Optimal. Leaf size=119 \[ \frac {a^2 (3 A-2 C) \sin (c+d x)}{2 d}-\frac {(A-2 C) \sin (c+d x) \left (a^2 \sec (c+d x)+a^2\right )}{2 d}+\frac {1}{2} a^2 x (3 A+2 C)+\frac {2 a^2 C \tanh ^{-1}(\sin (c+d x))}{d}+\frac {A \sin (c+d x) \cos (c+d x) (a \sec (c+d x)+a)^2}{2 d} \]
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Rubi [A] time = 0.29, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {4087, 4018, 3996, 3770} \[ \frac {a^2 (3 A-2 C) \sin (c+d x)}{2 d}-\frac {(A-2 C) \sin (c+d x) \left (a^2 \sec (c+d x)+a^2\right )}{2 d}+\frac {1}{2} a^2 x (3 A+2 C)+\frac {2 a^2 C \tanh ^{-1}(\sin (c+d x))}{d}+\frac {A \sin (c+d x) \cos (c+d x) (a \sec (c+d x)+a)^2}{2 d} \]
Antiderivative was successfully verified.
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Rule 3770
Rule 3996
Rule 4018
Rule 4087
Rubi steps
\begin {align*} \int \cos ^2(c+d x) (a+a \sec (c+d x))^2 \left (A+C \sec ^2(c+d x)\right ) \, dx &=\frac {A \cos (c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{2 d}+\frac {\int \cos (c+d x) (a+a \sec (c+d x))^2 (2 a A-a (A-2 C) \sec (c+d x)) \, dx}{2 a}\\ &=\frac {A \cos (c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{2 d}-\frac {(A-2 C) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{2 d}+\frac {\int \cos (c+d x) (a+a \sec (c+d x)) \left (a^2 (3 A-2 C)+4 a^2 C \sec (c+d x)\right ) \, dx}{2 a}\\ &=\frac {a^2 (3 A-2 C) \sin (c+d x)}{2 d}+\frac {A \cos (c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{2 d}-\frac {(A-2 C) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{2 d}-\frac {\int \left (-a^3 (3 A+2 C)-4 a^3 C \sec (c+d x)\right ) \, dx}{2 a}\\ &=\frac {1}{2} a^2 (3 A+2 C) x+\frac {a^2 (3 A-2 C) \sin (c+d x)}{2 d}+\frac {A \cos (c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{2 d}-\frac {(A-2 C) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{2 d}+\left (2 a^2 C\right ) \int \sec (c+d x) \, dx\\ &=\frac {1}{2} a^2 (3 A+2 C) x+\frac {2 a^2 C \tanh ^{-1}(\sin (c+d x))}{d}+\frac {a^2 (3 A-2 C) \sin (c+d x)}{2 d}+\frac {A \cos (c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{2 d}-\frac {(A-2 C) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{2 d}\\ \end {align*}
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Mathematica [B] time = 1.25, size = 292, normalized size = 2.45 \[ -\frac {a^2 \sec ^2\left (\frac {1}{2} (c+d x)\right ) \left (4 \cos (d x) \left (3 A d x-4 C \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+4 C \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )+2 C d x\right )+4 \cos (2 c+d x) \left (3 A d x-4 C \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+4 C \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )+2 C d x\right )+A \sin (2 c+d x)+8 A \sin (c+2 d x)+8 A \sin (3 c+2 d x)+A \sin (2 c+3 d x)+A \sin (4 c+3 d x)+A \sin (d x)+16 C \sin (d x)\right )}{16 d \left (\cos \left (\frac {c}{2}\right )-\sin \left (\frac {c}{2}\right )\right ) \left (\sin \left (\frac {c}{2}\right )+\cos \left (\frac {c}{2}\right )\right ) \left (\tan \left (\frac {1}{2} (c+d x)\right )-1\right ) \left (\tan \left (\frac {1}{2} (c+d x)\right )+1\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 116, normalized size = 0.97 \[ \frac {{\left (3 \, A + 2 \, C\right )} a^{2} d x \cos \left (d x + c\right ) + 2 \, C a^{2} \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) - 2 \, C a^{2} \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) + {\left (A a^{2} \cos \left (d x + c\right )^{2} + 4 \, A a^{2} \cos \left (d x + c\right ) + 2 \, C a^{2}\right )} \sin \left (d x + c\right )}{2 \, d \cos \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 143, normalized size = 1.20 \[ \frac {4 \, C a^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 4 \, C a^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {4 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1} + {\left (3 \, A a^{2} + 2 \, C a^{2}\right )} {\left (d x + c\right )} + \frac {2 \, {\left (3 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 5 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.90, size = 107, normalized size = 0.90 \[ \frac {a^{2} A \cos \left (d x +c \right ) \sin \left (d x +c \right )}{2 d}+\frac {3 a^{2} A x}{2}+\frac {3 A \,a^{2} c}{2 d}+a^{2} C x +\frac {C \,a^{2} c}{d}+\frac {2 a^{2} A \sin \left (d x +c \right )}{d}+\frac {2 a^{2} C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {a^{2} C \tan \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 101, normalized size = 0.85 \[ \frac {{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{2} + 4 \, {\left (d x + c\right )} A a^{2} + 4 \, {\left (d x + c\right )} C a^{2} + 4 \, C a^{2} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 8 \, A a^{2} \sin \left (d x + c\right ) + 4 \, C a^{2} \tan \left (d x + c\right )}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.64, size = 152, normalized size = 1.28 \[ \frac {2\,A\,a^2\,\sin \left (c+d\,x\right )}{d}+\frac {3\,A\,a^2\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {2\,C\,a^2\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {4\,C\,a^2\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {C\,a^2\,\sin \left (c+d\,x\right )}{d\,\cos \left (c+d\,x\right )}+\frac {A\,a^2\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )}{2\,d} \]
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 \[ a^{2} \left (\int A \cos ^{2}{\left (c + d x \right )}\, dx + \int 2 A \cos ^{2}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int A \cos ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int C \cos ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int 2 C \cos ^{2}{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx + \int C \cos ^{2}{\left (c + d x \right )} \sec ^{4}{\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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