Optimal. Leaf size=103 \[ \frac {\left (2 C \left (a^2+b^2\right )+3 A b^2\right ) \sin (c+d x)}{3 d}+\frac {a^2 A \tanh ^{-1}(\sin (c+d x))}{d}+a b x (2 A+C)+\frac {a b C \sin (c+d x) \cos (c+d x)}{3 d}+\frac {C \sin (c+d x) (a+b \cos (c+d x))^2}{3 d} \]
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Rubi [A] time = 0.28, antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {3050, 3033, 3023, 2735, 3770} \[ \frac {\left (2 C \left (a^2+b^2\right )+3 A b^2\right ) \sin (c+d x)}{3 d}+\frac {a^2 A \tanh ^{-1}(\sin (c+d x))}{d}+a b x (2 A+C)+\frac {a b C \sin (c+d x) \cos (c+d x)}{3 d}+\frac {C \sin (c+d x) (a+b \cos (c+d x))^2}{3 d} \]
Antiderivative was successfully verified.
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Rule 2735
Rule 3023
Rule 3033
Rule 3050
Rule 3770
Rubi steps
\begin {align*} \int (a+b \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx &=\frac {C (a+b \cos (c+d x))^2 \sin (c+d x)}{3 d}+\frac {1}{3} \int (a+b \cos (c+d x)) \left (3 a A+b (3 A+2 C) \cos (c+d x)+2 a C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx\\ &=\frac {a b C \cos (c+d x) \sin (c+d x)}{3 d}+\frac {C (a+b \cos (c+d x))^2 \sin (c+d x)}{3 d}+\frac {1}{6} \int \left (6 a^2 A+6 a b (2 A+C) \cos (c+d x)+2 \left (3 A b^2+2 \left (a^2+b^2\right ) C\right ) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx\\ &=\frac {\left (3 A b^2+2 \left (a^2+b^2\right ) C\right ) \sin (c+d x)}{3 d}+\frac {a b C \cos (c+d x) \sin (c+d x)}{3 d}+\frac {C (a+b \cos (c+d x))^2 \sin (c+d x)}{3 d}+\frac {1}{6} \int \left (6 a^2 A+6 a b (2 A+C) \cos (c+d x)\right ) \sec (c+d x) \, dx\\ &=a b (2 A+C) x+\frac {\left (3 A b^2+2 \left (a^2+b^2\right ) C\right ) \sin (c+d x)}{3 d}+\frac {a b C \cos (c+d x) \sin (c+d x)}{3 d}+\frac {C (a+b \cos (c+d x))^2 \sin (c+d x)}{3 d}+\left (a^2 A\right ) \int \sec (c+d x) \, dx\\ &=a b (2 A+C) x+\frac {a^2 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac {\left (3 A b^2+2 \left (a^2+b^2\right ) C\right ) \sin (c+d x)}{3 d}+\frac {a b C \cos (c+d x) \sin (c+d x)}{3 d}+\frac {C (a+b \cos (c+d x))^2 \sin (c+d x)}{3 d}\\ \end {align*}
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Mathematica [A] time = 0.25, size = 145, normalized size = 1.41 \[ \frac {3 \left (4 a^2 C+4 A b^2+3 b^2 C\right ) \sin (c+d x)-12 a^2 A \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+12 a^2 A \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )+24 a A b c+24 a A b d x+6 a b C \sin (2 (c+d x))+12 a b c C+12 a b C d x+b^2 C \sin (3 (c+d x))}{12 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.98, size = 99, normalized size = 0.96 \[ \frac {6 \, {\left (2 \, A + C\right )} a b d x + 3 \, A a^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, A a^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (C b^{2} \cos \left (d x + c\right )^{2} + 3 \, C a b \cos \left (d x + c\right ) + 3 \, C a^{2} + {\left (3 \, A + 2 \, C\right )} b^{2}\right )} \sin \left (d x + c\right )}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.93, size = 256, normalized size = 2.49 \[ \frac {3 \, A a^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 3 \, A a^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + 3 \, {\left (2 \, A a b + C a b\right )} {\left (d x + c\right )} + \frac {2 \, {\left (3 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 3 \, C a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3 \, A b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3 \, C b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 6 \, A b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2 \, C b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, C a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, A b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, C b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3}}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.20, size = 137, normalized size = 1.33 \[ \frac {a^{2} A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {a^{2} C \sin \left (d x +c \right )}{d}+2 A x a b +\frac {2 A a b c}{d}+\frac {a b C \cos \left (d x +c \right ) \sin \left (d x +c \right )}{d}+a b C x +\frac {C a b c}{d}+\frac {A \,b^{2} \sin \left (d x +c \right )}{d}+\frac {C \sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right ) b^{2}}{3 d}+\frac {2 b^{2} C \sin \left (d x +c \right )}{3 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.38, size = 105, normalized size = 1.02 \[ \frac {12 \, {\left (d x + c\right )} A a b + 3 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a b - 2 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C b^{2} + 6 \, A a^{2} \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 6 \, C a^{2} \sin \left (d x + c\right ) + 6 \, A b^{2} \sin \left (d x + c\right )}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.64, size = 170, normalized size = 1.65 \[ \frac {A\,b^2\,\sin \left (c+d\,x\right )}{d}+\frac {C\,a^2\,\sin \left (c+d\,x\right )}{d}+\frac {3\,C\,b^2\,\sin \left (c+d\,x\right )}{4\,d}+\frac {2\,A\,a^2\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {C\,b^2\,\sin \left (3\,c+3\,d\,x\right )}{12\,d}+\frac {4\,A\,a\,b\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {2\,C\,a\,b\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {C\,a\,b\,\sin \left (2\,c+2\,d\,x\right )}{2\,d} \]
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 \[ \int \left (A + C \cos ^{2}{\left (c + d x \right )}\right ) \left (a + b \cos {\left (c + d x \right )}\right )^{2} \sec {\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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