3.1.30 \(\int \frac {e^{4 \tanh ^{-1}(a x)}}{x} \, dx\) [30]

Optimal. Leaf size=13 \[ \frac {4}{1-a x}+\log (x) \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6261, 90} \begin {gather*} \frac {4}{1-a x}+\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(4*ArcTanh[a*x])/x,x]

[Out]

4/(1 - a*x) + Log[x]

Rule 90

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

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 \frac {e^{4 \tanh ^{-1}(a x)}}{x} \, dx &=\int \frac {(1+a x)^2}{x (1-a x)^2} \, dx\\ &=\int \left (\frac {1}{x}+\frac {4 a}{(-1+a x)^2}\right ) \, dx\\ &=\frac {4}{1-a x}+\log (x)\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 13, normalized size = 1.00 \begin {gather*} \frac {4}{1-a x}+\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(4*ArcTanh[a*x])/x,x]

[Out]

4/(1 - a*x) + Log[x]

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Maple [A]
time = 0.90, size = 13, normalized size = 1.00

method	result	size
default	\(-\frac {4}{a x -1}+\ln \left (x \right )\)	\(13\)
risch	\(-\frac {4}{a x -1}+\ln \left (-x \right )\)	\(15\)
norman	\(\frac {-4 a^{2} x^{2}-4 a x}{a^{2} x^{2}-1}+\ln \left (x \right )\)	\(29\)
meijerg	\(\frac {7 a^{2} x^{2}}{2 \left (-a^{2} x^{2}+1\right )}-\frac {2 a \left (\frac {x \left (-a^{2}\right )^{\frac {3}{2}}}{a^{2} \left (-a^{2} x^{2}+1\right )}-\frac {\left (-a^{2}\right )^{\frac {3}{2}} \arctanh \left (a x \right )}{a^{3}}\right )}{\sqrt {-a^{2}}}+\frac {2 a \left (\frac {2 x \sqrt {-a^{2}}}{-2 a^{2} x^{2}+2}+\frac {\sqrt {-a^{2}}\, \arctanh \left (a x \right )}{a}\right )}{\sqrt {-a^{2}}}+\frac {a^{2} x^{2}}{-2 a^{2} x^{2}+2}+\frac {1}{2}+\ln \left (x \right )+\frac {\ln \left (-a^{2}\right )}{2}\)	\(151\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x+1)^4/(-a^2*x^2+1)^2/x,x,method=_RETURNVERBOSE)

[Out]

-4/(a*x-1)+ln(x)

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Maxima [A]
time = 0.26, size = 12, normalized size = 0.92 \begin {gather*} -\frac {4}{a x - 1} + \log \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^4/(-a^2*x^2+1)^2/x,x, algorithm="maxima")

[Out]

-4/(a*x - 1) + log(x)

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Fricas [A]
time = 0.34, size = 18, normalized size = 1.38 \begin {gather*} \frac {{\left (a x - 1\right )} \log \left (x\right ) - 4}{a x - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^4/(-a^2*x^2+1)^2/x,x, algorithm="fricas")

[Out]

((a*x - 1)*log(x) - 4)/(a*x - 1)

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Sympy [A]
time = 0.08, size = 8, normalized size = 0.62 \begin {gather*} \log {\left (x \right )} - \frac {4}{a x - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)**4/(-a**2*x**2+1)**2/x,x)

[Out]

log(x) - 4/(a*x - 1)

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Giac [A]
time = 0.40, size = 13, normalized size = 1.00 \begin {gather*} -\frac {4}{a x - 1} + \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^4/(-a^2*x^2+1)^2/x,x, algorithm="giac")

[Out]

-4/(a*x - 1) + log(abs(x))

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Mupad [B]
time = 0.04, size = 12, normalized size = 0.92 \begin {gather*} \ln \left (x\right )-\frac {4}{a\,x-1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x) - 4/(a*x - 1)

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