Integrand size = 19, antiderivative size = 150 \[ \int (d+e x) \sin ^2\left (a+b x+c x^2\right ) \, dx=\frac {(d+e x)^2}{4 e}-\frac {(2 c d-b e) \sqrt {\pi } \cos \left (2 a-\frac {b^2}{2 c}\right ) \operatorname {FresnelC}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {\pi }}\right )}{8 c^{3/2}}+\frac {(2 c d-b e) \sqrt {\pi } \operatorname {FresnelS}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {\pi }}\right ) \sin \left (2 a-\frac {b^2}{2 c}\right )}{8 c^{3/2}}-\frac {e \sin \left (2 a+2 b x+2 c x^2\right )}{8 c} \]
1/4*(e*x+d)^2/e-1/8*e*sin(2*c*x^2+2*b*x+2*a)/c-1/8*(-b*e+2*c*d)*cos(2*a-1/ 2*b^2/c)*FresnelC((2*c*x+b)/c^(1/2)/Pi^(1/2))*Pi^(1/2)/c^(3/2)+1/8*(-b*e+2 *c*d)*FresnelS((2*c*x+b)/c^(1/2)/Pi^(1/2))*sin(2*a-1/2*b^2/c)*Pi^(1/2)/c^( 3/2)
Time = 0.33 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.93 \[ \int (d+e x) \sin ^2\left (a+b x+c x^2\right ) \, dx=\frac {-\left ((2 c d-b e) \sqrt {\pi } \cos \left (2 a-\frac {b^2}{2 c}\right ) \operatorname {FresnelC}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {\pi }}\right )\right )+(2 c d-b e) \sqrt {\pi } \operatorname {FresnelS}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {\pi }}\right ) \sin \left (2 a-\frac {b^2}{2 c}\right )+\sqrt {c} (2 c x (2 d+e x)-e \sin (2 (a+x (b+c x))))}{8 c^{3/2}} \]
(-((2*c*d - b*e)*Sqrt[Pi]*Cos[2*a - b^2/(2*c)]*FresnelC[(b + 2*c*x)/(Sqrt[ c]*Sqrt[Pi])]) + (2*c*d - b*e)*Sqrt[Pi]*FresnelS[(b + 2*c*x)/(Sqrt[c]*Sqrt [Pi])]*Sin[2*a - b^2/(2*c)] + Sqrt[c]*(2*c*x*(2*d + e*x) - e*Sin[2*(a + x* (b + c*x))]))/(8*c^(3/2))
Time = 0.27 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {3948, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int (d+e x) \sin ^2\left (a+b x+c x^2\right ) \, dx\) |
| \(\Big \downarrow \) 3948 |
| \(\displaystyle \int \left (\frac {1}{2} (d+e x)-\frac {1}{2} (d+e x) \cos \left (2 a+2 b x+2 c x^2\right )\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\sqrt {\pi } \cos \left (2 a-\frac {b^2}{2 c}\right ) (2 c d-b e) \operatorname {FresnelC}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {\pi }}\right )}{8 c^{3/2}}+\frac {\sqrt {\pi } \sin \left (2 a-\frac {b^2}{2 c}\right ) (2 c d-b e) \operatorname {FresnelS}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {\pi }}\right )}{8 c^{3/2}}-\frac {e \sin \left (2 a+2 b x+2 c x^2\right )}{8 c}+\frac {(d+e x)^2}{4 e}\) |
(d + e*x)^2/(4*e) - ((2*c*d - b*e)*Sqrt[Pi]*Cos[2*a - b^2/(2*c)]*FresnelC[ (b + 2*c*x)/(Sqrt[c]*Sqrt[Pi])])/(8*c^(3/2)) + ((2*c*d - b*e)*Sqrt[Pi]*Fre snelS[(b + 2*c*x)/(Sqrt[c]*Sqrt[Pi])]*Sin[2*a - b^2/(2*c)])/(8*c^(3/2)) - (e*Sin[2*a + 2*b*x + 2*c*x^2])/(8*c)
3.1.34.3.1 Defintions of rubi rules used
Int[((d_.) + (e_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]^(n_), x_Symbol] :> Int[ExpandTrigReduce[(d + e*x)^m, Sin[a + b*x + c*x^2]^n, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[n, 1]
Time = 0.44 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.13
| method | result | size |
| default | \(-\frac {e \sin \left (2 c \,x^{2}+2 b x +2 a \right )}{8 c}+\frac {e b \sqrt {\pi }\, \left (\cos \left (\frac {-4 a c +b^{2}}{2 c}\right ) \operatorname {C}\left (\frac {2 c x +b}{\sqrt {c}\, \sqrt {\pi }}\right )+\sin \left (\frac {-4 a c +b^{2}}{2 c}\right ) \operatorname {S}\left (\frac {2 c x +b}{\sqrt {c}\, \sqrt {\pi }}\right )\right )}{8 c^{\frac {3}{2}}}-\frac {\sqrt {\pi }\, d \left (\cos \left (\frac {-4 a c +b^{2}}{2 c}\right ) \operatorname {C}\left (\frac {2 c x +b}{\sqrt {c}\, \sqrt {\pi }}\right )+\sin \left (\frac {-4 a c +b^{2}}{2 c}\right ) \operatorname {S}\left (\frac {2 c x +b}{\sqrt {c}\, \sqrt {\pi }}\right )\right )}{4 \sqrt {c}}+\frac {d x}{2}+\frac {e \,x^{2}}{4}\) | \(170\) |
| risch | \(-\frac {\operatorname {erf}\left (\sqrt {2}\, \sqrt {i c}\, x +\frac {i b \sqrt {2}}{2 \sqrt {i c}}\right ) \sqrt {2}\, \sqrt {\pi }\, d \,{\mathrm e}^{-\frac {i \left (4 a c -b^{2}\right )}{2 c}}}{16 \sqrt {i c}}+\frac {e \,\operatorname {erf}\left (\sqrt {2}\, \sqrt {i c}\, x +\frac {i b \sqrt {2}}{2 \sqrt {i c}}\right ) \sqrt {2}\, \sqrt {\pi }\, b \,{\mathrm e}^{-\frac {i \left (4 a c -b^{2}\right )}{2 c}}}{32 \sqrt {i c}\, c}+\frac {\operatorname {erf}\left (-\sqrt {-2 i c}\, x +\frac {i b}{\sqrt {-2 i c}}\right ) \sqrt {\pi }\, d \,{\mathrm e}^{\frac {i \left (4 a c -b^{2}\right )}{2 c}}}{8 \sqrt {-2 i c}}-\frac {e \,\operatorname {erf}\left (-\sqrt {-2 i c}\, x +\frac {i b}{\sqrt {-2 i c}}\right ) \sqrt {\pi }\, b \,{\mathrm e}^{\frac {i \left (4 a c -b^{2}\right )}{2 c}}}{16 \sqrt {-2 i c}\, c}+\frac {d x}{2}+\frac {e \,x^{2}}{4}-\frac {e \sin \left (2 c \,x^{2}+2 b x +2 a \right )}{8 c}\) | \(257\) |
-1/8*e*sin(2*c*x^2+2*b*x+2*a)/c+1/8*e*b/c^(3/2)*Pi^(1/2)*(cos(1/2*(-4*a*c+ b^2)/c)*FresnelC((2*c*x+b)/c^(1/2)/Pi^(1/2))+sin(1/2*(-4*a*c+b^2)/c)*Fresn elS((2*c*x+b)/c^(1/2)/Pi^(1/2)))-1/4*Pi^(1/2)/c^(1/2)*d*(cos(1/2*(-4*a*c+b ^2)/c)*FresnelC((2*c*x+b)/c^(1/2)/Pi^(1/2))+sin(1/2*(-4*a*c+b^2)/c)*Fresne lS((2*c*x+b)/c^(1/2)/Pi^(1/2)))+1/2*d*x+1/4*e*x^2
Time = 0.31 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.99 \[ \int (d+e x) \sin ^2\left (a+b x+c x^2\right ) \, dx=\frac {2 \, c^{2} e x^{2} - \pi {\left (2 \, c d - b e\right )} \sqrt {\frac {c}{\pi }} \cos \left (-\frac {b^{2} - 4 \, a c}{2 \, c}\right ) \operatorname {C}\left (\frac {{\left (2 \, c x + b\right )} \sqrt {\frac {c}{\pi }}}{c}\right ) + \pi {\left (2 \, c d - b e\right )} \sqrt {\frac {c}{\pi }} \operatorname {S}\left (\frac {{\left (2 \, c x + b\right )} \sqrt {\frac {c}{\pi }}}{c}\right ) \sin \left (-\frac {b^{2} - 4 \, a c}{2 \, c}\right ) + 4 \, c^{2} d x - 2 \, c e \cos \left (c x^{2} + b x + a\right ) \sin \left (c x^{2} + b x + a\right )}{8 \, c^{2}} \]
1/8*(2*c^2*e*x^2 - pi*(2*c*d - b*e)*sqrt(c/pi)*cos(-1/2*(b^2 - 4*a*c)/c)*f resnel_cos((2*c*x + b)*sqrt(c/pi)/c) + pi*(2*c*d - b*e)*sqrt(c/pi)*fresnel _sin((2*c*x + b)*sqrt(c/pi)/c)*sin(-1/2*(b^2 - 4*a*c)/c) + 4*c^2*d*x - 2*c *e*cos(c*x^2 + b*x + a)*sin(c*x^2 + b*x + a))/c^2
\[ \int (d+e x) \sin ^2\left (a+b x+c x^2\right ) \, dx=\int \left (d + e x\right ) \sin ^{2}{\left (a + b x + c x^{2} \right )}\, dx \]
Result contains complex when optimal does not.
Time = 0.58 (sec) , antiderivative size = 738, normalized size of antiderivative = 4.92 \[ \int (d+e x) \sin ^2\left (a+b x+c x^2\right ) \, dx=\frac {{\left (4^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } {\left ({\left (\left (i - 1\right ) \, \cos \left (-\frac {b^{2} - 4 \, a c}{2 \, c}\right ) + \left (i + 1\right ) \, \sin \left (-\frac {b^{2} - 4 \, a c}{2 \, c}\right )\right )} \operatorname {erf}\left (\frac {2 i \, c x + i \, b}{\sqrt {2 i \, c}}\right ) + {\left (\left (i + 1\right ) \, \cos \left (-\frac {b^{2} - 4 \, a c}{2 \, c}\right ) + \left (i - 1\right ) \, \sin \left (-\frac {b^{2} - 4 \, a c}{2 \, c}\right )\right )} \operatorname {erf}\left (\frac {2 i \, c x + i \, b}{\sqrt {-2 i \, c}}\right )\right )} c^{\frac {3}{2}} + 16 \, c^{2} x\right )} d}{32 \, c^{2}} + \frac {\sqrt {2} {\left ({\left (-\left (i - 1\right ) \, \sqrt {2} \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {\frac {1}{2}} \sqrt {\frac {4 i \, c^{2} x^{2} + 4 i \, b c x + i \, b^{2}}{c}}\right ) - 1\right )} + \left (i + 1\right ) \, \sqrt {2} \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {\frac {1}{2}} \sqrt {-\frac {4 i \, c^{2} x^{2} + 4 i \, b c x + i \, b^{2}}{c}}\right ) - 1\right )}\right )} b^{2} \cos \left (-\frac {b^{2} - 4 \, a c}{2 \, c}\right ) + {\left (-\left (i + 1\right ) \, \sqrt {2} \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {\frac {1}{2}} \sqrt {\frac {4 i \, c^{2} x^{2} + 4 i \, b c x + i \, b^{2}}{c}}\right ) - 1\right )} + \left (i - 1\right ) \, \sqrt {2} \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {\frac {1}{2}} \sqrt {-\frac {4 i \, c^{2} x^{2} + 4 i \, b c x + i \, b^{2}}{c}}\right ) - 1\right )}\right )} b^{2} \sin \left (-\frac {b^{2} - 4 \, a c}{2 \, c}\right ) - 2 \, {\left ({\left (\left (i - 1\right ) \, \sqrt {2} \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {\frac {1}{2}} \sqrt {\frac {4 i \, c^{2} x^{2} + 4 i \, b c x + i \, b^{2}}{c}}\right ) - 1\right )} - \left (i + 1\right ) \, \sqrt {2} \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {\frac {1}{2}} \sqrt {-\frac {4 i \, c^{2} x^{2} + 4 i \, b c x + i \, b^{2}}{c}}\right ) - 1\right )}\right )} b c \cos \left (-\frac {b^{2} - 4 \, a c}{2 \, c}\right ) + {\left (\left (i + 1\right ) \, \sqrt {2} \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {\frac {1}{2}} \sqrt {\frac {4 i \, c^{2} x^{2} + 4 i \, b c x + i \, b^{2}}{c}}\right ) - 1\right )} - \left (i - 1\right ) \, \sqrt {2} \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {\frac {1}{2}} \sqrt {-\frac {4 i \, c^{2} x^{2} + 4 i \, b c x + i \, b^{2}}{c}}\right ) - 1\right )}\right )} b c \sin \left (-\frac {b^{2} - 4 \, a c}{2 \, c}\right )\right )} x + 2 \, \sqrt {2} {\left (4 \, c^{2} x^{2} - c {\left (-i \, e^{\left (\frac {4 i \, c^{2} x^{2} + 4 i \, b c x + i \, b^{2}}{2 \, c}\right )} + i \, e^{\left (-\frac {4 i \, c^{2} x^{2} + 4 i \, b c x + i \, b^{2}}{2 \, c}\right )}\right )} \cos \left (-\frac {b^{2} - 4 \, a c}{2 \, c}\right ) - c {\left (e^{\left (\frac {4 i \, c^{2} x^{2} + 4 i \, b c x + i \, b^{2}}{2 \, c}\right )} + e^{\left (-\frac {4 i \, c^{2} x^{2} + 4 i \, b c x + i \, b^{2}}{2 \, c}\right )}\right )} \sin \left (-\frac {b^{2} - 4 \, a c}{2 \, c}\right )\right )} \sqrt {\frac {4 \, c^{2} x^{2} + 4 \, b c x + b^{2}}{c}}\right )} e}{64 \, c^{2} \sqrt {\frac {4 \, c^{2} x^{2} + 4 \, b c x + b^{2}}{c}}} \]
1/32*(4^(1/4)*sqrt(2)*sqrt(pi)*(((I - 1)*cos(-1/2*(b^2 - 4*a*c)/c) + (I + 1)*sin(-1/2*(b^2 - 4*a*c)/c))*erf((2*I*c*x + I*b)/sqrt(2*I*c)) + ((I + 1)* cos(-1/2*(b^2 - 4*a*c)/c) + (I - 1)*sin(-1/2*(b^2 - 4*a*c)/c))*erf((2*I*c* x + I*b)/sqrt(-2*I*c)))*c^(3/2) + 16*c^2*x)*d/c^2 + 1/64*sqrt(2)*((-(I - 1 )*sqrt(2)*sqrt(pi)*(erf(sqrt(1/2)*sqrt((4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c )) - 1) + (I + 1)*sqrt(2)*sqrt(pi)*(erf(sqrt(1/2)*sqrt(-(4*I*c^2*x^2 + 4*I *b*c*x + I*b^2)/c)) - 1))*b^2*cos(-1/2*(b^2 - 4*a*c)/c) + (-(I + 1)*sqrt(2 )*sqrt(pi)*(erf(sqrt(1/2)*sqrt((4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c)) - 1) + (I - 1)*sqrt(2)*sqrt(pi)*(erf(sqrt(1/2)*sqrt(-(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c)) - 1))*b^2*sin(-1/2*(b^2 - 4*a*c)/c) - 2*(((I - 1)*sqrt(2)*sqrt (pi)*(erf(sqrt(1/2)*sqrt((4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c)) - 1) - (I + 1)*sqrt(2)*sqrt(pi)*(erf(sqrt(1/2)*sqrt(-(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2 )/c)) - 1))*b*c*cos(-1/2*(b^2 - 4*a*c)/c) + ((I + 1)*sqrt(2)*sqrt(pi)*(erf (sqrt(1/2)*sqrt((4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c)) - 1) - (I - 1)*sqrt( 2)*sqrt(pi)*(erf(sqrt(1/2)*sqrt(-(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c)) - 1 ))*b*c*sin(-1/2*(b^2 - 4*a*c)/c))*x + 2*sqrt(2)*(4*c^2*x^2 - c*(-I*e^(1/2* (4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c) + I*e^(-1/2*(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c))*cos(-1/2*(b^2 - 4*a*c)/c) - c*(e^(1/2*(4*I*c^2*x^2 + 4*I*b*c* x + I*b^2)/c) + e^(-1/2*(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c))*sin(-1/2*(b^ 2 - 4*a*c)/c))*sqrt((4*c^2*x^2 + 4*b*c*x + b^2)/c))*e/(c^2*sqrt((4*c^2*...
Result contains complex when optimal does not.
Time = 0.36 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.30 \[ \int (d+e x) \sin ^2\left (a+b x+c x^2\right ) \, dx=\frac {1}{4} \, e x^{2} + \frac {1}{2} \, d x - \frac {-i \, e e^{\left (2 i \, c x^{2} + 2 i \, b x + 2 i \, a\right )} - \frac {\sqrt {\pi } {\left (-2 i \, c d + i \, b e\right )} \operatorname {erf}\left (-\frac {1}{2} i \, \sqrt {c} {\left (2 \, x + \frac {b}{c}\right )} {\left (\frac {i \, c}{{\left | c \right |}} + 1\right )}\right ) e^{\left (-\frac {i \, b^{2} - 4 i \, a c}{2 \, c}\right )}}{\sqrt {c} {\left (\frac {i \, c}{{\left | c \right |}} + 1\right )}}}{16 \, c} - \frac {i \, e e^{\left (-2 i \, c x^{2} - 2 i \, b x - 2 i \, a\right )} - \frac {\sqrt {\pi } {\left (2 i \, c d - i \, b e\right )} \operatorname {erf}\left (\frac {1}{2} i \, \sqrt {c} {\left (2 \, x + \frac {b}{c}\right )} {\left (-\frac {i \, c}{{\left | c \right |}} + 1\right )}\right ) e^{\left (-\frac {-i \, b^{2} + 4 i \, a c}{2 \, c}\right )}}{\sqrt {c} {\left (-\frac {i \, c}{{\left | c \right |}} + 1\right )}}}{16 \, c} \]
1/4*e*x^2 + 1/2*d*x - 1/16*(-I*e*e^(2*I*c*x^2 + 2*I*b*x + 2*I*a) - sqrt(pi )*(-2*I*c*d + I*b*e)*erf(-1/2*I*sqrt(c)*(2*x + b/c)*(I*c/abs(c) + 1))*e^(- 1/2*(I*b^2 - 4*I*a*c)/c)/(sqrt(c)*(I*c/abs(c) + 1)))/c - 1/16*(I*e*e^(-2*I *c*x^2 - 2*I*b*x - 2*I*a) - sqrt(pi)*(2*I*c*d - I*b*e)*erf(1/2*I*sqrt(c)*( 2*x + b/c)*(-I*c/abs(c) + 1))*e^(-1/2*(-I*b^2 + 4*I*a*c)/c)/(sqrt(c)*(-I*c /abs(c) + 1)))/c
Timed out. \[ \int (d+e x) \sin ^2\left (a+b x+c x^2\right ) \, dx=\int {\sin \left (c\,x^2+b\,x+a\right )}^2\,\left (d+e\,x\right ) \,d x \]