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

Optimal. Leaf size=250 \[ -\frac{\sqrt{b} \sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{2 (-a)^{3/2}}+\frac{\sqrt{b} \sin \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 (-a)^{3/2}}-\frac{\sqrt{b} \cos \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 (-a)^{3/2}}-\frac{\sqrt{b} \cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{2 (-a)^{3/2}}+\frac{d \cos (c) \text{CosIntegral}(d x)}{a}-\frac{d \sin (c) \text{Si}(d x)}{a}-\frac{\sin (c+d x)}{a x} \]

[Out]

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

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Rubi [A]  time = 0.487328, antiderivative size = 250, normalized size of antiderivative = 1., number of steps used = 14, 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} \[ -\frac{\sqrt{b} \sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{2 (-a)^{3/2}}+\frac{\sqrt{b} \sin \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 (-a)^{3/2}}-\frac{\sqrt{b} \cos \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 (-a)^{3/2}}-\frac{\sqrt{b} \cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{2 (-a)^{3/2}}+\frac{d \cos (c) \text{CosIntegral}(d x)}{a}-\frac{d \sin (c) \text{Si}(d x)}{a}-\frac{\sin (c+d x)}{a x} \]

Antiderivative was successfully verified.

[In]

Int[Sin[c + d*x]/(x^2*(a + b*x^2)),x]

[Out]

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

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]

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

Mathematica [C]  time = 0.521921, size = 238, normalized size = 0.95 \[ \frac{d \cos (c) \text{CosIntegral}(d x)}{a}-\frac{i \left (\sqrt{b} x \sin \left (c-\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (d \left (x+\frac{i \sqrt{a}}{\sqrt{b}}\right )\right )-\sqrt{b} x \sin \left (c+\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (d \left (x-\frac{i \sqrt{a}}{\sqrt{b}}\right )\right )+\sqrt{b} x \cos \left (c-\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{Si}\left (d \left (x+\frac{i \sqrt{a}}{\sqrt{b}}\right )\right )+\sqrt{b} x \cos \left (c+\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right )-2 i \sqrt{a} d x \sin (c) \text{Si}(d x)-2 i \sqrt{a} \sin (c+d x)\right )}{2 a^{3/2} x} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(d*Cos[c]*CosIntegral[d*x])/a - ((I/2)*(Sqrt[b]*x*CosIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)]*Sin[c - (I*Sqrt[a]*
d)/Sqrt[b]] - Sqrt[b]*x*CosIntegral[d*(((-I)*Sqrt[a])/Sqrt[b] + x)]*Sin[c + (I*Sqrt[a]*d)/Sqrt[b]] - (2*I)*Sqr
t[a]*Sin[c + d*x] - (2*I)*Sqrt[a]*d*x*Sin[c]*SinIntegral[d*x] + Sqrt[b]*x*Cos[c - (I*Sqrt[a]*d)/Sqrt[b]]*SinIn
tegral[d*((I*Sqrt[a])/Sqrt[b] + x)] + Sqrt[b]*x*Cos[c + (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[
b] - d*x]))/(a^(3/2)*x)

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Maple [A]  time = 0.013, size = 270, normalized size = 1.1 \begin{align*} d \left ( -{\frac{b}{a} \left ({\frac{1}{2\,b} \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 ) \left ({\frac{1}{b} \left ( d\sqrt{-ab}+cb \right ) }-c \right ) ^{-1}}+{\frac{1}{2\,b} \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 ) \left ( -{\frac{1}{b} \left ( d\sqrt{-ab}-cb \right ) }-c \right ) ^{-1}} \right ) }+{\frac{1}{a} \left ( -{\frac{\sin \left ( dx+c \right ) }{dx}}-{\it Si} \left ( dx \right ) \sin \left ( c \right ) +{\it Ci} \left ( dx \right ) \cos \left ( c \right ) \right ) } \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d*(-1/a*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
*(-sin(d*x+c)/x/d-Si(d*x)*sin(c)+Ci(d*x)*cos(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^{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),x, algorithm="maxima")

[Out]

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

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Fricas [C]  time = 1.96774, size = 516, normalized size = 2.06 \begin{align*} \frac{2 \, a d^{2} x{\rm Ei}\left (i \, d x\right ) e^{\left (i \, c\right )} + 2 \, a d^{2} x{\rm Ei}\left (-i \, d x\right ) e^{\left (-i \, c\right )} - \sqrt{\frac{a d^{2}}{b}} b x{\rm Ei}\left (i \, d x - \sqrt{\frac{a d^{2}}{b}}\right ) e^{\left (i \, c + \sqrt{\frac{a d^{2}}{b}}\right )} + \sqrt{\frac{a d^{2}}{b}} b x{\rm Ei}\left (i \, d x + \sqrt{\frac{a d^{2}}{b}}\right ) e^{\left (i \, c - \sqrt{\frac{a d^{2}}{b}}\right )} - \sqrt{\frac{a d^{2}}{b}} b x{\rm Ei}\left (-i \, d x - \sqrt{\frac{a d^{2}}{b}}\right ) e^{\left (-i \, c + \sqrt{\frac{a d^{2}}{b}}\right )} + \sqrt{\frac{a d^{2}}{b}} b x{\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 d \sin \left (d x + c\right )}{4 \, a^{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),x, algorithm="fricas")

[Out]

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

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

[Out]

Integral(sin(c + d*x)/(x**2*(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^{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),x, algorithm="giac")

[Out]

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