3.65 \(\int \frac{x^4 \sin (c+d x)}{(a+b x^2)^2} \, dx\)

Optimal. Leaf size=450 \[ -\frac{3 \sqrt{-a} \sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{4 b^{5/2}}+\frac{3 \sqrt{-a} \sin \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{4 b^{5/2}}-\frac{a d \cos \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{4 b^3}-\frac{a d \cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{4 b^3}-\frac{a d \sin \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{4 b^3}+\frac{a d \sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{4 b^3}-\frac{3 \sqrt{-a} \cos \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{4 b^{5/2}}-\frac{3 \sqrt{-a} \cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{4 b^{5/2}}-\frac{x^3 \sin (c+d x)}{2 b \left (a+b x^2\right )}+\frac{x \sin (c+d x)}{2 b^2}-\frac{\cos (c+d x)}{b^2 d} \]

[Out]

-(Cos[c + d*x]/(b^2*d)) - (a*d*Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(4*b^3)
- (a*d*Cos[c - (Sqrt[-a]*d)/Sqrt[b]]*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(4*b^3) - (3*Sqrt[-a]*CosIntegra
l[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sin[c - (Sqrt[-a]*d)/Sqrt[b]])/(4*b^(5/2)) + (3*Sqrt[-a]*CosIntegral[(Sqrt[-a]*d
)/Sqrt[b] - d*x]*Sin[c + (Sqrt[-a]*d)/Sqrt[b]])/(4*b^(5/2)) + (x*Sin[c + d*x])/(2*b^2) - (x^3*Sin[c + d*x])/(2
*b*(a + b*x^2)) - (3*Sqrt[-a]*Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(4*b^(5/2
)) - (a*d*Sin[c + (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(4*b^3) - (3*Sqrt[-a]*Cos[c -
(Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(4*b^(5/2)) + (a*d*Sin[c - (Sqrt[-a]*d)/Sqrt[b
]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(4*b^3)

________________________________________________________________________________________

Rubi [A]  time = 0.782861, antiderivative size = 450, normalized size of antiderivative = 1., number of steps used = 24, number of rules used = 9, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.474, Rules used = {3343, 3345, 2638, 3333, 3303, 3299, 3302, 3346, 3296} \[ -\frac{3 \sqrt{-a} \sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{4 b^{5/2}}+\frac{3 \sqrt{-a} \sin \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{4 b^{5/2}}-\frac{a d \cos \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{4 b^3}-\frac{a d \cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{4 b^3}-\frac{a d \sin \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{4 b^3}+\frac{a d \sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{4 b^3}-\frac{3 \sqrt{-a} \cos \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{4 b^{5/2}}-\frac{3 \sqrt{-a} \cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{4 b^{5/2}}-\frac{x^3 \sin (c+d x)}{2 b \left (a+b x^2\right )}+\frac{x \sin (c+d x)}{2 b^2}-\frac{\cos (c+d x)}{b^2 d} \]

Antiderivative was successfully verified.

[In]

Int[(x^4*Sin[c + d*x])/(a + b*x^2)^2,x]

[Out]

-(Cos[c + d*x]/(b^2*d)) - (a*d*Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(4*b^3)
- (a*d*Cos[c - (Sqrt[-a]*d)/Sqrt[b]]*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(4*b^3) - (3*Sqrt[-a]*CosIntegra
l[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sin[c - (Sqrt[-a]*d)/Sqrt[b]])/(4*b^(5/2)) + (3*Sqrt[-a]*CosIntegral[(Sqrt[-a]*d
)/Sqrt[b] - d*x]*Sin[c + (Sqrt[-a]*d)/Sqrt[b]])/(4*b^(5/2)) + (x*Sin[c + d*x])/(2*b^2) - (x^3*Sin[c + d*x])/(2
*b*(a + b*x^2)) - (3*Sqrt[-a]*Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(4*b^(5/2
)) - (a*d*Sin[c + (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(4*b^3) - (3*Sqrt[-a]*Cos[c -
(Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(4*b^(5/2)) + (a*d*Sin[c - (Sqrt[-a]*d)/Sqrt[b
]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(4*b^3)

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 3345

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 2638

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

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]

Rule 3296

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> -Simp[((c + d*x)^m*Cos[e + f*x])/f, x] +
Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rubi steps

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

Mathematica [C]  time = 1.15837, size = 632, normalized size = 1.4 \[ -\frac{-a^2 d^2 \sin \left (c-\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{Si}\left (d \left (x+\frac{i \sqrt{a}}{\sqrt{b}}\right )\right )+a^2 d^2 \sin \left (c+\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right )+3 i a^{3/2} \sqrt{b} d \cos \left (c-\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{Si}\left (d \left (x+\frac{i \sqrt{a}}{\sqrt{b}}\right )\right )+3 i a^{3/2} \sqrt{b} d \cos \left (c+\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right )+3 i \sqrt{a} b^{3/2} d x^2 \cos \left (c-\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{Si}\left (d \left (x+\frac{i \sqrt{a}}{\sqrt{b}}\right )\right )+3 i \sqrt{a} b^{3/2} d x^2 \cos \left (c+\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right )+\sqrt{a} d \left (a+b x^2\right ) \text{CosIntegral}\left (d \left (x+\frac{i \sqrt{a}}{\sqrt{b}}\right )\right ) \left (3 i \sqrt{b} \sin \left (c-\frac{i \sqrt{a} d}{\sqrt{b}}\right )+\sqrt{a} d \cos \left (c-\frac{i \sqrt{a} d}{\sqrt{b}}\right )\right )+\sqrt{a} d \left (a+b x^2\right ) \text{CosIntegral}\left (d \left (x-\frac{i \sqrt{a}}{\sqrt{b}}\right )\right ) \left (\sqrt{a} d \cos \left (c+\frac{i \sqrt{a} d}{\sqrt{b}}\right )-3 i \sqrt{b} \sin \left (c+\frac{i \sqrt{a} d}{\sqrt{b}}\right )\right )-a b d^2 x^2 \sin \left (c-\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{Si}\left (d \left (x+\frac{i \sqrt{a}}{\sqrt{b}}\right )\right )+a b d^2 x^2 \sin \left (c+\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right )-2 a b d x \sin (c+d x)+4 a b \cos (c+d x)+4 b^2 x^2 \cos (c+d x)}{4 b^3 d \left (a+b x^2\right )} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(x^4*Sin[c + d*x])/(a + b*x^2)^2,x]

[Out]

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

________________________________________________________________________________________

Maple [B]  time = 0.092, size = 3453, normalized size = 7.7 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*sin(d*x+c)/(b*x^2+a)^2,x)

[Out]

1/d^5*(-d^4/b^2*cos(d*x+c)+sin(d*x+c)*(1/2*d^2*(a^2*d^4-6*a*b*c^2*d^2+b^2*c^4)/a*(d*x+c)+1/2*c*d^2*(3*a^2*d^4+
2*a*b*c^2*d^2-b^2*c^4)/a)/b^2/((d*x+c)^2*b-2*(d*x+c)*b*c+a*d^2+c^2*b)+1/4*d^2*(8*(d*(-a*b)^(1/2)+c*b)*a*c*d^2-
3*a^2*d^4-2*a*b*c^2*d^2+b^2*c^4)/a/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))+1/4*d^2*(-8*(d*(-a*b)^(1/2)-c*
b)*a*c*d^2-3*a^2*d^4-2*a*b*c^2*d^2+b^2*c^4)/a/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))-1/4*d^2*((d*(-a*b)
^(1/2)+c*b)/b*a^2*d^4-6*(d*(-a*b)^(1/2)+c*b)*a*c^2*d^2+(d*(-a*b)^(1/2)+c*b)*b*c^4+3*a^2*c*d^4+2*a*b*c^3*d^2-b^
2*c^5)/a/b^3/((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^2*d^4+6*(d*(-a*b)^(1
/2)-c*b)*a*c^2*d^2-(d*(-a*b)^(1/2)-c*b)*b*c^4+3*a^2*c*d^4+2*a*b*c^3*d^2-b^2*c^5)/a/b^3/(-(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)*(2*c^2*d^2*(3*a*d^2-b*c^2)/a/b*(d*x+c)-2*c*d^2*(a^2*d^4-b^2*c^4)/a/b^2)/((d*x+c)
^2*b-2*(d*x+c)*b*c+a*d^2+c^2*b)-c*d^2*(2*(d*(-a*b)^(1/2)+c*b)/b*a*d^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)*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))-c*d^2*(-2*(d*(-a*b)^(1/2)-c*b)/b*a*d^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)*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))-c*d^2*(3*(d*(-a*b)^(1/2)+c*b)*a*c*d^2-(d*(-a*b)^(1/2)+c*b)*b*c^3-a^2*d^4+b^2*c^4)/a/b^3/((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))-c*d^2*(-3*(d*(-a*b)^(1/2)-c*b)*a*c*d^2+(d*(-a*b)^(1/2)-c*b)*b*c^3-a^2*d^4+b^
2*c^4)/a/b^3/(-(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)*(-3*c^2*d^2*(a*d^2-b*c^2)/a/b*(d*x+c)-3*c^3*
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)+3/2*c^2*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))+3/2*c^2*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))+3/2*c^2*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))+3/2*c^2*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)*(-2*c^4*d^2/a*(d*x+c)+2*c^3*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)-c^4*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))-c^4*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))+c^3*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
))+c^3*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^4*(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^4*sin(d*x+c)/(b*x^2+a)^2,x, algorithm="maxima")

[Out]

Timed out

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Fricas [C]  time = 1.92142, size = 714, normalized size = 1.59 \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} + 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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*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 + 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)*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)*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)*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)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4} \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**4*sin(d*x+c)/(b*x**2+a)**2,x)

[Out]

Integral(x**4*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^{4} \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^4*sin(d*x+c)/(b*x^2+a)^2,x, algorithm="giac")

[Out]

integrate(x^4*sin(d*x + c)/(b*x^2 + a)^2, x)