\(\int \frac {\sin (c+d x)}{x^2 (a+b x^2)^3} \, dx\) [77]

Optimal result

Integrand size = 19, antiderivative size = 875 \[ \int \frac {\sin (c+d x)}{x^2 \left (a+b x^2\right )^3} \, dx=\frac {d \cos (c+d x)}{16 (-a)^{5/2} \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {d \cos (c+d x)}{16 (-a)^{5/2} \left (\sqrt {-a}+\sqrt {b} x\right )}+\frac {d \cos (c) \operatorname {CosIntegral}(d x)}{a^3}+\frac {7 d \cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 a^3}+\frac {7 d \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 a^3}-\frac {15 \sqrt {b} \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right ) \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 (-a)^{7/2}}+\frac {d^2 \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right ) \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 (-a)^{5/2} \sqrt {b}}+\frac {15 \sqrt {b} \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 (-a)^{7/2}}-\frac {d^2 \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 (-a)^{5/2} \sqrt {b}}-\frac {\sin (c+d x)}{a^3 x}-\frac {\sqrt {b} \sin (c+d x)}{16 (-a)^{5/2} \left (\sqrt {-a}-\sqrt {b} x\right )^2}+\frac {7 \sqrt {b} \sin (c+d x)}{16 a^3 \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\sqrt {b} \sin (c+d x)}{16 (-a)^{5/2} \left (\sqrt {-a}+\sqrt {b} x\right )^2}-\frac {7 \sqrt {b} \sin (c+d x)}{16 a^3 \left (\sqrt {-a}+\sqrt {b} x\right )}-\frac {d \sin (c) \text {Si}(d x)}{a^3}-\frac {15 \sqrt {b} \cos \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 {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)^{5/2} \sqrt {b}}+\frac {7 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^3}-\frac {15 \sqrt {b} \cos \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 {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)^{5/2} \sqrt {b}}-\frac {7 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^3} \]

[Out]

d*Ci(d*x)*cos(c)/a^3+7/16*d*Ci(d*x+d*(-a)^(1/2)/b^(1/2))*cos(c-d*(-a)^(1/2)/b^(1/2))/a^3+7/16*d*Ci(-d*x+d*(-a)
^(1/2)/b^(1/2))*cos(c+d*(-a)^(1/2)/b^(1/2))/a^3-d*Si(d*x)*sin(c)/a^3-sin(d*x+c)/a^3/x-7/16*d*Si(d*x+d*(-a)^(1/
2)/b^(1/2))*sin(c-d*(-a)^(1/2)/b^(1/2))/a^3-7/16*d*Si(d*x-d*(-a)^(1/2)/b^(1/2))*sin(c+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)^(5/2)/b^(1/2)+1/16*d^2*cos(c-d*(-a)^(
1/2)/b^(1/2))*Si(d*x+d*(-a)^(1/2)/b^(1/2))/(-a)^(5/2)/b^(1/2)+1/16*d^2*Ci(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^2*Ci(-d*x+d*(-a)^(1/2)/b^(1/2))*sin(c+d*(-a)^(1/2)/b^(1/2))/(-a)^(
5/2)/b^(1/2)+15/16*cos(c+d*(-a)^(1/2)/b^(1/2))*Si(d*x-d*(-a)^(1/2)/b^(1/2))*b^(1/2)/(-a)^(7/2)-15/16*cos(c-d*(
-a)^(1/2)/b^(1/2))*Si(d*x+d*(-a)^(1/2)/b^(1/2))*b^(1/2)/(-a)^(7/2)-15/16*Ci(d*x+d*(-a)^(1/2)/b^(1/2))*sin(c-d*
(-a)^(1/2)/b^(1/2))*b^(1/2)/(-a)^(7/2)+15/16*Ci(-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*sin(d*x+c)*b^(1/2)/(-a)^(5/2)/((-a)^(1/2)-x*b^(1/2))^2+1/16*d*cos(d*x+c)/(-a)^(5/2)/((-a)^(1/
2)-x*b^(1/2))+7/16*sin(d*x+c)*b^(1/2)/a^3/((-a)^(1/2)-x*b^(1/2))+1/16*sin(d*x+c)*b^(1/2)/(-a)^(5/2)/((-a)^(1/2
)+x*b^(1/2))^2+1/16*d*cos(d*x+c)/(-a)^(5/2)/((-a)^(1/2)+x*b^(1/2))-7/16*sin(d*x+c)*b^(1/2)/a^3/((-a)^(1/2)+x*b
^(1/2))

Rubi [A] (verified)

Time = 1.70 (sec) , antiderivative size = 875, normalized size of antiderivative = 1.00, number of steps used = 60, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {3426, 3378, 3384, 3380, 3383, 3414} \[ \int \frac {\sin (c+d x)}{x^2 \left (a+b x^2\right )^3} \, dx=\frac {\operatorname {CosIntegral}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) d^2}{16 (-a)^{5/2} \sqrt {b}}-\frac {\operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) d^2}{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 ) d^2}{16 (-a)^{5/2} \sqrt {b}}+\frac {\cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) d^2}{16 (-a)^{5/2} \sqrt {b}}+\frac {\cos (c+d x) d}{16 (-a)^{5/2} \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\cos (c+d x) d}{16 (-a)^{5/2} \left (\sqrt {b} x+\sqrt {-a}\right )}+\frac {\cos (c) \operatorname {CosIntegral}(d x) d}{a^3}+\frac {7 \cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) d}{16 a^3}+\frac {7 \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \operatorname {CosIntegral}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) d}{16 a^3}-\frac {\sin (c) \text {Si}(d x) d}{a^3}+\frac {7 \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) d}{16 a^3}-\frac {7 \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) d}{16 a^3}-\frac {15 \sqrt {b} \operatorname {CosIntegral}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 (-a)^{7/2}}+\frac {15 \sqrt {b} \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 (-a)^{7/2}}-\frac {\sin (c+d x)}{a^3 x}+\frac {7 \sqrt {b} \sin (c+d x)}{16 a^3 \left (\sqrt {-a}-\sqrt {b} x\right )}-\frac {7 \sqrt {b} \sin (c+d x)}{16 a^3 \left (\sqrt {b} x+\sqrt {-a}\right )}-\frac {\sqrt {b} \sin (c+d x)}{16 (-a)^{5/2} \left (\sqrt {-a}-\sqrt {b} x\right )^2}+\frac {\sqrt {b} \sin (c+d x)}{16 (-a)^{5/2} \left (\sqrt {b} x+\sqrt {-a}\right )^2}-\frac {15 \sqrt {b} \cos \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 {15 \sqrt {b} \cos \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}} \]

[In]

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

[Out]

(d*Cos[c + d*x])/(16*(-a)^(5/2)*(Sqrt[-a] - Sqrt[b]*x)) + (d*Cos[c + d*x])/(16*(-a)^(5/2)*(Sqrt[-a] + Sqrt[b]*
x)) + (d*Cos[c]*CosIntegral[d*x])/a^3 + (7*d*Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] -
d*x])/(16*a^3) + (7*d*Cos[c - (Sqrt[-a]*d)/Sqrt[b]]*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*a^3) - (15*Sq
rt[b]*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sin[c - (Sqrt[-a]*d)/Sqrt[b]])/(16*(-a)^(7/2)) + (d^2*CosIntegra
l[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sin[c - (Sqrt[-a]*d)/Sqrt[b]])/(16*(-a)^(5/2)*Sqrt[b]) + (15*Sqrt[b]*CosIntegral
[(Sqrt[-a]*d)/Sqrt[b] - d*x]*Sin[c + (Sqrt[-a]*d)/Sqrt[b]])/(16*(-a)^(7/2)) - (d^2*CosIntegral[(Sqrt[-a]*d)/Sq
rt[b] - d*x]*Sin[c + (Sqrt[-a]*d)/Sqrt[b]])/(16*(-a)^(5/2)*Sqrt[b]) - Sin[c + d*x]/(a^3*x) - (Sqrt[b]*Sin[c +
d*x])/(16*(-a)^(5/2)*(Sqrt[-a] - Sqrt[b]*x)^2) + (7*Sqrt[b]*Sin[c + d*x])/(16*a^3*(Sqrt[-a] - Sqrt[b]*x)) + (S
qrt[b]*Sin[c + d*x])/(16*(-a)^(5/2)*(Sqrt[-a] + Sqrt[b]*x)^2) - (7*Sqrt[b]*Sin[c + d*x])/(16*a^3*(Sqrt[-a] + S
qrt[b]*x)) - (d*Sin[c]*SinIntegral[d*x])/a^3 - (15*Sqrt[b]*Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]
*d)/Sqrt[b] - d*x])/(16*(-a)^(7/2)) + (d^2*Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*
x])/(16*(-a)^(5/2)*Sqrt[b]) + (7*d*Sin[c + (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*
a^3) - (15*Sqrt[b]*Cos[c - (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*(-a)^(7/2)) + (d
^2*Cos[c - (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*(-a)^(5/2)*Sqrt[b]) - (7*d*Sin[c
- (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*a^3)

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 3414

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])

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*} \text {integral}& = \int \left (\frac {\sin (c+d x)}{a^3 x^2}-\frac {b \sin (c+d x)}{a \left (a+b x^2\right )^3}-\frac {b \sin (c+d x)}{a^2 \left (a+b x^2\right )^2}-\frac {b \sin (c+d x)}{a^3 \left (a+b x^2\right )}\right ) \, dx \\ & = \frac {\int \frac {\sin (c+d x)}{x^2} \, dx}{a^3}-\frac {b \int \frac {\sin (c+d x)}{a+b x^2} \, dx}{a^3}-\frac {b \int \frac {\sin (c+d x)}{\left (a+b x^2\right )^2} \, dx}{a^2}-\frac {b \int \frac {\sin (c+d x)}{\left (a+b x^2\right )^3} \, dx}{a} \\ & = -\frac {\sin (c+d x)}{a^3 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^3}-\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^2}-\frac {b \int \left (-\frac {b^{3/2} \sin (c+d x)}{8 (-a)^{3/2} \left (\sqrt {-a} \sqrt {b}-b x\right )^3}-\frac {3 b \sin (c+d x)}{16 a^2 \left (\sqrt {-a} \sqrt {b}-b x\right )^2}-\frac {b^{3/2} \sin (c+d x)}{8 (-a)^{3/2} \left (\sqrt {-a} \sqrt {b}+b x\right )^3}-\frac {3 b \sin (c+d x)}{16 a^2 \left (\sqrt {-a} \sqrt {b}+b x\right )^2}-\frac {3 b \sin (c+d x)}{8 a^2 \left (-a b-b^2 x^2\right )}\right ) \, dx}{a}+\frac {d \int \frac {\cos (c+d x)}{x} \, dx}{a^3} \\ & = -\frac {\sin (c+d x)}{a^3 x}-\frac {b \int \frac {\sin (c+d x)}{\sqrt {-a}-\sqrt {b} x} \, dx}{2 (-a)^{7/2}}-\frac {b \int \frac {\sin (c+d x)}{\sqrt {-a}+\sqrt {b} x} \, dx}{2 (-a)^{7/2}}+\frac {\left (3 b^2\right ) \int \frac {\sin (c+d x)}{\left (\sqrt {-a} \sqrt {b}-b x\right )^2} \, dx}{16 a^3}+\frac {\left (3 b^2\right ) \int \frac {\sin (c+d x)}{\left (\sqrt {-a} \sqrt {b}+b x\right )^2} \, dx}{16 a^3}+\frac {b^2 \int \frac {\sin (c+d x)}{\left (\sqrt {-a} \sqrt {b}-b x\right )^2} \, dx}{4 a^3}+\frac {b^2 \int \frac {\sin (c+d x)}{\left (\sqrt {-a} \sqrt {b}+b x\right )^2} \, dx}{4 a^3}+\frac {\left (3 b^2\right ) \int \frac {\sin (c+d x)}{-a b-b^2 x^2} \, dx}{8 a^3}+\frac {b^2 \int \frac {\sin (c+d x)}{-a b-b^2 x^2} \, dx}{2 a^3}-\frac {b^{5/2} \int \frac {\sin (c+d x)}{\left (\sqrt {-a} \sqrt {b}-b x\right )^3} \, dx}{8 (-a)^{5/2}}-\frac {b^{5/2} \int \frac {\sin (c+d x)}{\left (\sqrt {-a} \sqrt {b}+b x\right )^3} \, dx}{8 (-a)^{5/2}}+\frac {(d \cos (c)) \int \frac {\cos (d x)}{x} \, dx}{a^3}-\frac {(d \sin (c)) \int \frac {\sin (d x)}{x} \, dx}{a^3} \\ & = \frac {d \cos (c) \operatorname {CosIntegral}(d x)}{a^3}-\frac {\sin (c+d x)}{a^3 x}-\frac {\sqrt {b} \sin (c+d x)}{16 (-a)^{5/2} \left (\sqrt {-a}-\sqrt {b} x\right )^2}+\frac {7 \sqrt {b} \sin (c+d x)}{16 a^3 \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\sqrt {b} \sin (c+d x)}{16 (-a)^{5/2} \left (\sqrt {-a}+\sqrt {b} x\right )^2}-\frac {7 \sqrt {b} \sin (c+d x)}{16 a^3 \left (\sqrt {-a}+\sqrt {b} x\right )}-\frac {d \sin (c) \text {Si}(d x)}{a^3}+\frac {\left (3 b^2\right ) \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}{8 a^3}+\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^3}-\frac {(3 b d) \int \frac {\cos (c+d x)}{\sqrt {-a} \sqrt {b}-b x} \, dx}{16 a^3}+\frac {(3 b d) \int \frac {\cos (c+d x)}{\sqrt {-a} \sqrt {b}+b x} \, dx}{16 a^3}-\frac {(b d) \int \frac {\cos (c+d x)}{\sqrt {-a} \sqrt {b}-b x} \, dx}{4 a^3}+\frac {(b d) \int \frac {\cos (c+d x)}{\sqrt {-a} \sqrt {b}+b x} \, dx}{4 a^3}+\frac {\left (b^{3/2} d\right ) \int \frac {\cos (c+d x)}{\left (\sqrt {-a} \sqrt {b}-b x\right )^2} \, dx}{16 (-a)^{5/2}}-\frac {\left (b^{3/2} d\right ) \int \frac {\cos (c+d x)}{\left (\sqrt {-a} \sqrt {b}+b x\right )^2} \, dx}{16 (-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)^{7/2}}+\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)^{7/2}}-\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)^{7/2}}-\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)^{7/2}} \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 4.04 (sec) , antiderivative size = 593, normalized size of antiderivative = 0.68 \[ \int \frac {\sin (c+d x)}{x^2 \left (a+b x^2\right )^3} \, dx=\frac {8 \sqrt {b} e^{-i c-\frac {\sqrt {a} d}{\sqrt {b}}} \left (-e^{\frac {2 \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (-\frac {\sqrt {a} d}{\sqrt {b}}-i d x\right )+\operatorname {ExpIntegralEi}\left (\frac {\sqrt {a} d}{\sqrt {b}}-i d x\right )\right )+\frac {e^{-i c-\frac {\sqrt {a} d}{\sqrt {b}}} \left (-\left (\left (7 b-7 \sqrt {a} \sqrt {b} d+a d^2\right ) e^{\frac {2 \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (-\frac {\sqrt {a} d}{\sqrt {b}}-i d x\right )\right )+\left (7 b+7 \sqrt {a} \sqrt {b} d+a d^2\right ) \operatorname {ExpIntegralEi}\left (\frac {\sqrt {a} d}{\sqrt {b}}-i d x\right )\right )}{\sqrt {b}}+8 \sqrt {b} e^{i c-\frac {\sqrt {a} d}{\sqrt {b}}} \left (-e^{\frac {2 \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (-\frac {\sqrt {a} d}{\sqrt {b}}+i d x\right )+\operatorname {ExpIntegralEi}\left (\frac {\sqrt {a} d}{\sqrt {b}}+i d x\right )\right )+\frac {e^{i c-\frac {\sqrt {a} d}{\sqrt {b}}} \left (-\left (\left (7 b-7 \sqrt {a} \sqrt {b} d+a d^2\right ) e^{\frac {2 \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (-\frac {\sqrt {a} d}{\sqrt {b}}+i d x\right )\right )+\left (7 b+7 \sqrt {a} \sqrt {b} d+a d^2\right ) \operatorname {ExpIntegralEi}\left (\frac {\sqrt {a} d}{\sqrt {b}}+i d x\right )\right )}{\sqrt {b}}-\frac {4 \sqrt {a} \cos (d x) \left (a d x \left (a+b x^2\right ) \cos (c)+\left (8 a^2+25 a b x^2+15 b^2 x^4\right ) \sin (c)\right )}{x \left (a+b x^2\right )^2}+\frac {4 \sqrt {a} \left (-\left (\left (8 a^2+25 a b x^2+15 b^2 x^4\right ) \cos (c)\right )+a d x \left (a+b x^2\right ) \sin (c)\right ) \sin (d x)}{x \left (a+b x^2\right )^2}+32 \sqrt {a} d (\cos (c) \operatorname {CosIntegral}(d x)-\sin (c) \text {Si}(d x))}{32 a^{7/2}} \]

[In]

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

[Out]

(8*Sqrt[b]*E^((-I)*c - (Sqrt[a]*d)/Sqrt[b])*(-(E^((2*Sqrt[a]*d)/Sqrt[b])*ExpIntegralEi[-((Sqrt[a]*d)/Sqrt[b])
- I*d*x]) + ExpIntegralEi[(Sqrt[a]*d)/Sqrt[b] - I*d*x]) + (E^((-I)*c - (Sqrt[a]*d)/Sqrt[b])*(-((7*b - 7*Sqrt[a
]*Sqrt[b]*d + a*d^2)*E^((2*Sqrt[a]*d)/Sqrt[b])*ExpIntegralEi[-((Sqrt[a]*d)/Sqrt[b]) - I*d*x]) + (7*b + 7*Sqrt[
a]*Sqrt[b]*d + a*d^2)*ExpIntegralEi[(Sqrt[a]*d)/Sqrt[b] - I*d*x]))/Sqrt[b] + 8*Sqrt[b]*E^(I*c - (Sqrt[a]*d)/Sq
rt[b])*(-(E^((2*Sqrt[a]*d)/Sqrt[b])*ExpIntegralEi[-((Sqrt[a]*d)/Sqrt[b]) + I*d*x]) + ExpIntegralEi[(Sqrt[a]*d)
/Sqrt[b] + I*d*x]) + (E^(I*c - (Sqrt[a]*d)/Sqrt[b])*(-((7*b - 7*Sqrt[a]*Sqrt[b]*d + a*d^2)*E^((2*Sqrt[a]*d)/Sq
rt[b])*ExpIntegralEi[-((Sqrt[a]*d)/Sqrt[b]) + I*d*x]) + (7*b + 7*Sqrt[a]*Sqrt[b]*d + a*d^2)*ExpIntegralEi[(Sqr
t[a]*d)/Sqrt[b] + I*d*x]))/Sqrt[b] - (4*Sqrt[a]*Cos[d*x]*(a*d*x*(a + b*x^2)*Cos[c] + (8*a^2 + 25*a*b*x^2 + 15*
b^2*x^4)*Sin[c]))/(x*(a + b*x^2)^2) + (4*Sqrt[a]*(-((8*a^2 + 25*a*b*x^2 + 15*b^2*x^4)*Cos[c]) + a*d*x*(a + b*x
^2)*Sin[c])*Sin[d*x])/(x*(a + b*x^2)^2) + 32*Sqrt[a]*d*(Cos[c]*CosIntegral[d*x] - Sin[c]*SinIntegral[d*x]))/(3
2*a^(7/2))

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.80 (sec) , antiderivative size = 910, normalized size of antiderivative = 1.04

method	result	size
risch	\(\frac {d^{2} {\mathrm e}^{\frac {i c b +d \sqrt {a b}}{b}} \operatorname {Ei}_{1}\left (\frac {i c b +d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right )}{32 a^{2} \sqrt {a b}}-\frac {d^{2} {\mathrm e}^{\frac {i c b -d \sqrt {a b}}{b}} \operatorname {Ei}_{1}\left (-\frac {-i c b +d \sqrt {a b}+b \left (i d x +i c \right )}{b}\right )}{32 a^{2} \sqrt {a b}}-\frac {7 d \,{\mathrm e}^{\frac {i c b +d \sqrt {a b}}{b}} \operatorname {Ei}_{1}\left (\frac {i c b +d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right )}{32 a^{3}}-\frac {7 d \,{\mathrm e}^{\frac {i c b -d \sqrt {a b}}{b}} \operatorname {Ei}_{1}\left (-\frac {-i c b +d \sqrt {a b}+b \left (i d x +i c \right )}{b}\right )}{32 a^{3}}+\frac {15 \,{\mathrm e}^{\frac {i c b +d \sqrt {a b}}{b}} \operatorname {Ei}_{1}\left (\frac {i c b +d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right ) b}{32 a^{3} \sqrt {a b}}-\frac {15 \,{\mathrm e}^{\frac {i c b -d \sqrt {a b}}{b}} \operatorname {Ei}_{1}\left (-\frac {-i c b +d \sqrt {a b}+b \left (i d x +i c \right )}{b}\right ) b}{32 a^{3} \sqrt {a b}}-\frac {d \,\operatorname {Ei}_{1}\left (-i d x \right ) {\mathrm e}^{i c}}{2 a^{3}}-\frac {d^{2} {\mathrm e}^{-\frac {i c b +d \sqrt {a b}}{b}} \operatorname {Ei}_{1}\left (-\frac {i c b +d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right )}{32 a^{2} \sqrt {a b}}+\frac {d^{2} {\mathrm e}^{-\frac {i c b -d \sqrt {a b}}{b}} \operatorname {Ei}_{1}\left (\frac {-i c b +d \sqrt {a b}+b \left (i d x +i c \right )}{b}\right )}{32 a^{2} \sqrt {a b}}-\frac {7 d \,{\mathrm e}^{-\frac {i c b +d \sqrt {a b}}{b}} \operatorname {Ei}_{1}\left (-\frac {i c b +d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right )}{32 a^{3}}-\frac {7 d \,{\mathrm e}^{-\frac {i c b -d \sqrt {a b}}{b}} \operatorname {Ei}_{1}\left (\frac {-i c b +d \sqrt {a b}+b \left (i d x +i c \right )}{b}\right )}{32 a^{3}}-\frac {15 \,{\mathrm e}^{-\frac {i c b +d \sqrt {a b}}{b}} \operatorname {Ei}_{1}\left (-\frac {i c b +d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right ) b}{32 a^{3} \sqrt {a b}}+\frac {15 \,{\mathrm e}^{-\frac {i c b -d \sqrt {a b}}{b}} \operatorname {Ei}_{1}\left (\frac {-i c b +d \sqrt {a b}+b \left (i d x +i c \right )}{b}\right ) b}{32 a^{3} \sqrt {a b}}-\frac {d \,\operatorname {Ei}_{1}\left (i d x \right ) {\mathrm e}^{-i c}}{2 a^{3}}+\frac {d^{2} \left (d^{3} x^{3} b +a \,d^{3} x \right ) \cos \left (d x +c \right )}{8 a^{2} x \left (-b^{2} x^{4} d^{4}-2 a b \,d^{4} x^{2}-a^{2} d^{4}\right )}-\frac {\left (-15 b^{2} x^{4} d^{4}-25 a b \,d^{4} x^{2}-8 a^{2} d^{4}\right ) \sin \left (d x +c \right )}{8 a^{3} x \left (-b^{2} x^{4} d^{4}-2 a b \,d^{4} x^{2}-a^{2} d^{4}\right )}\)	\(910\)
derivativedivides	\(\text {Expression too large to display}\)	\(1363\)
default	\(\text {Expression too large to display}\)	\(1363\)

[In]

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

[Out]

1/32/a^2*d^2/(a*b)^(1/2)*exp((I*c*b+d*(a*b)^(1/2))/b)*Ei(1,(I*c*b+d*(a*b)^(1/2)-b*(I*d*x+I*c))/b)-1/32/a^2*d^2
/(a*b)^(1/2)*exp((I*c*b-d*(a*b)^(1/2))/b)*Ei(1,-(-I*c*b+d*(a*b)^(1/2)+b*(I*d*x+I*c))/b)-7/32*d/a^3*exp((I*c*b+
d*(a*b)^(1/2))/b)*Ei(1,(I*c*b+d*(a*b)^(1/2)-b*(I*d*x+I*c))/b)-7/32*d/a^3*exp((I*c*b-d*(a*b)^(1/2))/b)*Ei(1,-(-
I*c*b+d*(a*b)^(1/2)+b*(I*d*x+I*c))/b)+15/32/a^3/(a*b)^(1/2)*exp((I*c*b+d*(a*b)^(1/2))/b)*Ei(1,(I*c*b+d*(a*b)^(
1/2)-b*(I*d*x+I*c))/b)*b-15/32/a^3/(a*b)^(1/2)*exp((I*c*b-d*(a*b)^(1/2))/b)*Ei(1,-(-I*c*b+d*(a*b)^(1/2)+b*(I*d
*x+I*c))/b)*b-1/2*d/a^3*Ei(1,-I*d*x)*exp(I*c)-1/32/a^2*d^2/(a*b)^(1/2)*exp(-(I*c*b+d*(a*b)^(1/2))/b)*Ei(1,-(I*
c*b+d*(a*b)^(1/2)-b*(I*d*x+I*c))/b)+1/32/a^2*d^2/(a*b)^(1/2)*exp(-(I*c*b-d*(a*b)^(1/2))/b)*Ei(1,(-I*c*b+d*(a*b
)^(1/2)+b*(I*d*x+I*c))/b)-7/32*d/a^3*exp(-(I*c*b+d*(a*b)^(1/2))/b)*Ei(1,-(I*c*b+d*(a*b)^(1/2)-b*(I*d*x+I*c))/b
)-7/32*d/a^3*exp(-(I*c*b-d*(a*b)^(1/2))/b)*Ei(1,(-I*c*b+d*(a*b)^(1/2)+b*(I*d*x+I*c))/b)-15/32/a^3/(a*b)^(1/2)*
exp(-(I*c*b+d*(a*b)^(1/2))/b)*Ei(1,-(I*c*b+d*(a*b)^(1/2)-b*(I*d*x+I*c))/b)*b+15/32/a^3/(a*b)^(1/2)*exp(-(I*c*b
-d*(a*b)^(1/2))/b)*Ei(1,(-I*c*b+d*(a*b)^(1/2)+b*(I*d*x+I*c))/b)*b-1/2*d/a^3*Ei(1,I*d*x)*exp(-I*c)+1/8/a^2*d^2*
(b*d^3*x^3+a*d^3*x)/x/(-b^2*d^4*x^4-2*a*b*d^4*x^2-a^2*d^4)*cos(d*x+c)-1/8*(-15*b^2*d^4*x^4-25*a*b*d^4*x^2-8*a^
2*d^4)/a^3/x/(-b^2*d^4*x^4-2*a*b*d^4*x^2-a^2*d^4)*sin(d*x+c)

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.30 (sec) , antiderivative size = 714, normalized size of antiderivative = 0.82 \[ \int \frac {\sin (c+d x)}{x^2 \left (a+b x^2\right )^3} \, dx=\frac {32 \, {\left (a b^{2} d^{2} x^{5} + 2 \, a^{2} b d^{2} x^{3} + a^{3} d^{2} x\right )} \cos \left (c\right ) \operatorname {Ci}\left (d x\right ) + {\left (7 \, a b^{2} d^{2} x^{5} + 14 \, a^{2} b d^{2} x^{3} + 7 \, a^{3} d^{2} x - {\left ({\left (a b^{2} d^{2} + 15 \, b^{3}\right )} x^{5} + 2 \, {\left (a^{2} b d^{2} + 15 \, a b^{2}\right )} x^{3} + {\left (a^{3} d^{2} + 15 \, a^{2} b\right )} 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 (7 \, a b^{2} d^{2} x^{5} + 14 \, a^{2} b d^{2} x^{3} + 7 \, a^{3} d^{2} x + {\left ({\left (a b^{2} d^{2} + 15 \, b^{3}\right )} x^{5} + 2 \, {\left (a^{2} b d^{2} + 15 \, a b^{2}\right )} x^{3} + {\left (a^{3} d^{2} + 15 \, a^{2} b\right )} 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 (7 \, a b^{2} d^{2} x^{5} + 14 \, a^{2} b d^{2} x^{3} + 7 \, a^{3} d^{2} x - {\left ({\left (a b^{2} d^{2} + 15 \, b^{3}\right )} x^{5} + 2 \, {\left (a^{2} b d^{2} + 15 \, a b^{2}\right )} x^{3} + {\left (a^{3} d^{2} + 15 \, a^{2} b\right )} 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 (7 \, a b^{2} d^{2} x^{5} + 14 \, a^{2} b d^{2} x^{3} + 7 \, a^{3} d^{2} x + {\left ({\left (a b^{2} d^{2} + 15 \, b^{3}\right )} x^{5} + 2 \, {\left (a^{2} b d^{2} + 15 \, a b^{2}\right )} x^{3} + {\left (a^{3} d^{2} + 15 \, a^{2} b\right )} 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 )} - 32 \, {\left (a b^{2} d^{2} x^{5} + 2 \, a^{2} b d^{2} x^{3} + a^{3} d^{2} x\right )} \sin \left (c\right ) \operatorname {Si}\left (d x\right ) - 4 \, {\left (a^{2} b d^{2} x^{3} + a^{3} d^{2} x\right )} \cos \left (d x + c\right ) - 4 \, {\left (15 \, a b^{2} d x^{4} + 25 \, a^{2} b d x^{2} + 8 \, a^{3} d\right )} \sin \left (d x + c\right )}{32 \, {\left (a^{4} b^{2} d x^{5} + 2 \, a^{5} b d x^{3} + a^{6} d x\right )}} \]

[In]

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

[Out]

1/32*(32*(a*b^2*d^2*x^5 + 2*a^2*b*d^2*x^3 + a^3*d^2*x)*cos(c)*cos_integral(d*x) + (7*a*b^2*d^2*x^5 + 14*a^2*b*
d^2*x^3 + 7*a^3*d^2*x - ((a*b^2*d^2 + 15*b^3)*x^5 + 2*(a^2*b*d^2 + 15*a*b^2)*x^3 + (a^3*d^2 + 15*a^2*b)*x)*sqr
t(a*d^2/b))*Ei(I*d*x - sqrt(a*d^2/b))*e^(I*c + sqrt(a*d^2/b)) + (7*a*b^2*d^2*x^5 + 14*a^2*b*d^2*x^3 + 7*a^3*d^
2*x + ((a*b^2*d^2 + 15*b^3)*x^5 + 2*(a^2*b*d^2 + 15*a*b^2)*x^3 + (a^3*d^2 + 15*a^2*b)*x)*sqrt(a*d^2/b))*Ei(I*d
*x + sqrt(a*d^2/b))*e^(I*c - sqrt(a*d^2/b)) + (7*a*b^2*d^2*x^5 + 14*a^2*b*d^2*x^3 + 7*a^3*d^2*x - ((a*b^2*d^2
+ 15*b^3)*x^5 + 2*(a^2*b*d^2 + 15*a*b^2)*x^3 + (a^3*d^2 + 15*a^2*b)*x)*sqrt(a*d^2/b))*Ei(-I*d*x - sqrt(a*d^2/b
))*e^(-I*c + sqrt(a*d^2/b)) + (7*a*b^2*d^2*x^5 + 14*a^2*b*d^2*x^3 + 7*a^3*d^2*x + ((a*b^2*d^2 + 15*b^3)*x^5 +
2*(a^2*b*d^2 + 15*a*b^2)*x^3 + (a^3*d^2 + 15*a^2*b)*x)*sqrt(a*d^2/b))*Ei(-I*d*x + sqrt(a*d^2/b))*e^(-I*c - sqr
t(a*d^2/b)) - 32*(a*b^2*d^2*x^5 + 2*a^2*b*d^2*x^3 + a^3*d^2*x)*sin(c)*sin_integral(d*x) - 4*(a^2*b*d^2*x^3 + a
^3*d^2*x)*cos(d*x + c) - 4*(15*a*b^2*d*x^4 + 25*a^2*b*d*x^2 + 8*a^3*d)*sin(d*x + c))/(a^4*b^2*d*x^5 + 2*a^5*b*
d*x^3 + a^6*d*x)

Sympy [F(-1)]

Timed out. \[ \int \frac {\sin (c+d x)}{x^2 \left (a+b x^2\right )^3} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\sin (c+d x)}{x^2 \left (a+b x^2\right )^3} \, dx=\int \frac {\sin \left (c+d\,x\right )}{x^2\,{\left (b\,x^2+a\right )}^3} \,d x \]

[In]

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

[Out]

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