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

Optimal. Leaf size=501 \[ \frac{d \cos \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{4 a^2}+\frac{d \cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{4 a^2}+\frac{d \sin \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{4 a^2}-\frac{d \sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{4 a^2}+\frac{\sqrt{b} \sin (c+d x)}{4 a^2 \left (\sqrt{-a}-\sqrt{b} x\right )}-\frac{\sqrt{b} \sin (c+d x)}{4 a^2 \left (\sqrt{-a}+\sqrt{b} x\right )}+\frac{d \cos (c) \text{CosIntegral}(d x)}{a^2}-\frac{d \sin (c) \text{Si}(d x)}{a^2}-\frac{\sin (c+d x)}{a^2 x}+\frac{3 \sqrt{b} \sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{4 (-a)^{5/2}}-\frac{3 \sqrt{b} \sin \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{4 (-a)^{5/2}}+\frac{3 \sqrt{b} \cos \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{4 (-a)^{5/2}}+\frac{3 \sqrt{b} \cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{4 (-a)^{5/2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 1.3129, antiderivative size = 501, normalized size of antiderivative = 1., number of steps used = 32, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316, Rules used = {3345, 3297, 3303, 3299, 3302, 3333} \[ \frac{d \cos \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{4 a^2}+\frac{d \cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{4 a^2}+\frac{d \sin \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{4 a^2}-\frac{d \sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{4 a^2}+\frac{\sqrt{b} \sin (c+d x)}{4 a^2 \left (\sqrt{-a}-\sqrt{b} x\right )}-\frac{\sqrt{b} \sin (c+d x)}{4 a^2 \left (\sqrt{-a}+\sqrt{b} x\right )}+\frac{d \cos (c) \text{CosIntegral}(d x)}{a^2}-\frac{d \sin (c) \text{Si}(d x)}{a^2}-\frac{\sin (c+d x)}{a^2 x}+\frac{3 \sqrt{b} \sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{4 (-a)^{5/2}}-\frac{3 \sqrt{b} \sin \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{4 (-a)^{5/2}}+\frac{3 \sqrt{b} \cos \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{4 (-a)^{5/2}}+\frac{3 \sqrt{b} \cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{4 (-a)^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(d*Cos[c]*CosIntegral[d*x])/a^2 + (d*Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(4
*a^2) + (d*Cos[c - (Sqrt[-a]*d)/Sqrt[b]]*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(4*a^2) + (3*Sqrt[b]*CosInte
gral[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sin[c - (Sqrt[-a]*d)/Sqrt[b]])/(4*(-a)^(5/2)) - (3*Sqrt[b]*CosIntegral[(Sqrt[
-a]*d)/Sqrt[b] - d*x]*Sin[c + (Sqrt[-a]*d)/Sqrt[b]])/(4*(-a)^(5/2)) - Sin[c + d*x]/(a^2*x) + (Sqrt[b]*Sin[c +
d*x])/(4*a^2*(Sqrt[-a] - Sqrt[b]*x)) - (Sqrt[b]*Sin[c + d*x])/(4*a^2*(Sqrt[-a] + Sqrt[b]*x)) - (d*Sin[c]*SinIn
tegral[d*x])/a^2 + (3*Sqrt[b]*Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(4*(-a)^(
5/2)) + (d*Sin[c + (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(4*a^2) + (3*Sqrt[b]*Cos[c -
(Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(4*(-a)^(5/2)) - (d*Sin[c - (Sqrt[-a]*d)/Sqrt[
b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(4*a^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 3333

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*Sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Int[ExpandIntegrand[Sin[c + d*x], (a +
b*x^n)^p, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[p, 0] && IGtQ[n, 0] && (EqQ[n, 2] || EqQ[p, -1])

Rubi steps

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

Mathematica [C]  time = 1.09659, size = 768, normalized size = 1.53 \[ \frac{a^{3/2} d x \cos \left (c-\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (d \left (x+\frac{i \sqrt{a}}{\sqrt{b}}\right )\right )-a^{3/2} d x \sin \left (c-\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{Si}\left (d \left (x+\frac{i \sqrt{a}}{\sqrt{b}}\right )\right )+a^{3/2} d x \sin \left (c+\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right )-4 a^{3/2} d x \sin (c) \text{Si}(d x)-4 a^{3/2} \sin (c+d x)-3 i b^{3/2} x^3 \sin \left (c-\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (d \left (x+\frac{i \sqrt{a}}{\sqrt{b}}\right )\right )-3 i b^{3/2} x^3 \cos \left (c-\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{Si}\left (d \left (x+\frac{i \sqrt{a}}{\sqrt{b}}\right )\right )-3 i b^{3/2} x^3 \cos \left (c+\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right )+\sqrt{a} b d x^3 \cos \left (c-\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (d \left (x+\frac{i \sqrt{a}}{\sqrt{b}}\right )\right )+4 \sqrt{a} d x \cos (c) \left (a+b x^2\right ) \text{CosIntegral}(d x)+x \left (a+b x^2\right ) \text{CosIntegral}\left (d \left (x-\frac{i \sqrt{a}}{\sqrt{b}}\right )\right ) \left (3 i \sqrt{b} \sin \left (c+\frac{i \sqrt{a} d}{\sqrt{b}}\right )+\sqrt{a} d \cos \left (c+\frac{i \sqrt{a} d}{\sqrt{b}}\right )\right )-3 i a \sqrt{b} x \sin \left (c-\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (d \left (x+\frac{i \sqrt{a}}{\sqrt{b}}\right )\right )-4 \sqrt{a} b d x^3 \sin (c) \text{Si}(d x)-\sqrt{a} b d x^3 \sin \left (c-\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{Si}\left (d \left (x+\frac{i \sqrt{a}}{\sqrt{b}}\right )\right )+\sqrt{a} b d x^3 \sin \left (c+\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right )-3 i a \sqrt{b} x \cos \left (c-\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{Si}\left (d \left (x+\frac{i \sqrt{a}}{\sqrt{b}}\right )\right )-3 i a \sqrt{b} x \cos \left (c+\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right )-6 \sqrt{a} b x^2 \sin (c+d x)}{4 a^{5/2} x \left (a+b x^2\right )} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(4*Sqrt[a]*d*x*(a + b*x^2)*Cos[c]*CosIntegral[d*x] + a^(3/2)*d*x*Cos[c - (I*Sqrt[a]*d)/Sqrt[b]]*CosIntegral[d*
((I*Sqrt[a])/Sqrt[b] + x)] + Sqrt[a]*b*d*x^3*Cos[c - (I*Sqrt[a]*d)/Sqrt[b]]*CosIntegral[d*((I*Sqrt[a])/Sqrt[b]
+ x)] - (3*I)*a*Sqrt[b]*x*CosIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)]*Sin[c - (I*Sqrt[a]*d)/Sqrt[b]] - (3*I)*b^(
3/2)*x^3*CosIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)]*Sin[c - (I*Sqrt[a]*d)/Sqrt[b]] + x*(a + b*x^2)*CosIntegral[d
*(((-I)*Sqrt[a])/Sqrt[b] + x)]*(Sqrt[a]*d*Cos[c + (I*Sqrt[a]*d)/Sqrt[b]] + (3*I)*Sqrt[b]*Sin[c + (I*Sqrt[a]*d)
/Sqrt[b]]) - 4*a^(3/2)*Sin[c + d*x] - 6*Sqrt[a]*b*x^2*Sin[c + d*x] - 4*a^(3/2)*d*x*Sin[c]*SinIntegral[d*x] - 4
*Sqrt[a]*b*d*x^3*Sin[c]*SinIntegral[d*x] - (3*I)*a*Sqrt[b]*x*Cos[c - (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*
Sqrt[a])/Sqrt[b] + x)] - (3*I)*b^(3/2)*x^3*Cos[c - (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] +
x)] - a^(3/2)*d*x*Sin[c - (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] - Sqrt[a]*b*d*x^3*S
in[c - (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] - (3*I)*a*Sqrt[b]*x*Cos[c + (I*Sqrt[a]*
d)/Sqrt[b]]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x] - (3*I)*b^(3/2)*x^3*Cos[c + (I*Sqrt[a]*d)/Sqrt[b]]*SinInt
egral[(I*Sqrt[a]*d)/Sqrt[b] - d*x] + a^(3/2)*d*x*Sin[c + (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(I*Sqrt[a]*d)/Sqrt
[b] - d*x] + Sqrt[a]*b*d*x^3*Sin[c + (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x])/(4*a^(5/
2)*x*(a + b*x^2))

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Maple [A]  time = 0.029, size = 769, normalized size = 1.5 \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^2/(b*x^2+a)^2,x)

[Out]

d*(-1/a^2*b*(1/2/((d*(-a*b)^(1/2)+c*b)/b-c)/b*(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/(-(d*(-a*b)^(1/2)-c*b)/b-c)/b*(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^2*(-sin(d*x+c)/x/d-Si(d*x)*sin(c)+Ci(d*x)*cos(c))-1/a*b*d^2*(sin(d*x+c)*(1/2/a/d^2*(d*x+c)-1/2*c/a/d^2)/((d
*x+c)^2*b-2*(d*x+c)*b*c+a*d^2+c^2*b)+1/4/a/d^2/b/((d*(-a*b)^(1/2)+c*b)/b-c)*(Si(d*x+c-(d*(-a*b)^(1/2)+c*b)/b)*
cos((d*(-a*b)^(1/2)+c*b)/b)+Ci(d*x+c-(d*(-a*b)^(1/2)+c*b)/b)*sin((d*(-a*b)^(1/2)+c*b)/b))+1/4/a/d^2/b/(-(d*(-a
*b)^(1/2)-c*b)/b-c)*(Si(d*x+c+(d*(-a*b)^(1/2)-c*b)/b)*cos((d*(-a*b)^(1/2)-c*b)/b)-Ci(d*x+c+(d*(-a*b)^(1/2)-c*b
)/b)*sin((d*(-a*b)^(1/2)-c*b)/b))-1/4/a/b/d^2*(-Si(d*x+c-(d*(-a*b)^(1/2)+c*b)/b)*sin((d*(-a*b)^(1/2)+c*b)/b)+C
i(d*x+c-(d*(-a*b)^(1/2)+c*b)/b)*cos((d*(-a*b)^(1/2)+c*b)/b))-1/4/a/b/d^2*(Si(d*x+c+(d*(-a*b)^(1/2)-c*b)/b)*sin
((d*(-a*b)^(1/2)-c*b)/b)+Ci(d*x+c+(d*(-a*b)^(1/2)-c*b)/b)*cos((d*(-a*b)^(1/2)-c*b)/b))))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [C]  time = 2.09078, size = 848, normalized size = 1.69 \begin{align*} \frac{4 \,{\left (a b d^{2} x^{3} + a^{2} d^{2} x\right )}{\rm Ei}\left (i \, d x\right ) e^{\left (i \, c\right )} + 4 \,{\left (a b d^{2} x^{3} + a^{2} d^{2} x\right )}{\rm Ei}\left (-i \, d x\right ) e^{\left (-i \, c\right )} +{\left (a b d^{2} x^{3} + a^{2} d^{2} x - 3 \,{\left (b^{2} x^{3} + a b x\right )} \sqrt{\frac{a d^{2}}{b}}\right )}{\rm Ei}\left (i \, d x - \sqrt{\frac{a d^{2}}{b}}\right ) e^{\left (i \, c + \sqrt{\frac{a d^{2}}{b}}\right )} +{\left (a b d^{2} x^{3} + a^{2} d^{2} x + 3 \,{\left (b^{2} x^{3} + a b x\right )} \sqrt{\frac{a d^{2}}{b}}\right )}{\rm Ei}\left (i \, d x + \sqrt{\frac{a d^{2}}{b}}\right ) e^{\left (i \, c - \sqrt{\frac{a d^{2}}{b}}\right )} +{\left (a b d^{2} x^{3} + a^{2} d^{2} x - 3 \,{\left (b^{2} x^{3} + a b x\right )} \sqrt{\frac{a d^{2}}{b}}\right )}{\rm Ei}\left (-i \, d x - \sqrt{\frac{a d^{2}}{b}}\right ) e^{\left (-i \, c + \sqrt{\frac{a d^{2}}{b}}\right )} +{\left (a b d^{2} x^{3} + a^{2} d^{2} x + 3 \,{\left (b^{2} x^{3} + a b x\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 )} - 4 \,{\left (3 \, a b d x^{2} + 2 \, a^{2} d\right )} \sin \left (d x + c\right )}{8 \,{\left (a^{3} b d x^{3} + a^{4} d x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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**2/(b*x**2+a)**2,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 )}^{2} x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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