3.1.7 \(\int \frac {a+b \tanh ^{-1}(c x)}{x} \, dx\) [7]

Optimal. Leaf size=26 \[ a \log (x)-\frac {1}{2} b \text {PolyLog}(2,-c x)+\frac {1}{2} b \text {PolyLog}(2,c x) \]

[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 = 0.01, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {6031} \begin {gather*} a \log (x)-\frac {1}{2} b \text {Li}_2(-c x)+\frac {1}{2} b \text {Li}_2(c x) \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]

Rubi steps

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

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Mathematica [A]
time = 0.01, size = 24, normalized size = 0.92 \begin {gather*} a \log (x)+\frac {1}{2} b (-\text {PolyLog}(2,-c x)+\text {PolyLog}(2,c x)) \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. \(46\) vs. \(2(22)=44\).
time = 0.02, size = 47, normalized size = 1.81

method	result	size
risch	\(a \ln \left (-c x \right )+\frac {b \dilog \left (-c x +1\right )}{2}-\frac {b \dilog \left (c x +1\right )}{2}\)	\(28\)
derivativedivides	\(a \ln \left (c x \right )+b \ln \left (c x \right ) \arctanh \left (c x \right )-\frac {b \dilog \left (c x \right )}{2}-\frac {b \dilog \left (c x +1\right )}{2}-\frac {b \ln \left (c x \right ) \ln \left (c x +1\right )}{2}\)	\(47\)
default	\(a \ln \left (c x \right )+b \ln \left (c x \right ) \arctanh \left (c x \right )-\frac {b \dilog \left (c x \right )}{2}-\frac {b \dilog \left (c x +1\right )}{2}-\frac {b \ln \left (c x \right ) \ln \left (c x +1\right )}{2}\)	\(47\)

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

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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 (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 = -0.04 \begin {gather*} \int \frac {a+b\,\mathrm {atanh}\left (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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