Optimal. Leaf size=165 \[ \frac {5 a^4 (2 A+B) \sin (c+d x)}{2 d}+\frac {a^4 (A+4 B) \tanh ^{-1}(\sin (c+d x))}{d}-\frac {(8 A-3 B) \sin (c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{6 d}+\frac {1}{2} a^4 x (12 A+13 B)+\frac {(2 A+B) \sin (c+d x) \cos (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{2 d}+\frac {a A \sin (c+d x) \cos ^2(c+d x) (a \sec (c+d x)+a)^3}{3 d} \]
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Rubi [A] time = 0.41, antiderivative size = 165, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {4017, 4018, 3996, 3770} \[ \frac {5 a^4 (2 A+B) \sin (c+d x)}{2 d}+\frac {a^4 (A+4 B) \tanh ^{-1}(\sin (c+d x))}{d}-\frac {(8 A-3 B) \sin (c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{6 d}+\frac {(2 A+B) \sin (c+d x) \cos (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{2 d}+\frac {1}{2} a^4 x (12 A+13 B)+\frac {a A \sin (c+d x) \cos ^2(c+d x) (a \sec (c+d x)+a)^3}{3 d} \]
Antiderivative was successfully verified.
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Rule 3770
Rule 3996
Rule 4017
Rule 4018
Rubi steps
\begin {align*} \int \cos ^3(c+d x) (a+a \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx &=\frac {a A \cos ^2(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{3 d}+\frac {1}{3} \int \cos ^2(c+d x) (a+a \sec (c+d x))^3 (3 a (2 A+B)-a (A-3 B) \sec (c+d x)) \, dx\\ &=\frac {a A \cos ^2(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{3 d}+\frac {(2 A+B) \cos (c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{2 d}+\frac {1}{6} \int \cos (c+d x) (a+a \sec (c+d x))^2 \left (2 a^2 (11 A+9 B)-a^2 (8 A-3 B) \sec (c+d x)\right ) \, dx\\ &=\frac {a A \cos ^2(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{3 d}+\frac {(2 A+B) \cos (c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{2 d}-\frac {(8 A-3 B) \left (a^4+a^4 \sec (c+d x)\right ) \sin (c+d x)}{6 d}+\frac {1}{6} \int \cos (c+d x) (a+a \sec (c+d x)) \left (15 a^3 (2 A+B)+6 a^3 (A+4 B) \sec (c+d x)\right ) \, dx\\ &=\frac {5 a^4 (2 A+B) \sin (c+d x)}{2 d}+\frac {a A \cos ^2(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{3 d}+\frac {(2 A+B) \cos (c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{2 d}-\frac {(8 A-3 B) \left (a^4+a^4 \sec (c+d x)\right ) \sin (c+d x)}{6 d}-\frac {1}{6} \int \left (-3 a^4 (12 A+13 B)-6 a^4 (A+4 B) \sec (c+d x)\right ) \, dx\\ &=\frac {1}{2} a^4 (12 A+13 B) x+\frac {5 a^4 (2 A+B) \sin (c+d x)}{2 d}+\frac {a A \cos ^2(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{3 d}+\frac {(2 A+B) \cos (c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{2 d}-\frac {(8 A-3 B) \left (a^4+a^4 \sec (c+d x)\right ) \sin (c+d x)}{6 d}+\left (a^4 (A+4 B)\right ) \int \sec (c+d x) \, dx\\ &=\frac {1}{2} a^4 (12 A+13 B) x+\frac {a^4 (A+4 B) \tanh ^{-1}(\sin (c+d x))}{d}+\frac {5 a^4 (2 A+B) \sin (c+d x)}{2 d}+\frac {a A \cos ^2(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{3 d}+\frac {(2 A+B) \cos (c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{2 d}-\frac {(8 A-3 B) \left (a^4+a^4 \sec (c+d x)\right ) \sin (c+d x)}{6 d}\\ \end {align*}
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Mathematica [B] time = 2.01, size = 342, normalized size = 2.07 \[ \frac {a^4 \cos ^5(c+d x) \sec ^8\left (\frac {1}{2} (c+d x)\right ) (\sec (c+d x)+1)^4 (A+B \sec (c+d x)) \left (\frac {3 (27 A+16 B) \sin (c) \cos (d x)}{d}+\frac {3 (4 A+B) \sin (2 c) \cos (2 d x)}{d}+\frac {3 (27 A+16 B) \cos (c) \sin (d x)}{d}+\frac {3 (4 A+B) \cos (2 c) \sin (2 d x)}{d}-\frac {12 (A+4 B) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{d}+\frac {12 (A+4 B) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )}{d}+\frac {A \sin (3 c) \cos (3 d x)}{d}+\frac {A \cos (3 c) \sin (3 d x)}{d}+72 A x+\frac {12 B \sin \left (\frac {d x}{2}\right )}{d \left (\cos \left (\frac {c}{2}\right )-\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {12 B \sin \left (\frac {d x}{2}\right )}{d \left (\sin \left (\frac {c}{2}\right )+\cos \left (\frac {c}{2}\right )\right ) \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )}+78 B x\right )}{192 (A \cos (c+d x)+B)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 150, normalized size = 0.91 \[ \frac {3 \, {\left (12 \, A + 13 \, B\right )} a^{4} d x \cos \left (d x + c\right ) + 3 \, {\left (A + 4 \, B\right )} a^{4} \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (A + 4 \, B\right )} a^{4} \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) + {\left (2 \, A a^{4} \cos \left (d x + c\right )^{3} + 3 \, {\left (4 \, A + B\right )} a^{4} \cos \left (d x + c\right )^{2} + 8 \, {\left (5 \, A + 3 \, B\right )} a^{4} \cos \left (d x + c\right ) + 6 \, B a^{4}\right )} \sin \left (d x + c\right )}{6 \, d \cos \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.58, size = 226, normalized size = 1.37 \[ -\frac {\frac {12 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1} - 3 \, {\left (12 \, A a^{4} + 13 \, B a^{4}\right )} {\left (d x + c\right )} - 6 \, {\left (A a^{4} + 4 \, B a^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) + 6 \, {\left (A a^{4} + 4 \, B a^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (30 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 21 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 76 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 48 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 54 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 27 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.14, size = 190, normalized size = 1.15 \[ \frac {A \sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right ) a^{4}}{3 d}+\frac {20 A \,a^{4} \sin \left (d x +c \right )}{3 d}+\frac {a^{4} B \cos \left (d x +c \right ) \sin \left (d x +c \right )}{2 d}+\frac {13 a^{4} B x}{2}+\frac {13 a^{4} B c}{2 d}+\frac {2 A \,a^{4} \cos \left (d x +c \right ) \sin \left (d x +c \right )}{d}+6 A \,a^{4} x +\frac {6 A \,a^{4} c}{d}+\frac {4 a^{4} B \sin \left (d x +c \right )}{d}+\frac {4 a^{4} B \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {A \,a^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {a^{4} B \tan \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 187, normalized size = 1.13 \[ -\frac {4 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{4} - 12 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} - 48 \, {\left (d x + c\right )} A a^{4} - 3 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{4} - 72 \, {\left (d x + c\right )} B a^{4} - 6 \, A a^{4} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 24 \, B a^{4} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 72 \, A a^{4} \sin \left (d x + c\right ) - 48 \, B a^{4} \sin \left (d x + c\right ) - 12 \, B a^{4} \tan \left (d x + c\right )}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.23, size = 242, normalized size = 1.47 \[ \frac {20\,A\,a^4\,\sin \left (c+d\,x\right )}{3\,d}+\frac {4\,B\,a^4\,\sin \left (c+d\,x\right )}{d}+\frac {12\,A\,a^4\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {2\,A\,a^4\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {13\,B\,a^4\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {8\,B\,a^4\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {A\,a^4\,{\cos \left (c+d\,x\right )}^2\,\sin \left (c+d\,x\right )}{3\,d}+\frac {B\,a^4\,\sin \left (c+d\,x\right )}{d\,\cos \left (c+d\,x\right )}+\frac {2\,A\,a^4\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )}{d}+\frac {B\,a^4\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )}{2\,d} \]
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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