3.1.13 \(\int e^{2 \tanh ^{-1}(a x)} x \, dx\) [13]

Optimal. Leaf size=26 \[ -\frac {2 x}{a}-\frac {x^2}{2}-\frac {2 \log (1-a x)}{a^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.02, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6261, 78} \begin {gather*} -\frac {2 \log (1-a x)}{a^2}-\frac {2 x}{a}-\frac {x^2}{2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(2*ArcTanh[a*x])*x,x]

[Out]

(-2*x)/a - x^2/2 - (2*Log[1 - a*x])/a^2

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rule 6261

Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*(x_)^(m_.), x_Symbol] :> Int[x^m*((1 + a*x)^(n/2)/(1 - a*x)^(n/2)), x] /; Fre
eQ[{a, m, n}, x] &&  !IntegerQ[(n - 1)/2]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]
time = 0.01, size = 26, normalized size = 1.00 \begin {gather*} -\frac {2 x}{a}-\frac {x^2}{2}-\frac {2 \log (1-a x)}{a^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*x)/a - x^2/2 - (2*Log[1 - a*x])/a^2

________________________________________________________________________________________

Maple [A]
time = 0.74, size = 28, normalized size = 1.08

method	result	size
norman	\(-\frac {x^{2}}{2}-\frac {2 x}{a}-\frac {2 \ln \left (a x -1\right )}{a^{2}}\)	\(24\)
risch	\(-\frac {x^{2}}{2}-\frac {2 x}{a}-\frac {2 \ln \left (a x -1\right )}{a^{2}}\)	\(24\)
default	\(-\frac {\frac {1}{2} a \,x^{2}+2 x}{a}-\frac {2 \ln \left (a x -1\right )}{a^{2}}\)	\(28\)
meijerg	\(\frac {-a^{2} x^{2}-\ln \left (-a^{2} x^{2}+1\right )}{2 a^{2}}-\frac {-\frac {2 x \left (-a^{2}\right )^{\frac {3}{2}}}{a^{2}}+\frac {2 \left (-a^{2}\right )^{\frac {3}{2}} \arctanh \left (a x \right )}{a^{3}}}{a \sqrt {-a^{2}}}-\frac {\ln \left (-a^{2} x^{2}+1\right )}{2 a^{2}}\)	\(87\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.25, size = 26, normalized size = 1.00 \begin {gather*} -\frac {a x^{2} + 4 \, x}{2 \, a} - \frac {2 \, \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)^2/(-a^2*x^2+1)*x,x, algorithm="maxima")

[Out]

-1/2*(a*x^2 + 4*x)/a - 2*log(a*x - 1)/a^2

________________________________________________________________________________________

Fricas [A]
time = 0.38, size = 25, normalized size = 0.96 \begin {gather*} -\frac {a^{2} x^{2} + 4 \, a x + 4 \, \log \left (a x - 1\right )}{2 \, a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(a^2*x^2 + 4*a*x + 4*log(a*x - 1))/a^2

________________________________________________________________________________________

Sympy [A]
time = 0.10, size = 22, normalized size = 0.85 \begin {gather*} - \frac {x^{2}}{2} - \frac {2 x}{a} - \frac {2 \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)**2/(-a**2*x**2+1)*x,x)

[Out]

-x**2/2 - 2*x/a - 2*log(a*x - 1)/a**2

________________________________________________________________________________________

Giac [A]
time = 0.41, size = 30, normalized size = 1.15 \begin {gather*} -\frac {a^{2} x^{2} + 4 \, a x}{2 \, a^{2}} - \frac {2 \, \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)^2/(-a^2*x^2+1)*x,x, algorithm="giac")

[Out]

-1/2*(a^2*x^2 + 4*a*x)/a^2 - 2*log(abs(a*x - 1))/a^2

________________________________________________________________________________________

Mupad [B]
time = 0.80, size = 23, normalized size = 0.88 \begin {gather*} -\frac {2\,\ln \left (a\,x-1\right )}{a^2}-\frac {2\,x}{a}-\frac {x^2}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x*(a*x + 1)^2)/(a^2*x^2 - 1),x)

[Out]

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

________________________________________________________________________________________