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

Optimal. Leaf size=746 \[ \frac{d^2 \sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{16 \sqrt{-a} b^{5/2}}-\frac{d^2 \sin \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 \sqrt{-a} b^{5/2}}+\frac{\sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{16 (-a)^{3/2} b^{3/2}}-\frac{\sin \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 (-a)^{3/2} b^{3/2}}-\frac{d \cos \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 a b^2}-\frac{d \cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{16 a b^2}+\frac{d^2 \cos \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 \sqrt{-a} b^{5/2}}+\frac{d^2 \cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{16 \sqrt{-a} b^{5/2}}-\frac{d \sin \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 a b^2}+\frac{d \sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{16 a b^2}+\frac{\cos \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 (-a)^{3/2} b^{3/2}}+\frac{\cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{16 (-a)^{3/2} b^{3/2}}-\frac{d \cos (c+d x)}{8 b^2 \left (a+b x^2\right )}-\frac{\sin (c+d x)}{16 a b^{3/2} \left (\sqrt{-a}-\sqrt{b} x\right )}+\frac{\sin (c+d x)}{16 a b^{3/2} \left (\sqrt{-a}+\sqrt{b} x\right )}-\frac{x \sin (c+d x)}{4 b \left (a+b x^2\right )^2} \]

[Out]

-(d*Cos[c + d*x])/(8*b^2*(a + b*x^2)) - (d*Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*
x])/(16*a*b^2) - (d*Cos[c - (Sqrt[-a]*d)/Sqrt[b]]*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*a*b^2) + (CosIn
tegral[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sin[c - (Sqrt[-a]*d)/Sqrt[b]])/(16*(-a)^(3/2)*b^(3/2)) + (d^2*CosIntegral[(
Sqrt[-a]*d)/Sqrt[b] + d*x]*Sin[c - (Sqrt[-a]*d)/Sqrt[b]])/(16*Sqrt[-a]*b^(5/2)) - (CosIntegral[(Sqrt[-a]*d)/Sq
rt[b] - d*x]*Sin[c + (Sqrt[-a]*d)/Sqrt[b]])/(16*(-a)^(3/2)*b^(3/2)) - (d^2*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] -
d*x]*Sin[c + (Sqrt[-a]*d)/Sqrt[b]])/(16*Sqrt[-a]*b^(5/2)) - Sin[c + d*x]/(16*a*b^(3/2)*(Sqrt[-a] - Sqrt[b]*x))
+ Sin[c + d*x]/(16*a*b^(3/2)*(Sqrt[-a] + Sqrt[b]*x)) - (x*Sin[c + d*x])/(4*b*(a + b*x^2)^2) + (Cos[c + (Sqrt[
-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*(-a)^(3/2)*b^(3/2)) + (d^2*Cos[c + (Sqrt[-a]*d)/S
qrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*Sqrt[-a]*b^(5/2)) - (d*Sin[c + (Sqrt[-a]*d)/Sqrt[b]]*SinI
ntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*a*b^2) + (Cos[c - (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt
[b] + d*x])/(16*(-a)^(3/2)*b^(3/2)) + (d^2*Cos[c - (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*
x])/(16*Sqrt[-a]*b^(5/2)) + (d*Sin[c - (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*a*b^
2)

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Rubi [A]  time = 1.13539, antiderivative size = 746, normalized size of antiderivative = 1., number of steps used = 28, number of rules used = 7, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.368, Rules used = {3343, 3333, 3297, 3303, 3299, 3302, 3342} \[ \frac{d^2 \sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{16 \sqrt{-a} b^{5/2}}-\frac{d^2 \sin \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 \sqrt{-a} b^{5/2}}+\frac{\sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{16 (-a)^{3/2} b^{3/2}}-\frac{\sin \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 (-a)^{3/2} b^{3/2}}-\frac{d \cos \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 a b^2}-\frac{d \cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{16 a b^2}+\frac{d^2 \cos \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 \sqrt{-a} b^{5/2}}+\frac{d^2 \cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{16 \sqrt{-a} b^{5/2}}-\frac{d \sin \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 a b^2}+\frac{d \sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{16 a b^2}+\frac{\cos \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 (-a)^{3/2} b^{3/2}}+\frac{\cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{16 (-a)^{3/2} b^{3/2}}-\frac{d \cos (c+d x)}{8 b^2 \left (a+b x^2\right )}-\frac{\sin (c+d x)}{16 a b^{3/2} \left (\sqrt{-a}-\sqrt{b} x\right )}+\frac{\sin (c+d x)}{16 a b^{3/2} \left (\sqrt{-a}+\sqrt{b} x\right )}-\frac{x \sin (c+d x)}{4 b \left (a+b x^2\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(d*Cos[c + d*x])/(8*b^2*(a + b*x^2)) - (d*Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*
x])/(16*a*b^2) - (d*Cos[c - (Sqrt[-a]*d)/Sqrt[b]]*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*a*b^2) + (CosIn
tegral[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sin[c - (Sqrt[-a]*d)/Sqrt[b]])/(16*(-a)^(3/2)*b^(3/2)) + (d^2*CosIntegral[(
Sqrt[-a]*d)/Sqrt[b] + d*x]*Sin[c - (Sqrt[-a]*d)/Sqrt[b]])/(16*Sqrt[-a]*b^(5/2)) - (CosIntegral[(Sqrt[-a]*d)/Sq
rt[b] - d*x]*Sin[c + (Sqrt[-a]*d)/Sqrt[b]])/(16*(-a)^(3/2)*b^(3/2)) - (d^2*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] -
d*x]*Sin[c + (Sqrt[-a]*d)/Sqrt[b]])/(16*Sqrt[-a]*b^(5/2)) - Sin[c + d*x]/(16*a*b^(3/2)*(Sqrt[-a] - Sqrt[b]*x))
+ Sin[c + d*x]/(16*a*b^(3/2)*(Sqrt[-a] + Sqrt[b]*x)) - (x*Sin[c + d*x])/(4*b*(a + b*x^2)^2) + (Cos[c + (Sqrt[
-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*(-a)^(3/2)*b^(3/2)) + (d^2*Cos[c + (Sqrt[-a]*d)/S
qrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*Sqrt[-a]*b^(5/2)) - (d*Sin[c + (Sqrt[-a]*d)/Sqrt[b]]*SinI
ntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*a*b^2) + (Cos[c - (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt
[b] + d*x])/(16*(-a)^(3/2)*b^(3/2)) + (d^2*Cos[c - (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*
x])/(16*Sqrt[-a]*b^(5/2)) + (d*Sin[c - (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*a*b^
2)

Rule 3343

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 3333

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*Sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Int[ExpandIntegrand[Sin[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 3297

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 3303

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 3299

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 3302

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 3342

Int[Cos[(c_.) + (d_.)*(x_)]*((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(e^m*(a + b*x^
n)^(p + 1)*Cos[c + d*x])/(b*n*(p + 1)), x] + Dist[(d*e^m)/(b*n*(p + 1)), Int[(a + b*x^n)^(p + 1)*Sin[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*} \int \frac{x^2 \sin (c+d x)}{\left (a+b x^2\right )^3} \, dx &=-\frac{x \sin (c+d x)}{4 b \left (a+b x^2\right )^2}+\frac{\int \frac{\sin (c+d x)}{\left (a+b x^2\right )^2} \, dx}{4 b}+\frac{d \int \frac{x \cos (c+d x)}{\left (a+b x^2\right )^2} \, dx}{4 b}\\ &=-\frac{d \cos (c+d x)}{8 b^2 \left (a+b x^2\right )}-\frac{x \sin (c+d x)}{4 b \left (a+b x^2\right )^2}+\frac{\int \left (-\frac{b \sin (c+d x)}{4 a \left (\sqrt{-a} \sqrt{b}-b x\right )^2}-\frac{b \sin (c+d x)}{4 a \left (\sqrt{-a} \sqrt{b}+b x\right )^2}-\frac{b \sin (c+d x)}{2 a \left (-a b-b^2 x^2\right )}\right ) \, dx}{4 b}-\frac{d^2 \int \frac{\sin (c+d x)}{a+b x^2} \, dx}{8 b^2}\\ &=-\frac{d \cos (c+d x)}{8 b^2 \left (a+b x^2\right )}-\frac{x \sin (c+d x)}{4 b \left (a+b x^2\right )^2}-\frac{\int \frac{\sin (c+d x)}{\left (\sqrt{-a} \sqrt{b}-b x\right )^2} \, dx}{16 a}-\frac{\int \frac{\sin (c+d x)}{\left (\sqrt{-a} \sqrt{b}+b x\right )^2} \, dx}{16 a}-\frac{\int \frac{\sin (c+d x)}{-a b-b^2 x^2} \, dx}{8 a}-\frac{d^2 \int \left (\frac{\sqrt{-a} \sin (c+d x)}{2 a \left (\sqrt{-a}-\sqrt{b} x\right )}+\frac{\sqrt{-a} \sin (c+d x)}{2 a \left (\sqrt{-a}+\sqrt{b} x\right )}\right ) \, dx}{8 b^2}\\ &=-\frac{d \cos (c+d x)}{8 b^2 \left (a+b x^2\right )}-\frac{\sin (c+d x)}{16 a b^{3/2} \left (\sqrt{-a}-\sqrt{b} x\right )}+\frac{\sin (c+d x)}{16 a b^{3/2} \left (\sqrt{-a}+\sqrt{b} x\right )}-\frac{x \sin (c+d x)}{4 b \left (a+b x^2\right )^2}-\frac{\int \left (-\frac{\sqrt{-a} \sin (c+d x)}{2 a b \left (\sqrt{-a}-\sqrt{b} x\right )}-\frac{\sqrt{-a} \sin (c+d x)}{2 a b \left (\sqrt{-a}+\sqrt{b} x\right )}\right ) \, dx}{8 a}+\frac{d \int \frac{\cos (c+d x)}{\sqrt{-a} \sqrt{b}-b x} \, dx}{16 a b}-\frac{d \int \frac{\cos (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} x} \, dx}{16 \sqrt{-a} b^2}+\frac{d^2 \int \frac{\sin (c+d x)}{\sqrt{-a}+\sqrt{b} x} \, dx}{16 \sqrt{-a} b^2}\\ &=-\frac{d \cos (c+d x)}{8 b^2 \left (a+b x^2\right )}-\frac{\sin (c+d x)}{16 a b^{3/2} \left (\sqrt{-a}-\sqrt{b} x\right )}+\frac{\sin (c+d x)}{16 a b^{3/2} \left (\sqrt{-a}+\sqrt{b} x\right )}-\frac{x \sin (c+d x)}{4 b \left (a+b x^2\right )^2}+\frac{\int \frac{\sin (c+d x)}{\sqrt{-a}-\sqrt{b} x} \, dx}{16 (-a)^{3/2} b}+\frac{\int \frac{\sin (c+d x)}{\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}+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} x} \, dx}{16 \sqrt{-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}-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} x} \, dx}{16 \sqrt{-a} b^2}+\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}+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} x} \, dx}{16 \sqrt{-a} b^2}+\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}-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} x} \, dx}{16 \sqrt{-a} b^2}\\ &=-\frac{d \cos (c+d x)}{8 b^2 \left (a+b x^2\right )}-\frac{d \cos \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Ci}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 a b^2}-\frac{d \cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Ci}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{16 a b^2}+\frac{d^2 \text{Ci}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right ) \sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{16 \sqrt{-a} b^{5/2}}-\frac{d^2 \text{Ci}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right ) \sin \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{16 \sqrt{-a} b^{5/2}}-\frac{\sin (c+d x)}{16 a b^{3/2} \left (\sqrt{-a}-\sqrt{b} x\right )}+\frac{\sin (c+d x)}{16 a b^{3/2} \left (\sqrt{-a}+\sqrt{b} x\right )}-\frac{x \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 \sqrt{-a} b^{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 )}{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 \sqrt{-a} b^{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 )}{16 a 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}{16 (-a)^{3/2} b}-\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}{16 (-a)^{3/2} b}+\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}{16 (-a)^{3/2} b}+\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}{16 (-a)^{3/2} b}\\ &=-\frac{d \cos (c+d x)}{8 b^2 \left (a+b x^2\right )}-\frac{d \cos \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Ci}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 a b^2}-\frac{d \cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Ci}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{16 a b^2}+\frac{\text{Ci}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right ) \sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{16 (-a)^{3/2} b^{3/2}}+\frac{d^2 \text{Ci}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right ) \sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{16 \sqrt{-a} b^{5/2}}-\frac{\text{Ci}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right ) \sin \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{16 (-a)^{3/2} b^{3/2}}-\frac{d^2 \text{Ci}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right ) \sin \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{16 \sqrt{-a} b^{5/2}}-\frac{\sin (c+d x)}{16 a b^{3/2} \left (\sqrt{-a}-\sqrt{b} x\right )}+\frac{\sin (c+d x)}{16 a b^{3/2} \left (\sqrt{-a}+\sqrt{b} x\right )}-\frac{x \sin (c+d x)}{4 b \left (a+b x^2\right )^2}+\frac{\cos \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 \sqrt{-a} b^{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 )}{16 a 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 )}{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 \sqrt{-a} b^{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 )}{16 a b^2}\\ \end{align*}

Mathematica [C]  time = 2.72722, size = 927, normalized size = 1.24 \[ \frac{\frac{2 \sqrt{a} b^2 \cos (d x) \sin (c) x^3}{\left (b x^2+a\right )^2}+\frac{2 \sqrt{a} b^2 \cos (c) \sin (d x) x^3}{\left (b x^2+a\right )^2}-\frac{2 a^{3/2} b d \cos (c) \cos (d x) x^2}{\left (b x^2+a\right )^2}+\frac{2 a^{3/2} b d \sin (c) \sin (d x) x^2}{\left (b x^2+a\right )^2}-\frac{2 a^{3/2} b \cos (d x) \sin (c) x}{\left (b x^2+a\right )^2}-\frac{2 a^{3/2} b \cos (c) \sin (d x) x}{\left (b x^2+a\right )^2}-\frac{2 a^{5/2} d \cos (c) \cos (d x)}{\left (b x^2+a\right )^2}+\frac{\text{CosIntegral}\left (d \left (x+\frac{i \sqrt{a}}{\sqrt{b}}\right )\right ) \left (i \left (b-a d^2\right ) \sin \left (c-\frac{i \sqrt{a} d}{\sqrt{b}}\right )-\sqrt{a} \sqrt{b} d \cos \left (c-\frac{i \sqrt{a} d}{\sqrt{b}}\right )\right )}{\sqrt{b}}+\frac{i \text{CosIntegral}\left (d \left (x-\frac{i \sqrt{a}}{\sqrt{b}}\right )\right ) \left (i \sqrt{a} \sqrt{b} d \cos \left (c+\frac{i \sqrt{a} d}{\sqrt{b}}\right )+\left (a d^2-b\right ) \sin \left (c+\frac{i \sqrt{a} d}{\sqrt{b}}\right )\right )}{\sqrt{b}}+\frac{2 a^{5/2} d \sin (c) \sin (d x)}{\left (b x^2+a\right )^2}-\frac{i a d^2 \cos (c) \cosh \left (\frac{\sqrt{a} d}{\sqrt{b}}\right ) \text{Si}\left (d \left (x+\frac{i \sqrt{a}}{\sqrt{b}}\right )\right )}{\sqrt{b}}+i \sqrt{b} \cos (c) \cosh \left (\frac{\sqrt{a} d}{\sqrt{b}}\right ) \text{Si}\left (d \left (x+\frac{i \sqrt{a}}{\sqrt{b}}\right )\right )+\sqrt{a} d \cosh \left (\frac{\sqrt{a} d}{\sqrt{b}}\right ) \sin (c) \text{Si}\left (d \left (x+\frac{i \sqrt{a}}{\sqrt{b}}\right )\right )-i \sqrt{a} d \cos (c) \sinh \left (\frac{\sqrt{a} d}{\sqrt{b}}\right ) \text{Si}\left (d \left (x+\frac{i \sqrt{a}}{\sqrt{b}}\right )\right )+\frac{a d^2 \sin (c) \sinh \left (\frac{\sqrt{a} d}{\sqrt{b}}\right ) \text{Si}\left (d \left (x+\frac{i \sqrt{a}}{\sqrt{b}}\right )\right )}{\sqrt{b}}-\sqrt{b} \sin (c) \sinh \left (\frac{\sqrt{a} d}{\sqrt{b}}\right ) \text{Si}\left (d \left (x+\frac{i \sqrt{a}}{\sqrt{b}}\right )\right )-\frac{i a d^2 \cos (c) \cosh \left (\frac{\sqrt{a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right )}{\sqrt{b}}+i \sqrt{b} \cos (c) \cosh \left (\frac{\sqrt{a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right )-\sqrt{a} d \cosh \left (\frac{\sqrt{a} d}{\sqrt{b}}\right ) \sin (c) \text{Si}\left (\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right )-i \sqrt{a} d \cos (c) \sinh \left (\frac{\sqrt{a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right )-\frac{a d^2 \sin (c) \sinh \left (\frac{\sqrt{a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right )}{\sqrt{b}}+\sqrt{b} \sin (c) \sinh \left (\frac{\sqrt{a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right )}{16 a^{3/2} b^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((-2*a^(5/2)*d*Cos[c]*Cos[d*x])/(a + b*x^2)^2 - (2*a^(3/2)*b*d*x^2*Cos[c]*Cos[d*x])/(a + b*x^2)^2 - (2*a^(3/2)
*b*x*Cos[d*x]*Sin[c])/(a + b*x^2)^2 + (2*Sqrt[a]*b^2*x^3*Cos[d*x]*Sin[c])/(a + b*x^2)^2 + (CosIntegral[d*((I*S
qrt[a])/Sqrt[b] + x)]*(-(Sqrt[a]*Sqrt[b]*d*Cos[c - (I*Sqrt[a]*d)/Sqrt[b]]) + I*(b - a*d^2)*Sin[c - (I*Sqrt[a]*
d)/Sqrt[b]]))/Sqrt[b] + (I*CosIntegral[d*(((-I)*Sqrt[a])/Sqrt[b] + x)]*(I*Sqrt[a]*Sqrt[b]*d*Cos[c + (I*Sqrt[a]
*d)/Sqrt[b]] + (-b + a*d^2)*Sin[c + (I*Sqrt[a]*d)/Sqrt[b]]))/Sqrt[b] - (2*a^(3/2)*b*x*Cos[c]*Sin[d*x])/(a + b*
x^2)^2 + (2*Sqrt[a]*b^2*x^3*Cos[c]*Sin[d*x])/(a + b*x^2)^2 + (2*a^(5/2)*d*Sin[c]*Sin[d*x])/(a + b*x^2)^2 + (2*
a^(3/2)*b*d*x^2*Sin[c]*Sin[d*x])/(a + b*x^2)^2 + I*Sqrt[b]*Cos[c]*Cosh[(Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*
Sqrt[a])/Sqrt[b] + x)] - (I*a*d^2*Cos[c]*Cosh[(Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)])/S
qrt[b] + Sqrt[a]*d*Cosh[(Sqrt[a]*d)/Sqrt[b]]*Sin[c]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] - I*Sqrt[a]*d*Cos
[c]*Sinh[(Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] - Sqrt[b]*Sin[c]*Sinh[(Sqrt[a]*d)/Sqrt[
b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] + (a*d^2*Sin[c]*Sinh[(Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*Sqrt[
a])/Sqrt[b] + x)])/Sqrt[b] + I*Sqrt[b]*Cos[c]*Cosh[(Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*
x] - (I*a*d^2*Cos[c]*Cosh[(Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x])/Sqrt[b] - Sqrt[a]*d*C
osh[(Sqrt[a]*d)/Sqrt[b]]*Sin[c]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x] - I*Sqrt[a]*d*Cos[c]*Sinh[(Sqrt[a]*d)
/Sqrt[b]]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x] + Sqrt[b]*Sin[c]*Sinh[(Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(I*S
qrt[a]*d)/Sqrt[b] - d*x] - (a*d^2*Sin[c]*Sinh[(Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x])/S
qrt[b])/(16*a^(3/2)*b^2)

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Maple [B]  time = 0.085, size = 2310, normalized size = 3.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d^3*(1/8*sin(d*x+c)*d^2*((d*x+c)^3*a*b*d^2+3*(d*x+c)^3*b^2*c^2-3*(d*x+c)^2*a*b*c*d^2-9*(d*x+c)^2*b^2*c^3-(d*
x+c)*a^2*d^4+8*(d*x+c)*a*b*c^2*d^2+9*(d*x+c)*b^2*c^4-3*a^2*c*d^4-6*a*b*c^3*d^2-3*b^2*c^5)/a^2/b/((d*x+c)^2*b-2
*(d*x+c)*b*c+a*d^2+c^2*b)^2+1/8*cos(d*x+c)*d^4*(2*(d*x+c)*b*c-a*d^2-c^2*b)/a/b^2/((d*x+c)^2*b-2*(d*x+c)*b*c+a*
d^2+c^2*b)+1/16*d^2*(2*(d*(-a*b)^(1/2)+c*b)*a*c*d^2-a^2*d^4-a*b*c^2*d^2+a*b*d^2+3*c^2*b^2)/a^2/b^3/((d*(-a*b)^
(1/2)+c*b)/b-c)*(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))+1/16*d^2*(-2*(d*(-a*b)^(1/2)-c*b)*a*c*d^2-a^2*d^4-a*b*c^2*d^2+a*b*d^2+3*c^2*b^2)
/a^2/b^3/(-(d*(-a*b)^(1/2)-c*b)/b-c)*(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))-1/16*d^2*((d*(-a*b)^(1/2)+c*b)/b*a*d^2+3*(d*(-a*b)^(1/2)+c*
b)*c^2-3*a*c*d^2-3*c^3*b)/a^2/b^2/((d*(-a*b)^(1/2)+c*b)/b-c)*(-Si(d*x+c-(d*(-a*b)^(1/2)+c*b)/b)*sin((d*(-a*b)^
(1/2)+c*b)/b)+Ci(d*x+c-(d*(-a*b)^(1/2)+c*b)/b)*cos((d*(-a*b)^(1/2)+c*b)/b))-1/16*d^2*(-(d*(-a*b)^(1/2)-c*b)/b*
a*d^2-3*(d*(-a*b)^(1/2)-c*b)*c^2-3*a*c*d^2-3*c^3*b)/a^2/b^2/(-(d*(-a*b)^(1/2)-c*b)/b-c)*(Si(d*x+c+(d*(-a*b)^(1
/2)-c*b)/b)*sin((d*(-a*b)^(1/2)-c*b)/b)+Ci(d*x+c+(d*(-a*b)^(1/2)-c*b)/b)*cos((d*(-a*b)^(1/2)-c*b)/b))-1/4*sin(
d*x+c)*c*d^2*(3*c*(d*x+c)^3*b^2-9*b^2*c^2*(d*x+c)^2+5*(d*x+c)*a*b*c*d^2+9*(d*x+c)*b^2*c^3-2*a^2*d^4-5*a*b*c^2*
d^2-3*b^2*c^4)/a^2/b/((d*x+c)^2*b-2*(d*x+c)*b*c+a*d^2+c^2*b)^2-1/4*cos(d*x+c)*c*d^4/a/b*(d*x+c)/((d*x+c)^2*b-2
*(d*x+c)*b*c+a*d^2+c^2*b)-1/8*c*d^2*((d*(-a*b)^(1/2)+c*b)/b*a*d^2+3*c*b)/a^2/b^2/((d*(-a*b)^(1/2)+c*b)/b-c)*(S
i(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))-1/8*c*d^2*(-(d*(-a*b)^(1/2)-c*b)/b*a*d^2+3*c*b)/a^2/b^2/(-(d*(-a*b)^(1/2)-c*b)/b-c)*(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))+
1/8*c*d^2*(3*(d*(-a*b)^(1/2)+c*b)*c-a*d^2-3*c^2*b)/a^2/b^2/((d*(-a*b)^(1/2)+c*b)/b-c)*(-Si(d*x+c-(d*(-a*b)^(1/
2)+c*b)/b)*sin((d*(-a*b)^(1/2)+c*b)/b)+Ci(d*x+c-(d*(-a*b)^(1/2)+c*b)/b)*cos((d*(-a*b)^(1/2)+c*b)/b))+1/8*c*d^2
*(-3*(d*(-a*b)^(1/2)-c*b)*c-a*d^2-3*c^2*b)/a^2/b^2/(-(d*(-a*b)^(1/2)-c*b)/b-c)*(Si(d*x+c+(d*(-a*b)^(1/2)-c*b)/
b)*sin((d*(-a*b)^(1/2)-c*b)/b)+Ci(d*x+c+(d*(-a*b)^(1/2)-c*b)/b)*cos((d*(-a*b)^(1/2)-c*b)/b))+d^6*c^2*(1/8*sin(
d*x+c)*(3*(d*x+c)^3*b-9*c*(d*x+c)^2*b+5*(d*x+c)*a*d^2+9*(d*x+c)*b*c^2-5*a*c*d^2-3*c^3*b)/a^2/d^4/((d*x+c)^2*b-
2*(d*x+c)*b*c+a*d^2+c^2*b)^2+1/8*cos(d*x+c)/a/b/d^2/((d*x+c)^2*b-2*(d*x+c)*b*c+a*d^2+c^2*b)+1/16*(a*d^2+3*b)/a
^2/b^2/d^4/((d*(-a*b)^(1/2)+c*b)/b-c)*(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))+1/16*(a*d^2+3*b)/a^2/b^2/d^4/(-(d*(-a*b)^(1/2)-c*b)/b-c)*(
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))-3/16/a^2/b/d^4*(-Si(d*x+c-(d*(-a*b)^(1/2)+c*b)/b)*sin((d*(-a*b)^(1/2)+c*b)/b)+Ci(d*x+c-(d*(-a*b)^
(1/2)+c*b)/b)*cos((d*(-a*b)^(1/2)+c*b)/b))-3/16/a^2/b/d^4*(Si(d*x+c+(d*(-a*b)^(1/2)-c*b)/b)*sin((d*(-a*b)^(1/2
)-c*b)/b)+Ci(d*x+c+(d*(-a*b)^(1/2)-c*b)/b)*cos((d*(-a*b)^(1/2)-c*b)/b))))

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Maxima [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Fricas [C]  time = 2.18829, size = 1176, normalized size = 1.58 \begin{align*} -\frac{{\left (a b^{2} d^{2} x^{4} + 2 \, a^{2} b d^{2} x^{2} + a^{3} d^{2} +{\left (a^{3} d^{2} +{\left (a b^{2} d^{2} - b^{3}\right )} x^{4} - a^{2} b + 2 \,{\left (a^{2} b d^{2} - a b^{2}\right )} x^{2}\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^{2} d^{2} x^{4} + 2 \, a^{2} b d^{2} x^{2} + a^{3} d^{2} -{\left (a^{3} d^{2} +{\left (a b^{2} d^{2} - b^{3}\right )} x^{4} - a^{2} b + 2 \,{\left (a^{2} b d^{2} - a b^{2}\right )} x^{2}\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^{2} d^{2} x^{4} + 2 \, a^{2} b d^{2} x^{2} + a^{3} d^{2} +{\left (a^{3} d^{2} +{\left (a b^{2} d^{2} - b^{3}\right )} x^{4} - a^{2} b + 2 \,{\left (a^{2} b d^{2} - a b^{2}\right )} x^{2}\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^{2} d^{2} x^{4} + 2 \, a^{2} b d^{2} x^{2} + a^{3} d^{2} -{\left (a^{3} d^{2} +{\left (a b^{2} d^{2} - b^{3}\right )} x^{4} - a^{2} b + 2 \,{\left (a^{2} b d^{2} - a b^{2}\right )} x^{2}\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^{2} b d^{2} x^{2} + a^{3} d^{2}\right )} \cos \left (d x + c\right ) - 4 \,{\left (a b^{2} d x^{3} - a^{2} b d x\right )} \sin \left (d x + c\right )}{32 \,{\left (a^{2} b^{4} d x^{4} + 2 \, a^{3} b^{3} d x^{2} + a^{4} b^{2} d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2} \sin \left (d x + c\right )}{{\left (b x^{2} + a\right )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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