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

Optimal. Leaf size=39 \[ -2 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{1-a^2 x^2}}{\sqrt{c-a c x}}\right ) \]

[Out]

-2*Sqrt[c]*ArcTanh[(Sqrt[c]*Sqrt[1 - a^2*x^2])/Sqrt[c - a*c*x]]

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Rubi [A]  time = 0.0399762, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069, Rules used = {875, 208} \[ -2 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{1-a^2 x^2}}{\sqrt{c-a c x}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-2*Sqrt[c]*ArcTanh[(Sqrt[c]*Sqrt[1 - a^2*x^2])/Sqrt[c - a*c*x]]

Rule 875

Int[Sqrt[(d_) + (e_.)*(x_)]/(((f_.) + (g_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[2*e^2, Subst[I
nt[1/(c*(e*f + d*g) + e^2*g*x^2), x], x, Sqrt[a + c*x^2]/Sqrt[d + e*x]], x] /; FreeQ[{a, c, d, e, f, g}, x] &&
NeQ[e*f - d*g, 0] && EqQ[c*d^2 + a*e^2, 0]

Rule 208

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

Rubi steps

\begin{align*} \int \frac{\sqrt{c-a c x}}{x \sqrt{1-a^2 x^2}} \, dx &=\left (2 a^2 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{-a^2 c+a^2 c^2 x^2} \, dx,x,\frac{\sqrt{1-a^2 x^2}}{\sqrt{c-a c x}}\right )\\ &=-2 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{1-a^2 x^2}}{\sqrt{c-a c x}}\right )\\ \end{align*}

Mathematica [A]  time = 0.033967, size = 67, normalized size = 1.72 \[ -\frac{2 \sqrt{c} \sqrt{\frac{a x}{c}+\frac{1}{c}} \sqrt{c-a c x} \tanh ^{-1}\left (\sqrt{c} \sqrt{\frac{a x}{c}+\frac{1}{c}}\right )}{\sqrt{1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.141, size = 58, normalized size = 1.5 \begin{align*} 2\,{\frac{\sqrt{-c \left ( ax-1 \right ) }\sqrt{-{a}^{2}{x}^{2}+1}\sqrt{c}}{ \left ( ax-1 \right ) \sqrt{c \left ( ax+1 \right ) }}{\it Artanh} \left ({\frac{\sqrt{c \left ( ax+1 \right ) }}{\sqrt{c}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(-c*(a*x-1))^(1/2)*(-a^2*x^2+1)^(1/2)/(a*x-1)/(c*(a*x+1))^(1/2)*c^(1/2)*arctanh((c*(a*x+1))^(1/2)/c^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-a c x + c}}{\sqrt{-a^{2} x^{2} + 1} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.59948, size = 252, normalized size = 6.46 \begin{align*} \left [\sqrt{c} \log \left (-\frac{a^{2} c x^{2} + a c x + 2 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{c} - 2 \, c}{a x^{2} - x}\right ), -2 \, \sqrt{-c} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right )\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[sqrt(c)*log(-(a^2*c*x^2 + a*c*x + 2*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(c) - 2*c)/(a*x^2 - x)), -2*sqrt(
-c)*arctan(sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(-c)/(a^2*c*x^2 - c))]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.1336, size = 69, normalized size = 1.77 \begin{align*} -\frac{2 \, c^{2}{\left (\frac{\arctan \left (\frac{\sqrt{2} \sqrt{c}}{\sqrt{-c}}\right )}{\sqrt{-c}} - \frac{\arctan \left (\frac{\sqrt{a c x + c}}{\sqrt{-c}}\right )}{\sqrt{-c}}\right )}}{{\left | c \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*c^2*(arctan(sqrt(2)*sqrt(c)/sqrt(-c))/sqrt(-c) - arctan(sqrt(a*c*x + c)/sqrt(-c))/sqrt(-c))/abs(c)