3.10.24 \(\int \frac {e^{\tanh ^{-1}(a x)}}{\sqrt {1-a^2 x^2}} \, dx\) [924]

Optimal. Leaf size=12 \[ -\frac {\log (1-a x)}{a} \]

[Out]

-ln(-a*x+1)/a

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Rubi [A]
time = 0.02, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {6275, 31} \begin {gather*} -\frac {\log (1-a x)}{a} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^ArcTanh[a*x]/Sqrt[1 - a^2*x^2],x]

[Out]

-(Log[1 - a*x]/a)

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 6275

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

Rubi steps

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

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Mathematica [A]
time = 0.00, size = 12, normalized size = 1.00 \begin {gather*} -\frac {\log (1-a x)}{a} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^ArcTanh[a*x]/Sqrt[1 - a^2*x^2],x]

[Out]

-(Log[1 - a*x]/a)

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Maple [A]
time = 0.05, size = 12, normalized size = 1.00

method	result	size
default	\(-\frac {\ln \left (a x -1\right )}{a}\)	\(12\)
norman	\(-\frac {\ln \left (a x -1\right )}{a}\)	\(12\)
risch	\(-\frac {\ln \left (a x -1\right )}{a}\)	\(12\)
meijerg	\(-\frac {\ln \left (-a^{2} x^{2}+1\right )}{2 a}+\frac {\arctanh \left (a x \right )}{a}\)	\(26\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/a*ln(a*x-1)

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Maxima [A]
time = 0.27, size = 11, normalized size = 0.92 \begin {gather*} -\frac {\log \left (a x - 1\right )}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(a*x - 1)/a

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Fricas [A]
time = 0.38, size = 11, normalized size = 0.92 \begin {gather*} -\frac {\log \left (a x - 1\right )}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(a*x - 1)/a

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Sympy [A]
time = 0.01, size = 8, normalized size = 0.67 \begin {gather*} - \frac {\log {\left (a x - 1 \right )}}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(a*x - 1)/a

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Giac [A]
time = 0.42, size = 12, normalized size = 1.00 \begin {gather*} -\frac {\log \left ({\left | a x - 1 \right |}\right )}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(abs(a*x - 1))/a

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Mupad [B]
time = 0.02, size = 11, normalized size = 0.92 \begin {gather*} -\frac {\ln \left (a\,x-1\right )}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(a*x + 1)/(a^2*x^2 - 1),x)

[Out]

-log(a*x - 1)/a

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