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

Optimal. Leaf size=374 \[ -\frac{b^2 \left (\left (3 c^2+1\right ) f^2-6 c d e f+3 d^2 e^2\right ) \text{PolyLog}\left (2,-\frac{c+d x+1}{-c-d x+1}\right )}{3 d^3}-\frac{(d e-c f) \left (\left (c^2+3\right ) f^2-2 c d e f+d^2 e^2\right ) \left (a+b \tanh ^{-1}(c+d x)\right )^2}{3 d^3 f}+\frac{\left (\left (3 c^2+1\right ) f^2-6 c d e f+3 d^2 e^2\right ) \left (a+b \tanh ^{-1}(c+d x)\right )^2}{3 d^3}-\frac{2 b \left (\left (3 c^2+1\right ) f^2-6 c d e f+3 d^2 e^2\right ) \log \left (\frac{2}{-c-d x+1}\right ) \left (a+b \tanh ^{-1}(c+d x)\right )}{3 d^3}+\frac{2 a b f x (d e-c f)}{d^2}+\frac{b f^2 (c+d x)^2 \left (a+b \tanh ^{-1}(c+d x)\right )}{3 d^3}+\frac{(e+f x)^3 \left (a+b \tanh ^{-1}(c+d x)\right )^2}{3 f}+\frac{b^2 f (d e-c f) \log \left (1-(c+d x)^2\right )}{d^3}+\frac{2 b^2 f (c+d x) (d e-c f) \tanh ^{-1}(c+d x)}{d^3}-\frac{b^2 f^2 \tanh ^{-1}(c+d x)}{3 d^3}+\frac{b^2 f^2 x}{3 d^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.639586, antiderivative size = 374, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 13, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.65, Rules used = {6111, 5928, 5910, 260, 5916, 321, 206, 6048, 5948, 5984, 5918, 2402, 2315} \[ -\frac{b^2 \left (\left (3 c^2+1\right ) f^2-6 c d e f+3 d^2 e^2\right ) \text{PolyLog}\left (2,-\frac{c+d x+1}{-c-d x+1}\right )}{3 d^3}-\frac{(d e-c f) \left (\left (c^2+3\right ) f^2-2 c d e f+d^2 e^2\right ) \left (a+b \tanh ^{-1}(c+d x)\right )^2}{3 d^3 f}+\frac{\left (\left (3 c^2+1\right ) f^2-6 c d e f+3 d^2 e^2\right ) \left (a+b \tanh ^{-1}(c+d x)\right )^2}{3 d^3}-\frac{2 b \left (\left (3 c^2+1\right ) f^2-6 c d e f+3 d^2 e^2\right ) \log \left (\frac{2}{-c-d x+1}\right ) \left (a+b \tanh ^{-1}(c+d x)\right )}{3 d^3}+\frac{2 a b f x (d e-c f)}{d^2}+\frac{b f^2 (c+d x)^2 \left (a+b \tanh ^{-1}(c+d x)\right )}{3 d^3}+\frac{(e+f x)^3 \left (a+b \tanh ^{-1}(c+d x)\right )^2}{3 f}+\frac{b^2 f (d e-c f) \log \left (1-(c+d x)^2\right )}{d^3}+\frac{2 b^2 f (c+d x) (d e-c f) \tanh ^{-1}(c+d x)}{d^3}-\frac{b^2 f^2 \tanh ^{-1}(c+d x)}{3 d^3}+\frac{b^2 f^2 x}{3 d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6111

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

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 (e+f x)^2 \left (a+b \tanh ^{-1}(c+d x)\right )^2 \, dx &=\frac{\operatorname{Subst}\left (\int \left (\frac{d e-c f}{d}+\frac{f x}{d}\right )^2 \left (a+b \tanh ^{-1}(x)\right )^2 \, dx,x,c+d x\right )}{d}\\ &=\frac{(e+f x)^3 \left (a+b \tanh ^{-1}(c+d x)\right )^2}{3 f}-\frac{(2 b) \operatorname{Subst}\left (\int \left (-\frac{3 f^2 (d e-c f) \left (a+b \tanh ^{-1}(x)\right )}{d^3}-\frac{f^3 x \left (a+b \tanh ^{-1}(x)\right )}{d^3}+\frac{\left ((d e-c f) \left (d^2 e^2-2 c d e f+3 f^2+c^2 f^2\right )+f \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) x\right ) \left (a+b \tanh ^{-1}(x)\right )}{d^3 \left (1-x^2\right )}\right ) \, dx,x,c+d x\right )}{3 f}\\ &=\frac{(e+f x)^3 \left (a+b \tanh ^{-1}(c+d x)\right )^2}{3 f}-\frac{(2 b) \operatorname{Subst}\left (\int \frac{\left ((d e-c f) \left (d^2 e^2-2 c d e f+3 f^2+c^2 f^2\right )+f \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) x\right ) \left (a+b \tanh ^{-1}(x)\right )}{1-x^2} \, dx,x,c+d x\right )}{3 d^3 f}+\frac{\left (2 b f^2\right ) \operatorname{Subst}\left (\int x \left (a+b \tanh ^{-1}(x)\right ) \, dx,x,c+d x\right )}{3 d^3}+\frac{(2 b f (d e-c f)) \operatorname{Subst}\left (\int \left (a+b \tanh ^{-1}(x)\right ) \, dx,x,c+d x\right )}{d^3}\\ &=\frac{2 a b f (d e-c f) x}{d^2}+\frac{b f^2 (c+d x)^2 \left (a+b \tanh ^{-1}(c+d x)\right )}{3 d^3}+\frac{(e+f x)^3 \left (a+b \tanh ^{-1}(c+d x)\right )^2}{3 f}-\frac{(2 b) \operatorname{Subst}\left (\int \left (\frac{(d e-c f) \left (d^2 e^2-2 c d e f+\left (3+c^2\right ) f^2\right ) \left (a+b \tanh ^{-1}(x)\right )}{1-x^2}+\frac{f \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) x \left (a+b \tanh ^{-1}(x)\right )}{1-x^2}\right ) \, dx,x,c+d x\right )}{3 d^3 f}-\frac{\left (b^2 f^2\right ) \operatorname{Subst}\left (\int \frac{x^2}{1-x^2} \, dx,x,c+d x\right )}{3 d^3}+\frac{\left (2 b^2 f (d e-c f)\right ) \operatorname{Subst}\left (\int \tanh ^{-1}(x) \, dx,x,c+d x\right )}{d^3}\\ &=\frac{b^2 f^2 x}{3 d^2}+\frac{2 a b f (d e-c f) x}{d^2}+\frac{2 b^2 f (d e-c f) (c+d x) \tanh ^{-1}(c+d x)}{d^3}+\frac{b f^2 (c+d x)^2 \left (a+b \tanh ^{-1}(c+d x)\right )}{3 d^3}+\frac{(e+f x)^3 \left (a+b \tanh ^{-1}(c+d x)\right )^2}{3 f}-\frac{\left (b^2 f^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,c+d x\right )}{3 d^3}-\frac{\left (2 b^2 f (d e-c f)\right ) \operatorname{Subst}\left (\int \frac{x}{1-x^2} \, dx,x,c+d x\right )}{d^3}-\frac{\left (2 b (d e-c f) \left (d^2 e^2-2 c d e f+\left (3+c^2\right ) f^2\right )\right ) \operatorname{Subst}\left (\int \frac{a+b \tanh ^{-1}(x)}{1-x^2} \, dx,x,c+d x\right )}{3 d^3 f}-\frac{\left (2 b \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right )\right ) \operatorname{Subst}\left (\int \frac{x \left (a+b \tanh ^{-1}(x)\right )}{1-x^2} \, dx,x,c+d x\right )}{3 d^3}\\ &=\frac{b^2 f^2 x}{3 d^2}+\frac{2 a b f (d e-c f) x}{d^2}-\frac{b^2 f^2 \tanh ^{-1}(c+d x)}{3 d^3}+\frac{2 b^2 f (d e-c f) (c+d x) \tanh ^{-1}(c+d x)}{d^3}+\frac{b f^2 (c+d x)^2 \left (a+b \tanh ^{-1}(c+d x)\right )}{3 d^3}-\frac{(d e-c f) \left (d^2 e^2-2 c d e f+\left (3+c^2\right ) f^2\right ) \left (a+b \tanh ^{-1}(c+d x)\right )^2}{3 d^3 f}+\frac{\left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) \left (a+b \tanh ^{-1}(c+d x)\right )^2}{3 d^3}+\frac{(e+f x)^3 \left (a+b \tanh ^{-1}(c+d x)\right )^2}{3 f}+\frac{b^2 f (d e-c f) \log \left (1-(c+d x)^2\right )}{d^3}-\frac{\left (2 b \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right )\right ) \operatorname{Subst}\left (\int \frac{a+b \tanh ^{-1}(x)}{1-x} \, dx,x,c+d x\right )}{3 d^3}\\ &=\frac{b^2 f^2 x}{3 d^2}+\frac{2 a b f (d e-c f) x}{d^2}-\frac{b^2 f^2 \tanh ^{-1}(c+d x)}{3 d^3}+\frac{2 b^2 f (d e-c f) (c+d x) \tanh ^{-1}(c+d x)}{d^3}+\frac{b f^2 (c+d x)^2 \left (a+b \tanh ^{-1}(c+d x)\right )}{3 d^3}-\frac{(d e-c f) \left (d^2 e^2-2 c d e f+\left (3+c^2\right ) f^2\right ) \left (a+b \tanh ^{-1}(c+d x)\right )^2}{3 d^3 f}+\frac{\left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) \left (a+b \tanh ^{-1}(c+d x)\right )^2}{3 d^3}+\frac{(e+f x)^3 \left (a+b \tanh ^{-1}(c+d x)\right )^2}{3 f}-\frac{2 b \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) \left (a+b \tanh ^{-1}(c+d x)\right ) \log \left (\frac{2}{1-c-d x}\right )}{3 d^3}+\frac{b^2 f (d e-c f) \log \left (1-(c+d x)^2\right )}{d^3}+\frac{\left (2 b^2 \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right )\right ) \operatorname{Subst}\left (\int \frac{\log \left (\frac{2}{1-x}\right )}{1-x^2} \, dx,x,c+d x\right )}{3 d^3}\\ &=\frac{b^2 f^2 x}{3 d^2}+\frac{2 a b f (d e-c f) x}{d^2}-\frac{b^2 f^2 \tanh ^{-1}(c+d x)}{3 d^3}+\frac{2 b^2 f (d e-c f) (c+d x) \tanh ^{-1}(c+d x)}{d^3}+\frac{b f^2 (c+d x)^2 \left (a+b \tanh ^{-1}(c+d x)\right )}{3 d^3}-\frac{(d e-c f) \left (d^2 e^2-2 c d e f+\left (3+c^2\right ) f^2\right ) \left (a+b \tanh ^{-1}(c+d x)\right )^2}{3 d^3 f}+\frac{\left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) \left (a+b \tanh ^{-1}(c+d x)\right )^2}{3 d^3}+\frac{(e+f x)^3 \left (a+b \tanh ^{-1}(c+d x)\right )^2}{3 f}-\frac{2 b \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) \left (a+b \tanh ^{-1}(c+d x)\right ) \log \left (\frac{2}{1-c-d x}\right )}{3 d^3}+\frac{b^2 f (d e-c f) \log \left (1-(c+d x)^2\right )}{d^3}-\frac{\left (2 b^2 \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right )\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-c-d x}\right )}{3 d^3}\\ &=\frac{b^2 f^2 x}{3 d^2}+\frac{2 a b f (d e-c f) x}{d^2}-\frac{b^2 f^2 \tanh ^{-1}(c+d x)}{3 d^3}+\frac{2 b^2 f (d e-c f) (c+d x) \tanh ^{-1}(c+d x)}{d^3}+\frac{b f^2 (c+d x)^2 \left (a+b \tanh ^{-1}(c+d x)\right )}{3 d^3}-\frac{(d e-c f) \left (d^2 e^2-2 c d e f+\left (3+c^2\right ) f^2\right ) \left (a+b \tanh ^{-1}(c+d x)\right )^2}{3 d^3 f}+\frac{\left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) \left (a+b \tanh ^{-1}(c+d x)\right )^2}{3 d^3}+\frac{(e+f x)^3 \left (a+b \tanh ^{-1}(c+d x)\right )^2}{3 f}-\frac{2 b \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) \left (a+b \tanh ^{-1}(c+d x)\right ) \log \left (\frac{2}{1-c-d x}\right )}{3 d^3}+\frac{b^2 f (d e-c f) \log \left (1-(c+d x)^2\right )}{d^3}-\frac{b^2 \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) \text{Li}_2\left (1-\frac{2}{1-c-d x}\right )}{3 d^3}\\ \end{align*}

Mathematica [B]  time = 2.98149, size = 795, normalized size = 2.13 \[ \frac{1}{3} a^2 f^2 x^3+a^2 e f x^2+a^2 e^2 x+\frac{1}{3} a b \left (2 x \left (3 e^2+3 f x e+f^2 x^2\right ) \tanh ^{-1}(c+d x)+\frac{d f x (6 d e-4 c f+d f x)-(c-1) \left (3 d^2 e^2-3 (c-1) d f e+(c-1)^2 f^2\right ) \log (-c-d x+1)+(c+1) \left (3 d^2 e^2-3 (c+1) d f e+(c+1)^2 f^2\right ) \log (c+d x+1)}{d^3}\right )+\frac{b^2 e^2 \left (\tanh ^{-1}(c+d x) \left ((c+d x-1) \tanh ^{-1}(c+d x)-2 \log \left (1+e^{-2 \tanh ^{-1}(c+d x)}\right )\right )+\text{PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c+d x)}\right )\right )}{d}+\frac{b^2 e f \left (\left (-c^2+2 c+d^2 x^2-1\right ) \tanh ^{-1}(c+d x)^2+2 \left (2 \log \left (1+e^{-2 \tanh ^{-1}(c+d x)}\right ) c+c+d x\right ) \tanh ^{-1}(c+d x)-2 \log \left (\frac{1}{\sqrt{1-(c+d x)^2}}\right )-2 c \text{PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c+d x)}\right )\right )}{d^2}-\frac{b^2 f^2 \left (1-(c+d x)^2\right )^{3/2} \left (-\frac{3 (c+d x) \tanh ^{-1}(c+d x)^2 c^2}{\sqrt{1-(c+d x)^2}}+3 \tanh ^{-1}(c+d x)^2 \cosh \left (3 \tanh ^{-1}(c+d x)\right ) c^2+6 \tanh ^{-1}(c+d x) \cosh \left (3 \tanh ^{-1}(c+d x)\right ) \log \left (1+e^{-2 \tanh ^{-1}(c+d x)}\right ) c^2-3 \tanh ^{-1}(c+d x)^2 \sinh \left (3 \tanh ^{-1}(c+d x)\right ) c^2+\frac{6 (c+d x) \tanh ^{-1}(c+d x) c}{\sqrt{1-(c+d x)^2}}-6 \cosh \left (3 \tanh ^{-1}(c+d x)\right ) \log \left (\frac{1}{\sqrt{1-(c+d x)^2}}\right ) c+6 \tanh ^{-1}(c+d x) \sinh \left (3 \tanh ^{-1}(c+d x)\right ) c+\frac{3 (c+d x) \tanh ^{-1}(c+d x)^2}{\sqrt{1-(c+d x)^2}}+\tanh ^{-1}(c+d x)^2 \cosh \left (3 \tanh ^{-1}(c+d x)\right )+2 \tanh ^{-1}(c+d x) \cosh \left (3 \tanh ^{-1}(c+d x)\right ) \log \left (1+e^{-2 \tanh ^{-1}(c+d x)}\right )+\frac{3 \left (3 c^2-4 c+1\right ) \tanh ^{-1}(c+d x)^2+2 \left (\left (9 c^2+3\right ) \log \left (1+e^{-2 \tanh ^{-1}(c+d x)}\right )+2\right ) \tanh ^{-1}(c+d x)-18 c \log \left (\frac{1}{\sqrt{1-(c+d x)^2}}\right )}{\sqrt{1-(c+d x)^2}}-\frac{4 \left (3 c^2+1\right ) \text{PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c+d x)}\right )}{\left (1-(c+d x)^2\right )^{3/2}}-\tanh ^{-1}(c+d x)^2 \sinh \left (3 \tanh ^{-1}(c+d x)\right )-\sinh \left (3 \tanh ^{-1}(c+d x)\right )-\frac{c+d x}{\sqrt{1-(c+d x)^2}}\right )}{12 d^3} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

a^2*e^2*x + a^2*e*f*x^2 + (a^2*f^2*x^3)/3 + (a*b*(2*x*(3*e^2 + 3*e*f*x + f^2*x^2)*ArcTanh[c + d*x] + (d*f*x*(6
*d*e - 4*c*f + d*f*x) - (-1 + c)*(3*d^2*e^2 - 3*(-1 + c)*d*e*f + (-1 + c)^2*f^2)*Log[1 - c - d*x] + (1 + c)*(3
*d^2*e^2 - 3*(1 + c)*d*e*f + (1 + c)^2*f^2)*Log[1 + c + d*x])/d^3))/3 + (b^2*e^2*(ArcTanh[c + d*x]*((-1 + c +
d*x)*ArcTanh[c + d*x] - 2*Log[1 + E^(-2*ArcTanh[c + d*x])]) + PolyLog[2, -E^(-2*ArcTanh[c + d*x])]))/d + (b^2*
e*f*((-1 + 2*c - c^2 + d^2*x^2)*ArcTanh[c + d*x]^2 + 2*ArcTanh[c + d*x]*(c + d*x + 2*c*Log[1 + E^(-2*ArcTanh[c
+ d*x])]) - 2*Log[1/Sqrt[1 - (c + d*x)^2]] - 2*c*PolyLog[2, -E^(-2*ArcTanh[c + d*x])]))/d^2 - (b^2*f^2*(1 - (
c + d*x)^2)^(3/2)*(-((c + d*x)/Sqrt[1 - (c + d*x)^2]) + (6*c*(c + d*x)*ArcTanh[c + d*x])/Sqrt[1 - (c + d*x)^2]
+ (3*(c + d*x)*ArcTanh[c + d*x]^2)/Sqrt[1 - (c + d*x)^2] - (3*c^2*(c + d*x)*ArcTanh[c + d*x]^2)/Sqrt[1 - (c +
d*x)^2] + ArcTanh[c + d*x]^2*Cosh[3*ArcTanh[c + d*x]] + 3*c^2*ArcTanh[c + d*x]^2*Cosh[3*ArcTanh[c + d*x]] + 2
*ArcTanh[c + d*x]*Cosh[3*ArcTanh[c + d*x]]*Log[1 + E^(-2*ArcTanh[c + d*x])] + 6*c^2*ArcTanh[c + d*x]*Cosh[3*Ar
cTanh[c + d*x]]*Log[1 + E^(-2*ArcTanh[c + d*x])] - 6*c*Cosh[3*ArcTanh[c + d*x]]*Log[1/Sqrt[1 - (c + d*x)^2]] +
(3*(1 - 4*c + 3*c^2)*ArcTanh[c + d*x]^2 + 2*ArcTanh[c + d*x]*(2 + (3 + 9*c^2)*Log[1 + E^(-2*ArcTanh[c + d*x])
]) - 18*c*Log[1/Sqrt[1 - (c + d*x)^2]])/Sqrt[1 - (c + d*x)^2] - (4*(1 + 3*c^2)*PolyLog[2, -E^(-2*ArcTanh[c + d
*x])])/(1 - (c + d*x)^2)^(3/2) - Sinh[3*ArcTanh[c + d*x]] + 6*c*ArcTanh[c + d*x]*Sinh[3*ArcTanh[c + d*x]] - Ar
cTanh[c + d*x]^2*Sinh[3*ArcTanh[c + d*x]] - 3*c^2*ArcTanh[c + d*x]^2*Sinh[3*ArcTanh[c + d*x]]))/(12*d^3)

________________________________________________________________________________________

Maple [B]  time = 0.069, size = 2694, normalized size = 7.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/d^2*b^2*f*ln(-1/2*d*x-1/2*c+1/2)*ln(d*x+c+1)*c^2*e-2/d^2*a*b*f*ln(d*x+c-1)*c*e-1/d^2*a*b*f*ln(d*x+c+1)*c^
2*e-1/d^2*b^2*f*ln(-1/2*d*x-1/2*c+1/2)*ln(d*x+c+1)*c*e+2*a*b/d*x*e*f-1/2/d*b^2*ln(1/2+1/2*d*x+1/2*c)*ln(-1/2*d
*x-1/2*c+1/2)*c*e^2+1/2/d*b^2*ln(-1/2*d*x-1/2*c+1/2)*ln(d*x+c+1)*c*e^2+1/6/d^3*b^2*f^2*ln(-1/2*d*x-1/2*c+1/2)*
ln(d*x+c+1)*c^3+1/2/d^3*b^2*f^2*ln(-1/2*d*x-1/2*c+1/2)*ln(d*x+c+1)*c^2+1/2/d^2*b^2*f*ln(1/2+1/2*d*x+1/2*c)*ln(
-1/2*d*x-1/2*c+1/2)*e-1/d^3*b^2*f^2*arctanh(d*x+c)*ln(d*x+c-1)*c+1/3/d^3*b^2*f^2*arctanh(d*x+c)*ln(d*x+c+1)*c^
3+1/d^3*b^2*f^2*arctanh(d*x+c)*ln(d*x+c+1)*c^2+1/d^3*b^2*f^2*arctanh(d*x+c)*ln(d*x+c+1)*c-1/3/d^3*b^2*f^2*arct
anh(d*x+c)*ln(d*x+c-1)*c^3+1/d^3*b^2*f^2*arctanh(d*x+c)*ln(d*x+c-1)*c^2+1/6/d^3*b^2*f^2*ln(1/2+1/2*d*x+1/2*c)*
ln(d*x+c-1)*c^3-1/d^3*a*b*f^2*ln(d*x+c-1)*c+1/d^3*a*b*f^2*ln(d*x+c+1)*c-1/d^2*b^2*f*arctanh(d*x+c)*ln(d*x+c+1)
*e-1/2/d^2*b^2*f*ln(1/2+1/2*d*x+1/2*c)*ln(d*x+c-1)*e-1/6/d^3*b^2*f^2*ln(1/2+1/2*d*x+1/2*c)*ln(-1/2*d*x-1/2*c+1
/2)*c^3-1/2/d^3*b^2*f^2*ln(1/2+1/2*d*x+1/2*c)*ln(-1/2*d*x-1/2*c+1/2)*c^2-1/2/d^3*b^2*f^2*ln(1/2+1/2*d*x+1/2*c)
*ln(-1/2*d*x-1/2*c+1/2)*c-1/2/d^2*b^2*f*ln(-1/2*d*x-1/2*c+1/2)*ln(d*x+c+1)*e+2/d^2*b^2*f*dilog(1/2+1/2*d*x+1/2
*c)*c*e+1/4/d^2*b^2*f*ln(d*x+c+1)^2*c^2*e+1/2/d^2*b^2*f*ln(d*x+c+1)^2*c*e-1/2/d^2*b^2*f*ln(d*x+c-1)^2*c*e+1/4/
d^2*b^2*f*ln(d*x+c-1)^2*c^2*e-4/3/d^2*b^2*f^2*arctanh(d*x+c)*x*c+2/d*b^2*f*arctanh(d*x+c)*e*x+2*arctanh(d*x+c)
*x*a*b*e^2+2/3*a*b*f^2*arctanh(d*x+c)*x^3+2/3*a*b/f*arctanh(d*x+c)*e^3+1/6*b^2/f*ln(1/2+1/2*d*x+1/2*c)*ln(-1/2
*d*x-1/2*c+1/2)*e^3-1/6*b^2/f*ln(-1/2*d*x-1/2*c+1/2)*ln(d*x+c+1)*e^3+1/3*a*b/f*ln(d*x+c-1)*e^3-1/3*a*b/f*ln(d*
x+c+1)*e^3-1/6*b^2/f*ln(1/2+1/2*d*x+1/2*c)*ln(d*x+c-1)*e^3-1/3*b^2/f*arctanh(d*x+c)*ln(d*x+c+1)*e^3+1/3*b^2/f*
arctanh(d*x+c)*ln(d*x+c-1)*e^3+1/3/d*a*b*f^2*x^2+1/d*e^2*a*b*ln(d*x+c-1)+1/3*a^2*f^2*x^3+a^2*x*e^2+1/d^2*b^2*f
*arctanh(d*x+c)*ln(d*x+c-1)*c^2*e-2/d^2*a*b*f*ln(d*x+c+1)*c*e+1/d^2*a*b*f*ln(d*x+c-1)*c^2*e-2/d^2*b^2*f*arctan
h(d*x+c)*ln(d*x+c-1)*c*e-1/d^2*b^2*f*arctanh(d*x+c)*ln(d*x+c+1)*c^2*e+1/d^2*b^2*f*ln(1/2+1/2*d*x+1/2*c)*ln(d*x
+c-1)*c*e+1/2/d^2*b^2*f*ln(1/2+1/2*d*x+1/2*c)*ln(-1/2*d*x-1/2*c+1/2)*c^2*e+1/d^2*b^2*f*ln(1/2+1/2*d*x+1/2*c)*l
n(-1/2*d*x-1/2*c+1/2)*c*e-1/2/d^2*b^2*f*ln(1/2+1/2*d*x+1/2*c)*ln(d*x+c-1)*c^2*e-2/d^2*b^2*f*arctanh(d*x+c)*ln(
d*x+c+1)*c*e+1/3*b^2*f^2*arctanh(d*x+c)^2*x^3+arctanh(d*x+c)^2*x*b^2*e^2+1/12*b^2/f*ln(d*x+c+1)^2*e^3-1/d*b^2*
dilog(1/2+1/2*d*x+1/2*c)*e^2-1/3/d^3*b^2*f^2*dilog(1/2+1/2*d*x+1/2*c)-1/12/d^3*b^2*f^2*ln(d*x+c+1)^2+1/6/d^3*b
^2*f^2*ln(d*x+c-1)-1/6/d^3*b^2*f^2*ln(d*x+c+1)+1/12/d^3*b^2*f^2*ln(d*x+c-1)^2-1/4/d*b^2*ln(d*x+c+1)^2*e^2+1/4/
d*b^2*ln(d*x+c-1)^2*e^2+1/12*b^2/f*ln(d*x+c-1)^2*e^3+1/3*b^2/f*arctanh(d*x+c)^2*e^3+a^2*f*x^2*e+1/3/d^3*b^2*f^
2*c+1/3*a^2/f*e^3-5/3/d^3*a*b*f^2*c^2+b^2*f*arctanh(d*x+c)^2*e*x^2+1/3*b^2*f^2*x/d^2-1/d^3*b^2*f^2*ln(d*x+c+1)
*c+1/4/d^2*b^2*f*ln(d*x+c+1)^2*e+1/d^2*b^2*f*ln(d*x+c-1)*e+1/d^2*b^2*f*ln(d*x+c+1)*e+1/d*e^2*a*b*ln(d*x+c+1)+1
/3/d^3*a*b*f^2*ln(d*x+c-1)+1/3/d^3*a*b*f^2*ln(d*x+c+1)-1/6/d^3*b^2*f^2*ln(1/2+1/2*d*x+1/2*c)*ln(-1/2*d*x-1/2*c
+1/2)+1/6/d^3*b^2*f^2*ln(-1/2*d*x-1/2*c+1/2)*ln(d*x+c+1)+1/4/d^2*b^2*f*ln(d*x+c-1)^2*e-1/12/d^3*b^2*f^2*ln(d*x
+c-1)^2*c^3-1/4/d^3*b^2*f^2*ln(d*x+c+1)^2*c^2-1/4/d^3*b^2*f^2*ln(d*x+c+1)^2*c-1/d^3*b^2*f^2*dilog(1/2+1/2*d*x+
1/2*c)*c^2+1/d*b^2*arctanh(d*x+c)*ln(d*x+c+1)*e^2-1/2/d*b^2*ln(1/2+1/2*d*x+1/2*c)*ln(d*x+c-1)*e^2-1/2/d*b^2*ln
(1/2+1/2*d*x+1/2*c)*ln(-1/2*d*x-1/2*c+1/2)*e^2+1/d*b^2*arctanh(d*x+c)*ln(d*x+c-1)*e^2+1/2/d*b^2*ln(-1/2*d*x-1/
2*c+1/2)*ln(d*x+c+1)*e^2+1/4/d^3*b^2*f^2*ln(d*x+c-1)^2*c^2-1/4/d^3*b^2*f^2*ln(d*x+c-1)^2*c+1/3/d^3*b^2*f^2*arc
tanh(d*x+c)*ln(d*x+c-1)+1/3/d^3*b^2*f^2*arctanh(d*x+c)*ln(d*x+c+1)-1/6/d^3*b^2*f^2*ln(1/2+1/2*d*x+1/2*c)*ln(d*
x+c-1)+1/3/d*b^2*f^2*arctanh(d*x+c)*x^2-5/3/d^3*b^2*f^2*arctanh(d*x+c)*c^2-1/4/d*b^2*ln(d*x+c-1)^2*c*e^2-1/4/d
*b^2*ln(d*x+c+1)^2*c*e^2-1/12/d^3*b^2*f^2*ln(d*x+c+1)^2*c^3-1/d^3*b^2*f^2*ln(d*x+c-1)*c-4/3*a*b/d^2*x*c*f^2+2/
d^2*b^2*f*arctanh(d*x+c)*e*c-1/3/d^3*a*b*f^2*ln(d*x+c-1)*c^3+1/d^3*a*b*f^2*ln(d*x+c-1)*c^2+1/3/d^3*a*b*f^2*ln(
d*x+c+1)*c^3+1/d^3*a*b*f^2*ln(d*x+c+1)*c^2+1/d^2*a*b*f*ln(d*x+c-1)*e-1/d^2*a*b*f*ln(d*x+c+1)*e-1/d*a*b*ln(d*x+
c-1)*c*e^2+1/d*a*b*ln(d*x+c+1)*c*e^2+1/d*b^2*arctanh(d*x+c)*ln(d*x+c+1)*c*e^2+1/2/d*b^2*ln(1/2+1/2*d*x+1/2*c)*
ln(d*x+c-1)*c*e^2-1/d*b^2*arctanh(d*x+c)*ln(d*x+c-1)*c*e^2+2*a*b*f*arctanh(d*x+c)*e*x^2-1/2/d^3*b^2*f^2*ln(1/2
+1/2*d*x+1/2*c)*ln(d*x+c-1)*c^2+1/2/d^3*b^2*f^2*ln(1/2+1/2*d*x+1/2*c)*ln(d*x+c-1)*c+1/d^2*b^2*f*arctanh(d*x+c)
*ln(d*x+c-1)*e+1/2/d^3*b^2*f^2*ln(-1/2*d*x-1/2*c+1/2)*ln(d*x+c+1)*c+2/d^2*a*b*f*c*e

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Maxima [B]  time = 1.95469, size = 1088, normalized size = 2.91 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a^2*f^2*x^3 + a^2*e*f*x^2 + (2*x^2*arctanh(d*x + c) + d*(2*x/d^2 - (c^2 + 2*c + 1)*log(d*x + c + 1)/d^3 +
(c^2 - 2*c + 1)*log(d*x + c - 1)/d^3))*a*b*e*f + 1/3*(2*x^3*arctanh(d*x + c) + d*((d*x^2 - 4*c*x)/d^3 + (c^3 +
3*c^2 + 3*c + 1)*log(d*x + c + 1)/d^4 - (c^3 - 3*c^2 + 3*c - 1)*log(d*x + c - 1)/d^4))*a*b*f^2 + a^2*e^2*x +
(2*(d*x + c)*arctanh(d*x + c) + log(-(d*x + c)^2 + 1))*a*b*e^2/d + 1/3*(3*d^2*e^2 - 6*c*d*e*f + 3*c^2*f^2 + f^
2)*(log(d*x + c + 1)*log(-1/2*d*x - 1/2*c + 1/2) + dilog(1/2*d*x + 1/2*c + 1/2))*b^2/d^3 - 1/6*(5*c^2*f^2 - 6*
d*e*f - 6*(d*e*f - f^2)*c + f^2)*b^2*log(d*x + c + 1)/d^3 + 1/6*(5*c^2*f^2 + 6*d*e*f - 6*(d*e*f + f^2)*c + f^2
)*b^2*log(d*x + c - 1)/d^3 + 1/12*(4*b^2*d*f^2*x + (b^2*d^3*f^2*x^3 + 3*b^2*d^3*e*f*x^2 + 3*b^2*d^3*e^2*x + (c
^3*f^2 + 3*d^2*e^2 - 3*(d*e*f - f^2)*c^2 - 3*d*e*f + 3*(d^2*e^2 - 2*d*e*f + f^2)*c + f^2)*b^2)*log(d*x + c + 1
)^2 + (b^2*d^3*f^2*x^3 + 3*b^2*d^3*e*f*x^2 + 3*b^2*d^3*e^2*x + (c^3*f^2 - 3*d^2*e^2 - 3*(d*e*f + f^2)*c^2 - 3*
d*e*f + 3*(d^2*e^2 + 2*d*e*f + f^2)*c - f^2)*b^2)*log(-d*x - c + 1)^2 + 2*(b^2*d^2*f^2*x^2 + 2*(3*d^2*e*f - 2*
c*d*f^2)*b^2*x)*log(d*x + c + 1) - 2*(b^2*d^2*f^2*x^2 + 2*(3*d^2*e*f - 2*c*d*f^2)*b^2*x + (b^2*d^3*f^2*x^3 + 3
*b^2*d^3*e*f*x^2 + 3*b^2*d^3*e^2*x + (c^3*f^2 + 3*d^2*e^2 - 3*(d*e*f - f^2)*c^2 - 3*d*e*f + 3*(d^2*e^2 - 2*d*e
*f + f^2)*c + f^2)*b^2)*log(d*x + c + 1))*log(-d*x - c + 1))/d^3

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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