3.1.38 \(\int (e+f x)^3 (a+b \tanh ^{-1}(c+d x))^2 \, dx\) [38]

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

[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 = 0.71, 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 = {6246, 6065, 6021, 266, 6037, 327, 212, 272, 45, 6195, 6095, 6131, 6055, 2449, 2352} \begin {gather*} \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} \end {gather*}

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 45

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 212

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

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 272

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 327

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 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 6065

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 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 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]

Rule 6195

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 6246

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 {\text {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 \text {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 \text {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 ) \text {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 ) \text {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 ) \text {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 \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 = 5.99, size = 1082, normalized size = 1.93 \begin {gather*} \frac {1}{12} \left (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 \left (6 x \left (4 e^3+6 e^2 f x+4 e f^2 x^2+f^3 x^3\right ) \tanh ^{-1}(c+d x)-\frac {-2 d f x \left (3 \left (1+3 c^2\right ) f^2-3 c d f (8 e+f x)+d^2 \left (18 e^2+6 e f x+f^2 x^2\right )\right )+3 (-1+c) \left (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\right ) \log (1-c-d x)+3 (1+c) \left (-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\right ) \log (1+c+d x)}{d^4}\right )+\frac {12 b^2 e^3 \left (\tanh ^{-1}(c+d x) \left ((-1+c+d x) \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 {18 b^2 e^2 f \left (\left (1-2 c+c^2-d^2 x^2\right ) \tanh ^{-1}(c+d x)^2-2 \tanh ^{-1}(c+d x) \left (c+d x+2 c \log \left (1+e^{-2 \tanh ^{-1}(c+d x)}\right )\right )+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^3 \left (-1-11 c^2-10 c d x+d^2 x^2-3 \left (1-4 c+6 c^2-4 c^3+c^4-d^4 x^4\right ) \tanh ^{-1}(c+d x)^2+2 \tanh ^{-1}(c+d x) \left (9 c+13 c^3+3 d x+9 c^2 d x-3 c d^2 x^2+d^3 x^3+12 \left (c+c^3\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 )-36 c^2 \log \left (\frac {1}{\sqrt {1-(c+d x)^2}}\right )-12 \left (c+c^3\right ) \text {PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c+d x)}\right )\right )}{d^4}-\frac {3 b^2 e f^2 \left (1-(c+d x)^2\right )^{3/2} \left (-\frac {c+d x}{\sqrt {1-(c+d x)^2}}+\frac {6 c (c+d x) \tanh ^{-1}(c+d x)}{\sqrt {1-(c+d x)^2}}+\frac {3 (c+d x) \tanh ^{-1}(c+d x)^2}{\sqrt {1-(c+d x)^2}}-\frac {3 c^2 (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 )+3 c^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 )+6 c^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 )-6 c \cosh \left (3 \tanh ^{-1}(c+d x)\right ) \log \left (\frac {1}{\sqrt {1-(c+d x)^2}}\right )+\frac {3 \left (1-4 c+3 c^2\right ) \tanh ^{-1}(c+d x)^2+2 \tanh ^{-1}(c+d x) \left (2+\left (3+9 c^2\right ) \log \left (1+e^{-2 \tanh ^{-1}(c+d x)}\right )\right )-18 c \log \left (\frac {1}{\sqrt {1-(c+d x)^2}}\right )}{\sqrt {1-(c+d x)^2}}-\frac {4 \left (1+3 c^2\right ) \text {PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c+d x)}\right )}{\left (1-(c+d x)^2\right )^{3/2}}-\sinh \left (3 \tanh ^{-1}(c+d x)\right )+6 c \tanh ^{-1}(c+d x) \sinh \left (3 \tanh ^{-1}(c+d x)\right )-\tanh ^{-1}(c+d x)^2 \sinh \left (3 \tanh ^{-1}(c+d x)\right )-3 c^2 \tanh ^{-1}(c+d x)^2 \sinh \left (3 \tanh ^{-1}(c+d x)\right )\right )}{d^3}\right ) \end {gather*}

Antiderivative was successfully verified.

[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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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(4768\) vs. \(2(548)=1096\).
time = 1.10, size = 4769, normalized size = 8.49

method	result	size
risch	\(\text {Expression too large to display}\)	\(4356\)
derivativedivides	\(\text {Expression too large to display}\)	\(4769\)
default	\(\text {Expression too large to display}\)	\(4769\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(1/4*e^3*b^2*ln(d*x+c-1)^2-1/4*e^3*b^2*ln(d*x+c+1)^2+e^3*b^2*arctanh(d*x+c)*ln(d*x+c-1)+e^3*b^2*arctanh(d*
x+c)*ln(d*x+c+1)-1/2*e^3*b^2*ln(d*x+c-1)*ln(1/2*d*x+1/2*c+1/2)+1/2*e^3*b^2*ln(-1/2*d*x-1/2*c+1/2)*ln(d*x+c+1)-
1/2*e^3*b^2*ln(-1/2*d*x-1/2*c+1/2)*ln(1/2*d*x+1/2*c+1/2)+e^3*a*b*ln(d*x+c-1)+e^3*a*b*ln(d*x+c+1)+1/4*a*b/d^3*f
^3*ln(d*x+c-1)*c^4+3/8*b^2/d*f*ln(d*x+c+1)^2*e^2-1/4*b^2/d^2*f^2*ln(d*x+c+1)^2*e-b^2/d^2*f^2*dilog(1/2*d*x+1/2
*c+1/2)*e-b^2*dilog(1/2*d*x+1/2*c+1/2)*e^3+1/12*b^2/d^3*f^3*(d*x+c)^2+1/3*b^2/d^3*f^3*ln(d*x+c-1)+1/3*b^2/d^3*
f^3*ln(d*x+c+1)+1/16*b^2/d^3*f^3*ln(d*x+c+1)^2+1/4*(c*f-d*e-f*(d*x+c))^4*a^2/d^3/f+1/16*b^2/d^3*f^3*ln(d*x+c-1
)^2-1/4*b^2*ln(d*x+c+1)^2*c*e^3-1/4*b^2*ln(d*x+c-1)^2*c*e^3-b^2*arctanh(d*x+c)^2*c*e^3+b^2*arctanh(d*x+c)^2*e^
3*(d*x+c)+1/6*a*b/d^3*f^3*(d*x+c)^3+1/4*a*b/d^3*f^3*ln(d*x+c-1)-1/4*a*b/d^3*f^3*ln(d*x+c+1)-a*b*ln(d*x+c-1)*c*
e^3+a*b*ln(d*x+c+1)*c*e^3-2*a*b*arctanh(d*x+c)*c*e^3+2*a*b*arctanh(d*x+c)*e^3*(d*x+c)+3/8*b^2/d*f*ln(d*x+c-1)^
2*e^2+1/4*b^2/d^2*f^2*ln(d*x+c-1)^2*e+3/8*b^2/d^3*f^3*ln(d*x+c-1)^2*c^2-1/4*b^2/d^3*f^3*ln(d*x+c-1)^2*c+1/4*b^
2/d^3*f^3*arctanh(d*x+c)*ln(d*x+c-1)-1/4*b^2/d^3*f^3*arctanh(d*x+c)*ln(d*x+c+1)+1/2*b^2/d^3*f^3*arctanh(d*x+c)
*(d*x+c)+1/6*b^2/d^3*f^3*arctanh(d*x+c)*(d*x+c)^3-1/8*b^2/d^3*f^3*ln(d*x+c+1)*ln(-1/2*d*x-1/2*c+1/2)+1/8*b^2/d
^3*f^3*ln(-1/2*d*x-1/2*c+1/2)*ln(1/2*d*x+1/2*c+1/2)+1/16*b^2/d^3*f^3*ln(d*x+c+1)^2*c^4+1/4*b^2/d^3*f^3*ln(d*x+
c+1)^2*c^3+3/8*b^2/d^3*f^3*ln(d*x+c+1)^2*c^2+1/4*b^2/d^3*f^3*ln(d*x+c+1)^2*c+b^2/d^3*f^3*dilog(1/2*d*x+1/2*c+1
/2)*c^3+b^2/d^3*f^3*dilog(1/2*d*x+1/2*c+1/2)*c-1/8*b^2/d^3*f^3*ln(d*x+c-1)*ln(1/2*d*x+1/2*c+1/2)+1/16*b^2/d^3*
f^3*ln(d*x+c-1)^2*c^4-1/4*b^2/d^3*f^3*ln(d*x+c-1)^2*c^3+1/4*b^2/d^3*f^3*arctanh(d*x+c)^2*c^4+1/4*b^2/d^3*f^3*a
rctanh(d*x+c)^2*(d*x+c)^4+3/2*b^2/d*f*ln(d*x+c+1)*e^2-1/2*b^2/d^2*f^2*ln(d*x+c+1)*e+3/2*b^2/d*f*ln(d*x+c-1)*e^
2+1/2*b^2/d^2*f^2*ln(d*x+c-1)*e+3/2*b^2/d^3*f^3*ln(d*x+c+1)*c^2+1/2*b^2/d^3*f^3*ln(d*x+c+1)*c+3/2*b^2/d^3*f^3*
ln(d*x+c-1)*c^2-b^2*arctanh(d*x+c)*ln(d*x+c-1)*c*e^3+1/2*b^2*ln(d*x+c+1)*ln(-1/2*d*x-1/2*c+1/2)*c*e^3-1/2*b^2*
ln(-1/2*d*x-1/2*c+1/2)*ln(1/2*d*x+1/2*c+1/2)*c*e^3+1/2*b^2*ln(d*x+c-1)*ln(1/2*d*x+1/2*c+1/2)*c*e^3+b^2*arctanh
(d*x+c)*ln(d*x+c+1)*c*e^3-b^2/d^3*f^3*c*(d*x+c)+b^2/d^2*f^2*e*(d*x+c)-1/2*b^2/d^3*f^3*ln(d*x+c-1)*c+1/4*b^2*d/
f*arctanh(d*x+c)^2*e^4+1/16*b^2*d/f*ln(d*x+c+1)^2*e^4+1/16*b^2*d/f*ln(d*x+c-1)^2*e^4+1/2*a*b/d^3*f^3*(d*x+c)-3
/2*b^2/d*f*arctanh(d*x+c)*ln(d*x+c+1)*e^2+b^2/d^2*f^2*arctanh(d*x+c)*ln(d*x+c+1)*e+3/2*b^2/d*f*arctanh(d*x+c)*
ln(d*x+c-1)*e^2+b^2/d^2*f^2*arctanh(d*x+c)*ln(d*x+c-1)*e+3*b^2/d*f*arctanh(d*x+c)*e^2*(d*x+c)-3/4*b^2/d*f*ln(d
*x+c+1)*ln(-1/2*d*x-1/2*c+1/2)*e^2+1/2*b^2/d^2*f^2*ln(d*x+c+1)*ln(-1/2*d*x-1/2*c+1/2)*e+3/4*b^2/d*f*ln(-1/2*d*
x-1/2*c+1/2)*ln(1/2*d*x+1/2*c+1/2)*e^2-1/2*b^2/d^2*f^2*ln(-1/2*d*x-1/2*c+1/2)*ln(1/2*d*x+1/2*c+1/2)*e-1/4*b^2/
d^2*f^2*ln(d*x+c+1)^2*c^3*e+3/8*b^2/d*f*ln(d*x+c+1)^2*c^2*e^2-3/4*b^2/d^2*f^2*ln(d*x+c+1)^2*c^2*e+3/4*b^2/d*f*
ln(d*x+c+1)^2*c*e^2-3/4*b^2/d^2*f^2*ln(d*x+c+1)^2*c*e-1/8*b^2*d/f*ln(d*x+c+1)*ln(-1/2*d*x-1/2*c+1/2)*e^4+1/8*b
^2*d/f*ln(-1/2*d*x-1/2*c+1/2)*ln(1/2*d*x+1/2*c+1/2)*e^4-1/8*b^2*d/f*ln(d*x+c-1)*ln(1/2*d*x+1/2*c+1/2)*e^4-1/4*
b^2*d/f*arctanh(d*x+c)*ln(d*x+c+1)*e^4+1/4*b^2*d/f*arctanh(d*x+c)*ln(d*x+c-1)*e^4-3*b^2/d^2*f^2*dilog(1/2*d*x+
1/2*c+1/2)*c^2*e+3*b^2/d*f*dilog(1/2*d*x+1/2*c+1/2)*c*e^2-3/4*b^2/d*f*ln(d*x+c-1)*ln(1/2*d*x+1/2*c+1/2)*e^2-1/
2*b^2/d^2*f^2*ln(d*x+c-1)*ln(1/2*d*x+1/2*c+1/2)*e-1/4*b^2/d^2*f^2*ln(d*x+c-1)^2*c^3*e+3/8*b^2/d*f*ln(d*x+c-1)^
2*c^2*e^2-b^2/d^2*f^2*arctanh(d*x+c)^2*c^3*e+3/2*b^2/d*f*arctanh(d*x+c)^2*c^2*e^2+3/2*b^2/d*f*arctanh(d*x+c)^2
*e^2*(d*x+c)^2+b^2/d^2*f^2*arctanh(d*x+c)^2*e*(d*x+c)^3+3/2*a*b/d*f*ln(d*x+c-1)*e^2+a*b/d^2*f^2*ln(d*x+c-1)*e-
1/4*a*b*d/f*ln(d*x+c+1)*e^4+1/4*a*b*d/f*ln(d*x+c-1)*e^4-1/4*a*b/d^3*f^3*ln(d*x+c+1)*c^4-a*b/d^3*f^3*ln(d*x+c+1
)*c^3-3/2*a*b/d^3*f^3*ln(d*x+c+1)*c^2-a*b/d^3*f^3*ln(d*x+c+1)*c-3*b^2/d^2*f^2*ln(d*x+c-1)*c*e-3*b^2/d^2*f^2*ln
(d*x+c+1)*c*e-b^2/d^3*f^3*arctanh(d*x+c)^2*c^3*(d*x+c)+3/2*b^2/d^3*f^3*arctanh(d*x+c)^2*c^2*(d*x+c)^2-b^2/d^3*
f^3*arctanh(d*x+c)^2*c*(d*x+c)^3-3/2*b^2/d^3*f^3*arctanh(d*x+c)*ln(d*x+c+1)*c^2-b^2/d^3*f^3*arctanh(d*x+c)*ln(
d*x+c+1)*c+1/4*b^2/d^3*f^3*arctanh(d*x+c)*ln(d*x+c-1)*c^4-b^2/d^3*f^3*arctanh(d*x+c)*ln(d*x+c-1)*c^3+3/2*b^2/d
^3*f^3*arctanh(d*x+c)*ln(d*x+c-1)*c^2-b^2/d^3*f^3*arctanh(d*x+c)*ln(d*x+c-1)*c-b^2/d^3*f^3*arctanh(d*x+c)*c*(d
*x+c)^2+3*b^2/d^3*f^3*arctanh(d*x+c)*c^2*(d*x+c)-1/8*b^2/d^3*f^3*ln(d*x+c+1)*ln(-1/2*d*x-1/2*c+1/2)*c^4-1/2*b^
2/d^3*f^3*ln(d*x+c+1)*ln(-1/2*d*x-1/2*c+1/2)*c^3-3/4*b^2/d^3*f^3*ln(d*x+c+1)*ln(-1/2*d*x-1/2*c+1/2)*c^2-1/2*b^
2/d^3*f^3*ln(d*x+c+1)*ln(-1/2*d*x-1/2*c+1/2)*c+1/8*b^2/d^3*f^3*ln(-1/2*d*x-1/2*c+1/2)*ln(1/2*d*x+1/2*c+1/2)*c^
4+1/2*b^2/d^3*f^3*ln(-1/2*d*x-1/2*c+1/2)*ln(1/2*d*x+1/2*c+1/2)*c^3+3/4*b^2/d^3*f^3*ln(-1/2*d*x-1/2*c+1/2)*ln(1
/2*d*x+1/2*c+1/2)*c^2+1/2*b^2/d^3*f^3*ln(-1/2*d*x-1/2*c+1/2)*ln(1/2*d*x+1/2*c+1/2)*c-1/8*b^2/d^3*f^3*ln(d*x+c-
1)*ln(1/2*d*x+1/2*c+1/2)*c^4+1/2*b^2/d^3*f^3*ln...

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1438 vs. \(2 (555) = 1110\).
time = 0.46, size = 1438, normalized size = 2.56 \begin {gather*} \frac {1}{4} \, a^{2} f^{3} x^{4} + a^{2} f^{2} x^{3} e + \frac {1}{12} \, {\left (6 \, x^{4} \operatorname {artanh}\left (d x + c\right ) + d {\left (\frac {2 \, {\left (d^{2} x^{3} - 3 \, c d x^{2} + 3 \, {\left (3 \, c^{2} + 1\right )} x\right )}}{d^{4}} - \frac {3 \, {\left (c^{4} + 4 \, c^{3} + 6 \, c^{2} + 4 \, c + 1\right )} \log \left (d x + c + 1\right )}{d^{5}} + \frac {3 \, {\left (c^{4} - 4 \, c^{3} + 6 \, c^{2} - 4 \, c + 1\right )} \log \left (d x + c - 1\right )}{d^{5}}\right )}\right )} a b f^{3} + \frac {3}{2} \, a^{2} f x^{2} e^{2} + {\left (2 \, x^{3} \operatorname {artanh}\left (d x + c\right ) + d {\left (\frac {d x^{2} - 4 \, c x}{d^{3}} + \frac {{\left (c^{3} + 3 \, c^{2} + 3 \, c + 1\right )} \log \left (d x + c + 1\right )}{d^{4}} - \frac {{\left (c^{3} - 3 \, c^{2} + 3 \, c - 1\right )} \log \left (d x + c - 1\right )}{d^{4}}\right )}\right )} a b f^{2} e + \frac {3}{2} \, {\left (2 \, x^{2} \operatorname {artanh}\left (d x + c\right ) + d {\left (\frac {2 \, x}{d^{2}} - \frac {{\left (c^{2} + 2 \, c + 1\right )} \log \left (d x + c + 1\right )}{d^{3}} + \frac {{\left (c^{2} - 2 \, c + 1\right )} \log \left (d x + c - 1\right )}{d^{3}}\right )}\right )} a b f e^{2} + a^{2} x e^{3} + \frac {{\left (2 \, {\left (d x + c\right )} \operatorname {artanh}\left (d x + c\right ) + \log \left (-{\left (d x + c\right )}^{2} + 1\right )\right )} a b e^{3}}{d} - \frac {{\left (3 \, b^{2} c d^{2} f e^{2} - b^{2} d^{3} e^{3} - {\left (3 \, c^{2} d f^{2} + d f^{2}\right )} b^{2} e + {\left (c^{3} f^{3} + c f^{3}\right )} b^{2}\right )} {\left (\log \left (d x + c + 1\right ) \log \left (-\frac {1}{2} \, d x - \frac {1}{2} \, c + \frac {1}{2}\right ) + {\rm Li}_2\left (\frac {1}{2} \, d x + \frac {1}{2} \, c + \frac {1}{2}\right )\right )}}{d^{4}} + \frac {{\left (18 \, {\left (c d^{2} f + d^{2} f\right )} b^{2} e^{2} - 6 \, {\left (5 \, c^{2} d f^{2} + 6 \, c d f^{2} + d f^{2}\right )} b^{2} e + {\left (13 \, c^{3} f^{3} + 18 \, c^{2} f^{3} + 9 \, c f^{3} + 4 \, f^{3}\right )} b^{2}\right )} \log \left (d x + c + 1\right )}{12 \, d^{4}} - \frac {{\left (18 \, {\left (c d^{2} f - d^{2} f\right )} b^{2} e^{2} - 6 \, {\left (5 \, c^{2} d f^{2} - 6 \, c d f^{2} + d f^{2}\right )} b^{2} e + {\left (13 \, c^{3} f^{3} - 18 \, c^{2} f^{3} + 9 \, c f^{3} - 4 \, f^{3}\right )} b^{2}\right )} \log \left (d x + c - 1\right )}{12 \, d^{4}} + \frac {4 \, b^{2} d^{2} f^{3} x^{2} + 3 \, {\left (b^{2} d^{4} f^{3} x^{4} + 4 \, b^{2} d^{4} f^{2} x^{3} e + 6 \, b^{2} d^{4} f x^{2} e^{2} + 4 \, b^{2} d^{4} x e^{3} + 4 \, {\left (c d^{3} + d^{3}\right )} b^{2} e^{3} - 6 \, {\left (c^{2} d^{2} f + 2 \, c d^{2} f + d^{2} f\right )} b^{2} e^{2} + 4 \, {\left (c^{3} d f^{2} + 3 \, c^{2} d f^{2} + 3 \, c d f^{2} + d f^{2}\right )} b^{2} e - {\left (c^{4} f^{3} + 4 \, c^{3} f^{3} + 6 \, c^{2} f^{3} + 4 \, c f^{3} + f^{3}\right )} b^{2}\right )} \log \left (d x + c + 1\right )^{2} + 3 \, {\left (b^{2} d^{4} f^{3} x^{4} + 4 \, b^{2} d^{4} f^{2} x^{3} e + 6 \, b^{2} d^{4} f x^{2} e^{2} + 4 \, b^{2} d^{4} x e^{3} + 4 \, {\left (c d^{3} - d^{3}\right )} b^{2} e^{3} - 6 \, {\left (c^{2} d^{2} f - 2 \, c d^{2} f + d^{2} f\right )} b^{2} e^{2} + 4 \, {\left (c^{3} d f^{2} - 3 \, c^{2} d f^{2} + 3 \, c d f^{2} - d f^{2}\right )} b^{2} e - {\left (c^{4} f^{3} - 4 \, c^{3} f^{3} + 6 \, c^{2} f^{3} - 4 \, c f^{3} + f^{3}\right )} b^{2}\right )} \log \left (-d x - c + 1\right )^{2} - 8 \, {\left (5 \, b^{2} c d f^{3} - 6 \, b^{2} d^{2} f^{2} e\right )} x + 4 \, {\left (b^{2} d^{3} f^{3} x^{3} - 3 \, {\left (b^{2} c d^{2} f^{3} - 2 \, b^{2} d^{3} f^{2} e\right )} x^{2} - 3 \, {\left (8 \, b^{2} c d^{2} f^{2} e - 6 \, b^{2} d^{3} f e^{2} - {\left (3 \, c^{2} d f^{3} + d f^{3}\right )} b^{2}\right )} x\right )} \log \left (d x + c + 1\right ) - 2 \, {\left (2 \, b^{2} d^{3} f^{3} x^{3} - 6 \, {\left (b^{2} c d^{2} f^{3} - 2 \, b^{2} d^{3} f^{2} e\right )} x^{2} - 6 \, {\left (8 \, b^{2} c d^{2} f^{2} e - 6 \, b^{2} d^{3} f e^{2} - {\left (3 \, c^{2} d f^{3} + d f^{3}\right )} b^{2}\right )} x + 3 \, {\left (b^{2} d^{4} f^{3} x^{4} + 4 \, b^{2} d^{4} f^{2} x^{3} e + 6 \, b^{2} d^{4} f x^{2} e^{2} + 4 \, b^{2} d^{4} x e^{3} + 4 \, {\left (c d^{3} + d^{3}\right )} b^{2} e^{3} - 6 \, {\left (c^{2} d^{2} f + 2 \, c d^{2} f + d^{2} f\right )} b^{2} e^{2} + 4 \, {\left (c^{3} d f^{2} + 3 \, c^{2} d f^{2} + 3 \, c d f^{2} + d f^{2}\right )} b^{2} e - {\left (c^{4} f^{3} + 4 \, c^{3} f^{3} + 6 \, c^{2} f^{3} + 4 \, c f^{3} + f^{3}\right )} b^{2}\right )} \log \left (d x + c + 1\right )\right )} \log \left (-d x - c + 1\right )}{48 \, d^{4}} \end {gather*}

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

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

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

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

[Out]

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

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

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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