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]

x*Si(x)*(b-a/x^2)^(1/2)/(-b*x^2+a)^(1/2)

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Rubi [A]  time = 0.47, antiderivative size = 28, normalized size of antiderivative = 1.00, 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 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]

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]

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*}

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Mathematica [C]  time = 0.70, size = 46, normalized size = 1.64 \[ \frac {i x (\text {Ei}(-i x)-\text {Ei}(i x)) \sqrt {b-\frac {a}{x^2}}}{2 \sqrt {a-b x^2}} \]

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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fricas [F]  time = 1.27, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-b x^{2} + a} \sqrt {\frac {b x^{2} - a}{x^{2}}} \sin \relax (x)}{b x^{2} - a}, x\right ) \]

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

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)

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maple [C]  time = 0.08, size = 72, normalized size = 2.57 \[ -\frac {\sqrt {-\frac {-b \,x^{2}+a}{x^{2}}}\, \left (b \,x^{2}-a \right ) x \sqrt {\frac {-b \,x^{2}+a}{b \,x^{2}-a}}\, \left (-i \Si \relax (x )+\frac {i \pi \,\mathrm {csgn}\relax (x )}{2}\right )}{\left (-b \,x^{2}+a \right )^{\frac {3}{2}}} \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {b - \frac {a}{x^{2}}} \sin \relax (x)}{\sqrt {-b x^{2} + a}}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {\sin \relax (x)\,\sqrt {b-\frac {a}{x^2}}}{\sqrt {a-b\,x^2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {- \frac {a}{x^{2}} + b} \sin {\relax (x )}}{\sqrt {a - b x^{2}}}\, dx \]

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]

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

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