Optimal. Leaf size=117 \[ \frac {a^3 (A+3 B) \tanh ^{-1}(\sin (c+d x))}{d}-\frac {(A-2 B) \sin (c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{2 d}+\frac {1}{2} a^3 x (7 A+6 B)+\frac {5 a^3 A \sin (c+d x)}{2 d}+\frac {a A \sin (c+d x) \cos (c+d x) (a \sec (c+d x)+a)^2}{2 d} \]
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Rubi [A] time = 0.26, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {4017, 4018, 3996, 3770} \[ \frac {a^3 (A+3 B) \tanh ^{-1}(\sin (c+d x))}{d}-\frac {(A-2 B) \sin (c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{2 d}+\frac {1}{2} a^3 x (7 A+6 B)+\frac {5 a^3 A \sin (c+d x)}{2 d}+\frac {a A \sin (c+d x) \cos (c+d x) (a \sec (c+d x)+a)^2}{2 d} \]
Antiderivative was successfully verified.
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Rule 3770
Rule 3996
Rule 4017
Rule 4018
Rubi steps
\begin {align*} \int \cos ^2(c+d x) (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx &=\frac {a A \cos (c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{2 d}+\frac {1}{2} \int \cos (c+d x) (a+a \sec (c+d x))^2 (2 a (2 A+B)-a (A-2 B) \sec (c+d x)) \, dx\\ &=\frac {a A \cos (c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{2 d}-\frac {(A-2 B) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{2 d}+\frac {1}{2} \int \cos (c+d x) (a+a \sec (c+d x)) \left (5 a^2 A+2 a^2 (A+3 B) \sec (c+d x)\right ) \, dx\\ &=\frac {5 a^3 A \sin (c+d x)}{2 d}+\frac {a A \cos (c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{2 d}-\frac {(A-2 B) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{2 d}-\frac {1}{2} \int \left (-a^3 (7 A+6 B)-2 a^3 (A+3 B) \sec (c+d x)\right ) \, dx\\ &=\frac {1}{2} a^3 (7 A+6 B) x+\frac {5 a^3 A \sin (c+d x)}{2 d}+\frac {a A \cos (c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{2 d}-\frac {(A-2 B) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{2 d}+\left (a^3 (A+3 B)\right ) \int \sec (c+d x) \, dx\\ &=\frac {1}{2} a^3 (7 A+6 B) x+\frac {a^3 (A+3 B) \tanh ^{-1}(\sin (c+d x))}{d}+\frac {5 a^3 A \sin (c+d x)}{2 d}+\frac {a A \cos (c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{2 d}-\frac {(A-2 B) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{2 d}\\ \end {align*}
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Mathematica [B] time = 5.09, size = 302, normalized size = 2.58 \[ \frac {a^3 \cos ^4(c+d x) \sec ^6\left (\frac {1}{2} (c+d x)\right ) (\sec (c+d x)+1)^3 (A+B \sec (c+d x)) \left (\frac {4 (3 A+B) \sin (c) \cos (d x)}{d}+\frac {4 (3 A+B) \cos (c) \sin (d x)}{d}-\frac {4 (A+3 B) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{d}+\frac {4 (A+3 B) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )}{d}+2 x (7 A+6 B)+\frac {A \sin (2 c) \cos (2 d x)}{d}+\frac {A \cos (2 c) \sin (2 d x)}{d}+\frac {4 B \sin \left (\frac {d x}{2}\right )}{d \left (\cos \left (\frac {c}{2}\right )-\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {4 B \sin \left (\frac {d x}{2}\right )}{d \left (\sin \left (\frac {c}{2}\right )+\cos \left (\frac {c}{2}\right )\right ) \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )}\right )}{32 (A \cos (c+d x)+B)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 127, normalized size = 1.09 \[ \frac {{\left (7 \, A + 6 \, B\right )} a^{3} d x \cos \left (d x + c\right ) + {\left (A + 3 \, B\right )} a^{3} \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left (A + 3 \, B\right )} a^{3} \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) + {\left (A a^{3} \cos \left (d x + c\right )^{2} + 2 \, {\left (3 \, A + B\right )} a^{3} \cos \left (d x + c\right ) + 2 \, B a^{3}\right )} \sin \left (d x + c\right )}{2 \, d \cos \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.34, size = 192, normalized size = 1.64 \[ -\frac {\frac {4 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1} - {\left (7 \, A a^{3} + 6 \, B a^{3}\right )} {\left (d x + c\right )} - 2 \, {\left (A a^{3} + 3 \, B a^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) + 2 \, {\left (A a^{3} + 3 \, B a^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (5 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 7 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.88, size = 145, normalized size = 1.24 \[ \frac {A \,a^{3} \cos \left (d x +c \right ) \sin \left (d x +c \right )}{2 d}+\frac {7 a^{3} A x}{2}+\frac {7 A \,a^{3} c}{2 d}+\frac {a^{3} B \sin \left (d x +c \right )}{d}+\frac {3 a^{3} A \sin \left (d x +c \right )}{d}+3 a^{3} B x +\frac {3 a^{3} B c}{d}+\frac {3 a^{3} B \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {A \,a^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {a^{3} B \tan \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.36, size = 140, normalized size = 1.20 \[ \frac {{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{3} + 12 \, {\left (d x + c\right )} A a^{3} + 12 \, {\left (d x + c\right )} B a^{3} + 2 \, A a^{3} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 6 \, B a^{3} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 12 \, A a^{3} \sin \left (d x + c\right ) + 4 \, B a^{3} \sin \left (d x + c\right ) + 4 \, B a^{3} \tan \left (d x + c\right )}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.08, size = 197, normalized size = 1.68 \[ \frac {3\,A\,a^3\,\sin \left (c+d\,x\right )}{d}+\frac {B\,a^3\,\sin \left (c+d\,x\right )}{d}+\frac {7\,A\,a^3\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {2\,A\,a^3\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {6\,B\,a^3\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {6\,B\,a^3\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {B\,a^3\,\sin \left (c+d\,x\right )}{d\,\cos \left (c+d\,x\right )}+\frac {A\,a^3\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )}{2\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ a^{3} \left (\int A \cos ^{2}{\left (c + d x \right )}\, dx + \int 3 A \cos ^{2}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int 3 A \cos ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int A \cos ^{2}{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx + \int B \cos ^{2}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int 3 B \cos ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int 3 B \cos ^{2}{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx + \int B \cos ^{2}{\left (c + d x \right )} \sec ^{4}{\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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