Integrand size = 19, antiderivative size = 270 \[ \int \frac {\sin (c+d x)}{x^3 \left (a+b x^2\right )} \, dx=-\frac {d \cos (c+d x)}{2 a x}-\frac {b \operatorname {CosIntegral}(d x) \sin (c)}{a^2}-\frac {d^2 \operatorname {CosIntegral}(d x) \sin (c)}{2 a}+\frac {b \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right ) \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 a^2}+\frac {b \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 a^2}-\frac {\sin (c+d x)}{2 a x^2}-\frac {b \cos (c) \text {Si}(d x)}{a^2}-\frac {d^2 \cos (c) \text {Si}(d x)}{2 a}-\frac {b \cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 a^2}+\frac {b \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{2 a^2} \]
-1/2*d*cos(d*x+c)/a/x-b*cos(c)*Si(d*x)/a^2-1/2*d^2*cos(c)*Si(d*x)/a+1/2*b* cos(c+d*(-a)^(1/2)/b^(1/2))*Si(d*x-d*(-a)^(1/2)/b^(1/2))/a^2+1/2*b*cos(c-d *(-a)^(1/2)/b^(1/2))*Si(d*x+d*(-a)^(1/2)/b^(1/2))/a^2-b*Ci(d*x)*sin(c)/a^2 -1/2*d^2*Ci(d*x)*sin(c)/a-1/2*sin(d*x+c)/a/x^2+1/2*b*Ci(d*x+d*(-a)^(1/2)/b ^(1/2))*sin(c-d*(-a)^(1/2)/b^(1/2))/a^2+1/2*b*Ci(-d*x+d*(-a)^(1/2)/b^(1/2) )*sin(c+d*(-a)^(1/2)/b^(1/2))/a^2
Result contains complex when optimal does not.
Time = 0.83 (sec) , antiderivative size = 240, normalized size of antiderivative = 0.89 \[ \int \frac {\sin (c+d x)}{x^3 \left (a+b x^2\right )} \, dx=\frac {i b e^{-i c-\frac {\sqrt {a} d}{\sqrt {b}}} \left (e^{\frac {2 \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (-\frac {\sqrt {a} d}{\sqrt {b}}-i d x\right )+\operatorname {ExpIntegralEi}\left (\frac {\sqrt {a} d}{\sqrt {b}}-i d x\right )\right )-i b e^{i c-\frac {\sqrt {a} d}{\sqrt {b}}} \left (e^{\frac {2 \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (-\frac {\sqrt {a} d}{\sqrt {b}}+i d x\right )+\operatorname {ExpIntegralEi}\left (\frac {\sqrt {a} d}{\sqrt {b}}+i d x\right )\right )-\frac {2 a \cos (d x) (d x \cos (c)+\sin (c))}{x^2}+\frac {2 a (-\cos (c)+d x \sin (c)) \sin (d x)}{x^2}-2 \left (2 b+a d^2\right ) (\operatorname {CosIntegral}(d x) \sin (c)+\cos (c) \text {Si}(d x))}{4 a^2} \]
(I*b*E^((-I)*c - (Sqrt[a]*d)/Sqrt[b])*(E^((2*Sqrt[a]*d)/Sqrt[b])*ExpIntegr alEi[-((Sqrt[a]*d)/Sqrt[b]) - I*d*x] + ExpIntegralEi[(Sqrt[a]*d)/Sqrt[b] - I*d*x]) - I*b*E^(I*c - (Sqrt[a]*d)/Sqrt[b])*(E^((2*Sqrt[a]*d)/Sqrt[b])*Ex pIntegralEi[-((Sqrt[a]*d)/Sqrt[b]) + I*d*x] + ExpIntegralEi[(Sqrt[a]*d)/Sq rt[b] + I*d*x]) - (2*a*Cos[d*x]*(d*x*Cos[c] + Sin[c]))/x^2 + (2*a*(-Cos[c] + d*x*Sin[c])*Sin[d*x])/x^2 - 2*(2*b + a*d^2)*(CosIntegral[d*x]*Sin[c] + Cos[c]*SinIntegral[d*x]))/(4*a^2)
Time = 0.66 (sec) , antiderivative size = 270, 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.
| \(\displaystyle \int \frac {\sin (c+d x)}{x^3 \left (a+b x^2\right )} \, dx\) |
| \(\Big \downarrow \) 3826 |
| \(\displaystyle \int \left (\frac {b^2 x \sin (c+d x)}{a^2 \left (a+b x^2\right )}-\frac {b \sin (c+d x)}{a^2 x}+\frac {\sin (c+d x)}{a x^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {b \sin (c) \operatorname {CosIntegral}(d x)}{a^2}+\frac {b \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \operatorname {CosIntegral}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 a^2}+\frac {b \sin \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 a^2}-\frac {b \cos (c) \text {Si}(d x)}{a^2}-\frac {b \cos \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 a^2}+\frac {b \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 a^2}-\frac {d^2 \sin (c) \operatorname {CosIntegral}(d x)}{2 a}-\frac {d^2 \cos (c) \text {Si}(d x)}{2 a}-\frac {\sin (c+d x)}{2 a x^2}-\frac {d \cos (c+d x)}{2 a x}\) |
-1/2*(d*Cos[c + d*x])/(a*x) - (b*CosIntegral[d*x]*Sin[c])/a^2 - (d^2*CosIn tegral[d*x]*Sin[c])/(2*a) + (b*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sin [c - (Sqrt[-a]*d)/Sqrt[b]])/(2*a^2) + (b*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x]*Sin[c + (Sqrt[-a]*d)/Sqrt[b]])/(2*a^2) - Sin[c + d*x]/(2*a*x^2) - ( b*Cos[c]*SinIntegral[d*x])/a^2 - (d^2*Cos[c]*SinIntegral[d*x])/(2*a) - (b* Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(2* a^2) + (b*Cos[c - (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(2*a^2)
3.1.64.3.1 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 = 0.38 (sec) , antiderivative size = 259, normalized size of antiderivative = 0.96
| method | result | size |
| derivativedivides | \(d^{2} \left (-\frac {\sin \left (d x +c \right )}{2 a \,d^{2} x^{2}}-\frac {\cos \left (d x +c \right )}{2 a d x}+\frac {b \left (\operatorname {Si}\left (d x +c -\frac {d \sqrt {-a b}+c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}+c b}{b}\right )+\operatorname {Ci}\left (d x +c -\frac {d \sqrt {-a b}+c b}{b}\right ) \sin \left (\frac {d \sqrt {-a b}+c b}{b}\right )\right )}{2 a^{2} d^{2}}+\frac {b \left (\operatorname {Si}\left (d x +c +\frac {d \sqrt {-a b}-c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}-c b}{b}\right )-\operatorname {Ci}\left (d x +c +\frac {d \sqrt {-a b}-c b}{b}\right ) \sin \left (\frac {d \sqrt {-a b}-c b}{b}\right )\right )}{2 a^{2} d^{2}}-\frac {\left (a \,d^{2}+2 b \right ) \left (\operatorname {Si}\left (d x \right ) \cos \left (c \right )+\operatorname {Ci}\left (d x \right ) \sin \left (c \right )\right )}{2 a^{2} d^{2}}\right )\) | \(259\) |
| default | \(d^{2} \left (-\frac {\sin \left (d x +c \right )}{2 a \,d^{2} x^{2}}-\frac {\cos \left (d x +c \right )}{2 a d x}+\frac {b \left (\operatorname {Si}\left (d x +c -\frac {d \sqrt {-a b}+c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}+c b}{b}\right )+\operatorname {Ci}\left (d x +c -\frac {d \sqrt {-a b}+c b}{b}\right ) \sin \left (\frac {d \sqrt {-a b}+c b}{b}\right )\right )}{2 a^{2} d^{2}}+\frac {b \left (\operatorname {Si}\left (d x +c +\frac {d \sqrt {-a b}-c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}-c b}{b}\right )-\operatorname {Ci}\left (d x +c +\frac {d \sqrt {-a b}-c b}{b}\right ) \sin \left (\frac {d \sqrt {-a b}-c b}{b}\right )\right )}{2 a^{2} d^{2}}-\frac {\left (a \,d^{2}+2 b \right ) \left (\operatorname {Si}\left (d x \right ) \cos \left (c \right )+\operatorname {Ci}\left (d x \right ) \sin \left (c \right )\right )}{2 a^{2} d^{2}}\right )\) | \(259\) |
| risch | \(\frac {i b \,{\mathrm e}^{\frac {i c b +d \sqrt {a b}}{b}} \operatorname {Ei}_{1}\left (\frac {i c b +d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right )}{4 a^{2}}+\frac {i b \,{\mathrm e}^{\frac {i c b -d \sqrt {a b}}{b}} \operatorname {Ei}_{1}\left (\frac {i c b -d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right )}{4 a^{2}}-\frac {i d^{2} {\mathrm e}^{i c} \operatorname {Ei}_{1}\left (-i d x \right )}{4 a}-\frac {i {\mathrm e}^{i c} \operatorname {Ei}_{1}\left (-i d x \right ) b}{2 a^{2}}-\frac {i b \,{\mathrm e}^{-\frac {i c b +d \sqrt {a b}}{b}} \operatorname {Ei}_{1}\left (-\frac {i c b +d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right )}{4 a^{2}}-\frac {i b \,{\mathrm e}^{-\frac {i c b -d \sqrt {a b}}{b}} \operatorname {Ei}_{1}\left (-\frac {i c b -d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right )}{4 a^{2}}+\frac {i d^{2} {\mathrm e}^{-i c} \operatorname {Ei}_{1}\left (i d x \right )}{4 a}+\frac {i {\mathrm e}^{-i c} \operatorname {Ei}_{1}\left (i d x \right ) b}{2 a^{2}}-\frac {d \cos \left (d x +c \right )}{2 a x}-\frac {\sin \left (d x +c \right )}{2 a \,x^{2}}\) | \(347\) |
d^2*(-1/2*sin(d*x+c)/a/d^2/x^2-1/2*cos(d*x+c)/a/d/x+1/2*b/a^2/d^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*b/a^2/d^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*(a*d^2+2*b)/d^2*(Si(d*x)*cos(c)+Ci (d*x)*sin(c)))
Result contains complex when optimal does not.
Time = 0.30 (sec) , antiderivative size = 225, normalized size of antiderivative = 0.83 \[ \int \frac {\sin (c+d x)}{x^3 \left (a+b x^2\right )} \, dx=\frac {-i \, b x^{2} {\rm Ei}\left (i \, d x - \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (i \, c + \sqrt {\frac {a d^{2}}{b}}\right )} - i \, b x^{2} {\rm Ei}\left (i \, d x + \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (i \, c - \sqrt {\frac {a d^{2}}{b}}\right )} + i \, b x^{2} {\rm Ei}\left (-i \, d x - \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (-i \, c + \sqrt {\frac {a d^{2}}{b}}\right )} + i \, b x^{2} {\rm Ei}\left (-i \, d x + \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (-i \, c - \sqrt {\frac {a d^{2}}{b}}\right )} - 2 \, {\left (a d^{2} + 2 \, b\right )} x^{2} \operatorname {Ci}\left (d x\right ) \sin \left (c\right ) - 2 \, {\left (a d^{2} + 2 \, b\right )} x^{2} \cos \left (c\right ) \operatorname {Si}\left (d x\right ) - 2 \, a d x \cos \left (d x + c\right ) - 2 \, a \sin \left (d x + c\right )}{4 \, a^{2} x^{2}} \]
1/4*(-I*b*x^2*Ei(I*d*x - sqrt(a*d^2/b))*e^(I*c + sqrt(a*d^2/b)) - I*b*x^2* Ei(I*d*x + sqrt(a*d^2/b))*e^(I*c - sqrt(a*d^2/b)) + I*b*x^2*Ei(-I*d*x - sq rt(a*d^2/b))*e^(-I*c + sqrt(a*d^2/b)) + I*b*x^2*Ei(-I*d*x + sqrt(a*d^2/b)) *e^(-I*c - sqrt(a*d^2/b)) - 2*(a*d^2 + 2*b)*x^2*cos_integral(d*x)*sin(c) - 2*(a*d^2 + 2*b)*x^2*cos(c)*sin_integral(d*x) - 2*a*d*x*cos(d*x + c) - 2*a *sin(d*x + c))/(a^2*x^2)
\[ \int \frac {\sin (c+d x)}{x^3 \left (a+b x^2\right )} \, dx=\int \frac {\sin {\left (c + d x \right )}}{x^{3} \left (a + b x^{2}\right )}\, dx \]
\[ \int \frac {\sin (c+d x)}{x^3 \left (a+b x^2\right )} \, dx=\int { \frac {\sin \left (d x + c\right )}{{\left (b x^{2} + a\right )} x^{3}} \,d x } \]
\[ \int \frac {\sin (c+d x)}{x^3 \left (a+b x^2\right )} \, dx=\int { \frac {\sin \left (d x + c\right )}{{\left (b x^{2} + a\right )} x^{3}} \,d x } \]
Timed out. \[ \int \frac {\sin (c+d x)}{x^3 \left (a+b x^2\right )} \, dx=\int \frac {\sin \left (c+d\,x\right )}{x^3\,\left (b\,x^2+a\right )} \,d x \]