Integrand size = 18, antiderivative size = 233 \[ \int (e+f x)^2 \sin \left (\frac {b}{(c+d x)^2}\right ) \, dx=\frac {2 b f^2 (c+d x) \cos \left (\frac {b}{(c+d x)^2}\right )}{3 d^3}-\frac {b f (d e-c f) \operatorname {CosIntegral}\left (\frac {b}{(c+d x)^2}\right )}{d^3}-\frac {\sqrt {b} (d e-c f)^2 \sqrt {2 \pi } \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{c+d x}\right )}{d^3}+\frac {2 b^{3/2} f^2 \sqrt {2 \pi } \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{c+d x}\right )}{3 d^3}+\frac {(d e-c f)^2 (c+d x) \sin \left (\frac {b}{(c+d x)^2}\right )}{d^3}+\frac {f (d e-c f) (c+d x)^2 \sin \left (\frac {b}{(c+d x)^2}\right )}{d^3}+\frac {f^2 (c+d x)^3 \sin \left (\frac {b}{(c+d x)^2}\right )}{3 d^3} \]
-b*f*(-c*f+d*e)*Ci(b/(d*x+c)^2)/d^3+2/3*b*f^2*(d*x+c)*cos(b/(d*x+c)^2)/d^3 +(-c*f+d*e)^2*(d*x+c)*sin(b/(d*x+c)^2)/d^3+f*(-c*f+d*e)*(d*x+c)^2*sin(b/(d *x+c)^2)/d^3+1/3*f^2*(d*x+c)^3*sin(b/(d*x+c)^2)/d^3+2/3*b^(3/2)*f^2*Fresne lS(b^(1/2)*2^(1/2)/Pi^(1/2)/(d*x+c))*2^(1/2)*Pi^(1/2)/d^3-(-c*f+d*e)^2*Fre snelC(b^(1/2)*2^(1/2)/Pi^(1/2)/(d*x+c))*b^(1/2)*2^(1/2)*Pi^(1/2)/d^3
Time = 0.88 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.14 \[ \int (e+f x)^2 \sin \left (\frac {b}{(c+d x)^2}\right ) \, dx=\frac {2 b c f^2 \cos \left (\frac {b}{(c+d x)^2}\right )+2 b d f^2 x \cos \left (\frac {b}{(c+d x)^2}\right )+3 b f (-d e+c f) \operatorname {CosIntegral}\left (\frac {b}{(c+d x)^2}\right )-3 \sqrt {b} (d e-c f)^2 \sqrt {2 \pi } \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{c+d x}\right )+2 b^{3/2} f^2 \sqrt {2 \pi } \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{c+d x}\right )+3 c d^2 e^2 \sin \left (\frac {b}{(c+d x)^2}\right )-3 c^2 d e f \sin \left (\frac {b}{(c+d x)^2}\right )+c^3 f^2 \sin \left (\frac {b}{(c+d x)^2}\right )+3 d^3 e^2 x \sin \left (\frac {b}{(c+d x)^2}\right )+3 d^3 e f x^2 \sin \left (\frac {b}{(c+d x)^2}\right )+d^3 f^2 x^3 \sin \left (\frac {b}{(c+d x)^2}\right )}{3 d^3} \]
(2*b*c*f^2*Cos[b/(c + d*x)^2] + 2*b*d*f^2*x*Cos[b/(c + d*x)^2] + 3*b*f*(-( d*e) + c*f)*CosIntegral[b/(c + d*x)^2] - 3*Sqrt[b]*(d*e - c*f)^2*Sqrt[2*Pi ]*FresnelC[(Sqrt[b]*Sqrt[2/Pi])/(c + d*x)] + 2*b^(3/2)*f^2*Sqrt[2*Pi]*Fres nelS[(Sqrt[b]*Sqrt[2/Pi])/(c + d*x)] + 3*c*d^2*e^2*Sin[b/(c + d*x)^2] - 3* c^2*d*e*f*Sin[b/(c + d*x)^2] + c^3*f^2*Sin[b/(c + d*x)^2] + 3*d^3*e^2*x*Si n[b/(c + d*x)^2] + 3*d^3*e*f*x^2*Sin[b/(c + d*x)^2] + d^3*f^2*x^3*Sin[b/(c + d*x)^2])/(3*d^3)
Time = 0.39 (sec) , antiderivative size = 216, 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.
| \(\displaystyle \int (e+f x)^2 \sin \left (\frac {b}{(c+d x)^2}\right ) \, dx\) |
| \(\Big \downarrow \) 3914 |
| \(\displaystyle \frac {\int \left (\sin \left (\frac {b}{(c+d x)^2}\right ) (d e-c f)^2+2 f (c+d x) \sin \left (\frac {b}{(c+d x)^2}\right ) (d e-c f)+f^2 (c+d x)^2 \sin \left (\frac {b}{(c+d x)^2}\right )\right )d(c+d x)}{d^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {2}{3} \sqrt {2 \pi } b^{3/2} f^2 \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{c+d x}\right )-b f (d e-c f) \operatorname {CosIntegral}\left (\frac {b}{(c+d x)^2}\right )-\sqrt {2 \pi } \sqrt {b} (d e-c f)^2 \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{c+d x}\right )+f (c+d x)^2 (d e-c f) \sin \left (\frac {b}{(c+d x)^2}\right )+(c+d x) (d e-c f)^2 \sin \left (\frac {b}{(c+d x)^2}\right )+\frac {1}{3} f^2 (c+d x)^3 \sin \left (\frac {b}{(c+d x)^2}\right )+\frac {2}{3} b f^2 (c+d x) \cos \left (\frac {b}{(c+d x)^2}\right )}{d^3}\) |
((2*b*f^2*(c + d*x)*Cos[b/(c + d*x)^2])/3 - b*f*(d*e - c*f)*CosIntegral[b/ (c + d*x)^2] - Sqrt[b]*(d*e - c*f)^2*Sqrt[2*Pi]*FresnelC[(Sqrt[b]*Sqrt[2/P i])/(c + d*x)] + (2*b^(3/2)*f^2*Sqrt[2*Pi]*FresnelS[(Sqrt[b]*Sqrt[2/Pi])/( c + d*x)])/3 + (d*e - c*f)^2*(c + d*x)*Sin[b/(c + d*x)^2] + f*(d*e - c*f)* (c + d*x)^2*Sin[b/(c + d*x)^2] + (f^2*(c + d*x)^3*Sin[b/(c + d*x)^2])/3)/d ^3
3.2.60.3.1 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 = 0.80 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.84
| method | result | size |
| derivativedivides | \(-\frac {-\left (c f -d e \right )^{2} \left (d x +c \right ) \sin \left (\frac {b}{\left (d x +c \right )^{2}}\right )+\left (c f -d e \right )^{2} \sqrt {b}\, \sqrt {2}\, \sqrt {\pi }\, \operatorname {C}\left (\frac {\sqrt {b}\, \sqrt {2}}{\sqrt {\pi }\, \left (d x +c \right )}\right )+f \left (c f -d e \right ) \left (d x +c \right )^{2} \sin \left (\frac {b}{\left (d x +c \right )^{2}}\right )-f \left (c f -d e \right ) b \,\operatorname {Ci}\left (\frac {b}{\left (d x +c \right )^{2}}\right )-\frac {f^{2} \left (d x +c \right )^{3} \sin \left (\frac {b}{\left (d x +c \right )^{2}}\right )}{3}+\frac {2 f^{2} b \left (-\left (d x +c \right ) \cos \left (\frac {b}{\left (d x +c \right )^{2}}\right )-\sqrt {b}\, \sqrt {2}\, \sqrt {\pi }\, \operatorname {S}\left (\frac {\sqrt {b}\, \sqrt {2}}{\sqrt {\pi }\, \left (d x +c \right )}\right )\right )}{3}}{d^{3}}\) | \(196\) |
| default | \(-\frac {-\left (c f -d e \right )^{2} \left (d x +c \right ) \sin \left (\frac {b}{\left (d x +c \right )^{2}}\right )+\left (c f -d e \right )^{2} \sqrt {b}\, \sqrt {2}\, \sqrt {\pi }\, \operatorname {C}\left (\frac {\sqrt {b}\, \sqrt {2}}{\sqrt {\pi }\, \left (d x +c \right )}\right )+f \left (c f -d e \right ) \left (d x +c \right )^{2} \sin \left (\frac {b}{\left (d x +c \right )^{2}}\right )-f \left (c f -d e \right ) b \,\operatorname {Ci}\left (\frac {b}{\left (d x +c \right )^{2}}\right )-\frac {f^{2} \left (d x +c \right )^{3} \sin \left (\frac {b}{\left (d x +c \right )^{2}}\right )}{3}+\frac {2 f^{2} b \left (-\left (d x +c \right ) \cos \left (\frac {b}{\left (d x +c \right )^{2}}\right )-\sqrt {b}\, \sqrt {2}\, \sqrt {\pi }\, \operatorname {S}\left (\frac {\sqrt {b}\, \sqrt {2}}{\sqrt {\pi }\, \left (d x +c \right )}\right )\right )}{3}}{d^{3}}\) | \(196\) |
| risch | \(-\frac {b \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {-i b}}{d x +c}\right ) c^{2} f^{2}}{2 d^{3} \sqrt {-i b}}+\frac {b \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {-i b}}{d x +c}\right ) c e f}{d^{2} \sqrt {-i b}}-\frac {b \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {-i b}}{d x +c}\right ) e^{2}}{2 d \sqrt {-i b}}-\frac {b \,\operatorname {Ei}_{1}\left (-\frac {i b}{\left (d x +c \right )^{2}}\right ) c \,f^{2}}{2 d^{3}}+\frac {b \,\operatorname {Ei}_{1}\left (-\frac {i b}{\left (d x +c \right )^{2}}\right ) e f}{2 d^{2}}-\frac {i b^{2} f^{2} \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {-i b}}{d x +c}\right )}{3 d^{3} \sqrt {-i b}}-\frac {b \,\operatorname {erf}\left (\frac {\sqrt {i b}}{d x +c}\right ) \sqrt {\pi }\, c^{2} f^{2}}{2 d^{3} \sqrt {i b}}+\frac {b \,\operatorname {erf}\left (\frac {\sqrt {i b}}{d x +c}\right ) \sqrt {\pi }\, c e f}{d^{2} \sqrt {i b}}-\frac {b \,\operatorname {erf}\left (\frac {\sqrt {i b}}{d x +c}\right ) \sqrt {\pi }\, e^{2}}{2 d \sqrt {i b}}-\frac {b \,\operatorname {Ei}_{1}\left (\frac {i b}{\left (d x +c \right )^{2}}\right ) c \,f^{2}}{2 d^{3}}+\frac {b \,\operatorname {Ei}_{1}\left (\frac {i b}{\left (d x +c \right )^{2}}\right ) e f}{2 d^{2}}+\frac {i b^{2} f^{2} \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {i b}}{d x +c}\right )}{3 d^{3} \sqrt {i b}}+\frac {2 b \,f^{2} \left (d x +c \right ) \cos \left (\frac {b}{\left (d x +c \right )^{2}}\right )}{3 d^{3}}+\frac {\left (f^{2} x^{3} d^{3}+3 x^{2} d^{3} e f +3 x \,d^{3} e^{2}+c^{3} f^{2}-3 c^{2} d e f +3 c \,d^{2} e^{2}\right ) \sin \left (\frac {b}{\left (d x +c \right )^{2}}\right )}{3 d^{3}}\) | \(457\) |
| parts | \(-\frac {\sqrt {\pi }\, \sqrt {b}\, \sqrt {2}\, \operatorname {C}\left (\frac {\sqrt {b}\, \sqrt {2}}{\sqrt {\pi }\, \left (d x +c \right )}\right ) f^{2} x^{2}}{d}+\sin \left (\frac {b}{\left (d x +c \right )^{2}}\right ) f^{2} x^{3}-\frac {2 \sqrt {\pi }\, \sqrt {b}\, \sqrt {2}\, \operatorname {C}\left (\frac {\sqrt {b}\, \sqrt {2}}{\sqrt {\pi }\, \left (d x +c \right )}\right ) e f x}{d}+\frac {\sin \left (\frac {b}{\left (d x +c \right )^{2}}\right ) c \,f^{2} x^{2}}{d}+2 \sin \left (\frac {b}{\left (d x +c \right )^{2}}\right ) e f \,x^{2}-\frac {\sqrt {\pi }\, \sqrt {b}\, \sqrt {2}\, \operatorname {C}\left (\frac {\sqrt {b}\, \sqrt {2}}{\sqrt {\pi }\, \left (d x +c \right )}\right ) e^{2}}{d}+\frac {2 \sin \left (\frac {b}{\left (d x +c \right )^{2}}\right ) c e f x}{d}+\sin \left (\frac {b}{\left (d x +c \right )^{2}}\right ) e^{2} x +\frac {\sin \left (\frac {b}{\left (d x +c \right )^{2}}\right ) c \,e^{2}}{d}+\frac {2 f \left (-\frac {2 b \left (-\frac {\operatorname {C}\left (\frac {\sqrt {b}\, \sqrt {2}}{\sqrt {\pi }\, \left (d x +c \right )}\right ) \left (-\frac {\pi \left (c f -d e \right ) \sqrt {2}\, \left (d x +c \right )}{2 \sqrt {b}}+\frac {\sqrt {2}\, f \pi \left (d x +c \right )^{2}}{4 \sqrt {b}}\right )}{\sqrt {\pi }\, d}+\frac {-\frac {\left (2 \sqrt {\pi }\, c f -2 \sqrt {\pi }\, d e \right ) \operatorname {Ci}\left (\frac {b}{\left (d x +c \right )^{2}}\right )}{2}-f \sqrt {\pi }\, \left (d x +c \right ) \cos \left (\frac {b}{\left (d x +c \right )^{2}}\right )-\sqrt {b}\, \sqrt {2}\, f \pi \,\operatorname {S}\left (\frac {\sqrt {b}\, \sqrt {2}}{\sqrt {\pi }\, \left (d x +c \right )}\right )}{2 \sqrt {\pi }\, d}\right )}{d}-\frac {-\frac {\left (c f -d e \right ) \left (d x +c \right )^{2} \sin \left (\frac {b}{\left (d x +c \right )^{2}}\right )}{2}+\frac {\left (c f -d e \right ) b \,\operatorname {Ci}\left (\frac {b}{\left (d x +c \right )^{2}}\right )}{2}+\frac {f \left (d x +c \right )^{3} \sin \left (\frac {b}{\left (d x +c \right )^{2}}\right )}{3}-\frac {2 f b \left (-\left (d x +c \right ) \cos \left (\frac {b}{\left (d x +c \right )^{2}}\right )-\sqrt {b}\, \sqrt {2}\, \sqrt {\pi }\, \operatorname {S}\left (\frac {\sqrt {b}\, \sqrt {2}}{\sqrt {\pi }\, \left (d x +c \right )}\right )\right )}{3}}{d^{2}}\right )}{d}\) | \(507\) |
-1/d^3*(-(c*f-d*e)^2*(d*x+c)*sin(b/(d*x+c)^2)+(c*f-d*e)^2*b^(1/2)*2^(1/2)* Pi^(1/2)*FresnelC(b^(1/2)*2^(1/2)/Pi^(1/2)/(d*x+c))+f*(c*f-d*e)*(d*x+c)^2* sin(b/(d*x+c)^2)-f*(c*f-d*e)*b*Ci(b/(d*x+c)^2)-1/3*f^2*(d*x+c)^3*sin(b/(d* x+c)^2)+2/3*f^2*b*(-(d*x+c)*cos(b/(d*x+c)^2)-b^(1/2)*2^(1/2)*Pi^(1/2)*Fres nelS(b^(1/2)*2^(1/2)/Pi^(1/2)/(d*x+c))))
Time = 0.31 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.12 \[ \int (e+f x)^2 \sin \left (\frac {b}{(c+d x)^2}\right ) \, dx=\frac {2 \, \sqrt {2} \pi b d f^{2} \sqrt {\frac {b}{\pi d^{2}}} \operatorname {S}\left (\frac {\sqrt {2} d \sqrt {\frac {b}{\pi d^{2}}}}{d x + c}\right ) - 3 \, \sqrt {2} \pi {\left (d^{3} e^{2} - 2 \, c d^{2} e f + c^{2} d f^{2}\right )} \sqrt {\frac {b}{\pi d^{2}}} \operatorname {C}\left (\frac {\sqrt {2} d \sqrt {\frac {b}{\pi d^{2}}}}{d x + c}\right ) + 2 \, {\left (b d f^{2} x + b c f^{2}\right )} \cos \left (\frac {b}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) - 3 \, {\left (b d e f - b c f^{2}\right )} \operatorname {Ci}\left (\frac {b}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) + {\left (d^{3} f^{2} x^{3} + 3 \, d^{3} e f x^{2} + 3 \, d^{3} e^{2} x + 3 \, c d^{2} e^{2} - 3 \, c^{2} d e f + c^{3} f^{2}\right )} \sin \left (\frac {b}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}{3 \, d^{3}} \]
1/3*(2*sqrt(2)*pi*b*d*f^2*sqrt(b/(pi*d^2))*fresnel_sin(sqrt(2)*d*sqrt(b/(p i*d^2))/(d*x + c)) - 3*sqrt(2)*pi*(d^3*e^2 - 2*c*d^2*e*f + c^2*d*f^2)*sqrt (b/(pi*d^2))*fresnel_cos(sqrt(2)*d*sqrt(b/(pi*d^2))/(d*x + c)) + 2*(b*d*f^ 2*x + b*c*f^2)*cos(b/(d^2*x^2 + 2*c*d*x + c^2)) - 3*(b*d*e*f - b*c*f^2)*co s_integral(b/(d^2*x^2 + 2*c*d*x + c^2)) + (d^3*f^2*x^3 + 3*d^3*e*f*x^2 + 3 *d^3*e^2*x + 3*c*d^2*e^2 - 3*c^2*d*e*f + c^3*f^2)*sin(b/(d^2*x^2 + 2*c*d*x + c^2)))/d^3
\[ \int (e+f x)^2 \sin \left (\frac {b}{(c+d x)^2}\right ) \, dx=\int \left (e + f x\right )^{2} \sin {\left (\frac {b}{c^{2} + 2 c d x + d^{2} x^{2}} \right )}\, dx \]
\[ \int (e+f x)^2 \sin \left (\frac {b}{(c+d x)^2}\right ) \, dx=\int { {\left (f x + e\right )}^{2} \sin \left (\frac {b}{{\left (d x + c\right )}^{2}}\right ) \,d x } \]
1/3*(2*b*f^2*x*cos(b/(d^2*x^2 + 2*c*d*x + c^2)) - 3*d^2*integrate(1/3*(2*b ^2*d*f^2*x*sin(b/(d^2*x^2 + 2*c*d*x + c^2)) + (b*c^3*f^2 - 3*(b*d^3*e*f - b*c*d^2*f^2)*x^2 - 3*(b*d^3*e^2 - b*c^2*d*f^2)*x)*cos(b/(d^2*x^2 + 2*c*d*x + c^2)))/(d^5*x^3 + 3*c*d^4*x^2 + 3*c^2*d^3*x + c^3*d^2), x) - 3*d^2*inte grate(1/3*(2*b^2*d*f^2*x*sin(b/(d^2*x^2 + 2*c*d*x + c^2)) + (b*c^3*f^2 - 3 *(b*d^3*e*f - b*c*d^2*f^2)*x^2 - 3*(b*d^3*e^2 - b*c^2*d*f^2)*x)*cos(b/(d^2 *x^2 + 2*c*d*x + c^2)))/((d^5*x^3 + 3*c*d^4*x^2 + 3*c^2*d^3*x + c^3*d^2)*c os(b/(d^2*x^2 + 2*c*d*x + c^2))^2 + (d^5*x^3 + 3*c*d^4*x^2 + 3*c^2*d^3*x + c^3*d^2)*sin(b/(d^2*x^2 + 2*c*d*x + c^2))^2), x) + (d^2*f^2*x^3 + 3*d^2*e *f*x^2 + 3*d^2*e^2*x)*sin(b/(d^2*x^2 + 2*c*d*x + c^2)))/d^2
\[ \int (e+f x)^2 \sin \left (\frac {b}{(c+d x)^2}\right ) \, dx=\int { {\left (f x + e\right )}^{2} \sin \left (\frac {b}{{\left (d x + c\right )}^{2}}\right ) \,d x } \]
Timed out. \[ \int (e+f x)^2 \sin \left (\frac {b}{(c+d x)^2}\right ) \, dx=\int \sin \left (\frac {b}{{\left (c+d\,x\right )}^2}\right )\,{\left (e+f\,x\right )}^2 \,d x \]