3.1315 \(\int \frac {e^{n \tanh ^{-1}(a x)}}{c-a^2 c x^2} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.03, antiderivative size = 18, normalized size of antiderivative = 1.00, 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 = {6137} \[ \frac {e^{n \tanh ^{-1}(a x)}}{a c n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6137

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))/((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[E^(n*ArcTanh[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 \tanh ^{-1}(a x)}}{c-a^2 c x^2} \, dx &=\frac {e^{n \tanh ^{-1}(a x)}}{a c n}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 33, normalized size = 1.83 \[ \frac {(1-a x)^{-n/2} (a x+1)^{n/2}}{a c n} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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fricas [A]  time = 0.82, size = 27, normalized size = 1.50 \[ \frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{a c n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*arctanh(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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{a^{2} c x^{2} - c}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*arctanh(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)

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maple [A]  time = 0.03, size = 18, normalized size = 1.00 \[ \frac {{\mathrm e}^{n \arctanh \left (a x \right )}}{a c n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.33, size = 30, normalized size = 1.67 \[ \frac {e^{\left (\frac {1}{2} \, n \log \left (a x + 1\right ) - \frac {1}{2} \, n \log \left (a x - 1\right )\right )}}{a c n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*arctanh(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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mupad [B]  time = 1.11, size = 31, normalized size = 1.72 \[ \frac {{\left (a\,x+1\right )}^{n/2}}{a\,c\,n\,{\left (1-a\,x\right )}^{n/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \begin {cases} \tilde {\infty } x & \text {for}\: c = 0 \wedge n = 0 \\\tilde {\infty } \int e^{n \operatorname {atanh}{\left (a x \right )}}\, dx & \text {for}\: c = 0 \\- \frac {\log {\left (x - \frac {1}{a} \right )}}{2 a c} + \frac {\log {\left (x + \frac {1}{a} \right )}}{2 a c} & \text {for}\: n = 0 \\\frac {e^{n \operatorname {atanh}{\left (a x \right )}}}{a c n} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((zoo*x, Eq(c, 0) & Eq(n, 0)), (zoo*Integral(exp(n*atanh(a*x)), x), Eq(c, 0)), (-log(x - 1/a)/(2*a*c)
+ log(x + 1/a)/(2*a*c), Eq(n, 0)), (exp(n*atanh(a*x))/(a*c*n), True))

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