Optimal. Leaf size=81 \[ 2 x \left (3 a^2+b^2\right )+\frac{6 a^2 \cos (d+e x)}{e}+\frac{6 a b \sin (d+e x)}{e}+\frac{2 (a (-\sin (d+e x))+a+b \cos (d+e x)) (a \cos (d+e x)+b \sin (d+e x))}{e} \]
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Rubi [A] time = 0.0469208, 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} \[ 2 x \left (3 a^2+b^2\right )+\frac{6 a^2 \cos (d+e x)}{e}+\frac{6 a b \sin (d+e x)}{e}+\frac{2 (a (-\sin (d+e x))+a+b \cos (d+e x)) (a \cos (d+e x)+b \sin (d+e x))}{e} \]
Antiderivative was successfully verified.
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Rule 3120
Rule 2637
Rule 2638
Rubi steps
\begin{align*} \int (2 a+2 b \cos (d+e x)-2 a \sin (d+e x))^2 \, dx &=\frac{2 (a+b \cos (d+e x)-a \sin (d+e x)) (a \cos (d+e x)+b \sin (d+e x))}{e}+\frac{1}{2} \int \left (4 \left (3 a^2+b^2\right )+12 a b \cos (d+e x)-12 a^2 \sin (d+e x)\right ) \, dx\\ &=2 \left (3 a^2+b^2\right ) x+\frac{2 (a+b \cos (d+e x)-a \sin (d+e x)) (a \cos (d+e x)+b \sin (d+e x))}{e}-\left (6 a^2\right ) \int \sin (d+e x) \, dx+(6 a b) \int \cos (d+e x) \, dx\\ &=2 \left (3 a^2+b^2\right ) x+\frac{6 a^2 \cos (d+e x)}{e}+\frac{6 a b \sin (d+e x)}{e}+\frac{2 (a+b \cos (d+e x)-a \sin (d+e x)) (a \cos (d+e x)+b \sin (d+e x))}{e}\\ \end{align*}
Mathematica [A] time = 0.1516, size = 92, normalized size = 1.14 \[ 4 \left (\frac{\left (3 a^2+b^2\right ) (d+e x)}{2 e}-\frac{\left (a^2-b^2\right ) \sin (2 (d+e x))}{4 e}+\frac{2 a^2 \cos (d+e x)}{e}+\frac{2 a b \sin (d+e x)}{e}+\frac{a b \cos (2 (d+e x))}{2 e}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.055, size = 100, normalized size = 1.2 \begin{align*} 4\,{\frac{{a}^{2} \left ( ex+d \right ) +2\,ab\sin \left ( ex+d \right ) +2\,{a}^{2}\cos \left ( ex+d \right ) +{b}^{2} \left ( 1/2\,\sin \left ( ex+d \right ) \cos \left ( ex+d \right ) +1/2\,ex+d/2 \right ) + \left ( \cos \left ( ex+d \right ) \right ) ^{2}ab+{a}^{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 = 0.980257, size = 132, normalized size = 1.63 \begin{align*} 4 \, a^{2} x + \frac{4 \, a b \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 )} b^{2}}{e} + 8 \, a{\left (\frac{a \cos \left (e x + d\right )}{e} + \frac{b \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 = 2.14909, size = 161, normalized size = 1.99 \begin{align*} \frac{2 \,{\left (2 \, a b \cos \left (e x + d\right )^{2} +{\left (3 \, a^{2} + b^{2}\right )} e x + 4 \, a^{2} \cos \left (e x + d\right ) +{\left (4 \, a b -{\left (a^{2} - b^{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 = 0.367994, 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} \cos{\left (d + e x \right )}}{e} - \frac{4 a b \sin ^{2}{\left (d + e x \right )}}{e} + \frac{8 a b \sin{\left (d + e x \right )}}{e} + 2 b^{2} x \sin ^{2}{\left (d + e x \right )} + 2 b^{2} x \cos ^{2}{\left (d + e x \right )} + \frac{2 b^{2} \sin{\left (d + e x \right )} \cos{\left (d + e x \right )}}{e} & \text{for}\: e \neq 0 \\x \left (- 2 a \sin{\left (d \right )} + 2 a + 2 b \cos{\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 = 1.12162, size = 107, normalized size = 1.32 \begin{align*} 2 \, a b \cos \left (2 \, x e + 2 \, d\right ) e^{\left (-1\right )} + 8 \, a^{2} \cos \left (x e + d\right ) e^{\left (-1\right )} + 8 \, a b e^{\left (-1\right )} \sin \left (x e + d\right ) -{\left (a^{2} - b^{2}\right )} e^{\left (-1\right )} \sin \left (2 \, x e + 2 \, d\right ) + 2 \,{\left (3 \, a^{2} + b^{2}\right )} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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