3.2.54 \(\int \frac {x^2 (a+b \tanh ^{-1}(c x))^2}{d+e x} \, dx\) [154]

Optimal. Leaf size=385 \[ \frac {a b x}{c e}+\frac {b^2 x \tanh ^{-1}(c x)}{c e}-\frac {d \left (a+b \tanh ^{-1}(c x)\right )^2}{c e^2}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^2 e}-\frac {d x \left (a+b \tanh ^{-1}(c x)\right )^2}{e^2}+\frac {x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 e}+\frac {2 b d \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{c e^2}-\frac {d^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+c x}\right )}{e^3}+\frac {d^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{e^3}+\frac {b^2 \log \left (1-c^2 x^2\right )}{2 c^2 e}+\frac {b^2 d \text {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{c e^2}+\frac {b d^2 \left (a+b \tanh ^{-1}(c x)\right ) \text {PolyLog}\left (2,1-\frac {2}{1+c x}\right )}{e^3}-\frac {b d^2 \left (a+b \tanh ^{-1}(c x)\right ) \text {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{e^3}+\frac {b^2 d^2 \text {PolyLog}\left (3,1-\frac {2}{1+c x}\right )}{2 e^3}-\frac {b^2 d^2 \text {PolyLog}\left (3,1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{2 e^3} \]

[Out]

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

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Rubi [A]
time = 0.31, antiderivative size = 385, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 11, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.524, Rules used = {6087, 6021, 6131, 6055, 2449, 2352, 6037, 6127, 266, 6095, 6059} \begin {gather*} -\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^2 e}+\frac {b d^2 \text {Li}_2\left (1-\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{e^3}-\frac {b d^2 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+e) (c x+1)}\right )}{e^3}-\frac {d^2 \log \left (\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{e^3}+\frac {d^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2 c (d+e x)}{(c x+1) (c d+e)}\right )}{e^3}-\frac {d x \left (a+b \tanh ^{-1}(c x)\right )^2}{e^2}-\frac {d \left (a+b \tanh ^{-1}(c x)\right )^2}{c e^2}+\frac {2 b d \log \left (\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c e^2}+\frac {x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 e}+\frac {a b x}{c e}+\frac {b^2 \log \left (1-c^2 x^2\right )}{2 c^2 e}+\frac {b^2 d^2 \text {Li}_3\left (1-\frac {2}{c x+1}\right )}{2 e^3}-\frac {b^2 d^2 \text {Li}_3\left (1-\frac {2 c (d+e x)}{(c d+e) (c x+1)}\right )}{2 e^3}+\frac {b^2 d \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{c e^2}+\frac {b^2 x \tanh ^{-1}(c x)}{c e} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^2*(a + b*ArcTanh[c*x])^2)/(d + e*x),x]

[Out]

(a*b*x)/(c*e) + (b^2*x*ArcTanh[c*x])/(c*e) - (d*(a + b*ArcTanh[c*x])^2)/(c*e^2) - (a + b*ArcTanh[c*x])^2/(2*c^
2*e) - (d*x*(a + b*ArcTanh[c*x])^2)/e^2 + (x^2*(a + b*ArcTanh[c*x])^2)/(2*e) + (2*b*d*(a + b*ArcTanh[c*x])*Log
[2/(1 - c*x)])/(c*e^2) - (d^2*(a + b*ArcTanh[c*x])^2*Log[2/(1 + c*x)])/e^3 + (d^2*(a + b*ArcTanh[c*x])^2*Log[(
2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/e^3 + (b^2*Log[1 - c^2*x^2])/(2*c^2*e) + (b^2*d*PolyLog[2, 1 - 2/(1 - c
*x)])/(c*e^2) + (b*d^2*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 + c*x)])/e^3 - (b*d^2*(a + b*ArcTanh[c*x])*Pol
yLog[2, 1 - (2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/e^3 + (b^2*d^2*PolyLog[3, 1 - 2/(1 + c*x)])/(2*e^3) - (b^2
*d^2*PolyLog[3, 1 - (2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/(2*e^3)

Rule 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

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 6021

Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*ArcTanh[c*x^n])^p, x] - Dist[b
*c*n*p, Int[x^n*((a + b*ArcTanh[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 6037

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

Rule 6055

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcTanh[c*x])^p)
*(Log[2/(1 + e*(x/d))]/e), x] + Dist[b*c*(p/e), Int[(a + b*ArcTanh[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 6059

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^2/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcTanh[c*x])^2)*(Lo
g[2/(1 + c*x)]/e), x] + (Simp[(a + b*ArcTanh[c*x])^2*(Log[2*c*((d + e*x)/((c*d + e)*(1 + c*x)))]/e), x] + Simp
[b*(a + b*ArcTanh[c*x])*(PolyLog[2, 1 - 2/(1 + c*x)]/e), x] - Simp[b*(a + b*ArcTanh[c*x])*(PolyLog[2, 1 - 2*c*
((d + e*x)/((c*d + e)*(1 + c*x)))]/e), x] + Simp[b^2*(PolyLog[3, 1 - 2/(1 + c*x)]/(2*e)), x] - Simp[b^2*(PolyL
og[3, 1 - 2*c*((d + e*x)/((c*d + e)*(1 + c*x)))]/(2*e)), x]) /; FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d^2 - e^2
, 0]

Rule 6087

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Int[E
xpandIntegrand[(a + b*ArcTanh[c*x])^p, (f*x)^m*(d + e*x)^q, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[
p, 0] && IntegerQ[q] && (GtQ[q, 0] || NeQ[a, 0] || IntegerQ[m])

Rule 6095

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

Rule 6127

Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[f^2
/e, Int[(f*x)^(m - 2)*(a + b*ArcTanh[c*x])^p, x], x] - Dist[d*(f^2/e), Int[(f*x)^(m - 2)*((a + b*ArcTanh[c*x])
^p/(d + e*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && GtQ[m, 1]

Rule 6131

Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(a + b*ArcTanh[c
*x])^(p + 1)/(b*e*(p + 1)), x] + Dist[1/(c*d), Int[(a + b*ArcTanh[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 \frac {x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{d+e x} \, dx &=\int \left (-\frac {d \left (a+b \tanh ^{-1}(c x)\right )^2}{e^2}+\frac {x \left (a+b \tanh ^{-1}(c x)\right )^2}{e}+\frac {d^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{e^2 (d+e x)}\right ) \, dx\\ &=-\frac {d \int \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx}{e^2}+\frac {d^2 \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{d+e x} \, dx}{e^2}+\frac {\int x \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx}{e}\\ &=-\frac {d x \left (a+b \tanh ^{-1}(c x)\right )^2}{e^2}+\frac {x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 e}-\frac {d^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+c x}\right )}{e^3}+\frac {d^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{e^3}+\frac {b d^2 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{e^3}-\frac {b d^2 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{e^3}+\frac {b^2 d^2 \text {Li}_3\left (1-\frac {2}{1+c x}\right )}{2 e^3}-\frac {b^2 d^2 \text {Li}_3\left (1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{2 e^3}+\frac {(2 b c d) \int \frac {x \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{e^2}-\frac {(b c) \int \frac {x^2 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{e}\\ &=-\frac {d \left (a+b \tanh ^{-1}(c x)\right )^2}{c e^2}-\frac {d x \left (a+b \tanh ^{-1}(c x)\right )^2}{e^2}+\frac {x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 e}-\frac {d^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+c x}\right )}{e^3}+\frac {d^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{e^3}+\frac {b d^2 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{e^3}-\frac {b d^2 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{e^3}+\frac {b^2 d^2 \text {Li}_3\left (1-\frac {2}{1+c x}\right )}{2 e^3}-\frac {b^2 d^2 \text {Li}_3\left (1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{2 e^3}+\frac {(2 b d) \int \frac {a+b \tanh ^{-1}(c x)}{1-c x} \, dx}{e^2}+\frac {b \int \left (a+b \tanh ^{-1}(c x)\right ) \, dx}{c e}-\frac {b \int \frac {a+b \tanh ^{-1}(c x)}{1-c^2 x^2} \, dx}{c e}\\ &=\frac {a b x}{c e}-\frac {d \left (a+b \tanh ^{-1}(c x)\right )^2}{c e^2}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^2 e}-\frac {d x \left (a+b \tanh ^{-1}(c x)\right )^2}{e^2}+\frac {x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 e}+\frac {2 b d \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{c e^2}-\frac {d^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+c x}\right )}{e^3}+\frac {d^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{e^3}+\frac {b d^2 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{e^3}-\frac {b d^2 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{e^3}+\frac {b^2 d^2 \text {Li}_3\left (1-\frac {2}{1+c x}\right )}{2 e^3}-\frac {b^2 d^2 \text {Li}_3\left (1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{2 e^3}-\frac {\left (2 b^2 d\right ) \int \frac {\log \left (\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx}{e^2}+\frac {b^2 \int \tanh ^{-1}(c x) \, dx}{c e}\\ &=\frac {a b x}{c e}+\frac {b^2 x \tanh ^{-1}(c x)}{c e}-\frac {d \left (a+b \tanh ^{-1}(c x)\right )^2}{c e^2}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^2 e}-\frac {d x \left (a+b \tanh ^{-1}(c x)\right )^2}{e^2}+\frac {x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 e}+\frac {2 b d \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{c e^2}-\frac {d^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+c x}\right )}{e^3}+\frac {d^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{e^3}+\frac {b d^2 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{e^3}-\frac {b d^2 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{e^3}+\frac {b^2 d^2 \text {Li}_3\left (1-\frac {2}{1+c x}\right )}{2 e^3}-\frac {b^2 d^2 \text {Li}_3\left (1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{2 e^3}+\frac {\left (2 b^2 d\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-c x}\right )}{c e^2}-\frac {b^2 \int \frac {x}{1-c^2 x^2} \, dx}{e}\\ &=\frac {a b x}{c e}+\frac {b^2 x \tanh ^{-1}(c x)}{c e}-\frac {d \left (a+b \tanh ^{-1}(c x)\right )^2}{c e^2}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^2 e}-\frac {d x \left (a+b \tanh ^{-1}(c x)\right )^2}{e^2}+\frac {x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 e}+\frac {2 b d \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{c e^2}-\frac {d^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+c x}\right )}{e^3}+\frac {d^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{e^3}+\frac {b^2 \log \left (1-c^2 x^2\right )}{2 c^2 e}+\frac {b^2 d \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{c e^2}+\frac {b d^2 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{e^3}-\frac {b d^2 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{e^3}+\frac {b^2 d^2 \text {Li}_3\left (1-\frac {2}{1+c x}\right )}{2 e^3}-\frac {b^2 d^2 \text {Li}_3\left (1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{2 e^3}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 12.29, size = 1250, normalized size = 3.25 \begin {gather*} \frac {-6 a^2 d e x+3 a^2 e^2 x^2+6 a^2 d^2 \log (d+e x)+\frac {6 a b \left (c e^2 x+i c^2 d^2 \pi \tanh ^{-1}(c x)-2 c^2 d e x \tanh ^{-1}(c x)+e^2 \left (-1+c^2 x^2\right ) \tanh ^{-1}(c x)+2 c^2 d^2 \tanh ^{-1}\left (\frac {c d}{e}\right ) \tanh ^{-1}(c x)-c^2 d^2 \tanh ^{-1}(c x)^2+c d e \tanh ^{-1}(c x)^2-c d \sqrt {1-\frac {c^2 d^2}{e^2}} e e^{-\tanh ^{-1}\left (\frac {c d}{e}\right )} \tanh ^{-1}(c x)^2-2 c^2 d^2 \tanh ^{-1}(c x) \log \left (1+e^{-2 \tanh ^{-1}(c x)}\right )-i c^2 d^2 \pi \log \left (1+e^{2 \tanh ^{-1}(c x)}\right )+2 c^2 d^2 \tanh ^{-1}\left (\frac {c d}{e}\right ) \log \left (1-e^{-2 \left (\tanh ^{-1}\left (\frac {c d}{e}\right )+\tanh ^{-1}(c x)\right )}\right )+2 c^2 d^2 \tanh ^{-1}(c x) \log \left (1-e^{-2 \left (\tanh ^{-1}\left (\frac {c d}{e}\right )+\tanh ^{-1}(c x)\right )}\right )-c d e \log \left (1-c^2 x^2\right )-\frac {1}{2} i c^2 d^2 \pi \log \left (1-c^2 x^2\right )-2 c^2 d^2 \tanh ^{-1}\left (\frac {c d}{e}\right ) \log \left (i \sinh \left (\tanh ^{-1}\left (\frac {c d}{e}\right )+\tanh ^{-1}(c x)\right )\right )+c^2 d^2 \text {PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c x)}\right )-c^2 d^2 \text {PolyLog}\left (2,e^{-2 \left (\tanh ^{-1}\left (\frac {c d}{e}\right )+\tanh ^{-1}(c x)\right )}\right )\right )}{c^2}+\frac {b^2 \left (6 c e^2 x \tanh ^{-1}(c x)+6 c d e \tanh ^{-1}(c x)^2-6 c^2 d e x \tanh ^{-1}(c x)^2+3 e^2 \left (-1+c^2 x^2\right ) \tanh ^{-1}(c x)^2-2 c^2 d^2 \tanh ^{-1}(c x)^3+2 c d e \tanh ^{-1}(c x)^3+12 c d e \tanh ^{-1}(c x) \log \left (1+e^{-2 \tanh ^{-1}(c x)}\right )-6 c^2 d^2 \tanh ^{-1}(c x)^2 \log \left (1+e^{-2 \tanh ^{-1}(c x)}\right )+3 e^2 \log \left (1-c^2 x^2\right )+6 c d \left (-e+c d \tanh ^{-1}(c x)\right ) \text {PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c x)}\right )+3 c^2 d^2 \text {PolyLog}\left (3,-e^{-2 \tanh ^{-1}(c x)}\right )-\frac {6 c d (-c d+e) (c d+e) \left (-6 c d \tanh ^{-1}(c x)^3+2 e \tanh ^{-1}(c x)^3-4 \sqrt {1-\frac {c^2 d^2}{e^2}} e e^{-\tanh ^{-1}\left (\frac {c d}{e}\right )} \tanh ^{-1}(c x)^3-6 i c d \pi \tanh ^{-1}(c x) \log \left (\frac {1}{2} \left (e^{-\tanh ^{-1}(c x)}+e^{\tanh ^{-1}(c x)}\right )\right )-6 c d \tanh ^{-1}(c x)^2 \log \left (1+\frac {(c d+e) e^{2 \tanh ^{-1}(c x)}}{c d-e}\right )+6 c d \tanh ^{-1}(c x)^2 \log \left (1-e^{\tanh ^{-1}\left (\frac {c d}{e}\right )+\tanh ^{-1}(c x)}\right )+6 c d \tanh ^{-1}(c x)^2 \log \left (1+e^{\tanh ^{-1}\left (\frac {c d}{e}\right )+\tanh ^{-1}(c x)}\right )+6 c d \tanh ^{-1}(c x)^2 \log \left (1-e^{2 \left (\tanh ^{-1}\left (\frac {c d}{e}\right )+\tanh ^{-1}(c x)\right )}\right )+12 c d \tanh ^{-1}\left (\frac {c d}{e}\right ) \tanh ^{-1}(c x) \log \left (\frac {1}{2} i e^{-\tanh ^{-1}\left (\frac {c d}{e}\right )-\tanh ^{-1}(c x)} \left (-1+e^{2 \left (\tanh ^{-1}\left (\frac {c d}{e}\right )+\tanh ^{-1}(c x)\right )}\right )\right )+6 c d \tanh ^{-1}(c x)^2 \log \left (\frac {1}{2} e^{-\tanh ^{-1}(c x)} \left (e \left (-1+e^{2 \tanh ^{-1}(c x)}\right )+c d \left (1+e^{2 \tanh ^{-1}(c x)}\right )\right )\right )-6 c d \tanh ^{-1}(c x)^2 \log \left (\frac {c (d+e x)}{\sqrt {1-c^2 x^2}}\right )-3 i c d \pi \tanh ^{-1}(c x) \log \left (1-c^2 x^2\right )-12 c d \tanh ^{-1}\left (\frac {c d}{e}\right ) \tanh ^{-1}(c x) \log \left (i \sinh \left (\tanh ^{-1}\left (\frac {c d}{e}\right )+\tanh ^{-1}(c x)\right )\right )-6 c d \tanh ^{-1}(c x) \text {PolyLog}\left (2,-\frac {(c d+e) e^{2 \tanh ^{-1}(c x)}}{c d-e}\right )+12 c d \tanh ^{-1}(c x) \text {PolyLog}\left (2,-e^{\tanh ^{-1}\left (\frac {c d}{e}\right )+\tanh ^{-1}(c x)}\right )+12 c d \tanh ^{-1}(c x) \text {PolyLog}\left (2,e^{\tanh ^{-1}\left (\frac {c d}{e}\right )+\tanh ^{-1}(c x)}\right )+6 c d \tanh ^{-1}(c x) \text {PolyLog}\left (2,e^{2 \left (\tanh ^{-1}\left (\frac {c d}{e}\right )+\tanh ^{-1}(c x)\right )}\right )+3 c d \text {PolyLog}\left (3,-\frac {(c d+e) e^{2 \tanh ^{-1}(c x)}}{c d-e}\right )-12 c d \text {PolyLog}\left (3,-e^{\tanh ^{-1}\left (\frac {c d}{e}\right )+\tanh ^{-1}(c x)}\right )-12 c d \text {PolyLog}\left (3,e^{\tanh ^{-1}\left (\frac {c d}{e}\right )+\tanh ^{-1}(c x)}\right )-3 c d \text {PolyLog}\left (3,e^{2 \left (\tanh ^{-1}\left (\frac {c d}{e}\right )+\tanh ^{-1}(c x)\right )}\right )\right )}{6 c^2 d^2-6 e^2}\right )}{c^2}}{6 e^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-6*a^2*d*e*x + 3*a^2*e^2*x^2 + 6*a^2*d^2*Log[d + e*x] + (6*a*b*(c*e^2*x + I*c^2*d^2*Pi*ArcTanh[c*x] - 2*c^2*d
*e*x*ArcTanh[c*x] + e^2*(-1 + c^2*x^2)*ArcTanh[c*x] + 2*c^2*d^2*ArcTanh[(c*d)/e]*ArcTanh[c*x] - c^2*d^2*ArcTan
h[c*x]^2 + c*d*e*ArcTanh[c*x]^2 - (c*d*Sqrt[1 - (c^2*d^2)/e^2]*e*ArcTanh[c*x]^2)/E^ArcTanh[(c*d)/e] - 2*c^2*d^
2*ArcTanh[c*x]*Log[1 + E^(-2*ArcTanh[c*x])] - I*c^2*d^2*Pi*Log[1 + E^(2*ArcTanh[c*x])] + 2*c^2*d^2*ArcTanh[(c*
d)/e]*Log[1 - E^(-2*(ArcTanh[(c*d)/e] + ArcTanh[c*x]))] + 2*c^2*d^2*ArcTanh[c*x]*Log[1 - E^(-2*(ArcTanh[(c*d)/
e] + ArcTanh[c*x]))] - c*d*e*Log[1 - c^2*x^2] - (I/2)*c^2*d^2*Pi*Log[1 - c^2*x^2] - 2*c^2*d^2*ArcTanh[(c*d)/e]
*Log[I*Sinh[ArcTanh[(c*d)/e] + ArcTanh[c*x]]] + c^2*d^2*PolyLog[2, -E^(-2*ArcTanh[c*x])] - c^2*d^2*PolyLog[2,
E^(-2*(ArcTanh[(c*d)/e] + ArcTanh[c*x]))]))/c^2 + (b^2*(6*c*e^2*x*ArcTanh[c*x] + 6*c*d*e*ArcTanh[c*x]^2 - 6*c^
2*d*e*x*ArcTanh[c*x]^2 + 3*e^2*(-1 + c^2*x^2)*ArcTanh[c*x]^2 - 2*c^2*d^2*ArcTanh[c*x]^3 + 2*c*d*e*ArcTanh[c*x]
^3 + 12*c*d*e*ArcTanh[c*x]*Log[1 + E^(-2*ArcTanh[c*x])] - 6*c^2*d^2*ArcTanh[c*x]^2*Log[1 + E^(-2*ArcTanh[c*x])
] + 3*e^2*Log[1 - c^2*x^2] + 6*c*d*(-e + c*d*ArcTanh[c*x])*PolyLog[2, -E^(-2*ArcTanh[c*x])] + 3*c^2*d^2*PolyLo
g[3, -E^(-2*ArcTanh[c*x])] - (6*c*d*(-(c*d) + e)*(c*d + e)*(-6*c*d*ArcTanh[c*x]^3 + 2*e*ArcTanh[c*x]^3 - (4*Sq
rt[1 - (c^2*d^2)/e^2]*e*ArcTanh[c*x]^3)/E^ArcTanh[(c*d)/e] - (6*I)*c*d*Pi*ArcTanh[c*x]*Log[(E^(-ArcTanh[c*x])
+ E^ArcTanh[c*x])/2] - 6*c*d*ArcTanh[c*x]^2*Log[1 + ((c*d + e)*E^(2*ArcTanh[c*x]))/(c*d - e)] + 6*c*d*ArcTanh[
c*x]^2*Log[1 - E^(ArcTanh[(c*d)/e] + ArcTanh[c*x])] + 6*c*d*ArcTanh[c*x]^2*Log[1 + E^(ArcTanh[(c*d)/e] + ArcTa
nh[c*x])] + 6*c*d*ArcTanh[c*x]^2*Log[1 - E^(2*(ArcTanh[(c*d)/e] + ArcTanh[c*x]))] + 12*c*d*ArcTanh[(c*d)/e]*Ar
cTanh[c*x]*Log[(I/2)*E^(-ArcTanh[(c*d)/e] - ArcTanh[c*x])*(-1 + E^(2*(ArcTanh[(c*d)/e] + ArcTanh[c*x])))] + 6*
c*d*ArcTanh[c*x]^2*Log[(e*(-1 + E^(2*ArcTanh[c*x])) + c*d*(1 + E^(2*ArcTanh[c*x])))/(2*E^ArcTanh[c*x])] - 6*c*
d*ArcTanh[c*x]^2*Log[(c*(d + e*x))/Sqrt[1 - c^2*x^2]] - (3*I)*c*d*Pi*ArcTanh[c*x]*Log[1 - c^2*x^2] - 12*c*d*Ar
cTanh[(c*d)/e]*ArcTanh[c*x]*Log[I*Sinh[ArcTanh[(c*d)/e] + ArcTanh[c*x]]] - 6*c*d*ArcTanh[c*x]*PolyLog[2, -(((c
*d + e)*E^(2*ArcTanh[c*x]))/(c*d - e))] + 12*c*d*ArcTanh[c*x]*PolyLog[2, -E^(ArcTanh[(c*d)/e] + ArcTanh[c*x])]
+ 12*c*d*ArcTanh[c*x]*PolyLog[2, E^(ArcTanh[(c*d)/e] + ArcTanh[c*x])] + 6*c*d*ArcTanh[c*x]*PolyLog[2, E^(2*(A
rcTanh[(c*d)/e] + ArcTanh[c*x]))] + 3*c*d*PolyLog[3, -(((c*d + e)*E^(2*ArcTanh[c*x]))/(c*d - e))] - 12*c*d*Pol
yLog[3, -E^(ArcTanh[(c*d)/e] + ArcTanh[c*x])] - 12*c*d*PolyLog[3, E^(ArcTanh[(c*d)/e] + ArcTanh[c*x])] - 3*c*d
*PolyLog[3, E^(2*(ArcTanh[(c*d)/e] + ArcTanh[c*x]))]))/(6*c^2*d^2 - 6*e^2)))/c^2)/(6*e^3)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 25.63, size = 1729, normalized size = 4.49

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1729\)
default	\(\text {Expression too large to display}\)	\(1729\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c^3*(-1/2*b^2*c^4/e^3*d^3/(c*d+e)*polylog(3,(c*d+e)*(c*x+1)^2/(-c^2*x^2+1)/(-c*d+e))-1/2*b^2*c^3/e^2*d^2/(c*
d+e)*polylog(3,(c*d+e)*(c*x+1)^2/(-c^2*x^2+1)/(-c*d+e))-b^2*c^3/e^3*d^2*arctanh(c*x)^2*ln(d*c*(1+(c*x+1)^2/(-c
^2*x^2+1))+e*((c*x+1)^2/(-c^2*x^2+1)-1))+2*b^2*c^2/e^2*ln(1+I*(c*x+1)/(-c^2*x^2+1)^(1/2))*d*arctanh(c*x)+2*b^2
*c^2/e^2*ln(1-I*(c*x+1)/(-c^2*x^2+1)^(1/2))*d*arctanh(c*x)+b^2*c^3*arctanh(c*x)^2*d^2/e^3*ln(c*e*x+c*d)-b^2*c^
3*d^2/e^3*arctanh(c*x)*polylog(2,-(c*x+1)^2/(-c^2*x^2+1))-a*b*c^2/e^2*ln(-c*e*x-e)*d-a*b*c^2/e^2*ln(-c*e*x+e)*
d-a*b*c^3/e^3*d^2*dilog((c*e*x+e)/(-c*d+e))+a*b*c^3/e^3*d^2*dilog((c*e*x-e)/(-c*d-e))-a^2*c^3/e^2*d*x+a*b*c^2/
e*x+1/2*b^2*c^3*arctanh(c*x)^2/e*x^2+b^2*c^2*arctanh(c*x)/e*x+1/2*I*b^2*c^3/e^3*d^2*Pi*csgn(I*(d*c*(1+(c*x+1)^
2/(-c^2*x^2+1))+e*((c*x+1)^2/(-c^2*x^2+1)-1))/(1+(c*x+1)^2/(-c^2*x^2+1)))^3*arctanh(c*x)^2-2*a*b*c^3*arctanh(c
*x)/e^2*d*x-b^2*c^3*arctanh(c*x)^2/e^2*d*x+a*b*c^3*arctanh(c*x)/e*x^2+2*a*b*c^3*arctanh(c*x)*d^2/e^3*ln(c*e*x+
c*d)-a*b*c^3/e^3*d^2*ln(c*e*x+c*d)*ln((c*e*x+e)/(-c*d+e))+a*b*c^3/e^3*d^2*ln(c*e*x+c*d)*ln((c*e*x-e)/(-c*d-e))
+b^2*c^3/e^2*d^2/(c*d+e)*arctanh(c*x)*polylog(2,(c*d+e)*(c*x+1)^2/(-c^2*x^2+1)/(-c*d+e))+b^2*c^4/e^3*d^3/(c*d+
e)*arctanh(c*x)^2*ln(1-(c*d+e)*(c*x+1)^2/(-c^2*x^2+1)/(-c*d+e))+b^2*c^4/e^3*d^3/(c*d+e)*arctanh(c*x)*polylog(2
,(c*d+e)*(c*x+1)^2/(-c^2*x^2+1)/(-c*d+e))+b^2*c^3/e^2*d^2/(c*d+e)*arctanh(c*x)^2*ln(1-(c*d+e)*(c*x+1)^2/(-c^2*
x^2+1)/(-c*d+e))+1/2*I*b^2*c^3/e^3*Pi*d^2*csgn(I/(1+(c*x+1)^2/(-c^2*x^2+1)))*csgn(I*(d*c*(1+(c*x+1)^2/(-c^2*x^
2+1))+e*((c*x+1)^2/(-c^2*x^2+1)-1)))*csgn(I*(d*c*(1+(c*x+1)^2/(-c^2*x^2+1))+e*((c*x+1)^2/(-c^2*x^2+1)-1))/(1+(
c*x+1)^2/(-c^2*x^2+1)))*arctanh(c*x)^2-1/2*a*b*c/e*ln(-c*e*x-e)+1/2*a*b*c/e*ln(-c*e*x+e)+a*b*c^2/e^2*d+1/2*b^2
*c^3*d^2/e^3*polylog(3,-(c*x+1)^2/(-c^2*x^2+1))-b^2*c^2/e^2*d*arctanh(c*x)^2+2*b^2*c^2/e^2*dilog(1+I*(c*x+1)/(
-c^2*x^2+1)^(1/2))*d+2*b^2*c^2/e^2*dilog(1-I*(c*x+1)/(-c^2*x^2+1)^(1/2))*d+a^2*c^3*d^2/e^3*ln(c*e*x+c*d)+1/2*a
^2*c^3/e*x^2-1/2*b^2*c/e*arctanh(c*x)^2-b^2*c/e*ln(1+(c*x+1)^2/(-c^2*x^2+1))+b^2*c*arctanh(c*x)/e-1/2*I*b^2*c^
3/e^3*d^2*Pi*csgn(I*(d*c*(1+(c*x+1)^2/(-c^2*x^2+1))+e*((c*x+1)^2/(-c^2*x^2+1)-1)))*csgn(I*(d*c*(1+(c*x+1)^2/(-
c^2*x^2+1))+e*((c*x+1)^2/(-c^2*x^2+1)-1))/(1+(c*x+1)^2/(-c^2*x^2+1)))^2*arctanh(c*x)^2-1/2*I*b^2*c^3/e^3*Pi*d^
2*csgn(I/(1+(c*x+1)^2/(-c^2*x^2+1)))*csgn(I*(d*c*(1+(c*x+1)^2/(-c^2*x^2+1))+e*((c*x+1)^2/(-c^2*x^2+1)-1))/(1+(
c*x+1)^2/(-c^2*x^2+1)))^2*arctanh(c*x)^2)

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

[Out]

1/8*(b^2*x^2*e - 2*b^2*d*x)*e^(-2)*log(-c*x + 1)^2 + 1/2*(2*d^2*e^(-3)*log(x*e + d) + (x^2*e - 2*d*x)*e^(-2))*
a^2 - integrate(-1/4*((b^2*c*x^3*e^2 - b^2*x^2*e^2)*log(c*x + 1)^2 + 4*(a*b*c*x^3*e^2 - a*b*x^2*e^2)*log(c*x +
1) + (2*b^2*c*d^2*x - (4*a*b*c + b^2*c)*x^3*e^2 + (b^2*c*d*e + 4*a*b*e^2)*x^2 - 2*(b^2*c*x^3*e^2 - b^2*x^2*e^
2)*log(c*x + 1))*log(-c*x + 1))/(c*x^2*e^3 + (c*d*e^2 - e^3)*x - d*e^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(x^2*(a+b*arctanh(c*x))^2/(e*x+d),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2*(a + b*atanh(c*x))^2)/(d + e*x),x)

[Out]

int((x^2*(a + b*atanh(c*x))^2)/(d + e*x), x)

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