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

Optimal. Leaf size=512 \[ \frac{d^2 \sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{16 a b^2}+\frac{d^2 \sin \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 a b^2}-\frac{d \cos \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 (-a)^{3/2} b^{3/2}}+\frac{d \cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{16 (-a)^{3/2} b^{3/2}}-\frac{d^2 \cos \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 a b^2}+\frac{d^2 \cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{16 a b^2}-\frac{d \sin \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 (-a)^{3/2} b^{3/2}}-\frac{d \sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{16 (-a)^{3/2} b^{3/2}}-\frac{d \cos (c+d x)}{16 a b^{3/2} \left (\sqrt{-a}-\sqrt{b} x\right )}+\frac{d \cos (c+d x)}{16 a b^{3/2} \left (\sqrt{-a}+\sqrt{b} x\right )}-\frac{\sin (c+d x)}{4 b \left (a+b x^2\right )^2} \]

[Out]

-(d*Cos[c + d*x])/(16*a*b^(3/2)*(Sqrt[-a] - Sqrt[b]*x)) + (d*Cos[c + d*x])/(16*a*b^(3/2)*(Sqrt[-a] + Sqrt[b]*x
)) - (d*Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*(-a)^(3/2)*b^(3/2)) + (d*Co
s[c - (Sqrt[-a]*d)/Sqrt[b]]*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*(-a)^(3/2)*b^(3/2)) + (d^2*CosIntegra
l[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sin[c - (Sqrt[-a]*d)/Sqrt[b]])/(16*a*b^2) + (d^2*CosIntegral[(Sqrt[-a]*d)/Sqrt[b
] - d*x]*Sin[c + (Sqrt[-a]*d)/Sqrt[b]])/(16*a*b^2) - Sin[c + d*x]/(4*b*(a + b*x^2)^2) - (d^2*Cos[c + (Sqrt[-a]
*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*a*b^2) - (d*Sin[c + (Sqrt[-a]*d)/Sqrt[b]]*SinIntegra
l[(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*a*b^2) - (d*Sin[c - (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/
(16*(-a)^(3/2)*b^(3/2))

________________________________________________________________________________________

Rubi [A]  time = 0.769319, antiderivative size = 512, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 6, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.353, Rules used = {3341, 3334, 3297, 3303, 3299, 3302} \[ \frac{d^2 \sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{16 a b^2}+\frac{d^2 \sin \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 a b^2}-\frac{d \cos \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 (-a)^{3/2} b^{3/2}}+\frac{d \cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{16 (-a)^{3/2} b^{3/2}}-\frac{d^2 \cos \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 a b^2}+\frac{d^2 \cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{16 a b^2}-\frac{d \sin \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 (-a)^{3/2} b^{3/2}}-\frac{d \sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{16 (-a)^{3/2} b^{3/2}}-\frac{d \cos (c+d x)}{16 a b^{3/2} \left (\sqrt{-a}-\sqrt{b} x\right )}+\frac{d \cos (c+d x)}{16 a b^{3/2} \left (\sqrt{-a}+\sqrt{b} x\right )}-\frac{\sin (c+d x)}{4 b \left (a+b x^2\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[(x*Sin[c + d*x])/(a + b*x^2)^3,x]

[Out]

-(d*Cos[c + d*x])/(16*a*b^(3/2)*(Sqrt[-a] - Sqrt[b]*x)) + (d*Cos[c + d*x])/(16*a*b^(3/2)*(Sqrt[-a] + Sqrt[b]*x
)) - (d*Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*(-a)^(3/2)*b^(3/2)) + (d*Co
s[c - (Sqrt[-a]*d)/Sqrt[b]]*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*(-a)^(3/2)*b^(3/2)) + (d^2*CosIntegra
l[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sin[c - (Sqrt[-a]*d)/Sqrt[b]])/(16*a*b^2) + (d^2*CosIntegral[(Sqrt[-a]*d)/Sqrt[b
] - d*x]*Sin[c + (Sqrt[-a]*d)/Sqrt[b]])/(16*a*b^2) - Sin[c + d*x]/(4*b*(a + b*x^2)^2) - (d^2*Cos[c + (Sqrt[-a]
*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*a*b^2) - (d*Sin[c + (Sqrt[-a]*d)/Sqrt[b]]*SinIntegra
l[(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*a*b^2) - (d*Sin[c - (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/
(16*(-a)^(3/2)*b^(3/2))

Rule 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 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]

Rubi steps

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

Mathematica [C]  time = 1.78729, size = 634, normalized size = 1.24 \[ \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 (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{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{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{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 a \sin (c)\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 a \cos (c)\right )}{\left (a+b x^2\right )^2}}{16 a b} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(x*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*Sin[c]))/(a + b*x^2)^2 - (2*(2*a*Cos[c] + d*x*(a + b*x^2)*Sin[c])*S
in[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[(Sqrt[a]*d)/Sqrt[b]]*(SinIntegra
l[d*((I*Sqrt[a])/Sqrt[b] + x)] - SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x])))/b + (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])/Sqrt[b] + x)] + SinIntegral[(I*Sqrt
[a]*d)/Sqrt[b] - d*x])))/(Sqrt[a]*Sqrt[b]) - (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)/Sqrt[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*a*b)

________________________________________________________________________________________

Maple [B]  time = 0.055, size = 1374, normalized size = 2.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d^2*(1/8*sin(d*x+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+1/8*cos(d*x+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)+1/16*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))+1/16*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))-1/16*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
/16*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*(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*sin(d*x+c)/(b*x^2+a)^3,x, algorithm="maxima")

[Out]

Timed out

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Fricas [C]  time = 2.0049, size = 1040, normalized size = 2.03 \begin{align*} -\frac{16 \, a^{2} b \sin \left (d x + c\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 (i \, b^{3} x^{4} + 2 i \, a b^{2} x^{2} + i \, a^{2} b\right )} \sqrt{\frac{a d^{2}}{b}}\right )}{\rm Ei}\left (i \, d x - \sqrt{\frac{a d^{2}}{b}}\right ) e^{\left (i \, c + \sqrt{\frac{a d^{2}}{b}}\right )} -{\left (-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 (-i \, b^{3} x^{4} - 2 i \, a b^{2} x^{2} - i \, a^{2} b\right )} \sqrt{\frac{a d^{2}}{b}}\right )}{\rm Ei}\left (i \, d x + \sqrt{\frac{a d^{2}}{b}}\right ) e^{\left (i \, c - \sqrt{\frac{a d^{2}}{b}}\right )} -{\left (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 (-i \, b^{3} x^{4} - 2 i \, a b^{2} x^{2} - i \, a^{2} b\right )} \sqrt{\frac{a d^{2}}{b}}\right )}{\rm Ei}\left (-i \, d x - \sqrt{\frac{a d^{2}}{b}}\right ) e^{\left (-i \, c + \sqrt{\frac{a d^{2}}{b}}\right )} -{\left (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 (i \, b^{3} x^{4} + 2 i \, a b^{2} x^{2} + i \, a^{2} b\right )} \sqrt{\frac{a d^{2}}{b}}\right )}{\rm Ei}\left (-i \, d x + \sqrt{\frac{a d^{2}}{b}}\right ) e^{\left (-i \, c - \sqrt{\frac{a d^{2}}{b}}\right )} - 8 \,{\left (a b^{2} d x^{3} + a^{2} b d x\right )} \cos \left (d x + c\right )}{64 \,{\left (a^{2} b^{4} x^{4} + 2 \, a^{3} b^{3} x^{2} + a^{4} b^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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