\(\int \frac {x^3 \sin (c+d x)}{(a+b x^2)^2} \, dx\) [66]

Optimal result

Integrand size = 19, antiderivative size = 431 \[ \int \frac {x^3 \sin (c+d x)}{\left (a+b x^2\right )^2} \, dx=\frac {\sqrt {-a} d \cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 b^{5/2}}-\frac {\sqrt {-a} d \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{4 b^{5/2}}+\frac {\operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right ) \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 b^2}+\frac {\operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 b^2}+\frac {\sin (c+d x)}{2 b^2}-\frac {x^2 \sin (c+d x)}{2 b \left (a+b x^2\right )}-\frac {\cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 b^2}+\frac {\sqrt {-a} d \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 b^{5/2}}+\frac {\cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{2 b^2}+\frac {\sqrt {-a} d \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{4 b^{5/2}} \]

[Out]

1/2*cos(c+d*(-a)^(1/2)/b^(1/2))*Si(d*x-d*(-a)^(1/2)/b^(1/2))/b^2+1/2*cos(c-d*(-a)^(1/2)/b^(1/2))*Si(d*x+d*(-a)
^(1/2)/b^(1/2))/b^2+1/2*sin(d*x+c)/b^2-1/2*x^2*sin(d*x+c)/b/(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))/b^2+1/2*Ci(-d*x+d*(-a)^(1/2)/b^(1/2))*sin(c+d*(-a)^(1/2)/b^(1/2))/b^2-1/4*d*Ci(d*x+d*(-a)
^(1/2)/b^(1/2))*cos(c-d*(-a)^(1/2)/b^(1/2))*(-a)^(1/2)/b^(5/2)+1/4*d*Ci(-d*x+d*(-a)^(1/2)/b^(1/2))*cos(c+d*(-a
)^(1/2)/b^(1/2))*(-a)^(1/2)/b^(5/2)+1/4*d*Si(d*x+d*(-a)^(1/2)/b^(1/2))*sin(c-d*(-a)^(1/2)/b^(1/2))*(-a)^(1/2)/
b^(5/2)-1/4*d*Si(d*x-d*(-a)^(1/2)/b^(1/2))*sin(c+d*(-a)^(1/2)/b^(1/2))*(-a)^(1/2)/b^(5/2)

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 431, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {3424, 3426, 3384, 3380, 3383, 3427, 2717, 3415} \[ \int \frac {x^3 \sin (c+d x)}{\left (a+b x^2\right )^2} \, dx=\frac {\sqrt {-a} d \cos \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 b^{5/2}}-\frac {\sqrt {-a} d \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \operatorname {CosIntegral}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 b^{5/2}}+\frac {\sqrt {-a} d \sin \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 b^{5/2}}+\frac {\sqrt {-a} d \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 b^{5/2}}+\frac {\sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \operatorname {CosIntegral}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 b^2}+\frac {\sin \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 b^2}-\frac {\cos \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 b^2}+\frac {\cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 b^2}-\frac {x^2 \sin (c+d x)}{2 b \left (a+b x^2\right )}+\frac {\sin (c+d x)}{2 b^2} \]

[In]

Int[(x^3*Sin[c + d*x])/(a + b*x^2)^2,x]

[Out]

(Sqrt[-a]*d*Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(4*b^(5/2)) - (Sqrt[-a]*d*C
os[c - (Sqrt[-a]*d)/Sqrt[b]]*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(4*b^(5/2)) + (CosIntegral[(Sqrt[-a]*d)/
Sqrt[b] + d*x]*Sin[c - (Sqrt[-a]*d)/Sqrt[b]])/(2*b^2) + (CosIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x]*Sin[c + (Sqrt
[-a]*d)/Sqrt[b]])/(2*b^2) + Sin[c + d*x]/(2*b^2) - (x^2*Sin[c + d*x])/(2*b*(a + b*x^2)) - (Cos[c + (Sqrt[-a]*d
)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(2*b^2) + (Sqrt[-a]*d*Sin[c + (Sqrt[-a]*d)/Sqrt[b]]*SinInt
egral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(4*b^(5/2)) + (Cos[c - (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[
b] + d*x])/(2*b^2) + (Sqrt[-a]*d*Sin[c - (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(4*b^(
5/2))

Rule 2717

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3380

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinIntegral[e + f*x]/d, x] /; FreeQ[{c, d,
e, f}, x] && EqQ[d*e - c*f, 0]

Rule 3383

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosIntegral[e - Pi/2 + f*x]/d, x] /; FreeQ
[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - c*f, 0]

Rule 3384

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[c*(f/d) + f*x]
/(c + d*x), x], x] + Dist[Sin[(d*e - c*f)/d], Int[Cos[c*(f/d) + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f},
x] && NeQ[d*e - c*f, 0]

Rule 3415

Int[Cos[(c_.) + (d_.)*(x_)]*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[Cos[c + d*x], (a +
b*x^n)^p, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[p, 0] && IGtQ[n, 0] && (EqQ[n, 2] || EqQ[p, -1])

Rule 3424

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*Sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[x^(m - n + 1)*(a + b*x
^n)^(p + 1)*(Sin[c + d*x]/(b*n*(p + 1))), x] + (-Dist[(m - n + 1)/(b*n*(p + 1)), Int[x^(m - n)*(a + b*x^n)^(p
+ 1)*Sin[c + d*x], x], x] - Dist[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]

Rule 3426

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*Sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Int[ExpandIntegrand[Sin[c +
d*x], x^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && ILtQ[p, 0] && IGtQ[n, 0] && (EqQ[n, 2] || EqQ
[p, -1]) && IntegerQ[m]

Rule 3427

Int[Cos[(c_.) + (d_.)*(x_)]*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[Cos[c +
d*x], x^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && ILtQ[p, 0] && IGtQ[n, 0] && (EqQ[n, 2] || EqQ
[p, -1]) && IntegerQ[m]

Rubi steps \begin{align*} \text {integral}& = -\frac {x^2 \sin (c+d x)}{2 b \left (a+b x^2\right )}+\frac {\int \frac {x \sin (c+d x)}{a+b x^2} \, dx}{b}+\frac {d \int \frac {x^2 \cos (c+d x)}{a+b x^2} \, dx}{2 b} \\ & = -\frac {x^2 \sin (c+d x)}{2 b \left (a+b x^2\right )}+\frac {\int \left (-\frac {\sin (c+d x)}{2 \sqrt {b} \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\sin (c+d x)}{2 \sqrt {b} \left (\sqrt {-a}+\sqrt {b} x\right )}\right ) \, dx}{b}+\frac {d \int \left (\frac {\cos (c+d x)}{b}-\frac {a \cos (c+d x)}{b \left (a+b x^2\right )}\right ) \, dx}{2 b} \\ & = -\frac {x^2 \sin (c+d x)}{2 b \left (a+b x^2\right )}-\frac {\int \frac {\sin (c+d x)}{\sqrt {-a}-\sqrt {b} x} \, dx}{2 b^{3/2}}+\frac {\int \frac {\sin (c+d x)}{\sqrt {-a}+\sqrt {b} x} \, dx}{2 b^{3/2}}+\frac {d \int \cos (c+d x) \, dx}{2 b^2}-\frac {(a d) \int \frac {\cos (c+d x)}{a+b x^2} \, dx}{2 b^2} \\ & = \frac {\sin (c+d x)}{2 b^2}-\frac {x^2 \sin (c+d x)}{2 b \left (a+b x^2\right )}-\frac {(a d) \int \left (\frac {\sqrt {-a} \cos (c+d x)}{2 a \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\sqrt {-a} \cos (c+d x)}{2 a \left (\sqrt {-a}+\sqrt {b} x\right )}\right ) \, dx}{2 b^2}+\frac {\cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \int \frac {\sin \left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{\sqrt {-a}+\sqrt {b} x} \, dx}{2 b^{3/2}}+\frac {\cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \int \frac {\sin \left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{\sqrt {-a}-\sqrt {b} x} \, dx}{2 b^{3/2}}+\frac {\sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \int \frac {\cos \left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{\sqrt {-a}+\sqrt {b} x} \, dx}{2 b^{3/2}}-\frac {\sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \int \frac {\cos \left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{\sqrt {-a}-\sqrt {b} x} \, dx}{2 b^{3/2}} \\ & = \frac {\operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right ) \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 b^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 b^2}+\frac {\sin (c+d x)}{2 b^2}-\frac {x^2 \sin (c+d x)}{2 b \left (a+b x^2\right )}-\frac {\cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 b^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 b^2}-\frac {\left (\sqrt {-a} d\right ) \int \frac {\cos (c+d x)}{\sqrt {-a}-\sqrt {b} x} \, dx}{4 b^2}-\frac {\left (\sqrt {-a} d\right ) \int \frac {\cos (c+d x)}{\sqrt {-a}+\sqrt {b} x} \, dx}{4 b^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 b^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 b^2}+\frac {\sin (c+d x)}{2 b^2}-\frac {x^2 \sin (c+d x)}{2 b \left (a+b x^2\right )}-\frac {\cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 b^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 b^2}-\frac {\left (\sqrt {-a} d \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )\right ) \int \frac {\cos \left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{\sqrt {-a}+\sqrt {b} x} \, dx}{4 b^2}-\frac {\left (\sqrt {-a} d \cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )\right ) \int \frac {\cos \left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{\sqrt {-a}-\sqrt {b} x} \, dx}{4 b^2}+\frac {\left (\sqrt {-a} d \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )\right ) \int \frac {\sin \left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{\sqrt {-a}+\sqrt {b} x} \, dx}{4 b^2}-\frac {\left (\sqrt {-a} d \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )\right ) \int \frac {\sin \left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{\sqrt {-a}-\sqrt {b} x} \, dx}{4 b^2} \\ & = \frac {\sqrt {-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^{5/2}}-\frac {\sqrt {-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^{5/2}}+\frac {\operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right ) \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 b^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 b^2}+\frac {\sin (c+d x)}{2 b^2}-\frac {x^2 \sin (c+d x)}{2 b \left (a+b x^2\right )}-\frac {\cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 b^2}+\frac {\sqrt {-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^{5/2}}+\frac {\cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{2 b^2}+\frac {\sqrt {-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^{5/2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.82 (sec) , antiderivative size = 286, normalized size of antiderivative = 0.66 \[ \int \frac {x^3 \sin (c+d x)}{\left (a+b x^2\right )^2} \, dx=\frac {i e^{-i c-\frac {\sqrt {a} d}{\sqrt {b}}} \left (\left (2 \sqrt {b}+\sqrt {a} d\right ) e^{\frac {2 \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (-\frac {\sqrt {a} d}{\sqrt {b}}-i d x\right )+\left (2 \sqrt {b}-\sqrt {a} d\right ) \operatorname {ExpIntegralEi}\left (\frac {\sqrt {a} d}{\sqrt {b}}-i d x\right )\right )-i e^{i c-\frac {\sqrt {a} d}{\sqrt {b}}} \left (\left (2 \sqrt {b}+\sqrt {a} d\right ) e^{\frac {2 \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (-\frac {\sqrt {a} d}{\sqrt {b}}+i d x\right )+\left (2 \sqrt {b}-\sqrt {a} d\right ) \operatorname {ExpIntegralEi}\left (\frac {\sqrt {a} d}{\sqrt {b}}+i d x\right )\right )+\frac {4 a \sqrt {b} \cos (d x) \sin (c)}{a+b x^2}+\frac {4 a \sqrt {b} \cos (c) \sin (d x)}{a+b x^2}}{8 b^{5/2}} \]

[In]

Integrate[(x^3*Sin[c + d*x])/(a + b*x^2)^2,x]

[Out]

(I*E^((-I)*c - (Sqrt[a]*d)/Sqrt[b])*((2*Sqrt[b] + Sqrt[a]*d)*E^((2*Sqrt[a]*d)/Sqrt[b])*ExpIntegralEi[-((Sqrt[a
]*d)/Sqrt[b]) - I*d*x] + (2*Sqrt[b] - Sqrt[a]*d)*ExpIntegralEi[(Sqrt[a]*d)/Sqrt[b] - I*d*x]) - I*E^(I*c - (Sqr
t[a]*d)/Sqrt[b])*((2*Sqrt[b] + Sqrt[a]*d)*E^((2*Sqrt[a]*d)/Sqrt[b])*ExpIntegralEi[-((Sqrt[a]*d)/Sqrt[b]) + I*d
*x] + (2*Sqrt[b] - Sqrt[a]*d)*ExpIntegralEi[(Sqrt[a]*d)/Sqrt[b] + I*d*x]) + (4*a*Sqrt[b]*Cos[d*x]*Sin[c])/(a +
b*x^2) + (4*a*Sqrt[b]*Cos[c]*Sin[d*x])/(a + b*x^2))/(8*b^(5/2))

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.63 (sec) , antiderivative size = 852, normalized size of antiderivative = 1.98

method	result	size
risch	\(\frac {i \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 ) d}{8 b^{3}}-\frac {i \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 ) d}{8 b^{3}}+\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 b^{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 b^{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 ) \sqrt {a b}\, d}{8 b^{3}}-\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 ) \sqrt {a b}\, d}{8 b^{3}}-\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 b^{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 b^{2}}+\frac {\left (\frac {\left (3 i \left (i d x +i c \right ) a b c \,d^{2}-i \left (i d x +i c \right ) b^{2} c^{3}+a^{2} d^{4}-b^{2} c^{4}\right ) d^{2}}{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 ) b^{2} a}+\frac {c^{3} d^{3} x}{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 ) a}-\frac {3 c^{2} d^{2} \left (i \left (i d x +i c \right ) b c +a \,d^{2}+c^{2} b \right )}{2 a b \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 )}-\frac {3 i c \,d^{2} \left (-i a c \,d^{2}-i b \,c^{3}-\left (i d x +i c \right ) a \,d^{2}+\left (i d x +i c \right ) b \,c^{2}\right )}{2 a b \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 )}\right ) \sin \left (d x +c \right )}{d^{4}}\)	\(852\)
derivativedivides	\(\text {Expression too large to display}\)	\(2531\)
default	\(\text {Expression too large to display}\)	\(2531\)

[In]

int(x^3*sin(d*x+c)/(b*x^2+a)^2,x,method=_RETURNVERBOSE)

[Out]

1/8*I/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)*d-1/8*I/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)*d+1/4*I/b^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)+1/4*I/b^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*I/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*b)^(1/2)*d-1/8*I/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*b)^
(1/2)*d-1/4*I/b^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)-1/4*I/b^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/d^4*(1/2/(2*I*(I*d*x+I*c)*b*c-b*(I*d*x+I
*c)^2+a*d^2+c^2*b)/b^2/a*(3*I*(I*d*x+I*c)*a*b*c*d^2-I*(I*d*x+I*c)*b^2*c^3+a^2*d^4-b^2*c^4)*d^2+1/2*c^3*d^3*x/(
-2*I*(I*d*x+I*c)*b*c+b*(I*d*x+I*c)^2-a*d^2-c^2*b)/a-3/2*c^2*d^2*(I*(I*d*x+I*c)*b*c+a*d^2+c^2*b)/a/b/(2*I*(I*d*
x+I*c)*b*c-b*(I*d*x+I*c)^2+a*d^2+c^2*b)-3/2*I*c*d^2*(-I*a*c*d^2-I*b*c^3-(I*d*x+I*c)*a*d^2+(I*d*x+I*c)*b*c^2)/a
/b/(-2*I*(I*d*x+I*c)*b*c+b*(I*d*x+I*c)^2-a*d^2-c^2*b))*sin(d*x+c)

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.32 (sec) , antiderivative size = 291, normalized size of antiderivative = 0.68 \[ \int \frac {x^3 \sin (c+d x)}{\left (a+b x^2\right )^2} \, dx=-\frac {{\left (2 i \, b x^{2} - {\left (-i \, b x^{2} - i \, a\right )} \sqrt {\frac {a d^{2}}{b}} + 2 i \, a\right )} {\rm Ei}\left (i \, d x - \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (i \, c + \sqrt {\frac {a d^{2}}{b}}\right )} + {\left (2 i \, b x^{2} - {\left (i \, b x^{2} + i \, a\right )} \sqrt {\frac {a d^{2}}{b}} + 2 i \, a\right )} {\rm Ei}\left (i \, d x + \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (i \, c - \sqrt {\frac {a d^{2}}{b}}\right )} + {\left (-2 i \, b x^{2} - {\left (i \, b x^{2} + i \, a\right )} \sqrt {\frac {a d^{2}}{b}} - 2 i \, a\right )} {\rm Ei}\left (-i \, d x - \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (-i \, c + \sqrt {\frac {a d^{2}}{b}}\right )} + {\left (-2 i \, b x^{2} - {\left (-i \, b x^{2} - i \, a\right )} \sqrt {\frac {a d^{2}}{b}} - 2 i \, a\right )} {\rm Ei}\left (-i \, d x + \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (-i \, c - \sqrt {\frac {a d^{2}}{b}}\right )} - 4 \, a \sin \left (d x + c\right )}{8 \, {\left (b^{3} x^{2} + a b^{2}\right )}} \]

[In]

integrate(x^3*sin(d*x+c)/(b*x^2+a)^2,x, algorithm="fricas")

[Out]

-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 - 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)) + (-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)) - 4*a*sin(d*
x + c))/(b^3*x^2 + a*b^2)

Sympy [F]

\[ \int \frac {x^3 \sin (c+d x)}{\left (a+b x^2\right )^2} \, dx=\int \frac {x^{3} \sin {\left (c + d x \right )}}{\left (a + b x^{2}\right )^{2}}\, dx \]

[In]

integrate(x**3*sin(d*x+c)/(b*x**2+a)**2,x)

[Out]

Integral(x**3*sin(c + d*x)/(a + b*x**2)**2, x)

Maxima [F]

\[ \int \frac {x^3 \sin (c+d x)}{\left (a+b x^2\right )^2} \, dx=\int { \frac {x^{3} \sin \left (d x + c\right )}{{\left (b x^{2} + a\right )}^{2}} \,d x } \]

[In]

integrate(x^3*sin(d*x+c)/(b*x^2+a)^2,x, algorithm="maxima")

[Out]

-1/2*((cos(c)^2 + sin(c)^2)*d*x^2*sin(d*x + c) + ((d^2*x^3*cos(c) - d*x^2*sin(c) - 2*x*cos(c))*cos(d*x + c)^2
+ (d^2*x^3*cos(c) - d*x^2*sin(c) - 2*x*cos(c))*sin(d*x + c)^2)*cos(d*x + 2*c) + ((cos(c)^2 + sin(c)^2)*d^2*x^3
- 2*(cos(c)^2 + sin(c)^2)*x)*cos(d*x + c) - 2*(((b^2*cos(c)^2 + b^2*sin(c)^2)*d^3*x^4 + 2*(a*b*cos(c)^2 + a*b
*sin(c)^2)*d^3*x^2 + (a^2*cos(c)^2 + a^2*sin(c)^2)*d^3)*cos(d*x + c)^2 + ((b^2*cos(c)^2 + b^2*sin(c)^2)*d^3*x^
4 + 2*(a*b*cos(c)^2 + a*b*sin(c)^2)*d^3*x^2 + (a^2*cos(c)^2 + a^2*sin(c)^2)*d^3)*sin(d*x + c)^2)*integrate((2*
a*d*x*sin(d*x + c) + ((2*a*d^2 + 3*b)*x^2 - a)*cos(d*x + c))/(b^3*d^3*x^6 + 3*a*b^2*d^3*x^4 + 3*a^2*b*d^3*x^2
+ a^3*d^3), x) - 2*(((b^2*cos(c)^2 + b^2*sin(c)^2)*d^3*x^4 + 2*(a*b*cos(c)^2 + a*b*sin(c)^2)*d^3*x^2 + (a^2*co
s(c)^2 + a^2*sin(c)^2)*d^3)*cos(d*x + c)^2 + ((b^2*cos(c)^2 + b^2*sin(c)^2)*d^3*x^4 + 2*(a*b*cos(c)^2 + a*b*si
n(c)^2)*d^3*x^2 + (a^2*cos(c)^2 + a^2*sin(c)^2)*d^3)*sin(d*x + c)^2)*integrate((2*a*d*x*sin(d*x + c) + ((2*a*d
^2 + 3*b)*x^2 - a)*cos(d*x + c))/((b^3*d^3*x^6 + 3*a*b^2*d^3*x^4 + 3*a^2*b*d^3*x^2 + a^3*d^3)*cos(d*x + c)^2 +
(b^3*d^3*x^6 + 3*a*b^2*d^3*x^4 + 3*a^2*b*d^3*x^2 + a^3*d^3)*sin(d*x + c)^2), x) + ((d^2*x^3*sin(c) + d*x^2*co
s(c) - 2*x*sin(c))*cos(d*x + c)^2 + (d^2*x^3*sin(c) + d*x^2*cos(c) - 2*x*sin(c))*sin(d*x + c)^2)*sin(d*x + 2*c
))/(((b^2*cos(c)^2 + b^2*sin(c)^2)*d^3*x^4 + 2*(a*b*cos(c)^2 + a*b*sin(c)^2)*d^3*x^2 + (a^2*cos(c)^2 + a^2*sin
(c)^2)*d^3)*cos(d*x + c)^2 + ((b^2*cos(c)^2 + b^2*sin(c)^2)*d^3*x^4 + 2*(a*b*cos(c)^2 + a*b*sin(c)^2)*d^3*x^2
+ (a^2*cos(c)^2 + a^2*sin(c)^2)*d^3)*sin(d*x + c)^2)

Giac [F]

\[ \int \frac {x^3 \sin (c+d x)}{\left (a+b x^2\right )^2} \, dx=\int { \frac {x^{3} \sin \left (d x + c\right )}{{\left (b x^{2} + a\right )}^{2}} \,d x } \]

[In]

integrate(x^3*sin(d*x+c)/(b*x^2+a)^2,x, algorithm="giac")

[Out]

integrate(x^3*sin(d*x + c)/(b*x^2 + a)^2, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^3 \sin (c+d x)}{\left (a+b x^2\right )^2} \, dx=\int \frac {x^3\,\sin \left (c+d\,x\right )}{{\left (b\,x^2+a\right )}^2} \,d x \]

[In]

int((x^3*sin(c + d*x))/(a + b*x^2)^2,x)

[Out]

int((x^3*sin(c + d*x))/(a + b*x^2)^2, x)