Optimal. Leaf size=120 \[ \frac {a^2 (2 A+3 B+2 C) \sin (c+d x)}{2 d}+\frac {1}{2} a^2 x (4 A+3 B+2 C)+\frac {a^2 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac {(3 B+2 C) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )}{6 d}+\frac {C \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d} \]
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Rubi [A] time = 0.34, antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3045, 2976, 2968, 3023, 2735, 3770} \[ \frac {a^2 (2 A+3 B+2 C) \sin (c+d x)}{2 d}+\frac {1}{2} a^2 x (4 A+3 B+2 C)+\frac {a^2 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac {(3 B+2 C) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )}{6 d}+\frac {C \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d} \]
Antiderivative was successfully verified.
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Rule 2735
Rule 2968
Rule 2976
Rule 3023
Rule 3045
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 (c+d x) \, dx &=\frac {C (a+a \cos (c+d x))^2 \sin (c+d x)}{3 d}+\frac {\int (a+a \cos (c+d x))^2 (3 a A+a (3 B+2 C) \cos (c+d x)) \sec (c+d x) \, dx}{3 a}\\ &=\frac {C (a+a \cos (c+d x))^2 \sin (c+d x)}{3 d}+\frac {(3 B+2 C) \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{6 d}+\frac {\int (a+a \cos (c+d x)) \left (6 a^2 A+3 a^2 (2 A+3 B+2 C) \cos (c+d x)\right ) \sec (c+d x) \, dx}{6 a}\\ &=\frac {C (a+a \cos (c+d x))^2 \sin (c+d x)}{3 d}+\frac {(3 B+2 C) \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{6 d}+\frac {\int \left (6 a^3 A+\left (6 a^3 A+3 a^3 (2 A+3 B+2 C)\right ) \cos (c+d x)+3 a^3 (2 A+3 B+2 C) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx}{6 a}\\ &=\frac {a^2 (2 A+3 B+2 C) \sin (c+d x)}{2 d}+\frac {C (a+a \cos (c+d x))^2 \sin (c+d x)}{3 d}+\frac {(3 B+2 C) \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{6 d}+\frac {\int \left (6 a^3 A+3 a^3 (4 A+3 B+2 C) \cos (c+d x)\right ) \sec (c+d x) \, dx}{6 a}\\ &=\frac {1}{2} a^2 (4 A+3 B+2 C) x+\frac {a^2 (2 A+3 B+2 C) \sin (c+d x)}{2 d}+\frac {C (a+a \cos (c+d x))^2 \sin (c+d x)}{3 d}+\frac {(3 B+2 C) \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{6 d}+\left (a^2 A\right ) \int \sec (c+d x) \, dx\\ &=\frac {1}{2} a^2 (4 A+3 B+2 C) x+\frac {a^2 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac {a^2 (2 A+3 B+2 C) \sin (c+d x)}{2 d}+\frac {C (a+a \cos (c+d x))^2 \sin (c+d x)}{3 d}+\frac {(3 B+2 C) \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{6 d}\\ \end {align*}
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Mathematica [A] time = 0.32, size = 121, normalized size = 1.01 \[ \frac {a^2 \left (3 (4 A+8 B+7 C) \sin (c+d x)-12 A \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+12 A \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )+24 A d x+3 (B+2 C) \sin (2 (c+d x))+18 B d x+C \sin (3 (c+d x))+12 C d x\right )}{12 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 108, normalized size = 0.90 \[ \frac {3 \, {\left (4 \, A + 3 \, B + 2 \, C\right )} a^{2} d x + 3 \, A a^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, A a^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) + {\left (2 \, C a^{2} \cos \left (d x + c\right )^{2} + 3 \, {\left (B + 2 \, C\right )} a^{2} \cos \left (d x + c\right ) + 2 \, {\left (3 \, A + 6 \, B + 5 \, C\right )} a^{2}\right )} \sin \left (d x + c\right )}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.49, size = 235, normalized size = 1.96 \[ \frac {6 \, A a^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 6 \, A a^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + 3 \, {\left (4 \, A a^{2} + 3 \, B a^{2} + 2 \, C a^{2}\right )} {\left (d x + c\right )} + \frac {2 \, {\left (6 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 9 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 12 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 24 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 16 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 6 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 15 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 18 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.27, size = 181, normalized size = 1.51 \[ \frac {a^{2} A \sin \left (d x +c \right )}{d}+\frac {B \,a^{2} \cos \left (d x +c \right ) \sin \left (d x +c \right )}{2 d}+\frac {3 a^{2} B x}{2}+\frac {3 B \,a^{2} c}{2 d}+\frac {C \sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right ) a^{2}}{3 d}+\frac {5 a^{2} C \sin \left (d x +c \right )}{3 d}+2 a^{2} A x +\frac {2 A \,a^{2} c}{d}+\frac {2 B \,a^{2} \sin \left (d x +c \right )}{d}+\frac {a^{2} C \cos \left (d x +c \right ) \sin \left (d x +c \right )}{d}+a^{2} C x +\frac {a^{2} C c}{d}+\frac {a^{2} A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 153, normalized size = 1.28 \[ \frac {24 \, {\left (d x + c\right )} A a^{2} + 3 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{2} + 12 \, {\left (d x + c\right )} B a^{2} - 4 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{2} + 6 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{2} + 12 \, A a^{2} \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 12 \, A a^{2} \sin \left (d x + c\right ) + 24 \, B a^{2} \sin \left (d x + c\right ) + 12 \, C a^{2} \sin \left (d x + c\right )}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.63, size = 226, normalized size = 1.88 \[ \frac {A\,a^2\,\sin \left (c+d\,x\right )}{d}+\frac {2\,B\,a^2\,\sin \left (c+d\,x\right )}{d}+\frac {7\,C\,a^2\,\sin \left (c+d\,x\right )}{4\,d}+\frac {4\,A\,a^2\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {3\,B\,a^2\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {2\,C\,a^2\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {B\,a^2\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {C\,a^2\,\sin \left (2\,c+2\,d\,x\right )}{2\,d}+\frac {C\,a^2\,\sin \left (3\,c+3\,d\,x\right )}{12\,d}-\frac {A\,a^2\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,2{}\mathrm {i}}{d} \]
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 \[ a^{2} \left (\int A \sec {\left (c + d x \right )}\, dx + \int 2 A \cos {\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int A \cos ^{2}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int B \cos {\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int 2 B \cos ^{2}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int B \cos ^{3}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int C \cos ^{2}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int 2 C \cos ^{3}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int C \cos ^{4}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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