\(\int e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x \, dx\) [882]

Optimal result

Integrand size = 23, antiderivative size = 71 \[ \int e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x \, dx=\frac {\sqrt {c-\frac {c}{a^2 x^2}} x}{a \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {c-\frac {c}{a^2 x^2}} x^2}{2 \sqrt {1-\frac {1}{a^2 x^2}}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {6332, 6328} \[ \int e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x \, dx=\frac {x^2 \sqrt {c-\frac {c}{a^2 x^2}}}{2 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {x \sqrt {c-\frac {c}{a^2 x^2}}}{a \sqrt {1-\frac {1}{a^2 x^2}}} \]

[In]

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

[Out]

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

Rule 6328

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_)^2)^(p_.), x_Symbol] :> Dist[c^p/a^(2*p), Int[(u/x^(
2*p))*(-1 + a*x)^(p - n/2)*(1 + a*x)^(p + n/2), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c + a^2*d, 0] &&  !
IntegerQ[n/2] && (IntegerQ[p] || GtQ[c, 0]) && IntegersQ[2*p, p + n/2]

Rule 6332

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_)^2)^(p_), x_Symbol] :> Dist[c^IntPart[p]*((c + d/x^2
)^FracPart[p]/(1 - 1/(a^2*x^2))^FracPart[p]), Int[u*(1 - 1/(a^2*x^2))^p*E^(n*ArcCoth[a*x]), x], x] /; FreeQ[{a
, c, d, n, p}, x] && EqQ[c + a^2*d, 0] &&  !IntegerQ[n/2] &&  !(IntegerQ[p] || GtQ[c, 0])

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {c-\frac {c}{a^2 x^2}} \int e^{\coth ^{-1}(a x)} \sqrt {1-\frac {1}{a^2 x^2}} x \, dx}{\sqrt {1-\frac {1}{a^2 x^2}}} \\ & = \frac {\sqrt {c-\frac {c}{a^2 x^2}} \int (1+a x) \, dx}{a \sqrt {1-\frac {1}{a^2 x^2}}} \\ & = \frac {\sqrt {c-\frac {c}{a^2 x^2}} x}{a \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {c-\frac {c}{a^2 x^2}} x^2}{2 \sqrt {1-\frac {1}{a^2 x^2}}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.61 \[ \int e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x \, dx=\frac {\sqrt {c-\frac {c}{a^2 x^2}} \left (\frac {x}{a}+\frac {x^2}{2}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.73

method	result	size
gosper	\(\frac {x^{2} \left (a x +2\right ) \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}}{2 \left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}\)	\(52\)
default	\(\frac {x^{2} \left (a x +2\right ) \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}}{2 \left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}\)	\(52\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.30 \[ \int e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x \, dx=\frac {\sqrt {a^{2} c} {\left (a x^{2} + 2 \, x\right )}}{2 \, a^{2}} \]

[In]

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

[Out]

1/2*sqrt(a^2*c)*(a*x^2 + 2*x)/a^2

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 4.06 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.63 \[ \int e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x \, dx=\frac {x^2\,\sqrt {c-\frac {c}{a^2\,x^2}}\,\left (a\,x+2\right )\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{2\,\left (a\,x-1\right )} \]

[In]

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

[Out]

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