3.831 \(\int \frac {e^{2 \tanh ^{-1}(a+b x)}}{x} \, dx\)

Optimal. Leaf size=33 \[ \frac {(a+1) \log (x)}{1-a}-\frac {2 \log (-a-b x+1)}{1-a} \]

[Out]

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

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Rubi [A]  time = 0.04, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {6163, 72} \[ \frac {(a+1) \log (x)}{1-a}-\frac {2 \log (-a-b x+1)}{1-a} \]

Antiderivative was successfully verified.

[In]

Int[E^(2*ArcTanh[a + b*x])/x,x]

[Out]

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

Rule 72

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]

Rule 6163

Int[E^(ArcTanh[(c_.)*((a_) + (b_.)*(x_))]*(n_.))*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Int[((d + e*x)^m*(1
+ a*c + b*c*x)^(n/2))/(1 - a*c - b*c*x)^(n/2), x] /; FreeQ[{a, b, c, d, e, m, n}, x]

Rubi steps

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

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Mathematica [A]  time = 0.02, size = 26, normalized size = 0.79 \[ \frac {2 \log (-a-b x+1)-(a+1) \log (x)}{a-1} \]

Antiderivative was successfully verified.

[In]

Integrate[E^(2*ArcTanh[a + b*x])/x,x]

[Out]

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

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fricas [A]  time = 0.46, size = 23, normalized size = 0.70 \[ -\frac {{\left (a + 1\right )} \log \relax (x) - 2 \, \log \left (b x + a - 1\right )}{a - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((a + 1)*log(x) - 2*log(b*x + a - 1))/(a - 1)

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giac [A]  time = 0.19, size = 34, normalized size = 1.03 \[ \frac {2 \, b \log \left ({\left | b x + a - 1 \right |}\right )}{a b - b} - \frac {{\left (a + 1\right )} \log \left ({\left | x \right |}\right )}{a - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.03, size = 35, normalized size = 1.06 \[ -\frac {\ln \relax (x )}{a -1}-\frac {\ln \relax (x ) a}{a -1}+\frac {2 \ln \left (b x +a -1\right )}{a -1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.31, size = 27, normalized size = 0.82 \[ -\frac {{\left (a + 1\right )} \log \relax (x)}{a - 1} + \frac {2 \, \log \left (b x + a - 1\right )}{a - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(a + 1)*log(x)/(a - 1) + 2*log(b*x + a - 1)/(a - 1)

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mupad [B]  time = 0.12, size = 28, normalized size = 0.85 \[ \frac {2\,\ln \left (a+b\,x-1\right )}{a-1}-\frac {2\,\ln \relax (x)}{a-1}-\ln \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*log(a + b*x - 1))/(a - 1) - (2*log(x))/(a - 1) - log(x)

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sympy [B]  time = 0.46, size = 88, normalized size = 2.67 \[ - \frac {\left (a + 1\right ) \log {\left (x + \frac {a^{2} - \frac {a^{2} \left (a + 1\right )}{a - 1} + \frac {2 a \left (a + 1\right )}{a - 1} - 1 - \frac {a + 1}{a - 1}}{a b + 3 b} \right )}}{a - 1} + \frac {2 \log {\left (x + \frac {a^{2} + \frac {2 a^{2}}{a - 1} - \frac {4 a}{a - 1} - 1 + \frac {2}{a - 1}}{a b + 3 b} \right )}}{a - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(a + 1)*log(x + (a**2 - a**2*(a + 1)/(a - 1) + 2*a*(a + 1)/(a - 1) - 1 - (a + 1)/(a - 1))/(a*b + 3*b))/(a - 1
) + 2*log(x + (a**2 + 2*a**2/(a - 1) - 4*a/(a - 1) - 1 + 2/(a - 1))/(a*b + 3*b))/(a - 1)

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