3.10.23 \(\int \frac {e^{\tanh ^{-1}(a x)} x}{\sqrt {1-a^2 x^2}} \, dx\) [923]

Optimal. Leaf size=19 \[ -\frac {x}{a}-\frac {\log (1-a x)}{a^2} \]

[Out]

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

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Rubi [A]
time = 0.04, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {6285, 45} \begin {gather*} -\frac {\log (1-a x)}{a^2}-\frac {x}{a} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(x/a) - Log[1 - a*x]/a^2

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 6285

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[c^p, Int[x^m*(1 -
a*x)^(p - n/2)*(1 + a*x)^(p + n/2), x], x] /; FreeQ[{a, c, d, m, 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)} x}{\sqrt {1-a^2 x^2}} \, dx &=\int \frac {x}{1-a x} \, dx\\ &=\int \left (-\frac {1}{a}-\frac {1}{a (-1+a x)}\right ) \, dx\\ &=-\frac {x}{a}-\frac {\log (1-a x)}{a^2}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 19, normalized size = 1.00 \begin {gather*} -\frac {x}{a}-\frac {\log (1-a x)}{a^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(x/a) - Log[1 - a*x]/a^2

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Maple [A]
time = 0.06, size = 19, normalized size = 1.00

method	result	size
default	\(-\frac {x}{a}-\frac {\ln \left (a x -1\right )}{a^{2}}\)	\(19\)
norman	\(-\frac {x}{a}-\frac {\ln \left (a x -1\right )}{a^{2}}\)	\(19\)
risch	\(-\frac {x}{a}-\frac {\ln \left (a x -1\right )}{a^{2}}\)	\(19\)
meijerg	\(-\frac {-\frac {2 x \left (-a^{2}\right )^{\frac {3}{2}}}{a^{2}}+\frac {2 \left (-a^{2}\right )^{\frac {3}{2}} \arctanh \left (a x \right )}{a^{3}}}{2 a \sqrt {-a^{2}}}-\frac {\ln \left (-a^{2} x^{2}+1\right )}{2 a^{2}}\)	\(60\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.26, size = 18, normalized size = 0.95 \begin {gather*} -\frac {x}{a} - \frac {\log \left (a x - 1\right )}{a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x/a - log(a*x - 1)/a^2

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Fricas [A]
time = 0.36, size = 15, normalized size = 0.79 \begin {gather*} -\frac {a x + \log \left (a x - 1\right )}{a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(a*x + log(a*x - 1))/a^2

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Sympy [A]
time = 0.03, size = 14, normalized size = 0.74 \begin {gather*} - \frac {x}{a} - \frac {\log {\left (a x - 1 \right )}}{a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x/a - log(a*x - 1)/a**2

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Giac [A]
time = 0.42, size = 19, normalized size = 1.00 \begin {gather*} -\frac {x}{a} - \frac {\log \left ({\left | a x - 1 \right |}\right )}{a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x/a - log(abs(a*x - 1))/a^2

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Mupad [B]
time = 0.86, size = 15, normalized size = 0.79 \begin {gather*} -\frac {\ln \left (a\,x-1\right )+a\,x}{a^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x*(a*x + 1))/(a^2*x^2 - 1),x)

[Out]

-(log(a*x - 1) + a*x)/a^2

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