos (c+d x)) \sec ^2(c+d x) \, dx\\ &=\frac {(5 A+2 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \tan (c+d x)}{2 d}+\frac {a A (a+a \cos (c+d x))^3 \sec (c+d x) \tan (c+d x)}{2 d}+\frac {1}{2} \int (a+a \cos (c+d x))^2 \left (a^2 (13 A+8 B)-2 a^2 (6 A+B) \cos (c+d x)\right ) \sec (c+d x) \, dx\\ &=-\frac {(6 A+B) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{2 d}+\frac {(5 A+2 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \tan (c+d x)}{2 d}+\frac {a A (a+a \cos (c+d x))^3 \sec (c+d x) \tan (c+d x)}{2 d}+\frac {1}{4} \int (a+a \cos (c+d x)) \left (2 a^3 (13 A+8 B)-10 a^3 (A-B) \cos (c+d x)\right ) \sec (c+d x) \, dx\\ &=-\frac {(6 A+B) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{2 d}+\frac {(5 A+2 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \tan (c+d x)}{2 d}+\frac {a A (a+a \cos (c+d x))^3 \sec (c+d x) \tan (c+d x)}{2 d}+\frac {1}{4} \int \left (2 a^4 (13 A+8 B)+\left (-10 a^4 (A-B)+2 a^4 (13 A+8 B)\right ) \cos (c+d x)-10 a^4 (A-B) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx\\ &=-\frac {5 a^4 (A-B) \sin (c+d x)}{2 d}-\frac {(6 A+B) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{2 d}+\frac {(5 A+2 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \tan (c+d x)}{2 d}+\frac {a A (a+a \cos (c+d x))^3 \sec (c+d x) \tan (c+d x)}{2 d}+\frac {1}{4} \int \left (2 a^4 (13 A+8 B)+2 a^4 (8 A+13 B) \cos (c+d x)\right ) \sec (c+d x) \, dx\\ &=\frac {1}{2} a^4 (8 A+13 B) x-\frac {5 a^4 (A-B) \sin (c+d x)}{2 d}-\frac {(6 A+B) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{2 d}+\frac {(5 A+2 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \tan (c+d x)}{2 d}+\frac {a A (a+a \cos (c+d x))^3 \sec (c+d x) \tan (c+d x)}{2 d}+\frac {1}{2} \left (a^4 (13 A+8 B)\right ) \int \sec (c+d x) \, dx\\ &=\frac {1}{2} a^4 (8 A+13 B) x+\frac {a^4 (13 A+8 B) \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac {5 a^4 (A-B) \sin (c+d x)}{2 d}-\frac {(6 A+B) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{2 d}+\frac {(5 A+2 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \tan (c+d x)}{2 d}+\frac {a A (a+a \cos (c+d x))^3 \sec (c+d x) \tan (c+d x)}{2 d}\\ \end {align*}
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Mathematica [B] time = 4.67, size = 343, normalized size = 2.12 \[ \frac {1}{64} a^4 (\cos (c+d x)+1)^4 \sec ^8\left (\frac {1}{2} (c+d x)\right ) \left (\frac {4 (A+4 B) \sin (c) \cos (d x)}{d}+\frac {4 (A+4 B) \cos (c) \sin (d x)}{d}+\frac {4 (4 A+B) \sin \left (\frac {d x}{2}\right )}{d \left (\cos \left (\frac {c}{2}\right )-\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {4 (4 A+B) \sin \left (\frac {d x}{2}\right )}{d \left (\sin \left (\frac {c}{2}\right )+\cos \left (\frac {c}{2}\right )\right ) \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )}-\frac {2 (13 A+8 B) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{d}+\frac {2 (13 A+8 B) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )}{d}+2 x (8 A+13 B)+\frac {A}{d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}-\frac {A}{d \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {B \sin (2 c) \cos (2 d x)}{d}+\frac {B \cos (2 c) \sin (2 d x)}{d}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 156, normalized size = 0.96 \[ \frac {2 \, {\left (8 \, A + 13 \, B\right )} a^{4} d x \cos \left (d x + c\right )^{2} + {\left (13 \, A + 8 \, B\right )} a^{4} \cos \left (d x + c\right )^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left (13 \, A + 8 \, B\right )} a^{4} \cos \left (d x + c\right )^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (B a^{4} \cos \left (d x + c\right )^{3} + 2 \, {\left (A + 4 \, B\right )} a^{4} \cos \left (d x + c\right )^{2} + 2 \, {\left (4 \, A + B\right )} a^{4} \cos \left (d x + c\right ) + A a^{4}\right )} \sin \left (d x + c\right )}{4 \, d \cos \left (d x + c\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.94, size = 230, normalized size = 1.42 \[ \frac {{\left (8 \, A a^{4} + 13 \, B a^{4}\right )} {\left (d x + c\right )} + {\left (13 \, A a^{4} + 8 \, B a^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - {\left (13 \, A a^{4} + 8 \, B a^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (5 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 5 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 7 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 7 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 9 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 9 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 11 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 11 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 1\right )}^{2}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.17, size = 182, normalized size = 1.12 \[ \frac {A \,a^{4} \sin \left (d x +c \right )}{d}+\frac {a^{4} B \cos \left (d x +c \right ) \sin \left (d x +c \right )}{2 d}+\frac {13 a^{4} B x}{2}+\frac {13 a^{4} B c}{2 d}+4 A \,a^{4} x +\frac {4 A \,a^{4} c}{d}+\frac {4 a^{4} B \sin \left (d x +c \right )}{d}+\frac {13 A \,a^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2 d}+\frac {4 A \,a^{4} \tan \left (d x +c \right )}{d}+\frac {4 a^{4} B \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {A \,a^{4} \sec \left (d x +c \right ) \tan \left (d x +c \right )}{2 d}+\frac {a^{4} B \tan \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.58, size = 199, normalized size = 1.23 \[ \frac {16 \, {\left (d x + c\right )} A a^{4} + {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{4} + 24 \, {\left (d x + c\right )} B a^{4} - A a^{4} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 12 \, A a^{4} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 8 \, B a^{4} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 4 \, A a^{4} \sin \left (d x + c\right ) + 16 \, B a^{4} \sin \left (d x + c\right ) + 16 \, A a^{4} \tan \left (d x + c\right ) + 4 \, B a^{4} \tan \left (d x + c\right )}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.40, size = 243, normalized size = 1.50 \[ \frac {A\,a^4\,\sin \left (c+d\,x\right )}{d}+\frac {4\,B\,a^4\,\sin \left (c+d\,x\right )}{d}+\frac {8\,A\,a^4\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {13\,A\,a^4\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {13\,B\,a^4\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {8\,B\,a^4\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {4\,A\,a^4\,\sin \left (c+d\,x\right )}{d\,\cos \left (c+d\,x\right )}+\frac {A\,a^4\,\sin \left (c+d\,x\right )}{2\,d\,{\cos \left (c+d\,x\right )}^2}+\frac {B\,a^4\,\sin \left (c+d\,x\right )}{d\,\cos \left (c+d\,x\right )}+\frac {B\,a^4\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )}{2\,d} \]
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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