3.2.46 \(\int (a+b \tanh ^{-1}(\frac {c}{x}))^2 \, dx\) [146]

Optimal. Leaf size=74 \[ c \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^2+x \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^2-2 b c \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right ) \log \left (\frac {2 c}{c-x}\right )-b^2 c \text {PolyLog}\left (2,-\frac {c+x}{c-x}\right ) \]

[Out]

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

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Rubi [A]
time = 0.08, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6025, 6022, 6132, 6056, 2449, 2352} \begin {gather*} c \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^2+x \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^2-2 b c \log \left (\frac {2 c}{c-x}\right ) \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )+b^2 (-c) \text {Li}_2\left (-\frac {c+x}{c-x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcTanh[c/x])^2,x]

[Out]

c*(a + b*ArcCoth[x/c])^2 + x*(a + b*ArcCoth[x/c])^2 - 2*b*c*(a + b*ArcCoth[x/c])*Log[(2*c)/(c - x)] - b^2*c*Po
lyLog[2, -((c + x)/(c - x))]

Rule 2352

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLog[2, 1 - c*x], x] /; FreeQ[{c, d, e
}, x] && EqQ[e + c*d, 0]

Rule 2449

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Dist[-e/g, Subst[Int[Log[2*d*x]/(1 - 2*
d*x), x], x, 1/(d + e*x)], x] /; FreeQ[{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]

Rule 6022

Int[((a_.) + ArcCoth[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*ArcCoth[c*x^n])^p, x] - Dist[b
*c*n*p, Int[x^n*((a + b*ArcCoth[c*x^n])^(p - 1)/(1 - c^2*x^(2*n))), x], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[p
, 0] && (EqQ[n, 1] || EqQ[p, 1])

Rule 6025

Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_)]*(b_.))^(p_), x_Symbol] :> Int[(a + b*ArcCoth[1/(x^n*c)])^p, x] /; FreeQ[
{a, b, c}, x] && IGtQ[p, 1] && ILtQ[n, 0]

Rule 6056

Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcCoth[c*x])^p)
*(Log[2/(1 + e*(x/d))]/e), x] + Dist[b*c*(p/e), Int[(a + b*ArcCoth[c*x])^(p - 1)*(Log[2/(1 + e*(x/d))]/(1 - c^
2*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 - e^2, 0]

Rule 6132

Int[(((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(a + b*ArcCoth[c
*x])^(p + 1)/(b*e*(p + 1)), x] + Dist[1/(c*d), Int[(a + b*ArcCoth[c*x])^p/(1 - c*x), x], x] /; FreeQ[{a, b, c,
d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]

Rubi steps

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

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Mathematica [A]
time = 0.08, size = 97, normalized size = 1.31 \begin {gather*} b^2 (-c+x) \tanh ^{-1}\left (\frac {c}{x}\right )^2+2 b \tanh ^{-1}\left (\frac {c}{x}\right ) \left (a x-b c \log \left (1-e^{-2 \tanh ^{-1}\left (\frac {c}{x}\right )}\right )\right )+a \left (a x+b c \log \left (1-\frac {c^2}{x^2}\right )-2 b c \log \left (\frac {c}{x}\right )\right )+b^2 c \text {PolyLog}\left (2,e^{-2 \tanh ^{-1}\left (\frac {c}{x}\right )}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcTanh[c/x])^2,x]

[Out]

b^2*(-c + x)*ArcTanh[c/x]^2 + 2*b*ArcTanh[c/x]*(a*x - b*c*Log[1 - E^(-2*ArcTanh[c/x])]) + a*(a*x + b*c*Log[1 -
c^2/x^2] - 2*b*c*Log[c/x]) + b^2*c*PolyLog[2, E^(-2*ArcTanh[c/x])]

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(286\) vs. \(2(77)=154\).
time = 0.22, size = 287, normalized size = 3.88

method	result	size
derivativedivides	\(-c \left (-\frac {a^{2} x}{c}-\frac {b^{2} x \arctanh \left (\frac {c}{x}\right )^{2}}{c}-b^{2} \arctanh \left (\frac {c}{x}\right ) \ln \left (1+\frac {c}{x}\right )+2 b^{2} \ln \left (\frac {c}{x}\right ) \arctanh \left (\frac {c}{x}\right )-b^{2} \arctanh \left (\frac {c}{x}\right ) \ln \left (\frac {c}{x}-1\right )-\frac {b^{2} \ln \left (\frac {c}{x}-1\right )^{2}}{4}+b^{2} \dilog \left (\frac {c}{2 x}+\frac {1}{2}\right )+\frac {b^{2} \ln \left (\frac {c}{x}-1\right ) \ln \left (\frac {c}{2 x}+\frac {1}{2}\right )}{2}+\frac {b^{2} \ln \left (1+\frac {c}{x}\right )^{2}}{4}+\frac {b^{2} \ln \left (-\frac {c}{2 x}+\frac {1}{2}\right ) \ln \left (\frac {c}{2 x}+\frac {1}{2}\right )}{2}-\frac {b^{2} \ln \left (-\frac {c}{2 x}+\frac {1}{2}\right ) \ln \left (1+\frac {c}{x}\right )}{2}-b^{2} \dilog \left (1+\frac {c}{x}\right )-b^{2} \ln \left (\frac {c}{x}\right ) \ln \left (1+\frac {c}{x}\right )-b^{2} \dilog \left (\frac {c}{x}\right )-\frac {2 a b x \arctanh \left (\frac {c}{x}\right )}{c}-a b \ln \left (1+\frac {c}{x}\right )+2 a b \ln \left (\frac {c}{x}\right )-a b \ln \left (\frac {c}{x}-1\right )\right )\)	\(287\)
default	\(-c \left (-\frac {a^{2} x}{c}-\frac {b^{2} x \arctanh \left (\frac {c}{x}\right )^{2}}{c}-b^{2} \arctanh \left (\frac {c}{x}\right ) \ln \left (1+\frac {c}{x}\right )+2 b^{2} \ln \left (\frac {c}{x}\right ) \arctanh \left (\frac {c}{x}\right )-b^{2} \arctanh \left (\frac {c}{x}\right ) \ln \left (\frac {c}{x}-1\right )-\frac {b^{2} \ln \left (\frac {c}{x}-1\right )^{2}}{4}+b^{2} \dilog \left (\frac {c}{2 x}+\frac {1}{2}\right )+\frac {b^{2} \ln \left (\frac {c}{x}-1\right ) \ln \left (\frac {c}{2 x}+\frac {1}{2}\right )}{2}+\frac {b^{2} \ln \left (1+\frac {c}{x}\right )^{2}}{4}+\frac {b^{2} \ln \left (-\frac {c}{2 x}+\frac {1}{2}\right ) \ln \left (\frac {c}{2 x}+\frac {1}{2}\right )}{2}-\frac {b^{2} \ln \left (-\frac {c}{2 x}+\frac {1}{2}\right ) \ln \left (1+\frac {c}{x}\right )}{2}-b^{2} \dilog \left (1+\frac {c}{x}\right )-b^{2} \ln \left (\frac {c}{x}\right ) \ln \left (1+\frac {c}{x}\right )-b^{2} \dilog \left (\frac {c}{x}\right )-\frac {2 a b x \arctanh \left (\frac {c}{x}\right )}{c}-a b \ln \left (1+\frac {c}{x}\right )+2 a b \ln \left (\frac {c}{x}\right )-a b \ln \left (\frac {c}{x}-1\right )\right )\)	\(287\)
risch	\(\text {Expression too large to display}\)	\(5973\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-c*(-a^2/c*x-b^2/c*x*arctanh(c/x)^2-b^2*arctanh(c/x)*ln(1+c/x)+2*b^2*ln(c/x)*arctanh(c/x)-b^2*arctanh(c/x)*ln(
c/x-1)-1/4*b^2*ln(c/x-1)^2+b^2*dilog(1/2*c/x+1/2)+1/2*b^2*ln(c/x-1)*ln(1/2*c/x+1/2)+1/4*b^2*ln(1+c/x)^2+1/2*b^
2*ln(-1/2*c/x+1/2)*ln(1/2*c/x+1/2)-1/2*b^2*ln(-1/2*c/x+1/2)*ln(1+c/x)-b^2*dilog(1+c/x)-b^2*ln(c/x)*ln(1+c/x)-b
^2*dilog(c/x)-2*a*b/c*x*arctanh(c/x)-a*b*ln(1+c/x)+2*a*b*ln(c/x)-a*b*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))^2,x, algorithm="maxima")

[Out]

(2*x*arctanh(c/x) + c*log(-c^2 + x^2))*a*b + 1/4*(x*log(c + x)^2 - 2*(c + x)*log(c + x)*log(-c + x) - (c - x)*
log(-c + x)^2 + integrate(-2*(c^2 + 3*c*x)*log(c + x)/(c^2 - x^2), x))*b^2 + a^2*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))^2,x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

integrate((b*arctanh(c/x) + a)^2, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (a+b\,\mathrm {atanh}\left (\frac {c}{x}\right )\right )}^2 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*atanh(c/x))^2,x)

[Out]

int((a + b*atanh(c/x))^2, x)

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