3.2.52 \(\int \frac {x}{(1+a x) \sqrt {1-a^2 x^2}} \, dx\)

Optimal. Leaf size=34 \[ \frac {\sqrt {1-a^2 x^2}}{a^2 (a x+1)}+\frac {\sin ^{-1}(a x)}{a^2} \]

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Rubi [A]  time = 0.02, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {793, 216} \begin {gather*} \frac {\sqrt {1-a^2 x^2}}{a^2 (a x+1)}+\frac {\sin ^{-1}(a x)}{a^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x/((1 + a*x)*Sqrt[1 - a^2*x^2]),x]

[Out]

Sqrt[1 - a^2*x^2]/(a^2*(1 + a*x)) + ArcSin[a*x]/a^2

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rule 793

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d*g - e*f)*(
d + e*x)^m*(a + c*x^2)^(p + 1))/(2*c*d*(m + p + 1)), x] + Dist[(m*(g*c*d + c*e*f) + 2*e*c*f*(p + 1))/(e*(2*c*d
)*(m + p + 1)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && EqQ[c*d^2
+ a*e^2, 0] && ((LtQ[m, -1] &&  !IGtQ[m + p + 1, 0]) || (LtQ[m, 0] && LtQ[p, -1]) || EqQ[m + 2*p + 2, 0]) &&
NeQ[m + p + 1, 0]

Rubi steps

\begin {align*} \int \frac {x}{(1+a x) \sqrt {1-a^2 x^2}} \, dx &=\frac {\sqrt {1-a^2 x^2}}{a^2 (1+a x)}+\frac {\int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{a}\\ &=\frac {\sqrt {1-a^2 x^2}}{a^2 (1+a x)}+\frac {\sin ^{-1}(a x)}{a^2}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 31, normalized size = 0.91 \begin {gather*} \frac {\frac {\sqrt {1-a^2 x^2}}{a x+1}+\sin ^{-1}(a x)}{a^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x/((1 + a*x)*Sqrt[1 - a^2*x^2]),x]

[Out]

(Sqrt[1 - a^2*x^2]/(1 + a*x) + ArcSin[a*x])/a^2

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IntegrateAlgebraic [A]  time = 0.34, size = 67, normalized size = 1.97 \begin {gather*} \frac {\sqrt {1-a^2 x^2}}{a^2 (a x+1)}+\frac {\sqrt {-a^2} \log \left (\sqrt {1-a^2 x^2}-\sqrt {-a^2} x\right )}{a^3} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x/((1 + a*x)*Sqrt[1 - a^2*x^2]),x]

[Out]

Sqrt[1 - a^2*x^2]/(a^2*(1 + a*x)) + (Sqrt[-a^2]*Log[-(Sqrt[-a^2]*x) + Sqrt[1 - a^2*x^2]])/a^3

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fricas [A]  time = 0.41, size = 58, normalized size = 1.71 \begin {gather*} \frac {a x - 2 \, {\left (a x + 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + \sqrt {-a^{2} x^{2} + 1} + 1}{a^{3} x + a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a*x - 2*(a*x + 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + sqrt(-a^2*x^2 + 1) + 1)/(a^3*x + a^2)

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giac [A]  time = 0.22, size = 52, normalized size = 1.53 \begin {gather*} \frac {\arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{a {\left | a \right |}} - \frac {2}{a {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} + 1\right )} {\left | a \right |}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arcsin(a*x)*sgn(a)/(a*abs(a)) - 2/(a*((sqrt(-a^2*x^2 + 1)*abs(a) + a)/(a^2*x) + 1)*abs(a))

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maple [A]  time = 0.01, size = 65, normalized size = 1.91 \begin {gather*} \frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}\, a}+\frac {\sqrt {-\left (x +\frac {1}{a}\right )^{2} a^{2}+2 \left (x +\frac {1}{a}\right ) a}}{\left (x +\frac {1}{a}\right ) a^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/a/(a^2)^(1/2)*arctan((a^2)^(1/2)/(-a^2*x^2+1)^(1/2)*x)+1/a^3/(x+1/a)*(-(x+1/a)^2*a^2+2*(x+1/a)*a)^(1/2)

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maxima [A]  time = 0.99, size = 33, normalized size = 0.97 \begin {gather*} \frac {\sqrt {-a^{2} x^{2} + 1}}{a^{3} x + a^{2}} + \frac {\arcsin \left (a x\right )}{a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(-a^2*x^2 + 1)/(a^3*x + a^2) + arcsin(a*x)/a^2

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mupad [B]  time = 2.60, size = 57, normalized size = 1.68 \begin {gather*} \frac {1}{a^2\,\sqrt {1-a^2\,x^2}}-\frac {x}{a\,\sqrt {1-a^2\,x^2}}-\frac {\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}}{a^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/(a^2*(1 - a^2*x^2)^(1/2)) - x/(a*(1 - a^2*x^2)^(1/2)) - (asinh(x*(-a^2)^(1/2))*(-a^2)^(1/2))/a^3

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )} \left (a x + 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(a*x+1)/(-a**2*x**2+1)**(1/2),x)

[Out]

Integral(x/(sqrt(-(a*x - 1)*(a*x + 1))*(a*x + 1)), x)

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