3.2.91 \(\int \frac {a+b \tanh ^{-1}(c \sqrt {x})}{x} \, dx\) [191]

Optimal. Leaf size=29 \[ a \log (x)-b \text {PolyLog}\left (2,-c \sqrt {x}\right )+b \text {PolyLog}\left (2,c \sqrt {x}\right ) \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6035, 6031} \begin {gather*} a \log (x)-b \text {Li}_2\left (-c \sqrt {x}\right )+b \text {Li}_2\left (c \sqrt {x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcTanh[c*Sqrt[x]])/x,x]

[Out]

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

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

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Mathematica [A]
time = 0.02, size = 29, normalized size = 1.00 \begin {gather*} a \log (x)-b \text {PolyLog}\left (2,-c \sqrt {x}\right )+b \text {PolyLog}\left (2,c \sqrt {x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcTanh[c*Sqrt[x]])/x,x]

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(62\) vs. \(2(25)=50\).
time = 0.08, size = 63, normalized size = 2.17

method	result	size
derivativedivides	\(2 a \ln \left (c \sqrt {x}\right )+2 b \ln \left (c \sqrt {x}\right ) \arctanh \left (c \sqrt {x}\right )-b \dilog \left (c \sqrt {x}\right )-b \dilog \left (1+c \sqrt {x}\right )-b \ln \left (c \sqrt {x}\right ) \ln \left (1+c \sqrt {x}\right )\)	\(63\)
default	\(2 a \ln \left (c \sqrt {x}\right )+2 b \ln \left (c \sqrt {x}\right ) \arctanh \left (c \sqrt {x}\right )-b \dilog \left (c \sqrt {x}\right )-b \dilog \left (1+c \sqrt {x}\right )-b \ln \left (c \sqrt {x}\right ) \ln \left (1+c \sqrt {x}\right )\)	\(63\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arctanh(c*x^(1/2)))/x,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (23) = 46\).
time = 0.36, size = 61, normalized size = 2.10 \begin {gather*} -{\left (\log \left (c \sqrt {x}\right ) \log \left (-c \sqrt {x} + 1\right ) + {\rm Li}_2\left (-c \sqrt {x} + 1\right )\right )} b + {\left (\log \left (c \sqrt {x} + 1\right ) \log \left (-c \sqrt {x}\right ) + {\rm Li}_2\left (c \sqrt {x} + 1\right )\right )} b + a \log \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctanh(c*x^(1/2)))/x,x, algorithm="maxima")

[Out]

-(log(c*sqrt(x))*log(-c*sqrt(x) + 1) + dilog(-c*sqrt(x) + 1))*b + (log(c*sqrt(x) + 1)*log(-c*sqrt(x)) + dilog(
c*sqrt(x) + 1))*b + 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^(1/2)))/x,x, algorithm="fricas")

[Out]

integral((b*arctanh(c*sqrt(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 \sqrt {x} \right )}}{x}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*atanh(c*sqrt(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^(1/2)))/x,x, algorithm="giac")

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*atanh(c*x^(1/2)))/x,x)

[Out]

int((a + b*atanh(c*x^(1/2)))/x, x)

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