3.2.39 \(\int \frac {a+b \tanh ^{-1}(\frac {c}{x})}{x} \, dx\) [139]

Optimal. Leaf size=30 \[ a \log (x)+\frac {1}{2} b \text {PolyLog}\left (2,-\frac {c}{x}\right )-\frac {1}{2} b \text {PolyLog}\left (2,\frac {c}{x}\right ) \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 30, 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 {1}{2} b \text {Li}_2\left (-\frac {c}{x}\right )-\frac {1}{2} b \text {Li}_2\left (\frac {c}{x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcTanh[c/x])/x,x]

[Out]

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

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

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Mathematica [A]
time = 0.01, size = 28, normalized size = 0.93 \begin {gather*} a \log (x)+\frac {1}{2} b \left (\text {PolyLog}\left (2,-\frac {c}{x}\right )-\text {PolyLog}\left (2,\frac {c}{x}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcTanh[c/x])/x,x]

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(62\) vs. \(2(26)=52\).
time = 0.11, size = 63, normalized size = 2.10

method	result	size
derivativedivides	\(-a \ln \left (\frac {c}{x}\right )-b \ln \left (\frac {c}{x}\right ) \arctanh \left (\frac {c}{x}\right )+\frac {b \dilog \left (1+\frac {c}{x}\right )}{2}+\frac {b \ln \left (\frac {c}{x}\right ) \ln \left (1+\frac {c}{x}\right )}{2}+\frac {b \dilog \left (\frac {c}{x}\right )}{2}\)	\(63\)
default	\(-a \ln \left (\frac {c}{x}\right )-b \ln \left (\frac {c}{x}\right ) \arctanh \left (\frac {c}{x}\right )+\frac {b \dilog \left (1+\frac {c}{x}\right )}{2}+\frac {b \ln \left (\frac {c}{x}\right ) \ln \left (1+\frac {c}{x}\right )}{2}+\frac {b \dilog \left (\frac {c}{x}\right )}{2}\)	\(63\)
risch	\(\frac {b \ln \left (x \right ) \ln \left (x +c \right )}{2}-\frac {i \pi \ln \left (-x \right ) b \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i \left (c -x \right )}{x}\right )^{2}}{4}+\frac {i \pi \ln \left (-x \right ) b \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (i \left (c -x \right )\right ) \mathrm {csgn}\left (\frac {i \left (c -x \right )}{x}\right )}{4}-\frac {i \pi \ln \left (-x \right ) b \mathrm {csgn}\left (\frac {i \left (x +c \right )}{x}\right )^{3}}{4}-\frac {i \pi \ln \left (-x \right ) b \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (i \left (x +c \right )\right ) \mathrm {csgn}\left (\frac {i \left (x +c \right )}{x}\right )}{4}-\frac {i \pi \ln \left (-x \right ) b \,\mathrm {csgn}\left (i \left (c -x \right )\right ) \mathrm {csgn}\left (\frac {i \left (c -x \right )}{x}\right )^{2}}{4}+\frac {i \pi \ln \left (-x \right ) b \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i \left (x +c \right )}{x}\right )^{2}}{4}+\frac {i \pi \ln \left (-x \right ) b \,\mathrm {csgn}\left (i \left (x +c \right )\right ) \mathrm {csgn}\left (\frac {i \left (x +c \right )}{x}\right )^{2}}{4}-\frac {i \pi \ln \left (-x \right ) b}{2}+\frac {i \pi \ln \left (-x \right ) b \mathrm {csgn}\left (\frac {i \left (c -x \right )}{x}\right )^{2}}{2}-\frac {i \pi \ln \left (-x \right ) b \mathrm {csgn}\left (\frac {i \left (c -x \right )}{x}\right )^{3}}{4}+\ln \left (-x \right ) a -\frac {\ln \left (\frac {x}{c}\right ) \ln \left (c -x \right ) b}{2}-\frac {\dilog \left (\frac {x}{c}\right ) b}{2}-\frac {\ln \left (x \right ) \ln \left (\frac {x +c}{c}\right ) b}{2}-\frac {\dilog \left (\frac {x +c}{c}\right ) b}{2}\)	\(329\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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))/x,x, algorithm="maxima")

[Out]

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

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Fricas [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))/x,x, algorithm="fricas")

[Out]

integral((b*arctanh(c/x) + a)/x, x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a + b \operatorname {atanh}{\left (\frac {c}{x} \right )}}{x}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*atanh(c/x))/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))/x,x, algorithm="giac")

[Out]

integrate((b*arctanh(c/x) + 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 (\frac {c}{x}\right )}{x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*atanh(c/x))/x,x)

[Out]

int((a + b*atanh(c/x))/x, x)

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