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

Optimal. Leaf size=87 \[ \frac{b^2 \text{PolyLog}\left (2,1-\frac{2}{1-\frac{c}{x}}\right )}{c}-\frac{\left (a+b \coth ^{-1}\left (\frac{x}{c}\right )\right )^2}{c}-\frac{\left (a+b \coth ^{-1}\left (\frac{x}{c}\right )\right )^2}{x}+\frac{2 b \log \left (\frac{2}{1-\frac{c}{x}}\right ) \left (a+b \coth ^{-1}\left (\frac{x}{c}\right )\right )}{c} \]

[Out]

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

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Rubi [B]  time = 0.506839, antiderivative size = 205, normalized size of antiderivative = 2.36, number of steps used = 28, number of rules used = 12, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.75, Rules used = {6099, 2454, 2389, 2296, 2295, 6715, 2430, 43, 2416, 2394, 2393, 2391} \[ \frac{b^2 \text{PolyLog}\left (2,-\frac{c-x}{2 x}\right )}{2 c}-\frac{b^2 \text{PolyLog}\left (2,\frac{c+x}{2 x}\right )}{2 c}-\frac{b \log \left (\frac{c+x}{2 x}\right ) \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )}{2 c}-\frac{b \log \left (\frac{c+x}{x}\right ) \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )}{2 x}+\frac{\left (1-\frac{c}{x}\right ) \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2}{4 c}-\frac{b^2 \left (\frac{c}{x}+1\right ) \log ^2\left (\frac{c+x}{x}\right )}{4 c}-\frac{b^2 \log \left (-\frac{c-x}{2 x}\right ) \log \left (\frac{c+x}{x}\right )}{2 c} \]

Warning: Unable to verify antiderivative.

[In]

Int[(a + b*ArcTanh[c/x])^2/x^2,x]

[Out]

((1 - c/x)*(2*a - b*Log[1 - c/x])^2)/(4*c) - (b*(2*a - b*Log[1 - c/x])*Log[(c + x)/(2*x)])/(2*c) - (b*(2*a - b
*Log[1 - c/x])*Log[(c + x)/x])/(2*x) - (b^2*Log[-(c - x)/(2*x)]*Log[(c + x)/x])/(2*c) - (b^2*(1 + c/x)*Log[(c
+ x)/x]^2)/(4*c) + (b^2*PolyLog[2, -(c - x)/(2*x)])/(2*c) - (b^2*PolyLog[2, (c + x)/(2*x)])/(2*c)

Rule 6099

Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(d*x)^
m*(a + (b*Log[1 + c*x^n])/2 - (b*Log[1 - c*x^n])/2)^p, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && IGtQ[p, 0] &&
IntegerQ[m] && IntegerQ[n]

Rule 2454

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[I
nt[x^(Simplify[(m + 1)/n] - 1)*(a + b*Log[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p,
q}, x] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0]) &&  !(EqQ[q, 1] && ILtQ[n, 0] &&
IGtQ[m, 0])

Rule 2389

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] :> Dist[1/e, Subst[Int[(a + b*Log[c*
x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, n, p}, x]

Rule 2296

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

Rule 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 6715

Int[(u_)*(x_)^(m_.), x_Symbol] :> Dist[1/(m + 1), Subst[Int[SubstFor[x^(m + 1), u, x], x], x, x^(m + 1)], x] /
; FreeQ[m, x] && NeQ[m, -1] && FunctionOfQ[x^(m + 1), u, x]

Rule 2430

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + Log[(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*
(g_.)), x_Symbol] :> Simp[x*(a + b*Log[c*(d + e*x)^n])^p*(f + g*Log[h*(i + j*x)^m]), x] + (-Dist[g*j*m, Int[(x
*(a + b*Log[c*(d + e*x)^n])^p)/(i + j*x), x], x] - Dist[b*e*n*p, Int[(x*(a + b*Log[c*(d + e*x)^n])^(p - 1)*(f
+ g*Log[h*(i + j*x)^m]))/(d + e*x), x], x]) /; FreeQ[{a, b, c, d, e, f, g, h, i, j, m, n}, x] && IGtQ[p, 0]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2416

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((h_.)*(x_))^(m_.)*((f_) + (g_.)*(x_)^(r_.))^(q
_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (h*x)^m*(f + g*x^r)^q, x], x] /; FreeQ[{a,
b, c, d, e, f, g, h, m, n, p, q, r}, x] && IntegerQ[m] && IntegerQ[q]

Rule 2394

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[(Log[(e*(f +
g*x))/(e*f - d*g)]*(a + b*Log[c*(d + e*x)^n]))/g, x] - Dist[(b*e*n)/g, Int[Log[(e*(f + g*x))/(e*f - d*g)]/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0]

Rule 2393

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

Rule 2391

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

Rubi steps

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

Mathematica [A]  time = 0.0965751, size = 101, normalized size = 1.16 \[ \frac{-b^2 x \text{PolyLog}\left (2,-e^{-2 \tanh ^{-1}\left (\frac{c}{x}\right )}\right )+a \left (2 b x \log \left (\frac{1}{\sqrt{1-\frac{c^2}{x^2}}}\right )-a c\right )+2 b \tanh ^{-1}\left (\frac{c}{x}\right ) \left (b x \log \left (e^{-2 \tanh ^{-1}\left (\frac{c}{x}\right )}+1\right )-a c\right )+b^2 (x-c) \tanh ^{-1}\left (\frac{c}{x}\right )^2}{c x} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [A]  time = 0.004, size = 144, normalized size = 1.7 \begin{align*} -{\frac{{a}^{2}}{x}}-{\frac{{b}^{2}}{x} \left ({\it Artanh} \left ({\frac{c}{x}} \right ) \right ) ^{2}}-{\frac{{b}^{2}}{c} \left ({\it Artanh} \left ({\frac{c}{x}} \right ) \right ) ^{2}}+2\,{\frac{{b}^{2}}{c}{\it Artanh} \left ({\frac{c}{x}} \right ) \ln \left ({ \left ( 1+{\frac{c}{x}} \right ) ^{2} \left ( 1-{\frac{{c}^{2}}{{x}^{2}}} \right ) ^{-1}}+1 \right ) }+{\frac{{b}^{2}}{c}{\it polylog} \left ( 2,-{ \left ( 1+{\frac{c}{x}} \right ) ^{2} \left ( 1-{\frac{{c}^{2}}{{x}^{2}}} \right ) ^{-1}} \right ) }-2\,{\frac{ab}{x}{\it Artanh} \left ({\frac{c}{x}} \right ) }-{\frac{ab}{c}\ln \left ( 1-{\frac{{c}^{2}}{{x}^{2}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a^2/x-arctanh(c/x)^2/x*b^2-1/c*b^2*arctanh(c/x)^2+2/c*arctanh(c/x)*ln((1+c/x)^2/(1-c^2/x^2)+1)*b^2+1/c*polylo
g(2,-(1+c/x)^2/(1-c^2/x^2))*b^2-2*a*b/x*arctanh(c/x)-1/c*a*b*ln(1-c^2/x^2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{4} \,{\left (c^{3} \int -\frac{\log \left (x\right )^{2}}{c^{3} x^{2} - c x^{4}}\,{d x} + c^{2}{\left (\frac{\log \left (-c^{2} + x^{2}\right )}{c^{3}} - \frac{\log \left (x^{2}\right )}{c^{3}}\right )} - 4 \, c^{2} \int -\frac{x \log \left (c + x\right )}{c^{3} x^{2} - c x^{4}}\,{d x} + 2 \, c^{2} \int -\frac{x \log \left (x\right )}{c^{3} x^{2} - c x^{4}}\,{d x} + 2 \, c{\left (\frac{\log \left (-c + x\right )}{c^{2}} - \frac{\log \left (x\right )}{c^{2}} + \frac{1}{c x}\right )} \log \left (-\frac{c}{x} + 1\right ) - c{\left (\frac{\log \left (c + x\right )}{c^{2}} - \frac{\log \left (-c + x\right )}{c^{2}}\right )} - c \int -\frac{x^{2} \log \left (x\right )^{2}}{c^{3} x^{2} - c x^{4}}\,{d x} - 2 \, c \int -\frac{x^{2} \log \left (c + x\right )}{c^{3} x^{2} - c x^{4}}\,{d x} + 4 \, c \int -\frac{x^{2} \log \left (x\right )}{c^{3} x^{2} - c x^{4}}\,{d x} - \frac{\log \left (-\frac{c}{x} + 1\right )^{2}}{x} - \frac{c \log \left (c + x\right )^{2} - 2 \,{\left ({\left (c + x\right )} \log \left (c + x\right ) -{\left (c + x\right )} \log \left (x\right ) - c\right )} \log \left (-c + x\right )}{c x} - \frac{x \log \left (-c + x\right )^{2} + x \log \left (x\right )^{2} - 2 \,{\left (x \log \left (x\right ) - x\right )} \log \left (-c + x\right ) - 2 \, x \log \left (x\right ) + 2 \, c}{c x} - 2 \, \int -\frac{x^{3} \log \left (c + x\right )}{c^{3} x^{2} - c x^{4}}\,{d x} + 2 \, \int -\frac{x^{3} \log \left (x\right )}{c^{3} x^{2} - c x^{4}}\,{d x}\right )} b^{2} - \frac{a b{\left (\frac{2 \, c \operatorname{artanh}\left (\frac{c}{x}\right )}{x} + \log \left (-\frac{c^{2}}{x^{2}} + 1\right )\right )}}{c} - \frac{a^{2}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(c^3*integrate(-log(x)^2/(c^3*x^2 - c*x^4), x) + c^2*(log(-c^2 + x^2)/c^3 - log(x^2)/c^3) - 4*c^2*integrat
e(-x*log(c + x)/(c^3*x^2 - c*x^4), x) + 2*c^2*integrate(-x*log(x)/(c^3*x^2 - c*x^4), x) + 2*c*(log(-c + x)/c^2
- log(x)/c^2 + 1/(c*x))*log(-c/x + 1) - c*(log(c + x)/c^2 - log(-c + x)/c^2) - c*integrate(-x^2*log(x)^2/(c^3
*x^2 - c*x^4), x) - 2*c*integrate(-x^2*log(c + x)/(c^3*x^2 - c*x^4), x) + 4*c*integrate(-x^2*log(x)/(c^3*x^2 -
c*x^4), x) - log(-c/x + 1)^2/x - (c*log(c + x)^2 - 2*((c + x)*log(c + x) - (c + x)*log(x) - c)*log(-c + x))/(
c*x) - (x*log(-c + x)^2 + x*log(x)^2 - 2*(x*log(x) - x)*log(-c + x) - 2*x*log(x) + 2*c)/(c*x) - 2*integrate(-x
^3*log(c + x)/(c^3*x^2 - c*x^4), x) + 2*integrate(-x^3*log(x)/(c^3*x^2 - c*x^4), x))*b^2 - a*b*(2*c*arctanh(c/
x)/x + log(-c^2/x^2 + 1))/c - a^2/x

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b^{2} \operatorname{artanh}\left (\frac{c}{x}\right )^{2} + 2 \, a b \operatorname{artanh}\left (\frac{c}{x}\right ) + a^{2}}{x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \operatorname{artanh}\left (\frac{c}{x}\right ) + a\right )}^{2}}{x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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