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

Optimal. Leaf size=476 \[ -\frac{d^2 \sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{16 b^3}-\frac{d^2 \sin \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 b^3}+\frac{3 d \cos \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 \sqrt{-a} b^{5/2}}-\frac{3 d \cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{16 \sqrt{-a} b^{5/2}}+\frac{d^2 \cos \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 b^3}-\frac{d^2 \cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{16 b^3}+\frac{3 d \sin \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 \sqrt{-a} b^{5/2}}+\frac{3 d \sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{16 \sqrt{-a} b^{5/2}}-\frac{\sin (c+d x)}{4 b^2 \left (a+b x^2\right )}-\frac{d x \cos (c+d x)}{8 b^2 \left (a+b x^2\right )}-\frac{x^2 \sin (c+d x)}{4 b \left (a+b x^2\right )^2} \]

[Out]

-(d*x*Cos[c + d*x])/(8*b^2*(a + b*x^2)) + (3*d*Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*CosIntegral[(Sqrt[-a]*d)/Sqrt[b]
- d*x])/(16*Sqrt[-a]*b^(5/2)) - (3*d*Cos[c - (Sqrt[-a]*d)/Sqrt[b]]*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(1
6*Sqrt[-a]*b^(5/2)) - (d^2*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sin[c - (Sqrt[-a]*d)/Sqrt[b]])/(16*b^3) - (
d^2*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x]*Sin[c + (Sqrt[-a]*d)/Sqrt[b]])/(16*b^3) - (x^2*Sin[c + d*x])/(4*b*
(a + b*x^2)^2) - Sin[c + d*x]/(4*b^2*(a + b*x^2)) + (d^2*Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d
)/Sqrt[b] - d*x])/(16*b^3) + (3*d*Sin[c + (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*S
qrt[-a]*b^(5/2)) - (d^2*Cos[c - (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*b^3) + (3*d
*Sin[c - (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*Sqrt[-a]*b^(5/2))

________________________________________________________________________________________

Rubi [A]  time = 1.0076, antiderivative size = 476, normalized size of antiderivative = 1., number of steps used = 27, number of rules used = 8, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.421, Rules used = {3343, 3341, 3334, 3303, 3299, 3302, 3344, 3345} \[ -\frac{d^2 \sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{16 b^3}-\frac{d^2 \sin \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 b^3}+\frac{3 d \cos \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 \sqrt{-a} b^{5/2}}-\frac{3 d \cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{16 \sqrt{-a} b^{5/2}}+\frac{d^2 \cos \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 b^3}-\frac{d^2 \cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{16 b^3}+\frac{3 d \sin \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 \sqrt{-a} b^{5/2}}+\frac{3 d \sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{16 \sqrt{-a} b^{5/2}}-\frac{\sin (c+d x)}{4 b^2 \left (a+b x^2\right )}-\frac{d x \cos (c+d x)}{8 b^2 \left (a+b x^2\right )}-\frac{x^2 \sin (c+d x)}{4 b \left (a+b x^2\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(d*x*Cos[c + d*x])/(8*b^2*(a + b*x^2)) + (3*d*Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*CosIntegral[(Sqrt[-a]*d)/Sqrt[b]
- d*x])/(16*Sqrt[-a]*b^(5/2)) - (3*d*Cos[c - (Sqrt[-a]*d)/Sqrt[b]]*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(1
6*Sqrt[-a]*b^(5/2)) - (d^2*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sin[c - (Sqrt[-a]*d)/Sqrt[b]])/(16*b^3) - (
d^2*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x]*Sin[c + (Sqrt[-a]*d)/Sqrt[b]])/(16*b^3) - (x^2*Sin[c + d*x])/(4*b*
(a + b*x^2)^2) - Sin[c + d*x]/(4*b^2*(a + b*x^2)) + (d^2*Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d
)/Sqrt[b] - d*x])/(16*b^3) + (3*d*Sin[c + (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*S
qrt[-a]*b^(5/2)) - (d^2*Cos[c - (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*b^3) + (3*d
*Sin[c - (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*Sqrt[-a]*b^(5/2))

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 3341

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])

Rule 3334

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 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 3344

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

Rubi steps

\begin{align*} \int \frac{x^3 \sin (c+d x)}{\left (a+b x^2\right )^3} \, dx &=-\frac{x^2 \sin (c+d x)}{4 b \left (a+b x^2\right )^2}+\frac{\int \frac{x \sin (c+d x)}{\left (a+b x^2\right )^2} \, dx}{2 b}+\frac{d \int \frac{x^2 \cos (c+d x)}{\left (a+b x^2\right )^2} \, dx}{4 b}\\ &=-\frac{d x \cos (c+d x)}{8 b^2 \left (a+b x^2\right )}-\frac{x^2 \sin (c+d x)}{4 b \left (a+b x^2\right )^2}-\frac{\sin (c+d x)}{4 b^2 \left (a+b x^2\right )}+\frac{d \int \frac{\cos (c+d x)}{a+b x^2} \, dx}{8 b^2}+\frac{d \int \frac{\cos (c+d x)}{a+b x^2} \, dx}{4 b^2}-\frac{d^2 \int \frac{x \sin (c+d x)}{a+b x^2} \, dx}{8 b^2}\\ &=-\frac{d x \cos (c+d x)}{8 b^2 \left (a+b x^2\right )}-\frac{x^2 \sin (c+d x)}{4 b \left (a+b x^2\right )^2}-\frac{\sin (c+d x)}{4 b^2 \left (a+b x^2\right )}+\frac{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}{8 b^2}+\frac{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}{4 b^2}-\frac{d^2 \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}{8 b^2}\\ &=-\frac{d x \cos (c+d x)}{8 b^2 \left (a+b x^2\right )}-\frac{x^2 \sin (c+d x)}{4 b \left (a+b x^2\right )^2}-\frac{\sin (c+d x)}{4 b^2 \left (a+b x^2\right )}-\frac{d \int \frac{\cos (c+d x)}{\sqrt{-a}-\sqrt{b} x} \, dx}{16 \sqrt{-a} b^2}-\frac{d \int \frac{\cos (c+d x)}{\sqrt{-a}+\sqrt{b} x} \, dx}{16 \sqrt{-a} b^2}-\frac{d \int \frac{\cos (c+d x)}{\sqrt{-a}-\sqrt{b} x} \, dx}{8 \sqrt{-a} b^2}-\frac{d \int \frac{\cos (c+d x)}{\sqrt{-a}+\sqrt{b} x} \, dx}{8 \sqrt{-a} b^2}+\frac{d^2 \int \frac{\sin (c+d x)}{\sqrt{-a}-\sqrt{b} x} \, dx}{16 b^{5/2}}-\frac{d^2 \int \frac{\sin (c+d x)}{\sqrt{-a}+\sqrt{b} x} \, dx}{16 b^{5/2}}\\ &=-\frac{d x \cos (c+d x)}{8 b^2 \left (a+b x^2\right )}-\frac{x^2 \sin (c+d x)}{4 b \left (a+b x^2\right )^2}-\frac{\sin (c+d x)}{4 b^2 \left (a+b x^2\right )}-\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 \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} x} \, dx}{8 \sqrt{-a} b^2}-\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 b^{5/2}}-\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 \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} x} \, dx}{8 \sqrt{-a} b^2}-\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 b^{5/2}}+\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} 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} x} \, dx}{8 \sqrt{-a} b^2}-\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 b^{5/2}}-\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} 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} x} \, dx}{8 \sqrt{-a} b^2}+\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 b^{5/2}}\\ &=-\frac{d x \cos (c+d x)}{8 b^2 \left (a+b x^2\right )}+\frac{3 d \cos \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Ci}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 \sqrt{-a} b^{5/2}}-\frac{3 d \cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Ci}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\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 b^3}-\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 b^3}-\frac{x^2 \sin (c+d x)}{4 b \left (a+b x^2\right )^2}-\frac{\sin (c+d x)}{4 b^2 \left (a+b x^2\right )}+\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 b^3}+\frac{3 d \sin \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^2 \cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{16 b^3}+\frac{3 d \sin \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}}\\ \end{align*}

Mathematica [C]  time = 1.94547, size = 647, normalized size = 1.36 \[ \frac{\frac{d^2 \cos (c) \left (-i \sinh \left (\frac{\sqrt{a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (d \left (x-\frac{i \sqrt{a}}{\sqrt{b}}\right )\right )+i \sinh \left (\frac{\sqrt{a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (d \left (x+\frac{i \sqrt{a}}{\sqrt{b}}\right )\right )+\cosh \left (\frac{\sqrt{a} d}{\sqrt{b}}\right ) \left (\text{Si}\left (\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right )-\text{Si}\left (d \left (x+\frac{i \sqrt{a}}{\sqrt{b}}\right )\right )\right )\right )}{b}-\frac{d^2 \sin (c) \left (\cosh \left (\frac{\sqrt{a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (d \left (x-\frac{i \sqrt{a}}{\sqrt{b}}\right )\right )+\cosh \left (\frac{\sqrt{a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (d \left (x+\frac{i \sqrt{a}}{\sqrt{b}}\right )\right )+i \sinh \left (\frac{\sqrt{a} d}{\sqrt{b}}\right ) \left (\text{Si}\left (d \left (x+\frac{i \sqrt{a}}{\sqrt{b}}\right )\right )+\text{Si}\left (\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right )\right )\right )}{b}+\frac{3 d \cos (c) \left (-i \cosh \left (\frac{\sqrt{a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (d \left (x-\frac{i \sqrt{a}}{\sqrt{b}}\right )\right )+i \cosh \left (\frac{\sqrt{a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (d \left (x+\frac{i \sqrt{a}}{\sqrt{b}}\right )\right )+\sinh \left (\frac{\sqrt{a} d}{\sqrt{b}}\right ) \left (\text{Si}\left (\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right )-\text{Si}\left (d \left (x+\frac{i \sqrt{a}}{\sqrt{b}}\right )\right )\right )\right )}{\sqrt{a} \sqrt{b}}-\frac{3 d \sin (c) \left (\sinh \left (\frac{\sqrt{a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (d \left (x-\frac{i \sqrt{a}}{\sqrt{b}}\right )\right )+\sinh \left (\frac{\sqrt{a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (d \left (x+\frac{i \sqrt{a}}{\sqrt{b}}\right )\right )+i \cosh \left (\frac{\sqrt{a} d}{\sqrt{b}}\right ) \left (\text{Si}\left (d \left (x+\frac{i \sqrt{a}}{\sqrt{b}}\right )\right )+\text{Si}\left (\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right )\right )\right )}{\sqrt{a} \sqrt{b}}-\frac{2 \cos (d x) \left (d x \cos (c) \left (a+b x^2\right )+2 \sin (c) \left (a+2 b x^2\right )\right )}{\left (a+b x^2\right )^2}+\frac{2 \sin (d x) \left (d x \sin (c) \left (a+b x^2\right )-2 \cos (c) \left (a+2 b x^2\right )\right )}{\left (a+b x^2\right )^2}}{16 b^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((-2*Cos[d*x]*(d*x*(a + b*x^2)*Cos[c] + 2*(a + 2*b*x^2)*Sin[c]))/(a + b*x^2)^2 + (2*(-2*(a + 2*b*x^2)*Cos[c] +
 d*x*(a + b*x^2)*Sin[c])*Sin[d*x])/(a + b*x^2)^2 + (d^2*Cos[c]*((-I)*CosIntegral[d*(((-I)*Sqrt[a])/Sqrt[b] + x
)]*Sinh[(Sqrt[a]*d)/Sqrt[b]] + I*CosIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)]*Sinh[(Sqrt[a]*d)/Sqrt[b]] + Cosh[(Sq
rt[a]*d)/Sqrt[b]]*(-SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] + SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x])))/b +
 (3*d*Cos[c]*((-I)*Cosh[(Sqrt[a]*d)/Sqrt[b]]*CosIntegral[d*(((-I)*Sqrt[a])/Sqrt[b] + x)] + I*Cosh[(Sqrt[a]*d)/
Sqrt[b]]*CosIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] + Sinh[(Sqrt[a]*d)/Sqrt[b]]*(-SinIntegral[d*((I*Sqrt[a])/Sqr
t[b] + x)] + SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x])))/(Sqrt[a]*Sqrt[b]) - (3*d*Sin[c]*(CosIntegral[d*(((-I)
*Sqrt[a])/Sqrt[b] + x)]*Sinh[(Sqrt[a]*d)/Sqrt[b]] + CosIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)]*Sinh[(Sqrt[a]*d)/
Sqrt[b]] + I*Cosh[(Sqrt[a]*d)/Sqrt[b]]*(SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] + SinIntegral[(I*Sqrt[a]*d)/S
qrt[b] - d*x])))/(Sqrt[a]*Sqrt[b]) - (d^2*Sin[c]*(Cosh[(Sqrt[a]*d)/Sqrt[b]]*CosIntegral[d*(((-I)*Sqrt[a])/Sqrt
[b] + x)] + Cosh[(Sqrt[a]*d)/Sqrt[b]]*CosIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] + I*Sinh[(Sqrt[a]*d)/Sqrt[b]]*(
SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] + SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x])))/b)/(16*b^2)

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Maple [B]  time = 0.099, size = 3391, normalized size = 7.1 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d^4*(1/8*sin(d*x+c)*d^2*(3*(d*x+c)^3*a*b^2*c*d^2+3*(d*x+c)^3*b^3*c^3-4*(d*x+c)^2*a^2*b*d^4-9*(d*x+c)^2*a*b^2
*c^2*d^2-9*(d*x+c)^2*b^3*c^4+5*(d*x+c)*a^2*b*c*d^4+14*(d*x+c)*a*b^2*c^3*d^2+9*(d*x+c)*b^3*c^5-2*a^3*d^6-7*a^2*
b*c^2*d^4-8*a*b^2*c^4*d^2-3*b^3*c^6)/a^2/b^2/((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*((d*
x+c)*a*d^2-3*(d*x+c)*b*c^2+2*a*c*d^2+2*c^3*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*((d*(-a*b
)^(1/2)+c*b)/b*a^2*d^4-3*(d*(-a*b)^(1/2)+c*b)*a*c^2*d^2+2*a^2*c*d^4+2*a*b*c^3*d^2-3*a*b*c*d^2-3*b^2*c^3)/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^2*d^4+3*(d*(-a*b)^(1/2)-c*b)*a
*c^2*d^2+2*a^2*c*d^4+2*a*b*c^3*d^2-3*a*b*c*d^2-3*b^2*c^3)/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))-3/
16*d^2*((d*(-a*b)^(1/2)+c*b)*a*c*d^2+(d*(-a*b)^(1/2)+c*b)*b*c^3-a^2*d^4-2*a*b*c^2*d^2-b^2*c^4)/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)*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*d^2*(-(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-
2*a*b*c^2*d^2-b^2*c^4)/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)*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))-3/8*sin(d*x+c)*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)^2-3/8*cos(d*x+c)
*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)-3/16*c*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))-3/16*c*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))+3/16*c*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))+3/16*c*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))+3/8*sin(d*x+c)*c^2*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+3/8*cos(d*x+c)*c^2*d^4/a/b*(d*x+c)/((d*x+c)^2*b-2*(d*x+c)*b*c+a*d^2+c^2*b)+3/
16*c^2*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))+3/16*c^2*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)*co
s((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*c^2*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))-3/16*c^2*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^3*(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))-3/1
6/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)*co
s((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^3*sin(d*x+c)/(b*x^2+a)^3,x, algorithm="maxima")

[Out]

Timed out

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Fricas [C]  time = 2.09553, size = 1079, normalized size = 2.27 \begin{align*} \frac{{\left (2 i \, a b^{2} d^{2} x^{4} + 4 i \, a^{2} b d^{2} x^{2} + 2 i \, a^{3} d^{2} + 2 \,{\left (3 i \, b^{3} x^{4} + 6 i \, a b^{2} x^{2} + 3 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 (2 i \, a b^{2} d^{2} x^{4} + 4 i \, a^{2} b d^{2} x^{2} + 2 i \, a^{3} d^{2} + 2 \,{\left (-3 i \, b^{3} x^{4} - 6 i \, a b^{2} x^{2} - 3 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 (-2 i \, a b^{2} d^{2} x^{4} - 4 i \, a^{2} b d^{2} x^{2} - 2 i \, a^{3} d^{2} + 2 \,{\left (-3 i \, b^{3} x^{4} - 6 i \, a b^{2} x^{2} - 3 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 (-2 i \, a b^{2} d^{2} x^{4} - 4 i \, a^{2} b d^{2} x^{2} - 2 i \, a^{3} d^{2} + 2 \,{\left (3 i \, b^{3} x^{4} + 6 i \, a b^{2} x^{2} + 3 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 )} - 8 \,{\left (a b^{2} d x^{3} + a^{2} b d x\right )} \cos \left (d x + c\right ) - 16 \,{\left (2 \, a b^{2} x^{2} + a^{2} b\right )} \sin \left (d x + c\right )}{64 \,{\left (a b^{5} x^{4} + 2 \, a^{2} b^{4} x^{2} + a^{3} b^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/64*((2*I*a*b^2*d^2*x^4 + 4*I*a^2*b*d^2*x^2 + 2*I*a^3*d^2 + 2*(3*I*b^3*x^4 + 6*I*a*b^2*x^2 + 3*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)) + (2*I*a*b^2*d^2*x^4 + 4*I*a^2*b*d^2*x^2 + 2*I*a^3
*d^2 + 2*(-3*I*b^3*x^4 - 6*I*a*b^2*x^2 - 3*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)) + (-2*I*a*b^2*d^2*x^4 - 4*I*a^2*b*d^2*x^2 - 2*I*a^3*d^2 + 2*(-3*I*b^3*x^4 - 6*I*a*b^2*x^2 - 3*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)) + (-2*I*a*b^2*d^2*x^4 - 4*I*a^2*b*d^2*x^2 -
 2*I*a^3*d^2 + 2*(3*I*b^3*x^4 + 6*I*a*b^2*x^2 + 3*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)) - 8*(a*b^2*d*x^3 + a^2*b*d*x)*cos(d*x + c) - 16*(2*a*b^2*x^2 + a^2*b)*sin(d*x + c))/(a*b^5*x^4
 + 2*a^2*b^4*x^2 + a^3*b^3)

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

[Out]

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