Optimal. Leaf size=81 \[ 2 x \left (3 a^2+c^2\right )+\frac{6 a^2 \sin (d+e x)}{e}-\frac{6 a c \cos (d+e x)}{e}-\frac{2 (c \cos (d+e x)-a \sin (d+e x)) (a \cos (d+e x)+a+c \sin (d+e x))}{e} \]
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Rubi [A] time = 0.0495218, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {3120, 2637, 2638} \[ 2 x \left (3 a^2+c^2\right )+\frac{6 a^2 \sin (d+e x)}{e}-\frac{6 a c \cos (d+e x)}{e}-\frac{2 (c \cos (d+e x)-a \sin (d+e x)) (a \cos (d+e x)+a+c \sin (d+e x))}{e} \]
Antiderivative was successfully verified.
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Rule 3120
Rule 2637
Rule 2638
Rubi steps
\begin{align*} \int (2 a+2 a \cos (d+e x)+2 c \sin (d+e x))^2 \, dx &=-\frac{2 (c \cos (d+e x)-a \sin (d+e x)) (a+a \cos (d+e x)+c \sin (d+e x))}{e}+\frac{1}{2} \int \left (4 \left (3 a^2+c^2\right )+12 a^2 \cos (d+e x)+12 a c \sin (d+e x)\right ) \, dx\\ &=2 \left (3 a^2+c^2\right ) x-\frac{2 (c \cos (d+e x)-a \sin (d+e x)) (a+a \cos (d+e x)+c \sin (d+e x))}{e}+\left (6 a^2\right ) \int \cos (d+e x) \, dx+(6 a c) \int \sin (d+e x) \, dx\\ &=2 \left (3 a^2+c^2\right ) x-\frac{6 a c \cos (d+e x)}{e}+\frac{6 a^2 \sin (d+e x)}{e}-\frac{2 (c \cos (d+e x)-a \sin (d+e x)) (a+a \cos (d+e x)+c \sin (d+e x))}{e}\\ \end{align*}
Mathematica [A] time = 0.145591, size = 92, normalized size = 1.14 \[ 4 \left (\frac{\left (3 a^2+c^2\right ) (d+e x)}{2 e}+\frac{\left (a^2-c^2\right ) \sin (2 (d+e x))}{4 e}+\frac{2 a^2 \sin (d+e x)}{e}-\frac{2 a c \cos (d+e x)}{e}-\frac{a c \cos (2 (d+e x))}{2 e}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.054, size = 101, normalized size = 1.3 \begin{align*} 4\,{\frac{{a}^{2} \left ( ex+d \right ) +2\,{a}^{2}\sin \left ( ex+d \right ) -2\,ac\cos \left ( ex+d \right ) +{a}^{2} \left ( 1/2\,\sin \left ( ex+d \right ) \cos \left ( ex+d \right ) +1/2\,ex+d/2 \right ) -ac \left ( \cos \left ( ex+d \right ) \right ) ^{2}+{c}^{2} \left ( -1/2\,\sin \left ( ex+d \right ) \cos \left ( ex+d \right ) +1/2\,ex+d/2 \right ) }{e}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01777, size = 134, normalized size = 1.65 \begin{align*} 4 \, a^{2} x - \frac{4 \, a c \cos \left (e x + d\right )^{2}}{e} + \frac{{\left (2 \, e x + 2 \, d + \sin \left (2 \, e x + 2 \, d\right )\right )} a^{2}}{e} + \frac{{\left (2 \, e x + 2 \, d - \sin \left (2 \, e x + 2 \, d\right )\right )} c^{2}}{e} - 8 \, a{\left (\frac{c \cos \left (e x + d\right )}{e} - \frac{a \sin \left (e x + d\right )}{e}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.08, size = 162, normalized size = 2. \begin{align*} -\frac{2 \,{\left (2 \, a c \cos \left (e x + d\right )^{2} -{\left (3 \, a^{2} + c^{2}\right )} e x + 4 \, a c \cos \left (e x + d\right ) -{\left (4 \, a^{2} +{\left (a^{2} - c^{2}\right )} \cos \left (e x + d\right )\right )} \sin \left (e x + d\right )\right )}}{e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.376701, size = 170, normalized size = 2.1 \begin{align*} \begin{cases} 2 a^{2} x \sin ^{2}{\left (d + e x \right )} + 2 a^{2} x \cos ^{2}{\left (d + e x \right )} + 4 a^{2} x + \frac{2 a^{2} \sin{\left (d + e x \right )} \cos{\left (d + e x \right )}}{e} + \frac{8 a^{2} \sin{\left (d + e x \right )}}{e} + \frac{4 a c \sin ^{2}{\left (d + e x \right )}}{e} - \frac{8 a c \cos{\left (d + e x \right )}}{e} + 2 c^{2} x \sin ^{2}{\left (d + e x \right )} + 2 c^{2} x \cos ^{2}{\left (d + e x \right )} - \frac{2 c^{2} \sin{\left (d + e x \right )} \cos{\left (d + e x \right )}}{e} & \text{for}\: e \neq 0 \\x \left (2 a \cos{\left (d \right )} + 2 a + 2 c \sin{\left (d \right )}\right )^{2} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17128, size = 105, normalized size = 1.3 \begin{align*} -2 \, a c \cos \left (2 \, x e + 2 \, d\right ) e^{\left (-1\right )} - 8 \, a c \cos \left (x e + d\right ) e^{\left (-1\right )} + 8 \, a^{2} e^{\left (-1\right )} \sin \left (x e + d\right ) +{\left (a^{2} - c^{2}\right )} e^{\left (-1\right )} \sin \left (2 \, x e + 2 \, d\right ) + 2 \,{\left (3 \, a^{2} + c^{2}\right )} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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