### 3.740 $$\int \frac{e^{n \coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx$$

Optimal. Leaf size=18 $\frac{e^{n \coth ^{-1}(a x)}}{a c n}$

[Out]

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

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Rubi [A]  time = 0.0345959, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 22, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.045, Rules used = {6183} $\frac{e^{n \coth ^{-1}(a x)}}{a c n}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Rule 6183

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

Mathematica [A]  time = 0.0493782, size = 18, normalized size = 1. $\frac{e^{n \coth ^{-1}(a x)}}{a c n}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.04, size = 18, normalized size = 1. \begin{align*}{\frac{{{\rm e}^{n{\rm arccoth} \left (ax\right )}}}{can}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(exp(n*arccoth(a*x))/(-a^2*c*x^2+c),x)

[Out]

exp(n*arccoth(a*x))/a/c/n

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Maxima [A]  time = 1.09939, size = 42, normalized size = 2.33 \begin{align*} -\frac{e^{\left (-\frac{1}{2} \, n \log \left (a x + 1\right ) + \frac{1}{2} \, n \log \left (a x - 1\right )\right )}}{a c n} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.67133, size = 54, normalized size = 3. \begin{align*} -\frac{\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{2} \, n}}{a c n} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*acoth(a*x))/(-a**2*c*x**2+c),x)

[Out]

Exception raised: TypeError

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{2} \, n}}{a^{2} c x^{2} - c}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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