3.34 \(\int \frac{\sqrt{b-\frac{a}{x^2}} \sin (x)}{\sqrt{a-b x^2}} \, dx\)

Optimal. Leaf size=28 \[ \frac{x \text{Si}(x) \sqrt{b-\frac{a}{x^2}}}{\sqrt{a-b x^2}} \]

[Out]

(Sqrt[b - a/x^2]*x*SinIntegral[x])/Sqrt[a - b*x^2]

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Rubi [A]  time = 0.470982, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {6721, 23, 3299} \[ \frac{x \text{Si}(x) \sqrt{b-\frac{a}{x^2}}}{\sqrt{a-b x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(Sqrt[b - a/x^2]*Sin[x])/Sqrt[a - b*x^2],x]

[Out]

(Sqrt[b - a/x^2]*x*SinIntegral[x])/Sqrt[a - b*x^2]

Rule 6721

Int[(u_.)*((a_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[(b^IntPart[p]*(a + b*x^n)^FracPart[p])/(x^(n*FracP
art[p])*(1 + a/(x^n*b))^FracPart[p]), Int[u*x^(n*p)*(1 + a/(x^n*b))^p, x], x] /; FreeQ[{a, b, p}, x] &&  !Inte
gerQ[p] && ILtQ[n, 0] &&  !RationalFunctionQ[u, x] && IntegerQ[p + 1/2]

Rule 23

Int[(u_.)*((a_) + (b_.)*(v_))^(m_)*((c_) + (d_.)*(v_))^(n_), x_Symbol] :> Dist[(a + b*v)^m/(c + d*v)^m, Int[u*
(c + d*v)^(m + n), x], x] /; FreeQ[{a, b, c, d, m, n}, x] && EqQ[b*c - a*d, 0] &&  !(IntegerQ[m] || IntegerQ[n
] || GtQ[b/d, 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]

Rubi steps

\begin{align*} \int \frac{\sqrt{b-\frac{a}{x^2}} \sin (x)}{\sqrt{a-b x^2}} \, dx &=\frac{\left (\sqrt{b-\frac{a}{x^2}} x\right ) \int \frac{\sqrt{1-\frac{b x^2}{a}} \sin (x)}{x \sqrt{a-b x^2}} \, dx}{\sqrt{1-\frac{b x^2}{a}}}\\ &=\frac{\left (\sqrt{b-\frac{a}{x^2}} x\right ) \int \frac{\sin (x)}{x} \, dx}{\sqrt{a-b x^2}}\\ &=\frac{\sqrt{b-\frac{a}{x^2}} x \text{Si}(x)}{\sqrt{a-b x^2}}\\ \end{align*}

Mathematica [C]  time = 0.708935, size = 46, normalized size = 1.64 \[ \frac{i x (\text{ExpIntegralEi}(-i x)-\text{ExpIntegralEi}(i x)) \sqrt{b-\frac{a}{x^2}}}{2 \sqrt{a-b x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(Sqrt[b - a/x^2]*Sin[x])/Sqrt[a - b*x^2],x]

[Out]

((I/2)*Sqrt[b - a/x^2]*x*(ExpIntegralEi[(-I)*x] - ExpIntegralEi[I*x]))/Sqrt[a - b*x^2]

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Maple [C]  time = 0.046, size = 72, normalized size = 2.6 \begin{align*} -{ \left ( b{x}^{2}-a \right ) x \left ( -i{\it Si} \left ( x \right ) +{\frac{i}{2}}\pi \,{\it csgn} \left ( x \right ) \right ) \sqrt{-{\frac{-b{x}^{2}+a}{{x}^{2}}}}\sqrt{{\frac{-b{x}^{2}+a}{b{x}^{2}-a}}} \left ( -b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)*(b-a/x^2)^(1/2)/(-b*x^2+a)^(1/2),x)

[Out]

-(-(-b*x^2+a)/x^2)^(1/2)*(b*x^2-a)/(-b*x^2+a)^(3/2)*x*(1/(b*x^2-a)*(-b*x^2+a))^(1/2)*(-I*Si(x)+1/2*I*Pi*csgn(x
))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)*(b-a/x^2)^(1/2)/(-b*x^2+a)^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(b - a/x^2)*sin(x)/sqrt(-b*x^2 + a), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-b x^{2} + a} \sqrt{\frac{b x^{2} - a}{x^{2}}} \sin \left (x\right )}{b x^{2} - a}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)*(b-a/x^2)^(1/2)/(-b*x^2+a)^(1/2),x, algorithm="fricas")

[Out]

integral(-sqrt(-b*x^2 + a)*sqrt((b*x^2 - a)/x^2)*sin(x)/(b*x^2 - a), x)

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Sympy [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(sin(x)*(b-a/x**2)**(1/2)/(-b*x**2+a)**(1/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)*(b-a/x^2)^(1/2)/(-b*x^2+a)^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(b - a/x^2)*sin(x)/sqrt(-b*x^2 + a), x)