Integrand size = 19, antiderivative size = 435 \[ \int \frac {\sin (c+d x)}{x \left (a+b x^2\right )^2} \, dx=\frac {d \cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 (-a)^{3/2} \sqrt {b}}-\frac {d \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{4 (-a)^{3/2} \sqrt {b}}+\frac {\operatorname {CosIntegral}(d x) \sin (c)}{a^2}-\frac {\operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right ) \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 a^2}-\frac {\operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 a^2}+\frac {\sin (c+d x)}{2 a \left (a+b x^2\right )}+\frac {\cos (c) \text {Si}(d x)}{a^2}+\frac {\cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 a^2}+\frac {d \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 (-a)^{3/2} \sqrt {b}}-\frac {\cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{2 a^2}+\frac {d \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{4 (-a)^{3/2} \sqrt {b}} \]
cos(c)*Si(d*x)/a^2-1/2*cos(c+d*(-a)^(1/2)/b^(1/2))*Si(d*x-d*(-a)^(1/2)/b^( 1/2))/a^2-1/2*cos(c-d*(-a)^(1/2)/b^(1/2))*Si(d*x+d*(-a)^(1/2)/b^(1/2))/a^2 +Ci(d*x)*sin(c)/a^2+1/2*sin(d*x+c)/a/(b*x^2+a)-1/2*Ci(d*x+d*(-a)^(1/2)/b^( 1/2))*sin(c-d*(-a)^(1/2)/b^(1/2))/a^2-1/2*Ci(-d*x+d*(-a)^(1/2)/b^(1/2))*si n(c+d*(-a)^(1/2)/b^(1/2))/a^2-1/4*d*Ci(d*x+d*(-a)^(1/2)/b^(1/2))*cos(c-d*( -a)^(1/2)/b^(1/2))/(-a)^(3/2)/b^(1/2)+1/4*d*Ci(-d*x+d*(-a)^(1/2)/b^(1/2))* cos(c+d*(-a)^(1/2)/b^(1/2))/(-a)^(3/2)/b^(1/2)+1/4*d*Si(d*x+d*(-a)^(1/2)/b ^(1/2))*sin(c-d*(-a)^(1/2)/b^(1/2))/(-a)^(3/2)/b^(1/2)-1/4*d*Si(d*x-d*(-a) ^(1/2)/b^(1/2))*sin(c+d*(-a)^(1/2)/b^(1/2))/(-a)^(3/2)/b^(1/2)
Result contains complex when optimal does not.
Time = 0.95 (sec) , antiderivative size = 419, normalized size of antiderivative = 0.96 \[ \int \frac {\sin (c+d x)}{x \left (a+b x^2\right )^2} \, dx=\frac {\frac {i \sqrt {a} d e^{-i c-\frac {\sqrt {a} d}{\sqrt {b}}} \left (e^{\frac {2 \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (-\frac {\sqrt {a} d}{\sqrt {b}}-i d x\right )-\operatorname {ExpIntegralEi}\left (\frac {\sqrt {a} d}{\sqrt {b}}-i d x\right )\right )}{\sqrt {b}}-2 i e^{-i c-\frac {\sqrt {a} d}{\sqrt {b}}} \left (e^{\frac {2 \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (-\frac {\sqrt {a} d}{\sqrt {b}}-i d x\right )+\operatorname {ExpIntegralEi}\left (\frac {\sqrt {a} d}{\sqrt {b}}-i d x\right )\right )-\frac {i \sqrt {a} d e^{i c-\frac {\sqrt {a} d}{\sqrt {b}}} \left (e^{\frac {2 \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (-\frac {\sqrt {a} d}{\sqrt {b}}+i d x\right )-\operatorname {ExpIntegralEi}\left (\frac {\sqrt {a} d}{\sqrt {b}}+i d x\right )\right )}{\sqrt {b}}+2 i e^{i c-\frac {\sqrt {a} d}{\sqrt {b}}} \left (e^{\frac {2 \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (-\frac {\sqrt {a} d}{\sqrt {b}}+i d x\right )+\operatorname {ExpIntegralEi}\left (\frac {\sqrt {a} d}{\sqrt {b}}+i d x\right )\right )+\frac {4 a \cos (d x) \sin (c)}{a+b x^2}+\frac {4 a \cos (c) \sin (d x)}{a+b x^2}+8 (\operatorname {CosIntegral}(d x) \sin (c)+\cos (c) \text {Si}(d x))}{8 a^2} \]
((I*Sqrt[a]*d*E^((-I)*c - (Sqrt[a]*d)/Sqrt[b])*(E^((2*Sqrt[a]*d)/Sqrt[b])* ExpIntegralEi[-((Sqrt[a]*d)/Sqrt[b]) - I*d*x] - ExpIntegralEi[(Sqrt[a]*d)/ Sqrt[b] - I*d*x]))/Sqrt[b] - (2*I)*E^((-I)*c - (Sqrt[a]*d)/Sqrt[b])*(E^((2 *Sqrt[a]*d)/Sqrt[b])*ExpIntegralEi[-((Sqrt[a]*d)/Sqrt[b]) - I*d*x] + ExpIn tegralEi[(Sqrt[a]*d)/Sqrt[b] - I*d*x]) - (I*Sqrt[a]*d*E^(I*c - (Sqrt[a]*d) /Sqrt[b])*(E^((2*Sqrt[a]*d)/Sqrt[b])*ExpIntegralEi[-((Sqrt[a]*d)/Sqrt[b]) + I*d*x] - ExpIntegralEi[(Sqrt[a]*d)/Sqrt[b] + I*d*x]))/Sqrt[b] + (2*I)*E^ (I*c - (Sqrt[a]*d)/Sqrt[b])*(E^((2*Sqrt[a]*d)/Sqrt[b])*ExpIntegralEi[-((Sq rt[a]*d)/Sqrt[b]) + I*d*x] + ExpIntegralEi[(Sqrt[a]*d)/Sqrt[b] + I*d*x]) + (4*a*Cos[d*x]*Sin[c])/(a + b*x^2) + (4*a*Cos[c]*Sin[d*x])/(a + b*x^2) + 8 *(CosIntegral[d*x]*Sin[c] + Cos[c]*SinIntegral[d*x]))/(8*a^2)
Time = 0.99 (sec) , antiderivative size = 435, 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 = {3826, 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.
| \(\displaystyle \int \frac {\sin (c+d x)}{x \left (a+b x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 3826 |
| \(\displaystyle \int \left (-\frac {b x \sin (c+d x)}{a^2 \left (a+b x^2\right )}+\frac {\sin (c+d x)}{a^2 x}-\frac {b x \sin (c+d x)}{a \left (a+b x^2\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\sin \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 a^2}-\frac {\sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \operatorname {CosIntegral}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 a^2}+\frac {\cos \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 a^2}-\frac {\cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 a^2}+\frac {\sin (c) \operatorname {CosIntegral}(d x)}{a^2}+\frac {\cos (c) \text {Si}(d x)}{a^2}+\frac {d \cos \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 (-a)^{3/2} \sqrt {b}}-\frac {d \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \operatorname {CosIntegral}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 (-a)^{3/2} \sqrt {b}}+\frac {d \sin \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 (-a)^{3/2} \sqrt {b}}+\frac {d \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 (-a)^{3/2} \sqrt {b}}+\frac {\sin (c+d x)}{2 a \left (a+b x^2\right )}\) |
(d*Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/ (4*(-a)^(3/2)*Sqrt[b]) - (d*Cos[c - (Sqrt[-a]*d)/Sqrt[b]]*CosIntegral[(Sqr t[-a]*d)/Sqrt[b] + d*x])/(4*(-a)^(3/2)*Sqrt[b]) + (CosIntegral[d*x]*Sin[c] )/a^2 - (CosIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sin[c - (Sqrt[-a]*d)/Sqrt [b]])/(2*a^2) - (CosIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x]*Sin[c + (Sqrt[-a] *d)/Sqrt[b]])/(2*a^2) + Sin[c + d*x]/(2*a*(a + b*x^2)) + (Cos[c]*SinIntegr al[d*x])/a^2 + (Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqr t[b] - d*x])/(2*a^2) + (d*Sin[c + (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[ -a]*d)/Sqrt[b] - d*x])/(4*(-a)^(3/2)*Sqrt[b]) - (Cos[c - (Sqrt[-a]*d)/Sqrt [b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(2*a^2) + (d*Sin[c - (Sqrt[- a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(4*(-a)^(3/2)*Sqrt [b])
3.1.70.3.1 Defintions of rubi rules used
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*Sin[(c_.) + (d_.)*(x_)], x_Sym bol] :> Int[ExpandIntegrand[Sin[c + d*x], x^m*(a + b*x^n)^p, x], x] /; Free Q[{a, b, c, d, m}, x] && ILtQ[p, 0] && IGtQ[n, 0] && (EqQ[n, 2] || EqQ[p, - 1]) && IntegerQ[m]
Time = 0.40 (sec) , antiderivative size = 478, normalized size of antiderivative = 1.10
| method | result | size |
| derivativedivides | \(\frac {\sin \left (d x +c \right ) d^{2}}{2 a \left (a \,d^{2}+c^{2} b -2 b c \left (d x +c \right )+b \left (d x +c \right )^{2}\right )}+\frac {\operatorname {Si}\left (d x \right ) \cos \left (c \right )+\operatorname {Ci}\left (d x \right ) \sin \left (c \right )}{a^{2}}-\frac {\operatorname {Si}\left (d x +c -\frac {d \sqrt {-a b}+c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}+c b}{b}\right )+\operatorname {Ci}\left (d x +c -\frac {d \sqrt {-a b}+c b}{b}\right ) \sin \left (\frac {d \sqrt {-a b}+c b}{b}\right )}{2 a^{2}}-\frac {\operatorname {Si}\left (d x +c +\frac {d \sqrt {-a b}-c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}-c b}{b}\right )-\operatorname {Ci}\left (d x +c +\frac {d \sqrt {-a b}-c b}{b}\right ) \sin \left (\frac {d \sqrt {-a b}-c b}{b}\right )}{2 a^{2}}+\frac {d^{2} \left (-\operatorname {Si}\left (d x +c -\frac {d \sqrt {-a b}+c b}{b}\right ) \sin \left (\frac {d \sqrt {-a b}+c b}{b}\right )+\operatorname {Ci}\left (d x +c -\frac {d \sqrt {-a b}+c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}+c b}{b}\right )\right )}{4 a b \left (-\frac {d \sqrt {-a b}+c b}{b}+c \right )}+\frac {d^{2} \left (\operatorname {Si}\left (d x +c +\frac {d \sqrt {-a b}-c b}{b}\right ) \sin \left (\frac {d \sqrt {-a b}-c b}{b}\right )+\operatorname {Ci}\left (d x +c +\frac {d \sqrt {-a b}-c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}-c b}{b}\right )\right )}{4 a b \left (\frac {d \sqrt {-a b}-c b}{b}+c \right )}\) | \(478\) |
| default | \(\frac {\sin \left (d x +c \right ) d^{2}}{2 a \left (a \,d^{2}+c^{2} b -2 b c \left (d x +c \right )+b \left (d x +c \right )^{2}\right )}+\frac {\operatorname {Si}\left (d x \right ) \cos \left (c \right )+\operatorname {Ci}\left (d x \right ) \sin \left (c \right )}{a^{2}}-\frac {\operatorname {Si}\left (d x +c -\frac {d \sqrt {-a b}+c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}+c b}{b}\right )+\operatorname {Ci}\left (d x +c -\frac {d \sqrt {-a b}+c b}{b}\right ) \sin \left (\frac {d \sqrt {-a b}+c b}{b}\right )}{2 a^{2}}-\frac {\operatorname {Si}\left (d x +c +\frac {d \sqrt {-a b}-c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}-c b}{b}\right )-\operatorname {Ci}\left (d x +c +\frac {d \sqrt {-a b}-c b}{b}\right ) \sin \left (\frac {d \sqrt {-a b}-c b}{b}\right )}{2 a^{2}}+\frac {d^{2} \left (-\operatorname {Si}\left (d x +c -\frac {d \sqrt {-a b}+c b}{b}\right ) \sin \left (\frac {d \sqrt {-a b}+c b}{b}\right )+\operatorname {Ci}\left (d x +c -\frac {d \sqrt {-a b}+c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}+c b}{b}\right )\right )}{4 a b \left (-\frac {d \sqrt {-a b}+c b}{b}+c \right )}+\frac {d^{2} \left (\operatorname {Si}\left (d x +c +\frac {d \sqrt {-a b}-c b}{b}\right ) \sin \left (\frac {d \sqrt {-a b}-c b}{b}\right )+\operatorname {Ci}\left (d x +c +\frac {d \sqrt {-a b}-c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}-c b}{b}\right )\right )}{4 a b \left (\frac {d \sqrt {-a b}-c b}{b}+c \right )}\) | \(478\) |
| risch | \(\frac {i {\mathrm e}^{\frac {i c b +d \sqrt {a b}}{b}} \operatorname {Ei}_{1}\left (\frac {i c b +d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right ) d}{8 a \sqrt {a b}}-\frac {i {\mathrm e}^{\frac {i c b -d \sqrt {a b}}{b}} \operatorname {Ei}_{1}\left (\frac {i c b -d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right ) d}{8 a \sqrt {a b}}-\frac {i {\mathrm e}^{\frac {i c b +d \sqrt {a b}}{b}} \operatorname {Ei}_{1}\left (\frac {i c b +d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right )}{4 a^{2}}-\frac {i {\mathrm e}^{\frac {i c b -d \sqrt {a b}}{b}} \operatorname {Ei}_{1}\left (\frac {i c b -d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right )}{4 a^{2}}+\frac {i {\mathrm e}^{i c} \operatorname {Ei}_{1}\left (-i d x \right )}{2 a^{2}}+\frac {i {\mathrm e}^{-\frac {i c b +d \sqrt {a b}}{b}} \operatorname {Ei}_{1}\left (-\frac {i c b +d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right ) d}{8 a \sqrt {a b}}-\frac {i {\mathrm e}^{-\frac {i c b -d \sqrt {a b}}{b}} \operatorname {Ei}_{1}\left (-\frac {i c b -d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right ) d}{8 a \sqrt {a b}}+\frac {i {\mathrm e}^{-\frac {i c b +d \sqrt {a b}}{b}} \operatorname {Ei}_{1}\left (-\frac {i c b +d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right )}{4 a^{2}}+\frac {i {\mathrm e}^{-\frac {i c b -d \sqrt {a b}}{b}} \operatorname {Ei}_{1}\left (-\frac {i c b -d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right )}{4 a^{2}}-\frac {{\mathrm e}^{-i c} \pi \,\operatorname {csgn}\left (d x \right )}{2 a^{2}}+\frac {{\mathrm e}^{-i c} \operatorname {Si}\left (d x \right )}{a^{2}}-\frac {i {\mathrm e}^{-i c} \operatorname {Ei}_{1}\left (-i d x \right )}{2 a^{2}}+\frac {d^{2} \sin \left (d x +c \right )}{2 a \left (d^{2} x^{2} b +a \,d^{2}\right )}\) | \(584\) |
1/2*sin(d*x+c)*d^2/a/(a*d^2+c^2*b-2*b*c*(d*x+c)+b*(d*x+c)^2)+1/a^2*(Si(d*x )*cos(c)+Ci(d*x)*sin(c))-1/2/a^2*(Si(d*x+c-(d*(-a*b)^(1/2)+c*b)/b)*cos((d* (-a*b)^(1/2)+c*b)/b)+Ci(d*x+c-(d*(-a*b)^(1/2)+c*b)/b)*sin((d*(-a*b)^(1/2)+ c*b)/b))-1/2/a^2*(Si(d*x+c+(d*(-a*b)^(1/2)-c*b)/b)*cos((d*(-a*b)^(1/2)-c*b )/b)-Ci(d*x+c+(d*(-a*b)^(1/2)-c*b)/b)*sin((d*(-a*b)^(1/2)-c*b)/b))+1/4*d^2 /a/b/(-(d*(-a*b)^(1/2)+c*b)/b+c)*(-Si(d*x+c-(d*(-a*b)^(1/2)+c*b)/b)*sin((d *(-a*b)^(1/2)+c*b)/b)+Ci(d*x+c-(d*(-a*b)^(1/2)+c*b)/b)*cos((d*(-a*b)^(1/2) +c*b)/b))+1/4*d^2/a/b/((d*(-a*b)^(1/2)-c*b)/b+c)*(Si(d*x+c+(d*(-a*b)^(1/2) -c*b)/b)*sin((d*(-a*b)^(1/2)-c*b)/b)+Ci(d*x+c+(d*(-a*b)^(1/2)-c*b)/b)*cos( (d*(-a*b)^(1/2)-c*b)/b))
Result contains complex when optimal does not.
Time = 0.31 (sec) , antiderivative size = 320, normalized size of antiderivative = 0.74 \[ \int \frac {\sin (c+d x)}{x \left (a+b x^2\right )^2} \, dx=-\frac {{\left (-2 i \, b x^{2} - {\left (-i \, b x^{2} - i \, a\right )} \sqrt {\frac {a d^{2}}{b}} - 2 i \, a\right )} {\rm Ei}\left (i \, d x - \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (i \, c + \sqrt {\frac {a d^{2}}{b}}\right )} + {\left (-2 i \, b x^{2} - {\left (i \, b x^{2} + i \, a\right )} \sqrt {\frac {a d^{2}}{b}} - 2 i \, a\right )} {\rm Ei}\left (i \, d x + \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (i \, c - \sqrt {\frac {a d^{2}}{b}}\right )} + {\left (2 i \, b x^{2} - {\left (i \, b x^{2} + i \, a\right )} \sqrt {\frac {a d^{2}}{b}} + 2 i \, a\right )} {\rm Ei}\left (-i \, d x - \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (-i \, c + \sqrt {\frac {a d^{2}}{b}}\right )} + {\left (2 i \, b x^{2} - {\left (-i \, b x^{2} - i \, a\right )} \sqrt {\frac {a d^{2}}{b}} + 2 i \, a\right )} {\rm Ei}\left (-i \, d x + \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (-i \, c - \sqrt {\frac {a d^{2}}{b}}\right )} - 8 \, {\left (b x^{2} + a\right )} \operatorname {Ci}\left (d x\right ) \sin \left (c\right ) - 8 \, {\left (b x^{2} + a\right )} \cos \left (c\right ) \operatorname {Si}\left (d x\right ) - 4 \, a \sin \left (d x + c\right )}{8 \, {\left (a^{2} b x^{2} + a^{3}\right )}} \]
-1/8*((-2*I*b*x^2 - (-I*b*x^2 - I*a)*sqrt(a*d^2/b) - 2*I*a)*Ei(I*d*x - sqr t(a*d^2/b))*e^(I*c + sqrt(a*d^2/b)) + (-2*I*b*x^2 - (I*b*x^2 + I*a)*sqrt(a *d^2/b) - 2*I*a)*Ei(I*d*x + sqrt(a*d^2/b))*e^(I*c - sqrt(a*d^2/b)) + (2*I* b*x^2 - (I*b*x^2 + I*a)*sqrt(a*d^2/b) + 2*I*a)*Ei(-I*d*x - sqrt(a*d^2/b))* e^(-I*c + sqrt(a*d^2/b)) + (2*I*b*x^2 - (-I*b*x^2 - I*a)*sqrt(a*d^2/b) + 2 *I*a)*Ei(-I*d*x + sqrt(a*d^2/b))*e^(-I*c - sqrt(a*d^2/b)) - 8*(b*x^2 + a)* cos_integral(d*x)*sin(c) - 8*(b*x^2 + a)*cos(c)*sin_integral(d*x) - 4*a*si n(d*x + c))/(a^2*b*x^2 + a^3)
\[ \int \frac {\sin (c+d x)}{x \left (a+b x^2\right )^2} \, dx=\int \frac {\sin {\left (c + d x \right )}}{x \left (a + b x^{2}\right )^{2}}\, dx \]
\[ \int \frac {\sin (c+d x)}{x \left (a+b x^2\right )^2} \, dx=\int { \frac {\sin \left (d x + c\right )}{{\left (b x^{2} + a\right )}^{2} x} \,d x } \]
\[ \int \frac {\sin (c+d x)}{x \left (a+b x^2\right )^2} \, dx=\int { \frac {\sin \left (d x + c\right )}{{\left (b x^{2} + a\right )}^{2} x} \,d x } \]
Timed out. \[ \int \frac {\sin (c+d x)}{x \left (a+b x^2\right )^2} \, dx=\int \frac {\sin \left (c+d\,x\right )}{x\,{\left (b\,x^2+a\right )}^2} \,d x \]