Integrand size = 19, antiderivative size = 450 \[ \int \frac {x^4 \sin (c+d x)}{\left (a+b x^2\right )^2} \, dx=-\frac {\cos (c+d x)}{b^2 d}-\frac {a d \cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 b^3}-\frac {a d \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{4 b^3}-\frac {3 \sqrt {-a} \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right ) \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 b^{5/2}}+\frac {3 \sqrt {-a} \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 b^{5/2}}+\frac {x \sin (c+d x)}{2 b^2}-\frac {x^3 \sin (c+d x)}{2 b \left (a+b x^2\right )}-\frac {3 \sqrt {-a} \cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 b^{5/2}}-\frac {a d \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 b^3}-\frac {3 \sqrt {-a} \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{4 b^{5/2}}+\frac {a d \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{4 b^3} \]
-cos(d*x+c)/b^2/d-1/4*a*d*Ci(d*x+d*(-a)^(1/2)/b^(1/2))*cos(c-d*(-a)^(1/2)/ b^(1/2))/b^3-1/4*a*d*Ci(-d*x+d*(-a)^(1/2)/b^(1/2))*cos(c+d*(-a)^(1/2)/b^(1 /2))/b^3+1/2*x*sin(d*x+c)/b^2-1/2*x^3*sin(d*x+c)/b/(b*x^2+a)+1/4*a*d*Si(d* x+d*(-a)^(1/2)/b^(1/2))*sin(c-d*(-a)^(1/2)/b^(1/2))/b^3+1/4*a*d*Si(d*x-d*( -a)^(1/2)/b^(1/2))*sin(c+d*(-a)^(1/2)/b^(1/2))/b^3+3/4*cos(c+d*(-a)^(1/2)/ b^(1/2))*Si(d*x-d*(-a)^(1/2)/b^(1/2))*(-a)^(1/2)/b^(5/2)-3/4*cos(c-d*(-a)^ (1/2)/b^(1/2))*Si(d*x+d*(-a)^(1/2)/b^(1/2))*(-a)^(1/2)/b^(5/2)-3/4*Ci(d*x+ d*(-a)^(1/2)/b^(1/2))*sin(c-d*(-a)^(1/2)/b^(1/2))*(-a)^(1/2)/b^(5/2)+3/4*C i(-d*x+d*(-a)^(1/2)/b^(1/2))*sin(c+d*(-a)^(1/2)/b^(1/2))*(-a)^(1/2)/b^(5/2 )
Result contains complex when optimal does not.
Time = 3.15 (sec) , antiderivative size = 298, normalized size of antiderivative = 0.66 \[ \int \frac {x^4 \sin (c+d x)}{\left (a+b x^2\right )^2} \, dx=-\frac {\sqrt {a} e^{-i c-\frac {\sqrt {a} d}{\sqrt {b}}} \left (\left (3 \sqrt {b}+\sqrt {a} d\right ) e^{\frac {2 \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (-\frac {\sqrt {a} d}{\sqrt {b}}-i d x\right )+\left (-3 \sqrt {b}+\sqrt {a} d\right ) \operatorname {ExpIntegralEi}\left (\frac {\sqrt {a} d}{\sqrt {b}}-i d x\right )\right )+\sqrt {a} e^{i c-\frac {\sqrt {a} d}{\sqrt {b}}} \left (\left (3 \sqrt {b}+\sqrt {a} d\right ) e^{\frac {2 \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (-\frac {\sqrt {a} d}{\sqrt {b}}+i d x\right )+\left (-3 \sqrt {b}+\sqrt {a} d\right ) \operatorname {ExpIntegralEi}\left (\frac {\sqrt {a} d}{\sqrt {b}}+i d x\right )\right )-4 b \cos (d x) \left (-\frac {2 \cos (c)}{d}+\frac {a x \sin (c)}{a+b x^2}\right )-4 b \left (\frac {a x \cos (c)}{a+b x^2}+\frac {2 \sin (c)}{d}\right ) \sin (d x)}{8 b^3} \]
-1/8*(Sqrt[a]*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]) + Sqrt[ a]*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] + Sq rt[a]*d)*ExpIntegralEi[(Sqrt[a]*d)/Sqrt[b] + I*d*x]) - 4*b*Cos[d*x]*((-2*C os[c])/d + (a*x*Sin[c])/(a + b*x^2)) - 4*b*((a*x*Cos[c])/(a + b*x^2) + (2* Sin[c])/d)*Sin[d*x])/b^3
Time = 1.17 (sec) , antiderivative size = 476, normalized size of antiderivative = 1.06, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {3824, 3826, 2009, 3827, 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.
| \(\displaystyle \int \frac {x^4 \sin (c+d x)}{\left (a+b x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 3824 |
| \(\displaystyle \frac {3 \int \frac {x^2 \sin (c+d x)}{b x^2+a}dx}{2 b}+\frac {d \int \frac {x^3 \cos (c+d x)}{b x^2+a}dx}{2 b}-\frac {x^3 \sin (c+d x)}{2 b \left (a+b x^2\right )}\) |
| \(\Big \downarrow \) 3826 |
| \(\displaystyle \frac {3 \int \left (\frac {\sin (c+d x)}{b}-\frac {a \sin (c+d x)}{b \left (b x^2+a\right )}\right )dx}{2 b}+\frac {d \int \frac {x^3 \cos (c+d x)}{b x^2+a}dx}{2 b}-\frac {x^3 \sin (c+d x)}{2 b \left (a+b x^2\right )}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {d \int \frac {x^3 \cos (c+d x)}{b x^2+a}dx}{2 b}+\frac {3 \left (-\frac {\sqrt {-a} \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \operatorname {CosIntegral}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 b^{3/2}}+\frac {\sqrt {-a} \sin \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 b^{3/2}}-\frac {\sqrt {-a} \cos \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 b^{3/2}}-\frac {\sqrt {-a} \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 b^{3/2}}-\frac {\cos (c+d x)}{b d}\right )}{2 b}-\frac {x^3 \sin (c+d x)}{2 b \left (a+b x^2\right )}\) |
| \(\Big \downarrow \) 3827 |
| \(\displaystyle \frac {d \int \left (\frac {x \cos (c+d x)}{b}-\frac {a x \cos (c+d x)}{b \left (b x^2+a\right )}\right )dx}{2 b}+\frac {3 \left (-\frac {\sqrt {-a} \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \operatorname {CosIntegral}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 b^{3/2}}+\frac {\sqrt {-a} \sin \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 b^{3/2}}-\frac {\sqrt {-a} \cos \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 b^{3/2}}-\frac {\sqrt {-a} \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 b^{3/2}}-\frac {\cos (c+d x)}{b d}\right )}{2 b}-\frac {x^3 \sin (c+d x)}{2 b \left (a+b x^2\right )}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3 \left (-\frac {\sqrt {-a} \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \operatorname {CosIntegral}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 b^{3/2}}+\frac {\sqrt {-a} \sin \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 b^{3/2}}-\frac {\sqrt {-a} \cos \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 b^{3/2}}-\frac {\sqrt {-a} \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 b^{3/2}}-\frac {\cos (c+d x)}{b d}\right )}{2 b}+\frac {d \left (-\frac {a \cos \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 b^2}-\frac {a \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \operatorname {CosIntegral}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 b^2}-\frac {a \sin \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 b^2}+\frac {a \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 b^2}+\frac {\cos (c+d x)}{b d^2}+\frac {x \sin (c+d x)}{b d}\right )}{2 b}-\frac {x^3 \sin (c+d x)}{2 b \left (a+b x^2\right )}\) |
-1/2*(x^3*Sin[c + d*x])/(b*(a + b*x^2)) + (3*(-(Cos[c + d*x]/(b*d)) - (Sqr t[-a]*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sin[c - (Sqrt[-a]*d)/Sqrt[b] ])/(2*b^(3/2)) + (Sqrt[-a]*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x]*Sin[c + (Sqrt[-a]*d)/Sqrt[b]])/(2*b^(3/2)) - (Sqrt[-a]*Cos[c + (Sqrt[-a]*d)/Sqrt[ b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(2*b^(3/2)) - (Sqrt[-a]*Cos[c - (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(2*b^(3/ 2))))/(2*b) + (d*(Cos[c + d*x]/(b*d^2) - (a*Cos[c + (Sqrt[-a]*d)/Sqrt[b]]* CosIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(2*b^2) - (a*Cos[c - (Sqrt[-a]*d) /Sqrt[b]]*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(2*b^2) + (x*Sin[c + d* x])/(b*d) - (a*Sin[c + (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt [b] - d*x])/(2*b^2) + (a*Sin[c - (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[- a]*d)/Sqrt[b] + d*x])/(2*b^2)))/(2*b)
3.1.65.3.1 Defintions of rubi rules used
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*Sin[(c_.) + (d_.)*(x_)], x_Sym bol] :> Simp[x^(m - n + 1)*(a + b*x^n)^(p + 1)*(Sin[c + d*x]/(b*n*(p + 1))) , x] + (-Simp[(m - n + 1)/(b*n*(p + 1)) Int[x^(m - n)*(a + b*x^n)^(p + 1) *Sin[c + d*x], x], x] - Simp[d/(b*n*(p + 1)) Int[x^(m - n + 1)*(a + b*x^n )^(p + 1)*Cos[c + d*x], x], x]) /; FreeQ[{a, b, c, d, m}, x] && ILtQ[p, -1] && IGtQ[n, 0] && (GtQ[m - n + 1, 0] || GtQ[n, 2]) && RationalQ[m]
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]
Int[Cos[(c_.) + (d_.)*(x_)]*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Sym bol] :> Int[ExpandIntegrand[Cos[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 = 0.78 (sec) , antiderivative size = 524, normalized size of antiderivative = 1.16
| method | result | size |
| risch | \(\frac {{\mathrm e}^{\frac {i c b +d \sqrt {a b}}{b}} \operatorname {Ei}_{1}\left (\frac {i c b +d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right ) a d}{8 b^{3}}+\frac {{\mathrm e}^{\frac {i c b -d \sqrt {a b}}{b}} \operatorname {Ei}_{1}\left (-\frac {-i c b +d \sqrt {a b}+b \left (i d x +i c \right )}{b}\right ) a d}{8 b^{3}}+\frac {3 \sqrt {a b}\, {\mathrm e}^{\frac {i c b +d \sqrt {a b}}{b}} \operatorname {Ei}_{1}\left (\frac {i c b +d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right )}{8 b^{3}}-\frac {3 \sqrt {a b}\, {\mathrm e}^{\frac {i c b -d \sqrt {a b}}{b}} \operatorname {Ei}_{1}\left (-\frac {-i c b +d \sqrt {a b}+b \left (i d x +i c \right )}{b}\right )}{8 b^{3}}+\frac {{\mathrm e}^{-\frac {i c b +d \sqrt {a b}}{b}} \operatorname {Ei}_{1}\left (-\frac {i c b +d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right ) a d}{8 b^{3}}+\frac {{\mathrm e}^{-\frac {i c b -d \sqrt {a b}}{b}} \operatorname {Ei}_{1}\left (\frac {-i c b +d \sqrt {a b}+b \left (i d x +i c \right )}{b}\right ) a d}{8 b^{3}}-\frac {3 \sqrt {a b}\, {\mathrm e}^{-\frac {i c b +d \sqrt {a b}}{b}} \operatorname {Ei}_{1}\left (-\frac {i c b +d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right )}{8 b^{3}}+\frac {3 \sqrt {a b}\, {\mathrm e}^{-\frac {i c b -d \sqrt {a b}}{b}} \operatorname {Ei}_{1}\left (\frac {-i c b +d \sqrt {a b}+b \left (i d x +i c \right )}{b}\right )}{8 b^{3}}-\frac {\cos \left (d x +c \right )}{b^{2} d}+\frac {d^{2} a x \sin \left (d x +c \right )}{2 b^{2} \left (d^{2} x^{2} b +a \,d^{2}\right )}\) | \(524\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(3411\) |
| default | \(\text {Expression too large to display}\) | \(3411\) |
1/8/b^3*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)*a*d+1/8/b^3*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)*a*d+3/8/b^3*(a*b)^(1/2)*exp((I*c*b+d*(a*b)^(1/2))/b)*E i(1,(I*c*b+d*(a*b)^(1/2)-b*(I*d*x+I*c))/b)-3/8/b^3*(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)+1/8/b^3*ex p(-(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)*a *d+1/8/b^3*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)*a*d-3/8/b^3*(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)+3/8/b^3*(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)-cos(d*x+c)/b^2/d +1/2*d^2*a*x/b^2/(b*d^2*x^2+a*d^2)*sin(d*x+c)
Result contains complex when optimal does not.
Time = 0.32 (sec) , antiderivative size = 351, normalized size of antiderivative = 0.78 \[ \int \frac {x^4 \sin (c+d x)}{\left (a+b x^2\right )^2} \, dx=\frac {4 \, a b d x \sin \left (d x + c\right ) - {\left (a b d^{2} x^{2} + a^{2} d^{2} + 3 \, {\left (b^{2} x^{2} + a b\right )} \sqrt {\frac {a d^{2}}{b}}\right )} {\rm Ei}\left (i \, d x - \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (i \, c + \sqrt {\frac {a d^{2}}{b}}\right )} - {\left (a b d^{2} x^{2} + a^{2} d^{2} - 3 \, {\left (b^{2} x^{2} + a b\right )} \sqrt {\frac {a d^{2}}{b}}\right )} {\rm Ei}\left (i \, d x + \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (i \, c - \sqrt {\frac {a d^{2}}{b}}\right )} - {\left (a b d^{2} x^{2} + a^{2} d^{2} + 3 \, {\left (b^{2} x^{2} + a b\right )} \sqrt {\frac {a d^{2}}{b}}\right )} {\rm Ei}\left (-i \, d x - \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (-i \, c + \sqrt {\frac {a d^{2}}{b}}\right )} - {\left (a b d^{2} x^{2} + a^{2} d^{2} - 3 \, {\left (b^{2} x^{2} + a b\right )} \sqrt {\frac {a d^{2}}{b}}\right )} {\rm Ei}\left (-i \, d x + \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (-i \, c - \sqrt {\frac {a d^{2}}{b}}\right )} - 8 \, {\left (b^{2} x^{2} + a b\right )} \cos \left (d x + c\right )}{8 \, {\left (b^{4} d x^{2} + a b^{3} d\right )}} \]
1/8*(4*a*b*d*x*sin(d*x + c) - (a*b*d^2*x^2 + a^2*d^2 + 3*(b^2*x^2 + a*b)*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^2 + a^2*d^2 - 3*(b^2*x^2 + a*b)*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^2 + a^2*d^2 + 3*(b^2*x^2 + a*b)*sqr t(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^2 + a^2*d^2 - 3*(b^2*x^2 + a*b)*sqrt(a*d^2/b))*Ei(-I*d*x + sqrt(a*d^2/b ))*e^(-I*c - sqrt(a*d^2/b)) - 8*(b^2*x^2 + a*b)*cos(d*x + c))/(b^4*d*x^2 + a*b^3*d)
\[ \int \frac {x^4 \sin (c+d x)}{\left (a+b x^2\right )^2} \, dx=\int \frac {x^{4} \sin {\left (c + d x \right )}}{\left (a + b x^{2}\right )^{2}}\, dx \]
\[ \int \frac {x^4 \sin (c+d x)}{\left (a+b x^2\right )^2} \, dx=\int { \frac {x^{4} \sin \left (d x + c\right )}{{\left (b x^{2} + a\right )}^{2}} \,d x } \]
-1/2*((b*cos(c)^2 + b*sin(c)^2)*d*x^4*cos(d*x + c) - 4*(a*cos(c)^2 + a*sin (c)^2)*x*sin(d*x + c) + ((b*d*x^4*cos(c) + 4*a*x*sin(c))*cos(d*x + c)^2 + (b*d*x^4*cos(c) + 4*a*x*sin(c))*sin(d*x + c)^2)*cos(d*x + 2*c) - 2*(((b^3* cos(c)^2 + b^3*sin(c)^2)*d^2*x^4 + 2*(a*b^2*cos(c)^2 + a*b^2*sin(c)^2)*d^2 *x^2 + (a^2*b*cos(c)^2 + a^2*b*sin(c)^2)*d^2)*cos(d*x + c)^2 + ((b^3*cos(c )^2 + b^3*sin(c)^2)*d^2*x^4 + 2*(a*b^2*cos(c)^2 + a*b^2*sin(c)^2)*d^2*x^2 + (a^2*b*cos(c)^2 + a^2*b*sin(c)^2)*d^2)*sin(d*x + c)^2)*integrate(-2*(a^2 *d*x*cos(d*x + c) - (3*a*b*x^2 - a^2)*sin(d*x + c))/(b^4*d^2*x^6 + 3*a*b^3 *d^2*x^4 + 3*a^2*b^2*d^2*x^2 + a^3*b*d^2), x) - 2*(((b^3*cos(c)^2 + b^3*si n(c)^2)*d^2*x^4 + 2*(a*b^2*cos(c)^2 + a*b^2*sin(c)^2)*d^2*x^2 + (a^2*b*cos (c)^2 + a^2*b*sin(c)^2)*d^2)*cos(d*x + c)^2 + ((b^3*cos(c)^2 + b^3*sin(c)^ 2)*d^2*x^4 + 2*(a*b^2*cos(c)^2 + a*b^2*sin(c)^2)*d^2*x^2 + (a^2*b*cos(c)^2 + a^2*b*sin(c)^2)*d^2)*sin(d*x + c)^2)*integrate(-2*(a^2*d*x*cos(d*x + c) - (3*a*b*x^2 - a^2)*sin(d*x + c))/((b^4*d^2*x^6 + 3*a*b^3*d^2*x^4 + 3*a^2 *b^2*d^2*x^2 + a^3*b*d^2)*cos(d*x + c)^2 + (b^4*d^2*x^6 + 3*a*b^3*d^2*x^4 + 3*a^2*b^2*d^2*x^2 + a^3*b*d^2)*sin(d*x + c)^2), x) + ((b*d*x^4*sin(c) - 4*a*x*cos(c))*cos(d*x + c)^2 + (b*d*x^4*sin(c) - 4*a*x*cos(c))*sin(d*x + c )^2)*sin(d*x + 2*c))/(((b^3*cos(c)^2 + b^3*sin(c)^2)*d^2*x^4 + 2*(a*b^2*co s(c)^2 + a*b^2*sin(c)^2)*d^2*x^2 + (a^2*b*cos(c)^2 + a^2*b*sin(c)^2)*d^2)* cos(d*x + c)^2 + ((b^3*cos(c)^2 + b^3*sin(c)^2)*d^2*x^4 + 2*(a*b^2*cos(...
\[ \int \frac {x^4 \sin (c+d x)}{\left (a+b x^2\right )^2} \, dx=\int { \frac {x^{4} \sin \left (d x + c\right )}{{\left (b x^{2} + a\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {x^4 \sin (c+d x)}{\left (a+b x^2\right )^2} \, dx=\int \frac {x^4\,\sin \left (c+d\,x\right )}{{\left (b\,x^2+a\right )}^2} \,d x \]