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

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

[Out]

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

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Rubi [A]  time = 0.25, antiderivative size = 177, normalized size of antiderivative = 1.00, 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 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 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 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]

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*}

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Mathematica [C]  time = 0.22, size = 163, normalized size = 0.92 \[ \frac {\sin \left (c-\frac {i \sqrt {a} d}{\sqrt {b}}\right ) \text {Ci}\left (d \left (x+\frac {i \sqrt {a}}{\sqrt {b}}\right )\right )+\sin \left (c+\frac {i \sqrt {a} d}{\sqrt {b}}\right ) \text {Ci}\left (d \left (x-\frac {i \sqrt {a}}{\sqrt {b}}\right )\right )+\cos \left (c-\frac {i \sqrt {a} d}{\sqrt {b}}\right ) \text {Si}\left (d \left (x+\frac {i \sqrt {a}}{\sqrt {b}}\right )\right )-\cos \left (c+\frac {i \sqrt {a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )}{2 b} \]

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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fricas [C]  time = 0.65, size = 146, normalized size = 0.82 \[ \frac {-i \, {\rm Ei}\left (i \, d x - \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (i \, c + \sqrt {\frac {a d^{2}}{b}}\right )} - i \, {\rm Ei}\left (i \, d x + \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (i \, c - \sqrt {\frac {a d^{2}}{b}}\right )} + i \, {\rm Ei}\left (-i \, d x - \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (-i \, c + \sqrt {\frac {a d^{2}}{b}}\right )} + i \, {\rm Ei}\left (-i \, d x + \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (-i \, c - \sqrt {\frac {a d^{2}}{b}}\right )}}{4 \, b} \]

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

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)

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maple [B]  time = 0.04, size = 494, normalized size = 2.79 \[ \frac {\frac {d^{2} \left (d \sqrt {-a b}+c b \right ) \left (\Si \left (d x +c -\frac {d \sqrt {-a b}+c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}+c b}{b}\right )+\Ci \left (d x +c -\frac {d \sqrt {-a b}+c b}{b}\right ) \sin \left (\frac {d \sqrt {-a b}+c b}{b}\right )\right )}{2 b^{2} \left (\frac {d \sqrt {-a b}+c b}{b}-c \right )}-\frac {d^{2} \left (d \sqrt {-a b}-c b \right ) \left (\Si \left (d x +c +\frac {d \sqrt {-a b}-c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}-c b}{b}\right )-\Ci \left (d x +c +\frac {d \sqrt {-a b}-c b}{b}\right ) \sin \left (\frac {d \sqrt {-a b}-c b}{b}\right )\right )}{2 b^{2} \left (-\frac {d \sqrt {-a b}-c b}{b}-c \right )}-d^{2} c \left (\frac {\Si \left (d x +c -\frac {d \sqrt {-a b}+c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}+c b}{b}\right )+\Ci \left (d x +c -\frac {d \sqrt {-a b}+c b}{b}\right ) \sin \left (\frac {d \sqrt {-a b}+c b}{b}\right )}{2 \left (\frac {d \sqrt {-a b}+c b}{b}-c \right ) b}+\frac {\Si \left (d x +c +\frac {d \sqrt {-a b}-c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}-c b}{b}\right )-\Ci \left (d x +c +\frac {d \sqrt {-a b}-c b}{b}\right ) \sin \left (\frac {d \sqrt {-a b}-c b}{b}\right )}{2 \left (-\frac {d \sqrt {-a b}-c b}{b}-c \right ) b}\right )}{d^{2}} \]

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))-d^2*c*(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 = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x\,\sin \left (c+d\,x\right )}{b\,x^2+a} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*sin(c + d*x))/(a + b*x^2),x)

[Out]

int((x*sin(c + d*x))/(a + b*x^2), x)

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

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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