Optimal. Leaf size=168 \[ \frac{a^2 d (c+d x) \sin ^2(e+f x)}{2 f^2}+\frac{4 a^2 d (c+d x) \sin (e+f x)}{f^2}-\frac{2 a^2 (c+d x)^2 \cos (e+f x)}{f}-\frac{a^2 (c+d x)^2 \sin (e+f x) \cos (e+f x)}{2 f}+\frac{a^2 (c+d x)^3}{2 d}+\frac{4 a^2 d^2 \cos (e+f x)}{f^3}+\frac{a^2 d^2 \sin (e+f x) \cos (e+f x)}{4 f^3}-\frac{a^2 d^2 x}{4 f^2} \]
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Rubi [A] time = 0.19208, antiderivative size = 168, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35, Rules used = {3317, 3296, 2638, 3311, 32, 2635, 8} \[ \frac{a^2 d (c+d x) \sin ^2(e+f x)}{2 f^2}+\frac{4 a^2 d (c+d x) \sin (e+f x)}{f^2}-\frac{2 a^2 (c+d x)^2 \cos (e+f x)}{f}-\frac{a^2 (c+d x)^2 \sin (e+f x) \cos (e+f x)}{2 f}+\frac{a^2 (c+d x)^3}{2 d}+\frac{4 a^2 d^2 \cos (e+f x)}{f^3}+\frac{a^2 d^2 \sin (e+f x) \cos (e+f x)}{4 f^3}-\frac{a^2 d^2 x}{4 f^2} \]
Antiderivative was successfully verified.
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Rule 3317
Rule 3296
Rule 2638
Rule 3311
Rule 32
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int (c+d x)^2 (a+a \sin (e+f x))^2 \, dx &=\int \left (a^2 (c+d x)^2+2 a^2 (c+d x)^2 \sin (e+f x)+a^2 (c+d x)^2 \sin ^2(e+f x)\right ) \, dx\\ &=\frac{a^2 (c+d x)^3}{3 d}+a^2 \int (c+d x)^2 \sin ^2(e+f x) \, dx+\left (2 a^2\right ) \int (c+d x)^2 \sin (e+f x) \, dx\\ &=\frac{a^2 (c+d x)^3}{3 d}-\frac{2 a^2 (c+d x)^2 \cos (e+f x)}{f}-\frac{a^2 (c+d x)^2 \cos (e+f x) \sin (e+f x)}{2 f}+\frac{a^2 d (c+d x) \sin ^2(e+f x)}{2 f^2}+\frac{1}{2} a^2 \int (c+d x)^2 \, dx-\frac{\left (a^2 d^2\right ) \int \sin ^2(e+f x) \, dx}{2 f^2}+\frac{\left (4 a^2 d\right ) \int (c+d x) \cos (e+f x) \, dx}{f}\\ &=\frac{a^2 (c+d x)^3}{2 d}-\frac{2 a^2 (c+d x)^2 \cos (e+f x)}{f}+\frac{4 a^2 d (c+d x) \sin (e+f x)}{f^2}+\frac{a^2 d^2 \cos (e+f x) \sin (e+f x)}{4 f^3}-\frac{a^2 (c+d x)^2 \cos (e+f x) \sin (e+f x)}{2 f}+\frac{a^2 d (c+d x) \sin ^2(e+f x)}{2 f^2}-\frac{\left (a^2 d^2\right ) \int 1 \, dx}{4 f^2}-\frac{\left (4 a^2 d^2\right ) \int \sin (e+f x) \, dx}{f^2}\\ &=-\frac{a^2 d^2 x}{4 f^2}+\frac{a^2 (c+d x)^3}{2 d}+\frac{4 a^2 d^2 \cos (e+f x)}{f^3}-\frac{2 a^2 (c+d x)^2 \cos (e+f x)}{f}+\frac{4 a^2 d (c+d x) \sin (e+f x)}{f^2}+\frac{a^2 d^2 \cos (e+f x) \sin (e+f x)}{4 f^3}-\frac{a^2 (c+d x)^2 \cos (e+f x) \sin (e+f x)}{2 f}+\frac{a^2 d (c+d x) \sin ^2(e+f x)}{2 f^2}\\ \end{align*}
Mathematica [A] time = 0.613693, size = 182, normalized size = 1.08 \[ \frac{a^2 \left (-16 \left (c^2 f^2+2 c d f^2 x+d^2 \left (f^2 x^2-2\right )\right ) \cos (e+f x)-2 c^2 f^2 \sin (2 (e+f x))+12 c^2 f^3 x-4 c d f^2 x \sin (2 (e+f x))+32 c d f \sin (e+f x)-2 d f (c+d x) \cos (2 (e+f x))+12 c d f^3 x^2-2 d^2 f^2 x^2 \sin (2 (e+f x))+32 d^2 f x \sin (e+f x)+d^2 \sin (2 (e+f x))+4 d^2 f^3 x^3\right )}{8 f^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.02, size = 567, normalized size = 3.4 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.02159, size = 686, normalized size = 4.08 \begin{align*} \frac{6 \,{\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} c^{2} + 24 \,{\left (f x + e\right )} a^{2} c^{2} + \frac{8 \,{\left (f x + e\right )}^{3} a^{2} d^{2}}{f^{2}} - \frac{24 \,{\left (f x + e\right )}^{2} a^{2} d^{2} e}{f^{2}} + \frac{6 \,{\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} d^{2} e^{2}}{f^{2}} + \frac{24 \,{\left (f x + e\right )} a^{2} d^{2} e^{2}}{f^{2}} + \frac{24 \,{\left (f x + e\right )}^{2} a^{2} c d}{f} - \frac{12 \,{\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} c d e}{f} - \frac{48 \,{\left (f x + e\right )} a^{2} c d e}{f} - 48 \, a^{2} c^{2} \cos \left (f x + e\right ) - \frac{48 \, a^{2} d^{2} e^{2} \cos \left (f x + e\right )}{f^{2}} + \frac{96 \, a^{2} c d e \cos \left (f x + e\right )}{f} - \frac{6 \,{\left (2 \,{\left (f x + e\right )}^{2} - 2 \,{\left (f x + e\right )} \sin \left (2 \, f x + 2 \, e\right ) - \cos \left (2 \, f x + 2 \, e\right )\right )} a^{2} d^{2} e}{f^{2}} + \frac{96 \,{\left ({\left (f x + e\right )} \cos \left (f x + e\right ) - \sin \left (f x + e\right )\right )} a^{2} d^{2} e}{f^{2}} + \frac{6 \,{\left (2 \,{\left (f x + e\right )}^{2} - 2 \,{\left (f x + e\right )} \sin \left (2 \, f x + 2 \, e\right ) - \cos \left (2 \, f x + 2 \, e\right )\right )} a^{2} c d}{f} - \frac{96 \,{\left ({\left (f x + e\right )} \cos \left (f x + e\right ) - \sin \left (f x + e\right )\right )} a^{2} c d}{f} + \frac{{\left (4 \,{\left (f x + e\right )}^{3} - 6 \,{\left (f x + e\right )} \cos \left (2 \, f x + 2 \, e\right ) - 3 \,{\left (2 \,{\left (f x + e\right )}^{2} - 1\right )} \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} d^{2}}{f^{2}} - \frac{48 \,{\left ({\left ({\left (f x + e\right )}^{2} - 2\right )} \cos \left (f x + e\right ) - 2 \,{\left (f x + e\right )} \sin \left (f x + e\right )\right )} a^{2} d^{2}}{f^{2}}}{24 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.78484, size = 447, normalized size = 2.66 \begin{align*} \frac{2 \, a^{2} d^{2} f^{3} x^{3} + 6 \, a^{2} c d f^{3} x^{2} - 2 \,{\left (a^{2} d^{2} f x + a^{2} c d f\right )} \cos \left (f x + e\right )^{2} +{\left (6 \, a^{2} c^{2} f^{3} + a^{2} d^{2} f\right )} x - 8 \,{\left (a^{2} d^{2} f^{2} x^{2} + 2 \, a^{2} c d f^{2} x + a^{2} c^{2} f^{2} - 2 \, a^{2} d^{2}\right )} \cos \left (f x + e\right ) +{\left (16 \, a^{2} d^{2} f x + 16 \, a^{2} c d f -{\left (2 \, a^{2} d^{2} f^{2} x^{2} + 4 \, a^{2} c d f^{2} x + 2 \, a^{2} c^{2} f^{2} - a^{2} d^{2}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{4 \, f^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.15149, size = 456, normalized size = 2.71 \begin{align*} \begin{cases} \frac{a^{2} c^{2} x \sin ^{2}{\left (e + f x \right )}}{2} + \frac{a^{2} c^{2} x \cos ^{2}{\left (e + f x \right )}}{2} + a^{2} c^{2} x - \frac{a^{2} c^{2} \sin{\left (e + f x \right )} \cos{\left (e + f x \right )}}{2 f} - \frac{2 a^{2} c^{2} \cos{\left (e + f x \right )}}{f} + \frac{a^{2} c d x^{2} \sin ^{2}{\left (e + f x \right )}}{2} + \frac{a^{2} c d x^{2} \cos ^{2}{\left (e + f x \right )}}{2} + a^{2} c d x^{2} - \frac{a^{2} c d x \sin{\left (e + f x \right )} \cos{\left (e + f x \right )}}{f} - \frac{4 a^{2} c d x \cos{\left (e + f x \right )}}{f} + \frac{4 a^{2} c d \sin{\left (e + f x \right )}}{f^{2}} - \frac{a^{2} c d \cos ^{2}{\left (e + f x \right )}}{2 f^{2}} + \frac{a^{2} d^{2} x^{3} \sin ^{2}{\left (e + f x \right )}}{6} + \frac{a^{2} d^{2} x^{3} \cos ^{2}{\left (e + f x \right )}}{6} + \frac{a^{2} d^{2} x^{3}}{3} - \frac{a^{2} d^{2} x^{2} \sin{\left (e + f x \right )} \cos{\left (e + f x \right )}}{2 f} - \frac{2 a^{2} d^{2} x^{2} \cos{\left (e + f x \right )}}{f} + \frac{a^{2} d^{2} x \sin ^{2}{\left (e + f x \right )}}{4 f^{2}} + \frac{4 a^{2} d^{2} x \sin{\left (e + f x \right )}}{f^{2}} - \frac{a^{2} d^{2} x \cos ^{2}{\left (e + f x \right )}}{4 f^{2}} + \frac{a^{2} d^{2} \sin{\left (e + f x \right )} \cos{\left (e + f x \right )}}{4 f^{3}} + \frac{4 a^{2} d^{2} \cos{\left (e + f x \right )}}{f^{3}} & \text{for}\: f \neq 0 \\\left (a \sin{\left (e \right )} + a\right )^{2} \left (c^{2} x + c d x^{2} + \frac{d^{2} x^{3}}{3}\right ) & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14398, size = 279, normalized size = 1.66 \begin{align*} \frac{1}{2} \, a^{2} d^{2} x^{3} + \frac{3}{2} \, a^{2} c d x^{2} + \frac{3}{2} \, a^{2} c^{2} x - \frac{{\left (a^{2} d^{2} f x + a^{2} c d f\right )} \cos \left (2 \, f x + 2 \, e\right )}{4 \, f^{3}} - \frac{2 \,{\left (a^{2} d^{2} f^{2} x^{2} + 2 \, a^{2} c d f^{2} x + a^{2} c^{2} f^{2} - 2 \, a^{2} d^{2}\right )} \cos \left (f x + e\right )}{f^{3}} - \frac{{\left (2 \, a^{2} d^{2} f^{2} x^{2} + 4 \, a^{2} c d f^{2} x + 2 \, a^{2} c^{2} f^{2} - a^{2} d^{2}\right )} \sin \left (2 \, f x + 2 \, e\right )}{8 \, f^{3}} + \frac{4 \,{\left (a^{2} d^{2} f x + a^{2} c d f\right )} \sin \left (f x + e\right )}{f^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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