Optimal. Leaf size=121 \[ -\frac {a^2 (2 A-2 B-3 C) \sin (c+d x)}{2 d}+\frac {a^2 (2 A+B) \tanh ^{-1}(\sin (c+d x))}{d}+\frac {1}{2} a^2 x (2 A+4 B+3 C)-\frac {(2 A-C) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )}{2 d}+\frac {A \tan (c+d x) (a \cos (c+d x)+a)^2}{d} \]
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Rubi [A] time = 0.38, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.146, Rules used = {3043, 2976, 2968, 3023, 2735, 3770} \[ -\frac {a^2 (2 A-2 B-3 C) \sin (c+d x)}{2 d}+\frac {a^2 (2 A+B) \tanh ^{-1}(\sin (c+d x))}{d}+\frac {1}{2} a^2 x (2 A+4 B+3 C)-\frac {(2 A-C) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )}{2 d}+\frac {A \tan (c+d x) (a \cos (c+d x)+a)^2}{d} \]
Antiderivative was successfully verified.
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Rule 2735
Rule 2968
Rule 2976
Rule 3023
Rule 3043
Rule 3770
Rubi steps
\begin {align*} \int (a+a \cos (c+d x))^2 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx &=\frac {A (a+a \cos (c+d x))^2 \tan (c+d x)}{d}+\frac {\int (a+a \cos (c+d x))^2 (a (2 A+B)-a (2 A-C) \cos (c+d x)) \sec (c+d x) \, dx}{a}\\ &=-\frac {(2 A-C) \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{2 d}+\frac {A (a+a \cos (c+d x))^2 \tan (c+d x)}{d}+\frac {\int (a+a \cos (c+d x)) \left (2 a^2 (2 A+B)-a^2 (2 A-2 B-3 C) \cos (c+d x)\right ) \sec (c+d x) \, dx}{2 a}\\ &=-\frac {(2 A-C) \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{2 d}+\frac {A (a+a \cos (c+d x))^2 \tan (c+d x)}{d}+\frac {\int \left (2 a^3 (2 A+B)+\left (2 a^3 (2 A+B)-a^3 (2 A-2 B-3 C)\right ) \cos (c+d x)-a^3 (2 A-2 B-3 C) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx}{2 a}\\ &=-\frac {a^2 (2 A-2 B-3 C) \sin (c+d x)}{2 d}-\frac {(2 A-C) \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{2 d}+\frac {A (a+a \cos (c+d x))^2 \tan (c+d x)}{d}+\frac {\int \left (2 a^3 (2 A+B)+a^3 (2 A+4 B+3 C) \cos (c+d x)\right ) \sec (c+d x) \, dx}{2 a}\\ &=\frac {1}{2} a^2 (2 A+4 B+3 C) x-\frac {a^2 (2 A-2 B-3 C) \sin (c+d x)}{2 d}-\frac {(2 A-C) \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{2 d}+\frac {A (a+a \cos (c+d x))^2 \tan (c+d x)}{d}+\left (a^2 (2 A+B)\right ) \int \sec (c+d x) \, dx\\ &=\frac {1}{2} a^2 (2 A+4 B+3 C) x+\frac {a^2 (2 A+B) \tanh ^{-1}(\sin (c+d x))}{d}-\frac {a^2 (2 A-2 B-3 C) \sin (c+d x)}{2 d}-\frac {(2 A-C) \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{2 d}+\frac {A (a+a \cos (c+d x))^2 \tan (c+d x)}{d}\\ \end {align*}
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Mathematica [A] time = 0.60, size = 174, normalized size = 1.44 \[ \frac {a^2 \left (4 A \tan (c+d x)-8 A \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+8 A \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )+4 A c+4 A d x+4 (B+2 C) \sin (c+d x)-4 B \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+4 B \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )+8 B c+8 B d x+C \sin (2 (c+d x))+6 c C+6 C d x\right )}{4 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 130, normalized size = 1.07 \[ \frac {{\left (2 \, A + 4 \, B + 3 \, C\right )} a^{2} d x \cos \left (d x + c\right ) + {\left (2 \, A + B\right )} a^{2} \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left (2 \, A + B\right )} a^{2} \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) + {\left (C a^{2} \cos \left (d x + c\right )^{2} + 2 \, {\left (B + 2 \, C\right )} a^{2} \cos \left (d x + c\right ) + 2 \, A a^{2}\right )} \sin \left (d x + c\right )}{2 \, d \cos \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.96, size = 198, normalized size = 1.64 \[ -\frac {\frac {4 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1} - {\left (2 \, A a^{2} + 4 \, B a^{2} + 3 \, C a^{2}\right )} {\left (d x + c\right )} - 2 \, {\left (2 \, A a^{2} + B a^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) + 2 \, {\left (2 \, A a^{2} + B a^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (2 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 5 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.28, size = 160, normalized size = 1.32 \[ a^{2} A x +\frac {A \,a^{2} c}{d}+\frac {B \,a^{2} \sin \left (d x +c \right )}{d}+\frac {a^{2} C \cos \left (d x +c \right ) \sin \left (d x +c \right )}{2 d}+\frac {3 a^{2} C x}{2}+\frac {3 a^{2} C c}{2 d}+\frac {2 a^{2} A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+2 a^{2} B x +\frac {2 B \,a^{2} c}{d}+\frac {2 a^{2} C \sin \left (d x +c \right )}{d}+\frac {a^{2} A \tan \left (d x +c \right )}{d}+\frac {B \,a^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 151, normalized size = 1.25 \[ \frac {4 \, {\left (d x + c\right )} A a^{2} + 8 \, {\left (d x + c\right )} B a^{2} + {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{2} + 4 \, {\left (d x + c\right )} C a^{2} + 4 \, A a^{2} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 2 \, B a^{2} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 4 \, B a^{2} \sin \left (d x + c\right ) + 8 \, C a^{2} \sin \left (d x + c\right ) + 4 \, A a^{2} \tan \left (d x + c\right )}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.81, size = 232, normalized size = 1.92 \[ \frac {2\,A\,a^2\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )+4\,B\,a^2\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )+3\,C\,a^2\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )-A\,a^2\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,4{}\mathrm {i}-B\,a^2\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,2{}\mathrm {i}}{d}+\frac {\frac {B\,a^2\,\sin \left (2\,c+2\,d\,x\right )}{2}+C\,a^2\,\sin \left (2\,c+2\,d\,x\right )+\frac {C\,a^2\,\sin \left (3\,c+3\,d\,x\right )}{8}+A\,a^2\,\sin \left (c+d\,x\right )+\frac {C\,a^2\,\sin \left (c+d\,x\right )}{8}}{d\,\cos \left (c+d\,x\right )} \]
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 \[ a^{2} \left (\int A \sec ^{2}{\left (c + d x \right )}\, dx + \int 2 A \cos {\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int A \cos ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int B \cos {\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int 2 B \cos ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int B \cos ^{3}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int C \cos ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int 2 C \cos ^{3}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int C \cos ^{4}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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