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

Optimal. Leaf size=270 \[ -\frac{b \sin (c) \text{CosIntegral}(d x)}{a^2}+\frac{b \sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{2 a^2}+\frac{b \sin \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 a^2}-\frac{b \cos (c) \text{Si}(d x)}{a^2}-\frac{b \cos \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 a^2}+\frac{b \cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{2 a^2}-\frac{d^2 \sin (c) \text{CosIntegral}(d x)}{2 a}-\frac{d^2 \cos (c) \text{Si}(d x)}{2 a}-\frac{\sin (c+d x)}{2 a x^2}-\frac{d \cos (c+d x)}{2 a x} \]

[Out]

-(d*Cos[c + d*x])/(2*a*x) - (b*CosIntegral[d*x]*Sin[c])/a^2 - (d^2*CosIntegral[d*x]*Sin[c])/(2*a) + (b*CosInte
gral[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sin[c - (Sqrt[-a]*d)/Sqrt[b]])/(2*a^2) + (b*CosIntegral[(Sqrt[-a]*d)/Sqrt[b]
- d*x]*Sin[c + (Sqrt[-a]*d)/Sqrt[b]])/(2*a^2) - Sin[c + d*x]/(2*a*x^2) - (b*Cos[c]*SinIntegral[d*x])/a^2 - (d^
2*Cos[c]*SinIntegral[d*x])/(2*a) - (b*Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(
2*a^2) + (b*Cos[c - (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(2*a^2)

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Rubi [A]  time = 0.507786, antiderivative size = 270, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {3345, 3297, 3303, 3299, 3302} \[ -\frac{b \sin (c) \text{CosIntegral}(d x)}{a^2}+\frac{b \sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{2 a^2}+\frac{b \sin \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 a^2}-\frac{b \cos (c) \text{Si}(d x)}{a^2}-\frac{b \cos \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 a^2}+\frac{b \cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{2 a^2}-\frac{d^2 \sin (c) \text{CosIntegral}(d x)}{2 a}-\frac{d^2 \cos (c) \text{Si}(d x)}{2 a}-\frac{\sin (c+d x)}{2 a x^2}-\frac{d \cos (c+d x)}{2 a x} \]

Antiderivative was successfully verified.

[In]

Int[Sin[c + d*x]/(x^3*(a + b*x^2)),x]

[Out]

-(d*Cos[c + d*x])/(2*a*x) - (b*CosIntegral[d*x]*Sin[c])/a^2 - (d^2*CosIntegral[d*x]*Sin[c])/(2*a) + (b*CosInte
gral[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sin[c - (Sqrt[-a]*d)/Sqrt[b]])/(2*a^2) + (b*CosIntegral[(Sqrt[-a]*d)/Sqrt[b]
- d*x]*Sin[c + (Sqrt[-a]*d)/Sqrt[b]])/(2*a^2) - Sin[c + d*x]/(2*a*x^2) - (b*Cos[c]*SinIntegral[d*x])/a^2 - (d^
2*Cos[c]*SinIntegral[d*x])/(2*a) - (b*Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(
2*a^2) + (b*Cos[c - (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(2*a^2)

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]

Rubi steps

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

Mathematica [C]  time = 0.678373, size = 247, normalized size = 0.91 \[ -\frac{x^2 \sin (c) \left (a d^2+2 b\right ) \text{CosIntegral}(d x)-b x^2 \sin \left (c-\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (d \left (x+\frac{i \sqrt{a}}{\sqrt{b}}\right )\right )-b x^2 \sin \left (c+\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (d \left (x-\frac{i \sqrt{a}}{\sqrt{b}}\right )\right )-b x^2 \cos \left (c-\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{Si}\left (d \left (x+\frac{i \sqrt{a}}{\sqrt{b}}\right )\right )+b x^2 \cos \left (c+\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right )+a d^2 x^2 \cos (c) \text{Si}(d x)+a \sin (c+d x)+a d x \cos (c+d x)+2 b x^2 \cos (c) \text{Si}(d x)}{2 a^2 x^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(a*d*x*Cos[c + d*x] + (2*b + a*d^2)*x^2*CosIntegral[d*x]*Sin[c] - b*x^2*CosIntegral[d*((I*Sqrt[a])/Sqrt[b] +
x)]*Sin[c - (I*Sqrt[a]*d)/Sqrt[b]] - b*x^2*CosIntegral[d*(((-I)*Sqrt[a])/Sqrt[b] + x)]*Sin[c + (I*Sqrt[a]*d)/S
qrt[b]] + a*Sin[c + d*x] + 2*b*x^2*Cos[c]*SinIntegral[d*x] + a*d^2*x^2*Cos[c]*SinIntegral[d*x] - b*x^2*Cos[c -
(I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] + b*x^2*Cos[c + (I*Sqrt[a]*d)/Sqrt[b]]*SinInt
egral[(I*Sqrt[a]*d)/Sqrt[b] - d*x])/(2*a^2*x^2)

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Maple [A]  time = 0.028, size = 259, normalized size = 1. \begin{align*}{d}^{2} \left ( -{\frac{\sin \left ( dx+c \right ) }{2\,a{x}^{2}{d}^{2}}}-{\frac{\cos \left ( dx+c \right ) }{2\,axd}}+{\frac{b}{2\,{d}^{2}{a}^{2}} \left ({\it Si} \left ( dx+c-{\frac{1}{b} \left ( d\sqrt{-ab}+cb \right ) } \right ) \cos \left ({\frac{1}{b} \left ( d\sqrt{-ab}+cb \right ) } \right ) +{\it Ci} \left ( dx+c-{\frac{1}{b} \left ( d\sqrt{-ab}+cb \right ) } \right ) \sin \left ({\frac{1}{b} \left ( d\sqrt{-ab}+cb \right ) } \right ) \right ) }+{\frac{b}{2\,{d}^{2}{a}^{2}} \left ({\it Si} \left ( dx+c+{\frac{1}{b} \left ( d\sqrt{-ab}-cb \right ) } \right ) \cos \left ({\frac{1}{b} \left ( d\sqrt{-ab}-cb \right ) } \right ) -{\it Ci} \left ( dx+c+{\frac{1}{b} \left ( d\sqrt{-ab}-cb \right ) } \right ) \sin \left ({\frac{1}{b} \left ( d\sqrt{-ab}-cb \right ) } \right ) \right ) }-{\frac{ \left ( a{d}^{2}+2\,b \right ) \left ({\it Si} \left ( dx \right ) \cos \left ( c \right ) +{\it Ci} \left ( dx \right ) \sin \left ( c \right ) \right ) }{2\,{d}^{2}{a}^{2}}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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 )} x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [C]  time = 1.89675, size = 517, normalized size = 1.91 \begin{align*} \frac{i \,{\left (a d^{2} + 2 \, b\right )} x^{2}{\rm Ei}\left (i \, d x\right ) e^{\left (i \, c\right )} - i \,{\left (a d^{2} + 2 \, b\right )} x^{2}{\rm Ei}\left (-i \, d x\right ) e^{\left (-i \, c\right )} - i \, b x^{2}{\rm Ei}\left (i \, d x - \sqrt{\frac{a d^{2}}{b}}\right ) e^{\left (i \, c + \sqrt{\frac{a d^{2}}{b}}\right )} - i \, b x^{2}{\rm Ei}\left (i \, d x + \sqrt{\frac{a d^{2}}{b}}\right ) e^{\left (i \, c - \sqrt{\frac{a d^{2}}{b}}\right )} + i \, b x^{2}{\rm Ei}\left (-i \, d x - \sqrt{\frac{a d^{2}}{b}}\right ) e^{\left (-i \, c + \sqrt{\frac{a d^{2}}{b}}\right )} + i \, b x^{2}{\rm Ei}\left (-i \, d x + \sqrt{\frac{a d^{2}}{b}}\right ) e^{\left (-i \, c - \sqrt{\frac{a d^{2}}{b}}\right )} - 2 \, a d x \cos \left (d x + c\right ) - 2 \, a \sin \left (d x + c\right )}{4 \, a^{2} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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 )} x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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