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

Optimal. Leaf size=20 \[ \text{Unintegrable}\left (\frac{(d x)^m}{\left (a+b \tanh ^{-1}\left (c x^n\right )\right )^2},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0264206, 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 = {} \[ \int \frac{(d x)^m}{\left (a+b \tanh ^{-1}\left (c x^n\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 1.7533, size = 0, normalized size = 0. \[ \int \frac{(d x)^m}{\left (a+b \tanh ^{-1}\left (c x^n\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.275, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( dx \right ) ^{m}}{ \left ( a+b{\it Artanh} \left ( c{x}^{n} \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^n))^2,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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^{n}\right )^{2} + 2 \, a b \operatorname{artanh}\left (c x^{n}\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^n))^2,x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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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^{n}\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^n))^2,x, algorithm="giac")

[Out]

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