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

Optimal. Leaf size=58 \[ -\frac{b c n x^{n+1} \text{Hypergeometric2F1}\left (1,\frac{n+1}{2 n},\frac{1}{2} \left (\frac{1}{n}+3\right ),c^2 x^{2 n}\right )}{n+1}+a x+b x \tanh ^{-1}\left (c x^n\right ) \]

[Out]

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

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Rubi [A]  time = 0.0279346, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {6091, 364} \[ a x-\frac{b c n x^{n+1} \, _2F_1\left (1,\frac{n+1}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );c^2 x^{2 n}\right )}{n+1}+b x \tanh ^{-1}\left (c x^n\right ) \]

Antiderivative was successfully verified.

[In]

Int[a + b*ArcTanh[c*x^n],x]

[Out]

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

Rule 6091

Int[ArcTanh[(c_.)*(x_)^(n_)], x_Symbol] :> Simp[x*ArcTanh[c*x^n], x] - Dist[c*n, Int[x^n/(1 - c^2*x^(2*n)), x]
, x] /; FreeQ[{c, n}, x]

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

Mathematica [A]  time = 0.0316411, size = 58, normalized size = 1. \[ -\frac{b c n x^{n+1} \text{Hypergeometric2F1}\left (1,\frac{n+1}{2 n},\frac{1}{2} \left (\frac{1}{n}+3\right ),c^2 x^{2 n}\right )}{n+1}+a x+b x \tanh ^{-1}\left (c x^n\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[a + b*ArcTanh[c*x^n],x]

[Out]

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

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Maple [F]  time = 0.191, size = 0, normalized size = 0. \begin{align*} \int a+b{\it Artanh} \left ( c{x}^{n} \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(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} \,{\left (n \int \frac{1}{c x^{n} + 1}\,{d x} + n \int \frac{1}{c x^{n} - 1}\,{d x} + x \log \left (c x^{n} + 1\right ) - x \log \left (-c x^{n} + 1\right )\right )} b + a x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(b*arctanh(c*x^n) + a, x)

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

[Out]

Integral(a + b*atanh(c*x**n), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(b*arctanh(c*x^n) + a, x)