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

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

[Out]

-((a + b*ArcTanh[c*x^n])/x) - (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 = 0.032358, antiderivative size = 67, 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} \[ -\frac{a+b \tanh ^{-1}\left (c x^n\right )}{x}-\frac{b c n x^{n-1} \, _2F_1\left (1,-\frac{1-n}{2 n};\frac{1}{2} \left (3-\frac{1}{n}\right );c^2 x^{2 n}\right )}{1-n} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcTanh[c*x^n])/x^2,x]

[Out]

-((a + b*ArcTanh[c*x^n])/x) - (b*c*n*x^(-1 + n)*Hypergeometric2F1[1, -(1 - n)/(2*n), (3 - n^(-1))/2, c^2*x^(2*
n)])/(1 - 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 \frac{a+b \tanh ^{-1}\left (c x^n\right )}{x^2} \, dx &=-\frac{a+b \tanh ^{-1}\left (c x^n\right )}{x}+(b c n) \int \frac{x^{-2+n}}{1-c^2 x^{2 n}} \, dx\\ &=-\frac{a+b \tanh ^{-1}\left (c x^n\right )}{x}-\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 = 0.0609311, size = 66, normalized size = 0.99 \[ \frac{b c n x^{n-1} \text{Hypergeometric2F1}\left (1,\frac{n-1}{2 n},\frac{n-1}{2 n}+1,c^2 x^{2 n}\right )}{n-1}-\frac{a}{x}-\frac{b \tanh ^{-1}\left (c x^n\right )}{x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.102, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b{\it Artanh} \left ( c{x}^{n} \right ) }{{x}^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((a+b*arctanh(c*x^n))/x^2,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^{2} x^{n} + x^{2}}\,{d x} + n \int \frac{1}{c x^{2} x^{n} - x^{2}}\,{d x} + \frac{\log \left (c x^{n} + 1\right ) - \log \left (-c x^{n} + 1\right )}{x}\right )} b - \frac{a}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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