3.3.27 \(\int \frac {a+b \tanh ^{-1}(c x^n)}{x} \, dx\) [227]

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

[Out]

a*ln(x)-1/2*b*polylog(2,-c*x^n)/n+1/2*b*polylog(2,c*x^n)/n

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Rubi [A]
time = 0.02, antiderivative size = 36, normalized size of antiderivative = 1.00, 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 = {6035, 6031} \begin {gather*} 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 {gather*}

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 6031

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

Rule 6035

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]

Rubi steps

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

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 0.06, size = 39, normalized size = 1.08 \begin {gather*} \frac {b c x^n \, _3F_2\left (\frac {1}{2},\frac {1}{2},1;\frac {3}{2},\frac {3}{2};c^2 x^{2 n}\right )}{n}+a \log (x) \end {gather*}

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 [A]
time = 0.14, size = 65, normalized size = 1.81

method	result	size
risch	\(a \ln \left (x \right )+\frac {b \dilog \left (1-c \,x^{n}\right )}{2 n}-\frac {b \dilog \left (c \,x^{n}+1\right )}{2 n}\)	\(35\)
derivativedivides	\(\frac {a \ln \left (c \,x^{n}\right )+b \ln \left (c \,x^{n}\right ) \arctanh \left (c \,x^{n}\right )-\frac {b \dilog \left (c \,x^{n}+1\right )}{2}-\frac {b \ln \left (c \,x^{n}\right ) \ln \left (c \,x^{n}+1\right )}{2}-\frac {b \dilog \left (c \,x^{n}\right )}{2}}{n}\)	\(65\)
default	\(\frac {a \ln \left (c \,x^{n}\right )+b \ln \left (c \,x^{n}\right ) \arctanh \left (c \,x^{n}\right )-\frac {b \dilog \left (c \,x^{n}+1\right )}{2}-\frac {b \ln \left (c \,x^{n}\right ) \ln \left (c \,x^{n}+1\right )}{2}-\frac {b \dilog \left (c \,x^{n}\right )}{2}}{n}\)	\(65\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 141 vs. \(2 (30) = 60\).
time = 0.38, size = 141, normalized size = 3.92 \begin {gather*} -\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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a + b \operatorname {atanh}{\left (c x^{n} \right )}}{x}\, dx \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {a+b\,\mathrm {atanh}\left (c\,x^n\right )}{x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*atanh(c*x^n))/x,x)

[Out]

int((a + b*atanh(c*x^n))/x, x)

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