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

Optimal. Leaf size=730 \[ -\frac{d^2 \sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{16 a^2 b}-\frac{d^2 \sin \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 a^2 b}-\frac{\sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{2 a^3}-\frac{\sin \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 a^3}+\frac{d^2 \cos \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 a^2 b}-\frac{d^2 \cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{16 a^2 b}+\frac{\cos \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 a^3}-\frac{\cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{2 a^3}+\frac{\sin (c+d x)}{2 a^2 \left (a+b x^2\right )}+\frac{d \cos (c+d x)}{16 a^2 \sqrt{b} \left (\sqrt{-a}-\sqrt{b} x\right )}-\frac{d \cos (c+d x)}{16 a^2 \sqrt{b} \left (\sqrt{-a}+\sqrt{b} x\right )}+\frac{\sin (c) \text{CosIntegral}(d x)}{a^3}+\frac{\cos (c) \text{Si}(d x)}{a^3}-\frac{5 d \cos \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 (-a)^{5/2} \sqrt{b}}+\frac{5 d \cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{16 (-a)^{5/2} \sqrt{b}}-\frac{5 d \sin \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 (-a)^{5/2} \sqrt{b}}-\frac{5 d \sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{16 (-a)^{5/2} \sqrt{b}}+\frac{\sin (c+d x)}{4 a \left (a+b x^2\right )^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 1.83024, antiderivative size = 730, normalized size of antiderivative = 1., number of steps used = 41, number of rules used = 7, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.368, Rules used = {3345, 3303, 3299, 3302, 3341, 3334, 3297} \[ -\frac{d^2 \sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{16 a^2 b}-\frac{d^2 \sin \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 a^2 b}-\frac{\sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{2 a^3}-\frac{\sin \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 a^3}+\frac{d^2 \cos \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 a^2 b}-\frac{d^2 \cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{16 a^2 b}+\frac{\cos \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 a^3}-\frac{\cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{2 a^3}+\frac{\sin (c+d x)}{2 a^2 \left (a+b x^2\right )}+\frac{d \cos (c+d x)}{16 a^2 \sqrt{b} \left (\sqrt{-a}-\sqrt{b} x\right )}-\frac{d \cos (c+d x)}{16 a^2 \sqrt{b} \left (\sqrt{-a}+\sqrt{b} x\right )}+\frac{\sin (c) \text{CosIntegral}(d x)}{a^3}+\frac{\cos (c) \text{Si}(d x)}{a^3}-\frac{5 d \cos \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 (-a)^{5/2} \sqrt{b}}+\frac{5 d \cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{16 (-a)^{5/2} \sqrt{b}}-\frac{5 d \sin \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 (-a)^{5/2} \sqrt{b}}-\frac{5 d \sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{16 (-a)^{5/2} \sqrt{b}}+\frac{\sin (c+d x)}{4 a \left (a+b x^2\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[Sin[c + d*x]/(x*(a + b*x^2)^3),x]

[Out]

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

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 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])

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]

Rubi steps

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

Mathematica [C]  time = 7.93089, size = 1384, normalized size = 1.9 \[ \text{result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Cos[c]*(SinIntegral[d*x]/a^3 + (((I*Sqrt[a]*Sqrt[b]*d + b*d*x)*Cos[d*x] + b*Sin[d*x])/(Sqrt[a] - I*Sqrt[b]*x)^
2 + I*d^2*CosIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)]*Sinh[(Sqrt[a]*d)/Sqrt[b]] - d^2*Cosh[(Sqrt[a]*d)/Sqrt[b]]*S
inIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)])/(16*a^2*b) - (((5*I)/16)*Sqrt[b]*(-(Sin[d*x]/(I*Sqrt[a]*Sqrt[b] + b*x
)) + (d*(Cosh[(Sqrt[a]*d)/Sqrt[b]]*CosIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] + I*Sinh[(Sqrt[a]*d)/Sqrt[b]]*SinI
ntegral[d*((I*Sqrt[a])/Sqrt[b] + x)]))/b))/a^(5/2) - (I*CosIntegral[((-I)*Sqrt[a]*d)/Sqrt[b] + d*x]*Sinh[(Sqrt
[a]*d)/Sqrt[b]] - Cosh[(Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x])/(2*a^3) + ((((-I)*Sqrt[a
]*Sqrt[b]*d + b*d*x)*Cos[d*x] + b*Sin[d*x])/(Sqrt[a] + I*Sqrt[b]*x)^2 - I*d^2*CosIntegral[d*(((-I)*Sqrt[a])/Sq
rt[b] + x)]*Sinh[(Sqrt[a]*d)/Sqrt[b]] + d^2*Cosh[(Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x]
)/(16*a^2*b) + (((5*I)/16)*Sqrt[b]*(-(Sin[d*x]/((-I)*Sqrt[a]*Sqrt[b] + b*x)) + (d*(Cosh[(Sqrt[a]*d)/Sqrt[b]]*C
osIntegral[d*(((-I)*Sqrt[a])/Sqrt[b] + x)] + I*Sinh[(Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d
*x]))/b))/a^(5/2) - ((-I)*CosIntegral[(I*Sqrt[a]*d)/Sqrt[b] + d*x]*Sinh[(Sqrt[a]*d)/Sqrt[b]] + Cosh[(Sqrt[a]*d
)/Sqrt[b]]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] + d*x])/(2*a^3)) + Sin[c]*(CosIntegral[d*x]/a^3 + (-(d^2*Cosh[(Sq
rt[a]*d)/Sqrt[b]]*CosIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)]) + (b*Cos[d*x] + ((-I)*Sqrt[a]*Sqrt[b]*d - b*d*x)*S
in[d*x])/(Sqrt[a] - I*Sqrt[b]*x)^2 - I*d^2*Sinh[(Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)])
/(16*a^2*b) - (((5*I)/16)*Sqrt[b]*(-(Cos[d*x]/(I*Sqrt[a]*Sqrt[b] + b*x)) + (I*d*(CosIntegral[d*((I*Sqrt[a])/Sq
rt[b] + x)]*Sinh[(Sqrt[a]*d)/Sqrt[b]] + I*Cosh[(Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)]))
/b))/a^(5/2) - (Cosh[(Sqrt[a]*d)/Sqrt[b]]*CosIntegral[((-I)*Sqrt[a]*d)/Sqrt[b] + d*x] + I*Sinh[(Sqrt[a]*d)/Sqr
t[b]]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x])/(2*a^3) + (-(d^2*Cosh[(Sqrt[a]*d)/Sqrt[b]]*CosIntegral[d*(((-I
)*Sqrt[a])/Sqrt[b] + x)]) + (b*Cos[d*x] + I*Sqrt[a]*Sqrt[b]*d*Sin[d*x] - b*d*x*Sin[d*x])/(Sqrt[a] + I*Sqrt[b]*
x)^2 - I*d^2*Sinh[(Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x])/(16*a^2*b) + (((5*I)/16)*Sqrt
[b]*(-(Cos[d*x]/((-I)*Sqrt[a]*Sqrt[b] + b*x)) - (d*(I*CosIntegral[d*(((-I)*Sqrt[a])/Sqrt[b] + x)]*Sinh[(Sqrt[a
]*d)/Sqrt[b]] - Cosh[(Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x]))/b))/a^(5/2) - (Cosh[(Sqrt
[a]*d)/Sqrt[b]]*CosIntegral[(I*Sqrt[a]*d)/Sqrt[b] + d*x] + I*Sinh[(Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(I*Sqrt[a]*
d)/Sqrt[b] + d*x])/(2*a^3))

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Maple [A]  time = 0.042, size = 584, normalized size = 0.8 \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/(b*x^2+a)^3,x)

[Out]

1/4*sin(d*x+c)*d^2*(2*(d*x+c)^2*b-4*(d*x+c)*b*c+3*a*d^2+2*c^2*b)/a^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^3*x/((d*x+c)^2*b-2*(d*x+c)*b*c+a*d^2+c^2*b)/a^2-1/16*(a*d^2+8*b)/b/a^3*(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+8*b)/b/a^3*(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/a^3*(Si(d*x)*cos(c)+Ci(d*x)*sin(c))-5/16*d^2/a^2/b/((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))-5/16*d^2/a^2/b/(-(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}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [C]  time = 2.12876, size = 1455, normalized size = 1.99 \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/(b*x^2+a)^3,x, algorithm="fricas")

[Out]

1/64*((-32*I*b^3*x^4 - 64*I*a*b^2*x^2 - 32*I*a^2*b)*Ei(I*d*x)*e^(I*c) + (32*I*b^3*x^4 + 64*I*a*b^2*x^2 + 32*I*
a^2*b)*Ei(-I*d*x)*e^(-I*c) + (2*I*a^3*d^2 + 2*I*(a*b^2*d^2 + 8*b^3)*x^4 + 16*I*a^2*b + 4*I*(a^2*b*d^2 + 8*a*b^
2)*x^2 + 2*(-5*I*b^3*x^4 - 10*I*a*b^2*x^2 - 5*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^3*d^2 + 2*I*(a*b^2*d^2 + 8*b^3)*x^4 + 16*I*a^2*b + 4*I*(a^2*b*d^2 + 8*a*b^2)*x^2 + 2*(5*I*b
^3*x^4 + 10*I*a*b^2*x^2 + 5*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^3*d^2 - 2*I*(a*b^2*d^2 + 8*b^3)*x^4 - 16*I*a^2*b - 4*I*(a^2*b*d^2 + 8*a*b^2)*x^2 + 2*(5*I*b^3*x^4 + 10*I*a*b
^2*x^2 + 5*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^3*d^2 - 2*I*(
a*b^2*d^2 + 8*b^3)*x^4 - 16*I*a^2*b - 4*I*(a^2*b*d^2 + 8*a*b^2)*x^2 + 2*(-5*I*b^3*x^4 - 10*I*a*b^2*x^2 - 5*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 + 3*a^2*b)*sin(d*x + c))/(a^3*b^3*x^4 + 2*a^4*b^2*x^2 + a^5*b)

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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/(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}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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