Optimal. Leaf size=52 \[ \frac{(a C+b B) \sin (c+d x)}{d}+\frac{1}{2} x (a B+2 b C)+\frac{a B \sin (c+d x) \cos (c+d x)}{2 d} \]
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Rubi [A] time = 0.141558, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.132, Rules used = {4072, 3996, 3787, 2637, 8} \[ \frac{(a C+b B) \sin (c+d x)}{d}+\frac{1}{2} x (a B+2 b C)+\frac{a B \sin (c+d x) \cos (c+d x)}{2 d} \]
Antiderivative was successfully verified.
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Rule 4072
Rule 3996
Rule 3787
Rule 2637
Rule 8
Rubi steps
\begin{align*} \int \cos ^3(c+d x) (a+b \sec (c+d x)) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\int \cos ^2(c+d x) (a+b \sec (c+d x)) (B+C \sec (c+d x)) \, dx\\ &=\frac{a B \cos (c+d x) \sin (c+d x)}{2 d}-\frac{1}{2} \int \cos (c+d x) (-2 (b B+a C)-(a B+2 b C) \sec (c+d x)) \, dx\\ &=\frac{a B \cos (c+d x) \sin (c+d x)}{2 d}-(-b B-a C) \int \cos (c+d x) \, dx-\frac{1}{2} (-a B-2 b C) \int 1 \, dx\\ &=\frac{1}{2} (a B+2 b C) x+\frac{(b B+a C) \sin (c+d x)}{d}+\frac{a B \cos (c+d x) \sin (c+d x)}{2 d}\\ \end{align*}
Mathematica [A] time = 0.0866127, size = 51, normalized size = 0.98 \[ \frac{4 (a C+b B) \sin (c+d x)+a B \sin (2 (c+d x))+2 a B c+2 a B d x+4 b C d x}{4 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.053, size = 57, normalized size = 1.1 \begin{align*}{\frac{1}{d} \left ( 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 = 0.954808, size = 74, normalized size = 1.42 \begin{align*} \frac{{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a + 4 \,{\left (d x + c\right )} C b + 4 \, C a \sin \left (d x + c\right ) + 4 \, B b \sin \left (d x + c\right )}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.482725, size = 104, normalized size = 2. \begin{align*} \frac{{\left (B a + 2 \, C b\right )} d x +{\left (B a \cos \left (d x + c\right ) + 2 \, C a + 2 \, B b\right )} \sin \left (d x + c\right )}{2 \, 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 = 1.17867, size = 163, normalized size = 3.13 \begin{align*} \frac{{\left (B a + 2 \, C b\right )}{\left (d x + c\right )} - \frac{2 \,{\left (B a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 2 \, C a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 2 \, B b \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 ) - 2 \, C a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 2 \, 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 )}^{2}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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