Optimal. Leaf size=82 \[ \frac {a^2 (2 B+3 C) \sin (c+d x)}{2 d}+\frac {a^2 B \tanh ^{-1}(\sin (c+d x))}{d}+\frac {1}{2} a^2 x (4 B+3 C)+\frac {C \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )}{2 d} \]
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Rubi [A] time = 0.27, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {3029, 2976, 2968, 3023, 2735, 3770} \[ \frac {a^2 (2 B+3 C) \sin (c+d x)}{2 d}+\frac {a^2 B \tanh ^{-1}(\sin (c+d x))}{d}+\frac {1}{2} a^2 x (4 B+3 C)+\frac {C \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )}{2 d} \]
Antiderivative was successfully verified.
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Rule 2735
Rule 2968
Rule 2976
Rule 3023
Rule 3029
Rule 3770
Rubi steps
\begin {align*} \int (a+a \cos (c+d x))^2 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx &=\int (a+a \cos (c+d x))^2 (B+C \cos (c+d x)) \sec (c+d x) \, dx\\ &=\frac {C \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{2 d}+\frac {1}{2} \int (a+a \cos (c+d x)) (2 a B+a (2 B+3 C) \cos (c+d x)) \sec (c+d x) \, dx\\ &=\frac {C \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{2 d}+\frac {1}{2} \int \left (2 a^2 B+\left (2 a^2 B+a^2 (2 B+3 C)\right ) \cos (c+d x)+a^2 (2 B+3 C) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx\\ &=\frac {a^2 (2 B+3 C) \sin (c+d x)}{2 d}+\frac {C \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{2 d}+\frac {1}{2} \int \left (2 a^2 B+a^2 (4 B+3 C) \cos (c+d x)\right ) \sec (c+d x) \, dx\\ &=\frac {1}{2} a^2 (4 B+3 C) x+\frac {a^2 (2 B+3 C) \sin (c+d x)}{2 d}+\frac {C \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{2 d}+\left (a^2 B\right ) \int \sec (c+d x) \, dx\\ &=\frac {1}{2} a^2 (4 B+3 C) x+\frac {a^2 B \tanh ^{-1}(\sin (c+d x))}{d}+\frac {a^2 (2 B+3 C) \sin (c+d x)}{2 d}+\frac {C \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{2 d}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 96, normalized size = 1.17 \[ \frac {a^2 \left (4 (B+2 C) \sin (c+d x)-4 B \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+4 B \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )+8 B d x+C \sin (2 (c+d x))+6 C d x\right )}{4 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 79, normalized size = 0.96 \[ \frac {{\left (4 \, B + 3 \, C\right )} a^{2} d x + B a^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) - B a^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) + {\left (C a^{2} \cos \left (d x + c\right ) + 2 \, {\left (B + 2 \, C\right )} a^{2}\right )} \sin \left (d x + c\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.69, size = 145, normalized size = 1.77 \[ \frac {2 \, B a^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 2 \, B a^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + {\left (4 \, B a^{2} + 3 \, C a^{2}\right )} {\left (d x + c\right )} + \frac {2 \, {\left (2 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 5 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.23, size = 108, normalized size = 1.32 \[ \frac {a^{2} C \cos \left (d x +c \right ) \sin \left (d x +c \right )}{2 d}+\frac {3 a^{2} C x}{2}+\frac {3 a^{2} C c}{2 d}+\frac {B \,a^{2} \sin \left (d x +c \right )}{d}+\frac {2 a^{2} C \sin \left (d x +c \right )}{d}+2 a^{2} B x +\frac {2 B \,a^{2} c}{d}+\frac {B \,a^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.53, size = 101, normalized size = 1.23 \[ \frac {8 \, {\left (d x + c\right )} B a^{2} + {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{2} + 4 \, {\left (d x + c\right )} C a^{2} + 2 \, B a^{2} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 4 \, B a^{2} \sin \left (d x + c\right ) + 8 \, C a^{2} \sin \left (d x + c\right )}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.18, size = 141, normalized size = 1.72 \[ \frac {B\,a^2\,\sin \left (c+d\,x\right )}{d}+\frac {2\,C\,a^2\,\sin \left (c+d\,x\right )}{d}+\frac {4\,B\,a^2\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {2\,B\,a^2\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {3\,C\,a^2\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {C\,a^2\,\sin \left (2\,c+2\,d\,x\right )}{4\,d} \]
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 \[ a^{2} \left (\int B \cos {\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int 2 B \cos ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int B \cos ^{3}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int C \cos ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int 2 C \cos ^{3}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int C \cos ^{4}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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