Optimal. Leaf size=30 \[ \frac{(a \sin (c+d x)+b \sec (c+d x))^{n+1}}{d (n+1)} \]
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Rubi [A] time = 0.0592225, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.023, Rules used = {4385} \[ \frac{(a \sin (c+d x)+b \sec (c+d x))^{n+1}}{d (n+1)} \]
Antiderivative was successfully verified.
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Rule 4385
Rubi steps
\begin{align*} \int (b \sec (c+d x)+a \sin (c+d x))^n (a \cos (c+d x)+b \sec (c+d x) \tan (c+d x)) \, dx &=\frac{(b \sec (c+d x)+a \sin (c+d x))^{1+n}}{d (1+n)}\\ \end{align*}
Mathematica [A] time = 1.12582, size = 51, normalized size = 1.7 \[ \frac{\sec (c+d x) (a \sin (2 (c+d x))+2 b) (a \sin (c+d x)+b \sec (c+d x))^n}{2 d (n+1)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.078, size = 31, normalized size = 1. \begin{align*}{\frac{ \left ( b\sec \left ( dx+c \right ) +a\sin \left ( dx+c \right ) \right ) ^{1+n}}{d \left ( 1+n \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 3.09156, size = 150, normalized size = 5. \begin{align*} \frac{{\left (a \cos \left (d x + c\right ) \sin \left (d x + c\right ) + b\right )} \left (\frac{a \cos \left (d x + c\right ) \sin \left (d x + c\right ) + b}{\cos \left (d x + c\right )}\right )^{n}}{{\left (d n + d\right )} \cos \left (d x + c\right )} \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \sec \left (d x + c\right ) \tan \left (d x + c\right ) + a \cos \left (d x + c\right )\right )}{\left (b \sec \left (d x + c\right ) + a \sin \left (d x + c\right )\right )}^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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