3.47 \(\int \frac{(d x)^m}{(a+b \tanh ^{-1}(c x))^2} \, dx\)

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0263568, 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 = {} \[ \int \frac{(d x)^m}{\left (a+b \tanh ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(d*x)^m/(a + b*ArcTanh[c*x])^2,x]

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.499509, size = 0, normalized size = 0. \[ \int \frac{(d x)^m}{\left (a+b \tanh ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.658, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( dx \right ) ^{m}}{ \left ( a+b{\it Artanh} \left ( cx \right ) \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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