\(\int \frac {x \sin (c+d x)}{(a+b x^2)^3} \, dx\) [74]

Optimal result

Integrand size = 17, antiderivative size = 512 \[ \int \frac {x \sin (c+d x)}{\left (a+b x^2\right )^3} \, dx=-\frac {d \cos (c+d x)}{16 a b^{3/2} \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {d \cos (c+d x)}{16 a b^{3/2} \left (\sqrt {-a}+\sqrt {b} x\right )}-\frac {d \cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 (-a)^{3/2} b^{3/2}}+\frac {d \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 (-a)^{3/2} b^{3/2}}+\frac {d^2 \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right ) \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 a b^2}+\frac {d^2 \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 a b^2}-\frac {\sin (c+d x)}{4 b \left (a+b x^2\right )^2}-\frac {d^2 \cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 a b^2}-\frac {d \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 (-a)^{3/2} b^{3/2}}+\frac {d^2 \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 a b^2}-\frac {d \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 (-a)^{3/2} b^{3/2}} \]

[Out]

1/16*d*Ci(d*x+d*(-a)^(1/2)/b^(1/2))*cos(c-d*(-a)^(1/2)/b^(1/2))/(-a)^(3/2)/b^(3/2)-1/16*d*Ci(-d*x+d*(-a)^(1/2)
/b^(1/2))*cos(c+d*(-a)^(1/2)/b^(1/2))/(-a)^(3/2)/b^(3/2)+1/16*d^2*cos(c+d*(-a)^(1/2)/b^(1/2))*Si(d*x-d*(-a)^(1
/2)/b^(1/2))/a/b^2+1/16*d^2*cos(c-d*(-a)^(1/2)/b^(1/2))*Si(d*x+d*(-a)^(1/2)/b^(1/2))/a/b^2-1/4*sin(d*x+c)/b/(b
*x^2+a)^2+1/16*d^2*Ci(d*x+d*(-a)^(1/2)/b^(1/2))*sin(c-d*(-a)^(1/2)/b^(1/2))/a/b^2-1/16*d*Si(d*x+d*(-a)^(1/2)/b
^(1/2))*sin(c-d*(-a)^(1/2)/b^(1/2))/(-a)^(3/2)/b^(3/2)+1/16*d^2*Ci(-d*x+d*(-a)^(1/2)/b^(1/2))*sin(c+d*(-a)^(1/
2)/b^(1/2))/a/b^2+1/16*d*Si(d*x-d*(-a)^(1/2)/b^(1/2))*sin(c+d*(-a)^(1/2)/b^(1/2))/(-a)^(3/2)/b^(3/2)-1/16*d*co
s(d*x+c)/a/b^(3/2)/((-a)^(1/2)-x*b^(1/2))+1/16*d*cos(d*x+c)/a/b^(3/2)/((-a)^(1/2)+x*b^(1/2))

Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 512, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {3422, 3415, 3378, 3384, 3380, 3383} \[ \int \frac {x \sin (c+d x)}{\left (a+b x^2\right )^3} \, dx=-\frac {d \cos \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 (-a)^{3/2} b^{3/2}}+\frac {d \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \operatorname {CosIntegral}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 (-a)^{3/2} b^{3/2}}-\frac {d \sin \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 (-a)^{3/2} b^{3/2}}-\frac {d \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 (-a)^{3/2} b^{3/2}}-\frac {d \cos (c+d x)}{16 a b^{3/2} \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {d \cos (c+d x)}{16 a b^{3/2} \left (\sqrt {-a}+\sqrt {b} x\right )}+\frac {d^2 \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \operatorname {CosIntegral}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 a b^2}+\frac {d^2 \sin \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 a b^2}-\frac {d^2 \cos \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 a b^2}+\frac {d^2 \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 a b^2}-\frac {\sin (c+d x)}{4 b \left (a+b x^2\right )^2} \]

[In]

Int[(x*Sin[c + d*x])/(a + b*x^2)^3,x]

[Out]

-1/16*(d*Cos[c + d*x])/(a*b^(3/2)*(Sqrt[-a] - Sqrt[b]*x)) + (d*Cos[c + d*x])/(16*a*b^(3/2)*(Sqrt[-a] + Sqrt[b]
*x)) - (d*Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*(-a)^(3/2)*b^(3/2)) + (d*
Cos[c - (Sqrt[-a]*d)/Sqrt[b]]*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*(-a)^(3/2)*b^(3/2)) + (d^2*CosInteg
ral[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sin[c - (Sqrt[-a]*d)/Sqrt[b]])/(16*a*b^2) + (d^2*CosIntegral[(Sqrt[-a]*d)/Sqrt
[b] - d*x]*Sin[c + (Sqrt[-a]*d)/Sqrt[b]])/(16*a*b^2) - Sin[c + d*x]/(4*b*(a + b*x^2)^2) - (d^2*Cos[c + (Sqrt[-
a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*a*b^2) - (d*Sin[c + (Sqrt[-a]*d)/Sqrt[b]]*SinInteg
ral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*(-a)^(3/2)*b^(3/2)) + (d^2*Cos[c - (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqr
t[-a]*d)/Sqrt[b] + d*x])/(16*a*b^2) - (d*Sin[c - (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x]
)/(16*(-a)^(3/2)*b^(3/2))

Rule 3378

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(c + d*x)^(m + 1)*(Sin[e + f*x]/(d*(m
+ 1))), x] - Dist[f/(d*(m + 1)), Int[(c + d*x)^(m + 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && LtQ[
m, -1]

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 3422

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

Rubi steps \begin{align*} \text {integral}& = -\frac {\sin (c+d x)}{4 b \left (a+b x^2\right )^2}+\frac {d \int \frac {\cos (c+d x)}{\left (a+b x^2\right )^2} \, dx}{4 b} \\ & = -\frac {\sin (c+d x)}{4 b \left (a+b x^2\right )^2}+\frac {d \int \left (-\frac {b \cos (c+d x)}{4 a \left (\sqrt {-a} \sqrt {b}-b x\right )^2}-\frac {b \cos (c+d x)}{4 a \left (\sqrt {-a} \sqrt {b}+b x\right )^2}-\frac {b \cos (c+d x)}{2 a \left (-a b-b^2 x^2\right )}\right ) \, dx}{4 b} \\ & = -\frac {\sin (c+d x)}{4 b \left (a+b x^2\right )^2}-\frac {d \int \frac {\cos (c+d x)}{\left (\sqrt {-a} \sqrt {b}-b x\right )^2} \, dx}{16 a}-\frac {d \int \frac {\cos (c+d x)}{\left (\sqrt {-a} \sqrt {b}+b x\right )^2} \, dx}{16 a}-\frac {d \int \frac {\cos (c+d x)}{-a b-b^2 x^2} \, dx}{8 a} \\ & = -\frac {d \cos (c+d x)}{16 a b^{3/2} \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {d \cos (c+d x)}{16 a b^{3/2} \left (\sqrt {-a}+\sqrt {b} x\right )}-\frac {\sin (c+d x)}{4 b \left (a+b x^2\right )^2}-\frac {d \int \left (-\frac {\sqrt {-a} \cos (c+d x)}{2 a b \left (\sqrt {-a}-\sqrt {b} x\right )}-\frac {\sqrt {-a} \cos (c+d x)}{2 a b \left (\sqrt {-a}+\sqrt {b} x\right )}\right ) \, dx}{8 a}-\frac {d^2 \int \frac {\sin (c+d x)}{\sqrt {-a} \sqrt {b}-b x} \, dx}{16 a b}+\frac {d^2 \int \frac {\sin (c+d x)}{\sqrt {-a} \sqrt {b}+b x} \, dx}{16 a b} \\ & = -\frac {d \cos (c+d x)}{16 a b^{3/2} \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {d \cos (c+d x)}{16 a b^{3/2} \left (\sqrt {-a}+\sqrt {b} x\right )}-\frac {\sin (c+d x)}{4 b \left (a+b x^2\right )^2}+\frac {d \int \frac {\cos (c+d x)}{\sqrt {-a}-\sqrt {b} x} \, dx}{16 (-a)^{3/2} b}+\frac {d \int \frac {\cos (c+d x)}{\sqrt {-a}+\sqrt {b} x} \, dx}{16 (-a)^{3/2} b}+\frac {\left (d^2 \cos \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}+b x} \, dx}{16 a b}+\frac {\left (d^2 \cos \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}-b x} \, dx}{16 a b}+\frac {\left (d^2 \sin \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}+b x} \, dx}{16 a b}-\frac {\left (d^2 \sin \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}-b x} \, dx}{16 a b} \\ & = -\frac {d \cos (c+d x)}{16 a b^{3/2} \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {d \cos (c+d x)}{16 a b^{3/2} \left (\sqrt {-a}+\sqrt {b} x\right )}+\frac {d^2 \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right ) \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 a b^2}+\frac {d^2 \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 a b^2}-\frac {\sin (c+d x)}{4 b \left (a+b x^2\right )^2}-\frac {d^2 \cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 a b^2}+\frac {d^2 \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 a b^2}+\frac {\left (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}{16 (-a)^{3/2} b}+\frac {\left (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}{16 (-a)^{3/2} b}-\frac {\left (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}{16 (-a)^{3/2} b}+\frac {\left (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}{16 (-a)^{3/2} b} \\ & = -\frac {d \cos (c+d x)}{16 a b^{3/2} \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {d \cos (c+d x)}{16 a b^{3/2} \left (\sqrt {-a}+\sqrt {b} x\right )}-\frac {d \cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 (-a)^{3/2} b^{3/2}}+\frac {d \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 (-a)^{3/2} b^{3/2}}+\frac {d^2 \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right ) \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 a b^2}+\frac {d^2 \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 a b^2}-\frac {\sin (c+d x)}{4 b \left (a+b x^2\right )^2}-\frac {d^2 \cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 a b^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 )}{16 (-a)^{3/2} b^{3/2}}+\frac {d^2 \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 a b^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 )}{16 (-a)^{3/2} b^{3/2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 2.53 (sec) , antiderivative size = 317, normalized size of antiderivative = 0.62 \[ \int \frac {x \sin (c+d x)}{\left (a+b x^2\right )^3} \, dx=\frac {i d e^{-i c-\frac {\sqrt {a} d}{\sqrt {b}}} \left (-\left (\left (\sqrt {b}-\sqrt {a} d\right ) e^{\frac {2 \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (-\frac {\sqrt {a} d}{\sqrt {b}}-i d x\right )\right )+\left (\sqrt {b}+\sqrt {a} d\right ) \operatorname {ExpIntegralEi}\left (\frac {\sqrt {a} d}{\sqrt {b}}-i d x\right )\right )-i d e^{i c-\frac {\sqrt {a} d}{\sqrt {b}}} \left (-\left (\left (\sqrt {b}-\sqrt {a} d\right ) e^{\frac {2 \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (-\frac {\sqrt {a} d}{\sqrt {b}}+i d x\right )\right )+\left (\sqrt {b}+\sqrt {a} d\right ) \operatorname {ExpIntegralEi}\left (\frac {\sqrt {a} d}{\sqrt {b}}+i d x\right )\right )+\frac {4 \sqrt {a} b \cos (d x) \left (d x \left (a+b x^2\right ) \cos (c)-2 a \sin (c)\right )}{\left (a+b x^2\right )^2}-\frac {4 \sqrt {a} b \left (2 a \cos (c)+d x \left (a+b x^2\right ) \sin (c)\right ) \sin (d x)}{\left (a+b x^2\right )^2}}{32 a^{3/2} b^2} \]

[In]

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

[Out]

(I*d*E^((-I)*c - (Sqrt[a]*d)/Sqrt[b])*(-((Sqrt[b] - Sqrt[a]*d)*E^((2*Sqrt[a]*d)/Sqrt[b])*ExpIntegralEi[-((Sqrt
[a]*d)/Sqrt[b]) - I*d*x]) + (Sqrt[b] + Sqrt[a]*d)*ExpIntegralEi[(Sqrt[a]*d)/Sqrt[b] - I*d*x]) - I*d*E^(I*c - (
Sqrt[a]*d)/Sqrt[b])*(-((Sqrt[b] - Sqrt[a]*d)*E^((2*Sqrt[a]*d)/Sqrt[b])*ExpIntegralEi[-((Sqrt[a]*d)/Sqrt[b]) +
I*d*x]) + (Sqrt[b] + Sqrt[a]*d)*ExpIntegralEi[(Sqrt[a]*d)/Sqrt[b] + I*d*x]) + (4*Sqrt[a]*b*Cos[d*x]*(d*x*(a +
b*x^2)*Cos[c] - 2*a*Sin[c]))/(a + b*x^2)^2 - (4*Sqrt[a]*b*(2*a*Cos[c] + d*x*(a + b*x^2)*Sin[c])*Sin[d*x])/(a +
b*x^2)^2)/(32*a^(3/2)*b^2)

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.68 (sec) , antiderivative size = 628, normalized size of antiderivative = 1.23

method	result	size
risch	\(\frac {i d^{2} {\mathrm e}^{\frac {i c b +d \sqrt {a b}}{b}} \operatorname {Ei}_{1}\left (\frac {i c b +d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right )}{32 a \,b^{2}}+\frac {i d^{2} {\mathrm e}^{\frac {i c b -d \sqrt {a b}}{b}} \operatorname {Ei}_{1}\left (-\frac {-i c b +d \sqrt {a b}+b \left (i d x +i c \right )}{b}\right )}{32 a \,b^{2}}-\frac {i d \,{\mathrm e}^{\frac {i c b +d \sqrt {a b}}{b}} \sqrt {a b}\, \operatorname {Ei}_{1}\left (\frac {i c b +d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right )}{32 a^{2} b^{2}}+\frac {i d \sqrt {a b}\, {\mathrm e}^{\frac {i c b -d \sqrt {a b}}{b}} \operatorname {Ei}_{1}\left (-\frac {-i c b +d \sqrt {a b}+b \left (i d x +i c \right )}{b}\right )}{32 a^{2} b^{2}}-\frac {i d^{2} {\mathrm e}^{-\frac {i c b -d \sqrt {a b}}{b}} \operatorname {Ei}_{1}\left (\frac {-i c b +d \sqrt {a b}+b \left (i d x +i c \right )}{b}\right )}{32 a \,b^{2}}-\frac {i d^{2} \operatorname {Ei}_{1}\left (-\frac {i c b +d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right ) {\mathrm e}^{-\frac {i c b +d \sqrt {a b}}{b}}}{32 a \,b^{2}}+\frac {i d \sqrt {a b}\, {\mathrm e}^{-\frac {i c b -d \sqrt {a b}}{b}} \operatorname {Ei}_{1}\left (\frac {-i c b +d \sqrt {a b}+b \left (i d x +i c \right )}{b}\right )}{32 a^{2} b^{2}}-\frac {i d \sqrt {a b}\, \operatorname {Ei}_{1}\left (-\frac {i c b +d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right ) {\mathrm e}^{-\frac {i c b +d \sqrt {a b}}{b}}}{32 a^{2} b^{2}}-\frac {d^{3} x \left (d^{2} x^{2} b +a \,d^{2}\right ) \cos \left (d x +c \right )}{8 a b \left (-b^{2} x^{4} d^{4}-2 a b \,d^{4} x^{2}-a^{2} d^{4}\right )}+\frac {d^{4} \sin \left (d x +c \right )}{4 b \left (-b^{2} x^{4} d^{4}-2 a b \,d^{4} x^{2}-a^{2} d^{4}\right )}\)	\(628\)
derivativedivides	\(\text {Expression too large to display}\)	\(1360\)
default	\(\text {Expression too large to display}\)	\(1360\)

[In]

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

[Out]

1/32*I*d^2/a/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)+1/32*I*d^2/a/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)-1/32*I*d/a^2/b^2*exp((I*c*b+d*(a*b)^(1
/2))/b)*(a*b)^(1/2)*Ei(1,(I*c*b+d*(a*b)^(1/2)-b*(I*d*x+I*c))/b)+1/32*I*d/a^2/b^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)-1/32*I*d^2/a/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)-1/32*I*d^2/a/b^2*Ei(1,-(I*c*b+d*(a*b)^(1/2)-b*(I*d*x+I*c))/b)*exp(-(I*
c*b+d*(a*b)^(1/2))/b)+1/32*I*d/a^2/b^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)-1/32*I*d/a^2/b^2*(a*b)^(1/2)*Ei(1,-(I*c*b+d*(a*b)^(1/2)-b*(I*d*x+I*c))/b)*exp(-(I*c*b+d*(a*b)^
(1/2))/b)-1/8*d^3/a*x*(b*d^2*x^2+a*d^2)/b/(-b^2*d^4*x^4-2*a*b*d^4*x^2-a^2*d^4)*cos(d*x+c)+1/4*d^4/b/(-b^2*d^4*
x^4-2*a*b*d^4*x^2-a^2*d^4)*sin(d*x+c)

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.32 (sec) , antiderivative size = 483, normalized size of antiderivative = 0.94 \[ \int \frac {x \sin (c+d x)}{\left (a+b x^2\right )^3} \, dx=-\frac {8 \, a^{2} b \sin \left (d x + c\right ) + {\left (i \, a b^{2} d^{2} x^{4} + 2 i \, a^{2} b d^{2} x^{2} + i \, a^{3} d^{2} - {\left (i \, b^{3} x^{4} + 2 i \, a b^{2} x^{2} + i \, a^{2} b\right )} \sqrt {\frac {a d^{2}}{b}}\right )} {\rm Ei}\left (i \, d x - \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (i \, c + \sqrt {\frac {a d^{2}}{b}}\right )} + {\left (i \, a b^{2} d^{2} x^{4} + 2 i \, a^{2} b d^{2} x^{2} + i \, a^{3} d^{2} - {\left (-i \, b^{3} x^{4} - 2 i \, a b^{2} x^{2} - i \, a^{2} b\right )} \sqrt {\frac {a d^{2}}{b}}\right )} {\rm Ei}\left (i \, d x + \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (i \, c - \sqrt {\frac {a d^{2}}{b}}\right )} + {\left (-i \, a b^{2} d^{2} x^{4} - 2 i \, a^{2} b d^{2} x^{2} - i \, a^{3} d^{2} - {\left (-i \, b^{3} x^{4} - 2 i \, a b^{2} x^{2} - i \, a^{2} b\right )} \sqrt {\frac {a d^{2}}{b}}\right )} {\rm Ei}\left (-i \, d x - \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (-i \, c + \sqrt {\frac {a d^{2}}{b}}\right )} + {\left (-i \, a b^{2} d^{2} x^{4} - 2 i \, a^{2} b d^{2} x^{2} - i \, a^{3} d^{2} - {\left (i \, b^{3} x^{4} + 2 i \, a b^{2} x^{2} + i \, a^{2} b\right )} \sqrt {\frac {a d^{2}}{b}}\right )} {\rm Ei}\left (-i \, d x + \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (-i \, c - \sqrt {\frac {a d^{2}}{b}}\right )} - 4 \, {\left (a b^{2} d x^{3} + a^{2} b d x\right )} \cos \left (d x + c\right )}{32 \, {\left (a^{2} b^{4} x^{4} + 2 \, a^{3} b^{3} x^{2} + a^{4} b^{2}\right )}} \]

[In]

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

[Out]

-1/32*(8*a^2*b*sin(d*x + c) + (I*a*b^2*d^2*x^4 + 2*I*a^2*b*d^2*x^2 + I*a^3*d^2 - (I*b^3*x^4 + 2*I*a*b^2*x^2 +
I*a^2*b)*sqrt(a*d^2/b))*Ei(I*d*x - sqrt(a*d^2/b))*e^(I*c + sqrt(a*d^2/b)) + (I*a*b^2*d^2*x^4 + 2*I*a^2*b*d^2*x
^2 + I*a^3*d^2 - (-I*b^3*x^4 - 2*I*a*b^2*x^2 - I*a^2*b)*sqrt(a*d^2/b))*Ei(I*d*x + sqrt(a*d^2/b))*e^(I*c - sqrt
(a*d^2/b)) + (-I*a*b^2*d^2*x^4 - 2*I*a^2*b*d^2*x^2 - I*a^3*d^2 - (-I*b^3*x^4 - 2*I*a*b^2*x^2 - I*a^2*b)*sqrt(a
*d^2/b))*Ei(-I*d*x - sqrt(a*d^2/b))*e^(-I*c + sqrt(a*d^2/b)) + (-I*a*b^2*d^2*x^4 - 2*I*a^2*b*d^2*x^2 - I*a^3*d
^2 - (I*b^3*x^4 + 2*I*a*b^2*x^2 + I*a^2*b)*sqrt(a*d^2/b))*Ei(-I*d*x + sqrt(a*d^2/b))*e^(-I*c - sqrt(a*d^2/b))
- 4*(a*b^2*d*x^3 + a^2*b*d*x)*cos(d*x + c))/(a^2*b^4*x^4 + 2*a^3*b^3*x^2 + a^4*b^2)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

-1/2*((cos(c)^2 + sin(c)^2)*x*cos(d*x + c) + (x*cos(d*x + c)^2*cos(c) + x*cos(c)*sin(d*x + c)^2)*cos(d*x + 2*c
) + 2*(((b^3*cos(c)^2 + b^3*sin(c)^2)*d*x^6 + 3*(a*b^2*cos(c)^2 + a*b^2*sin(c)^2)*d*x^4 + 3*(a^2*b*cos(c)^2 +
a^2*b*sin(c)^2)*d*x^2 + (a^3*cos(c)^2 + a^3*sin(c)^2)*d)*cos(d*x + c)^2 + ((b^3*cos(c)^2 + b^3*sin(c)^2)*d*x^6
+ 3*(a*b^2*cos(c)^2 + a*b^2*sin(c)^2)*d*x^4 + 3*(a^2*b*cos(c)^2 + a^2*b*sin(c)^2)*d*x^2 + (a^3*cos(c)^2 + a^3
*sin(c)^2)*d)*sin(d*x + c)^2)*integrate(1/2*(5*b*x^2 - a)*cos(d*x + c)/(b^4*d*x^8 + 4*a*b^3*d*x^6 + 6*a^2*b^2*
d*x^4 + 4*a^3*b*d*x^2 + a^4*d), x) + 2*(((b^3*cos(c)^2 + b^3*sin(c)^2)*d*x^6 + 3*(a*b^2*cos(c)^2 + a*b^2*sin(c
)^2)*d*x^4 + 3*(a^2*b*cos(c)^2 + a^2*b*sin(c)^2)*d*x^2 + (a^3*cos(c)^2 + a^3*sin(c)^2)*d)*cos(d*x + c)^2 + ((b
^3*cos(c)^2 + b^3*sin(c)^2)*d*x^6 + 3*(a*b^2*cos(c)^2 + a*b^2*sin(c)^2)*d*x^4 + 3*(a^2*b*cos(c)^2 + a^2*b*sin(
c)^2)*d*x^2 + (a^3*cos(c)^2 + a^3*sin(c)^2)*d)*sin(d*x + c)^2)*integrate(1/2*(5*b*x^2 - a)*cos(d*x + c)/((b^4*
d*x^8 + 4*a*b^3*d*x^6 + 6*a^2*b^2*d*x^4 + 4*a^3*b*d*x^2 + a^4*d)*cos(d*x + c)^2 + (b^4*d*x^8 + 4*a*b^3*d*x^6 +
6*a^2*b^2*d*x^4 + 4*a^3*b*d*x^2 + a^4*d)*sin(d*x + c)^2), x) + (x*cos(d*x + c)^2*sin(c) + x*sin(d*x + c)^2*si
n(c))*sin(d*x + 2*c))/(((b^3*cos(c)^2 + b^3*sin(c)^2)*d*x^6 + 3*(a*b^2*cos(c)^2 + a*b^2*sin(c)^2)*d*x^4 + 3*(a
^2*b*cos(c)^2 + a^2*b*sin(c)^2)*d*x^2 + (a^3*cos(c)^2 + a^3*sin(c)^2)*d)*cos(d*x + c)^2 + ((b^3*cos(c)^2 + b^3
*sin(c)^2)*d*x^6 + 3*(a*b^2*cos(c)^2 + a*b^2*sin(c)^2)*d*x^4 + 3*(a^2*b*cos(c)^2 + a^2*b*sin(c)^2)*d*x^2 + (a^
3*cos(c)^2 + a^3*sin(c)^2)*d)*sin(d*x + c)^2)

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int((x*sin(c + d*x))/(a + b*x^2)^3,x)

[Out]

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