3.144 \(\int \frac{x}{(c+a c x) \tanh ^{-1}(a x)^2} \, dx\)

Optimal. Leaf size=18 \[ \text{Unintegrable}\left (\frac{x}{(a c x+c) \tanh ^{-1}(a x)^2},x\right ) \]

[Out]

Unintegrable[x/((c + a*c*x)*ArcTanh[a*x]^2), x]

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Rubi [A]  time = 0.0390449, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{x}{(c+a c x) \tanh ^{-1}(a x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[x/((c + a*c*x)*ArcTanh[a*x]^2),x]

[Out]

Defer[Int][x/((c + a*c*x)*ArcTanh[a*x]^2), x]

Rubi steps

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

Mathematica [A]  time = 1.93592, size = 0, normalized size = 0. \[ \int \frac{x}{(c+a c x) \tanh ^{-1}(a x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[x/((c + a*c*x)*ArcTanh[a*x]^2),x]

[Out]

Integrate[x/((c + a*c*x)*ArcTanh[a*x]^2), x]

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Maple [A]  time = 0.252, size = 0, normalized size = 0. \begin{align*} \int{\frac{x}{ \left ( acx+c \right ) \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(a*c*x+c)/arctanh(a*x)^2,x)

[Out]

int(x/(a*c*x+c)/arctanh(a*x)^2,x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{2 \,{\left (a x^{2} - x\right )}}{a c \log \left (a x + 1\right ) - a c \log \left (-a x + 1\right )} + \int -\frac{2 \,{\left (2 \, a x - 1\right )}}{a c \log \left (a x + 1\right ) - a c \log \left (-a x + 1\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(a*c*x+c)/arctanh(a*x)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x}{{\left (a c x + c\right )} \operatorname{artanh}\left (a x\right )^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(a*c*x+c)/arctanh(a*x)^2,x, algorithm="fricas")

[Out]

integral(x/((a*c*x + c)*arctanh(a*x)^2), x)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{x}{a x \operatorname{atanh}^{2}{\left (a x \right )} + \operatorname{atanh}^{2}{\left (a x \right )}}\, dx}{c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(a*c*x+c)/atanh(a*x)**2,x)

[Out]

Integral(x/(a*x*atanh(a*x)**2 + atanh(a*x)**2), x)/c

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{{\left (a c x + c\right )} \operatorname{artanh}\left (a x\right )^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(a*c*x+c)/arctanh(a*x)^2,x, algorithm="giac")

[Out]

integrate(x/((a*c*x + c)*arctanh(a*x)^2), x)