Integrand size = 19, antiderivative size = 501 \[ \int \frac {\sin (c+d x)}{x^2 \left (a+b x^2\right )^2} \, dx=\frac {d \cos (c) \operatorname {CosIntegral}(d x)}{a^2}+\frac {d \cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 a^2}+\frac {d \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{4 a^2}+\frac {3 \sqrt {b} \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right ) \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 (-a)^{5/2}}-\frac {3 \sqrt {b} \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 (-a)^{5/2}}-\frac {\sin (c+d x)}{a^2 x}+\frac {\sqrt {b} \sin (c+d x)}{4 a^2 \left (\sqrt {-a}-\sqrt {b} x\right )}-\frac {\sqrt {b} \sin (c+d x)}{4 a^2 \left (\sqrt {-a}+\sqrt {b} x\right )}-\frac {d \sin (c) \text {Si}(d x)}{a^2}+\frac {3 \sqrt {b} \cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 (-a)^{5/2}}+\frac {d \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 a^2}+\frac {3 \sqrt {b} \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{4 (-a)^{5/2}}-\frac {d \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{4 a^2} \]
d*Ci(d*x)*cos(c)/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^2+1/4*d*Ci(-d*x+d*(-a)^(1/2)/b^(1/2))*cos(c+d*(-a)^(1/2)/b^(1/ 2))/a^2-d*Si(d*x)*sin(c)/a^2-sin(d*x+c)/a^2/x-1/4*d*Si(d*x+d*(-a)^(1/2)/b^ (1/2))*sin(c-d*(-a)^(1/2)/b^(1/2))/a^2-1/4*d*Si(d*x-d*(-a)^(1/2)/b^(1/2))* sin(c+d*(-a)^(1/2)/b^(1/2))/a^2-3/4*cos(c+d*(-a)^(1/2)/b^(1/2))*Si(d*x-d*( -a)^(1/2)/b^(1/2))*b^(1/2)/(-a)^(5/2)+3/4*cos(c-d*(-a)^(1/2)/b^(1/2))*Si(d *x+d*(-a)^(1/2)/b^(1/2))*b^(1/2)/(-a)^(5/2)+3/4*Ci(d*x+d*(-a)^(1/2)/b^(1/2 ))*sin(c-d*(-a)^(1/2)/b^(1/2))*b^(1/2)/(-a)^(5/2)-3/4*Ci(-d*x+d*(-a)^(1/2) /b^(1/2))*sin(c+d*(-a)^(1/2)/b^(1/2))*b^(1/2)/(-a)^(5/2)+1/4*sin(d*x+c)*b^ (1/2)/a^2/((-a)^(1/2)-x*b^(1/2))-1/4*sin(d*x+c)*b^(1/2)/a^2/((-a)^(1/2)+x* b^(1/2))
Result contains complex when optimal does not.
Time = 2.30 (sec) , antiderivative size = 330, normalized size of antiderivative = 0.66 \[ \int \frac {\sin (c+d x)}{x^2 \left (a+b x^2\right )^2} \, dx=\frac {e^{-i c-\frac {\sqrt {a} d}{\sqrt {b}}} \left (-\left (\left (3 \sqrt {b}-\sqrt {a} d\right ) e^{\frac {2 \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (-\frac {\sqrt {a} d}{\sqrt {b}}-i d x\right )\right )+\left (3 \sqrt {b}+\sqrt {a} d\right ) \operatorname {ExpIntegralEi}\left (\frac {\sqrt {a} d}{\sqrt {b}}-i d x\right )\right )+e^{i c-\frac {\sqrt {a} d}{\sqrt {b}}} \left (-\left (\left (3 \sqrt {b}-\sqrt {a} d\right ) e^{\frac {2 \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (-\frac {\sqrt {a} d}{\sqrt {b}}+i d x\right )\right )+\left (3 \sqrt {b}+\sqrt {a} d\right ) \operatorname {ExpIntegralEi}\left (\frac {\sqrt {a} d}{\sqrt {b}}+i d x\right )\right )-\frac {4 \sqrt {a} \left (2 a+3 b x^2\right ) \cos (d x) \sin (c)}{x \left (a+b x^2\right )}-\frac {4 \sqrt {a} \left (2 a+3 b x^2\right ) \cos (c) \sin (d x)}{x \left (a+b x^2\right )}+8 \sqrt {a} d (\cos (c) \operatorname {CosIntegral}(d x)-\sin (c) \text {Si}(d x))}{8 a^{5/2}} \]
(E^((-I)*c - (Sqrt[a]*d)/Sqrt[b])*(-((3*Sqrt[b] - Sqrt[a]*d)*E^((2*Sqrt[a] *d)/Sqrt[b])*ExpIntegralEi[-((Sqrt[a]*d)/Sqrt[b]) - I*d*x]) + (3*Sqrt[b] + Sqrt[a]*d)*ExpIntegralEi[(Sqrt[a]*d)/Sqrt[b] - I*d*x]) + E^(I*c - (Sqrt[a ]*d)/Sqrt[b])*(-((3*Sqrt[b] - Sqrt[a]*d)*E^((2*Sqrt[a]*d)/Sqrt[b])*ExpInte gralEi[-((Sqrt[a]*d)/Sqrt[b]) + I*d*x]) + (3*Sqrt[b] + Sqrt[a]*d)*ExpInteg ralEi[(Sqrt[a]*d)/Sqrt[b] + I*d*x]) - (4*Sqrt[a]*(2*a + 3*b*x^2)*Cos[d*x]* Sin[c])/(x*(a + b*x^2)) - (4*Sqrt[a]*(2*a + 3*b*x^2)*Cos[c]*Sin[d*x])/(x*( a + b*x^2)) + 8*Sqrt[a]*d*(Cos[c]*CosIntegral[d*x] - Sin[c]*SinIntegral[d* x]))/(8*a^(5/2))
Time = 1.36 (sec) , antiderivative size = 501, 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.
| \(\displaystyle \int \frac {\sin (c+d x)}{x^2 \left (a+b x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 3826 |
| \(\displaystyle \int \left (-\frac {b \sin (c+d x)}{a^2 \left (a+b x^2\right )}+\frac {\sin (c+d x)}{a^2 x^2}-\frac {b \sin (c+d x)}{a \left (a+b x^2\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {d \cos \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 a^2}+\frac {d \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \operatorname {CosIntegral}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 a^2}+\frac {d \sin \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 a^2}-\frac {d \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 a^2}+\frac {\sqrt {b} \sin (c+d x)}{4 a^2 \left (\sqrt {-a}-\sqrt {b} x\right )}-\frac {\sqrt {b} \sin (c+d x)}{4 a^2 \left (\sqrt {-a}+\sqrt {b} x\right )}+\frac {d \cos (c) \operatorname {CosIntegral}(d x)}{a^2}-\frac {d \sin (c) \text {Si}(d x)}{a^2}-\frac {\sin (c+d x)}{a^2 x}+\frac {3 \sqrt {b} \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \operatorname {CosIntegral}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 (-a)^{5/2}}-\frac {3 \sqrt {b} \sin \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 (-a)^{5/2}}+\frac {3 \sqrt {b} \cos \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 (-a)^{5/2}}+\frac {3 \sqrt {b} \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 (-a)^{5/2}}\) |
(d*Cos[c]*CosIntegral[d*x])/a^2 + (d*Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*CosInte gral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(4*a^2) + (d*Cos[c - (Sqrt[-a]*d)/Sqrt[b ]]*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(4*a^2) + (3*Sqrt[b]*CosIntegr al[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sin[c - (Sqrt[-a]*d)/Sqrt[b]])/(4*(-a)^(5/2 )) - (3*Sqrt[b]*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x]*Sin[c + (Sqrt[-a]* d)/Sqrt[b]])/(4*(-a)^(5/2)) - Sin[c + d*x]/(a^2*x) + (Sqrt[b]*Sin[c + d*x] )/(4*a^2*(Sqrt[-a] - Sqrt[b]*x)) - (Sqrt[b]*Sin[c + d*x])/(4*a^2*(Sqrt[-a] + Sqrt[b]*x)) - (d*Sin[c]*SinIntegral[d*x])/a^2 + (3*Sqrt[b]*Cos[c + (Sqr t[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(4*(-a)^(5/2)) + (d*Sin[c + (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x] )/(4*a^2) + (3*Sqrt[b]*Cos[c - (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a] *d)/Sqrt[b] + d*x])/(4*(-a)^(5/2)) - (d*Sin[c - (Sqrt[-a]*d)/Sqrt[b]]*SinI ntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(4*a^2)
3.1.71.3.1 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]
Result contains complex when optimal does not.
Time = 0.48 (sec) , antiderivative size = 622, normalized size of antiderivative = 1.24
| method | result | size |
| risch | \(-\frac {d \,{\mathrm e}^{\frac {i c b +d \sqrt {a b}}{b}} \operatorname {Ei}_{1}\left (\frac {i c b +d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right )}{8 a^{2}}-\frac {d \,{\mathrm e}^{\frac {i c b -d \sqrt {a b}}{b}} \operatorname {Ei}_{1}\left (\frac {i c b -d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right )}{8 a^{2}}+\frac {3 \,{\mathrm e}^{\frac {i c b +d \sqrt {a b}}{b}} \operatorname {Ei}_{1}\left (\frac {i c b +d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right ) b}{8 a^{2} \sqrt {a b}}-\frac {3 \,{\mathrm e}^{\frac {i c b -d \sqrt {a b}}{b}} \operatorname {Ei}_{1}\left (\frac {i c b -d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right ) b}{8 a^{2} \sqrt {a b}}-\frac {d \,\operatorname {Ei}_{1}\left (-i d x \right ) {\mathrm e}^{i c}}{2 a^{2}}-\frac {d \,{\mathrm e}^{-\frac {i c b -d \sqrt {a b}}{b}} \operatorname {Ei}_{1}\left (-\frac {i c b -d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right )}{8 a^{2}}-\frac {d \,{\mathrm e}^{-\frac {i c b +d \sqrt {a b}}{b}} \operatorname {Ei}_{1}\left (-\frac {i c b +d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right )}{8 a^{2}}+\frac {3 \,{\mathrm e}^{-\frac {i c b -d \sqrt {a b}}{b}} \operatorname {Ei}_{1}\left (-\frac {i c b -d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right ) b}{8 a^{2} \sqrt {a b}}-\frac {3 \,{\mathrm e}^{-\frac {i c b +d \sqrt {a b}}{b}} \operatorname {Ei}_{1}\left (-\frac {i c b +d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right ) b}{8 a^{2} \sqrt {a b}}-\frac {d \,\operatorname {Ei}_{1}\left (i d x \right ) {\mathrm e}^{-i c}}{2 a^{2}}-\frac {\left (6 i \left (i d x +i c \right ) b c -3 b \left (i d x +i c \right )^{2}+2 a \,d^{2}+3 c^{2} b \right ) \sin \left (d x +c \right )}{2 a^{2} \left (2 i \left (i d x +i c \right ) b c -b \left (i d x +i c \right )^{2}+a \,d^{2}+c^{2} b \right ) x}\) | \(622\) |
| derivativedivides | \(d \left (\frac {-\frac {\sin \left (d x +c \right )}{d x}-\operatorname {Si}\left (d x \right ) \sin \left (c \right )+\operatorname {Ci}\left (d x \right ) \cos \left (c \right )}{a^{2}}-\frac {b \left (-\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 b \left (-\frac {d \sqrt {-a b}+c b}{b}+c \right )}-\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 b \left (\frac {d \sqrt {-a b}-c b}{b}+c \right )}\right )}{a^{2}}-\frac {b \,d^{2} \left (\frac {\sin \left (d x +c \right ) \left (\frac {d x +c}{2 a \,d^{2}}-\frac {c}{2 a \,d^{2}}\right )}{a \,d^{2}+c^{2} b -2 b c \left (d x +c \right )+b \left (d x +c \right )^{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 )}{4 a \,d^{2} b \left (-\frac {d \sqrt {-a b}+c b}{b}+c \right )}-\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 )}{4 a \,d^{2} b \left (\frac {d \sqrt {-a b}-c b}{b}+c \right )}-\frac {-\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 )}{4 a b \,d^{2}}-\frac {\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 )}{4 a b \,d^{2}}\right )}{a}\right )\) | \(761\) |
| default | \(d \left (\frac {-\frac {\sin \left (d x +c \right )}{d x}-\operatorname {Si}\left (d x \right ) \sin \left (c \right )+\operatorname {Ci}\left (d x \right ) \cos \left (c \right )}{a^{2}}-\frac {b \left (-\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 b \left (-\frac {d \sqrt {-a b}+c b}{b}+c \right )}-\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 b \left (\frac {d \sqrt {-a b}-c b}{b}+c \right )}\right )}{a^{2}}-\frac {b \,d^{2} \left (\frac {\sin \left (d x +c \right ) \left (\frac {d x +c}{2 a \,d^{2}}-\frac {c}{2 a \,d^{2}}\right )}{a \,d^{2}+c^{2} b -2 b c \left (d x +c \right )+b \left (d x +c \right )^{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 )}{4 a \,d^{2} b \left (-\frac {d \sqrt {-a b}+c b}{b}+c \right )}-\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 )}{4 a \,d^{2} b \left (\frac {d \sqrt {-a b}-c b}{b}+c \right )}-\frac {-\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 )}{4 a b \,d^{2}}-\frac {\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 )}{4 a b \,d^{2}}\right )}{a}\right )\) | \(761\) |
-1/8*d/a^2*exp((I*c*b+d*(a*b)^(1/2))/b)*Ei(1,(I*c*b+d*(a*b)^(1/2)-b*(I*d*x +I*c))/b)-1/8*d/a^2*exp((I*c*b-d*(a*b)^(1/2))/b)*Ei(1,(I*c*b-d*(a*b)^(1/2) -b*(I*d*x+I*c))/b)+3/8/a^2/(a*b)^(1/2)*exp((I*c*b+d*(a*b)^(1/2))/b)*Ei(1,( I*c*b+d*(a*b)^(1/2)-b*(I*d*x+I*c))/b)*b-3/8/a^2/(a*b)^(1/2)*exp((I*c*b-d*( a*b)^(1/2))/b)*Ei(1,(I*c*b-d*(a*b)^(1/2)-b*(I*d*x+I*c))/b)*b-1/2*d/a^2*Ei( 1,-I*d*x)*exp(I*c)-1/8*d/a^2*exp(-(I*c*b-d*(a*b)^(1/2))/b)*Ei(1,-(I*c*b-d* (a*b)^(1/2)-b*(I*d*x+I*c))/b)-1/8*d/a^2*exp(-(I*c*b+d*(a*b)^(1/2))/b)*Ei(1 ,-(I*c*b+d*(a*b)^(1/2)-b*(I*d*x+I*c))/b)+3/8/a^2/(a*b)^(1/2)*exp(-(I*c*b-d *(a*b)^(1/2))/b)*Ei(1,-(I*c*b-d*(a*b)^(1/2)-b*(I*d*x+I*c))/b)*b-3/8/a^2/(a *b)^(1/2)*exp(-(I*c*b+d*(a*b)^(1/2))/b)*Ei(1,-(I*c*b+d*(a*b)^(1/2)-b*(I*d* x+I*c))/b)*b-1/2*d/a^2*Ei(1,I*d*x)*exp(-I*c)-1/2*(6*I*(I*d*x+I*c)*b*c-3*b* (I*d*x+I*c)^2+2*a*d^2+3*c^2*b)/a^2/(2*I*(I*d*x+I*c)*b*c-b*(I*d*x+I*c)^2+a* d^2+c^2*b)/x*sin(d*x+c)
Result contains complex when optimal does not.
Time = 0.34 (sec) , antiderivative size = 400, normalized size of antiderivative = 0.80 \[ \int \frac {\sin (c+d x)}{x^2 \left (a+b x^2\right )^2} \, dx=\frac {8 \, {\left (a b d^{2} x^{3} + a^{2} d^{2} x\right )} \cos \left (c\right ) \operatorname {Ci}\left (d x\right ) + {\left (a b d^{2} x^{3} + a^{2} d^{2} x - 3 \, {\left (b^{2} x^{3} + a b x\right )} \sqrt {\frac {a d^{2}}{b}}\right )} {\rm Ei}\left (i \, d x - \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (i \, c + \sqrt {\frac {a d^{2}}{b}}\right )} + {\left (a b d^{2} x^{3} + a^{2} d^{2} x + 3 \, {\left (b^{2} x^{3} + a b x\right )} \sqrt {\frac {a d^{2}}{b}}\right )} {\rm Ei}\left (i \, d x + \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (i \, c - \sqrt {\frac {a d^{2}}{b}}\right )} + {\left (a b d^{2} x^{3} + a^{2} d^{2} x - 3 \, {\left (b^{2} x^{3} + a b x\right )} \sqrt {\frac {a d^{2}}{b}}\right )} {\rm Ei}\left (-i \, d x - \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (-i \, c + \sqrt {\frac {a d^{2}}{b}}\right )} + {\left (a b d^{2} x^{3} + a^{2} d^{2} x + 3 \, {\left (b^{2} x^{3} + a b x\right )} \sqrt {\frac {a d^{2}}{b}}\right )} {\rm Ei}\left (-i \, d x + \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (-i \, c - \sqrt {\frac {a d^{2}}{b}}\right )} - 8 \, {\left (a b d^{2} x^{3} + a^{2} d^{2} x\right )} \sin \left (c\right ) \operatorname {Si}\left (d x\right ) - 4 \, {\left (3 \, a b d x^{2} + 2 \, a^{2} d\right )} \sin \left (d x + c\right )}{8 \, {\left (a^{3} b d x^{3} + a^{4} d x\right )}} \]
1/8*(8*(a*b*d^2*x^3 + a^2*d^2*x)*cos(c)*cos_integral(d*x) + (a*b*d^2*x^3 + a^2*d^2*x - 3*(b^2*x^3 + a*b*x)*sqrt(a*d^2/b))*Ei(I*d*x - sqrt(a*d^2/b))* e^(I*c + sqrt(a*d^2/b)) + (a*b*d^2*x^3 + a^2*d^2*x + 3*(b^2*x^3 + a*b*x)*s qrt(a*d^2/b))*Ei(I*d*x + sqrt(a*d^2/b))*e^(I*c - sqrt(a*d^2/b)) + (a*b*d^2 *x^3 + a^2*d^2*x - 3*(b^2*x^3 + a*b*x)*sqrt(a*d^2/b))*Ei(-I*d*x - sqrt(a*d ^2/b))*e^(-I*c + sqrt(a*d^2/b)) + (a*b*d^2*x^3 + a^2*d^2*x + 3*(b^2*x^3 + a*b*x)*sqrt(a*d^2/b))*Ei(-I*d*x + sqrt(a*d^2/b))*e^(-I*c - sqrt(a*d^2/b)) - 8*(a*b*d^2*x^3 + a^2*d^2*x)*sin(c)*sin_integral(d*x) - 4*(3*a*b*d*x^2 + 2*a^2*d)*sin(d*x + c))/(a^3*b*d*x^3 + a^4*d*x)
\[ \int \frac {\sin (c+d x)}{x^2 \left (a+b x^2\right )^2} \, dx=\int \frac {\sin {\left (c + d x \right )}}{x^{2} \left (a + b x^{2}\right )^{2}}\, dx \]
\[ \int \frac {\sin (c+d x)}{x^2 \left (a+b x^2\right )^2} \, dx=\int { \frac {\sin \left (d x + c\right )}{{\left (b x^{2} + a\right )}^{2} x^{2}} \,d x } \]
\[ \int \frac {\sin (c+d x)}{x^2 \left (a+b x^2\right )^2} \, dx=\int { \frac {\sin \left (d x + c\right )}{{\left (b x^{2} + a\right )}^{2} x^{2}} \,d x } \]
Timed out. \[ \int \frac {\sin (c+d x)}{x^2 \left (a+b x^2\right )^2} \, dx=\int \frac {\sin \left (c+d\,x\right )}{x^2\,{\left (b\,x^2+a\right )}^2} \,d x \]