3.1.78 \(\int \frac {\sin (c+d x)}{x^3 (a+b x^2)^3} \, dx\) [78]

Optimal. Leaf size=791 \[ -\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}} \]

[Out]

-1/2*d*cos(d*x+c)/a^3/x-3*b*cos(c)*Si(d*x)/a^4-1/2*d^2*cos(c)*Si(d*x)/a^3+3/2*b*cos(c+d*(-a)^(1/2)/b^(1/2))*Si
(d*x-d*(-a)^(1/2)/b^(1/2))/a^4+1/16*d^2*cos(c+d*(-a)^(1/2)/b^(1/2))*Si(d*x-d*(-a)^(1/2)/b^(1/2))/a^3+3/2*b*cos
(c-d*(-a)^(1/2)/b^(1/2))*Si(d*x+d*(-a)^(1/2)/b^(1/2))/a^4+1/16*d^2*cos(c-d*(-a)^(1/2)/b^(1/2))*Si(d*x+d*(-a)^(
1/2)/b^(1/2))/a^3-3*b*Ci(d*x)*sin(c)/a^4-1/2*d^2*Ci(d*x)*sin(c)/a^3-1/2*sin(d*x+c)/a^3/x^2-1/4*b*sin(d*x+c)/a^
2/(b*x^2+a)^2-b*sin(d*x+c)/a^3/(b*x^2+a)+3/2*b*Ci(d*x+d*(-a)^(1/2)/b^(1/2))*sin(c-d*(-a)^(1/2)/b^(1/2))/a^4+1/
16*d^2*Ci(d*x+d*(-a)^(1/2)/b^(1/2))*sin(c-d*(-a)^(1/2)/b^(1/2))/a^3+3/2*b*Ci(-d*x+d*(-a)^(1/2)/b^(1/2))*sin(c+
d*(-a)^(1/2)/b^(1/2))/a^4+1/16*d^2*Ci(-d*x+d*(-a)^(1/2)/b^(1/2))*sin(c+d*(-a)^(1/2)/b^(1/2))/a^3+9/16*d*Ci(d*x
+d*(-a)^(1/2)/b^(1/2))*cos(c-d*(-a)^(1/2)/b^(1/2))*b^(1/2)/(-a)^(7/2)-9/16*d*Ci(-d*x+d*(-a)^(1/2)/b^(1/2))*cos
(c+d*(-a)^(1/2)/b^(1/2))*b^(1/2)/(-a)^(7/2)-9/16*d*Si(d*x+d*(-a)^(1/2)/b^(1/2))*sin(c-d*(-a)^(1/2)/b^(1/2))*b^
(1/2)/(-a)^(7/2)+9/16*d*Si(d*x-d*(-a)^(1/2)/b^(1/2))*sin(c+d*(-a)^(1/2)/b^(1/2))*b^(1/2)/(-a)^(7/2)-1/16*d*cos
(d*x+c)*b^(1/2)/a^3/((-a)^(1/2)-x*b^(1/2))+1/16*d*cos(d*x+c)*b^(1/2)/a^3/((-a)^(1/2)+x*b^(1/2))

________________________________________________________________________________________

Rubi [A]
time = 1.38, antiderivative size = 791, normalized size of antiderivative = 1.00, 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 = {3426, 3378, 3384, 3380, 3383, 3422, 3415} \begin {gather*} -\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 {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 {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 {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 {b \sin (c+d x)}{a^3 \left (a+b x^2\right )}-\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 {b \sin (c+d x)}{4 a^2 \left (a+b x^2\right )^2}-\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}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*(d*Cos[c + d*x])/(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)/Sq
rt[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*CosIn
tegral[(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 + (Sqr
t[-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 - (Sqr
t[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*(-a)^(7/2))

Rule 3378

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 3380

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 3383

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 3384

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 3415

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 3422

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 3426

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^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*}

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Mathematica [C] Result contains complex when optimal does not.
time = 1.72, size = 995, normalized size = 1.26 \begin {gather*} \frac {-\frac {2 a \cos (d x) \left (d x \left (4 a^2+7 a b x^2+3 b^2 x^4\right ) \cos (c)+2 \left (2 a^2+9 a b x^2+6 b^2 x^4\right ) \sin (c)\right )}{x^2 \left (a+b x^2\right )^2}+\frac {2 a \left (-2 \left (2 a^2+9 a b x^2+6 b^2 x^4\right ) \cos (c)+d x \left (4 a^2+7 a b x^2+3 b^2 x^4\right ) \sin (c)\right ) \sin (d x)}{x^2 \left (a+b x^2\right )^2}-8 \left (6 b+a d^2\right ) (\text {Ci}(d x) \sin (c)+\cos (c) \text {Si}(d x))+24 b \cos (c) \left (i \text {Ci}\left (d \left (-\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right ) \sinh \left (\frac {\sqrt {a} d}{\sqrt {b}}\right )-i \text {Ci}\left (d \left (\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right ) \sinh \left (\frac {\sqrt {a} d}{\sqrt {b}}\right )+\cosh \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \left (\text {Si}\left (d \left (\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )-\text {Si}\left (\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )\right )\right )+a d^2 \cos (c) \left (i \text {Ci}\left (d \left (-\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right ) \sinh \left (\frac {\sqrt {a} d}{\sqrt {b}}\right )-i \text {Ci}\left (d \left (\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right ) \sinh \left (\frac {\sqrt {a} d}{\sqrt {b}}\right )+\cosh \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \left (\text {Si}\left (d \left (\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )-\text {Si}\left (\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )\right )\right )+9 \sqrt {a} \sqrt {b} d \cos (c) \left (-i \cosh \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \text {Ci}\left (d \left (-\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )+i \cosh \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \text {Ci}\left (d \left (\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )+\sinh \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \left (-\text {Si}\left (d \left (\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )+\text {Si}\left (\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )\right )\right )-9 \sqrt {a} \sqrt {b} d \sin (c) \left (\text {Ci}\left (d \left (-\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right ) \sinh \left (\frac {\sqrt {a} d}{\sqrt {b}}\right )+\text {Ci}\left (d \left (\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right ) \sinh \left (\frac {\sqrt {a} d}{\sqrt {b}}\right )+i \cosh \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \left (\text {Si}\left (d \left (\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )+\text {Si}\left (\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )\right )\right )+24 b \sin (c) \left (\cosh \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \text {Ci}\left (d \left (-\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )+\cosh \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \text {Ci}\left (d \left (\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )+i \sinh \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \left (\text {Si}\left (d \left (\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )+\text {Si}\left (\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )\right )\right )+a d^2 \sin (c) \left (\cosh \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \text {Ci}\left (d \left (-\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )+\cosh \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \text {Ci}\left (d \left (\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )+i \sinh \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \left (\text {Si}\left (d \left (\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )+\text {Si}\left (\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )\right )\right )}{16 a^4} \end {gather*}

Antiderivative was successfully verified.

[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 = 1.84, size = 697, normalized size = 0.88 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d^2*(-1/4*sin(d*x+c)*(2*a^2*d^4+9*a*b*c^2*d^2-18*a*b*c*d^2*(d*x+c)+9*a*b*d^2*(d*x+c)^2+6*b^2*c^4-24*b^2*c^3*(d
*x+c)+36*b^2*c^2*(d*x+c)^2-24*b^2*c*(d*x+c)^3+6*b^2*(d*x+c)^4)/a^3/d^2/x^2/(d^2*a+b*c^2-2*b*c*(d*x+c)+b*(d*x+c
)^2)^2-1/8*cos(d*x+c)*(4*d^2*a+3*b*c^2-6*b*c*(d*x+c)+3*b*(d*x+c)^2)/a^3/d/x/(d^2*a+b*c^2-2*b*c*(d*x+c)+b*(d*x+
c)^2)+1/16*(a*d^2+24*b)/d^2/a^4*(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)/d^2/a^4*(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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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] Result contains complex when optimal does not.
time = 0.47, size = 768, normalized size = 0.97 \begin {gather*} -\frac {8 \, {\left (-i \, {\left (a b^{2} d^{2} + 6 \, b^{3}\right )} x^{6} - 2 i \, {\left (a^{2} b d^{2} + 6 \, a b^{2}\right )} x^{4} - i \, {\left (a^{3} d^{2} + 6 \, a^{2} b\right )} x^{2}\right )} {\rm Ei}\left (i \, d x\right ) e^{\left (i \, c\right )} + 8 \, {\left (i \, {\left (a b^{2} d^{2} + 6 \, b^{3}\right )} x^{6} + 2 i \, {\left (a^{2} b d^{2} + 6 \, a b^{2}\right )} x^{4} + i \, {\left (a^{3} d^{2} + 6 \, a^{2} b\right )} x^{2}\right )} {\rm Ei}\left (-i \, d x\right ) e^{\left (-i \, c\right )} + {\left (i \, {\left (a b^{2} d^{2} + 24 \, b^{3}\right )} x^{6} + 2 i \, {\left (a^{2} b d^{2} + 24 \, a b^{2}\right )} x^{4} + i \, {\left (a^{3} d^{2} + 24 \, a^{2} b\right )} x^{2} + 9 \, {\left (-i \, b^{3} x^{6} - 2 i \, a b^{2} x^{4} - i \, a^{2} b x^{2}\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 (i \, {\left (a b^{2} d^{2} + 24 \, b^{3}\right )} x^{6} + 2 i \, {\left (a^{2} b d^{2} + 24 \, a b^{2}\right )} x^{4} + i \, {\left (a^{3} d^{2} + 24 \, a^{2} b\right )} x^{2} + 9 \, {\left (i \, b^{3} x^{6} + 2 i \, a b^{2} x^{4} + i \, a^{2} b x^{2}\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 (-i \, {\left (a b^{2} d^{2} + 24 \, b^{3}\right )} x^{6} - 2 i \, {\left (a^{2} b d^{2} + 24 \, a b^{2}\right )} x^{4} - i \, {\left (a^{3} d^{2} + 24 \, a^{2} b\right )} x^{2} + 9 \, {\left (i \, b^{3} x^{6} + 2 i \, a b^{2} x^{4} + i \, a^{2} b x^{2}\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 (-i \, {\left (a b^{2} d^{2} + 24 \, b^{3}\right )} x^{6} - 2 i \, {\left (a^{2} b d^{2} + 24 \, a b^{2}\right )} x^{4} - i \, {\left (a^{3} d^{2} + 24 \, a^{2} b\right )} x^{2} + 9 \, {\left (-i \, b^{3} x^{6} - 2 i \, a b^{2} x^{4} - i \, a^{2} b x^{2}\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^{2} d x^{5} + 7 \, a^{2} b d x^{3} + 4 \, a^{3} d x\right )} \cos \left (d x + c\right ) + 8 \, {\left (6 \, a b^{2} x^{4} + 9 \, a^{2} b x^{2} + 2 \, a^{3}\right )} \sin \left (d x + c\right )}{32 \, {\left (a^{4} b^{2} x^{6} + 2 \, a^{5} b x^{4} + a^{6} x^{2}\right )}} \end {gather*}

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]

-1/32*(8*(-I*(a*b^2*d^2 + 6*b^3)*x^6 - 2*I*(a^2*b*d^2 + 6*a*b^2)*x^4 - I*(a^3*d^2 + 6*a^2*b)*x^2)*Ei(I*d*x)*e^
(I*c) + 8*(I*(a*b^2*d^2 + 6*b^3)*x^6 + 2*I*(a^2*b*d^2 + 6*a*b^2)*x^4 + I*(a^3*d^2 + 6*a^2*b)*x^2)*Ei(-I*d*x)*e
^(-I*c) + (I*(a*b^2*d^2 + 24*b^3)*x^6 + 2*I*(a^2*b*d^2 + 24*a*b^2)*x^4 + I*(a^3*d^2 + 24*a^2*b)*x^2 + 9*(-I*b^
3*x^6 - 2*I*a*b^2*x^4 - 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)) + (I*(a*
b^2*d^2 + 24*b^3)*x^6 + 2*I*(a^2*b*d^2 + 24*a*b^2)*x^4 + I*(a^3*d^2 + 24*a^2*b)*x^2 + 9*(I*b^3*x^6 + 2*I*a*b^2
*x^4 + 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)) + (-I*(a*b^2*d^2 + 24*b^3
)*x^6 - 2*I*(a^2*b*d^2 + 24*a*b^2)*x^4 - I*(a^3*d^2 + 24*a^2*b)*x^2 + 9*(I*b^3*x^6 + 2*I*a*b^2*x^4 + 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)) + (-I*(a*b^2*d^2 + 24*b^3)*x^6 - 2*I*(a
^2*b*d^2 + 24*a*b^2)*x^4 - I*(a^3*d^2 + 24*a^2*b)*x^2 + 9*(-I*b^3*x^6 - 2*I*a*b^2*x^4 - 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)) + 4*(3*a*b^2*d*x^5 + 7*a^2*b*d*x^3 + 4*a^3*d*x)*cos(
d*x + c) + 8*(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)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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