Optimal. Leaf size=157 \[ \frac{4 b \left (15 a^2+4 b^2\right ) \sin (d+e x)}{3 e}-\frac{4 a \left (15 a^2+4 b^2\right ) \cos (d+e x)}{3 e}+4 a x \left (5 a^2+3 b^2\right )-\frac{20 \left (a^2 \cos (d+e x)-a b \sin (d+e x)\right ) (a \sin (d+e x)+a+b \cos (d+e x))}{3 e}-\frac{8 (a \cos (d+e x)-b \sin (d+e x)) (a \sin (d+e x)+a+b \cos (d+e x))^2}{3 e} \]
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Rubi [A] time = 0.14, antiderivative size = 157, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3120, 3146, 2637, 2638} \[ \frac{4 b \left (15 a^2+4 b^2\right ) \sin (d+e x)}{3 e}-\frac{4 a \left (15 a^2+4 b^2\right ) \cos (d+e x)}{3 e}+4 a x \left (5 a^2+3 b^2\right )-\frac{20 \left (a^2 \cos (d+e x)-a b \sin (d+e x)\right ) (a \sin (d+e x)+a+b \cos (d+e x))}{3 e}-\frac{8 (a \cos (d+e x)-b \sin (d+e x)) (a \sin (d+e x)+a+b \cos (d+e x))^2}{3 e} \]
Antiderivative was successfully verified.
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Rule 3120
Rule 3146
Rule 2637
Rule 2638
Rubi steps
\begin{align*} \int (2 a+2 b \cos (d+e x)+2 a \sin (d+e x))^3 \, dx &=-\frac{8 (a+b \cos (d+e x)+a \sin (d+e x))^2 (a \cos (d+e x)-b \sin (d+e x))}{3 e}+\frac{1}{3} \int (2 a+2 b \cos (d+e x)+2 a \sin (d+e x)) \left (4 \left (5 a^2+2 b^2\right )+20 a b \cos (d+e x)+20 a^2 \sin (d+e x)\right ) \, dx\\ &=-\frac{8 (a+b \cos (d+e x)+a \sin (d+e x))^2 (a \cos (d+e x)-b \sin (d+e x))}{3 e}-\frac{20 (a+b \cos (d+e x)+a \sin (d+e x)) \left (a^2 \cos (d+e x)-a b \sin (d+e x)\right )}{3 e}+\frac{\int \left (48 a^2 \left (5 a^2+3 b^2\right )+16 a b \left (15 a^2+4 b^2\right ) \cos (d+e x)+16 a^2 \left (15 a^2+4 b^2\right ) \sin (d+e x)\right ) \, dx}{12 a}\\ &=4 a \left (5 a^2+3 b^2\right ) x-\frac{8 (a+b \cos (d+e x)+a \sin (d+e x))^2 (a \cos (d+e x)-b \sin (d+e x))}{3 e}-\frac{20 (a+b \cos (d+e x)+a \sin (d+e x)) \left (a^2 \cos (d+e x)-a b \sin (d+e x)\right )}{3 e}+\frac{1}{3} \left (4 a \left (15 a^2+4 b^2\right )\right ) \int \sin (d+e x) \, dx+\frac{1}{3} \left (4 b \left (15 a^2+4 b^2\right )\right ) \int \cos (d+e x) \, dx\\ &=4 a \left (5 a^2+3 b^2\right ) x-\frac{4 a \left (15 a^2+4 b^2\right ) \cos (d+e x)}{3 e}+\frac{4 b \left (15 a^2+4 b^2\right ) \sin (d+e x)}{3 e}-\frac{8 (a+b \cos (d+e x)+a \sin (d+e x))^2 (a \cos (d+e x)-b \sin (d+e x))}{3 e}-\frac{20 (a+b \cos (d+e x)+a \sin (d+e x)) \left (a^2 \cos (d+e x)-a b \sin (d+e x)\right )}{3 e}\\ \end{align*}
Mathematica [A] time = 0.464093, size = 135, normalized size = 0.86 \[ \frac{2 \left (6 a \left (5 a^2+3 b^2\right ) (d+e x)-9 a \left (a^2-b^2\right ) \sin (2 (d+e x))+9 b \left (5 a^2+b^2\right ) \sin (d+e x)+b \left (b^2-3 a^2\right ) \sin (3 (d+e x))-9 a \left (5 a^2+b^2\right ) \cos (d+e x)+a \left (a^2-3 b^2\right ) \cos (3 (d+e x))-18 a^2 b \cos (2 (d+e x))\right )}{3 e} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.069, size = 177, normalized size = 1.1 \begin{align*} 8\,{\frac{{a}^{3} \left ( ex+d \right ) +3\,\sin \left ( ex+d \right ){a}^{2}b-3\,{a}^{3}\cos \left ( ex+d \right ) +3\,a{b}^{2} \left ( 1/2\,\sin \left ( ex+d \right ) \cos \left ( ex+d \right ) +1/2\,ex+d/2 \right ) -3\, \left ( \cos \left ( ex+d \right ) \right ) ^{2}{a}^{2}b+3\,{a}^{3} \left ( -1/2\,\sin \left ( ex+d \right ) \cos \left ( ex+d \right ) +1/2\,ex+d/2 \right ) +1/3\,{b}^{3} \left ( 2+ \left ( \cos \left ( ex+d \right ) \right ) ^{2} \right ) \sin \left ( ex+d \right ) - \left ( \cos \left ( ex+d \right ) \right ) ^{3}a{b}^{2}+{a}^{2}b \left ( \sin \left ( ex+d \right ) \right ) ^{3}-1/3\,{a}^{3} \left ( 2+ \left ( \sin \left ( ex+d \right ) \right ) ^{2} \right ) \cos \left ( ex+d \right ) }{e}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.996859, size = 258, normalized size = 1.64 \begin{align*} -\frac{8 \, a b^{2} \cos \left (e x + d\right )^{3}}{e} + \frac{8 \, a^{2} b \sin \left (e x + d\right )^{3}}{e} + 8 \, a^{3} x + \frac{8 \,{\left (\cos \left (e x + d\right )^{3} - 3 \, \cos \left (e x + d\right )\right )} a^{3}}{3 \, e} - \frac{8 \,{\left (\sin \left (e x + d\right )^{3} - 3 \, \sin \left (e x + d\right )\right )} b^{3}}{3 \, e} - 24 \, a^{2}{\left (\frac{a \cos \left (e x + d\right )}{e} - \frac{b \sin \left (e x + d\right )}{e}\right )} - 6 \,{\left (\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}\right )} a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.30833, size = 293, normalized size = 1.87 \begin{align*} -\frac{4 \,{\left (18 \, a^{2} b \cos \left (e x + d\right )^{2} + 24 \, a^{3} \cos \left (e x + d\right ) - 2 \,{\left (a^{3} - 3 \, a b^{2}\right )} \cos \left (e x + d\right )^{3} - 3 \,{\left (5 \, a^{3} + 3 \, a b^{2}\right )} e x -{\left (24 \, a^{2} b + 4 \, b^{3} - 2 \,{\left (3 \, a^{2} b - b^{3}\right )} \cos \left (e x + d\right )^{2} - 9 \,{\left (a^{3} - a b^{2}\right )} \cos \left (e x + d\right )\right )} \sin \left (e x + d\right )\right )}}{3 \, e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.935438, size = 291, normalized size = 1.85 \begin{align*} \begin{cases} 12 a^{3} x \sin ^{2}{\left (d + e x \right )} + 12 a^{3} x \cos ^{2}{\left (d + e x \right )} + 8 a^{3} x - \frac{8 a^{3} \sin ^{2}{\left (d + e x \right )} \cos{\left (d + e x \right )}}{e} - \frac{12 a^{3} \sin{\left (d + e x \right )} \cos{\left (d + e x \right )}}{e} - \frac{16 a^{3} \cos ^{3}{\left (d + e x \right )}}{3 e} - \frac{24 a^{3} \cos{\left (d + e x \right )}}{e} + \frac{8 a^{2} b \sin ^{3}{\left (d + e x \right )}}{e} + \frac{24 a^{2} b \sin ^{2}{\left (d + e x \right )}}{e} + \frac{24 a^{2} b \sin{\left (d + e x \right )}}{e} + 12 a b^{2} x \sin ^{2}{\left (d + e x \right )} + 12 a b^{2} x \cos ^{2}{\left (d + e x \right )} + \frac{12 a b^{2} \sin{\left (d + e x \right )} \cos{\left (d + e x \right )}}{e} - \frac{8 a b^{2} \cos ^{3}{\left (d + e x \right )}}{e} + \frac{16 b^{3} \sin ^{3}{\left (d + e x \right )}}{3 e} + \frac{8 b^{3} \sin{\left (d + e x \right )} \cos ^{2}{\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 )^{3} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10934, size = 204, normalized size = 1.3 \begin{align*} -12 \, a^{2} b \cos \left (2 \, x e + 2 \, d\right ) e^{\left (-1\right )} + \frac{2}{3} \,{\left (a^{3} - 3 \, a b^{2}\right )} \cos \left (3 \, x e + 3 \, d\right ) e^{\left (-1\right )} - 6 \,{\left (5 \, a^{3} + a b^{2}\right )} \cos \left (x e + d\right ) e^{\left (-1\right )} - \frac{2}{3} \,{\left (3 \, a^{2} b - b^{3}\right )} e^{\left (-1\right )} \sin \left (3 \, x e + 3 \, d\right ) - 6 \,{\left (a^{3} - a b^{2}\right )} e^{\left (-1\right )} \sin \left (2 \, x e + 2 \, d\right ) + 6 \,{\left (5 \, a^{2} b + b^{3}\right )} e^{\left (-1\right )} \sin \left (x e + d\right ) + 4 \,{\left (5 \, a^{3} + 3 \, a b^{2}\right )} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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