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

Optimal. Leaf size=416 \[ \frac{\sin \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{4 \sqrt{-a} b^{3/2}}-\frac{\sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{4 \sqrt{-a} b^{3/2}}+\frac{d \cos \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{4 b^2}+\frac{d \cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{4 b^2}+\frac{d \sin \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{4 b^2}-\frac{d \sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{4 b^2}-\frac{\cos \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{4 \sqrt{-a} b^{3/2}}-\frac{\cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{4 \sqrt{-a} b^{3/2}}-\frac{x \sin (c+d x)}{2 b \left (a+b x^2\right )} \]

[Out]

(d*Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(4*b^2) + (d*Cos[c - (Sqrt[-a]*d)/Sq
rt[b]]*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(4*b^2) - (CosIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sin[c - (Sq
rt[-a]*d)/Sqrt[b]])/(4*Sqrt[-a]*b^(3/2)) + (CosIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x]*Sin[c + (Sqrt[-a]*d)/Sqrt[
b]])/(4*Sqrt[-a]*b^(3/2)) - (x*Sin[c + d*x])/(2*b*(a + b*x^2)) - (Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(S
qrt[-a]*d)/Sqrt[b] - d*x])/(4*Sqrt[-a]*b^(3/2)) + (d*Sin[c + (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sq
rt[b] - d*x])/(4*b^2) - (Cos[c - (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(4*Sqrt[-a]*b^
(3/2)) - (d*Sin[c - (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(4*b^2)

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Rubi [A]  time = 0.572871, antiderivative size = 416, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316, Rules used = {3343, 3333, 3303, 3299, 3302, 3346} \[ \frac{\sin \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{4 \sqrt{-a} b^{3/2}}-\frac{\sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{4 \sqrt{-a} b^{3/2}}+\frac{d \cos \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{4 b^2}+\frac{d \cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{4 b^2}+\frac{d \sin \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{4 b^2}-\frac{d \sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{4 b^2}-\frac{\cos \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{4 \sqrt{-a} b^{3/2}}-\frac{\cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{4 \sqrt{-a} b^{3/2}}-\frac{x \sin (c+d x)}{2 b \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(d*Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(4*b^2) + (d*Cos[c - (Sqrt[-a]*d)/Sq
rt[b]]*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(4*b^2) - (CosIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sin[c - (Sq
rt[-a]*d)/Sqrt[b]])/(4*Sqrt[-a]*b^(3/2)) + (CosIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x]*Sin[c + (Sqrt[-a]*d)/Sqrt[
b]])/(4*Sqrt[-a]*b^(3/2)) - (x*Sin[c + d*x])/(2*b*(a + b*x^2)) - (Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(S
qrt[-a]*d)/Sqrt[b] - d*x])/(4*Sqrt[-a]*b^(3/2)) + (d*Sin[c + (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sq
rt[b] - d*x])/(4*b^2) - (Cos[c - (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(4*Sqrt[-a]*b^
(3/2)) - (d*Sin[c - (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(4*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 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 3346

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

Mathematica [C]  time = 0.814362, size = 583, normalized size = 1.4 \[ \frac{-a^{3/2} d \sin \left (c-\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{Si}\left (d \left (x+\frac{i \sqrt{a}}{\sqrt{b}}\right )\right )+a^{3/2} d \sin \left (c+\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right )+i b^{3/2} x^2 \cos \left (c-\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{Si}\left (d \left (x+\frac{i \sqrt{a}}{\sqrt{b}}\right )\right )+i b^{3/2} x^2 \cos \left (c+\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right )+\left (a+b x^2\right ) \text{CosIntegral}\left (d \left (x+\frac{i \sqrt{a}}{\sqrt{b}}\right )\right ) \left (i \sqrt{b} \sin \left (c-\frac{i \sqrt{a} d}{\sqrt{b}}\right )+\sqrt{a} d \cos \left (c-\frac{i \sqrt{a} d}{\sqrt{b}}\right )\right )+\left (a+b x^2\right ) \text{CosIntegral}\left (d \left (x-\frac{i \sqrt{a}}{\sqrt{b}}\right )\right ) \left (\sqrt{a} d \cos \left (c+\frac{i \sqrt{a} d}{\sqrt{b}}\right )-i \sqrt{b} \sin \left (c+\frac{i \sqrt{a} d}{\sqrt{b}}\right )\right )-\sqrt{a} b d x^2 \sin \left (c-\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{Si}\left (d \left (x+\frac{i \sqrt{a}}{\sqrt{b}}\right )\right )+\sqrt{a} b d x^2 \sin \left (c+\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right )+i a \sqrt{b} \cos \left (c-\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{Si}\left (d \left (x+\frac{i \sqrt{a}}{\sqrt{b}}\right )\right )+i a \sqrt{b} \cos \left (c+\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right )-2 \sqrt{a} b x \sin (c+d x)}{4 \sqrt{a} b^2 \left (a+b x^2\right )} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((a + b*x^2)*CosIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)]*(Sqrt[a]*d*Cos[c - (I*Sqrt[a]*d)/Sqrt[b]] + I*Sqrt[b]*Si
n[c - (I*Sqrt[a]*d)/Sqrt[b]]) + (a + b*x^2)*CosIntegral[d*(((-I)*Sqrt[a])/Sqrt[b] + x)]*(Sqrt[a]*d*Cos[c + (I*
Sqrt[a]*d)/Sqrt[b]] - I*Sqrt[b]*Sin[c + (I*Sqrt[a]*d)/Sqrt[b]]) - 2*Sqrt[a]*b*x*Sin[c + d*x] + I*a*Sqrt[b]*Cos
[c - (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] + I*b^(3/2)*x^2*Cos[c - (I*Sqrt[a]*d)/Sqr
t[b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] - a^(3/2)*d*Sin[c - (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*Sq
rt[a])/Sqrt[b] + x)] - Sqrt[a]*b*d*x^2*Sin[c - (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)]
+ I*a*Sqrt[b]*Cos[c + (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x] + I*b^(3/2)*x^2*Cos[c +
(I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x] + a^(3/2)*d*Sin[c + (I*Sqrt[a]*d)/Sqrt[b]]*Si
nIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x] + Sqrt[a]*b*d*x^2*Sin[c + (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(I*Sqrt[a]
*d)/Sqrt[b] - d*x])/(4*Sqrt[a]*b^2*(a + b*x^2))

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Maple [B]  time = 0.06, size = 1804, normalized size = 4.3 \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)^2,x)

[Out]

1/d^3*(sin(d*x+c)*(-1/2*d^2*(a*d^2-b*c^2)/a/b*(d*x+c)-1/2*c*d^2*(a*d^2+b*c^2)/a/b)/((d*x+c)^2*b-2*(d*x+c)*b*c+
a*d^2+c^2*b)+1/4*d^2*(a*d^2+b*c^2)/a/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))+1/4*d^2*(a*d^2+b*c^2)/a/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))+1/4*d^2*((d*(-a*b)^(1/2)+c*b)/b*a*d^2-(d*(-a*b)^(1/2)+c*b)*c^2+a*c*d
^2+c^3*b)/a/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*d^2*(-(d*(-a*b)^(1/2)-c*b)/b*a*d^2+(d*(-a*b)^(1/
2)-c*b)*c^2+a*c*d^2+c^3*b)/a/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))+sin(d*x+c)*(-c^2*d^2/a*(d*x+c)+c*d^
2*(a*d^2+b*c^2)/a/b)/((d*x+c)^2*b-2*(d*x+c)*b*c+a*d^2+c^2*b)-1/2*c^2*d^2/a/b/((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))-1/2*c^2*d^2/a/b/(-(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))+1/2*c*d^2*((d*(-a*b)^(1/2)+c*b)*c-a*d^2-c^2*
b)/a/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))+1/2*c*d^2*(-(d*(-a*b)^(1/2)-c*b)*c-a*d^2-c^2*b)/a/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^4*c^2*(sin(d*x+c)*(1/2/a/d^2*(d*x+c)-1/2*c/a/d^2)/((d*x+c)^2*b-2*(d*x+c
)*b*c+a*d^2+c^2*b)+1/4/a/d^2/b/((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))+1/4/a/d^2/b/(-(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))-1/4/a/b/d^2*(-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/a/b/d^2*(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)^2,x, algorithm="maxima")

[Out]

Timed out

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Fricas [C]  time = 1.88163, size = 666, normalized size = 1.6 \begin{align*} -\frac{4 \, a b d x \sin \left (d x + c\right ) -{\left (a b d^{2} x^{2} + a^{2} d^{2} +{\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} -{\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} +{\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} -{\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 (a b^{3} d x^{2} + a^{2} 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)^2,x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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 )}^{2}}\,{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)^2,x, algorithm="giac")

[Out]

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