Integrand size = 17, antiderivative size = 140 \[ \int (d+e x) \sin \left (a+b x+c x^2\right ) \, dx=-\frac {e \cos \left (a+b x+c x^2\right )}{2 c}+\frac {(2 c d-b e) \sqrt {\frac {\pi }{2}} \cos \left (a-\frac {b^2}{4 c}\right ) \operatorname {FresnelS}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right )}{2 c^{3/2}}+\frac {(2 c d-b e) \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right ) \sin \left (a-\frac {b^2}{4 c}\right )}{2 c^{3/2}} \] Output:
-1/2*e*cos(c*x^2+b*x+a)/c+1/4*(-b*e+2*c*d)*2^(1/2)*Pi^(1/2)*cos(a-1/4*b^2/ c)*FresnelS(1/2*(2*c*x+b)/c^(1/2)*2^(1/2)/Pi^(1/2))/c^(3/2)+1/4*(-b*e+2*c* d)*2^(1/2)*Pi^(1/2)*FresnelC(1/2*(2*c*x+b)/c^(1/2)*2^(1/2)/Pi^(1/2))*sin(a -1/4*b^2/c)/c^(3/2)
Time = 0.94 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.91 \[ \int (d+e x) \sin \left (a+b x+c x^2\right ) \, dx=\frac {-2 \sqrt {c} e \cos (a+x (b+c x))+(2 c d-b e) \sqrt {2 \pi } \cos \left (a-\frac {b^2}{4 c}\right ) \operatorname {FresnelS}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right )+(2 c d-b e) \sqrt {2 \pi } \operatorname {FresnelC}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right ) \sin \left (a-\frac {b^2}{4 c}\right )}{4 c^{3/2}} \] Input:
Integrate[(d + e*x)*Sin[a + b*x + c*x^2],x]
Output:
(-2*Sqrt[c]*e*Cos[a + x*(b + c*x)] + (2*c*d - b*e)*Sqrt[2*Pi]*Cos[a - b^2/ (4*c)]*FresnelS[(b + 2*c*x)/(Sqrt[c]*Sqrt[2*Pi])] + (2*c*d - b*e)*Sqrt[2*P i]*FresnelC[(b + 2*c*x)/(Sqrt[c]*Sqrt[2*Pi])]*Sin[a - b^2/(4*c)])/(4*c^(3/ 2))
Time = 0.34 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.95, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {3942, 3928, 3832, 3833}
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.
\(\displaystyle \int (d+e x) \sin \left (a+b x+c x^2\right ) \, dx\) |
\(\Big \downarrow \) 3942 |
\(\displaystyle \frac {(2 c d-b e) \int \sin \left (c x^2+b x+a\right )dx}{2 c}-\frac {e \cos \left (a+b x+c x^2\right )}{2 c}\) |
\(\Big \downarrow \) 3928 |
\(\displaystyle \frac {(2 c d-b e) \left (\sin \left (a-\frac {b^2}{4 c}\right ) \int \cos \left (\frac {(b+2 c x)^2}{4 c}\right )dx+\cos \left (a-\frac {b^2}{4 c}\right ) \int \sin \left (\frac {(b+2 c x)^2}{4 c}\right )dx\right )}{2 c}-\frac {e \cos \left (a+b x+c x^2\right )}{2 c}\) |
\(\Big \downarrow \) 3832 |
\(\displaystyle \frac {(2 c d-b e) \left (\sin \left (a-\frac {b^2}{4 c}\right ) \int \cos \left (\frac {(b+2 c x)^2}{4 c}\right )dx+\frac {\sqrt {\frac {\pi }{2}} \cos \left (a-\frac {b^2}{4 c}\right ) \operatorname {FresnelS}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right )}{\sqrt {c}}\right )}{2 c}-\frac {e \cos \left (a+b x+c x^2\right )}{2 c}\) |
\(\Big \downarrow \) 3833 |
\(\displaystyle \frac {(2 c d-b e) \left (\frac {\sqrt {\frac {\pi }{2}} \sin \left (a-\frac {b^2}{4 c}\right ) \operatorname {FresnelC}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right )}{\sqrt {c}}+\frac {\sqrt {\frac {\pi }{2}} \cos \left (a-\frac {b^2}{4 c}\right ) \operatorname {FresnelS}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right )}{\sqrt {c}}\right )}{2 c}-\frac {e \cos \left (a+b x+c x^2\right )}{2 c}\) |
Input:
Int[(d + e*x)*Sin[a + b*x + c*x^2],x]
Output:
-1/2*(e*Cos[a + b*x + c*x^2])/c + ((2*c*d - b*e)*((Sqrt[Pi/2]*Cos[a - b^2/ (4*c)]*FresnelS[(b + 2*c*x)/(Sqrt[c]*Sqrt[2*Pi])])/Sqrt[c] + (Sqrt[Pi/2]*F resnelC[(b + 2*c*x)/(Sqrt[c]*Sqrt[2*Pi])]*Sin[a - b^2/(4*c)])/Sqrt[c]))/(2 *c)
Int[Sin[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[ d, 2]))*FresnelS[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]
Int[Cos[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[ d, 2]))*FresnelC[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]
Int[Sin[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[Cos[(b^2 - 4* a*c)/(4*c)] Int[Sin[(b + 2*c*x)^2/(4*c)], x], x] - Simp[Sin[(b^2 - 4*a*c) /(4*c)] Int[Cos[(b + 2*c*x)^2/(4*c)], x], x] /; FreeQ[{a, b, c}, x] && Ne Q[b^2 - 4*a*c, 0]
Int[((d_.) + (e_.)*(x_))*Sin[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[(-e)*(Cos[a + b*x + c*x^2]/(2*c)), x] + Simp[(2*c*d - b*e)/(2*c) Int[Sin[a + b*x + c*x^2], x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0]
Time = 1.01 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.31
method | result | size |
default | \(-\frac {e \cos \left (c \,x^{2}+b x +a \right )}{2 c}-\frac {e b \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (\frac {\frac {b^{2}}{4}-a c}{c}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \left (c x +\frac {b}{2}\right )}{\sqrt {\pi }\, \sqrt {c}}\right )-\sin \left (\frac {\frac {b^{2}}{4}-a c}{c}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \left (c x +\frac {b}{2}\right )}{\sqrt {\pi }\, \sqrt {c}}\right )\right )}{4 c^{\frac {3}{2}}}+\frac {\sqrt {2}\, \sqrt {\pi }\, d \left (\cos \left (\frac {\frac {b^{2}}{4}-a c}{c}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \left (c x +\frac {b}{2}\right )}{\sqrt {\pi }\, \sqrt {c}}\right )-\sin \left (\frac {\frac {b^{2}}{4}-a c}{c}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \left (c x +\frac {b}{2}\right )}{\sqrt {\pi }\, \sqrt {c}}\right )\right )}{2 \sqrt {c}}\) | \(184\) |
risch | \(\frac {i \operatorname {erf}\left (-\sqrt {-i c}\, x +\frac {i b}{2 \sqrt {-i c}}\right ) \sqrt {\pi }\, d \,{\mathrm e}^{\frac {i \left (4 a c -b^{2}\right )}{4 c}}}{4 \sqrt {-i c}}-\frac {i e \,\operatorname {erf}\left (-\sqrt {-i c}\, x +\frac {i b}{2 \sqrt {-i c}}\right ) \sqrt {\pi }\, b \,{\mathrm e}^{\frac {i \left (4 a c -b^{2}\right )}{4 c}}}{8 \sqrt {-i c}\, c}+\frac {i \operatorname {erf}\left (\sqrt {i c}\, x +\frac {i b}{2 \sqrt {i c}}\right ) \sqrt {\pi }\, d \,{\mathrm e}^{-\frac {i \left (4 a c -b^{2}\right )}{4 c}}}{4 \sqrt {i c}}-\frac {i e \,\operatorname {erf}\left (\sqrt {i c}\, x +\frac {i b}{2 \sqrt {i c}}\right ) \sqrt {\pi }\, b \,{\mathrm e}^{-\frac {i \left (4 a c -b^{2}\right )}{4 c}}}{8 \sqrt {i c}\, c}-\frac {e \cos \left (c \,x^{2}+b x +a \right )}{2 c}\) | \(229\) |
parts | \(\frac {\sqrt {2}\, \sqrt {\pi }\, \cos \left (\frac {\frac {b^{2}}{4}-a c}{c}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \left (c x +\frac {b}{2}\right )}{\sqrt {\pi }\, \sqrt {c}}\right ) e x}{2 \sqrt {c}}-\frac {\sqrt {2}\, \sqrt {\pi }\, \sin \left (\frac {\frac {b^{2}}{4}-a c}{c}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \left (c x +\frac {b}{2}\right )}{\sqrt {\pi }\, \sqrt {c}}\right ) e x}{2 \sqrt {c}}+\frac {\sqrt {2}\, \sqrt {\pi }\, \cos \left (\frac {\frac {b^{2}}{4}-a c}{c}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \left (c x +\frac {b}{2}\right )}{\sqrt {\pi }\, \sqrt {c}}\right ) d}{2 \sqrt {c}}-\frac {\sqrt {2}\, \sqrt {\pi }\, \sin \left (\frac {\frac {b^{2}}{4}-a c}{c}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \left (c x +\frac {b}{2}\right )}{\sqrt {\pi }\, \sqrt {c}}\right ) d}{2 \sqrt {c}}-\frac {\sqrt {2}\, \sqrt {\pi }\, e \left (\frac {\cos \left (\frac {\frac {b^{2}}{4}-a c}{c}\right ) \sqrt {2}\, \sqrt {\pi }\, \left (\operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {c}\, x}{\sqrt {\pi }}+\frac {\sqrt {2}\, b}{2 \sqrt {\pi }\, \sqrt {c}}\right ) \left (\frac {\sqrt {2}\, \sqrt {c}\, x}{\sqrt {\pi }}+\frac {\sqrt {2}\, b}{2 \sqrt {\pi }\, \sqrt {c}}\right )+\frac {\cos \left (\frac {\pi \left (\frac {\sqrt {2}\, \sqrt {c}\, x}{\sqrt {\pi }}+\frac {\sqrt {2}\, b}{2 \sqrt {\pi }\, \sqrt {c}}\right )^{2}}{2}\right )}{\pi }\right )}{2 \sqrt {c}}-\frac {\sin \left (\frac {\frac {b^{2}}{4}-a c}{c}\right ) \sqrt {2}\, \sqrt {\pi }\, \left (\operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {c}\, x}{\sqrt {\pi }}+\frac {\sqrt {2}\, b}{2 \sqrt {\pi }\, \sqrt {c}}\right ) \left (\frac {\sqrt {2}\, \sqrt {c}\, x}{\sqrt {\pi }}+\frac {\sqrt {2}\, b}{2 \sqrt {\pi }\, \sqrt {c}}\right )-\frac {\sin \left (\frac {\pi \left (\frac {\sqrt {2}\, \sqrt {c}\, x}{\sqrt {\pi }}+\frac {\sqrt {2}\, b}{2 \sqrt {\pi }\, \sqrt {c}}\right )^{2}}{2}\right )}{\pi }\right )}{2 \sqrt {c}}\right )}{2 \sqrt {c}}\) | \(420\) |
Input:
int((e*x+d)*sin(c*x^2+b*x+a),x,method=_RETURNVERBOSE)
Output:
-1/2*e*cos(c*x^2+b*x+a)/c-1/4*e*b/c^(3/2)*2^(1/2)*Pi^(1/2)*(cos((1/4*b^2-a *c)/c)*FresnelS(2^(1/2)/Pi^(1/2)/c^(1/2)*(c*x+1/2*b))-sin((1/4*b^2-a*c)/c) *FresnelC(2^(1/2)/Pi^(1/2)/c^(1/2)*(c*x+1/2*b)))+1/2*2^(1/2)*Pi^(1/2)/c^(1 /2)*d*(cos((1/4*b^2-a*c)/c)*FresnelS(2^(1/2)/Pi^(1/2)/c^(1/2)*(c*x+1/2*b)) -sin((1/4*b^2-a*c)/c)*FresnelC(2^(1/2)/Pi^(1/2)/c^(1/2)*(c*x+1/2*b)))
Time = 0.08 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.96 \[ \int (d+e x) \sin \left (a+b x+c x^2\right ) \, dx=\frac {\sqrt {2} \pi {\left (2 \, c d - b e\right )} \sqrt {\frac {c}{\pi }} \cos \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right ) \operatorname {S}\left (\frac {\sqrt {2} {\left (2 \, c x + b\right )} \sqrt {\frac {c}{\pi }}}{2 \, c}\right ) + \sqrt {2} \pi {\left (2 \, c d - b e\right )} \sqrt {\frac {c}{\pi }} \operatorname {C}\left (\frac {\sqrt {2} {\left (2 \, c x + b\right )} \sqrt {\frac {c}{\pi }}}{2 \, c}\right ) \sin \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right ) - 2 \, c e \cos \left (c x^{2} + b x + a\right )}{4 \, c^{2}} \] Input:
integrate((e*x+d)*sin(c*x^2+b*x+a),x, algorithm="fricas")
Output:
1/4*(sqrt(2)*pi*(2*c*d - b*e)*sqrt(c/pi)*cos(-1/4*(b^2 - 4*a*c)/c)*fresnel _sin(1/2*sqrt(2)*(2*c*x + b)*sqrt(c/pi)/c) + sqrt(2)*pi*(2*c*d - b*e)*sqrt (c/pi)*fresnel_cos(1/2*sqrt(2)*(2*c*x + b)*sqrt(c/pi)/c)*sin(-1/4*(b^2 - 4 *a*c)/c) - 2*c*e*cos(c*x^2 + b*x + a))/c^2
\[ \int (d+e x) \sin \left (a+b x+c x^2\right ) \, dx=\int \left (d + e x\right ) \sin {\left (a + b x + c x^{2} \right )}\, dx \] Input:
integrate((e*x+d)*sin(c*x**2+b*x+a),x)
Output:
Integral((d + e*x)*sin(a + b*x + c*x**2), x)
Result contains complex when optimal does not.
Time = 0.39 (sec) , antiderivative size = 694, normalized size of antiderivative = 4.96 \[ \int (d+e x) \sin \left (a+b x+c x^2\right ) \, dx =\text {Too large to display} \] Input:
integrate((e*x+d)*sin(c*x^2+b*x+a),x, algorithm="maxima")
Output:
-1/8*sqrt(2)*sqrt(pi)*((-(I + 1)*cos(-1/4*(b^2 - 4*a*c)/c) + (I - 1)*sin(- 1/4*(b^2 - 4*a*c)/c))*erf(1/2*(2*I*c*x + I*b)/sqrt(I*c)) + (-(I - 1)*cos(- 1/4*(b^2 - 4*a*c)/c) + (I + 1)*sin(-1/4*(b^2 - 4*a*c)/c))*erf(1/2*(2*I*c*x + I*b)/sqrt(-I*c)))*d/sqrt(c) + 1/16*((-(I + 1)*sqrt(2)*sqrt(pi)*(erf(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(1/2*sqrt(-(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c)) - 1))*b^2*cos(-1/4* (b^2 - 4*a*c)/c) + ((I - 1)*sqrt(2)*sqrt(pi)*(erf(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(1/2*sqrt(-(4*I *c^2*x^2 + 4*I*b*c*x + I*b^2)/c)) - 1))*b^2*sin(-1/4*(b^2 - 4*a*c)/c) - 2* (((I + 1)*sqrt(2)*sqrt(pi)*(erf(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(1/2*sqrt(-(4*I*c^2*x^2 + 4*I*b*c *x + I*b^2)/c)) - 1))*b*c*cos(-1/4*(b^2 - 4*a*c)/c) + (-(I - 1)*sqrt(2)*sq rt(pi)*(erf(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(1/2*sqrt(-(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c)) - 1) )*b*c*sin(-1/4*(b^2 - 4*a*c)/c))*x - 4*(c*(e^(1/4*(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c) + e^(-1/4*(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c))*cos(-1/4*(b^2 - 4*a*c)/c) + c*(I*e^(1/4*(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c) - I*e^(-1/ 4*(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c))*sin(-1/4*(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*x^2 + 4*b*c*x + b^2)/c))
Result contains complex when optimal does not.
Time = 0.16 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.42 \[ \int (d+e x) \sin \left (a+b x+c x^2\right ) \, dx=-\frac {\frac {\sqrt {2} \sqrt {\pi } {\left (-2 i \, c d + i \, b e\right )} \operatorname {erf}\left (-\frac {1}{4} \, \sqrt {2} {\left (2 \, x + \frac {b}{c}\right )} {\left (-\frac {i \, c}{{\left | c \right |}} + 1\right )} \sqrt {{\left | c \right |}}\right ) e^{\left (-\frac {i \, b^{2} - 4 i \, a c}{4 \, c}\right )}}{{\left (-\frac {i \, c}{{\left | c \right |}} + 1\right )} \sqrt {{\left | c \right |}}} + 2 \, e e^{\left (i \, c x^{2} + i \, b x + i \, a\right )}}{8 \, c} - \frac {\frac {\sqrt {2} \sqrt {\pi } {\left (2 i \, c d - i \, b e\right )} \operatorname {erf}\left (-\frac {1}{4} \, \sqrt {2} {\left (2 \, x + \frac {b}{c}\right )} {\left (\frac {i \, c}{{\left | c \right |}} + 1\right )} \sqrt {{\left | c \right |}}\right ) e^{\left (-\frac {-i \, b^{2} + 4 i \, a c}{4 \, c}\right )}}{{\left (\frac {i \, c}{{\left | c \right |}} + 1\right )} \sqrt {{\left | c \right |}}} + 2 \, e e^{\left (-i \, c x^{2} - i \, b x - i \, a\right )}}{8 \, c} \] Input:
integrate((e*x+d)*sin(c*x^2+b*x+a),x, algorithm="giac")
Output:
-1/8*(sqrt(2)*sqrt(pi)*(-2*I*c*d + I*b*e)*erf(-1/4*sqrt(2)*(2*x + b/c)*(-I *c/abs(c) + 1)*sqrt(abs(c)))*e^(-1/4*(I*b^2 - 4*I*a*c)/c)/((-I*c/abs(c) + 1)*sqrt(abs(c))) + 2*e*e^(I*c*x^2 + I*b*x + I*a))/c - 1/8*(sqrt(2)*sqrt(pi )*(2*I*c*d - I*b*e)*erf(-1/4*sqrt(2)*(2*x + b/c)*(I*c/abs(c) + 1)*sqrt(abs (c)))*e^(-1/4*(-I*b^2 + 4*I*a*c)/c)/((I*c/abs(c) + 1)*sqrt(abs(c))) + 2*e* e^(-I*c*x^2 - I*b*x - I*a))/c
Timed out. \[ \int (d+e x) \sin \left (a+b x+c x^2\right ) \, dx=\int \sin \left (c\,x^2+b\,x+a\right )\,\left (d+e\,x\right ) \,d x \] Input:
int(sin(a + b*x + c*x^2)*(d + e*x),x)
Output:
int(sin(a + b*x + c*x^2)*(d + e*x), x)
\[ \int (d+e x) \sin \left (a+b x+c x^2\right ) \, dx=\left (\int \sin \left (c \,x^{2}+b x +a \right )d x \right ) d +\left (\int \sin \left (c \,x^{2}+b x +a \right ) x d x \right ) e \] Input:
int((e*x+d)*sin(c*x^2+b*x+a),x)
Output:
int(sin(a + b*x + c*x**2),x)*d + int(sin(a + b*x + c*x**2)*x,x)*e