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

Optimal. Leaf size=856 \[ \text{result too large to display} \]

[Out]

(d*Cos[c + d*x])/(16*(-a)^(3/2)*b*(Sqrt[-a] - Sqrt[b]*x)) + (d*Cos[c + d*x])/(16*(-a)^(3/2)*b*(Sqrt[-a] + Sqrt
[b]*x)) - (3*d*Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*a^2*b) - (3*d*Cos[c
- (Sqrt[-a]*d)/Sqrt[b]]*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*a^2*b) - (3*CosIntegral[(Sqrt[-a]*d)/Sqrt
[b] + d*x]*Sin[c - (Sqrt[-a]*d)/Sqrt[b]])/(16*(-a)^(5/2)*Sqrt[b]) + (d^2*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*
x]*Sin[c - (Sqrt[-a]*d)/Sqrt[b]])/(16*(-a)^(3/2)*b^(3/2)) + (3*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x]*Sin[c +
(Sqrt[-a]*d)/Sqrt[b]])/(16*(-a)^(5/2)*Sqrt[b]) - (d^2*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x]*Sin[c + (Sqrt[-
a]*d)/Sqrt[b]])/(16*(-a)^(3/2)*b^(3/2)) - Sin[c + d*x]/(16*(-a)^(3/2)*Sqrt[b]*(Sqrt[-a] - Sqrt[b]*x)^2) - (3*S
in[c + d*x])/(16*a^2*Sqrt[b]*(Sqrt[-a] - Sqrt[b]*x)) + Sin[c + d*x]/(16*(-a)^(3/2)*Sqrt[b]*(Sqrt[-a] + Sqrt[b]
*x)^2) + (3*Sin[c + d*x])/(16*a^2*Sqrt[b]*(Sqrt[-a] + Sqrt[b]*x)) - (3*Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*SinIntegr
al[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*(-a)^(5/2)*Sqrt[b]) + (d^2*Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt
[-a]*d)/Sqrt[b] - d*x])/(16*(-a)^(3/2)*b^(3/2)) - (3*d*Sin[c + (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/
Sqrt[b] - d*x])/(16*a^2*b) - (3*Cos[c - (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*(-a
)^(5/2)*Sqrt[b]) + (d^2*Cos[c - (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*(-a)^(3/2)*
b^(3/2)) + (3*d*Sin[c - (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*a^2*b)

________________________________________________________________________________________

Rubi [A]  time = 1.18125, antiderivative size = 856, normalized size of antiderivative = 1., number of steps used = 28, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312, Rules used = {3333, 3297, 3303, 3299, 3302} \[ \frac{\text{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)^{3/2} b^{3/2}}-\frac{\text{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)^{3/2} b^{3/2}}+\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)^{3/2} b^{3/2}}+\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)^{3/2} b^{3/2}}+\frac{\cos (c+d x) d}{16 (-a)^{3/2} b \left (\sqrt{-a}-\sqrt{b} x\right )}+\frac{\cos (c+d x) d}{16 (-a)^{3/2} b \left (\sqrt{b} x+\sqrt{-a}\right )}-\frac{3 \cos \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right ) d}{16 a^2 b}-\frac{3 \cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right ) d}{16 a^2 b}-\frac{3 \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^2 b}+\frac{3 \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^2 b}-\frac{3 \text{CosIntegral}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{16 (-a)^{5/2} \sqrt{b}}+\frac{3 \text{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{3 \sin (c+d x)}{16 a^2 \sqrt{b} \left (\sqrt{-a}-\sqrt{b} x\right )}+\frac{3 \sin (c+d x)}{16 a^2 \sqrt{b} \left (\sqrt{b} x+\sqrt{-a}\right )}-\frac{\sin (c+d x)}{16 (-a)^{3/2} \sqrt{b} \left (\sqrt{-a}-\sqrt{b} x\right )^2}+\frac{\sin (c+d x)}{16 (-a)^{3/2} \sqrt{b} \left (\sqrt{b} x+\sqrt{-a}\right )^2}-\frac{3 \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{3 \cos \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}} \]

Antiderivative was successfully verified.

[In]

Int[Sin[c + d*x]/(a + b*x^2)^3,x]

[Out]

(d*Cos[c + d*x])/(16*(-a)^(3/2)*b*(Sqrt[-a] - Sqrt[b]*x)) + (d*Cos[c + d*x])/(16*(-a)^(3/2)*b*(Sqrt[-a] + Sqrt
[b]*x)) - (3*d*Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*a^2*b) - (3*d*Cos[c
- (Sqrt[-a]*d)/Sqrt[b]]*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*a^2*b) - (3*CosIntegral[(Sqrt[-a]*d)/Sqrt
[b] + d*x]*Sin[c - (Sqrt[-a]*d)/Sqrt[b]])/(16*(-a)^(5/2)*Sqrt[b]) + (d^2*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*
x]*Sin[c - (Sqrt[-a]*d)/Sqrt[b]])/(16*(-a)^(3/2)*b^(3/2)) + (3*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x]*Sin[c +
(Sqrt[-a]*d)/Sqrt[b]])/(16*(-a)^(5/2)*Sqrt[b]) - (d^2*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x]*Sin[c + (Sqrt[-
a]*d)/Sqrt[b]])/(16*(-a)^(3/2)*b^(3/2)) - Sin[c + d*x]/(16*(-a)^(3/2)*Sqrt[b]*(Sqrt[-a] - Sqrt[b]*x)^2) - (3*S
in[c + d*x])/(16*a^2*Sqrt[b]*(Sqrt[-a] - Sqrt[b]*x)) + Sin[c + d*x]/(16*(-a)^(3/2)*Sqrt[b]*(Sqrt[-a] + Sqrt[b]
*x)^2) + (3*Sin[c + d*x])/(16*a^2*Sqrt[b]*(Sqrt[-a] + Sqrt[b]*x)) - (3*Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*SinIntegr
al[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*(-a)^(5/2)*Sqrt[b]) + (d^2*Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt
[-a]*d)/Sqrt[b] - d*x])/(16*(-a)^(3/2)*b^(3/2)) - (3*d*Sin[c + (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/
Sqrt[b] - d*x])/(16*a^2*b) - (3*Cos[c - (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*(-a
)^(5/2)*Sqrt[b]) + (d^2*Cos[c - (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*(-a)^(3/2)*
b^(3/2)) + (3*d*Sin[c - (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*a^2*b)

Rule 3333

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

Rule 3297

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[((c + d*x)^(m + 1)*Sin[e + f*x])/(d*(
m + 1)), x] - Dist[f/(d*(m + 1)), Int[(c + d*x)^(m + 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && LtQ[
m, -1]

Rule 3303

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[(c*f)/d + f*x]
/(c + d*x), x], x] + Dist[Sin[(d*e - c*f)/d], Int[Cos[(c*f)/d + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f},
x] && NeQ[d*e - c*f, 0]

Rule 3299

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinIntegral[e + f*x]/d, x] /; FreeQ[{c, d,
e, f}, x] && EqQ[d*e - c*f, 0]

Rule 3302

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosIntegral[e - Pi/2 + f*x]/d, x] /; FreeQ
[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - c*f, 0]

Rubi steps

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

Mathematica [C]  time = 2.42944, size = 932, normalized size = 1.09 \[ \frac{\frac{6 b^{5/2} \cos (d x) \sin (c) x^3}{\left (b x^2+a\right )^2}+\frac{6 b^{5/2} \cos (c) \sin (d x) x^3}{\left (b x^2+a\right )^2}+\frac{2 a b^{3/2} d \cos (c) \cos (d x) x^2}{\left (b x^2+a\right )^2}-\frac{2 a b^{3/2} d \sin (c) \sin (d x) x^2}{\left (b x^2+a\right )^2}+\frac{10 a b^{3/2} \cos (d x) \sin (c) x}{\left (b x^2+a\right )^2}+\frac{10 a b^{3/2} \cos (c) \sin (d x) x}{\left (b x^2+a\right )^2}+\frac{2 a^2 \sqrt{b} d \cos (c) \cos (d x)}{\left (b x^2+a\right )^2}+\frac{i \text{CosIntegral}\left (d \left (x+\frac{i \sqrt{a}}{\sqrt{b}}\right )\right ) \left (3 i \sqrt{a} \sqrt{b} d \cos \left (c-\frac{i \sqrt{a} d}{\sqrt{b}}\right )+\left (a d^2+3 b\right ) \sin \left (c-\frac{i \sqrt{a} d}{\sqrt{b}}\right )\right )}{\sqrt{a}}-\frac{i \text{CosIntegral}\left (d \left (x-\frac{i \sqrt{a}}{\sqrt{b}}\right )\right ) \left (\left (a d^2+3 b\right ) \sin \left (c+\frac{i \sqrt{a} d}{\sqrt{b}}\right )-3 i \sqrt{a} \sqrt{b} d \cos \left (c+\frac{i \sqrt{a} d}{\sqrt{b}}\right )\right )}{\sqrt{a}}-\frac{2 a^2 \sqrt{b} d \sin (c) \sin (d x)}{\left (b x^2+a\right )^2}+i \sqrt{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 )+\frac{3 i b \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 )}{\sqrt{a}}+3 \sqrt{b} 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 )-3 i \sqrt{b} 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{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 )-\frac{3 b \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 )}{\sqrt{a}}+i \sqrt{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 )+\frac{3 i b \cos (c) \cosh \left (\frac{\sqrt{a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right )}{\sqrt{a}}-3 \sqrt{b} 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 )-3 i \sqrt{b} 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{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 )+\frac{3 b \sin (c) \sinh \left (\frac{\sqrt{a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right )}{\sqrt{a}}}{16 a^2 b^{3/2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((2*a^2*Sqrt[b]*d*Cos[c]*Cos[d*x])/(a + b*x^2)^2 + (2*a*b^(3/2)*d*x^2*Cos[c]*Cos[d*x])/(a + b*x^2)^2 + (10*a*b
^(3/2)*x*Cos[d*x]*Sin[c])/(a + b*x^2)^2 + (6*b^(5/2)*x^3*Cos[d*x]*Sin[c])/(a + b*x^2)^2 + (I*CosIntegral[d*((I
*Sqrt[a])/Sqrt[b] + x)]*((3*I)*Sqrt[a]*Sqrt[b]*d*Cos[c - (I*Sqrt[a]*d)/Sqrt[b]] + (3*b + a*d^2)*Sin[c - (I*Sqr
t[a]*d)/Sqrt[b]]))/Sqrt[a] - (I*CosIntegral[d*(((-I)*Sqrt[a])/Sqrt[b] + x)]*((-3*I)*Sqrt[a]*Sqrt[b]*d*Cos[c +
(I*Sqrt[a]*d)/Sqrt[b]] + (3*b + a*d^2)*Sin[c + (I*Sqrt[a]*d)/Sqrt[b]]))/Sqrt[a] + (10*a*b^(3/2)*x*Cos[c]*Sin[d
*x])/(a + b*x^2)^2 + (6*b^(5/2)*x^3*Cos[c]*Sin[d*x])/(a + b*x^2)^2 - (2*a^2*Sqrt[b]*d*Sin[c]*Sin[d*x])/(a + b*
x^2)^2 - (2*a*b^(3/2)*d*x^2*Sin[c]*Sin[d*x])/(a + b*x^2)^2 + ((3*I)*b*Cos[c]*Cosh[(Sqrt[a]*d)/Sqrt[b]]*SinInte
gral[d*((I*Sqrt[a])/Sqrt[b] + x)])/Sqrt[a] + I*Sqrt[a]*d^2*Cos[c]*Cosh[(Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*
Sqrt[a])/Sqrt[b] + x)] + 3*Sqrt[b]*d*Cosh[(Sqrt[a]*d)/Sqrt[b]]*Sin[c]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)]
- (3*I)*Sqrt[b]*d*Cos[c]*Sinh[(Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] - (3*b*Sin[c]*Sin
h[(Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)])/Sqrt[a] - Sqrt[a]*d^2*Sin[c]*Sinh[(Sqrt[a]*d)
/Sqrt[b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] + ((3*I)*b*Cos[c]*Cosh[(Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(I*
Sqrt[a]*d)/Sqrt[b] - d*x])/Sqrt[a] + I*Sqrt[a]*d^2*Cos[c]*Cosh[(Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(I*Sqrt[a]*d)/
Sqrt[b] - d*x] - 3*Sqrt[b]*d*Cosh[(Sqrt[a]*d)/Sqrt[b]]*Sin[c]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x] - (3*I)
*Sqrt[b]*d*Cos[c]*Sinh[(Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x] + (3*b*Sin[c]*Sinh[(Sqrt[
a]*d)/Sqrt[b]]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x])/Sqrt[a] + Sqrt[a]*d^2*Sin[c]*Sinh[(Sqrt[a]*d)/Sqrt[b]
]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x])/(16*a^2*b^(3/2))

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Maple [A]  time = 0.032, size = 602, normalized size = 0.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d^5*(1/8*sin(d*x+c)*(3*(d*x+c)^3*b-9*c*(d*x+c)^2*b+5*(d*x+c)*a*d^2+9*(d*x+c)*b*c^2-5*a*c*d^2-3*c^3*b)/a^2/d^4/
((d*x+c)^2*b-2*(d*x+c)*b*c+a*d^2+c^2*b)^2+1/8*cos(d*x+c)/a/b/d^2/((d*x+c)^2*b-2*(d*x+c)*b*c+a*d^2+c^2*b)+1/16*
(a*d^2+3*b)/a^2/b^2/d^4/((d*(-a*b)^(1/2)+c*b)/b-c)*(Si(d*x+c-(d*(-a*b)^(1/2)+c*b)/b)*cos((d*(-a*b)^(1/2)+c*b)/
b)+Ci(d*x+c-(d*(-a*b)^(1/2)+c*b)/b)*sin((d*(-a*b)^(1/2)+c*b)/b))+1/16*(a*d^2+3*b)/a^2/b^2/d^4/(-(d*(-a*b)^(1/2
)-c*b)/b-c)*(Si(d*x+c+(d*(-a*b)^(1/2)-c*b)/b)*cos((d*(-a*b)^(1/2)-c*b)/b)-Ci(d*x+c+(d*(-a*b)^(1/2)-c*b)/b)*sin
((d*(-a*b)^(1/2)-c*b)/b))-3/16/a^2/b/d^4*(-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))-3/16/a^2/b/d^4*(Si(d*x+c+(d*(-a*b)^(1/2)-c*b)/b)*sin((
d*(-a*b)^(1/2)-c*b)/b)+Ci(d*x+c+(d*(-a*b)^(1/2)-c*b)/b)*cos((d*(-a*b)^(1/2)-c*b)/b)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin \left (d x + c\right )}{{\left (b x^{2} + a\right )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [C]  time = 2.20541, size = 1233, normalized size = 1.44 \begin{align*} -\frac{{\left (3 \, a b^{2} d^{2} x^{4} + 6 \, a^{2} b d^{2} x^{2} + 3 \, a^{3} d^{2} -{\left (a^{3} d^{2} +{\left (a b^{2} d^{2} + 3 \, b^{3}\right )} x^{4} + 3 \, a^{2} b + 2 \,{\left (a^{2} b d^{2} + 3 \, a b^{2}\right )} 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 (3 \, a b^{2} d^{2} x^{4} + 6 \, a^{2} b d^{2} x^{2} + 3 \, a^{3} d^{2} +{\left (a^{3} d^{2} +{\left (a b^{2} d^{2} + 3 \, b^{3}\right )} x^{4} + 3 \, a^{2} b + 2 \,{\left (a^{2} b d^{2} + 3 \, a b^{2}\right )} 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 (3 \, a b^{2} d^{2} x^{4} + 6 \, a^{2} b d^{2} x^{2} + 3 \, a^{3} d^{2} -{\left (a^{3} d^{2} +{\left (a b^{2} d^{2} + 3 \, b^{3}\right )} x^{4} + 3 \, a^{2} b + 2 \,{\left (a^{2} b d^{2} + 3 \, a b^{2}\right )} 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 (3 \, a b^{2} d^{2} x^{4} + 6 \, a^{2} b d^{2} x^{2} + 3 \, a^{3} d^{2} +{\left (a^{3} d^{2} +{\left (a b^{2} d^{2} + 3 \, b^{3}\right )} x^{4} + 3 \, a^{2} b + 2 \,{\left (a^{2} b d^{2} + 3 \, a b^{2}\right )} 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 (a^{2} b d^{2} x^{2} + a^{3} d^{2}\right )} \cos \left (d x + c\right ) - 4 \,{\left (3 \, a b^{2} d x^{3} + 5 \, a^{2} b d x\right )} \sin \left (d x + c\right )}{32 \,{\left (a^{3} b^{3} d x^{4} + 2 \, a^{4} b^{2} d x^{2} + a^{5} b d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/32*((3*a*b^2*d^2*x^4 + 6*a^2*b*d^2*x^2 + 3*a^3*d^2 - (a^3*d^2 + (a*b^2*d^2 + 3*b^3)*x^4 + 3*a^2*b + 2*(a^2*
b*d^2 + 3*a*b^2)*x^2)*sqrt(a*d^2/b))*Ei(I*d*x - sqrt(a*d^2/b))*e^(I*c + sqrt(a*d^2/b)) + (3*a*b^2*d^2*x^4 + 6*
a^2*b*d^2*x^2 + 3*a^3*d^2 + (a^3*d^2 + (a*b^2*d^2 + 3*b^3)*x^4 + 3*a^2*b + 2*(a^2*b*d^2 + 3*a*b^2)*x^2)*sqrt(a
*d^2/b))*Ei(I*d*x + sqrt(a*d^2/b))*e^(I*c - sqrt(a*d^2/b)) + (3*a*b^2*d^2*x^4 + 6*a^2*b*d^2*x^2 + 3*a^3*d^2 -
(a^3*d^2 + (a*b^2*d^2 + 3*b^3)*x^4 + 3*a^2*b + 2*(a^2*b*d^2 + 3*a*b^2)*x^2)*sqrt(a*d^2/b))*Ei(-I*d*x - sqrt(a*
d^2/b))*e^(-I*c + sqrt(a*d^2/b)) + (3*a*b^2*d^2*x^4 + 6*a^2*b*d^2*x^2 + 3*a^3*d^2 + (a^3*d^2 + (a*b^2*d^2 + 3*
b^3)*x^4 + 3*a^2*b + 2*(a^2*b*d^2 + 3*a*b^2)*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*(a^2*b*d^2*x^2 + a^3*d^2)*cos(d*x + c) - 4*(3*a*b^2*d*x^3 + 5*a^2*b*d*x)*sin(d*x + c))/(a^3*b^3*d*x
^4 + 2*a^4*b^2*d*x^2 + a^5*b*d)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin \left (d x + c\right )}{{\left (b x^{2} + a\right )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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