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

Optimal. Leaf size=14 \[ \text{Unintegrable}\left (\left (a+b \tanh ^{-1}\left (c x^n\right )\right )^2,x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0063323, 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 = {} \[ \int \left (a+b \tanh ^{-1}\left (c x^n\right )\right )^2 \, dx \]

Verification is Not applicable to the result.

[In]

Int[(a + b*ArcTanh[c*x^n])^2,x]

[Out]

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

Rubi steps

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

Mathematica [A]  time = 1.89704, size = 0, normalized size = 0. \[ \int \left (a+b \tanh ^{-1}\left (c x^n\right )\right )^2 \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(a + b*ArcTanh[c*x^n])^2,x]

[Out]

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

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Maple [A]  time = 0.187, size = 0, normalized size = 0. \begin{align*} \int \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((a+b*arctanh(c*x^n))^2,x)

[Out]

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

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

[Out]

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

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (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((a+b*arctanh(c*x^n))^2,x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*atanh(c*x**n))**2, x)

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\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((a+b*arctanh(c*x^n))^2,x, algorithm="giac")

[Out]

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