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

Optimal. Leaf size=67 \[ \frac{1}{2} x^2 \left (a+b \tanh ^{-1}\left (c x^n\right )\right )-\frac{b c n x^{n+2} \text{Hypergeometric2F1}\left (1,\frac{n+2}{2 n},\frac{1}{2} \left (\frac{2}{n}+3\right ),c^2 x^{2 n}\right )}{2 (n+2)} \]

[Out]

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

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Rubi [A]  time = 0.0243278, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {6097, 364} \[ \frac{1}{2} x^2 \left (a+b \tanh ^{-1}\left (c x^n\right )\right )-\frac{b c n x^{n+2} \, _2F_1\left (1,\frac{n+2}{2 n};\frac{1}{2} \left (3+\frac{2}{n}\right );c^2 x^{2 n}\right )}{2 (n+2)} \]

Antiderivative was successfully verified.

[In]

Int[x*(a + b*ArcTanh[c*x^n]),x]

[Out]

(x^2*(a + b*ArcTanh[c*x^n]))/2 - (b*c*n*x^(2 + n)*Hypergeometric2F1[1, (2 + n)/(2*n), (3 + 2/n)/2, c^2*x^(2*n)
])/(2*(2 + 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 \left (a+b \tanh ^{-1}\left (c x^n\right )\right ) \, dx &=\frac{1}{2} x^2 \left (a+b \tanh ^{-1}\left (c x^n\right )\right )-\frac{1}{2} (b c n) \int \frac{x^{1+n}}{1-c^2 x^{2 n}} \, dx\\ &=\frac{1}{2} x^2 \left (a+b \tanh ^{-1}\left (c x^n\right )\right )-\frac{b c n x^{2+n} \, _2F_1\left (1,\frac{2+n}{2 n};\frac{1}{2} \left (3+\frac{2}{n}\right );c^2 x^{2 n}\right )}{2 (2+n)}\\ \end{align*}

Mathematica [A]  time = 0.0445054, size = 73, normalized size = 1.09 \[ -\frac{b c n x^{n+2} \text{Hypergeometric2F1}\left (1,\frac{n+2}{2 n},\frac{n+2}{2 n}+1,c^2 x^{2 n}\right )}{2 (n+2)}+\frac{a x^2}{2}+\frac{1}{2} b x^2 \tanh ^{-1}\left (c x^n\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x*(a + b*ArcTanh[c*x^n]),x]

[Out]

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

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Maple [F]  time = 0.21, size = 0, normalized size = 0. \begin{align*} \int x \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*(a+b*arctanh(c*x^n)),x)

[Out]

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

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(b*x*arctanh(c*x^n) + a*x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x*(a + b*atanh(c*x**n)), x)

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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\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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