3.235 \(\int \frac{(a+b \tanh ^{-1}(c x^n))^2}{x^3} \, dx\)

Optimal. Leaf size=18 \[ \text{Unintegrable}\left (\frac{\left (a+b \tanh ^{-1}\left (c x^n\right )\right )^2}{x^3},x\right ) \]

[Out]

Unintegrable[(a + b*ArcTanh[c*x^n])^2/x^3, x]

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Rubi [A]  time = 0.0233473, 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 = {} \[ \int \frac{\left (a+b \tanh ^{-1}\left (c x^n\right )\right )^2}{x^3} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(a + b*ArcTanh[c*x^n])^2/x^3,x]

[Out]

Defer[Int][(a + b*ArcTanh[c*x^n])^2/x^3, x]

Rubi steps

\begin{align*} \int \frac{\left (a+b \tanh ^{-1}\left (c x^n\right )\right )^2}{x^3} \, dx &=\int \frac{\left (a+b \tanh ^{-1}\left (c x^n\right )\right )^2}{x^3} \, dx\\ \end{align*}

Mathematica [A]  time = 12.3165, size = 0, normalized size = 0. \[ \int \frac{\left (a+b \tanh ^{-1}\left (c x^n\right )\right )^2}{x^3} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(a + b*ArcTanh[c*x^n])^2/x^3,x]

[Out]

Integrate[(a + b*ArcTanh[c*x^n])^2/x^3, x]

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Maple [A]  time = 0.109, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b{\it Artanh} \left ( c{x}^{n} \right ) \right ) ^{2}}{{x}^{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arctanh(c*x^n))^2/x^3,x)

[Out]

int((a+b*arctanh(c*x^n))^2/x^3,x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{b^{2} \log \left (-c x^{n} + 1\right )^{2}}{8 \, x^{2}} - \frac{a^{2}}{2 \, x^{2}} - \int -\frac{{\left (b^{2} c x^{n} - b^{2}\right )} \log \left (c x^{n} + 1\right )^{2} + 4 \,{\left (a b c x^{n} - a b\right )} \log \left (c x^{n} + 1\right ) +{\left (4 \, a b +{\left (b^{2} c n - 4 \, a b c\right )} x^{n} - 2 \,{\left (b^{2} c x^{n} - b^{2}\right )} \log \left (c x^{n} + 1\right )\right )} \log \left (-c x^{n} + 1\right )}{4 \,{\left (c x^{3} x^{n} - x^{3}\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctanh(c*x^n))^2/x^3,x, algorithm="maxima")

[Out]

-1/8*b^2*log(-c*x^n + 1)^2/x^2 - 1/2*a^2/x^2 - integrate(-1/4*((b^2*c*x^n - b^2)*log(c*x^n + 1)^2 + 4*(a*b*c*x
^n - a*b)*log(c*x^n + 1) + (4*a*b + (b^2*c*n - 4*a*b*c)*x^n - 2*(b^2*c*x^n - b^2)*log(c*x^n + 1))*log(-c*x^n +
1))/(c*x^3*x^n - x^3), x)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b^{2} \operatorname{artanh}\left (c x^{n}\right )^{2} + 2 \, a b \operatorname{artanh}\left (c x^{n}\right ) + a^{2}}{x^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctanh(c*x^n))^2/x^3,x, algorithm="fricas")

[Out]

integral((b^2*arctanh(c*x^n)^2 + 2*a*b*arctanh(c*x^n) + a^2)/x^3, 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((a+b*atanh(c*x**n))**2/x**3,x)

[Out]

Timed out

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \operatorname{artanh}\left (c x^{n}\right ) + a\right )}^{2}}{x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctanh(c*x^n))^2/x^3,x, algorithm="giac")

[Out]

integrate((b*arctanh(c*x^n) + a)^2/x^3, x)