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

Optimal. Leaf size=16 \[ \text{Unintegrable}\left (x \left (a+b \tanh ^{-1}\left (c x^n\right )\right )^2,x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0149523, 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 = {} \[ \int x \left (a+b \tanh ^{-1}\left (c x^n\right )\right )^2 \, dx \]

Verification is Not applicable to the result.

[In]

Int[x*(a + b*ArcTanh[c*x^n])^2,x]

[Out]

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

Rubi steps

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

Mathematica [A]  time = 8.61828, size = 0, normalized size = 0. \[ \int x \left (a+b \tanh ^{-1}\left (c x^n\right )\right )^2 \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.162, size = 0, normalized size = 0. \begin{align*} \int x \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(x*(a+b*arctanh(c*x^n))^2,x)

[Out]

int(x*(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}{8} \, b^{2} x^{2} \log \left (-c x^{n} + 1\right )^{2} + \frac{1}{2} \, a^{2} x^{2} - \int -\frac{{\left (b^{2} c x x^{n} - b^{2} x\right )} \log \left (c x^{n} + 1\right )^{2} + 4 \,{\left (a b c x x^{n} - a b x\right )} \log \left (c x^{n} + 1\right ) +{\left (4 \, a b x -{\left (b^{2} c n + 4 \, a b c\right )} x x^{n} - 2 \,{\left (b^{2} c x x^{n} - b^{2} x\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(x*(a+b*arctanh(c*x^n))^2,x, algorithm="maxima")

[Out]

1/8*b^2*x^2*log(-c*x^n + 1)^2 + 1/2*a^2*x^2 - integrate(-1/4*((b^2*c*x*x^n - b^2*x)*log(c*x^n + 1)^2 + 4*(a*b*
c*x*x^n - a*b*x)*log(c*x^n + 1) + (4*a*b*x - (b^2*c*n + 4*a*b*c)*x*x^n - 2*(b^2*c*x*x^n - b^2*x)*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} x \operatorname{artanh}\left (c x^{n}\right )^{2} + 2 \, a b x \operatorname{artanh}\left (c x^{n}\right ) + a^{2} x, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(b^2*x*arctanh(c*x^n)^2 + 2*a*b*x*arctanh(c*x^n) + a^2*x, 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(x*(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{\left (b \operatorname{artanh}\left (c x^{n}\right ) + a\right )}^{2} x\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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