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

Optimal. Leaf size=791 \[ \text{result too large to display} \]

[Out]

-(d*Cos[c + d*x])/(2*a^3*x) - (Sqrt[b]*d*Cos[c + d*x])/(16*a^3*(Sqrt[-a] - Sqrt[b]*x)) + (Sqrt[b]*d*Cos[c + d*
x])/(16*a^3*(Sqrt[-a] + Sqrt[b]*x)) - (9*Sqrt[b]*d*Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*CosIntegral[(Sqrt[-a]*d)/Sqrt
[b] - d*x])/(16*(-a)^(7/2)) + (9*Sqrt[b]*d*Cos[c - (Sqrt[-a]*d)/Sqrt[b]]*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*
x])/(16*(-a)^(7/2)) - (3*b*CosIntegral[d*x]*Sin[c])/a^4 - (d^2*CosIntegral[d*x]*Sin[c])/(2*a^3) + (3*b*CosInte
gral[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sin[c - (Sqrt[-a]*d)/Sqrt[b]])/(2*a^4) + (d^2*CosIntegral[(Sqrt[-a]*d)/Sqrt[b
] + d*x]*Sin[c - (Sqrt[-a]*d)/Sqrt[b]])/(16*a^3) + (3*b*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x]*Sin[c + (Sqrt[
-a]*d)/Sqrt[b]])/(2*a^4) + (d^2*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x]*Sin[c + (Sqrt[-a]*d)/Sqrt[b]])/(16*a^3
) - Sin[c + d*x]/(2*a^3*x^2) - (b*Sin[c + d*x])/(4*a^2*(a + b*x^2)^2) - (b*Sin[c + d*x])/(a^3*(a + b*x^2)) - (
3*b*Cos[c]*SinIntegral[d*x])/a^4 - (d^2*Cos[c]*SinIntegral[d*x])/(2*a^3) - (3*b*Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*
SinIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(2*a^4) - (d^2*Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)
/Sqrt[b] - d*x])/(16*a^3) - (9*Sqrt[b]*d*Sin[c + (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x]
)/(16*(-a)^(7/2)) + (3*b*Cos[c - (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(2*a^4) + (d^2
*Cos[c - (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*a^3) - (9*Sqrt[b]*d*Sin[c - (Sqrt[
-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*(-a)^(7/2))

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Rubi [A]  time = 1.87678, antiderivative size = 791, normalized size of antiderivative = 1., number of steps used = 46, number of rules used = 7, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.368, Rules used = {3345, 3297, 3303, 3299, 3302, 3341, 3334} \[ \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^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 a^3}-\frac{3 b \sin (c) \text{CosIntegral}(d x)}{a^4}+\frac{3 b \sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{2 a^4}+\frac{3 b \sin \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 a^4}-\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^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 a^3}-\frac{3 b \cos (c) \text{Si}(d x)}{a^4}-\frac{3 b \cos \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 a^4}+\frac{3 b \cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{2 a^4}-\frac{b \sin (c+d x)}{a^3 \left (a+b x^2\right )}-\frac{b \sin (c+d x)}{4 a^2 \left (a+b x^2\right )^2}-\frac{\sqrt{b} d \cos (c+d x)}{16 a^3 \left (\sqrt{-a}-\sqrt{b} x\right )}+\frac{\sqrt{b} d \cos (c+d x)}{16 a^3 \left (\sqrt{-a}+\sqrt{b} x\right )}-\frac{d^2 \sin (c) \text{CosIntegral}(d x)}{2 a^3}-\frac{d^2 \cos (c) \text{Si}(d x)}{2 a^3}-\frac{\sin (c+d x)}{2 a^3 x^2}-\frac{d \cos (c+d x)}{2 a^3 x}-\frac{9 \sqrt{b} 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)^{7/2}}+\frac{9 \sqrt{b} 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)^{7/2}}-\frac{9 \sqrt{b} 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)^{7/2}}-\frac{9 \sqrt{b} 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)^{7/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(d*Cos[c + d*x])/(2*a^3*x) - (Sqrt[b]*d*Cos[c + d*x])/(16*a^3*(Sqrt[-a] - Sqrt[b]*x)) + (Sqrt[b]*d*Cos[c + d*
x])/(16*a^3*(Sqrt[-a] + Sqrt[b]*x)) - (9*Sqrt[b]*d*Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*CosIntegral[(Sqrt[-a]*d)/Sqrt
[b] - d*x])/(16*(-a)^(7/2)) + (9*Sqrt[b]*d*Cos[c - (Sqrt[-a]*d)/Sqrt[b]]*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*
x])/(16*(-a)^(7/2)) - (3*b*CosIntegral[d*x]*Sin[c])/a^4 - (d^2*CosIntegral[d*x]*Sin[c])/(2*a^3) + (3*b*CosInte
gral[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sin[c - (Sqrt[-a]*d)/Sqrt[b]])/(2*a^4) + (d^2*CosIntegral[(Sqrt[-a]*d)/Sqrt[b
] + d*x]*Sin[c - (Sqrt[-a]*d)/Sqrt[b]])/(16*a^3) + (3*b*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x]*Sin[c + (Sqrt[
-a]*d)/Sqrt[b]])/(2*a^4) + (d^2*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x]*Sin[c + (Sqrt[-a]*d)/Sqrt[b]])/(16*a^3
) - Sin[c + d*x]/(2*a^3*x^2) - (b*Sin[c + d*x])/(4*a^2*(a + b*x^2)^2) - (b*Sin[c + d*x])/(a^3*(a + b*x^2)) - (
3*b*Cos[c]*SinIntegral[d*x])/a^4 - (d^2*Cos[c]*SinIntegral[d*x])/(2*a^3) - (3*b*Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*
SinIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(2*a^4) - (d^2*Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)
/Sqrt[b] - d*x])/(16*a^3) - (9*Sqrt[b]*d*Sin[c + (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x]
)/(16*(-a)^(7/2)) + (3*b*Cos[c - (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(2*a^4) + (d^2
*Cos[c - (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*a^3) - (9*Sqrt[b]*d*Sin[c - (Sqrt[
-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*(-a)^(7/2))

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 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 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])

Rubi steps

\begin{align*} \int \frac{\sin (c+d x)}{x^3 \left (a+b x^2\right )^3} \, dx &=\int \left (\frac{\sin (c+d x)}{a^3 x^3}-\frac{3 b \sin (c+d x)}{a^4 x}+\frac{b^2 x \sin (c+d x)}{a^2 \left (a+b x^2\right )^3}+\frac{2 b^2 x \sin (c+d x)}{a^3 \left (a+b x^2\right )^2}+\frac{3 b^2 x \sin (c+d x)}{a^4 \left (a+b x^2\right )}\right ) \, dx\\ &=\frac{\int \frac{\sin (c+d x)}{x^3} \, dx}{a^3}-\frac{(3 b) \int \frac{\sin (c+d x)}{x} \, dx}{a^4}+\frac{\left (3 b^2\right ) \int \frac{x \sin (c+d x)}{a+b x^2} \, dx}{a^4}+\frac{\left (2 b^2\right ) \int \frac{x \sin (c+d x)}{\left (a+b x^2\right )^2} \, dx}{a^3}+\frac{b^2 \int \frac{x \sin (c+d x)}{\left (a+b x^2\right )^3} \, dx}{a^2}\\ &=-\frac{\sin (c+d x)}{2 a^3 x^2}-\frac{b \sin (c+d x)}{4 a^2 \left (a+b x^2\right )^2}-\frac{b \sin (c+d x)}{a^3 \left (a+b x^2\right )}+\frac{\left (3 b^2\right ) \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}{a^4}+\frac{d \int \frac{\cos (c+d x)}{x^2} \, dx}{2 a^3}+\frac{(b d) \int \frac{\cos (c+d x)}{a+b x^2} \, dx}{a^3}+\frac{(b d) \int \frac{\cos (c+d x)}{\left (a+b x^2\right )^2} \, dx}{4 a^2}-\frac{(3 b \cos (c)) \int \frac{\sin (d x)}{x} \, dx}{a^4}-\frac{(3 b \sin (c)) \int \frac{\cos (d x)}{x} \, dx}{a^4}\\ &=-\frac{d \cos (c+d x)}{2 a^3 x}-\frac{3 b \text{Ci}(d x) \sin (c)}{a^4}-\frac{\sin (c+d x)}{2 a^3 x^2}-\frac{b \sin (c+d x)}{4 a^2 \left (a+b x^2\right )^2}-\frac{b \sin (c+d x)}{a^3 \left (a+b x^2\right )}-\frac{3 b \cos (c) \text{Si}(d x)}{a^4}-\frac{\left (3 b^{3/2}\right ) \int \frac{\sin (c+d x)}{\sqrt{-a}-\sqrt{b} x} \, dx}{2 a^4}+\frac{\left (3 b^{3/2}\right ) \int \frac{\sin (c+d x)}{\sqrt{-a}+\sqrt{b} x} \, dx}{2 a^4}+\frac{(b 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}{a^3}+\frac{(b 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 a^2}-\frac{d^2 \int \frac{\sin (c+d x)}{x} \, dx}{2 a^3}\\ &=-\frac{d \cos (c+d x)}{2 a^3 x}-\frac{3 b \text{Ci}(d x) \sin (c)}{a^4}-\frac{\sin (c+d x)}{2 a^3 x^2}-\frac{b \sin (c+d x)}{4 a^2 \left (a+b x^2\right )^2}-\frac{b \sin (c+d x)}{a^3 \left (a+b x^2\right )}-\frac{3 b \cos (c) \text{Si}(d x)}{a^4}+\frac{(b d) \int \frac{\cos (c+d x)}{\sqrt{-a}-\sqrt{b} x} \, dx}{2 (-a)^{7/2}}+\frac{(b d) \int \frac{\cos (c+d x)}{\sqrt{-a}+\sqrt{b} x} \, dx}{2 (-a)^{7/2}}-\frac{\left (b^2 d\right ) \int \frac{\cos (c+d x)}{\left (\sqrt{-a} \sqrt{b}-b x\right )^2} \, dx}{16 a^3}-\frac{\left (b^2 d\right ) \int \frac{\cos (c+d x)}{\left (\sqrt{-a} \sqrt{b}+b x\right )^2} \, dx}{16 a^3}-\frac{\left (b^2 d\right ) \int \frac{\cos (c+d x)}{-a b-b^2 x^2} \, dx}{8 a^3}-\frac{\left (d^2 \cos (c)\right ) \int \frac{\sin (d x)}{x} \, dx}{2 a^3}+\frac{\left (3 b^{3/2} \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}{2 a^4}+\frac{\left (3 b^{3/2} \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}{2 a^4}-\frac{\left (d^2 \sin (c)\right ) \int \frac{\cos (d x)}{x} \, dx}{2 a^3}+\frac{\left (3 b^{3/2} \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}{2 a^4}-\frac{\left (3 b^{3/2} \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}{2 a^4}\\ &=-\frac{d \cos (c+d x)}{2 a^3 x}-\frac{\sqrt{b} d \cos (c+d x)}{16 a^3 \left (\sqrt{-a}-\sqrt{b} x\right )}+\frac{\sqrt{b} d \cos (c+d x)}{16 a^3 \left (\sqrt{-a}+\sqrt{b} x\right )}-\frac{3 b \text{Ci}(d x) \sin (c)}{a^4}-\frac{d^2 \text{Ci}(d x) \sin (c)}{2 a^3}+\frac{3 b \text{Ci}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right ) \sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{2 a^4}+\frac{3 b \text{Ci}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right ) \sin \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{2 a^4}-\frac{\sin (c+d x)}{2 a^3 x^2}-\frac{b \sin (c+d x)}{4 a^2 \left (a+b x^2\right )^2}-\frac{b \sin (c+d x)}{a^3 \left (a+b x^2\right )}-\frac{3 b \cos (c) \text{Si}(d x)}{a^4}-\frac{d^2 \cos (c) \text{Si}(d x)}{2 a^3}-\frac{3 b \cos \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 a^4}+\frac{3 b \cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{2 a^4}-\frac{\left (b^2 d\right ) \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^3}-\frac{\left (b d^2\right ) \int \frac{\sin (c+d x)}{\sqrt{-a} \sqrt{b}-b x} \, dx}{16 a^3}+\frac{\left (b d^2\right ) \int \frac{\sin (c+d x)}{\sqrt{-a} \sqrt{b}+b x} \, dx}{16 a^3}+\frac{\left (b 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}{2 (-a)^{7/2}}+\frac{\left (b 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}{2 (-a)^{7/2}}-\frac{\left (b 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}{2 (-a)^{7/2}}+\frac{\left (b 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}{2 (-a)^{7/2}}\\ &=-\frac{d \cos (c+d x)}{2 a^3 x}-\frac{\sqrt{b} d \cos (c+d x)}{16 a^3 \left (\sqrt{-a}-\sqrt{b} x\right )}+\frac{\sqrt{b} d \cos (c+d x)}{16 a^3 \left (\sqrt{-a}+\sqrt{b} x\right )}-\frac{\sqrt{b} d \cos \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Ci}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 (-a)^{7/2}}+\frac{\sqrt{b} d \cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Ci}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{2 (-a)^{7/2}}-\frac{3 b \text{Ci}(d x) \sin (c)}{a^4}-\frac{d^2 \text{Ci}(d x) \sin (c)}{2 a^3}+\frac{3 b \text{Ci}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right ) \sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{2 a^4}+\frac{3 b \text{Ci}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right ) \sin \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{2 a^4}-\frac{\sin (c+d x)}{2 a^3 x^2}-\frac{b \sin (c+d x)}{4 a^2 \left (a+b x^2\right )^2}-\frac{b \sin (c+d x)}{a^3 \left (a+b x^2\right )}-\frac{3 b \cos (c) \text{Si}(d x)}{a^4}-\frac{d^2 \cos (c) \text{Si}(d x)}{2 a^3}-\frac{3 b \cos \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 a^4}-\frac{\sqrt{b} d \sin \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 (-a)^{7/2}}+\frac{3 b \cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{2 a^4}-\frac{\sqrt{b} d \sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{2 (-a)^{7/2}}+\frac{(b d) \int \frac{\cos (c+d x)}{\sqrt{-a}-\sqrt{b} x} \, dx}{16 (-a)^{7/2}}+\frac{(b d) \int \frac{\cos (c+d x)}{\sqrt{-a}+\sqrt{b} x} \, dx}{16 (-a)^{7/2}}+\frac{\left (b 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^3}+\frac{\left (b 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^3}+\frac{\left (b 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^3}-\frac{\left (b 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^3}\\ &=-\frac{d \cos (c+d x)}{2 a^3 x}-\frac{\sqrt{b} d \cos (c+d x)}{16 a^3 \left (\sqrt{-a}-\sqrt{b} x\right )}+\frac{\sqrt{b} d \cos (c+d x)}{16 a^3 \left (\sqrt{-a}+\sqrt{b} x\right )}-\frac{\sqrt{b} d \cos \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Ci}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 (-a)^{7/2}}+\frac{\sqrt{b} d \cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Ci}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{2 (-a)^{7/2}}-\frac{3 b \text{Ci}(d x) \sin (c)}{a^4}-\frac{d^2 \text{Ci}(d x) \sin (c)}{2 a^3}+\frac{3 b \text{Ci}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right ) \sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{2 a^4}+\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^3}+\frac{3 b \text{Ci}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right ) \sin \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{2 a^4}+\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^3}-\frac{\sin (c+d x)}{2 a^3 x^2}-\frac{b \sin (c+d x)}{4 a^2 \left (a+b x^2\right )^2}-\frac{b \sin (c+d x)}{a^3 \left (a+b x^2\right )}-\frac{3 b \cos (c) \text{Si}(d x)}{a^4}-\frac{d^2 \cos (c) \text{Si}(d x)}{2 a^3}-\frac{3 b \cos \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 a^4}-\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^3}-\frac{\sqrt{b} d \sin \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 (-a)^{7/2}}+\frac{3 b \cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{2 a^4}+\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^3}-\frac{\sqrt{b} d \sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{2 (-a)^{7/2}}+\frac{\left (b 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)^{7/2}}+\frac{\left (b 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)^{7/2}}-\frac{\left (b 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)^{7/2}}+\frac{\left (b 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)^{7/2}}\\ &=-\frac{d \cos (c+d x)}{2 a^3 x}-\frac{\sqrt{b} d \cos (c+d x)}{16 a^3 \left (\sqrt{-a}-\sqrt{b} x\right )}+\frac{\sqrt{b} d \cos (c+d x)}{16 a^3 \left (\sqrt{-a}+\sqrt{b} x\right )}-\frac{9 \sqrt{b} 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)^{7/2}}+\frac{9 \sqrt{b} 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)^{7/2}}-\frac{3 b \text{Ci}(d x) \sin (c)}{a^4}-\frac{d^2 \text{Ci}(d x) \sin (c)}{2 a^3}+\frac{3 b \text{Ci}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right ) \sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{2 a^4}+\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^3}+\frac{3 b \text{Ci}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right ) \sin \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{2 a^4}+\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^3}-\frac{\sin (c+d x)}{2 a^3 x^2}-\frac{b \sin (c+d x)}{4 a^2 \left (a+b x^2\right )^2}-\frac{b \sin (c+d x)}{a^3 \left (a+b x^2\right )}-\frac{3 b \cos (c) \text{Si}(d x)}{a^4}-\frac{d^2 \cos (c) \text{Si}(d x)}{2 a^3}-\frac{3 b \cos \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 a^4}-\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^3}-\frac{9 \sqrt{b} 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)^{7/2}}+\frac{3 b \cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{2 a^4}+\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^3}-\frac{9 \sqrt{b} 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)^{7/2}}\\ \end{align*}

Mathematica [C]  time = 2.72224, size = 995, normalized size = 1.26 \[ \text{result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((-2*a*Cos[d*x]*(d*x*(4*a^2 + 7*a*b*x^2 + 3*b^2*x^4)*Cos[c] + 2*(2*a^2 + 9*a*b*x^2 + 6*b^2*x^4)*Sin[c]))/(x^2*
(a + b*x^2)^2) + (2*a*(-2*(2*a^2 + 9*a*b*x^2 + 6*b^2*x^4)*Cos[c] + d*x*(4*a^2 + 7*a*b*x^2 + 3*b^2*x^4)*Sin[c])
*Sin[d*x])/(x^2*(a + b*x^2)^2) - 8*(6*b + a*d^2)*(CosIntegral[d*x]*Sin[c] + Cos[c]*SinIntegral[d*x]) + 24*b*Co
s[c]*(I*CosIntegral[d*(((-I)*Sqrt[a])/Sqrt[b] + x)]*Sinh[(Sqrt[a]*d)/Sqrt[b]] - I*CosIntegral[d*((I*Sqrt[a])/S
qrt[b] + x)]*Sinh[(Sqrt[a]*d)/Sqrt[b]] + Cosh[(Sqrt[a]*d)/Sqrt[b]]*(SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] -
SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x])) + a*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]]*(SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] - SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x])) + 9*Sqrt[a]*
Sqrt[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])) - 9*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)/S
qrt[b] - d*x])) + 24*b*Sin[c]*(Cosh[(Sqrt[a]*d)/Sqrt[b]]*CosIntegral[d*(((-I)*Sqrt[a])/Sqrt[b] + x)] + Cosh[(S
qrt[a]*d)/Sqrt[b]]*CosIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] + I*Sinh[(Sqrt[a]*d)/Sqrt[b]]*(SinIntegral[d*((I*S
qrt[a])/Sqrt[b] + x)] + SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x])) + a*d^2*Sin[c]*(Cosh[(Sqrt[a]*d)/Sqrt[b]]*C
osIntegral[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])))/(16*a^4)

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Maple [A]  time = 0.055, size = 701, normalized size = 0.9 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d^2*(-1/4*sin(d*x+c)*(6*b^2*(d*x+c)^4-24*c*(d*x+c)^3*b^2+9*(d*x+c)^2*a*b*d^2+36*b^2*c^2*(d*x+c)^2-18*(d*x+c)*a
*b*c*d^2-24*(d*x+c)*b^2*c^3+2*a^2*d^4+9*a*b*c^2*d^2+6*b^2*c^4)/a^3/x^2/d^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)*(3*(d*x+c)^2*b-6*(d*x+c)*b*c+4*a*d^2+3*c^2*b)/a^3/x/d/((d*x+c)^2*b-2*(d*x+c)*b*c+a*d^2+c
^2*b)+1/16*(a*d^2+24*b)/a^4/d^2*(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+24*b)/a^4/d^2*(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/a^4*(a*d^2+6*b)/d^2*
(Si(d*x)*cos(c)+Ci(d*x)*sin(c))+9/16/a^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))+9/16/a^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)))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [C]  time = 2.40066, size = 1685, normalized size = 2.13 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/64*((16*I*(a*b^2*d^2 + 6*b^3)*x^6 + 32*I*(a^2*b*d^2 + 6*a*b^2)*x^4 + 16*I*(a^3*d^2 + 6*a^2*b)*x^2)*Ei(I*d*x)
*e^(I*c) + (-16*I*(a*b^2*d^2 + 6*b^3)*x^6 - 32*I*(a^2*b*d^2 + 6*a*b^2)*x^4 - 16*I*(a^3*d^2 + 6*a^2*b)*x^2)*Ei(
-I*d*x)*e^(-I*c) + (-2*I*(a*b^2*d^2 + 24*b^3)*x^6 - 4*I*(a^2*b*d^2 + 24*a*b^2)*x^4 - 2*I*(a^3*d^2 + 24*a^2*b)*
x^2 + 2*(9*I*b^3*x^6 + 18*I*a*b^2*x^4 + 9*I*a^2*b*x^2)*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 + 24*b^3)*x^6 - 4*I*(a^2*b*d^2 + 24*a*b^2)*x^4 - 2*I*(a^3*d^2 + 24*a^2*b)*x^2 + 2
*(-9*I*b^3*x^6 - 18*I*a*b^2*x^4 - 9*I*a^2*b*x^2)*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 + 24*b^3)*x^6 + 4*I*(a^2*b*d^2 + 24*a*b^2)*x^4 + 2*I*(a^3*d^2 + 24*a^2*b)*x^2 + 2*(-9*I*
b^3*x^6 - 18*I*a*b^2*x^4 - 9*I*a^2*b*x^2)*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 + 24*b^3)*x^6 + 4*I*(a^2*b*d^2 + 24*a*b^2)*x^4 + 2*I*(a^3*d^2 + 24*a^2*b)*x^2 + 2*(9*I*b^3*x^
6 + 18*I*a*b^2*x^4 + 9*I*a^2*b*x^2)*sqrt(a*d^2/b))*Ei(-I*d*x + sqrt(a*d^2/b))*e^(-I*c - sqrt(a*d^2/b)) - 8*(3*
a*b^2*d*x^5 + 7*a^2*b*d*x^3 + 4*a^3*d*x)*cos(d*x + c) - 16*(6*a*b^2*x^4 + 9*a^2*b*x^2 + 2*a^3)*sin(d*x + c))/(
a^4*b^2*x^6 + 2*a^5*b*x^4 + a^6*x^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(sin(d*x+c)/x**3/(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{\sin \left (d x + c\right )}{{\left (b x^{2} + a\right )}^{3} x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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