3.3.17 \(\int \frac {a+b \tanh ^{-1}(c x^{3/2})}{x} \, dx\) [217]

Optimal. Leaf size=34 \[ a \log (x)-\frac {1}{3} b \text {PolyLog}\left (2,-c x^{3/2}\right )+\frac {1}{3} b \text {PolyLog}\left (2,c x^{3/2}\right ) \]

[Out]

a*ln(x)-1/3*b*polylog(2,-c*x^(3/2))+1/3*b*polylog(2,c*x^(3/2))

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Rubi [A]
time = 0.02, antiderivative size = 34, 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)-\frac {1}{3} b \text {Li}_2\left (-c x^{3/2}\right )+\frac {1}{3} b \text {Li}_2\left (c x^{3/2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcTanh[c*x^(3/2)])/x,x]

[Out]

a*Log[x] - (b*PolyLog[2, -(c*x^(3/2))])/3 + (b*PolyLog[2, c*x^(3/2)])/3

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^{3/2}\right )}{x} \, dx &=\frac {2}{3} \text {Subst}\left (\int \frac {a+b \tanh ^{-1}(c x)}{x} \, dx,x,x^{3/2}\right )\\ &=a \log (x)-\frac {1}{3} b \text {Li}_2\left (-c x^{3/2}\right )+\frac {1}{3} b \text {Li}_2\left (c x^{3/2}\right )\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 32, normalized size = 0.94 \begin {gather*} a \log (x)+\frac {1}{3} b \left (-\text {PolyLog}\left (2,-c x^{3/2}\right )+\text {PolyLog}\left (2,c x^{3/2}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcTanh[c*x^(3/2)])/x,x]

[Out]

a*Log[x] + (b*(-PolyLog[2, -(c*x^(3/2))] + PolyLog[2, c*x^(3/2)]))/3

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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 = 0.12, size = 63, normalized size = 1.85

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arctanh(c*x^(3/2)))/x,x,method=_RETURNVERBOSE)

[Out]

2/3*a*ln(c*x^(3/2))+2/3*b*ln(c*x^(3/2))*arctanh(c*x^(3/2))-1/3*b*dilog(c*x^(3/2))-1/3*b*dilog(c*x^(3/2)+1)-1/3
*b*ln(c*x^(3/2))*ln(c*x^(3/2)+1)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 62 vs. \(2 (24) = 48\).
time = 0.37, size = 62, normalized size = 1.82 \begin {gather*} -\frac {1}{3} \, {\left (\log \left (c x^{\frac {3}{2}}\right ) \log \left (-c x^{\frac {3}{2}} + 1\right ) + {\rm Li}_2\left (-c x^{\frac {3}{2}} + 1\right )\right )} b + \frac {1}{3} \, {\left (\log \left (c x^{\frac {3}{2}} + 1\right ) \log \left (-c x^{\frac {3}{2}}\right ) + {\rm Li}_2\left (c x^{\frac {3}{2}} + 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^(3/2)))/x,x, algorithm="maxima")

[Out]

-1/3*(log(c*x^(3/2))*log(-c*x^(3/2) + 1) + dilog(-c*x^(3/2) + 1))*b + 1/3*(log(c*x^(3/2) + 1)*log(-c*x^(3/2))
+ dilog(c*x^(3/2) + 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^(3/2)))/x,x, algorithm="fricas")

[Out]

integral((b*arctanh(c*x^(3/2)) + a)/x, x)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*atanh(c*x**(3/2)))/x,x)

[Out]

Timed out

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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^(3/2)))/x,x, algorithm="giac")

[Out]

integrate((b*arctanh(c*x^(3/2)) + 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^{3/2}\right )}{x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*atanh(c*x^(3/2)))/x,x)

[Out]

int((a + b*atanh(c*x^(3/2)))/x, x)

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