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

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

[Out]

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

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Rubi [A]  time = 0.0263933, 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 = {} \[ \int \frac{(d x)^m}{\left (a+b \tanh ^{-1}\left (\frac{c}{x^2}\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.389505, size = 0, normalized size = 0. \[ \int \frac{(d x)^m}{\left (a+b \tanh ^{-1}\left (\frac{c}{x^2}\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.146, size = 0, normalized size = 0. \begin{align*} \int{ \left ( dx \right ) ^{m} \left ( a+b{\it Artanh} \left ({\frac{c}{{x}^{2}}} \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))^2,x)

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{{\left (d^{m} x^{4} - c^{2} d^{m}\right )} x^{m}}{b^{2} c x \log \left (x^{2} + c\right ) - b^{2} c x \log \left (x^{2} - c\right ) + 2 \, a b c x} + \int -\frac{{\left (d^{m}{\left (m + 3\right )} x^{4} - c^{2} d^{m}{\left (m - 1\right )}\right )} x^{m}}{b^{2} c x^{2} \log \left (x^{2} + c\right ) - b^{2} c x^{2} \log \left (x^{2} - c\right ) + 2 \, a b c x^{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))^2,x, algorithm="maxima")

[Out]

(d^m*x^4 - c^2*d^m)*x^m/(b^2*c*x*log(x^2 + c) - b^2*c*x*log(x^2 - c) + 2*a*b*c*x) + integrate(-(d^m*(m + 3)*x^
4 - c^2*d^m*(m - 1))*x^m/(b^2*c*x^2*log(x^2 + c) - b^2*c*x^2*log(x^2 - c) + 2*a*b*c*x^2), 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 (\frac{c}{x^{2}}\right )^{2} + 2 \, a b \operatorname{artanh}\left (\frac{c}{x^{2}}\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))^2,x, algorithm="fricas")

[Out]

integral((d*x)^m/(b^2*arctanh(c/x^2)^2 + 2*a*b*arctanh(c/x^2) + 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**2))**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 (\frac{c}{x^{2}}\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))^2,x, algorithm="giac")

[Out]

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