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

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

[Out]

Unintegrable(1/(a*c*x+c)/arctanh(a*x)^2,x)

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Rubi [A]  time = 0.02, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {1}{(c+a c x) \tanh ^{-1}(a x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[1/((c + a*c*x)*ArcTanh[a*x]^2),x]

[Out]

Defer[Int][1/((c + a*c*x)*ArcTanh[a*x]^2), x]

Rubi steps

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

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Mathematica [A]  time = 1.04, size = 0, normalized size = 0.00 \[ \int \frac {1}{(c+a c x) \tanh ^{-1}(a x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[1/((c + a*c*x)*ArcTanh[a*x]^2),x]

[Out]

Integrate[1/((c + a*c*x)*ArcTanh[a*x]^2), x]

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fricas [A]  time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{{\left (a c x + c\right )} \operatorname {artanh}\left (a x\right )^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(1/((a*c*x + c)*arctanh(a*x)^2), x)

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (a c x + c\right )} \operatorname {artanh}\left (a x\right )^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.26, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a c x +c \right ) \arctanh \left (a x \right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*c*x+c)/arctanh(a*x)^2,x)

[Out]

int(1/(a*c*x+c)/arctanh(a*x)^2,x)

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {2 \, {\left (a x - 1\right )}}{a c \log \left (a x + 1\right ) - a c \log \left (-a x + 1\right )} + 2 \, \int -\frac {1}{c \log \left (a x + 1\right ) - c \log \left (-a x + 1\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [A]  time = 0.00, size = -1, normalized size = -0.06 \[ \int \frac {1}{{\mathrm {atanh}\left (a\,x\right )}^2\,\left (c+a\,c\,x\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(atanh(a*x)^2*(c + a*c*x)),x)

[Out]

int(1/(atanh(a*x)^2*(c + a*c*x)), x)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {1}{a x \operatorname {atanh}^{2}{\left (a x \right )} + \operatorname {atanh}^{2}{\left (a x \right )}}\, dx}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*c*x+c)/atanh(a*x)**2,x)

[Out]

Integral(1/(a*x*atanh(a*x)**2 + atanh(a*x)**2), x)/c

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