\(\int (e+f x) \sin (\frac {b}{(c+d x)^2}) \, dx\) [161]

Optimal result

Integrand size = 16, antiderivative size = 120 \[ \int (e+f x) \sin \left (\frac {b}{(c+d x)^2}\right ) \, dx=-\frac {b f \operatorname {CosIntegral}\left (\frac {b}{(c+d x)^2}\right )}{2 d^2}-\frac {\sqrt {b} (d e-c f) \sqrt {2 \pi } \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{c+d x}\right )}{d^2}+\frac {(d e-c f) (c+d x) \sin \left (\frac {b}{(c+d x)^2}\right )}{d^2}+\frac {f (c+d x)^2 \sin \left (\frac {b}{(c+d x)^2}\right )}{2 d^2} \] Output:

-1/2*b*f*Ci(b/(d*x+c)^2)/d^2-b^(1/2)*(-c*f+d*e)*2^(1/2)*Pi^(1/2)*FresnelC( 
b^(1/2)*2^(1/2)/Pi^(1/2)/(d*x+c))/d^2+(-c*f+d*e)*(d*x+c)*sin(b/(d*x+c)^2)/ 
d^2+1/2*f*(d*x+c)^2*sin(b/(d*x+c)^2)/d^2
 

Mathematica [A] (verified)

Time = 0.52 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.79 \[ \int (e+f x) \sin \left (\frac {b}{(c+d x)^2}\right ) \, dx=-\frac {b f \operatorname {CosIntegral}\left (\frac {b}{(c+d x)^2}\right )+2 \sqrt {b} (d e-c f) \sqrt {2 \pi } \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{c+d x}\right )+(c+d x) (-2 d e+c f-d f x) \sin \left (\frac {b}{(c+d x)^2}\right )}{2 d^2} \] Input:

Integrate[(e + f*x)*Sin[b/(c + d*x)^2],x]
 

Output:

-1/2*(b*f*CosIntegral[b/(c + d*x)^2] + 2*Sqrt[b]*(d*e - c*f)*Sqrt[2*Pi]*Fr 
esnelC[(Sqrt[b]*Sqrt[2/Pi])/(c + d*x)] + (c + d*x)*(-2*d*e + c*f - d*f*x)* 
Sin[b/(c + d*x)^2])/d^2
 

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.93, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3914, 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.

\(\displaystyle \int (e+f x) \sin \left (\frac {b}{(c+d x)^2}\right ) \, dx\)

\(\Big \downarrow \) 3914

\(\displaystyle \frac {\int \left ((d e-c f) \sin \left (\frac {b}{(c+d x)^2}\right )+f (c+d x) \sin \left (\frac {b}{(c+d x)^2}\right )\right )d(c+d x)}{d^2}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {-\frac {1}{2} b f \operatorname {CosIntegral}\left (\frac {b}{(c+d x)^2}\right )-\sqrt {2 \pi } \sqrt {b} (d e-c f) \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{c+d x}\right )+(c+d x) (d e-c f) \sin \left (\frac {b}{(c+d x)^2}\right )+\frac {1}{2} f (c+d x)^2 \sin \left (\frac {b}{(c+d x)^2}\right )}{d^2}\)

Input:

Int[(e + f*x)*Sin[b/(c + d*x)^2],x]
 

Output:

(-1/2*(b*f*CosIntegral[b/(c + d*x)^2]) - Sqrt[b]*(d*e - c*f)*Sqrt[2*Pi]*Fr 
esnelC[(Sqrt[b]*Sqrt[2/Pi])/(c + d*x)] + (d*e - c*f)*(c + d*x)*Sin[b/(c + 
d*x)^2] + (f*(c + d*x)^2*Sin[b/(c + d*x)^2])/2)/d^2
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3914

Int[((g_.) + (h_.)*(x_))^(m_.)*((a_.) + (b_.)*Sin[(c_.) + (d_.)*((e_.) + (f 
_.)*(x_))^(n_)])^(p_.), x_Symbol] :> Module[{k = If[FractionQ[n], Denominat 
or[n], 1]}, Simp[k/f^(m + 1)   Subst[Int[ExpandIntegrand[(a + b*Sin[c + d*x 
^(k*n)])^p, x^(k - 1)*(f*g - e*h + h*x^k)^m, x], x], x, (e + f*x)^(1/k)], x 
]] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && IGtQ[p, 0] && IGtQ[m, 0]
 

Maple [A] (verified)

Time = 1.30 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.84

method	result	size
derivativedivides	\(\frac {-\left (c f -d e \right ) \left (d x +c \right ) \sin \left (\frac {b}{\left (d x +c \right )^{2}}\right )+\left (c f -d e \right ) \sqrt {b}\, \sqrt {2}\, \sqrt {\pi }\, \operatorname {FresnelC}\left (\frac {\sqrt {b}\, \sqrt {2}}{\sqrt {\pi }\, \left (d x +c \right )}\right )+\frac {f \left (d x +c \right )^{2} \sin \left (\frac {b}{\left (d x +c \right )^{2}}\right )}{2}-\frac {f b \,\operatorname {Ci}\left (\frac {b}{\left (d x +c \right )^{2}}\right )}{2}}{d^{2}}\)	\(101\)
default	\(\frac {-\left (c f -d e \right ) \left (d x +c \right ) \sin \left (\frac {b}{\left (d x +c \right )^{2}}\right )+\left (c f -d e \right ) \sqrt {b}\, \sqrt {2}\, \sqrt {\pi }\, \operatorname {FresnelC}\left (\frac {\sqrt {b}\, \sqrt {2}}{\sqrt {\pi }\, \left (d x +c \right )}\right )+\frac {f \left (d x +c \right )^{2} \sin \left (\frac {b}{\left (d x +c \right )^{2}}\right )}{2}-\frac {f b \,\operatorname {Ci}\left (\frac {b}{\left (d x +c \right )^{2}}\right )}{2}}{d^{2}}\)	\(101\)
risch	\(\frac {b \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {-i b}}{d x +c}\right ) c f}{2 d^{2} \sqrt {-i b}}-\frac {b \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {-i b}}{d x +c}\right ) e}{2 d \sqrt {-i b}}+\frac {b \,\operatorname {expIntegral}_{1}\left (-\frac {i b}{\left (d x +c \right )^{2}}\right ) f}{4 d^{2}}+\frac {b \,\operatorname {erf}\left (\frac {\sqrt {i b}}{d x +c}\right ) \sqrt {\pi }\, c f}{2 d^{2} \sqrt {i b}}-\frac {b \,\operatorname {erf}\left (\frac {\sqrt {i b}}{d x +c}\right ) \sqrt {\pi }\, e}{2 d \sqrt {i b}}+\frac {b \,\operatorname {expIntegral}_{1}\left (\frac {i b}{\left (d x +c \right )^{2}}\right ) f}{4 d^{2}}-\frac {\left (e \left (-d x -c \right )-\frac {f \left (d x +c \right )^{2}}{2 d}-\frac {c f \left (-d x -c \right )}{d}\right ) \sin \left (\frac {b}{\left (d x +c \right )^{2}}\right )}{d}\)	\(222\)
parts	\(-\frac {\sqrt {b}\, \sqrt {2}\, \operatorname {FresnelC}\left (\frac {\sqrt {b}\, \sqrt {2}}{\sqrt {\pi }\, \left (d x +c \right )}\right ) \sqrt {\pi }\, f x}{d}+\sin \left (\frac {b}{\left (d x +c \right )^{2}}\right ) f \,x^{2}-\frac {\sqrt {b}\, \sqrt {2}\, \operatorname {FresnelC}\left (\frac {\sqrt {b}\, \sqrt {2}}{\sqrt {\pi }\, \left (d x +c \right )}\right ) \sqrt {\pi }\, e}{d}+\frac {\sin \left (\frac {b}{\left (d x +c \right )^{2}}\right ) c f x}{d}+\sin \left (\frac {b}{\left (d x +c \right )^{2}}\right ) e x +\frac {\sin \left (\frac {b}{\left (d x +c \right )^{2}}\right ) c e}{d}+\frac {f \left (-\frac {2 b \left (-\frac {\operatorname {FresnelC}\left (\frac {\sqrt {b}\, \sqrt {2}}{\sqrt {\pi }\, \left (d x +c \right )}\right ) \sqrt {2}\, \sqrt {\pi }\, \left (d x +c \right )}{2 \sqrt {b}}+\frac {\operatorname {Ci}\left (\frac {b}{\left (d x +c \right )^{2}}\right )}{2}\right )}{d}+\frac {-\frac {\left (d x +c \right )^{2} \sin \left (\frac {b}{\left (d x +c \right )^{2}}\right )}{2}+\frac {b \,\operatorname {Ci}\left (\frac {b}{\left (d x +c \right )^{2}}\right )}{2}}{d}\right )}{d}\)	\(226\)

Input:

int((f*x+e)*sin(b/(d*x+c)^2),x,method=_RETURNVERBOSE)
 

Output:

1/d^2*(-(c*f-d*e)*(d*x+c)*sin(b/(d*x+c)^2)+(c*f-d*e)*b^(1/2)*2^(1/2)*Pi^(1 
/2)*FresnelC(b^(1/2)*2^(1/2)/Pi^(1/2)/(d*x+c))+1/2*f*(d*x+c)^2*sin(b/(d*x+ 
c)^2)-1/2*f*b*Ci(b/(d*x+c)^2))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.08 \[ \int (e+f x) \sin \left (\frac {b}{(c+d x)^2}\right ) \, dx=-\frac {2 \, \sqrt {2} \pi {\left (d^{2} e - c d f\right )} \sqrt {\frac {b}{\pi d^{2}}} \operatorname {C}\left (\frac {\sqrt {2} d \sqrt {\frac {b}{\pi d^{2}}}}{d x + c}\right ) + b f \operatorname {Ci}\left (\frac {b}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) - {\left (d^{2} f x^{2} + 2 \, d^{2} e x + 2 \, c d e - c^{2} f\right )} \sin \left (\frac {b}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}{2 \, d^{2}} \] Input:

integrate((f*x+e)*sin(b/(d*x+c)^2),x, algorithm="fricas")
 

Output:

-1/2*(2*sqrt(2)*pi*(d^2*e - c*d*f)*sqrt(b/(pi*d^2))*fresnel_cos(sqrt(2)*d* 
sqrt(b/(pi*d^2))/(d*x + c)) + b*f*cos_integral(b/(d^2*x^2 + 2*c*d*x + c^2) 
) - (d^2*f*x^2 + 2*d^2*e*x + 2*c*d*e - c^2*f)*sin(b/(d^2*x^2 + 2*c*d*x + c 
^2)))/d^2
 

Sympy [F]

\[ \int (e+f x) \sin \left (\frac {b}{(c+d x)^2}\right ) \, dx=\int \left (e + f x\right ) \sin {\left (\frac {b}{c^{2} + 2 c d x + d^{2} x^{2}} \right )}\, dx \] Input:

integrate((f*x+e)*sin(b/(d*x+c)**2),x)
 

Output:

Integral((e + f*x)*sin(b/(c**2 + 2*c*d*x + d**2*x**2)), x)
 

Maxima [F]

\[ \int (e+f x) \sin \left (\frac {b}{(c+d x)^2}\right ) \, dx=\int { {\left (f x + e\right )} \sin \left (\frac {b}{{\left (d x + c\right )}^{2}}\right ) \,d x } \] Input:

integrate((f*x+e)*sin(b/(d*x+c)^2),x, algorithm="maxima")
 

Output:

1/2*(f*x^2 + 2*e*x)*sin(b/(d^2*x^2 + 2*c*d*x + c^2)) + integrate(1/2*(b*d* 
f*x^2 + 2*b*d*e*x)*cos(b/(d^2*x^2 + 2*c*d*x + c^2))/(d^3*x^3 + 3*c*d^2*x^2 
 + 3*c^2*d*x + c^3), x) + integrate(1/2*(b*d*f*x^2 + 2*b*d*e*x)*cos(b/(d^2 
*x^2 + 2*c*d*x + c^2))/((d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3)*cos(b/(d 
^2*x^2 + 2*c*d*x + c^2))^2 + (d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3)*sin 
(b/(d^2*x^2 + 2*c*d*x + c^2))^2), x)
 

Giac [F]

\[ \int (e+f x) \sin \left (\frac {b}{(c+d x)^2}\right ) \, dx=\int { {\left (f x + e\right )} \sin \left (\frac {b}{{\left (d x + c\right )}^{2}}\right ) \,d x } \] Input:

integrate((f*x+e)*sin(b/(d*x+c)^2),x, algorithm="giac")
 

Output:

integrate((f*x + e)*sin(b/(d*x + c)^2), x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int (e+f x) \sin \left (\frac {b}{(c+d x)^2}\right ) \, dx=\int \sin \left (\frac {b}{{\left (c+d\,x\right )}^2}\right )\,\left (e+f\,x\right ) \,d x \] Input:

int(sin(b/(c + d*x)^2)*(e + f*x),x)
 

Output:

int(sin(b/(c + d*x)^2)*(e + f*x), x)
 

Reduce [F]

\[ \int (e+f x) \sin \left (\frac {b}{(c+d x)^2}\right ) \, dx=\left (\int \sin \left (\frac {b}{d^{2} x^{2}+2 c d x +c^{2}}\right )d x \right ) e +\left (\int \sin \left (\frac {b}{d^{2} x^{2}+2 c d x +c^{2}}\right ) x d x \right ) f \] Input:

int((f*x+e)*sin(b/(d*x+c)^2),x)
 

Output:

int(sin(b/(c**2 + 2*c*d*x + d**2*x**2)),x)*e + int(sin(b/(c**2 + 2*c*d*x + 
 d**2*x**2))*x,x)*f