3.38 \(\int (e+f x)^3 (a+b \tanh ^{-1}(c+d x))^2 \, dx\)

Optimal. Leaf size=562 \[ \frac {(d e-c f) \left (\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2\right ) \left (a+b \tanh ^{-1}(c+d x)\right )^2}{d^4}-\frac {2 b (d e-c f) \left (\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2\right ) \log \left (\frac {2}{-c-d x+1}\right ) \left (a+b \tanh ^{-1}(c+d x)\right )}{d^4}+\frac {a b f x \left (\left (6 c^2+1\right ) f^2-12 c d e f+6 d^2 e^2\right )}{2 d^3}-\frac {\left (6 \left (c^2+1\right ) d^2 e^2 f^2-4 c \left (c^2+3\right ) d e f^3+\left (c^4+6 c^2+1\right ) f^4-4 c d^3 e^3 f+d^4 e^4\right ) \left (a+b \tanh ^{-1}(c+d x)\right )^2}{4 d^4 f}+\frac {b f^2 (c+d x)^2 (d e-c f) \left (a+b \tanh ^{-1}(c+d x)\right )}{d^4}+\frac {b f^3 (c+d x)^3 \left (a+b \tanh ^{-1}(c+d x)\right )}{6 d^4}+\frac {(e+f x)^4 \left (a+b \tanh ^{-1}(c+d x)\right )^2}{4 f}-\frac {b^2 (d e-c f) \left (\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2\right ) \text {Li}_2\left (-\frac {c+d x+1}{-c-d x+1}\right )}{d^4}+\frac {b^2 f \left (\left (6 c^2+1\right ) f^2-12 c d e f+6 d^2 e^2\right ) \log \left (1-(c+d x)^2\right )}{4 d^4}+\frac {b^2 f (c+d x) \left (\left (6 c^2+1\right ) f^2-12 c d e f+6 d^2 e^2\right ) \tanh ^{-1}(c+d x)}{2 d^4}-\frac {b^2 f^2 (d e-c f) \tanh ^{-1}(c+d x)}{d^4}+\frac {b^2 f^3 (c+d x)^2}{12 d^4}+\frac {b^2 f^3 \log \left (1-(c+d x)^2\right )}{12 d^4}+\frac {b^2 f^2 x (d e-c f)}{d^3} \]

[Out]

b^2*f^2*(-c*f+d*e)*x/d^3+1/2*a*b*f*(6*d^2*e^2-12*c*d*e*f+(6*c^2+1)*f^2)*x/d^3+1/12*b^2*f^3*(d*x+c)^2/d^4-b^2*f
^2*(-c*f+d*e)*arctanh(d*x+c)/d^4+1/2*b^2*f*(6*d^2*e^2-12*c*d*e*f+(6*c^2+1)*f^2)*(d*x+c)*arctanh(d*x+c)/d^4+b*f
^2*(-c*f+d*e)*(d*x+c)^2*(a+b*arctanh(d*x+c))/d^4+1/6*b*f^3*(d*x+c)^3*(a+b*arctanh(d*x+c))/d^4+(-c*f+d*e)*(d^2*
e^2-2*c*d*e*f+(c^2+1)*f^2)*(a+b*arctanh(d*x+c))^2/d^4-1/4*(d^4*e^4-4*c*d^3*e^3*f+6*(c^2+1)*d^2*e^2*f^2-4*c*(c^
2+3)*d*e*f^3+(c^4+6*c^2+1)*f^4)*(a+b*arctanh(d*x+c))^2/d^4/f+1/4*(f*x+e)^4*(a+b*arctanh(d*x+c))^2/f-2*b*(-c*f+
d*e)*(d^2*e^2-2*c*d*e*f+(c^2+1)*f^2)*(a+b*arctanh(d*x+c))*ln(2/(-d*x-c+1))/d^4+1/12*b^2*f^3*ln(1-(d*x+c)^2)/d^
4+1/4*b^2*f*(6*d^2*e^2-12*c*d*e*f+(6*c^2+1)*f^2)*ln(1-(d*x+c)^2)/d^4-b^2*(-c*f+d*e)*(d^2*e^2-2*c*d*e*f+(c^2+1)
*f^2)*polylog(2,(-d*x-c-1)/(-d*x-c+1))/d^4

________________________________________________________________________________________

Rubi [A]  time = 1.04, antiderivative size = 562, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 15, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {6111, 5928, 5910, 260, 5916, 321, 206, 266, 43, 6048, 5948, 5984, 5918, 2402, 2315} \[ -\frac {b^2 (d e-c f) \left (\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2\right ) \text {PolyLog}\left (2,-\frac {c+d x+1}{-c-d x+1}\right )}{d^4}+\frac {a b f x \left (\left (6 c^2+1\right ) f^2-12 c d e f+6 d^2 e^2\right )}{2 d^3}+\frac {(d e-c f) \left (\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2\right ) \left (a+b \tanh ^{-1}(c+d x)\right )^2}{d^4}-\frac {\left (6 \left (c^2+1\right ) d^2 e^2 f^2-4 c \left (c^2+3\right ) d e f^3+\left (c^4+6 c^2+1\right ) f^4-4 c d^3 e^3 f+d^4 e^4\right ) \left (a+b \tanh ^{-1}(c+d x)\right )^2}{4 d^4 f}-\frac {2 b (d e-c f) \left (\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2\right ) \log \left (\frac {2}{-c-d x+1}\right ) \left (a+b \tanh ^{-1}(c+d x)\right )}{d^4}+\frac {b f^2 (c+d x)^2 (d e-c f) \left (a+b \tanh ^{-1}(c+d x)\right )}{d^4}+\frac {b f^3 (c+d x)^3 \left (a+b \tanh ^{-1}(c+d x)\right )}{6 d^4}+\frac {(e+f x)^4 \left (a+b \tanh ^{-1}(c+d x)\right )^2}{4 f}+\frac {b^2 f \left (\left (6 c^2+1\right ) f^2-12 c d e f+6 d^2 e^2\right ) \log \left (1-(c+d x)^2\right )}{4 d^4}+\frac {b^2 f (c+d x) \left (\left (6 c^2+1\right ) f^2-12 c d e f+6 d^2 e^2\right ) \tanh ^{-1}(c+d x)}{2 d^4}+\frac {b^2 f^2 x (d e-c f)}{d^3}-\frac {b^2 f^2 (d e-c f) \tanh ^{-1}(c+d x)}{d^4}+\frac {b^2 f^3 (c+d x)^2}{12 d^4}+\frac {b^2 f^3 \log \left (1-(c+d x)^2\right )}{12 d^4} \]

Antiderivative was successfully verified.

[In]

Int[(e + f*x)^3*(a + b*ArcTanh[c + d*x])^2,x]

[Out]

(b^2*f^2*(d*e - c*f)*x)/d^3 + (a*b*f*(6*d^2*e^2 - 12*c*d*e*f + (1 + 6*c^2)*f^2)*x)/(2*d^3) + (b^2*f^3*(c + d*x
)^2)/(12*d^4) - (b^2*f^2*(d*e - c*f)*ArcTanh[c + d*x])/d^4 + (b^2*f*(6*d^2*e^2 - 12*c*d*e*f + (1 + 6*c^2)*f^2)
*(c + d*x)*ArcTanh[c + d*x])/(2*d^4) + (b*f^2*(d*e - c*f)*(c + d*x)^2*(a + b*ArcTanh[c + d*x]))/d^4 + (b*f^3*(
c + d*x)^3*(a + b*ArcTanh[c + d*x]))/(6*d^4) + ((d*e - c*f)*(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2)*(a + b*ArcTa
nh[c + d*x])^2)/d^4 - ((d^4*e^4 - 4*c*d^3*e^3*f + 6*(1 + c^2)*d^2*e^2*f^2 - 4*c*(3 + c^2)*d*e*f^3 + (1 + 6*c^2
+ c^4)*f^4)*(a + b*ArcTanh[c + d*x])^2)/(4*d^4*f) + ((e + f*x)^4*(a + b*ArcTanh[c + d*x])^2)/(4*f) - (2*b*(d*
e - c*f)*(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2)*(a + b*ArcTanh[c + d*x])*Log[2/(1 - c - d*x)])/d^4 + (b^2*f^3*L
og[1 - (c + d*x)^2])/(12*d^4) + (b^2*f*(6*d^2*e^2 - 12*c*d*e*f + (1 + 6*c^2)*f^2)*Log[1 - (c + d*x)^2])/(4*d^4
) - (b^2*(d*e - c*f)*(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2)*PolyLog[2, -((1 + c + d*x)/(1 - c - d*x))])/d^4

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 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 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 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

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 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]

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 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 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 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 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 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 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 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]

Rubi steps

\begin {align*} \int (e+f x)^3 \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 )^3 \left (a+b \tanh ^{-1}(x)\right )^2 \, dx,x,c+d x\right )}{d}\\ &=\frac {(e+f x)^4 \left (a+b \tanh ^{-1}(c+d x)\right )^2}{4 f}-\frac {b \operatorname {Subst}\left (\int \left (-\frac {f^2 \left (6 d^2 e^2-12 c d e f+\left (1+6 c^2\right ) f^2\right ) \left (a+b \tanh ^{-1}(x)\right )}{d^4}-\frac {4 f^3 (d e-c f) x \left (a+b \tanh ^{-1}(x)\right )}{d^4}-\frac {f^4 x^2 \left (a+b \tanh ^{-1}(x)\right )}{d^4}+\frac {\left (d^4 e^4-4 c d^3 e^3 f+6 \left (1+c^2\right ) d^2 e^2 f^2-4 c \left (3+c^2\right ) d e f^3+\left (1+6 c^2+c^4\right ) f^4+4 f (d e-c f) \left (d^2 e^2-2 c d e f+f^2+c^2 f^2\right ) x\right ) \left (a+b \tanh ^{-1}(x)\right )}{d^4 \left (1-x^2\right )}\right ) \, dx,x,c+d x\right )}{2 f}\\ &=\frac {(e+f x)^4 \left (a+b \tanh ^{-1}(c+d x)\right )^2}{4 f}-\frac {b \operatorname {Subst}\left (\int \frac {\left (d^4 e^4-4 c d^3 e^3 f+6 \left (1+c^2\right ) d^2 e^2 f^2-4 c \left (3+c^2\right ) d e f^3+\left (1+6 c^2+c^4\right ) f^4+4 f (d e-c f) \left (d^2 e^2-2 c d e f+f^2+c^2 f^2\right ) x\right ) \left (a+b \tanh ^{-1}(x)\right )}{1-x^2} \, dx,x,c+d x\right )}{2 d^4 f}+\frac {\left (b f^3\right ) \operatorname {Subst}\left (\int x^2 \left (a+b \tanh ^{-1}(x)\right ) \, dx,x,c+d x\right )}{2 d^4}+\frac {\left (2 b f^2 (d e-c f)\right ) \operatorname {Subst}\left (\int x \left (a+b \tanh ^{-1}(x)\right ) \, dx,x,c+d x\right )}{d^4}+\frac {\left (b f \left (6 d^2 e^2-12 c d e f+\left (1+6 c^2\right ) f^2\right )\right ) \operatorname {Subst}\left (\int \left (a+b \tanh ^{-1}(x)\right ) \, dx,x,c+d x\right )}{2 d^4}\\ &=\frac {a b f \left (6 d^2 e^2-12 c d e f+\left (1+6 c^2\right ) f^2\right ) x}{2 d^3}+\frac {b f^2 (d e-c f) (c+d x)^2 \left (a+b \tanh ^{-1}(c+d x)\right )}{d^4}+\frac {b f^3 (c+d x)^3 \left (a+b \tanh ^{-1}(c+d x)\right )}{6 d^4}+\frac {(e+f x)^4 \left (a+b \tanh ^{-1}(c+d x)\right )^2}{4 f}-\frac {b \operatorname {Subst}\left (\int \left (\frac {d^4 e^4 \left (1+\frac {f \left (-4 c d^3 e^3+f \left (6 \left (1+c^2\right ) d^2 e^2-4 c \left (3+c^2\right ) d e f+\left (1+6 c^2+c^4\right ) f^2\right )\right )}{d^4 e^4}\right ) \left (a+b \tanh ^{-1}(x)\right )}{1-x^2}+\frac {4 f (d e-c f) \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right ) x \left (a+b \tanh ^{-1}(x)\right )}{1-x^2}\right ) \, dx,x,c+d x\right )}{2 d^4 f}-\frac {\left (b^2 f^3\right ) \operatorname {Subst}\left (\int \frac {x^3}{1-x^2} \, dx,x,c+d x\right )}{6 d^4}-\frac {\left (b^2 f^2 (d e-c f)\right ) \operatorname {Subst}\left (\int \frac {x^2}{1-x^2} \, dx,x,c+d x\right )}{d^4}+\frac {\left (b^2 f \left (6 d^2 e^2-12 c d e f+\left (1+6 c^2\right ) f^2\right )\right ) \operatorname {Subst}\left (\int \tanh ^{-1}(x) \, dx,x,c+d x\right )}{2 d^4}\\ &=\frac {b^2 f^2 (d e-c f) x}{d^3}+\frac {a b f \left (6 d^2 e^2-12 c d e f+\left (1+6 c^2\right ) f^2\right ) x}{2 d^3}+\frac {b^2 f \left (6 d^2 e^2-12 c d e f+\left (1+6 c^2\right ) f^2\right ) (c+d x) \tanh ^{-1}(c+d x)}{2 d^4}+\frac {b f^2 (d e-c f) (c+d x)^2 \left (a+b \tanh ^{-1}(c+d x)\right )}{d^4}+\frac {b f^3 (c+d x)^3 \left (a+b \tanh ^{-1}(c+d x)\right )}{6 d^4}+\frac {(e+f x)^4 \left (a+b \tanh ^{-1}(c+d x)\right )^2}{4 f}-\frac {\left (b^2 f^3\right ) \operatorname {Subst}\left (\int \frac {x}{1-x} \, dx,x,(c+d x)^2\right )}{12 d^4}-\frac {\left (b^2 f^2 (d e-c f)\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,c+d x\right )}{d^4}-\frac {\left (2 b (d e-c f) \left (d^2 e^2-2 c d e f+\left (1+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 )}{d^4}-\frac {\left (b^2 f \left (6 d^2 e^2-12 c d e f+\left (1+6 c^2\right ) f^2\right )\right ) \operatorname {Subst}\left (\int \frac {x}{1-x^2} \, dx,x,c+d x\right )}{2 d^4}-\frac {\left (b \left (d^4 e^4-4 c d^3 e^3 f+6 \left (1+c^2\right ) d^2 e^2 f^2-4 c \left (3+c^2\right ) d e f^3+\left (1+6 c^2+c^4\right ) f^4\right )\right ) \operatorname {Subst}\left (\int \frac {a+b \tanh ^{-1}(x)}{1-x^2} \, dx,x,c+d x\right )}{2 d^4 f}\\ &=\frac {b^2 f^2 (d e-c f) x}{d^3}+\frac {a b f \left (6 d^2 e^2-12 c d e f+\left (1+6 c^2\right ) f^2\right ) x}{2 d^3}-\frac {b^2 f^2 (d e-c f) \tanh ^{-1}(c+d x)}{d^4}+\frac {b^2 f \left (6 d^2 e^2-12 c d e f+\left (1+6 c^2\right ) f^2\right ) (c+d x) \tanh ^{-1}(c+d x)}{2 d^4}+\frac {b f^2 (d e-c f) (c+d x)^2 \left (a+b \tanh ^{-1}(c+d x)\right )}{d^4}+\frac {b f^3 (c+d x)^3 \left (a+b \tanh ^{-1}(c+d x)\right )}{6 d^4}+\frac {(d e-c f) \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right ) \left (a+b \tanh ^{-1}(c+d x)\right )^2}{d^4}-\frac {\left (d^4 e^4-4 c d^3 e^3 f+6 \left (1+c^2\right ) d^2 e^2 f^2-4 c \left (3+c^2\right ) d e f^3+\left (1+6 c^2+c^4\right ) f^4\right ) \left (a+b \tanh ^{-1}(c+d x)\right )^2}{4 d^4 f}+\frac {(e+f x)^4 \left (a+b \tanh ^{-1}(c+d x)\right )^2}{4 f}+\frac {b^2 f \left (6 d^2 e^2-12 c d e f+\left (1+6 c^2\right ) f^2\right ) \log \left (1-(c+d x)^2\right )}{4 d^4}-\frac {\left (b^2 f^3\right ) \operatorname {Subst}\left (\int \left (-1+\frac {1}{1-x}\right ) \, dx,x,(c+d x)^2\right )}{12 d^4}-\frac {\left (2 b (d e-c f) \left (d^2 e^2-2 c d e f+\left (1+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 )}{d^4}\\ &=\frac {b^2 f^2 (d e-c f) x}{d^3}+\frac {a b f \left (6 d^2 e^2-12 c d e f+\left (1+6 c^2\right ) f^2\right ) x}{2 d^3}+\frac {b^2 f^3 (c+d x)^2}{12 d^4}-\frac {b^2 f^2 (d e-c f) \tanh ^{-1}(c+d x)}{d^4}+\frac {b^2 f \left (6 d^2 e^2-12 c d e f+\left (1+6 c^2\right ) f^2\right ) (c+d x) \tanh ^{-1}(c+d x)}{2 d^4}+\frac {b f^2 (d e-c f) (c+d x)^2 \left (a+b \tanh ^{-1}(c+d x)\right )}{d^4}+\frac {b f^3 (c+d x)^3 \left (a+b \tanh ^{-1}(c+d x)\right )}{6 d^4}+\frac {(d e-c f) \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right ) \left (a+b \tanh ^{-1}(c+d x)\right )^2}{d^4}-\frac {\left (d^4 e^4-4 c d^3 e^3 f+6 \left (1+c^2\right ) d^2 e^2 f^2-4 c \left (3+c^2\right ) d e f^3+\left (1+6 c^2+c^4\right ) f^4\right ) \left (a+b \tanh ^{-1}(c+d x)\right )^2}{4 d^4 f}+\frac {(e+f x)^4 \left (a+b \tanh ^{-1}(c+d x)\right )^2}{4 f}-\frac {2 b (d e-c f) \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right ) \left (a+b \tanh ^{-1}(c+d x)\right ) \log \left (\frac {2}{1-c-d x}\right )}{d^4}+\frac {b^2 f^3 \log \left (1-(c+d x)^2\right )}{12 d^4}+\frac {b^2 f \left (6 d^2 e^2-12 c d e f+\left (1+6 c^2\right ) f^2\right ) \log \left (1-(c+d x)^2\right )}{4 d^4}+\frac {\left (2 b^2 (d e-c f) \left (d^2 e^2-2 c d e f+\left (1+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 )}{d^4}\\ &=\frac {b^2 f^2 (d e-c f) x}{d^3}+\frac {a b f \left (6 d^2 e^2-12 c d e f+\left (1+6 c^2\right ) f^2\right ) x}{2 d^3}+\frac {b^2 f^3 (c+d x)^2}{12 d^4}-\frac {b^2 f^2 (d e-c f) \tanh ^{-1}(c+d x)}{d^4}+\frac {b^2 f \left (6 d^2 e^2-12 c d e f+\left (1+6 c^2\right ) f^2\right ) (c+d x) \tanh ^{-1}(c+d x)}{2 d^4}+\frac {b f^2 (d e-c f) (c+d x)^2 \left (a+b \tanh ^{-1}(c+d x)\right )}{d^4}+\frac {b f^3 (c+d x)^3 \left (a+b \tanh ^{-1}(c+d x)\right )}{6 d^4}+\frac {(d e-c f) \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right ) \left (a+b \tanh ^{-1}(c+d x)\right )^2}{d^4}-\frac {\left (d^4 e^4-4 c d^3 e^3 f+6 \left (1+c^2\right ) d^2 e^2 f^2-4 c \left (3+c^2\right ) d e f^3+\left (1+6 c^2+c^4\right ) f^4\right ) \left (a+b \tanh ^{-1}(c+d x)\right )^2}{4 d^4 f}+\frac {(e+f x)^4 \left (a+b \tanh ^{-1}(c+d x)\right )^2}{4 f}-\frac {2 b (d e-c f) \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right ) \left (a+b \tanh ^{-1}(c+d x)\right ) \log \left (\frac {2}{1-c-d x}\right )}{d^4}+\frac {b^2 f^3 \log \left (1-(c+d x)^2\right )}{12 d^4}+\frac {b^2 f \left (6 d^2 e^2-12 c d e f+\left (1+6 c^2\right ) f^2\right ) \log \left (1-(c+d x)^2\right )}{4 d^4}-\frac {\left (2 b^2 (d e-c f) \left (d^2 e^2-2 c d e f+\left (1+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 )}{d^4}\\ &=\frac {b^2 f^2 (d e-c f) x}{d^3}+\frac {a b f \left (6 d^2 e^2-12 c d e f+\left (1+6 c^2\right ) f^2\right ) x}{2 d^3}+\frac {b^2 f^3 (c+d x)^2}{12 d^4}-\frac {b^2 f^2 (d e-c f) \tanh ^{-1}(c+d x)}{d^4}+\frac {b^2 f \left (6 d^2 e^2-12 c d e f+\left (1+6 c^2\right ) f^2\right ) (c+d x) \tanh ^{-1}(c+d x)}{2 d^4}+\frac {b f^2 (d e-c f) (c+d x)^2 \left (a+b \tanh ^{-1}(c+d x)\right )}{d^4}+\frac {b f^3 (c+d x)^3 \left (a+b \tanh ^{-1}(c+d x)\right )}{6 d^4}+\frac {(d e-c f) \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right ) \left (a+b \tanh ^{-1}(c+d x)\right )^2}{d^4}-\frac {\left (d^4 e^4-4 c d^3 e^3 f+6 \left (1+c^2\right ) d^2 e^2 f^2-4 c \left (3+c^2\right ) d e f^3+\left (1+6 c^2+c^4\right ) f^4\right ) \left (a+b \tanh ^{-1}(c+d x)\right )^2}{4 d^4 f}+\frac {(e+f x)^4 \left (a+b \tanh ^{-1}(c+d x)\right )^2}{4 f}-\frac {2 b (d e-c f) \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right ) \left (a+b \tanh ^{-1}(c+d x)\right ) \log \left (\frac {2}{1-c-d x}\right )}{d^4}+\frac {b^2 f^3 \log \left (1-(c+d x)^2\right )}{12 d^4}+\frac {b^2 f \left (6 d^2 e^2-12 c d e f+\left (1+6 c^2\right ) f^2\right ) \log \left (1-(c+d x)^2\right )}{4 d^4}-\frac {b^2 (d e-c f) \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right ) \text {Li}_2\left (1-\frac {2}{1-c-d x}\right )}{d^4}\\ \end {align*}

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Mathematica [A]  time = 7.48, size = 1082, normalized size = 1.93 \[ \frac {1}{12} \left (3 a^2 f^3 x^4+12 a^2 e f^2 x^3+18 a^2 e^2 f x^2+12 a^2 e^3 x+a b \left (6 x \left (4 e^3+6 f x e^2+4 f^2 x^2 e+f^3 x^3\right ) \tanh ^{-1}(c+d x)-\frac {-2 d f x \left (\left (18 e^2+6 f x e+f^2 x^2\right ) d^2-3 c f (8 e+f x) d+3 \left (3 c^2+1\right ) f^2\right )+3 (c-1) \left (4 d^3 e^3-6 (c-1) d^2 f e^2+4 (c-1)^2 d f^2 e-(c-1)^3 f^3\right ) \log (-c-d x+1)+3 (c+1) \left (-4 d^3 e^3+6 (c+1) d^2 f e^2-4 (c+1)^2 d f^2 e+(c+1)^3 f^3\right ) \log (c+d x+1)}{d^4}\right )+\frac {12 b^2 e^3 \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 {Li}_2\left (-e^{-2 \tanh ^{-1}(c+d x)}\right )\right )}{d}-\frac {18 b^2 e^2 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 {Li}_2\left (-e^{-2 \tanh ^{-1}(c+d x)}\right )\right )}{d^2}+\frac {b^2 f^3 \left (-36 \log \left (\frac {1}{\sqrt {1-(c+d x)^2}}\right ) c^2-11 c^2-10 d x c+d^2 x^2-3 \left (c^4-4 c^3+6 c^2-4 c-d^4 x^4+1\right ) \tanh ^{-1}(c+d x)^2+2 \tanh ^{-1}(c+d x) \left (13 c^3+9 d x c^2-3 d^2 x^2 c+9 c+d^3 x^3+3 d x+12 \left (c^3+c\right ) \log \left (1+e^{-2 \tanh ^{-1}(c+d x)}\right )\right )-8 \log \left (\frac {1}{\sqrt {1-(c+d x)^2}}\right )-12 \left (c^3+c\right ) \text {Li}_2\left (-e^{-2 \tanh ^{-1}(c+d x)}\right )-1\right )}{d^4}-\frac {3 b^2 e 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 {Li}_2\left (-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 )}{d^3}\right ) \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(e + f*x)^3*(a + b*ArcTanh[c + d*x])^2,x]

[Out]

(12*a^2*e^3*x + 18*a^2*e^2*f*x^2 + 12*a^2*e*f^2*x^3 + 3*a^2*f^3*x^4 + a*b*(6*x*(4*e^3 + 6*e^2*f*x + 4*e*f^2*x^
2 + f^3*x^3)*ArcTanh[c + d*x] - (-2*d*f*x*(3*(1 + 3*c^2)*f^2 - 3*c*d*f*(8*e + f*x) + d^2*(18*e^2 + 6*e*f*x + f
^2*x^2)) + 3*(-1 + c)*(4*d^3*e^3 - 6*(-1 + c)*d^2*e^2*f + 4*(-1 + c)^2*d*e*f^2 - (-1 + c)^3*f^3)*Log[1 - c - d
*x] + 3*(1 + c)*(-4*d^3*e^3 + 6*(1 + c)*d^2*e^2*f - 4*(1 + c)^2*d*e*f^2 + (1 + c)^3*f^3)*Log[1 + c + d*x])/d^4
) + (12*b^2*e^3*(ArcTanh[c + d*x]*((-1 + c + d*x)*ArcTanh[c + d*x] - 2*Log[1 + E^(-2*ArcTanh[c + d*x])]) + Pol
yLog[2, -E^(-2*ArcTanh[c + d*x])]))/d - (18*b^2*e^2*f*((1 - 2*c + c^2 - d^2*x^2)*ArcTanh[c + d*x]^2 - 2*ArcTan
h[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^3*(-1 - 11*c^2 - 10*c*d*x + d^2*x^2 - 3*(1 - 4*c + 6*c^2 - 4*c^3 + c^
4 - d^4*x^4)*ArcTanh[c + d*x]^2 + 2*ArcTanh[c + d*x]*(9*c + 13*c^3 + 3*d*x + 9*c^2*d*x - 3*c*d^2*x^2 + d^3*x^3
+ 12*(c + c^3)*Log[1 + E^(-2*ArcTanh[c + d*x])]) - 8*Log[1/Sqrt[1 - (c + d*x)^2]] - 36*c^2*Log[1/Sqrt[1 - (c
+ d*x)^2]] - 12*(c + c^3)*PolyLog[2, -E^(-2*ArcTanh[c + d*x])]))/d^4 - (3*b^2*e*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)*ArcT
anh[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]*Co
sh[3*ArcTanh[c + d*x]]*Log[1 + E^(-2*ArcTanh[c + d*x])] + 6*c^2*ArcTanh[c + d*x]*Cosh[3*ArcTanh[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]] - ArcTanh[c + d*x]^2*Sin
h[3*ArcTanh[c + d*x]] - 3*c^2*ArcTanh[c + d*x]^2*Sinh[3*ArcTanh[c + d*x]]))/d^3)/12

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fricas [F]  time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (a^{2} f^{3} x^{3} + 3 \, a^{2} e f^{2} x^{2} + 3 \, a^{2} e^{2} f x + a^{2} e^{3} + {\left (b^{2} f^{3} x^{3} + 3 \, b^{2} e f^{2} x^{2} + 3 \, b^{2} e^{2} f x + b^{2} e^{3}\right )} \operatorname {artanh}\left (d x + c\right )^{2} + 2 \, {\left (a b f^{3} x^{3} + 3 \, a b e f^{2} x^{2} + 3 \, a b e^{2} f x + a b e^{3}\right )} \operatorname {artanh}\left (d x + c\right ), x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)^3*(a+b*arctanh(d*x+c))^2,x, algorithm="fricas")

[Out]

integral(a^2*f^3*x^3 + 3*a^2*e*f^2*x^2 + 3*a^2*e^2*f*x + a^2*e^3 + (b^2*f^3*x^3 + 3*b^2*e*f^2*x^2 + 3*b^2*e^2*
f*x + b^2*e^3)*arctanh(d*x + c)^2 + 2*(a*b*f^3*x^3 + 3*a*b*e*f^2*x^2 + 3*a*b*e^2*f*x + a*b*e^3)*arctanh(d*x +
c), x)

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)^3*(a+b*arctanh(d*x+c))^2,x, algorithm="giac")

[Out]

Timed out

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maple [B]  time = 0.09, size = 4401, normalized size = 7.83 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x+e)^3*(a+b*arctanh(d*x+c))^2,x)

[Out]

1/4/d*e^3*b^2*ln(d*x+c-1)^2+1/d^3*b^2*f^2*c*e-1/d^3*b^2*f^2*arctanh(d*x+c)*ln(d*x+c-1)*c^3*e-3/4/d^2*b^2*f*ln(
d*x+c+1)*ln(-1/2*d*x-1/2*c+1/2)*c^2*e^2+3/2/d^3*b^2*f^2*ln(d*x+c+1)*ln(-1/2*d*x-1/2*c+1/2)*c^2*e-3/2/d^2*b^2*f
*ln(d*x+c+1)*ln(-1/2*d*x-1/2*c+1/2)*c*e^2+3/2/d^4*b^2*f^3*ln(d*x+c+1)*c^2+1/2/d^4*b^2*f^3*ln(d*x+c+1)*c-1/8*b^
2/f*ln(d*x+c+1)*ln(-1/2*d*x-1/2*c+1/2)*e^4+1/8*b^2/f*ln(-1/2*d*x-1/2*c+1/2)*ln(1/2+1/2*d*x+1/2*c)*e^4+1/4*b^2/
f*arctanh(d*x+c)*ln(d*x+c-1)*e^4+1/2*a*b/f*arctanh(d*x+c)*e^4+1/6/d*a*b*f^3*x^3+b^2/d^2*e*f^2*x-5/6*b^2/d^3*c*
f^3*x+1/2*a*b/d^3*f^3*x+1/d*e^3*a*b*ln(d*x+c+1)+1/d*e^3*b^2*arctanh(d*x+c)*ln(d*x+c+1)-1/2/d*e^3*b^2*ln(-1/2*d
*x-1/2*c+1/2)*ln(1/2+1/2*d*x+1/2*c)+1/2/d*e^3*b^2*ln(-1/2*d*x-1/2*c+1/2)*ln(d*x+c+1)-1/4/d^4*a*b*f^3*ln(d*x+c+
1)+3/2/d^2*b^2*f*ln(d*x+c+1)*e^2-1/8/d^4*b^2*f^3*ln(d*x+c+1)*ln(-1/2*d*x-1/2*c+1/2)-1/2/d^4*b^2*f^3*ln(d*x+c-1
)*c+1/16/d^4*b^2*f^3*ln(d*x+c-1)^2*c^4+1/6/d*b^2*f^3*arctanh(d*x+c)*x^3+13/6/d^4*b^2*f^3*arctanh(d*x+c)*c^3-1/
8*b^2/f*ln(d*x+c-1)*ln(1/2+1/2*d*x+1/2*c)*e^4+1/4*a*b/f*ln(d*x+c-1)*e^4-1/4*a*b/f*ln(d*x+c+1)*e^4+1/d*e^3*b^2*
arctanh(d*x+c)*ln(d*x+c-1)+1/d*e^3*a*b*ln(d*x+c-1)-1/4/d*b^2*ln(d*x+c+1)^2*c*e^3-1/4/d*b^2*ln(d*x+c-1)^2*c*e^3
+1/2/d^3*b^2*f^2*ln(d*x+c-1)*e+3/8/d^4*b^2*f^3*ln(d*x+c-1)^2*c^2-1/2/d^3*b^2*f^2*ln(d*x+c+1)*e-1/d^3*b^2*f^2*d
ilog(1/2+1/2*d*x+1/2*c)*e+3/2/d^2*b^2*f*ln(d*x+c-1)*e^2+3/2/d^4*b^2*f^3*ln(d*x+c-1)*c^2+1/8/d^4*b^2*f^3*ln(-1/
2*d*x-1/2*c+1/2)*ln(1/2+1/2*d*x+1/2*c)+1/4/d^4*b^2*f^3*arctanh(d*x+c)*ln(d*x+c-1)-1/4/d^4*b^2*f^3*arctanh(d*x+
c)*ln(d*x+c+1)-1/8/d^4*b^2*f^3*ln(d*x+c-1)*ln(1/2+1/2*d*x+1/2*c)+1/2/d^3*b^2*f^3*arctanh(d*x+c)*x+1/2/d^4*b^2*
f^3*arctanh(d*x+c)*c+3/8/d^2*b^2*f*ln(d*x+c-1)^2*e^2+1/4/d^3*b^2*f^2*ln(d*x+c-1)^2*e+1/4/d^4*b^2*f^3*ln(d*x+c+
1)^2*c^3-1/4/d^4*b^2*f^3*ln(d*x+c-1)^2*c^3+3/8/d^4*b^2*f^3*ln(d*x+c+1)^2*c^2+1/4/d^4*b^2*f^3*ln(d*x+c+1)^2*c+1
/d^4*b^2*f^3*dilog(1/2+1/2*d*x+1/2*c)*c-1/4*b^2/f*arctanh(d*x+c)*ln(d*x+c+1)*e^4+1/4/d^4*a*b*f^3*ln(d*x+c-1)+b
^2*f^2*arctanh(d*x+c)^2*e*x^3+3/2*b^2*f*arctanh(d*x+c)^2*e^2*x^2+2*arctanh(d*x+c)*x*a*b*e^3+1/16/d^4*b^2*f^3*l
n(d*x+c+1)^2*c^4-1/4/d^4*b^2*f^3*ln(d*x+c-1)^2*c+1/d^4*b^2*f^3*dilog(1/2+1/2*d*x+1/2*c)*c^3+3/8/d^2*b^2*f*ln(d
*x+c+1)^2*e^2-1/4/d^3*b^2*f^2*ln(d*x+c+1)^2*e-4*a*b/d^2*f^2*c*e*x+3/2/d^3*b^2*f^2*ln(d*x+c+1)*ln(-1/2*d*x-1/2*
c+1/2)*c*e+a^2*x*e^3+1/4*a^2*f^3*x^4+13/6/d^4*a*b*f^3*c^3+1/2/d^4*a*b*f^3*c-1/2/d*e^3*b^2*ln(d*x+c-1)*ln(1/2+1
/2*d*x+1/2*c)-3/4/d^2*b^2*f*ln(d*x+c-1)*ln(1/2+1/2*d*x+1/2*c)*e^2-1/2/d^3*b^2*f^2*ln(d*x+c-1)*ln(1/2+1/2*d*x+1
/2*c)*e-3/4/d^2*b^2*f*ln(d*x+c+1)*ln(-1/2*d*x-1/2*c+1/2)*e^2+1/2/d^3*b^2*f^2*ln(d*x+c+1)*ln(-1/2*d*x-1/2*c+1/2
)*e-1/2/d^2*b^2*f^3*arctanh(d*x+c)*x^2*c+3/2/d^3*b^2*f^3*arctanh(d*x+c)*c^2*x+1/d*b^2*f^2*arctanh(d*x+c)*e*x^2
-5/d^3*b^2*f^2*arctanh(d*x+c)*e*c^2+3/2*a*b/d^3*f^3*c^2*x+3*a*b/d*f*e^2*x-1/d^4*b^2*f^3*arctanh(d*x+c)*ln(d*x+
c-1)*c+3/4/d^2*b^2*f*ln(-1/2*d*x-1/2*c+1/2)*ln(1/2+1/2*d*x+1/2*c)*e^2-1/8/d^4*b^2*f^3*ln(d*x+c+1)*ln(-1/2*d*x-
1/2*c+1/2)*c^4-3/2/d^4*a*b*f^3*ln(d*x+c+1)*c^2-1/d^4*a*b*f^3*ln(d*x+c-1)*c-1/d^4*a*b*f^3*ln(d*x+c+1)*c-1/2/d^4
*b^2*f^3*ln(d*x+c+1)*ln(-1/2*d*x-1/2*c+1/2)*c^3-3/4/d^4*b^2*f^3*ln(d*x+c+1)*ln(-1/2*d*x-1/2*c+1/2)*c^2+1/2/d^4
*b^2*f^3*ln(d*x+c-1)*ln(1/2+1/2*d*x+1/2*c)*c-1/2/d^4*b^2*f^3*ln(d*x+c+1)*ln(-1/2*d*x-1/2*c+1/2)*c+1/8/d^4*b^2*
f^3*ln(-1/2*d*x-1/2*c+1/2)*ln(1/2+1/2*d*x+1/2*c)*c^4+1/2/d^4*b^2*f^3*ln(-1/2*d*x-1/2*c+1/2)*ln(1/2+1/2*d*x+1/2
*c)*c^3+3/4/d^4*b^2*f^3*ln(-1/2*d*x-1/2*c+1/2)*ln(1/2+1/2*d*x+1/2*c)*c^2+1/2/d^4*b^2*f^3*ln(-1/2*d*x-1/2*c+1/2
)*ln(1/2+1/2*d*x+1/2*c)*c-1/d^4*b^2*f^3*arctanh(d*x+c)*ln(d*x+c+1)*c^3-3/2/d^4*b^2*f^3*arctanh(d*x+c)*ln(d*x+c
+1)*c^2-1/d^4*b^2*f^3*arctanh(d*x+c)*ln(d*x+c+1)*c+1/4/d^4*b^2*f^3*arctanh(d*x+c)*ln(d*x+c-1)*c^4+3/d^3*a*b*f^
2*ln(d*x+c+1)*c*e+3/d^3*a*b*f^2*ln(d*x+c+1)*c^2*e-11/12/d^4*b^2*f^3*c^2+1/4*a^2/f*e^4-3/d^2*a*b*f*ln(d*x+c+1)*
c*e^2+3/d^3*b^2*f^2*arctanh(d*x+c)*ln(d*x+c+1)*c*e+1/3/d^4*b^2*f^3*ln(d*x+c-1)-1/4/d*e^3*b^2*ln(d*x+c+1)^2+a^2
*f^2*x^3*e+3/2*a^2*f*x^2*e^2+1/12/d^2*b^2*f^3*x^2+1/16*b^2/f*ln(d*x+c-1)^2*e^4+1/4*b^2/f*arctanh(d*x+c)^2*e^4+
1/16*b^2/f*ln(d*x+c+1)^2*e^4+1/4*b^2*f^3*arctanh(d*x+c)^2*x^4+arctanh(d*x+c)^2*x*b^2*e^3-1/d*b^2*dilog(1/2+1/2
*d*x+1/2*c)*e^3+1/3/d^4*b^2*f^3*ln(d*x+c+1)+1/16/d^4*b^2*f^3*ln(d*x+c-1)^2+1/16/d^4*b^2*f^3*ln(d*x+c+1)^2+3/4/
d^2*b^2*f*ln(d*x+c+1)^2*c*e^2-3/d^3*b^2*f^2*ln(d*x+c-1)*c*e-3/d^3*b^2*f^2*ln(d*x+c+1)*c*e-3/2/d^2*a*b*f*ln(d*x
+c+1)*e^2+1/4/d^4*a*b*f^3*ln(d*x+c-1)*c^4+1/d^3*a*b*f^2*ln(d*x+c+1)*e+3/2/d^2*a*b*f*ln(d*x+c-1)*e^2+3/2/d^4*a*
b*f^3*ln(d*x+c-1)*c^2-3/4/d^3*b^2*f^2*ln(d*x+c-1)^2*c*e+3/4/d^3*b^2*f^2*ln(d*x+c-1)^2*c^2*e+3/d*b^2*f*arctanh(
d*x+c)*e^2*x+3/d^2*b^2*f*arctanh(d*x+c)*e^2*c-1/d*a*b*ln(d*x+c-1)*c*e^3-3/d^3*b^2*f^2*dilog(1/2+1/2*d*x+1/2*c)
*c^2*e+3/d^2*b^2*f*dilog(1/2+1/2*d*x+1/2*c)*c*e^2-1/2/d^3*b^2*f^2*ln(-1/2*d*x-1/2*c+1/2)*ln(1/2+1/2*d*x+1/2*c)
*e+3/2/d^4*b^2*f^3*arctanh(d*x+c)*ln(d*x+c-1)*c^2-1/d^4*a*b*f^3*ln(d*x+c-1)*c^3-1/d^4*a*b*f^3*ln(d*x+c+1)*c^3-
3/4/d^3*b^2*f^2*ln(d*x+c+1)^2*c*e+3/8/d^2*b^2*f*ln(d*x+c+1)^2*c^2*e^2-1/4/d^3*b^2*f^2*ln(d*x+c+1)^2*c^3*e+1/d^
3*b^2*f^2*arctanh(d*x+c)*ln(d*x+c-1)*e+1/d^3*b^2*f^2*arctanh(d*x+c)*ln(d*x+c+1)*e+3/2/d^2*b^2*f*arctanh(d*x+c)
*ln(d*x+c-1)*e^2-3/2/d^2*b^2*f*arctanh(d*x+c)*ln(d*x+c+1)*e^2-1/8/d^4*b^2*f^3*ln(d*x+c-1)*ln(1/2+1/2*d*x+1/2*c
)*c^4+1/2/d^4*b^2*f^3*ln(d*x+c-1)*ln(1/2+1/2*d*x+1/2*c)*c^3-3/4/d^4*b^2*f^3*ln(d*x+c-1)*ln(1/2+1/2*d*x+1/2*c)*
c^2-3/4/d^2*b^2*f*ln(d*x+c-1)^2*c*e^2+1/d*a*b*ln(d*x+c+1)*c*e^3+2*a*b*f^2*arctanh(d*x+c)*e*x^3+3*a*b*f*arctanh
(d*x+c)*e^2*x^2-3/4/d^3*b^2*f^2*ln(d*x+c+1)^2*c^2*e-1/2/d*b^2*ln(-1/2*d*x-1/2*c+1/2)*ln(1/2+1/2*d*x+1/2*c)*c*e
^3+1/d*b^2*arctanh(d*x+c)*ln(d*x+c+1)*c*e^3-1/d*b^2*arctanh(d*x+c)*ln(d*x+c-1)*c*e^3-3/d^2*a*b*f*ln(d*x+c-1)*c
*e^2-4/d^2*b^2*f^2*arctanh(d*x+c)*e*x*c+3/d^3*b^2*f^2*arctanh(d*x+c)*ln(d*x+c+1)*c^2*e-3/d^2*b^2*f*arctanh(d*x
+c)*ln(d*x+c+1)*c*e^2+3/2/d^2*b^2*f*ln(-1/2*d*x-1/2*c+1/2)*ln(1/2+1/2*d*x+1/2*c)*c*e^2+1/2/d^3*b^2*f^2*ln(d*x+
c+1)*ln(-1/2*d*x-1/2*c+1/2)*c^3*e-3/2/d^2*b^2*f*arctanh(d*x+c)*ln(d*x+c+1)*c^2*e^2-3/2/d^3*b^2*f^2*ln(-1/2*d*x
-1/2*c+1/2)*ln(1/2+1/2*d*x+1/2*c)*c*e+1/d^3*b^2*f^2*arctanh(d*x+c)*ln(d*x+c+1)*c^3*e+1/2*a*b*f^3*arctanh(d*x+c
)*x^4+1/d^3*a*b*f^2*ln(d*x+c+1)*c^3*e+3/d^3*a*b*f^2*ln(d*x+c-1)*c^2*e+3/2/d^2*a*b*f*ln(d*x+c-1)*c^2*e^2-1/d^3*
a*b*f^2*ln(d*x+c-1)*c^3*e-1/2/d^3*b^2*f^2*ln(-1/2*d*x-1/2*c+1/2)*ln(1/2+1/2*d*x+1/2*c)*c^3*e-3/d^3*b^2*f^2*arc
tanh(d*x+c)*ln(d*x+c-1)*c*e-3/d^2*b^2*f*arctanh(d*x+c)*ln(d*x+c-1)*c*e^2+3/4/d^2*b^2*f*ln(-1/2*d*x-1/2*c+1/2)*
ln(1/2+1/2*d*x+1/2*c)*c^2*e^2-3/2/d^3*b^2*f^2*ln(-1/2*d*x-1/2*c+1/2)*ln(1/2+1/2*d*x+1/2*c)*c^2*e-3/d^3*a*b*f^2
*ln(d*x+c-1)*c*e-3/2/d^2*a*b*f*ln(d*x+c+1)*c^2*e^2+3/2/d^2*b^2*f*arctanh(d*x+c)*ln(d*x+c-1)*c^2*e^2+1/2/d^3*b^
2*f^2*ln(d*x+c-1)*ln(1/2+1/2*d*x+1/2*c)*c^3*e-3/4/d^2*b^2*f*ln(d*x+c-1)*ln(1/2+1/2*d*x+1/2*c)*c^2*e^2-3/2/d^3*
b^2*f^2*ln(d*x+c-1)*ln(1/2+1/2*d*x+1/2*c)*c^2*e+3/2/d^2*b^2*f*ln(d*x+c-1)*ln(1/2+1/2*d*x+1/2*c)*c*e^2+3/2/d^3*
b^2*f^2*ln(d*x+c-1)*ln(1/2+1/2*d*x+1/2*c)*c*e+1/2/d*b^2*ln(d*x+c-1)*ln(1/2+1/2*d*x+1/2*c)*c*e^3+1/2/d*b^2*ln(d
*x+c+1)*ln(-1/2*d*x-1/2*c+1/2)*c*e^3+1/d^3*a*b*f^2*ln(d*x+c-1)*e-1/4/d^4*a*b*f^3*ln(d*x+c+1)*c^4+3/8/d^2*b^2*f
*ln(d*x+c-1)^2*c^2*e^2-1/4/d^3*b^2*f^2*ln(d*x+c-1)^2*c^3*e-5/d^3*a*b*f^2*c^2*e+3/d^2*a*b*f*c*e^2+3/d^3*b^2*f^2
*arctanh(d*x+c)*ln(d*x+c-1)*c^2*e-1/2/d^2*a*b*f^3*c*x^2+1/d*a*b*f^2*e*x^2-1/4/d^4*b^2*f^3*arctanh(d*x+c)*ln(d*
x+c+1)*c^4-1/d^4*b^2*f^3*arctanh(d*x+c)*ln(d*x+c-1)*c^3

________________________________________________________________________________________

maxima [B]  time = 0.65, size = 1363, normalized size = 2.43 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)^3*(a+b*arctanh(d*x+c))^2,x, algorithm="maxima")

[Out]

1/4*a^2*f^3*x^4 + a^2*e*f^2*x^3 + 3/2*a^2*e^2*f*x^2 + 3/2*(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^2*f + (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*e*f^2 + 1/12*(6*x^4*arctanh(d*x + c) + d*(2*(d^2*x^3 - 3*c*d*x^2 + 3*(3*c^2 + 1)*x)/d^4 - 3*(c^4 + 4*c^3
+ 6*c^2 + 4*c + 1)*log(d*x + c + 1)/d^5 + 3*(c^4 - 4*c^3 + 6*c^2 - 4*c + 1)*log(d*x + c - 1)/d^5))*a*b*f^3 +
a^2*e^3*x + (2*(d*x + c)*arctanh(d*x + c) + log(-(d*x + c)^2 + 1))*a*b*e^3/d + (d^3*e^3 + 3*c^2*d*e*f^2 - c^3*
f^3 + d*e*f^2 - (3*d^2*e^2*f + f^3)*c)*(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^4 + 1/12*(13*c^3*f^3 + 18*d^2*e^2*f - 6*d*e*f^2 - 6*(5*d*e*f^2 - 3*f^3)*c^2 + 4*f^3 + 9*(2*d^2*e^
2*f - 4*d*e*f^2 + f^3)*c)*b^2*log(d*x + c + 1)/d^4 - 1/12*(13*c^3*f^3 - 18*d^2*e^2*f - 6*d*e*f^2 - 6*(5*d*e*f^
2 + 3*f^3)*c^2 - 4*f^3 + 9*(2*d^2*e^2*f + 4*d*e*f^2 + f^3)*c)*b^2*log(d*x + c - 1)/d^4 + 1/48*(4*b^2*d^2*f^3*x
^2 + 8*(6*d^2*e*f^2 - 5*c*d*f^3)*b^2*x + 3*(b^2*d^4*f^3*x^4 + 4*b^2*d^4*e*f^2*x^3 + 6*b^2*d^4*e^2*f*x^2 + 4*b^
2*d^4*e^3*x - (c^4*f^3 - 4*d^3*e^3 + 6*d^2*e^2*f - 4*(d*e*f^2 - f^3)*c^3 - 4*d*e*f^2 + 6*(d^2*e^2*f - 2*d*e*f^
2 + f^3)*c^2 + f^3 - 4*(d^3*e^3 - 3*d^2*e^2*f + 3*d*e*f^2 - f^3)*c)*b^2)*log(d*x + c + 1)^2 + 3*(b^2*d^4*f^3*x
^4 + 4*b^2*d^4*e*f^2*x^3 + 6*b^2*d^4*e^2*f*x^2 + 4*b^2*d^4*e^3*x - (c^4*f^3 + 4*d^3*e^3 + 6*d^2*e^2*f - 4*(d*e
*f^2 + f^3)*c^3 + 4*d*e*f^2 + 6*(d^2*e^2*f + 2*d*e*f^2 + f^3)*c^2 + f^3 - 4*(d^3*e^3 + 3*d^2*e^2*f + 3*d*e*f^2
+ f^3)*c)*b^2)*log(-d*x - c + 1)^2 + 4*(b^2*d^3*f^3*x^3 + 3*(2*d^3*e*f^2 - c*d^2*f^3)*b^2*x^2 + 3*(6*d^3*e^2*
f - 8*c*d^2*e*f^2 + 3*c^2*d*f^3 + d*f^3)*b^2*x)*log(d*x + c + 1) - 2*(2*b^2*d^3*f^3*x^3 + 6*(2*d^3*e*f^2 - c*d
^2*f^3)*b^2*x^2 + 6*(6*d^3*e^2*f - 8*c*d^2*e*f^2 + 3*c^2*d*f^3 + d*f^3)*b^2*x + 3*(b^2*d^4*f^3*x^4 + 4*b^2*d^4
*e*f^2*x^3 + 6*b^2*d^4*e^2*f*x^2 + 4*b^2*d^4*e^3*x - (c^4*f^3 - 4*d^3*e^3 + 6*d^2*e^2*f - 4*(d*e*f^2 - f^3)*c^
3 - 4*d*e*f^2 + 6*(d^2*e^2*f - 2*d*e*f^2 + f^3)*c^2 + f^3 - 4*(d^3*e^3 - 3*d^2*e^2*f + 3*d*e*f^2 - f^3)*c)*b^2
)*log(d*x + c + 1))*log(-d*x - c + 1))/d^4

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (e+f\,x\right )}^3\,{\left (a+b\,\mathrm {atanh}\left (c+d\,x\right )\right )}^2 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e + f*x)^3*(a + b*atanh(c + d*x))^2,x)

[Out]

int((e + f*x)^3*(a + b*atanh(c + d*x))^2, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \operatorname {atanh}{\left (c + d x \right )}\right )^{2} \left (e + f x\right )^{3}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)**3*(a+b*atanh(d*x+c))**2,x)

[Out]

Integral((a + b*atanh(c + d*x))**2*(e + f*x)**3, x)

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