Optimal. Leaf size=182 \[ \frac {a^2 (c+d x)^3}{3 d}+\frac {4 a b d (c+d x) \sin (e+f x)}{f^2}-\frac {2 a b (c+d x)^2 \cos (e+f x)}{f}+\frac {4 a b d^2 \cos (e+f x)}{f^3}+\frac {b^2 d (c+d x) \sin ^2(e+f x)}{2 f^2}-\frac {b^2 (c+d x)^2 \sin (e+f x) \cos (e+f x)}{2 f}+\frac {b^2 (c+d x)^3}{6 d}+\frac {b^2 d^2 \sin (e+f x) \cos (e+f x)}{4 f^3}-\frac {b^2 d^2 x}{4 f^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.19, antiderivative size = 182, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {3317, 3296, 2638, 3311, 32, 2635, 8} \[ \frac {a^2 (c+d x)^3}{3 d}+\frac {4 a b d (c+d x) \sin (e+f x)}{f^2}-\frac {2 a b (c+d x)^2 \cos (e+f x)}{f}+\frac {4 a b d^2 \cos (e+f x)}{f^3}+\frac {b^2 d (c+d x) \sin ^2(e+f x)}{2 f^2}-\frac {b^2 (c+d x)^2 \sin (e+f x) \cos (e+f x)}{2 f}+\frac {b^2 (c+d x)^3}{6 d}+\frac {b^2 d^2 \sin (e+f x) \cos (e+f x)}{4 f^3}-\frac {b^2 d^2 x}{4 f^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 32
Rule 2635
Rule 2638
Rule 3296
Rule 3311
Rule 3317
Rubi steps
\begin {align*} \int (c+d x)^2 (a+b \sin (e+f x))^2 \, dx &=\int \left (a^2 (c+d x)^2+2 a b (c+d x)^2 \sin (e+f x)+b^2 (c+d x)^2 \sin ^2(e+f x)\right ) \, dx\\ &=\frac {a^2 (c+d x)^3}{3 d}+(2 a b) \int (c+d x)^2 \sin (e+f x) \, dx+b^2 \int (c+d x)^2 \sin ^2(e+f x) \, dx\\ &=\frac {a^2 (c+d x)^3}{3 d}-\frac {2 a b (c+d x)^2 \cos (e+f x)}{f}-\frac {b^2 (c+d x)^2 \cos (e+f x) \sin (e+f x)}{2 f}+\frac {b^2 d (c+d x) \sin ^2(e+f x)}{2 f^2}+\frac {1}{2} b^2 \int (c+d x)^2 \, dx-\frac {\left (b^2 d^2\right ) \int \sin ^2(e+f x) \, dx}{2 f^2}+\frac {(4 a b d) \int (c+d x) \cos (e+f x) \, dx}{f}\\ &=\frac {a^2 (c+d x)^3}{3 d}+\frac {b^2 (c+d x)^3}{6 d}-\frac {2 a b (c+d x)^2 \cos (e+f x)}{f}+\frac {4 a b d (c+d x) \sin (e+f x)}{f^2}+\frac {b^2 d^2 \cos (e+f x) \sin (e+f x)}{4 f^3}-\frac {b^2 (c+d x)^2 \cos (e+f x) \sin (e+f x)}{2 f}+\frac {b^2 d (c+d x) \sin ^2(e+f x)}{2 f^2}-\frac {\left (4 a b d^2\right ) \int \sin (e+f x) \, dx}{f^2}-\frac {\left (b^2 d^2\right ) \int 1 \, dx}{4 f^2}\\ &=-\frac {b^2 d^2 x}{4 f^2}+\frac {a^2 (c+d x)^3}{3 d}+\frac {b^2 (c+d x)^3}{6 d}+\frac {4 a b d^2 \cos (e+f x)}{f^3}-\frac {2 a b (c+d x)^2 \cos (e+f x)}{f}+\frac {4 a b d (c+d x) \sin (e+f x)}{f^2}+\frac {b^2 d^2 \cos (e+f x) \sin (e+f x)}{4 f^3}-\frac {b^2 (c+d x)^2 \cos (e+f x) \sin (e+f x)}{2 f}+\frac {b^2 d (c+d x) \sin ^2(e+f x)}{2 f^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.90, size = 249, normalized size = 1.37 \[ \frac {24 a^2 c^2 f^3 x+24 a^2 c d f^3 x^2+8 a^2 d^2 f^3 x^3-48 a b \left (c^2 f^2+2 c d f^2 x+d^2 \left (f^2 x^2-2\right )\right ) \cos (e+f x)+96 a b c d f \sin (e+f x)+96 a b d^2 f x \sin (e+f x)-6 b^2 c^2 f^2 \sin (2 (e+f x))+12 b^2 c^2 f^3 x-12 b^2 c d f^2 x \sin (2 (e+f x))-6 b^2 d f (c+d x) \cos (2 (e+f x))+12 b^2 c d f^3 x^2-6 b^2 d^2 f^2 x^2 \sin (2 (e+f x))+3 b^2 d^2 \sin (2 (e+f x))+4 b^2 d^2 f^3 x^3}{24 f^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.63, size = 226, normalized size = 1.24 \[ \frac {2 \, {\left (2 \, a^{2} + b^{2}\right )} d^{2} f^{3} x^{3} + 6 \, {\left (2 \, a^{2} + b^{2}\right )} c d f^{3} x^{2} - 6 \, {\left (b^{2} d^{2} f x + b^{2} c d f\right )} \cos \left (f x + e\right )^{2} + 3 \, {\left (2 \, {\left (2 \, a^{2} + b^{2}\right )} c^{2} f^{3} + b^{2} d^{2} f\right )} x - 24 \, {\left (a b d^{2} f^{2} x^{2} + 2 \, a b c d f^{2} x + a b c^{2} f^{2} - 2 \, a b d^{2}\right )} \cos \left (f x + e\right ) + 3 \, {\left (16 \, a b d^{2} f x + 16 \, a b c d f - {\left (2 \, b^{2} d^{2} f^{2} x^{2} + 4 \, b^{2} c d f^{2} x + 2 \, b^{2} c^{2} f^{2} - b^{2} d^{2}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{12 \, f^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.34, size = 229, normalized size = 1.26 \[ \frac {1}{3} \, a^{2} d^{2} x^{3} + \frac {1}{6} \, b^{2} d^{2} x^{3} + a^{2} c d x^{2} + \frac {1}{2} \, b^{2} c d x^{2} + a^{2} c^{2} x + \frac {1}{2} \, b^{2} c^{2} x - \frac {{\left (b^{2} d^{2} f x + b^{2} c d f\right )} \cos \left (2 \, f x + 2 \, e\right )}{4 \, f^{3}} - \frac {2 \, {\left (a b d^{2} f^{2} x^{2} + 2 \, a b c d f^{2} x + a b c^{2} f^{2} - 2 \, a b d^{2}\right )} \cos \left (f x + e\right )}{f^{3}} - \frac {{\left (2 \, b^{2} d^{2} f^{2} x^{2} + 4 \, b^{2} c d f^{2} x + 2 \, b^{2} c^{2} f^{2} - b^{2} d^{2}\right )} \sin \left (2 \, f x + 2 \, e\right )}{8 \, f^{3}} + \frac {4 \, {\left (a b d^{2} f x + a b c d f\right )} \sin \left (f x + e\right )}{f^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.04, size = 561, normalized size = 3.08 \[ \frac {\frac {a^{2} d^{2} \left (f x +e \right )^{3}}{3 f^{2}}+\frac {a^{2} c d \left (f x +e \right )^{2}}{f}-\frac {a^{2} d^{2} e \left (f x +e \right )^{2}}{f^{2}}+a^{2} c^{2} \left (f x +e \right )-\frac {2 a^{2} c d e \left (f x +e \right )}{f}+\frac {a^{2} d^{2} e^{2} \left (f x +e \right )}{f^{2}}+\frac {2 a b \,d^{2} \left (-\left (f x +e \right )^{2} \cos \left (f x +e \right )+2 \cos \left (f x +e \right )+2 \left (f x +e \right ) \sin \left (f x +e \right )\right )}{f^{2}}+\frac {4 a b c d \left (\sin \left (f x +e \right )-\left (f x +e \right ) \cos \left (f x +e \right )\right )}{f}-\frac {4 a b \,d^{2} e \left (\sin \left (f x +e \right )-\left (f x +e \right ) \cos \left (f x +e \right )\right )}{f^{2}}-2 a b \,c^{2} \cos \left (f x +e \right )+\frac {4 a b c d e \cos \left (f x +e \right )}{f}-\frac {2 a b \,d^{2} e^{2} \cos \left (f x +e \right )}{f^{2}}+\frac {b^{2} d^{2} \left (\left (f x +e \right )^{2} \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {\left (f x +e \right ) \left (\cos ^{2}\left (f x +e \right )\right )}{2}+\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{4}+\frac {f x}{4}+\frac {e}{4}-\frac {\left (f x +e \right )^{3}}{3}\right )}{f^{2}}+\frac {2 b^{2} c d \left (\left (f x +e \right ) \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {\left (f x +e \right )^{2}}{4}+\frac {\left (\sin ^{2}\left (f x +e \right )\right )}{4}\right )}{f}-\frac {2 b^{2} d^{2} e \left (\left (f x +e \right ) \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {\left (f x +e \right )^{2}}{4}+\frac {\left (\sin ^{2}\left (f x +e \right )\right )}{4}\right )}{f^{2}}+b^{2} c^{2} \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {2 b^{2} c d e \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )}{f}+\frac {b^{2} d^{2} e^{2} \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )}{f^{2}}}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.44, size = 502, normalized size = 2.76 \[ \frac {24 \, {\left (f x + e\right )} a^{2} c^{2} + 6 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} b^{2} c^{2} + \frac {8 \, {\left (f x + e\right )}^{3} a^{2} d^{2}}{f^{2}} - \frac {24 \, {\left (f x + e\right )}^{2} a^{2} d^{2} e}{f^{2}} + \frac {24 \, {\left (f x + e\right )} a^{2} d^{2} e^{2}}{f^{2}} + \frac {6 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} b^{2} d^{2} e^{2}}{f^{2}} + \frac {24 \, {\left (f x + e\right )}^{2} a^{2} c d}{f} - \frac {48 \, {\left (f x + e\right )} a^{2} c d e}{f} - \frac {12 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} b^{2} c d e}{f} - 48 \, a b c^{2} \cos \left (f x + e\right ) - \frac {48 \, a b d^{2} e^{2} \cos \left (f x + e\right )}{f^{2}} + \frac {96 \, a b c d e \cos \left (f x + e\right )}{f} + \frac {96 \, {\left ({\left (f x + e\right )} \cos \left (f x + e\right ) - \sin \left (f x + e\right )\right )} a b d^{2} e}{f^{2}} - \frac {6 \, {\left (2 \, {\left (f x + e\right )}^{2} - 2 \, {\left (f x + e\right )} \sin \left (2 \, f x + 2 \, e\right ) - \cos \left (2 \, f x + 2 \, e\right )\right )} b^{2} d^{2} e}{f^{2}} - \frac {96 \, {\left ({\left (f x + e\right )} \cos \left (f x + e\right ) - \sin \left (f x + e\right )\right )} a b c d}{f} + \frac {6 \, {\left (2 \, {\left (f x + e\right )}^{2} - 2 \, {\left (f x + e\right )} \sin \left (2 \, f x + 2 \, e\right ) - \cos \left (2 \, f x + 2 \, e\right )\right )} b^{2} c d}{f} - \frac {48 \, {\left ({\left ({\left (f x + e\right )}^{2} - 2\right )} \cos \left (f x + e\right ) - 2 \, {\left (f x + e\right )} \sin \left (f x + e\right )\right )} a b d^{2}}{f^{2}} + \frac {{\left (4 \, {\left (f x + e\right )}^{3} - 6 \, {\left (f x + e\right )} \cos \left (2 \, f x + 2 \, e\right ) - 3 \, {\left (2 \, {\left (f x + e\right )}^{2} - 1\right )} \sin \left (2 \, f x + 2 \, e\right )\right )} b^{2} d^{2}}{f^{2}}}{24 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.15, size = 281, normalized size = 1.54 \[ a^2\,c^2\,x+\frac {b^2\,c^2\,x}{2}+\frac {a^2\,d^2\,x^3}{3}+\frac {b^2\,d^2\,x^3}{6}-\frac {b^2\,c^2\,\sin \left (2\,e+2\,f\,x\right )}{4\,f}+\frac {b^2\,d^2\,\sin \left (2\,e+2\,f\,x\right )}{8\,f^3}+a^2\,c\,d\,x^2+\frac {b^2\,c\,d\,x^2}{2}-\frac {2\,a\,b\,c^2\,\cos \left (e+f\,x\right )}{f}+\frac {4\,a\,b\,d^2\,\cos \left (e+f\,x\right )}{f^3}-\frac {b^2\,d^2\,x^2\,\sin \left (2\,e+2\,f\,x\right )}{4\,f}-\frac {b^2\,c\,d\,\cos \left (2\,e+2\,f\,x\right )}{4\,f^2}-\frac {b^2\,d^2\,x\,\cos \left (2\,e+2\,f\,x\right )}{4\,f^2}+\frac {4\,a\,b\,c\,d\,\sin \left (e+f\,x\right )}{f^2}+\frac {4\,a\,b\,d^2\,x\,\sin \left (e+f\,x\right )}{f^2}-\frac {2\,a\,b\,d^2\,x^2\,\cos \left (e+f\,x\right )}{f}-\frac {b^2\,c\,d\,x\,\sin \left (2\,e+2\,f\,x\right )}{2\,f}-\frac {4\,a\,b\,c\,d\,x\,\cos \left (e+f\,x\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 2.22, size = 456, normalized size = 2.51 \[ \begin {cases} a^{2} c^{2} x + a^{2} c d x^{2} + \frac {a^{2} d^{2} x^{3}}{3} - \frac {2 a b c^{2} \cos {\left (e + f x \right )}}{f} - \frac {4 a b c d x \cos {\left (e + f x \right )}}{f} + \frac {4 a b c d \sin {\left (e + f x \right )}}{f^{2}} - \frac {2 a b d^{2} x^{2} \cos {\left (e + f x \right )}}{f} + \frac {4 a b d^{2} x \sin {\left (e + f x \right )}}{f^{2}} + \frac {4 a b d^{2} \cos {\left (e + f x \right )}}{f^{3}} + \frac {b^{2} c^{2} x \sin ^{2}{\left (e + f x \right )}}{2} + \frac {b^{2} c^{2} x \cos ^{2}{\left (e + f x \right )}}{2} - \frac {b^{2} c^{2} \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} + \frac {b^{2} c d x^{2} \sin ^{2}{\left (e + f x \right )}}{2} + \frac {b^{2} c d x^{2} \cos ^{2}{\left (e + f x \right )}}{2} - \frac {b^{2} c d x \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {b^{2} c d \cos ^{2}{\left (e + f x \right )}}{2 f^{2}} + \frac {b^{2} d^{2} x^{3} \sin ^{2}{\left (e + f x \right )}}{6} + \frac {b^{2} d^{2} x^{3} \cos ^{2}{\left (e + f x \right )}}{6} - \frac {b^{2} d^{2} x^{2} \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} + \frac {b^{2} d^{2} x \sin ^{2}{\left (e + f x \right )}}{4 f^{2}} - \frac {b^{2} d^{2} x \cos ^{2}{\left (e + f x \right )}}{4 f^{2}} + \frac {b^{2} d^{2} \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{4 f^{3}} & \text {for}\: f \neq 0 \\\left (a + b \sin {\relax (e )}\right )^{2} \left (c^{2} x + c d x^{2} + \frac {d^{2} x^{3}}{3}\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________