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

Optimal. Leaf size=177 \[ \frac{\sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{2 b}+\frac{\sin \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 b}-\frac{\cos \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 b}+\frac{\cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{2 b} \]

[Out]

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

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Rubi [A]  time = 0.247684, antiderivative size = 177, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {3345, 3303, 3299, 3302} \[ \frac{\sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{2 b}+\frac{\sin \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 b}-\frac{\cos \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 b}+\frac{\cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{2 b} \]

Antiderivative was successfully verified.

[In]

Int[(x*Sin[c + d*x])/(a + b*x^2),x]

[Out]

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

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

Mathematica [C]  time = 0.223432, size = 163, normalized size = 0.92 \[ \frac{\sin \left (c-\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (d \left (x+\frac{i \sqrt{a}}{\sqrt{b}}\right )\right )+\sin \left (c+\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (d \left (x-\frac{i \sqrt{a}}{\sqrt{b}}\right )\right )+\cos \left (c-\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{Si}\left (d \left (x+\frac{i \sqrt{a}}{\sqrt{b}}\right )\right )-\cos \left (c+\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right )}{2 b} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [B]  time = 0.016, size = 494, normalized size = 2.8 \begin{align*}{\frac{1}{{d}^{2}} \left ({\frac{{d}^{2}}{2\,{b}^{2}} \left ( d\sqrt{-ab}+cb \right ) \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{{d}^{2}}{2\,{b}^{2}} \left ( d\sqrt{-ab}-cb \right ) \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}}-c{d}^{2} \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 ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d^2*(1/2*d^2*(d*(-a*b)^(1/2)+c*b)/b^2/((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))-1/2*d^2*(d*(-a*b)^(1/2)-c*b)/
b^2/(-(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))-c*d^2*(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))))

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

[Out]

Timed out

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Fricas [C]  time = 1.77579, size = 308, normalized size = 1.74 \begin{align*} \frac{-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 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(-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)))/b

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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