3.1.70 \(\int \frac {\sin (c+d x)}{x (a+b x^2)^2} \, dx\) [70]

Optimal. Leaf size=435 \[ \frac {d \cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Ci}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 (-a)^{3/2} \sqrt {b}}-\frac {d \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Ci}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{4 (-a)^{3/2} \sqrt {b}}+\frac {\text {Ci}(d x) \sin (c)}{a^2}-\frac {\text {Ci}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right ) \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 a^2}-\frac {\text {Ci}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 a^2}+\frac {\sin (c+d x)}{2 a \left (a+b x^2\right )}+\frac {\cos (c) \text {Si}(d x)}{a^2}+\frac {\cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 a^2}+\frac {d \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 (-a)^{3/2} \sqrt {b}}-\frac {\cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{2 a^2}+\frac {d \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{4 (-a)^{3/2} \sqrt {b}} \]

[Out]

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

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Rubi [A]
time = 0.63, antiderivative size = 435, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {3426, 3384, 3380, 3383, 3422, 3415} \begin {gather*} -\frac {\sin \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 a^2}-\frac {\sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{2 a^2}+\frac {\cos \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 a^2}-\frac {\cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 a^2}+\frac {\sin (c) \text {CosIntegral}(d x)}{a^2}+\frac {\cos (c) \text {Si}(d x)}{a^2}+\frac {d \cos \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 (-a)^{3/2} \sqrt {b}}-\frac {d \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{4 (-a)^{3/2} \sqrt {b}}+\frac {d \sin \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 (-a)^{3/2} \sqrt {b}}+\frac {d \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 (-a)^{3/2} \sqrt {b}}+\frac {\sin (c+d x)}{2 a \left (a+b x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sin[c + d*x]/(x*(a + b*x^2)^2),x]

[Out]

(d*Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(4*(-a)^(3/2)*Sqrt[b]) - (d*Cos[c -
(Sqrt[-a]*d)/Sqrt[b]]*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(4*(-a)^(3/2)*Sqrt[b]) + (CosIntegral[d*x]*Sin[
c])/a^2 - (CosIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sin[c - (Sqrt[-a]*d)/Sqrt[b]])/(2*a^2) - (CosIntegral[(Sqrt
[-a]*d)/Sqrt[b] - d*x]*Sin[c + (Sqrt[-a]*d)/Sqrt[b]])/(2*a^2) + Sin[c + d*x]/(2*a*(a + b*x^2)) + (Cos[c]*SinIn
tegral[d*x])/a^2 + (Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(2*a^2) + (d*Sin[c
+ (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(4*(-a)^(3/2)*Sqrt[b]) - (Cos[c - (Sqrt[-a]*d
)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(2*a^2) + (d*Sin[c - (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sq
rt[-a]*d)/Sqrt[b] + d*x])/(4*(-a)^(3/2)*Sqrt[b])

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

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Mathematica [C] Result contains complex when optimal does not.
time = 0.53, size = 650, normalized size = 1.49 \begin {gather*} \frac {4 a \sqrt {b} \text {Ci}(d x) \sin (c)+4 b^{3/2} x^2 \text {Ci}(d x) \sin (c)-i \left (a+b x^2\right ) \text {Ci}\left (d \left (\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right ) \left (\sqrt {a} d \cos \left (c-\frac {i \sqrt {a} d}{\sqrt {b}}\right )-2 i \sqrt {b} \sin \left (c-\frac {i \sqrt {a} d}{\sqrt {b}}\right )\right )+i \left (a+b x^2\right ) \text {Ci}\left (d \left (-\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right ) \left (\sqrt {a} d \cos \left (c+\frac {i \sqrt {a} d}{\sqrt {b}}\right )+2 i \sqrt {b} \sin \left (c+\frac {i \sqrt {a} d}{\sqrt {b}}\right )\right )+2 a \sqrt {b} \sin (c+d x)+4 a \sqrt {b} \cos (c) \text {Si}(d x)+4 b^{3/2} x^2 \cos (c) \text {Si}(d x)-2 a \sqrt {b} \cos \left (c-\frac {i \sqrt {a} d}{\sqrt {b}}\right ) \text {Si}\left (d \left (\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )-2 b^{3/2} x^2 \cos \left (c-\frac {i \sqrt {a} d}{\sqrt {b}}\right ) \text {Si}\left (d \left (\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )+i a^{3/2} d \sin \left (c-\frac {i \sqrt {a} d}{\sqrt {b}}\right ) \text {Si}\left (d \left (\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )+i \sqrt {a} b d x^2 \sin \left (c-\frac {i \sqrt {a} d}{\sqrt {b}}\right ) \text {Si}\left (d \left (\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )+2 a \sqrt {b} \cos \left (c+\frac {i \sqrt {a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )+2 b^{3/2} x^2 \cos \left (c+\frac {i \sqrt {a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )+i a^{3/2} d \sin \left (c+\frac {i \sqrt {a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )+i \sqrt {a} b d x^2 \sin \left (c+\frac {i \sqrt {a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )}{4 a^2 \sqrt {b} \left (a+b x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*a*Sqrt[b]*CosIntegral[d*x]*Sin[c] + 4*b^(3/2)*x^2*CosIntegral[d*x]*Sin[c] - I*(a + b*x^2)*CosIntegral[d*((I
*Sqrt[a])/Sqrt[b] + x)]*(Sqrt[a]*d*Cos[c - (I*Sqrt[a]*d)/Sqrt[b]] - (2*I)*Sqrt[b]*Sin[c - (I*Sqrt[a]*d)/Sqrt[b
]]) + I*(a + b*x^2)*CosIntegral[d*(((-I)*Sqrt[a])/Sqrt[b] + x)]*(Sqrt[a]*d*Cos[c + (I*Sqrt[a]*d)/Sqrt[b]] + (2
*I)*Sqrt[b]*Sin[c + (I*Sqrt[a]*d)/Sqrt[b]]) + 2*a*Sqrt[b]*Sin[c + d*x] + 4*a*Sqrt[b]*Cos[c]*SinIntegral[d*x] +
4*b^(3/2)*x^2*Cos[c]*SinIntegral[d*x] - 2*a*Sqrt[b]*Cos[c - (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*Sqrt[a])
/Sqrt[b] + x)] - 2*b^(3/2)*x^2*Cos[c - (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] + I*a^(
3/2)*d*Sin[c - (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] + I*Sqrt[a]*b*d*x^2*Sin[c - (I*
Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] + 2*a*Sqrt[b]*Cos[c + (I*Sqrt[a]*d)/Sqrt[b]]*SinI
ntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x] + 2*b^(3/2)*x^2*Cos[c + (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(I*Sqrt[a]*d)/
Sqrt[b] - d*x] + I*a^(3/2)*d*Sin[c + (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x] + I*Sqrt[
a]*b*d*x^2*Sin[c + (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x])/(4*a^2*Sqrt[b]*(a + b*x^2)
)

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Maple [A]
time = 0.16, size = 478, normalized size = 1.10

method	result	size
derivativedivides	\(\frac {\sin \left (d x +c \right ) d^{2}}{2 a \left (d^{2} a +b \,c^{2}-2 b c \left (d x +c \right )+b \left (d x +c \right )^{2}\right )}-\frac {\sinIntegral \left (d x +c -\frac {d \sqrt {-a b}+c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}+c b}{b}\right )+\cosineIntegral \left (d x +c -\frac {d \sqrt {-a b}+c b}{b}\right ) \sin \left (\frac {d \sqrt {-a b}+c b}{b}\right )}{2 a^{2}}-\frac {\sinIntegral \left (d x +c +\frac {d \sqrt {-a b}-c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}-c b}{b}\right )-\cosineIntegral \left (d x +c +\frac {d \sqrt {-a b}-c b}{b}\right ) \sin \left (\frac {d \sqrt {-a b}-c b}{b}\right )}{2 a^{2}}+\frac {\sinIntegral \left (d x \right ) \cos \left (c \right )+\cosineIntegral \left (d x \right ) \sin \left (c \right )}{a^{2}}+\frac {d^{2} \left (-\sinIntegral \left (d x +c -\frac {d \sqrt {-a b}+c b}{b}\right ) \sin \left (\frac {d \sqrt {-a b}+c b}{b}\right )+\cosineIntegral \left (d x +c -\frac {d \sqrt {-a b}+c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}+c b}{b}\right )\right )}{4 a b \left (-\frac {d \sqrt {-a b}+c b}{b}+c \right )}+\frac {d^{2} \left (\sinIntegral \left (d x +c +\frac {d \sqrt {-a b}-c b}{b}\right ) \sin \left (\frac {d \sqrt {-a b}-c b}{b}\right )+\cosineIntegral \left (d x +c +\frac {d \sqrt {-a b}-c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}-c b}{b}\right )\right )}{4 a b \left (\frac {d \sqrt {-a b}-c b}{b}+c \right )}\)	\(478\)
default	\(\frac {\sin \left (d x +c \right ) d^{2}}{2 a \left (d^{2} a +b \,c^{2}-2 b c \left (d x +c \right )+b \left (d x +c \right )^{2}\right )}-\frac {\sinIntegral \left (d x +c -\frac {d \sqrt {-a b}+c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}+c b}{b}\right )+\cosineIntegral \left (d x +c -\frac {d \sqrt {-a b}+c b}{b}\right ) \sin \left (\frac {d \sqrt {-a b}+c b}{b}\right )}{2 a^{2}}-\frac {\sinIntegral \left (d x +c +\frac {d \sqrt {-a b}-c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}-c b}{b}\right )-\cosineIntegral \left (d x +c +\frac {d \sqrt {-a b}-c b}{b}\right ) \sin \left (\frac {d \sqrt {-a b}-c b}{b}\right )}{2 a^{2}}+\frac {\sinIntegral \left (d x \right ) \cos \left (c \right )+\cosineIntegral \left (d x \right ) \sin \left (c \right )}{a^{2}}+\frac {d^{2} \left (-\sinIntegral \left (d x +c -\frac {d \sqrt {-a b}+c b}{b}\right ) \sin \left (\frac {d \sqrt {-a b}+c b}{b}\right )+\cosineIntegral \left (d x +c -\frac {d \sqrt {-a b}+c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}+c b}{b}\right )\right )}{4 a b \left (-\frac {d \sqrt {-a b}+c b}{b}+c \right )}+\frac {d^{2} \left (\sinIntegral \left (d x +c +\frac {d \sqrt {-a b}-c b}{b}\right ) \sin \left (\frac {d \sqrt {-a b}-c b}{b}\right )+\cosineIntegral \left (d x +c +\frac {d \sqrt {-a b}-c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}-c b}{b}\right )\right )}{4 a b \left (\frac {d \sqrt {-a b}-c b}{b}+c \right )}\)	\(478\)
risch	\(\frac {i {\mathrm e}^{\frac {i b c +d \sqrt {a b}}{b}} \expIntegral \left (1, \frac {i b c +d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right ) d}{8 a \sqrt {a b}}-\frac {i {\mathrm e}^{\frac {i b c -d \sqrt {a b}}{b}} \expIntegral \left (1, \frac {i b c -d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right ) d}{8 a \sqrt {a b}}-\frac {i {\mathrm e}^{\frac {i b c +d \sqrt {a b}}{b}} \expIntegral \left (1, \frac {i b c +d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right )}{4 a^{2}}-\frac {i {\mathrm e}^{\frac {i b c -d \sqrt {a b}}{b}} \expIntegral \left (1, \frac {i b c -d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right )}{4 a^{2}}+\frac {i {\mathrm e}^{i c} \expIntegral \left (1, -i d x \right )}{2 a^{2}}+\frac {i {\mathrm e}^{-\frac {i b c +d \sqrt {a b}}{b}} \expIntegral \left (1, -\frac {i b c +d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right ) d}{8 a \sqrt {a b}}-\frac {i {\mathrm e}^{-\frac {i b c -d \sqrt {a b}}{b}} \expIntegral \left (1, -\frac {i b c -d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right ) d}{8 a \sqrt {a b}}+\frac {i {\mathrm e}^{-\frac {i b c +d \sqrt {a b}}{b}} \expIntegral \left (1, -\frac {i b c +d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right )}{4 a^{2}}+\frac {i {\mathrm e}^{-\frac {i b c -d \sqrt {a b}}{b}} \expIntegral \left (1, -\frac {i b c -d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right )}{4 a^{2}}-\frac {{\mathrm e}^{-i c} \pi \,\mathrm {csgn}\left (d x \right )}{2 a^{2}}+\frac {{\mathrm e}^{-i c} \sinIntegral \left (d x \right )}{a^{2}}-\frac {i {\mathrm e}^{-i c} \expIntegral \left (1, -i d x \right )}{2 a^{2}}+\frac {d^{2} \sin \left (d x +c \right )}{2 a \left (d^{2} x^{2} b +d^{2} a \right )}\)	\(584\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \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/(b*x^2+a)^2,x, algorithm="maxima")

[Out]

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

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Fricas [C] Result contains complex when optimal does not.
time = 0.40, size = 332, normalized size = 0.76 \begin {gather*} -\frac {4 \, {\left (i \, b x^{2} + i \, a\right )} {\rm Ei}\left (i \, d x\right ) e^{\left (i \, c\right )} + 4 \, {\left (-i \, b x^{2} - i \, a\right )} {\rm Ei}\left (-i \, d x\right ) e^{\left (-i \, c\right )} + {\left (-2 i \, b x^{2} - {\left (-i \, b x^{2} - i \, a\right )} \sqrt {\frac {a d^{2}}{b}} - 2 i \, a\right )} {\rm Ei}\left (i \, d x - \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (i \, c + \sqrt {\frac {a d^{2}}{b}}\right )} + {\left (-2 i \, b x^{2} - {\left (i \, b x^{2} + i \, a\right )} \sqrt {\frac {a d^{2}}{b}} - 2 i \, a\right )} {\rm Ei}\left (i \, d x + \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (i \, c - \sqrt {\frac {a d^{2}}{b}}\right )} + {\left (2 i \, b x^{2} - {\left (i \, b x^{2} + i \, a\right )} \sqrt {\frac {a d^{2}}{b}} + 2 i \, a\right )} {\rm Ei}\left (-i \, d x - \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (-i \, c + \sqrt {\frac {a d^{2}}{b}}\right )} + {\left (2 i \, b x^{2} - {\left (-i \, b x^{2} - i \, a\right )} \sqrt {\frac {a d^{2}}{b}} + 2 i \, a\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 \, a \sin \left (d x + c\right )}{8 \, {\left (a^{2} b x^{2} + a^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(4*(I*b*x^2 + I*a)*Ei(I*d*x)*e^(I*c) + 4*(-I*b*x^2 - I*a)*Ei(-I*d*x)*e^(-I*c) + (-2*I*b*x^2 - (-I*b*x^2 -
I*a)*sqrt(a*d^2/b) - 2*I*a)*Ei(I*d*x - sqrt(a*d^2/b))*e^(I*c + sqrt(a*d^2/b)) + (-2*I*b*x^2 - (I*b*x^2 + I*a)
*sqrt(a*d^2/b) - 2*I*a)*Ei(I*d*x + sqrt(a*d^2/b))*e^(I*c - sqrt(a*d^2/b)) + (2*I*b*x^2 - (I*b*x^2 + I*a)*sqrt(
a*d^2/b) + 2*I*a)*Ei(-I*d*x - sqrt(a*d^2/b))*e^(-I*c + sqrt(a*d^2/b)) + (2*I*b*x^2 - (-I*b*x^2 - I*a)*sqrt(a*d
^2/b) + 2*I*a)*Ei(-I*d*x + sqrt(a*d^2/b))*e^(-I*c - sqrt(a*d^2/b)) - 4*a*sin(d*x + c))/(a^2*b*x^2 + a^3)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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/(b*x^2+a)^2,x, algorithm="giac")

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(c + d*x)/(x*(a + b*x^2)^2),x)

[Out]

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

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