3.10 \(\int (d+e x)^2 (a+b \tanh ^{-1}(c x))^2 \, dx\)

Optimal. Leaf size=257 \[ -\frac{b^2 \left (3 c^2 d^2+e^2\right ) \text{PolyLog}\left (2,1-\frac{2}{1-c x}\right )}{3 c^3}+\frac{\left (3 c^2 d^2+e^2\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{3 c^3}-\frac{d \left (\frac{3 e^2}{c^2}+d^2\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{3 e}-\frac{2 b \left (3 c^2 d^2+e^2\right ) \log \left (\frac{2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{3 c^3}+\frac{2 a b d e x}{c}+\frac{(d+e x)^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{3 e}+\frac{b e^2 x^2 \left (a+b \tanh ^{-1}(c x)\right )}{3 c}+\frac{b^2 d e \log \left (1-c^2 x^2\right )}{c^2}+\frac{b^2 e^2 x}{3 c^2}-\frac{b^2 e^2 \tanh ^{-1}(c x)}{3 c^3}+\frac{2 b^2 d e x \tanh ^{-1}(c x)}{c} \]

[Out]

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

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Rubi [A]  time = 0.408535, antiderivative size = 257, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 12, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667, Rules used = {5928, 5910, 260, 5916, 321, 206, 6048, 5948, 5984, 5918, 2402, 2315} \[ -\frac{b^2 \left (3 c^2 d^2+e^2\right ) \text{PolyLog}\left (2,1-\frac{2}{1-c x}\right )}{3 c^3}+\frac{\left (3 c^2 d^2+e^2\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{3 c^3}-\frac{d \left (\frac{3 e^2}{c^2}+d^2\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{3 e}-\frac{2 b \left (3 c^2 d^2+e^2\right ) \log \left (\frac{2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{3 c^3}+\frac{2 a b d e x}{c}+\frac{(d+e x)^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{3 e}+\frac{b e^2 x^2 \left (a+b \tanh ^{-1}(c x)\right )}{3 c}+\frac{b^2 d e \log \left (1-c^2 x^2\right )}{c^2}+\frac{b^2 e^2 x}{3 c^2}-\frac{b^2 e^2 \tanh ^{-1}(c x)}{3 c^3}+\frac{2 b^2 d e x \tanh ^{-1}(c x)}{c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 5928

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

Rule 5910

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

Rule 260

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 5916

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

Rule 321

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

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 6048

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

Rule 5948

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 5984

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]

Rule 5918

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 2402

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 2315

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

Rubi steps

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

Mathematica [A]  time = 0.635976, size = 319, normalized size = 1.24 \[ \frac{b^2 \left (3 c^2 d^2+e^2\right ) \text{PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c x)}\right )+3 a^2 c^3 d^2 x+3 a^2 c^3 d e x^2+a^2 c^3 e^2 x^3+b \tanh ^{-1}(c x) \left (2 a c^3 x \left (3 d^2+3 d e x+e^2 x^2\right )-2 b \left (3 c^2 d^2+e^2\right ) \log \left (e^{-2 \tanh ^{-1}(c x)}+1\right )+b e \left (6 c^2 d x+c^2 e x^2-e\right )\right )+3 a b c^2 d^2 \log \left (1-c^2 x^2\right )+6 a b c^2 d e x+a b c^2 e^2 x^2+a b e^2 \log \left (c^2 x^2-1\right )+3 a b c d e \log (1-c x)-3 a b c d e \log (c x+1)+b^2 (c x-1) \tanh ^{-1}(c x)^2 \left (c^2 \left (3 d^2+3 d e x+e^2 x^2\right )+c e (3 d+e x)+e^2\right )+3 b^2 c d e \log \left (1-c^2 x^2\right )+b^2 c e^2 x}{3 c^3} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(3*a^2*c^3*d^2*x + 6*a*b*c^2*d*e*x + b^2*c*e^2*x + 3*a^2*c^3*d*e*x^2 + a*b*c^2*e^2*x^2 + a^2*c^3*e^2*x^3 + b^2
*(-1 + c*x)*(e^2 + c*e*(3*d + e*x) + c^2*(3*d^2 + 3*d*e*x + e^2*x^2))*ArcTanh[c*x]^2 + b*ArcTanh[c*x]*(b*e*(-e
+ 6*c^2*d*x + c^2*e*x^2) + 2*a*c^3*x*(3*d^2 + 3*d*e*x + e^2*x^2) - 2*b*(3*c^2*d^2 + e^2)*Log[1 + E^(-2*ArcTan
h[c*x])]) + 3*a*b*c*d*e*Log[1 - c*x] - 3*a*b*c*d*e*Log[1 + c*x] + 3*a*b*c^2*d^2*Log[1 - c^2*x^2] + 3*b^2*c*d*e
*Log[1 - c^2*x^2] + a*b*e^2*Log[-1 + c^2*x^2] + b^2*(3*c^2*d^2 + e^2)*PolyLog[2, -E^(-2*ArcTanh[c*x])])/(3*c^3
)

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Maple [B]  time = 0.049, size = 1050, normalized size = 4.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3/c^3*b^2*e^2*arctanh(c*x)*ln(c*x-1)+1/6/c^3*b^2*e^2*ln(-1/2*c*x+1/2)*ln(c*x+1)+1/3/c^3*a*b*e^2*ln(c*x-1)+b^
2*e*arctanh(c*x)^2*x^2*d+2/3*a*b*e^2*arctanh(c*x)*x^3-1/6*b^2/e*ln(-1/2*c*x+1/2)*ln(c*x+1)*d^3-1/6*b^2/e*ln(1/
2+1/2*c*x)*ln(c*x-1)*d^3+1/3/c^3*b^2*e^2*arctanh(c*x)*ln(c*x+1)-1/6/c^3*b^2*e^2*ln(1/2+1/2*c*x)*ln(c*x-1)-1/6/
c^3*b^2*e^2*ln(-1/2*c*x+1/2)*ln(1/2+1/2*c*x)+1/2/c*b^2*ln(-1/2*c*x+1/2)*ln(c*x+1)*d^2+1/c*a*b*ln(c*x-1)*d^2+1/
c*a*b*ln(c*x+1)*d^2+1/3*a^2/e*d^3+1/4/c*b^2*ln(c*x-1)^2*d^2-1/c*b^2*dilog(1/2+1/2*c*x)*d^2-1/4/c*b^2*ln(c*x+1)
^2*d^2-1/12/c^3*b^2*e^2*ln(c*x+1)^2+1/6/c^3*b^2*e^2*ln(c*x-1)-1/6/c^3*b^2*e^2*ln(c*x+1)-1/3/c^3*b^2*e^2*dilog(
1/2+1/2*c*x)+1/12/c^3*b^2*e^2*ln(c*x-1)^2+1/3*b^2/e*arctanh(c*x)^2*d^3+1/3*b^2*e^2*arctanh(c*x)^2*x^3+a^2*e*x^
2*d-1/3*b^2/e*arctanh(c*x)*ln(c*x+1)*d^3+1/3*a^2*e^2*x^3+a^2*x*d^2-1/c^2*b^2*e*arctanh(c*x)*ln(c*x+1)*d-1/2/c^
2*b^2*e*ln(1/2+1/2*c*x)*ln(c*x-1)*d+1/2/c^2*b^2*e*ln(-1/2*c*x+1/2)*ln(1/2+1/2*c*x)*d+1/c^2*b^2*e*arctanh(c*x)*
ln(c*x-1)*d+2*a*b*e*arctanh(c*x)*x^2*d-1/c^2*a*b*e*ln(c*x+1)*d+1/c^2*a*b*e*ln(c*x-1)*d-1/2/c^2*b^2*e*ln(-1/2*c
*x+1/2)*ln(c*x+1)*d+1/3/c^3*a*b*e^2*ln(c*x+1)+1/4/c^2*b^2*e*ln(c*x+1)^2*d+1/c^2*b^2*e*ln(c*x+1)*d-1/3*a*b/e*ln
(c*x+1)*d^3+2/3*a*b/e*arctanh(c*x)*d^3+1/3*a*b/e*ln(c*x-1)*d^3+1/6*b^2/e*ln(-1/2*c*x+1/2)*ln(1/2+1/2*c*x)*d^3+
2*a*b*arctanh(c*x)*x*d^2+1/3*b^2/e*arctanh(c*x)*ln(c*x-1)*d^3+1/3/c*a*b*x^2*e^2+1/c^2*b^2*e*ln(c*x-1)*d+1/4/c^
2*b^2*e*ln(c*x-1)^2*d+1/3/c*b^2*e^2*arctanh(c*x)*x^2+1/c*b^2*arctanh(c*x)*ln(c*x-1)*d^2+1/c*b^2*arctanh(c*x)*l
n(c*x+1)*d^2-1/2/c*b^2*ln(1/2+1/2*c*x)*ln(c*x-1)*d^2-1/2/c*b^2*ln(-1/2*c*x+1/2)*ln(1/2+1/2*c*x)*d^2+b^2*arctan
h(c*x)^2*x*d^2+1/12*b^2/e*ln(c*x+1)^2*d^3+1/12*b^2/e*ln(c*x-1)^2*d^3+1/3*b^2*e^2*x/c^2+2*a*b*d*e*x/c+2*b^2*d*e
*x*arctanh(c*x)/c

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Maxima [B]  time = 1.79474, size = 707, normalized size = 2.75 \begin{align*} \frac{1}{3} \, a^{2} e^{2} x^{3} + a^{2} d e x^{2} +{\left (2 \, x^{2} \operatorname{artanh}\left (c x\right ) + c{\left (\frac{2 \, x}{c^{2}} - \frac{\log \left (c x + 1\right )}{c^{3}} + \frac{\log \left (c x - 1\right )}{c^{3}}\right )}\right )} a b d e + \frac{1}{3} \,{\left (2 \, x^{3} \operatorname{artanh}\left (c x\right ) + c{\left (\frac{x^{2}}{c^{2}} + \frac{\log \left (c^{2} x^{2} - 1\right )}{c^{4}}\right )}\right )} a b e^{2} + a^{2} d^{2} x + \frac{{\left (2 \, c x \operatorname{artanh}\left (c x\right ) + \log \left (-c^{2} x^{2} + 1\right )\right )} a b d^{2}}{c} + \frac{{\left (3 \, c^{2} d^{2} + e^{2}\right )}{\left (\log \left (c x + 1\right ) \log \left (-\frac{1}{2} \, c x + \frac{1}{2}\right ) +{\rm Li}_2\left (\frac{1}{2} \, c x + \frac{1}{2}\right )\right )} b^{2}}{3 \, c^{3}} + \frac{{\left (6 \, c d e - e^{2}\right )} b^{2} \log \left (c x + 1\right )}{6 \, c^{3}} + \frac{{\left (6 \, c d e + e^{2}\right )} b^{2} \log \left (c x - 1\right )}{6 \, c^{3}} + \frac{4 \, b^{2} c e^{2} x +{\left (b^{2} c^{3} e^{2} x^{3} + 3 \, b^{2} c^{3} d e x^{2} + 3 \, b^{2} c^{3} d^{2} x +{\left (3 \, c^{2} d^{2} - 3 \, c d e + e^{2}\right )} b^{2}\right )} \log \left (c x + 1\right )^{2} +{\left (b^{2} c^{3} e^{2} x^{3} + 3 \, b^{2} c^{3} d e x^{2} + 3 \, b^{2} c^{3} d^{2} x -{\left (3 \, c^{2} d^{2} + 3 \, c d e + e^{2}\right )} b^{2}\right )} \log \left (-c x + 1\right )^{2} + 2 \,{\left (b^{2} c^{2} e^{2} x^{2} + 6 \, b^{2} c^{2} d e x\right )} \log \left (c x + 1\right ) - 2 \,{\left (b^{2} c^{2} e^{2} x^{2} + 6 \, b^{2} c^{2} d e x +{\left (b^{2} c^{3} e^{2} x^{3} + 3 \, b^{2} c^{3} d e x^{2} + 3 \, b^{2} c^{3} d^{2} x +{\left (3 \, c^{2} d^{2} - 3 \, c d e + e^{2}\right )} b^{2}\right )} \log \left (c x + 1\right )\right )} \log \left (-c x + 1\right )}{12 \, c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a^2*e^2*x^3 + a^2*d*e*x^2 + (2*x^2*arctanh(c*x) + c*(2*x/c^2 - log(c*x + 1)/c^3 + log(c*x - 1)/c^3))*a*b*d
*e + 1/3*(2*x^3*arctanh(c*x) + c*(x^2/c^2 + log(c^2*x^2 - 1)/c^4))*a*b*e^2 + a^2*d^2*x + (2*c*x*arctanh(c*x) +
log(-c^2*x^2 + 1))*a*b*d^2/c + 1/3*(3*c^2*d^2 + e^2)*(log(c*x + 1)*log(-1/2*c*x + 1/2) + dilog(1/2*c*x + 1/2)
)*b^2/c^3 + 1/6*(6*c*d*e - e^2)*b^2*log(c*x + 1)/c^3 + 1/6*(6*c*d*e + e^2)*b^2*log(c*x - 1)/c^3 + 1/12*(4*b^2*
c*e^2*x + (b^2*c^3*e^2*x^3 + 3*b^2*c^3*d*e*x^2 + 3*b^2*c^3*d^2*x + (3*c^2*d^2 - 3*c*d*e + e^2)*b^2)*log(c*x +
1)^2 + (b^2*c^3*e^2*x^3 + 3*b^2*c^3*d*e*x^2 + 3*b^2*c^3*d^2*x - (3*c^2*d^2 + 3*c*d*e + e^2)*b^2)*log(-c*x + 1)
^2 + 2*(b^2*c^2*e^2*x^2 + 6*b^2*c^2*d*e*x)*log(c*x + 1) - 2*(b^2*c^2*e^2*x^2 + 6*b^2*c^2*d*e*x + (b^2*c^3*e^2*
x^3 + 3*b^2*c^3*d*e*x^2 + 3*b^2*c^3*d^2*x + (3*c^2*d^2 - 3*c*d*e + e^2)*b^2)*log(c*x + 1))*log(-c*x + 1))/c^3

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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