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

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

[Out]

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

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Rubi [A]  time = 0.03, 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 = {6137} \[ \frac {e^{\tanh ^{-1}(a x)}}{a c} \]

Antiderivative was successfully verified.

[In]

Int[E^ArcTanh[a*x]/(c - a^2*c*x^2),x]

[Out]

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

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

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Mathematica [A]  time = 0.01, size = 26, normalized size = 2.00 \[ \frac {\sqrt {a x+1}}{a c \sqrt {1-a x}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Sqrt[1 + a*x]/(a*c*Sqrt[1 - a*x])

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fricas [A]  time = 0.49, size = 33, normalized size = 2.54 \[ \frac {a x - \sqrt {-a^{2} x^{2} + 1} - 1}{a^{2} c x - a c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.21, size = 37, normalized size = 2.85 \[ \frac {2}{c {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} - 1\right )} {\left | a \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.03, size = 25, normalized size = 1.92 \[ \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}\, a c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.42, size = 125, normalized size = 9.62 \[ -\frac {1}{2} \, a {\left (\frac {\sqrt {-a^{2} x^{2} + 1} c}{a^{3} c^{2} x + a^{2} c^{2}} + \frac {\sqrt {-a^{2} x^{2} + 1} c}{a^{3} c^{2} x - a^{2} c^{2}} - \frac {\sqrt {-a^{2} x^{2} + 1}}{a^{3} c x + a^{2} c} + \frac {\sqrt {-a^{2} x^{2} + 1}}{a^{3} c x - a^{2} c}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.87, size = 47, normalized size = 3.62 \[ \frac {\sqrt {1-a^2\,x^2}}{c\,\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )\,\sqrt {-a^2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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