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

Optimal. Leaf size=36 \[ -\frac{b \text{PolyLog}\left (2,-c x^n\right )}{2 n}+\frac{b \text{PolyLog}\left (2,c x^n\right )}{2 n}+a \log (x) \]

[Out]

a*Log[x] - (b*PolyLog[2, -(c*x^n)])/(2*n) + (b*PolyLog[2, c*x^n])/(2*n)

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Rubi [A]  time = 0.0350596, antiderivative size = 36, 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 = {6095, 5912} \[ -\frac{b \text{PolyLog}\left (2,-c x^n\right )}{2 n}+\frac{b \text{PolyLog}\left (2,c x^n\right )}{2 n}+a \log (x) \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcTanh[c*x^n])/x,x]

[Out]

a*Log[x] - (b*PolyLog[2, -(c*x^n)])/(2*n) + (b*PolyLog[2, c*x^n])/(2*n)

Rule 6095

Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/n, Subst[Int[(a + b*ArcTanh[c*x])
^p/x, x], x, x^n], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[p, 0]

Rule 5912

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))/(x_), x_Symbol] :> Simp[a*Log[x], x] + (-Simp[(b*PolyLog[2, -(c*x)])/2
, x] + Simp[(b*PolyLog[2, c*x])/2, x]) /; FreeQ[{a, b, c}, x]

Rubi steps

\begin{align*} \int \frac{a+b \tanh ^{-1}\left (c x^n\right )}{x} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{a+b \tanh ^{-1}(c x)}{x} \, dx,x,x^n\right )}{n}\\ &=a \log (x)-\frac{b \text{Li}_2\left (-c x^n\right )}{2 n}+\frac{b \text{Li}_2\left (c x^n\right )}{2 n}\\ \end{align*}

Mathematica [C]  time = 0.0750161, size = 39, normalized size = 1.08 \[ \frac{b c x^n \text{HypergeometricPFQ}\left (\left \{\frac{1}{2},\frac{1}{2},1\right \},\left \{\frac{3}{2},\frac{3}{2}\right \},c^2 x^{2 n}\right )}{n}+a \log (x) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcTanh[c*x^n])/x,x]

[Out]

(b*c*x^n*HypergeometricPFQ[{1/2, 1/2, 1}, {3/2, 3/2}, c^2*x^(2*n)])/n + a*Log[x]

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Maple [B]  time = 0.036, size = 76, normalized size = 2.1 \begin{align*}{\frac{a\ln \left ( c{x}^{n} \right ) }{n}}+{\frac{b\ln \left ( c{x}^{n} \right ){\it Artanh} \left ( c{x}^{n} \right ) }{n}}-{\frac{b{\it dilog} \left ( c{x}^{n} \right ) }{2\,n}}-{\frac{b{\it dilog} \left ( c{x}^{n}+1 \right ) }{2\,n}}-{\frac{b\ln \left ( c{x}^{n} \right ) \ln \left ( c{x}^{n}+1 \right ) }{2\,n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/n*a*ln(c*x^n)+1/n*b*ln(c*x^n)*arctanh(c*x^n)-1/2/n*b*dilog(c*x^n)-1/2/n*b*dilog(c*x^n+1)-1/2/n*b*ln(c*x^n)*l
n(c*x^n+1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.81852, size = 456, normalized size = 12.67 \begin{align*} -\frac{b n \log \left (c \cosh \left (n \log \left (x\right )\right ) + c \sinh \left (n \log \left (x\right )\right ) + 1\right ) \log \left (x\right ) - b n \log \left (-c \cosh \left (n \log \left (x\right )\right ) - c \sinh \left (n \log \left (x\right )\right ) + 1\right ) \log \left (x\right ) - b n \log \left (x\right ) \log \left (-\frac{c \cosh \left (n \log \left (x\right )\right ) + c \sinh \left (n \log \left (x\right )\right ) + 1}{c \cosh \left (n \log \left (x\right )\right ) + c \sinh \left (n \log \left (x\right )\right ) - 1}\right ) - 2 \, a n \log \left (x\right ) - b{\rm Li}_2\left (c \cosh \left (n \log \left (x\right )\right ) + c \sinh \left (n \log \left (x\right )\right )\right ) + b{\rm Li}_2\left (-c \cosh \left (n \log \left (x\right )\right ) - c \sinh \left (n \log \left (x\right )\right )\right )}{2 \, n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(b*n*log(c*cosh(n*log(x)) + c*sinh(n*log(x)) + 1)*log(x) - b*n*log(-c*cosh(n*log(x)) - c*sinh(n*log(x)) +
1)*log(x) - b*n*log(x)*log(-(c*cosh(n*log(x)) + c*sinh(n*log(x)) + 1)/(c*cosh(n*log(x)) + c*sinh(n*log(x)) -
1)) - 2*a*n*log(x) - b*dilog(c*cosh(n*log(x)) + c*sinh(n*log(x))) + b*dilog(-c*cosh(n*log(x)) - c*sinh(n*log(x
))))/n

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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}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*atanh(c*x**n))/x, 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}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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