Optimal. Leaf size=92 \[ \frac{\sin (c+d x) (2 a A+3 a C+3 b B)}{3 d}+\frac{(a B+A b) \sin (c+d x) \cos (c+d x)}{2 d}+\frac{1}{2} x (a B+A b+2 b C)+\frac{a A \sin (c+d x) \cos ^2(c+d x)}{3 d} \]
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Rubi [A] time = 0.179562, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.128, Rules used = {4074, 4047, 2637, 4045, 8} \[ \frac{\sin (c+d x) (2 a A+3 a C+3 b B)}{3 d}+\frac{(a B+A b) \sin (c+d x) \cos (c+d x)}{2 d}+\frac{1}{2} x (a B+A b+2 b C)+\frac{a A \sin (c+d x) \cos ^2(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 4074
Rule 4047
Rule 2637
Rule 4045
Rule 8
Rubi steps
\begin{align*} \int \cos ^3(c+d x) (a+b \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\frac{a A \cos ^2(c+d x) \sin (c+d x)}{3 d}-\frac{1}{3} \int \cos ^2(c+d x) \left (-3 (A b+a B)-(2 a A+3 b B+3 a C) \sec (c+d x)-3 b C \sec ^2(c+d x)\right ) \, dx\\ &=\frac{a A \cos ^2(c+d x) \sin (c+d x)}{3 d}-\frac{1}{3} \int \cos ^2(c+d x) \left (-3 (A b+a B)-3 b C \sec ^2(c+d x)\right ) \, dx-\frac{1}{3} (-2 a A-3 b B-3 a C) \int \cos (c+d x) \, dx\\ &=\frac{(2 a A+3 b B+3 a C) \sin (c+d x)}{3 d}+\frac{(A b+a B) \cos (c+d x) \sin (c+d x)}{2 d}+\frac{a A \cos ^2(c+d x) \sin (c+d x)}{3 d}-\frac{1}{2} (-A b-a B-2 b C) \int 1 \, dx\\ &=\frac{1}{2} (A b+a B+2 b C) x+\frac{(2 a A+3 b B+3 a C) \sin (c+d x)}{3 d}+\frac{(A b+a B) \cos (c+d x) \sin (c+d x)}{2 d}+\frac{a A \cos ^2(c+d x) \sin (c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.183066, size = 85, normalized size = 0.92 \[ \frac{3 \sin (c+d x) (3 a A+4 a C+4 b B)+3 (a B+A b) \sin (2 (c+d x))+a A \sin (3 (c+d x))+6 a B c+6 a B d x+6 A b c+6 A b d x+12 b C d x}{12 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.061, size = 102, normalized size = 1.1 \begin{align*}{\frac{1}{d} \left ({\frac{Aa \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{3}}+Ab \left ({\frac{\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2}}+{\frac{dx}{2}}+{\frac{c}{2}} \right ) +Ba \left ({\frac{\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2}}+{\frac{dx}{2}}+{\frac{c}{2}} \right ) +Bb\sin \left ( dx+c \right ) +aC\sin \left ( dx+c \right ) +Cb \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0218, size = 132, normalized size = 1.43 \begin{align*} -\frac{4 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a - 3 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a - 3 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A b - 12 \,{\left (d x + c\right )} C b - 12 \, C a \sin \left (d x + c\right ) - 12 \, B b \sin \left (d x + c\right )}{12 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.509012, size = 173, normalized size = 1.88 \begin{align*} \frac{3 \,{\left (B a +{\left (A + 2 \, C\right )} b\right )} d x +{\left (2 \, A a \cos \left (d x + c\right )^{2} + 2 \,{\left (2 \, A + 3 \, C\right )} a + 6 \, B b + 3 \,{\left (B a + A b\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.17735, size = 306, normalized size = 3.33 \begin{align*} \frac{3 \,{\left (B a + A b + 2 \, C b\right )}{\left (d x + c\right )} + \frac{2 \,{\left (6 \, A a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 3 \, B a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 6 \, C a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 3 \, A b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 6 \, B b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 4 \, A a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 12 \, C a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 12 \, B b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 6 \, A a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 3 \, B a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 6 \, C a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 3 \, A b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 6 \, B b \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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