Integrand size = 18, antiderivative size = 198 \[ \int (e+f x) \sin \left (a+\frac {b}{(c+d x)^2}\right ) \, dx=-\frac {b f \cos (a) \operatorname {CosIntegral}\left (\frac {b}{(c+d x)^2}\right )}{2 d^2}-\frac {\sqrt {b} (d e-c f) \sqrt {2 \pi } \cos (a) \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{c+d x}\right )}{d^2}+\frac {\sqrt {b} (d e-c f) \sqrt {2 \pi } \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{c+d x}\right ) \sin (a)}{d^2}+\frac {(d e-c f) (c+d x) \sin \left (a+\frac {b}{(c+d x)^2}\right )}{d^2}+\frac {f (c+d x)^2 \sin \left (a+\frac {b}{(c+d x)^2}\right )}{2 d^2}+\frac {b f \sin (a) \text {Si}\left (\frac {b}{(c+d x)^2}\right )}{2 d^2} \]
-1/2*b*f*Ci(b/(d*x+c)^2)*cos(a)/d^2+1/2*b*f*Si(b/(d*x+c)^2)*sin(a)/d^2+(-c *f+d*e)*(d*x+c)*sin(a+b/(d*x+c)^2)/d^2+1/2*f*(d*x+c)^2*sin(a+b/(d*x+c)^2)/ d^2-(-c*f+d*e)*cos(a)*FresnelC(b^(1/2)*2^(1/2)/Pi^(1/2)/(d*x+c))*b^(1/2)*2 ^(1/2)*Pi^(1/2)/d^2+(-c*f+d*e)*FresnelS(b^(1/2)*2^(1/2)/Pi^(1/2)/(d*x+c))* sin(a)*b^(1/2)*2^(1/2)*Pi^(1/2)/d^2
Time = 0.69 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.22 \[ \int (e+f x) \sin \left (a+\frac {b}{(c+d x)^2}\right ) \, dx=\frac {-b f \cos (a) \operatorname {CosIntegral}\left (\frac {b}{(c+d x)^2}\right )-2 \sqrt {b} (d e-c f) \sqrt {2 \pi } \cos (a) \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{c+d x}\right )+2 \sqrt {b} d e \sqrt {2 \pi } \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{c+d x}\right ) \sin (a)-2 \sqrt {b} c f \sqrt {2 \pi } \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{c+d x}\right ) \sin (a)+2 c d e \sin \left (a+\frac {b}{(c+d x)^2}\right )-c^2 f \sin \left (a+\frac {b}{(c+d x)^2}\right )+2 d^2 e x \sin \left (a+\frac {b}{(c+d x)^2}\right )+d^2 f x^2 \sin \left (a+\frac {b}{(c+d x)^2}\right )+b f \sin (a) \text {Si}\left (\frac {b}{(c+d x)^2}\right )}{2 d^2} \]
(-(b*f*Cos[a]*CosIntegral[b/(c + d*x)^2]) - 2*Sqrt[b]*(d*e - c*f)*Sqrt[2*P i]*Cos[a]*FresnelC[(Sqrt[b]*Sqrt[2/Pi])/(c + d*x)] + 2*Sqrt[b]*d*e*Sqrt[2* Pi]*FresnelS[(Sqrt[b]*Sqrt[2/Pi])/(c + d*x)]*Sin[a] - 2*Sqrt[b]*c*f*Sqrt[2 *Pi]*FresnelS[(Sqrt[b]*Sqrt[2/Pi])/(c + d*x)]*Sin[a] + 2*c*d*e*Sin[a + b/( c + d*x)^2] - c^2*f*Sin[a + b/(c + d*x)^2] + 2*d^2*e*x*Sin[a + b/(c + d*x) ^2] + d^2*f*x^2*Sin[a + b/(c + d*x)^2] + b*f*Sin[a]*SinIntegral[b/(c + d*x )^2])/(2*d^2)
Time = 0.40 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.93, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, 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.
| \(\displaystyle \int (e+f x) \sin \left (a+\frac {b}{(c+d x)^2}\right ) \, dx\) |
| \(\Big \downarrow \) 3914 |
| \(\displaystyle \frac {\int \left ((d e-c f) \sin \left (a+\frac {b}{(c+d x)^2}\right )+f (c+d x) \sin \left (a+\frac {b}{(c+d x)^2}\right )\right )d(c+d x)}{d^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {1}{2} b f \cos (a) \operatorname {CosIntegral}\left (\frac {b}{(c+d x)^2}\right )-\sqrt {2 \pi } \sqrt {b} \cos (a) (d e-c f) \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{c+d x}\right )+\sqrt {2 \pi } \sqrt {b} \sin (a) (d e-c f) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{c+d x}\right )+(c+d x) (d e-c f) \sin \left (a+\frac {b}{(c+d x)^2}\right )+\frac {1}{2} b f \sin (a) \text {Si}\left (\frac {b}{(c+d x)^2}\right )+\frac {1}{2} f (c+d x)^2 \sin \left (a+\frac {b}{(c+d x)^2}\right )}{d^2}\) |
(-1/2*(b*f*Cos[a]*CosIntegral[b/(c + d*x)^2]) - Sqrt[b]*(d*e - c*f)*Sqrt[2 *Pi]*Cos[a]*FresnelC[(Sqrt[b]*Sqrt[2/Pi])/(c + d*x)] + Sqrt[b]*(d*e - c*f) *Sqrt[2*Pi]*FresnelS[(Sqrt[b]*Sqrt[2/Pi])/(c + d*x)]*Sin[a] + (d*e - c*f)* (c + d*x)*Sin[a + b/(c + d*x)^2] + (f*(c + d*x)^2*Sin[a + b/(c + d*x)^2])/ 2 + (b*f*Sin[a]*SinIntegral[b/(c + d*x)^2])/2)/d^2
3.2.78.3.1 Defintions of rubi rules used
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]
Time = 0.62 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.76
| method | result | size |
| derivativedivides | \(\frac {-\left (c f -d e \right ) \left (d x +c \right ) \sin \left (a +\frac {b}{\left (d x +c \right )^{2}}\right )+\left (c f -d e \right ) \sqrt {b}\, \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (a \right ) \operatorname {C}\left (\frac {\sqrt {b}\, \sqrt {2}}{\sqrt {\pi }\, \left (d x +c \right )}\right )-\sin \left (a \right ) \operatorname {S}\left (\frac {\sqrt {b}\, \sqrt {2}}{\sqrt {\pi }\, \left (d x +c \right )}\right )\right )+\frac {f \left (d x +c \right )^{2} \sin \left (a +\frac {b}{\left (d x +c \right )^{2}}\right )}{2}-f b \left (\frac {\cos \left (a \right ) \operatorname {Ci}\left (\frac {b}{\left (d x +c \right )^{2}}\right )}{2}-\frac {\sin \left (a \right ) \operatorname {Si}\left (\frac {b}{\left (d x +c \right )^{2}}\right )}{2}\right )}{d^{2}}\) | \(150\) |
| default | \(\frac {-\left (c f -d e \right ) \left (d x +c \right ) \sin \left (a +\frac {b}{\left (d x +c \right )^{2}}\right )+\left (c f -d e \right ) \sqrt {b}\, \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (a \right ) \operatorname {C}\left (\frac {\sqrt {b}\, \sqrt {2}}{\sqrt {\pi }\, \left (d x +c \right )}\right )-\sin \left (a \right ) \operatorname {S}\left (\frac {\sqrt {b}\, \sqrt {2}}{\sqrt {\pi }\, \left (d x +c \right )}\right )\right )+\frac {f \left (d x +c \right )^{2} \sin \left (a +\frac {b}{\left (d x +c \right )^{2}}\right )}{2}-f b \left (\frac {\cos \left (a \right ) \operatorname {Ci}\left (\frac {b}{\left (d x +c \right )^{2}}\right )}{2}-\frac {\sin \left (a \right ) \operatorname {Si}\left (\frac {b}{\left (d x +c \right )^{2}}\right )}{2}\right )}{d^{2}}\) | \(150\) |
| risch | \(\frac {{\mathrm e}^{i a} b \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {-i b}}{d x +c}\right ) c f}{2 d^{2} \sqrt {-i b}}-\frac {{\mathrm e}^{i a} b \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {-i b}}{d x +c}\right ) e}{2 d \sqrt {-i b}}+\frac {{\mathrm e}^{i a} b \,\operatorname {Ei}_{1}\left (-\frac {i b}{\left (d x +c \right )^{2}}\right ) f}{4 d^{2}}+\frac {{\mathrm e}^{-i a} b \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {i b}}{d x +c}\right ) c f}{2 d^{2} \sqrt {i b}}-\frac {{\mathrm e}^{-i a} b \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {i b}}{d x +c}\right ) e}{2 d \sqrt {i b}}+\frac {{\mathrm e}^{-i a} b \,\operatorname {Ei}_{1}\left (\frac {i b}{\left (d x +c \right )^{2}}\right ) f}{4 d^{2}}-\frac {\left (e \left (-d x -c \right )+\frac {f \left (-\frac {1}{2} d^{2} x^{2}-c d x -\frac {1}{2} c^{2}\right )}{d}-\frac {c f \left (-d x -c \right )}{d}\right ) \sin \left (\frac {a \,d^{2} x^{2}+2 a c d x +a \,c^{2}+b}{\left (d x +c \right )^{2}}\right )}{d}\) | \(283\) |
| parts | \(-\frac {\sqrt {\pi }\, \sqrt {b}\, \sqrt {2}\, \cos \left (a \right ) \operatorname {C}\left (\frac {\sqrt {b}\, \sqrt {2}}{\sqrt {\pi }\, \left (d x +c \right )}\right ) f x}{d}+\frac {\sqrt {\pi }\, \sqrt {b}\, \sqrt {2}\, \sin \left (a \right ) \operatorname {S}\left (\frac {\sqrt {b}\, \sqrt {2}}{\sqrt {\pi }\, \left (d x +c \right )}\right ) f x}{d}-\frac {\sqrt {\pi }\, \sqrt {b}\, \sqrt {2}\, \cos \left (a \right ) \operatorname {C}\left (\frac {\sqrt {b}\, \sqrt {2}}{\sqrt {\pi }\, \left (d x +c \right )}\right ) e}{d}+\frac {\sqrt {\pi }\, \sqrt {b}\, \sqrt {2}\, \sin \left (a \right ) \operatorname {S}\left (\frac {\sqrt {b}\, \sqrt {2}}{\sqrt {\pi }\, \left (d x +c \right )}\right ) e}{d}+\sin \left (a +\frac {b}{\left (d x +c \right )^{2}}\right ) f \,x^{2}+\frac {\sin \left (a +\frac {b}{\left (d x +c \right )^{2}}\right ) c f x}{d}+\sin \left (a +\frac {b}{\left (d x +c \right )^{2}}\right ) e x +\frac {\sin \left (a +\frac {b}{\left (d x +c \right )^{2}}\right ) c e}{d}+\frac {f \left (\sqrt {b}\, \sqrt {2}\, \sqrt {\pi }\, \left (-\frac {\cos \left (a \right ) \sqrt {b}\, \sqrt {2}\, \left (-\frac {\operatorname {C}\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 \sqrt {\pi }}+\frac {\sin \left (a \right ) \sqrt {b}\, \sqrt {2}\, \left (-\frac {\operatorname {S}\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 {Si}\left (\frac {b}{\left (d x +c \right )^{2}}\right )}{2}\right )}{d \sqrt {\pi }}\right )+\frac {-\frac {\left (d x +c \right )^{2} \sin \left (a +\frac {b}{\left (d x +c \right )^{2}}\right )}{2}+b \left (\frac {\cos \left (a \right ) \operatorname {Ci}\left (\frac {b}{\left (d x +c \right )^{2}}\right )}{2}-\frac {\sin \left (a \right ) \operatorname {Si}\left (\frac {b}{\left (d x +c \right )^{2}}\right )}{2}\right )}{d}\right )}{d}\) | \(410\) |
1/d^2*(-(c*f-d*e)*(d*x+c)*sin(a+b/(d*x+c)^2)+(c*f-d*e)*b^(1/2)*2^(1/2)*Pi^ (1/2)*(cos(a)*FresnelC(b^(1/2)*2^(1/2)/Pi^(1/2)/(d*x+c))-sin(a)*FresnelS(b ^(1/2)*2^(1/2)/Pi^(1/2)/(d*x+c)))+1/2*f*(d*x+c)^2*sin(a+b/(d*x+c)^2)-f*b*( 1/2*cos(a)*Ci(b/(d*x+c)^2)-1/2*sin(a)*Si(b/(d*x+c)^2)))
Time = 0.29 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.18 \[ \int (e+f x) \sin \left (a+\frac {b}{(c+d x)^2}\right ) \, dx=-\frac {2 \, \sqrt {2} \pi {\left (d^{2} e - c d f\right )} \sqrt {\frac {b}{\pi d^{2}}} \cos \left (a\right ) \operatorname {C}\left (\frac {\sqrt {2} d \sqrt {\frac {b}{\pi d^{2}}}}{d x + c}\right ) - 2 \, \sqrt {2} \pi {\left (d^{2} e - c d f\right )} \sqrt {\frac {b}{\pi d^{2}}} \operatorname {S}\left (\frac {\sqrt {2} d \sqrt {\frac {b}{\pi d^{2}}}}{d x + c}\right ) \sin \left (a\right ) + b f \cos \left (a\right ) \operatorname {Ci}\left (\frac {b}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) - b f \sin \left (a\right ) \operatorname {Si}\left (\frac {b}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) - {\left (d^{2} f x^{2} + 2 \, d^{2} e x + 2 \, c d e - c^{2} f\right )} \sin \left (\frac {a d^{2} x^{2} + 2 \, a c d x + a c^{2} + b}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}{2 \, d^{2}} \]
-1/2*(2*sqrt(2)*pi*(d^2*e - c*d*f)*sqrt(b/(pi*d^2))*cos(a)*fresnel_cos(sqr t(2)*d*sqrt(b/(pi*d^2))/(d*x + c)) - 2*sqrt(2)*pi*(d^2*e - c*d*f)*sqrt(b/( pi*d^2))*fresnel_sin(sqrt(2)*d*sqrt(b/(pi*d^2))/(d*x + c))*sin(a) + b*f*co s(a)*cos_integral(b/(d^2*x^2 + 2*c*d*x + c^2)) - b*f*sin(a)*sin_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)*si n((a*d^2*x^2 + 2*a*c*d*x + a*c^2 + b)/(d^2*x^2 + 2*c*d*x + c^2)))/d^2
\[ \int (e+f x) \sin \left (a+\frac {b}{(c+d x)^2}\right ) \, dx=\int \left (e + f x\right ) \sin {\left (a + \frac {b}{c^{2} + 2 c d x + d^{2} x^{2}} \right )}\, dx \]
\[ \int (e+f x) \sin \left (a+\frac {b}{(c+d x)^2}\right ) \, dx=\int { {\left (f x + e\right )} \sin \left (a + \frac {b}{{\left (d x + c\right )}^{2}}\right ) \,d x } \]
1/2*(f*x^2 + 2*e*x)*sin((a*d^2*x^2 + 2*a*c*d*x + a*c^2 + 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((a*d^2*x^2 + 2*a* c*d*x + a*c^2 + 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((a*d^2*x^2 + 2*a*c*d*x + a*c^2 + 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((a*d^2*x^2 + 2*a*c*d*x + a*c^2 + 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((a*d^2*x^2 + 2*a*c*d*x + a*c^2 + b)/(d^2*x^2 + 2*c*d*x + c^2))^2), x)
\[ \int (e+f x) \sin \left (a+\frac {b}{(c+d x)^2}\right ) \, dx=\int { {\left (f x + e\right )} \sin \left (a + \frac {b}{{\left (d x + c\right )}^{2}}\right ) \,d x } \]
Timed out. \[ \int (e+f x) \sin \left (a+\frac {b}{(c+d x)^2}\right ) \, dx=\int \sin \left (a+\frac {b}{{\left (c+d\,x\right )}^2}\right )\,\left (e+f\,x\right ) \,d x \]