Optimal. Leaf size=156 \[ -\frac{a^2 \left (12 c^2+16 c d+7 d^2\right ) \cos (e+f x)}{6 f}-\frac{a^2 \left (12 c^2+16 c d+7 d^2\right ) \sin (e+f x) \cos (e+f x)}{24 f}+\frac{1}{8} a^2 x \left (12 c^2+16 c d+7 d^2\right )-\frac{d (8 c-d) \cos (e+f x) (a \sin (e+f x)+a)^2}{12 f}-\frac{d^2 \cos (e+f x) (a \sin (e+f x)+a)^3}{4 a f} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.201897, antiderivative size = 156, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {2761, 2751, 2644} \[ -\frac{a^2 \left (12 c^2+16 c d+7 d^2\right ) \cos (e+f x)}{6 f}-\frac{a^2 \left (12 c^2+16 c d+7 d^2\right ) \sin (e+f x) \cos (e+f x)}{24 f}+\frac{1}{8} a^2 x \left (12 c^2+16 c d+7 d^2\right )-\frac{d (8 c-d) \cos (e+f x) (a \sin (e+f x)+a)^2}{12 f}-\frac{d^2 \cos (e+f x) (a \sin (e+f x)+a)^3}{4 a f} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2761
Rule 2751
Rule 2644
Rubi steps
\begin{align*} \int (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^2 \, dx &=-\frac{d^2 \cos (e+f x) (a+a \sin (e+f x))^3}{4 a f}+\frac{\int (a+a \sin (e+f x))^2 \left (a \left (4 c^2+3 d^2\right )+a (8 c-d) d \sin (e+f x)\right ) \, dx}{4 a}\\ &=-\frac{(8 c-d) d \cos (e+f x) (a+a \sin (e+f x))^2}{12 f}-\frac{d^2 \cos (e+f x) (a+a \sin (e+f x))^3}{4 a f}+\frac{1}{12} \left (12 c^2+16 c d+7 d^2\right ) \int (a+a \sin (e+f x))^2 \, dx\\ &=\frac{1}{8} a^2 \left (12 c^2+16 c d+7 d^2\right ) x-\frac{a^2 \left (12 c^2+16 c d+7 d^2\right ) \cos (e+f x)}{6 f}-\frac{a^2 \left (12 c^2+16 c d+7 d^2\right ) \cos (e+f x) \sin (e+f x)}{24 f}-\frac{(8 c-d) d \cos (e+f x) (a+a \sin (e+f x))^2}{12 f}-\frac{d^2 \cos (e+f x) (a+a \sin (e+f x))^3}{4 a f}\\ \end{align*}
Mathematica [A] time = 0.524634, size = 148, normalized size = 0.95 \[ -\frac{a^2 \cos (e+f x) \left (6 \left (12 c^2+16 c d+7 d^2\right ) \sin ^{-1}\left (\frac{\sqrt{1-\sin (e+f x)}}{\sqrt{2}}\right )+\sqrt{\cos ^2(e+f x)} \left (3 \left (4 c^2+16 c d+7 d^2\right ) \sin (e+f x)+16 \left (3 c^2+5 c d+2 d^2\right )+16 d (c+d) \sin ^2(e+f x)+6 d^2 \sin ^3(e+f x)\right )\right )}{24 f \sqrt{\cos ^2(e+f x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.039, size = 219, normalized size = 1.4 \begin{align*}{\frac{1}{f} \left ({a}^{2}{c}^{2} \left ( -{\frac{\sin \left ( fx+e \right ) \cos \left ( fx+e \right ) }{2}}+{\frac{fx}{2}}+{\frac{e}{2}} \right ) -{\frac{2\,{a}^{2}cd \left ( 2+ \left ( \sin \left ( fx+e \right ) \right ) ^{2} \right ) \cos \left ( fx+e \right ) }{3}}+{a}^{2}{d}^{2} \left ( -{\frac{\cos \left ( fx+e \right ) }{4} \left ( \left ( \sin \left ( fx+e \right ) \right ) ^{3}+{\frac{3\,\sin \left ( fx+e \right ) }{2}} \right ) }+{\frac{3\,fx}{8}}+{\frac{3\,e}{8}} \right ) -2\,{a}^{2}{c}^{2}\cos \left ( fx+e \right ) +4\,{a}^{2}cd \left ( -1/2\,\sin \left ( fx+e \right ) \cos \left ( fx+e \right ) +1/2\,fx+e/2 \right ) -{\frac{2\,{a}^{2}{d}^{2} \left ( 2+ \left ( \sin \left ( fx+e \right ) \right ) ^{2} \right ) \cos \left ( fx+e \right ) }{3}}+{a}^{2}{c}^{2} \left ( fx+e \right ) -2\,{a}^{2}cd\cos \left ( fx+e \right ) +{a}^{2}{d}^{2} \left ( -{\frac{\sin \left ( fx+e \right ) \cos \left ( fx+e \right ) }{2}}+{\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.29266, size = 285, normalized size = 1.83 \begin{align*} \frac{24 \,{\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} c^{2} + 96 \,{\left (f x + e\right )} a^{2} c^{2} + 64 \,{\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} a^{2} c d + 96 \,{\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} c d + 64 \,{\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} a^{2} d^{2} + 3 \,{\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} d^{2} + 24 \,{\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} d^{2} - 192 \, a^{2} c^{2} \cos \left (f x + e\right ) - 192 \, a^{2} c d \cos \left (f x + e\right )}{96 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.68755, size = 324, normalized size = 2.08 \begin{align*} \frac{16 \,{\left (a^{2} c d + a^{2} d^{2}\right )} \cos \left (f x + e\right )^{3} + 3 \,{\left (12 \, a^{2} c^{2} + 16 \, a^{2} c d + 7 \, a^{2} d^{2}\right )} f x - 48 \,{\left (a^{2} c^{2} + 2 \, a^{2} c d + a^{2} d^{2}\right )} \cos \left (f x + e\right ) + 3 \,{\left (2 \, a^{2} d^{2} \cos \left (f x + e\right )^{3} -{\left (4 \, a^{2} c^{2} + 16 \, a^{2} c d + 9 \, a^{2} d^{2}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{24 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 1.88581, size = 459, normalized size = 2.94 \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} + 2 a^{2} c d x \sin ^{2}{\left (e + f x \right )} + 2 a^{2} c d x \cos ^{2}{\left (e + f x \right )} - \frac{2 a^{2} c d \sin ^{2}{\left (e + f x \right )} \cos{\left (e + f x \right )}}{f} - \frac{2 a^{2} c d \sin{\left (e + f x \right )} \cos{\left (e + f x \right )}}{f} - \frac{4 a^{2} c d \cos ^{3}{\left (e + f x \right )}}{3 f} - \frac{2 a^{2} c d \cos{\left (e + f x \right )}}{f} + \frac{3 a^{2} d^{2} x \sin ^{4}{\left (e + f x \right )}}{8} + \frac{3 a^{2} d^{2} x \sin ^{2}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{4} + \frac{a^{2} d^{2} x \sin ^{2}{\left (e + f x \right )}}{2} + \frac{3 a^{2} d^{2} x \cos ^{4}{\left (e + f x \right )}}{8} + \frac{a^{2} d^{2} x \cos ^{2}{\left (e + f x \right )}}{2} - \frac{5 a^{2} d^{2} \sin ^{3}{\left (e + f x \right )} \cos{\left (e + f x \right )}}{8 f} - \frac{2 a^{2} d^{2} \sin ^{2}{\left (e + f x \right )} \cos{\left (e + f x \right )}}{f} - \frac{3 a^{2} d^{2} \sin{\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{8 f} - \frac{a^{2} d^{2} \sin{\left (e + f x \right )} \cos{\left (e + f x \right )}}{2 f} - \frac{4 a^{2} d^{2} \cos ^{3}{\left (e + f x \right )}}{3 f} & \text{for}\: f \neq 0 \\x \left (c + d \sin{\left (e \right )}\right )^{2} \left (a \sin{\left (e \right )} + a\right )^{2} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.34071, size = 281, normalized size = 1.8 \begin{align*} -\frac{2 \, a^{2} c d \cos \left (f x + e\right )}{f} + \frac{a^{2} d^{2} \sin \left (4 \, f x + 4 \, e\right )}{32 \, f} - \frac{a^{2} d^{2} \sin \left (2 \, f x + 2 \, e\right )}{4 \, f} + \frac{1}{8} \,{\left (4 \, a^{2} c^{2} + 16 \, a^{2} c d + 3 \, a^{2} d^{2}\right )} x + \frac{1}{2} \,{\left (2 \, a^{2} c^{2} + a^{2} d^{2}\right )} x + \frac{{\left (a^{2} c d + a^{2} d^{2}\right )} \cos \left (3 \, f x + 3 \, e\right )}{6 \, f} - \frac{{\left (4 \, a^{2} c^{2} + 3 \, a^{2} c d + 3 \, a^{2} d^{2}\right )} \cos \left (f x + e\right )}{2 \, f} - \frac{{\left (a^{2} c^{2} + 4 \, a^{2} c d + a^{2} d^{2}\right )} \sin \left (2 \, f x + 2 \, e\right )}{4 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]