3.6.60 \(\int \frac {e^{\coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx\) [560]

Optimal. Leaf size=13 \[ \frac {e^{\coth ^{-1}(a x)}}{a c} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {6318} \begin {gather*} \frac {e^{\coth ^{-1}(a x)}}{a c} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^ArcCoth[a*x]/(c - a^2*c*x^2),x]

[Out]

E^ArcCoth[a*x]/(a*c)

Rule 6318

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))/((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[E^(n*ArcCoth[a*x])/(a*c*n), x] /; F
reeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] &&  !IntegerQ[n/2]

Rubi steps

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

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Mathematica [A]
time = 0.03, size = 13, normalized size = 1.00 \begin {gather*} \frac {e^{\coth ^{-1}(a x)}}{a c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

E^ArcCoth[a*x]/(a*c)

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Maple [A]
time = 0.19, size = 23, normalized size = 1.77

method	result	size
gosper	\(\frac {1}{\sqrt {\frac {a x -1}{a x +1}}\, a c}\)	\(23\)
default	\(\frac {1}{\sqrt {\frac {a x -1}{a x +1}}\, a c}\)	\(23\)
trager	\(\frac {\left (a x +1\right ) \sqrt {-\frac {-a x +1}{a x +1}}}{a c \left (a x -1\right )}\)	\(37\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.26, size = 22, normalized size = 1.69 \begin {gather*} \frac {1}{a c \sqrt {\frac {a x - 1}{a x + 1}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/(a*c*sqrt((a*x - 1)/(a*x + 1)))

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Fricas [A]
time = 0.33, size = 34, normalized size = 2.62 \begin {gather*} \frac {{\left (a x + 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} c x - a c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \frac {\int \frac {1}{a^{2} x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}} - \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\, dx}{c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

undef

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Mupad [B]
time = 1.22, size = 22, normalized size = 1.69 \begin {gather*} \frac {1}{a\,c\,\sqrt {\frac {a\,x-1}{a\,x+1}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/(a*c*((a*x - 1)/(a*x + 1))^(1/2))

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