\(\int \frac {e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^2} \, dx\) [494]

Optimal result

Integrand size = 25, antiderivative size = 37 \[ \int \frac {e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^2} \, dx=-\frac {2 a c^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{3 \left (c-\frac {c}{a x}\right )^{3/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {6313, 663} \[ \int \frac {e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^2} \, dx=-\frac {2 a c^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{3 \left (c-\frac {c}{a x}\right )^{3/2}} \]

[In]

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

[Out]

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

Rule 663

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

Rule 6313

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_))^(p_.)*(x_)^(m_.), x_Symbol] :> Dist[-c^n, Subst[Int[(c +
d*x)^(p - n)*((1 - x^2/a^2)^(n/2)/x^(m + 2)), x], x, 1/x], x] /; FreeQ[{a, c, d, p}, x] && EqQ[c + a*d, 0] &&
IntegerQ[(n - 1)/2] && IntegerQ[m] && (IntegerQ[p] || EqQ[p, n/2] || EqQ[p, n/2 + 1] || LtQ[-5, m, -1]) && In
tegerQ[2*p]

Rubi steps \begin{align*} \text {integral}& = -\left (c \text {Subst}\left (\int \frac {\sqrt {1-\frac {x^2}{a^2}}}{\sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )\right ) \\ & = -\frac {2 a c^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{3 \left (c-\frac {c}{a x}\right )^{3/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.22 \[ \int \frac {e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^2} \, dx=-\frac {2 a \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {c-\frac {c}{a x}} (1+a x)}{-3+3 a x} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.05 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.11

method	result	size
gosper	\(-\frac {2 \left (a x +1\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}}{3 x \sqrt {\frac {a x -1}{a x +1}}}\)	\(41\)
default	\(-\frac {2 \left (a x +1\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}}{3 x \sqrt {\frac {a x -1}{a x +1}}}\)	\(41\)
risch	\(-\frac {2 \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \left (a^{2} x^{2}+2 a x +1\right )}{3 \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) x}\)	\(56\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.57 \[ \int \frac {e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^2} \, dx=-\frac {2 \, {\left (a^{2} x^{2} + 2 \, a x + 1\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{3 \, {\left (a x^{2} - x\right )}} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 4.44 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.27 \[ \int \frac {e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^2} \, dx=-\frac {2\,\sqrt {c-\frac {c}{a\,x}}\,{\left (a\,x+1\right )}^2\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{3\,x\,\left (a\,x-1\right )} \]

[In]

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

[Out]

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