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

Optimal. Leaf size=197 \[ -\frac{\sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{2 a}-\frac{\sin \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 a}+\frac{\cos \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 a}-\frac{\cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{2 a}+\frac{\sin (c) \text{CosIntegral}(d x)}{a}+\frac{\cos (c) \text{Si}(d x)}{a} \]

[Out]

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

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Rubi [A]  time = 0.382325, antiderivative size = 197, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {3345, 3303, 3299, 3302} \[ -\frac{\sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{2 a}-\frac{\sin \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 a}+\frac{\cos \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 a}-\frac{\cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{2 a}+\frac{\sin (c) \text{CosIntegral}(d x)}{a}+\frac{\cos (c) \text{Si}(d x)}{a} \]

Antiderivative was successfully verified.

[In]

Int[Sin[c + d*x]/(x*(a + b*x^2)),x]

[Out]

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

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

Mathematica [C]  time = 0.372245, size = 179, normalized size = 0.91 \[ -\frac{\sin \left (c-\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (d \left (x+\frac{i \sqrt{a}}{\sqrt{b}}\right )\right )+\sin \left (c+\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (d \left (x-\frac{i \sqrt{a}}{\sqrt{b}}\right )\right )+\cos \left (c-\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{Si}\left (d \left (x+\frac{i \sqrt{a}}{\sqrt{b}}\right )\right )-\cos \left (c+\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right )-2 \sin (c) \text{CosIntegral}(d x)-2 \cos (c) \text{Si}(d x)}{2 a} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(-2*CosIntegral[d*x]*Sin[c] + CosIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)]*Sin[c - (I*Sqrt[a]*d)/Sqrt[b]] + CosIn
tegral[d*(((-I)*Sqrt[a])/Sqrt[b] + x)]*Sin[c + (I*Sqrt[a]*d)/Sqrt[b]] - 2*Cos[c]*SinIntegral[d*x] + Cos[c - (I
*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] - Cos[c + (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(I*
Sqrt[a]*d)/Sqrt[b] - d*x])/(2*a)

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Maple [A]  time = 0.019, size = 200, normalized size = 1. \begin{align*} -{\frac{1}{2\,a} \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{1}{2\,a} \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{{\it Si} \left ( dx \right ) \cos \left ( c \right ) +{\it Ci} \left ( dx \right ) \sin \left ( c \right ) }{a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/a*(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*(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*(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}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [C]  time = 1.78218, size = 375, normalized size = 1.9 \begin{align*} \frac{-2 i \,{\rm Ei}\left (i \, d x\right ) e^{\left (i \, c\right )} + 2 i \,{\rm Ei}\left (-i \, d x\right ) e^{\left (-i \, c\right )} + i \,{\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 \,{\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 \,{\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 \,{\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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Integral(sin(c + d*x)/(x*(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}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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