Optimal. Leaf size=109 \[ \frac {a \left (a^2-6 b^2\right ) \sin (d+e x) \cos (d+e x)}{6 e}+\frac {1}{2} a x \left (a^2+4 b^2\right )-\frac {a^2 \cos (d+e x) (a+b \sin (d+e x))^2}{3 b e}+\frac {\left (a^4-8 a^2 b^2-3 b^4\right ) \cos (d+e x)}{3 b e} \]
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Rubi [A] time = 0.10, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {3023, 2734} \[ \frac {\left (-8 a^2 b^2+a^4-3 b^4\right ) \cos (d+e x)}{3 b e}+\frac {a \left (a^2-6 b^2\right ) \sin (d+e x) \cos (d+e x)}{6 e}+\frac {1}{2} a x \left (a^2+4 b^2\right )-\frac {a^2 \cos (d+e x) (a+b \sin (d+e x))^2}{3 b e} \]
Antiderivative was successfully verified.
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Rule 2734
Rule 3023
Rubi steps
\begin {align*} \int (a+b \sin (d+e x)) \left (b^2+2 a b \sin (d+e x)+a^2 \sin ^2(d+e x)\right ) \, dx &=-\frac {a^2 \cos (d+e x) (a+b \sin (d+e x))^2}{3 b e}+\frac {\int (a+b \sin (d+e x)) \left (b \left (2 a^2+3 b^2\right )-a \left (a^2-6 b^2\right ) \sin (d+e x)\right ) \, dx}{3 b}\\ &=\frac {1}{2} a \left (a^2+4 b^2\right ) x+\frac {\left (a^4-8 a^2 b^2-3 b^4\right ) \cos (d+e x)}{3 b e}+\frac {a \left (a^2-6 b^2\right ) \cos (d+e x) \sin (d+e x)}{6 e}-\frac {a^2 \cos (d+e x) (a+b \sin (d+e x))^2}{3 b e}\\ \end {align*}
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Mathematica [A] time = 0.29, size = 77, normalized size = 0.71 \[ \frac {a \left (6 \left (a^2+4 b^2\right ) (d+e x)-3 \left (a^2+2 b^2\right ) \sin (2 (d+e x))+a b \cos (3 (d+e x))\right )-3 b \left (11 a^2+4 b^2\right ) \cos (d+e x)}{12 e} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.74, size = 76, normalized size = 0.70 \[ \frac {2 \, a^{2} b \cos \left (e x + d\right )^{3} + 3 \, {\left (a^{3} + 4 \, a b^{2}\right )} e x - 3 \, {\left (a^{3} + 2 \, a b^{2}\right )} \cos \left (e x + d\right ) \sin \left (e x + d\right ) - 6 \, {\left (3 \, a^{2} b + b^{3}\right )} \cos \left (e x + d\right )}{6 \, e} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 79, normalized size = 0.72 \[ \frac {1}{12} \, a^{2} b \cos \left (3 \, x e + 3 \, d\right ) e^{\left (-1\right )} - \frac {1}{4} \, {\left (11 \, a^{2} b + 4 \, b^{3}\right )} \cos \left (x e + d\right ) e^{\left (-1\right )} - \frac {1}{4} \, {\left (a^{3} + 2 \, a b^{2}\right )} e^{\left (-1\right )} \sin \left (2 \, x e + 2 \, d\right ) + \frac {1}{2} \, {\left (a^{3} + 4 \, a b^{2}\right )} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 115, normalized size = 1.06 \[ \frac {-\frac {a^{2} b \left (2+\sin ^{2}\left (e x +d \right )\right ) \cos \left (e x +d \right )}{3}+a^{3} \left (-\frac {\sin \left (e x +d \right ) \cos \left (e x +d \right )}{2}+\frac {e x}{2}+\frac {d}{2}\right )+2 a \,b^{2} \left (-\frac {\sin \left (e x +d \right ) \cos \left (e x +d \right )}{2}+\frac {e x}{2}+\frac {d}{2}\right )-2 \cos \left (e x +d \right ) a^{2} b -b^{3} \cos \left (e x +d \right )+a \,b^{2} \left (e x +d \right )}{e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 112, normalized size = 1.03 \[ \frac {3 \, {\left (2 \, e x + 2 \, d - \sin \left (2 \, e x + 2 \, d\right )\right )} a^{3} + 4 \, {\left (\cos \left (e x + d\right )^{3} - 3 \, \cos \left (e x + d\right )\right )} a^{2} b + 6 \, {\left (2 \, e x + 2 \, d - \sin \left (2 \, e x + 2 \, d\right )\right )} a b^{2} + 12 \, {\left (e x + d\right )} a b^{2} - 24 \, a^{2} b \cos \left (e x + d\right ) - 12 \, b^{3} \cos \left (e x + d\right )}{12 \, e} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.90, size = 88, normalized size = 0.81 \[ -\frac {6\,b^3\,\cos \left (d+e\,x\right )+\frac {3\,a^3\,\sin \left (2\,d+2\,e\,x\right )}{2}-\frac {a^2\,b\,\cos \left (3\,d+3\,e\,x\right )}{2}+3\,a\,b^2\,\sin \left (2\,d+2\,e\,x\right )+\frac {33\,a^2\,b\,\cos \left (d+e\,x\right )}{2}-3\,a^3\,e\,x-12\,a\,b^2\,e\,x}{6\,e} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.67, size = 204, normalized size = 1.87 \[ \begin {cases} \frac {a^{3} x \sin ^{2}{\left (d + e x \right )}}{2} + \frac {a^{3} x \cos ^{2}{\left (d + e x \right )}}{2} - \frac {a^{3} \sin {\left (d + e x \right )} \cos {\left (d + e x \right )}}{2 e} - \frac {a^{2} b \sin ^{2}{\left (d + e x \right )} \cos {\left (d + e x \right )}}{e} - \frac {2 a^{2} b \cos ^{3}{\left (d + e x \right )}}{3 e} - \frac {2 a^{2} b \cos {\left (d + e x \right )}}{e} + a b^{2} x \sin ^{2}{\left (d + e x \right )} + a b^{2} x \cos ^{2}{\left (d + e x \right )} + a b^{2} x - \frac {a b^{2} \sin {\left (d + e x \right )} \cos {\left (d + e x \right )}}{e} - \frac {b^{3} \cos {\left (d + e x \right )}}{e} & \text {for}\: e \neq 0 \\x \left (a + b \sin {\relax (d )}\right ) \left (a^{2} \sin ^{2}{\relax (d )} + 2 a b \sin {\relax (d )} + b^{2}\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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