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

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

cos(c)*Si(d*x)/a^3-1/2*cos(c+d*(-a)^(1/2)/b^(1/2))*Si(d*x-d*(-a)^(1/2)/b^(1/2))/a^3-1/16*d^2*cos(c+d*(-a)^(1/2
)/b^(1/2))*Si(d*x-d*(-a)^(1/2)/b^(1/2))/a^2/b-1/2*cos(c-d*(-a)^(1/2)/b^(1/2))*Si(d*x+d*(-a)^(1/2)/b^(1/2))/a^3
-1/16*d^2*cos(c-d*(-a)^(1/2)/b^(1/2))*Si(d*x+d*(-a)^(1/2)/b^(1/2))/a^2/b+Ci(d*x)*sin(c)/a^3+1/4*sin(d*x+c)/a/(
b*x^2+a)^2+1/2*sin(d*x+c)/a^2/(b*x^2+a)-1/2*Ci(d*x+d*(-a)^(1/2)/b^(1/2))*sin(c-d*(-a)^(1/2)/b^(1/2))/a^3-1/16*
d^2*Ci(d*x+d*(-a)^(1/2)/b^(1/2))*sin(c-d*(-a)^(1/2)/b^(1/2))/a^2/b-1/2*Ci(-d*x+d*(-a)^(1/2)/b^(1/2))*sin(c+d*(
-a)^(1/2)/b^(1/2))/a^3-1/16*d^2*Ci(-d*x+d*(-a)^(1/2)/b^(1/2))*sin(c+d*(-a)^(1/2)/b^(1/2))/a^2/b+5/16*d*Ci(d*x+
d*(-a)^(1/2)/b^(1/2))*cos(c-d*(-a)^(1/2)/b^(1/2))/(-a)^(5/2)/b^(1/2)-5/16*d*Ci(-d*x+d*(-a)^(1/2)/b^(1/2))*cos(
c+d*(-a)^(1/2)/b^(1/2))/(-a)^(5/2)/b^(1/2)-5/16*d*Si(d*x+d*(-a)^(1/2)/b^(1/2))*sin(c-d*(-a)^(1/2)/b^(1/2))/(-a
)^(5/2)/b^(1/2)+5/16*d*Si(d*x-d*(-a)^(1/2)/b^(1/2))*sin(c+d*(-a)^(1/2)/b^(1/2))/(-a)^(5/2)/b^(1/2)+1/16*d*cos(
d*x+c)/a^2/b^(1/2)/((-a)^(1/2)-x*b^(1/2))-1/16*d*cos(d*x+c)/a^2/b^(1/2)/((-a)^(1/2)+x*b^(1/2))

________________________________________________________________________________________

Rubi [A]  time = 1.83, antiderivative size = 730, normalized size of antiderivative = 1.00, 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 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 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 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 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 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 3345

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

Rubi steps

\begin {align*} \int \frac {\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*}

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Mathematica [C]  time = 2.87, size = 924, normalized size = 1.27 \[ \frac {\frac {16 b^2 \text {Ci}(d x) \sin (c) x^4}{\left (b x^2+a\right )^2}+\frac {16 b^2 \cos (c) \text {Si}(d x) x^4}{\left (b x^2+a\right )^2}-\frac {2 a b d \cos (c+d x) x^3}{\left (b x^2+a\right )^2}+\frac {32 a b \text {Ci}(d x) \sin (c) x^2}{\left (b x^2+a\right )^2}+\frac {8 a b \sin (c+d x) x^2}{\left (b x^2+a\right )^2}+\frac {32 a b \cos (c) \text {Si}(d x) x^2}{\left (b x^2+a\right )^2}-\frac {2 a^2 d \cos (c+d x) x}{\left (b x^2+a\right )^2}+\frac {16 a^2 \text {Ci}(d x) \sin (c)}{\left (b x^2+a\right )^2}-\frac {\text {Ci}\left (d \left (x+\frac {i \sqrt {a}}{\sqrt {b}}\right )\right ) \left (5 i \sqrt {a} \sqrt {b} d \cos \left (c-\frac {i \sqrt {a} d}{\sqrt {b}}\right )+\left (a d^2+8 b\right ) \sin \left (c-\frac {i \sqrt {a} d}{\sqrt {b}}\right )\right )}{b}-\frac {\text {Ci}\left (d \left (x-\frac {i \sqrt {a}}{\sqrt {b}}\right )\right ) \left (\left (a d^2+8 b\right ) \sin \left (c+\frac {i \sqrt {a} d}{\sqrt {b}}\right )-5 i \sqrt {a} \sqrt {b} d \cos \left (c+\frac {i \sqrt {a} d}{\sqrt {b}}\right )\right )}{b}+\frac {12 a^2 \sin (c+d x)}{\left (b x^2+a\right )^2}+\frac {16 a^2 \cos (c) \text {Si}(d x)}{\left (b x^2+a\right )^2}-\frac {a d^2 \cos (c) \cosh \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \text {Si}\left (d \left (x+\frac {i \sqrt {a}}{\sqrt {b}}\right )\right )}{b}-8 \cos (c) \cosh \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \text {Si}\left (d \left (x+\frac {i \sqrt {a}}{\sqrt {b}}\right )\right )+\frac {5 i \sqrt {a} d \cosh \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \sin (c) \text {Si}\left (d \left (x+\frac {i \sqrt {a}}{\sqrt {b}}\right )\right )}{\sqrt {b}}+\frac {5 \sqrt {a} d \cos (c) \sinh \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \text {Si}\left (d \left (x+\frac {i \sqrt {a}}{\sqrt {b}}\right )\right )}{\sqrt {b}}-\frac {i a d^2 \sin (c) \sinh \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \text {Si}\left (d \left (x+\frac {i \sqrt {a}}{\sqrt {b}}\right )\right )}{b}-8 i \sin (c) \sinh \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \text {Si}\left (d \left (x+\frac {i \sqrt {a}}{\sqrt {b}}\right )\right )+\frac {a d^2 \cos (c) \cosh \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )}{b}+8 \cos (c) \cosh \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )+\frac {5 i \sqrt {a} d \cosh \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \sin (c) \text {Si}\left (\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )}{\sqrt {b}}-\frac {5 \sqrt {a} d \cos (c) \sinh \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )}{\sqrt {b}}-\frac {i a d^2 \sin (c) \sinh \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )}{b}-8 i \sin (c) \sinh \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )}{16 a^3} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((-2*a^2*d*x*Cos[c + d*x])/(a + b*x^2)^2 - (2*a*b*d*x^3*Cos[c + d*x])/(a + b*x^2)^2 + (16*a^2*CosIntegral[d*x]
*Sin[c])/(a + b*x^2)^2 + (32*a*b*x^2*CosIntegral[d*x]*Sin[c])/(a + b*x^2)^2 + (16*b^2*x^4*CosIntegral[d*x]*Sin
[c])/(a + b*x^2)^2 - (CosIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)]*((5*I)*Sqrt[a]*Sqrt[b]*d*Cos[c - (I*Sqrt[a]*d)/
Sqrt[b]] + (8*b + a*d^2)*Sin[c - (I*Sqrt[a]*d)/Sqrt[b]]))/b - (CosIntegral[d*(((-I)*Sqrt[a])/Sqrt[b] + x)]*((-
5*I)*Sqrt[a]*Sqrt[b]*d*Cos[c + (I*Sqrt[a]*d)/Sqrt[b]] + (8*b + a*d^2)*Sin[c + (I*Sqrt[a]*d)/Sqrt[b]]))/b + (12
*a^2*Sin[c + d*x])/(a + b*x^2)^2 + (8*a*b*x^2*Sin[c + d*x])/(a + b*x^2)^2 + (16*a^2*Cos[c]*SinIntegral[d*x])/(
a + b*x^2)^2 + (32*a*b*x^2*Cos[c]*SinIntegral[d*x])/(a + b*x^2)^2 + (16*b^2*x^4*Cos[c]*SinIntegral[d*x])/(a +
b*x^2)^2 - 8*Cos[c]*Cosh[(Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] - (a*d^2*Cos[c]*Cosh[(S
qrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)])/b + ((5*I)*Sqrt[a]*d*Cosh[(Sqrt[a]*d)/Sqrt[b]]*Si
n[c]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)])/Sqrt[b] + (5*Sqrt[a]*d*Cos[c]*Sinh[(Sqrt[a]*d)/Sqrt[b]]*SinInte
gral[d*((I*Sqrt[a])/Sqrt[b] + x)])/Sqrt[b] - (8*I)*Sin[c]*Sinh[(Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*Sqrt[a])
/Sqrt[b] + x)] - (I*a*d^2*Sin[c]*Sinh[(Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)])/b + 8*Cos
[c]*Cosh[(Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x] + (a*d^2*Cos[c]*Cosh[(Sqrt[a]*d)/Sqrt[b
]]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x])/b + ((5*I)*Sqrt[a]*d*Cosh[(Sqrt[a]*d)/Sqrt[b]]*Sin[c]*SinIntegral
[(I*Sqrt[a]*d)/Sqrt[b] - d*x])/Sqrt[b] - (5*Sqrt[a]*d*Cos[c]*Sinh[(Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(I*Sqrt[a]*
d)/Sqrt[b] - d*x])/Sqrt[b] - (8*I)*Sin[c]*Sinh[(Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x] -
(I*a*d^2*Sin[c]*Sinh[(Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x])/b)/(16*a^3)

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fricas [C]  time = 0.97, size = 645, normalized size = 0.88 \[ \frac {{\left (-32 i \, b^{3} x^{4} - 64 i \, a b^{2} x^{2} - 32 i \, a^{2} b\right )} {\rm Ei}\left (i \, d x\right ) e^{\left (i \, c\right )} + {\left (32 i \, b^{3} x^{4} + 64 i \, a b^{2} x^{2} + 32 i \, a^{2} b\right )} {\rm Ei}\left (-i \, d x\right ) e^{\left (-i \, c\right )} + {\left (2 i \, a^{3} d^{2} + 2 i \, {\left (a b^{2} d^{2} + 8 \, b^{3}\right )} x^{4} + 16 i \, a^{2} b + 4 i \, {\left (a^{2} b d^{2} + 8 \, a b^{2}\right )} x^{2} + 2 \, {\left (-5 i \, b^{3} x^{4} - 10 i \, a b^{2} x^{2} - 5 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^{3} d^{2} + 2 i \, {\left (a b^{2} d^{2} + 8 \, b^{3}\right )} x^{4} + 16 i \, a^{2} b + 4 i \, {\left (a^{2} b d^{2} + 8 \, a b^{2}\right )} x^{2} + 2 \, {\left (5 i \, b^{3} x^{4} + 10 i \, a b^{2} x^{2} + 5 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^{3} d^{2} - 2 i \, {\left (a b^{2} d^{2} + 8 \, b^{3}\right )} x^{4} - 16 i \, a^{2} b - 4 i \, {\left (a^{2} b d^{2} + 8 \, a b^{2}\right )} x^{2} + 2 \, {\left (5 i \, b^{3} x^{4} + 10 i \, a b^{2} x^{2} + 5 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^{3} d^{2} - 2 i \, {\left (a b^{2} d^{2} + 8 \, b^{3}\right )} x^{4} - 16 i \, a^{2} b - 4 i \, {\left (a^{2} b d^{2} + 8 \, a b^{2}\right )} x^{2} + 2 \, {\left (-5 i \, b^{3} x^{4} - 10 i \, a b^{2} x^{2} - 5 i \, a^{2} b\right )} \sqrt {\frac {a d^{2}}{b}}\right )} {\rm Ei}\left (-i \, d x + \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (-i \, c - \sqrt {\frac {a d^{2}}{b}}\right )} - 8 \, {\left (a b^{2} d x^{3} + a^{2} b d x\right )} \cos \left (d x + c\right ) + 16 \, {\left (2 \, a b^{2} x^{2} + 3 \, a^{2} b\right )} \sin \left (d x + c\right )}{64 \, {\left (a^{3} b^{3} x^{4} + 2 \, a^{4} b^{2} x^{2} + a^{5} b\right )}} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin \left (d x + c\right )}{{\left (b x^{2} + a\right )}^{3} x}\,{d x} \]

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)

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maple [A]  time = 0.07, size = 584, normalized size = 0.80 \[ \frac {\sin \left (d x +c \right ) d^{2} \left (2 \left (d x +c \right )^{2} b -4 \left (d x +c \right ) b c +3 a \,d^{2}+2 b \,c^{2}\right )}{4 a^{2} \left (\left (d x +c \right )^{2} b -2 \left (d x +c \right ) b c +a \,d^{2}+b \,c^{2}\right )^{2}}-\frac {\cos \left (d x +c \right ) d^{3} x}{8 a^{2} \left (\left (d x +c \right )^{2} b -2 \left (d x +c \right ) b c +a \,d^{2}+b \,c^{2}\right )}-\frac {\left (a \,d^{2}+8 b \right ) \left (\Si \left (d x +c -\frac {d \sqrt {-a b}+c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}+c b}{b}\right )+\Ci \left (d x +c -\frac {d \sqrt {-a b}+c b}{b}\right ) \sin \left (\frac {d \sqrt {-a b}+c b}{b}\right )\right )}{16 b \,a^{3}}-\frac {\left (a \,d^{2}+8 b \right ) \left (\Si \left (d x +c +\frac {d \sqrt {-a b}-c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}-c b}{b}\right )-\Ci \left (d x +c +\frac {d \sqrt {-a b}-c b}{b}\right ) \sin \left (\frac {d \sqrt {-a b}-c b}{b}\right )\right )}{16 b \,a^{3}}+\frac {\Si \left (d x \right ) \cos \relax (c )+\Ci \left (d x \right ) \sin \relax (c )}{a^{3}}-\frac {5 d^{2} \left (-\Si \left (d x +c -\frac {d \sqrt {-a b}+c b}{b}\right ) \sin \left (\frac {d \sqrt {-a b}+c b}{b}\right )+\Ci \left (d x +c -\frac {d \sqrt {-a b}+c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}+c b}{b}\right )\right )}{16 a^{2} b \left (\frac {d \sqrt {-a b}+c b}{b}-c \right )}-\frac {5 d^{2} \left (\Si \left (d x +c +\frac {d \sqrt {-a b}-c b}{b}\right ) \sin \left (\frac {d \sqrt {-a b}-c b}{b}\right )+\Ci \left (d x +c +\frac {d \sqrt {-a b}-c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}-c b}{b}\right )\right )}{16 a^{2} b \left (-\frac {d \sqrt {-a b}-c b}{b}-c \right )} \]

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*b*c^2)/a^2/((d*x+c)^2*b-2*(d*x+c)*b*c+a*d^2+b*c^2)^2
-1/8*cos(d*x+c)*d^3*x/a^2/((d*x+c)^2*b-2*(d*x+c)*b*c+a*d^2+b*c^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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin \left (d x + c\right )}{{\left (b x^{2} + a\right )}^{3} x}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\sin \left (c+d\,x\right )}{x\,{\left (b\,x^2+a\right )}^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(c + d*x)/(x*(a + b*x^2)^3),x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin {\left (c + d x \right )}}{x \left (a + b x^{2}\right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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