3.161 \(\int \frac {a+b \tanh ^{-1}(\frac {c}{x^2})}{x} \, dx\)

Optimal. Leaf size=30 \[ a \log (x)+\frac {1}{4} b \text {Li}_2\left (-\frac {c}{x^2}\right )-\frac {1}{4} b \text {Li}_2\left (\frac {c}{x^2}\right ) \]

[Out]

a*ln(x)+1/4*b*polylog(2,-c/x^2)-1/4*b*polylog(2,c/x^2)

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Rubi [A]  time = 0.03, 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 = {6095, 5912} \[ \frac {1}{4} b \text {PolyLog}\left (2,-\frac {c}{x^2}\right )-\frac {1}{4} b \text {PolyLog}\left (2,\frac {c}{x^2}\right )+a \log (x) \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcTanh[c/x^2])/x,x]

[Out]

a*Log[x] + (b*PolyLog[2, -(c/x^2)])/4 - (b*PolyLog[2, c/x^2])/4

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]

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]

Rubi steps

\begin {align*} \int \frac {a+b \tanh ^{-1}\left (\frac {c}{x^2}\right )}{x} \, dx &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \frac {a+b \tanh ^{-1}(c x)}{x} \, dx,x,\frac {1}{x^2}\right )\right )\\ &=a \log (x)+\frac {1}{4} b \text {Li}_2\left (-\frac {c}{x^2}\right )-\frac {1}{4} b \text {Li}_2\left (\frac {c}{x^2}\right )\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 28, normalized size = 0.93 \[ a \log (x)+\frac {1}{4} b \left (\text {Li}_2\left (-\frac {c}{x^2}\right )-\text {Li}_2\left (\frac {c}{x^2}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcTanh[c/x^2])/x,x]

[Out]

a*Log[x] + (b*(PolyLog[2, -(c/x^2)] - PolyLog[2, c/x^2]))/4

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fricas [F]  time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \operatorname {artanh}\left (\frac {c}{x^{2}}\right ) + a}{x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*arctanh(c/x^2) + a)/x, x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {b \operatorname {artanh}\left (\frac {c}{x^{2}}\right ) + a}{x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*arctanh(c/x^2) + a)/x, x)

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maple [B]  time = 0.06, size = 154, normalized size = 5.13 \[ -a \ln \left (\frac {1}{x}\right )-b \ln \left (\frac {1}{x}\right ) \arctanh \left (\frac {c}{x^{2}}\right )+\frac {b \ln \left (\frac {1}{x}\right ) \ln \left (1+\frac {\sqrt {-c}}{x}\right )}{2}+\frac {b \ln \left (\frac {1}{x}\right ) \ln \left (1-\frac {\sqrt {-c}}{x}\right )}{2}+\frac {b \dilog \left (1+\frac {\sqrt {-c}}{x}\right )}{2}+\frac {b \dilog \left (1-\frac {\sqrt {-c}}{x}\right )}{2}-\frac {b \ln \left (\frac {1}{x}\right ) \ln \left (1-\frac {\sqrt {c}}{x}\right )}{2}-\frac {b \ln \left (\frac {1}{x}\right ) \ln \left (1+\frac {\sqrt {c}}{x}\right )}{2}-\frac {b \dilog \left (1-\frac {\sqrt {c}}{x}\right )}{2}-\frac {b \dilog \left (1+\frac {\sqrt {c}}{x}\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arctanh(c/x^2))/x,x)

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{2} \, b \int \frac {\log \left (\frac {c}{x^{2}} + 1\right ) - \log \left (-\frac {c}{x^{2}} + 1\right )}{x}\,{d x} + a \log \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {a+b\,\mathrm {atanh}\left (\frac {c}{x^2}\right )}{x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*atanh(c/x^2))/x,x)

[Out]

int((a + b*atanh(c/x^2))/x, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {a + b \operatorname {atanh}{\left (\frac {c}{x^{2}} \right )}}{x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*atanh(c/x**2))/x, x)

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