\(\int \frac {\sqrt {c-a c x}}{x \sqrt {1-a^2 x^2}} \, dx\) [219]

Optimal result

Integrand size = 29, antiderivative size = 39 \[ \int \frac {\sqrt {c-a c x}}{x \sqrt {1-a^2 x^2}} \, dx=-2 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {1-a^2 x^2}}{\sqrt {c-a c x}}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {889, 214} \[ \int \frac {\sqrt {c-a c x}}{x \sqrt {1-a^2 x^2}} \, dx=-2 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {1-a^2 x^2}}{\sqrt {c-a c x}}\right ) \]

[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 214

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

Rule 889

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]

Rubi steps \begin{align*} \text {integral}& = \left (2 a^2 c^2\right ) \text {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] (verified)

Time = 0.24 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.21 \[ \int \frac {\sqrt {c-a c x}}{x \sqrt {1-a^2 x^2}} \, dx=-\frac {2 \sqrt {c-a c x} \arctan \left (\frac {\sqrt {-1+a x}}{\sqrt {1-a^2 x^2}}\right )}{\sqrt {-1+a x}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.36 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.49

method	result	size
default	\(\frac {2 \sqrt {-c \left (a x -1\right )}\, \sqrt {-a^{2} x^{2}+1}\, \sqrt {c}\, \operatorname {arctanh}\left (\frac {\sqrt {c \left (a x +1\right )}}{\sqrt {c}}\right )}{\left (a x -1\right ) \sqrt {c \left (a x +1\right )}}\)	\(58\)

[In]

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

[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))

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 110, normalized size of antiderivative = 2.82 \[ \int \frac {\sqrt {c-a c x}}{x \sqrt {1-a^2 x^2}} \, dx=\left [\sqrt {c} \log \left (-\frac {a^{2} c x^{2} + a c x + 2 \, \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} \sqrt {c} - 2 \, c}{a x^{2} - x}\right ), -2 \, \sqrt {-c} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} \sqrt {-c}}{a^{2} c x^{2} - c}\right )\right ] \]

[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))]

Sympy [F]

\[ \int \frac {\sqrt {c-a c x}}{x \sqrt {1-a^2 x^2}} \, dx=\int \frac {\sqrt {- c \left (a x - 1\right )}}{x \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]

[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)

Maxima [F]

\[ \int \frac {\sqrt {c-a c x}}{x \sqrt {1-a^2 x^2}} \, dx=\int { \frac {\sqrt {-a c x + c}}{\sqrt {-a^{2} x^{2} + 1} x} \,d x } \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.46 \[ \int \frac {\sqrt {c-a c x}}{x \sqrt {1-a^2 x^2}} \, dx=-\frac {2 \, c^{3} {\left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {c}}{\sqrt {-c}}\right )}{\sqrt {-c} c} - \frac {\arctan \left (\frac {\sqrt {a c x + c}}{\sqrt {-c}}\right )}{\sqrt {-c} c}\right )}}{{\left | c \right |}} \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {c-a c x}}{x \sqrt {1-a^2 x^2}} \, dx=\int \frac {\sqrt {c-a\,c\,x}}{x\,\sqrt {1-a^2\,x^2}} \,d x \]

[In]

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

[Out]

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