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

Optimal. Leaf size=66 \[ \frac{1}{3} x^3 \left (a+b \tanh ^{-1}\left (c x^n\right )\right )-\frac{b c n x^{n+3} \text{Hypergeometric2F1}\left (1,\frac{n+3}{2 n},\frac{3 (n+1)}{2 n},c^2 x^{2 n}\right )}{3 (n+3)} \]

[Out]

(x^3*(a + b*ArcTanh[c*x^n]))/3 - (b*c*n*x^(3 + n)*Hypergeometric2F1[1, (3 + n)/(2*n), (3*(1 + n))/(2*n), c^2*x
^(2*n)])/(3*(3 + n))

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Rubi [A]  time = 0.0305945, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {6097, 364} \[ \frac{1}{3} x^3 \left (a+b \tanh ^{-1}\left (c x^n\right )\right )-\frac{b c n x^{n+3} \, _2F_1\left (1,\frac{n+3}{2 n};\frac{3 (n+1)}{2 n};c^2 x^{2 n}\right )}{3 (n+3)} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*ArcTanh[c*x^n]),x]

[Out]

(x^3*(a + b*ArcTanh[c*x^n]))/3 - (b*c*n*x^(3 + n)*Hypergeometric2F1[1, (3 + n)/(2*n), (3*(1 + n))/(2*n), c^2*x
^(2*n)])/(3*(3 + n))

Rule 6097

Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcTa
nh[c*x^n]))/(d*(m + 1)), x] - Dist[(b*c*n)/(d*(m + 1)), Int[(x^(n - 1)*(d*x)^(m + 1))/(1 - c^2*x^(2*n)), x], x
] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1]

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rubi steps

\begin{align*} \int x^2 \left (a+b \tanh ^{-1}\left (c x^n\right )\right ) \, dx &=\frac{1}{3} x^3 \left (a+b \tanh ^{-1}\left (c x^n\right )\right )-\frac{1}{3} (b c n) \int \frac{x^{2+n}}{1-c^2 x^{2 n}} \, dx\\ &=\frac{1}{3} x^3 \left (a+b \tanh ^{-1}\left (c x^n\right )\right )-\frac{b c n x^{3+n} \, _2F_1\left (1,\frac{3+n}{2 n};\frac{3 (1+n)}{2 n};c^2 x^{2 n}\right )}{3 (3+n)}\\ \end{align*}

Mathematica [A]  time = 0.0519168, size = 73, normalized size = 1.11 \[ -\frac{b c n x^{n+3} \text{Hypergeometric2F1}\left (1,\frac{n+3}{2 n},\frac{n+3}{2 n}+1,c^2 x^{2 n}\right )}{3 (n+3)}+\frac{a x^3}{3}+\frac{1}{3} b x^3 \tanh ^{-1}\left (c x^n\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*x^3)/3 + (b*x^3*ArcTanh[c*x^n])/3 - (b*c*n*x^(3 + n)*Hypergeometric2F1[1, (3 + n)/(2*n), 1 + (3 + n)/(2*n),
c^2*x^(2*n)])/(3*(3 + n))

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Maple [F]  time = 0.12, size = 0, normalized size = 0. \begin{align*} \int{x}^{2} \left ( a+b{\it Artanh} \left ( c{x}^{n} \right ) \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{3} \, a x^{3} + \frac{1}{6} \,{\left (x^{3} \log \left (c x^{n} + 1\right ) - x^{3} \log \left (-c x^{n} + 1\right ) + 3 \, n \int \frac{x^{2}}{3 \,{\left (c x^{n} + 1\right )}}\,{d x} + 3 \, n \int \frac{x^{2}}{3 \,{\left (c x^{n} - 1\right )}}\,{d x}\right )} b \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a*x^3 + 1/6*(x^3*log(c*x^n + 1) - x^3*log(-c*x^n + 1) + 3*n*integrate(1/3*x^2/(c*x^n + 1), x) + 3*n*integr
ate(1/3*x^2/(c*x^n - 1), x))*b

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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