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

Optimal. Leaf size=875 \[ \text{result too large to display} \]

[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)

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Rubi [A]  time = 2.8462, antiderivative size = 875, normalized size of antiderivative = 1., number of steps used = 60, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316, Rules used = {3345, 3297, 3303, 3299, 3302, 3333} \[ \frac{\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)^{5/2} \sqrt{b}}-\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)^{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) \text{CosIntegral}(d x) d}{a^3}+\frac{7 \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^3}+\frac{7 \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^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} \text{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} \text{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}} \]

Antiderivative was successfully verified.

[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 3345

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

Rule 3297

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

Rule 3303

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

Rule 3299

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

Rule 3302

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

Rule 3333

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

Rubi steps

\begin{align*} \int \frac{\sin (c+d x)}{x^2 \left (a+b x^2\right )^3} \, dx &=\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) \text{Ci}(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}}\\ &=\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) \text{Ci}(d x)}{a^3}-\frac{\sqrt{b} \text{Ci}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right ) \sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{2 (-a)^{7/2}}+\frac{\sqrt{b} \text{Ci}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right ) \sin \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{2 (-a)^{7/2}}-\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{\sqrt{b} \cos \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 (-a)^{7/2}}-\frac{\sqrt{b} \cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{2 (-a)^{7/2}}-\frac{(3 b) \int \frac{\sin (c+d x)}{\sqrt{-a}-\sqrt{b} x} \, dx}{16 (-a)^{7/2}}-\frac{(3 b) \int \frac{\sin (c+d x)}{\sqrt{-a}+\sqrt{b} x} \, dx}{16 (-a)^{7/2}}-\frac{b \int \frac{\sin (c+d x)}{\sqrt{-a}-\sqrt{b} x} \, dx}{4 (-a)^{7/2}}-\frac{b \int \frac{\sin (c+d x)}{\sqrt{-a}+\sqrt{b} x} \, dx}{4 (-a)^{7/2}}+\frac{\left (\sqrt{b} d^2\right ) \int \frac{\sin (c+d x)}{\sqrt{-a} \sqrt{b}-b x} \, dx}{16 (-a)^{5/2}}+\frac{\left (\sqrt{b} d^2\right ) \int \frac{\sin (c+d x)}{\sqrt{-a} \sqrt{b}+b x} \, dx}{16 (-a)^{5/2}}+\frac{\left (3 b d \cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right )\right ) \int \frac{\cos \left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{\sqrt{-a} \sqrt{b}+b x} \, dx}{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}+b x} \, dx}{4 a^3}-\frac{\left (3 b d \cos \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right )\right ) \int \frac{\cos \left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{\sqrt{-a} \sqrt{b}-b x} \, dx}{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}-b x} \, dx}{4 a^3}-\frac{\left (3 b d \sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right )\right ) \int \frac{\sin \left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{\sqrt{-a} \sqrt{b}+b x} \, dx}{16 a^3}-\frac{\left (b d \sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right )\right ) \int \frac{\sin \left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{\sqrt{-a} \sqrt{b}+b x} \, dx}{4 a^3}-\frac{\left (3 b d \sin \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right )\right ) \int \frac{\sin \left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{\sqrt{-a} \sqrt{b}-b x} \, dx}{16 a^3}-\frac{\left (b d \sin \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right )\right ) \int \frac{\sin \left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{\sqrt{-a} \sqrt{b}-b x} \, dx}{4 a^3}\\ &=\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) \text{Ci}(d x)}{a^3}+\frac{7 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^3}+\frac{7 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^3}-\frac{\sqrt{b} \text{Ci}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right ) \sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{2 (-a)^{7/2}}+\frac{\sqrt{b} \text{Ci}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right ) \sin \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{2 (-a)^{7/2}}-\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{\sqrt{b} \cos \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 (-a)^{7/2}}+\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{\sqrt{b} \cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{2 (-a)^{7/2}}-\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{\left (3 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}{16 (-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}{4 (-a)^{7/2}}+\frac{\left (\sqrt{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)^{5/2}}+\frac{\left (3 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}{16 (-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}{4 (-a)^{7/2}}-\frac{\left (\sqrt{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)^{5/2}}-\frac{\left (3 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}{16 (-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}{4 (-a)^{7/2}}+\frac{\left (\sqrt{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)^{5/2}}-\frac{\left (3 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}{16 (-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}{4 (-a)^{7/2}}+\frac{\left (\sqrt{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)^{5/2}}\\ &=\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) \text{Ci}(d x)}{a^3}+\frac{7 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^3}+\frac{7 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^3}-\frac{15 \sqrt{b} \text{Ci}\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 \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{15 \sqrt{b} \text{Ci}\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 \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{\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}\\ \end{align*}

Mathematica [C]  time = 2.86386, size = 1673, normalized size = 1.91 \[ \text{result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((-I/16)*((-2*I)*a^(5/2)*Sqrt[b]*d*x*Cos[c + d*x] - (2*I)*a^(3/2)*b^(3/2)*d*x^3*Cos[c + d*x] + (16*I)*Sqrt[a]*
Sqrt[b]*d*x*(a + b*x^2)^2*Cos[c]*CosIntegral[d*x] + (7*I)*a^(5/2)*Sqrt[b]*d*x*Cos[c - (I*Sqrt[a]*d)/Sqrt[b]]*C
osIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] + (14*I)*a^(3/2)*b^(3/2)*d*x^3*Cos[c - (I*Sqrt[a]*d)/Sqrt[b]]*CosInteg
ral[d*((I*Sqrt[a])/Sqrt[b] + x)] + (7*I)*Sqrt[a]*b^(5/2)*d*x^5*Cos[c - (I*Sqrt[a]*d)/Sqrt[b]]*CosIntegral[d*((
I*Sqrt[a])/Sqrt[b] + x)] + 15*a^2*b*x*CosIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)]*Sin[c - (I*Sqrt[a]*d)/Sqrt[b]]
+ a^3*d^2*x*CosIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)]*Sin[c - (I*Sqrt[a]*d)/Sqrt[b]] + 30*a*b^2*x^3*CosIntegral
[d*((I*Sqrt[a])/Sqrt[b] + x)]*Sin[c - (I*Sqrt[a]*d)/Sqrt[b]] + 2*a^2*b*d^2*x^3*CosIntegral[d*((I*Sqrt[a])/Sqrt
[b] + x)]*Sin[c - (I*Sqrt[a]*d)/Sqrt[b]] + 15*b^3*x^5*CosIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)]*Sin[c - (I*Sqrt
[a]*d)/Sqrt[b]] + a*b^2*d^2*x^5*CosIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)]*Sin[c - (I*Sqrt[a]*d)/Sqrt[b]] - x*(a
+ b*x^2)^2*CosIntegral[d*(((-I)*Sqrt[a])/Sqrt[b] + x)]*((-7*I)*Sqrt[a]*Sqrt[b]*d*Cos[c + (I*Sqrt[a]*d)/Sqrt[b
]] + (15*b + a*d^2)*Sin[c + (I*Sqrt[a]*d)/Sqrt[b]]) - (16*I)*a^(5/2)*Sqrt[b]*Sin[c + d*x] - (50*I)*a^(3/2)*b^(
3/2)*x^2*Sin[c + d*x] - (30*I)*Sqrt[a]*b^(5/2)*x^4*Sin[c + d*x] - (16*I)*a^(5/2)*Sqrt[b]*d*x*Sin[c]*SinIntegra
l[d*x] - (32*I)*a^(3/2)*b^(3/2)*d*x^3*Sin[c]*SinIntegral[d*x] - (16*I)*Sqrt[a]*b^(5/2)*d*x^5*Sin[c]*SinIntegra
l[d*x] + 15*a^2*b*x*Cos[c - (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] + a^3*d^2*x*Cos[c
- (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] + 30*a*b^2*x^3*Cos[c - (I*Sqrt[a]*d)/Sqrt[b]
]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] + 2*a^2*b*d^2*x^3*Cos[c - (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*
Sqrt[a])/Sqrt[b] + x)] + 15*b^3*x^5*Cos[c - (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] +
a*b^2*d^2*x^5*Cos[c - (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] - (7*I)*a^(5/2)*Sqrt[b]*
d*x*Sin[c - (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] - (14*I)*a^(3/2)*b^(3/2)*d*x^3*Sin
[c - (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] - (7*I)*Sqrt[a]*b^(5/2)*d*x^5*Sin[c - (I*
Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] + 15*a^2*b*x*Cos[c + (I*Sqrt[a]*d)/Sqrt[b]]*SinIn
tegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x] + a^3*d^2*x*Cos[c + (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[
b] - d*x] + 30*a*b^2*x^3*Cos[c + (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x] + 2*a^2*b*d^2
*x^3*Cos[c + (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x] + 15*b^3*x^5*Cos[c + (I*Sqrt[a]*d
)/Sqrt[b]]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x] + a*b^2*d^2*x^5*Cos[c + (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral
[(I*Sqrt[a]*d)/Sqrt[b] - d*x] + (7*I)*a^(5/2)*Sqrt[b]*d*x*Sin[c + (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(I*Sqrt[a
]*d)/Sqrt[b] - d*x] + (14*I)*a^(3/2)*b^(3/2)*d*x^3*Sin[c + (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(I*Sqrt[a]*d)/Sq
rt[b] - d*x] + (7*I)*Sqrt[a]*b^(5/2)*d*x^5*Sin[c + (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] -
d*x]))/(a^(7/2)*Sqrt[b]*x*(a + b*x^2)^2)

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Maple [B]  time = 0.051, size = 1375, normalized size = 1.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [C]  time = 2.21259, size = 1493, normalized size = 1.71 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/32*(16*(a*b^2*d^2*x^5 + 2*a^2*b*d^2*x^3 + a^3*d^2*x)*Ei(I*d*x)*e^(I*c) + 16*(a*b^2*d^2*x^5 + 2*a^2*b*d^2*x^3
+ a^3*d^2*x)*Ei(-I*d*x)*e^(-I*c) + (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 + 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)) - 4*(a^2*b*d^2*x^3 + a^3*d^2*x)*co
s(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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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