3.2.3 \(\int \frac {x^4 (a+b \tanh ^{-1}(c x))^2}{(d+c d x)^2} \, dx\) [103]

Optimal. Leaf size=394 \[ -\frac {2 a b x}{c^4 d^2}+\frac {b^2 x}{3 c^4 d^2}-\frac {b^2}{2 c^5 d^2 (1+c x)}+\frac {b^2 \tanh ^{-1}(c x)}{6 c^5 d^2}-\frac {2 b^2 x \tanh ^{-1}(c x)}{c^4 d^2}+\frac {b x^2 \left (a+b \tanh ^{-1}(c x)\right )}{3 c^3 d^2}-\frac {b \left (a+b \tanh ^{-1}(c x)\right )}{c^5 d^2 (1+c x)}+\frac {29 \left (a+b \tanh ^{-1}(c x)\right )^2}{6 c^5 d^2}+\frac {3 x \left (a+b \tanh ^{-1}(c x)\right )^2}{c^4 d^2}-\frac {x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{c^3 d^2}+\frac {x^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{3 c^2 d^2}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{c^5 d^2 (1+c x)}-\frac {20 b \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{3 c^5 d^2}+\frac {4 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+c x}\right )}{c^5 d^2}-\frac {b^2 \log \left (1-c^2 x^2\right )}{c^5 d^2}-\frac {10 b^2 \text {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{3 c^5 d^2}-\frac {4 b \left (a+b \tanh ^{-1}(c x)\right ) \text {PolyLog}\left (2,1-\frac {2}{1+c x}\right )}{c^5 d^2}-\frac {2 b^2 \text {PolyLog}\left (3,1-\frac {2}{1+c x}\right )}{c^5 d^2} \]

[Out]

-2*a*b*x/c^4/d^2+1/3*b^2*x/c^4/d^2-1/2*b^2/c^5/d^2/(c*x+1)+1/6*b^2*arctanh(c*x)/c^5/d^2-2*b^2*x*arctanh(c*x)/c
^4/d^2+1/3*b*x^2*(a+b*arctanh(c*x))/c^3/d^2-b*(a+b*arctanh(c*x))/c^5/d^2/(c*x+1)+29/6*(a+b*arctanh(c*x))^2/c^5
/d^2+3*x*(a+b*arctanh(c*x))^2/c^4/d^2-x^2*(a+b*arctanh(c*x))^2/c^3/d^2+1/3*x^3*(a+b*arctanh(c*x))^2/c^2/d^2-(a
+b*arctanh(c*x))^2/c^5/d^2/(c*x+1)-20/3*b*(a+b*arctanh(c*x))*ln(2/(-c*x+1))/c^5/d^2+4*(a+b*arctanh(c*x))^2*ln(
2/(c*x+1))/c^5/d^2-b^2*ln(-c^2*x^2+1)/c^5/d^2-10/3*b^2*polylog(2,1-2/(-c*x+1))/c^5/d^2-4*b*(a+b*arctanh(c*x))*
polylog(2,1-2/(c*x+1))/c^5/d^2-2*b^2*polylog(3,1-2/(c*x+1))/c^5/d^2

________________________________________________________________________________________

Rubi [A]
time = 0.62, antiderivative size = 394, normalized size of antiderivative = 1.00, number of steps used = 33, number of rules used = 19, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.864, Rules used = {6087, 6021, 6131, 6055, 2449, 2352, 6037, 6127, 266, 6095, 327, 212, 6065, 6063, 641, 46, 213, 6203, 6745} \begin {gather*} -\frac {4 b \text {Li}_2\left (1-\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c^5 d^2}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{c^5 d^2 (c x+1)}+\frac {29 \left (a+b \tanh ^{-1}(c x)\right )^2}{6 c^5 d^2}-\frac {b \left (a+b \tanh ^{-1}(c x)\right )}{c^5 d^2 (c x+1)}-\frac {20 b \log \left (\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{3 c^5 d^2}+\frac {4 \log \left (\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{c^5 d^2}-\frac {2 a b x}{c^4 d^2}+\frac {3 x \left (a+b \tanh ^{-1}(c x)\right )^2}{c^4 d^2}-\frac {x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{c^3 d^2}+\frac {b x^2 \left (a+b \tanh ^{-1}(c x)\right )}{3 c^3 d^2}+\frac {x^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{3 c^2 d^2}-\frac {10 b^2 \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{3 c^5 d^2}-\frac {2 b^2 \text {Li}_3\left (1-\frac {2}{c x+1}\right )}{c^5 d^2}-\frac {b^2}{2 c^5 d^2 (c x+1)}+\frac {b^2 \tanh ^{-1}(c x)}{6 c^5 d^2}+\frac {b^2 x}{3 c^4 d^2}-\frac {2 b^2 x \tanh ^{-1}(c x)}{c^4 d^2}-\frac {b^2 \log \left (1-c^2 x^2\right )}{c^5 d^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^4*(a + b*ArcTanh[c*x])^2)/(d + c*d*x)^2,x]

[Out]

(-2*a*b*x)/(c^4*d^2) + (b^2*x)/(3*c^4*d^2) - b^2/(2*c^5*d^2*(1 + c*x)) + (b^2*ArcTanh[c*x])/(6*c^5*d^2) - (2*b
^2*x*ArcTanh[c*x])/(c^4*d^2) + (b*x^2*(a + b*ArcTanh[c*x]))/(3*c^3*d^2) - (b*(a + b*ArcTanh[c*x]))/(c^5*d^2*(1
+ c*x)) + (29*(a + b*ArcTanh[c*x])^2)/(6*c^5*d^2) + (3*x*(a + b*ArcTanh[c*x])^2)/(c^4*d^2) - (x^2*(a + b*ArcT
anh[c*x])^2)/(c^3*d^2) + (x^3*(a + b*ArcTanh[c*x])^2)/(3*c^2*d^2) - (a + b*ArcTanh[c*x])^2/(c^5*d^2*(1 + c*x))
- (20*b*(a + b*ArcTanh[c*x])*Log[2/(1 - c*x)])/(3*c^5*d^2) + (4*(a + b*ArcTanh[c*x])^2*Log[2/(1 + c*x)])/(c^5
*d^2) - (b^2*Log[1 - c^2*x^2])/(c^5*d^2) - (10*b^2*PolyLog[2, 1 - 2/(1 - c*x)])/(3*c^5*d^2) - (4*b*(a + b*ArcT
anh[c*x])*PolyLog[2, 1 - 2/(1 + c*x)])/(c^5*d^2) - (2*b^2*PolyLog[3, 1 - 2/(1 + c*x)])/(c^5*d^2)

Rule 46

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && Lt
Q[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 213

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(-1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])]
, x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[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 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 641

Int[((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a/d + (c/e)*x)^
p, x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[d, 0] && I
ntegerQ[m + p]))

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 6063

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

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 6087

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

Rule 6095

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

Rule 6127

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

Rule 6131

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

Rule 6203

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

Rule 6745

Int[(u_)*PolyLog[n_, v_], x_Symbol] :> With[{w = DerivativeDivides[v, u*v, x]}, Simp[w*PolyLog[n + 1, v], x] /
;  !FalseQ[w]] /; FreeQ[n, x]

Rubi steps

\begin {align*} \int \frac {x^4 \left (a+b \tanh ^{-1}(c x)\right )^2}{(d+c d x)^2} \, dx &=\int \left (\frac {3 \left (a+b \tanh ^{-1}(c x)\right )^2}{c^4 d^2}-\frac {2 x \left (a+b \tanh ^{-1}(c x)\right )^2}{c^3 d^2}+\frac {x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{c^2 d^2}+\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{c^4 d^2 (1+c x)^2}-\frac {4 \left (a+b \tanh ^{-1}(c x)\right )^2}{c^4 d^2 (1+c x)}\right ) \, dx\\ &=\frac {\int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{(1+c x)^2} \, dx}{c^4 d^2}+\frac {3 \int \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx}{c^4 d^2}-\frac {4 \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{1+c x} \, dx}{c^4 d^2}-\frac {2 \int x \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx}{c^3 d^2}+\frac {\int x^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx}{c^2 d^2}\\ &=\frac {3 x \left (a+b \tanh ^{-1}(c x)\right )^2}{c^4 d^2}-\frac {x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{c^3 d^2}+\frac {x^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{3 c^2 d^2}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{c^5 d^2 (1+c x)}+\frac {4 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+c x}\right )}{c^5 d^2}+\frac {(2 b) \int \left (\frac {a+b \tanh ^{-1}(c x)}{2 (1+c x)^2}-\frac {a+b \tanh ^{-1}(c x)}{2 \left (-1+c^2 x^2\right )}\right ) \, dx}{c^4 d^2}-\frac {(8 b) \int \frac {\left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1+c x}\right )}{1-c^2 x^2} \, dx}{c^4 d^2}-\frac {(6 b) \int \frac {x \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{c^3 d^2}+\frac {(2 b) \int \frac {x^2 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{c^2 d^2}-\frac {(2 b) \int \frac {x^3 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{3 c d^2}\\ &=\frac {3 \left (a+b \tanh ^{-1}(c x)\right )^2}{c^5 d^2}+\frac {3 x \left (a+b \tanh ^{-1}(c x)\right )^2}{c^4 d^2}-\frac {x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{c^3 d^2}+\frac {x^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{3 c^2 d^2}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{c^5 d^2 (1+c x)}+\frac {4 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+c x}\right )}{c^5 d^2}-\frac {4 b \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{c^5 d^2}+\frac {b \int \frac {a+b \tanh ^{-1}(c x)}{(1+c x)^2} \, dx}{c^4 d^2}-\frac {b \int \frac {a+b \tanh ^{-1}(c x)}{-1+c^2 x^2} \, dx}{c^4 d^2}-\frac {(2 b) \int \left (a+b \tanh ^{-1}(c x)\right ) \, dx}{c^4 d^2}+\frac {(2 b) \int \frac {a+b \tanh ^{-1}(c x)}{1-c^2 x^2} \, dx}{c^4 d^2}-\frac {(6 b) \int \frac {a+b \tanh ^{-1}(c x)}{1-c x} \, dx}{c^4 d^2}+\frac {\left (4 b^2\right ) \int \frac {\text {Li}_2\left (1-\frac {2}{1+c x}\right )}{1-c^2 x^2} \, dx}{c^4 d^2}+\frac {(2 b) \int x \left (a+b \tanh ^{-1}(c x)\right ) \, dx}{3 c^3 d^2}-\frac {(2 b) \int \frac {x \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{3 c^3 d^2}\\ &=-\frac {2 a b x}{c^4 d^2}+\frac {b x^2 \left (a+b \tanh ^{-1}(c x)\right )}{3 c^3 d^2}-\frac {b \left (a+b \tanh ^{-1}(c x)\right )}{c^5 d^2 (1+c x)}+\frac {29 \left (a+b \tanh ^{-1}(c x)\right )^2}{6 c^5 d^2}+\frac {3 x \left (a+b \tanh ^{-1}(c x)\right )^2}{c^4 d^2}-\frac {x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{c^3 d^2}+\frac {x^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{3 c^2 d^2}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{c^5 d^2 (1+c x)}-\frac {6 b \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{c^5 d^2}+\frac {4 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+c x}\right )}{c^5 d^2}-\frac {4 b \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{c^5 d^2}-\frac {2 b^2 \text {Li}_3\left (1-\frac {2}{1+c x}\right )}{c^5 d^2}-\frac {(2 b) \int \frac {a+b \tanh ^{-1}(c x)}{1-c x} \, dx}{3 c^4 d^2}+\frac {b^2 \int \frac {1}{(1+c x) \left (1-c^2 x^2\right )} \, dx}{c^4 d^2}-\frac {\left (2 b^2\right ) \int \tanh ^{-1}(c x) \, dx}{c^4 d^2}+\frac {\left (6 b^2\right ) \int \frac {\log \left (\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx}{c^4 d^2}-\frac {b^2 \int \frac {x^2}{1-c^2 x^2} \, dx}{3 c^2 d^2}\\ &=-\frac {2 a b x}{c^4 d^2}+\frac {b^2 x}{3 c^4 d^2}-\frac {2 b^2 x \tanh ^{-1}(c x)}{c^4 d^2}+\frac {b x^2 \left (a+b \tanh ^{-1}(c x)\right )}{3 c^3 d^2}-\frac {b \left (a+b \tanh ^{-1}(c x)\right )}{c^5 d^2 (1+c x)}+\frac {29 \left (a+b \tanh ^{-1}(c x)\right )^2}{6 c^5 d^2}+\frac {3 x \left (a+b \tanh ^{-1}(c x)\right )^2}{c^4 d^2}-\frac {x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{c^3 d^2}+\frac {x^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{3 c^2 d^2}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{c^5 d^2 (1+c x)}-\frac {20 b \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{3 c^5 d^2}+\frac {4 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+c x}\right )}{c^5 d^2}-\frac {4 b \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{c^5 d^2}-\frac {2 b^2 \text {Li}_3\left (1-\frac {2}{1+c x}\right )}{c^5 d^2}-\frac {\left (6 b^2\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-c x}\right )}{c^5 d^2}-\frac {b^2 \int \frac {1}{1-c^2 x^2} \, dx}{3 c^4 d^2}+\frac {\left (2 b^2\right ) \int \frac {\log \left (\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx}{3 c^4 d^2}+\frac {b^2 \int \frac {1}{(1-c x) (1+c x)^2} \, dx}{c^4 d^2}+\frac {\left (2 b^2\right ) \int \frac {x}{1-c^2 x^2} \, dx}{c^3 d^2}\\ &=-\frac {2 a b x}{c^4 d^2}+\frac {b^2 x}{3 c^4 d^2}-\frac {b^2 \tanh ^{-1}(c x)}{3 c^5 d^2}-\frac {2 b^2 x \tanh ^{-1}(c x)}{c^4 d^2}+\frac {b x^2 \left (a+b \tanh ^{-1}(c x)\right )}{3 c^3 d^2}-\frac {b \left (a+b \tanh ^{-1}(c x)\right )}{c^5 d^2 (1+c x)}+\frac {29 \left (a+b \tanh ^{-1}(c x)\right )^2}{6 c^5 d^2}+\frac {3 x \left (a+b \tanh ^{-1}(c x)\right )^2}{c^4 d^2}-\frac {x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{c^3 d^2}+\frac {x^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{3 c^2 d^2}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{c^5 d^2 (1+c x)}-\frac {20 b \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{3 c^5 d^2}+\frac {4 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+c x}\right )}{c^5 d^2}-\frac {b^2 \log \left (1-c^2 x^2\right )}{c^5 d^2}-\frac {3 b^2 \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{c^5 d^2}-\frac {4 b \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{c^5 d^2}-\frac {2 b^2 \text {Li}_3\left (1-\frac {2}{1+c x}\right )}{c^5 d^2}-\frac {\left (2 b^2\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-c x}\right )}{3 c^5 d^2}+\frac {b^2 \int \left (\frac {1}{2 (1+c x)^2}-\frac {1}{2 \left (-1+c^2 x^2\right )}\right ) \, dx}{c^4 d^2}\\ &=-\frac {2 a b x}{c^4 d^2}+\frac {b^2 x}{3 c^4 d^2}-\frac {b^2}{2 c^5 d^2 (1+c x)}-\frac {b^2 \tanh ^{-1}(c x)}{3 c^5 d^2}-\frac {2 b^2 x \tanh ^{-1}(c x)}{c^4 d^2}+\frac {b x^2 \left (a+b \tanh ^{-1}(c x)\right )}{3 c^3 d^2}-\frac {b \left (a+b \tanh ^{-1}(c x)\right )}{c^5 d^2 (1+c x)}+\frac {29 \left (a+b \tanh ^{-1}(c x)\right )^2}{6 c^5 d^2}+\frac {3 x \left (a+b \tanh ^{-1}(c x)\right )^2}{c^4 d^2}-\frac {x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{c^3 d^2}+\frac {x^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{3 c^2 d^2}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{c^5 d^2 (1+c x)}-\frac {20 b \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{3 c^5 d^2}+\frac {4 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+c x}\right )}{c^5 d^2}-\frac {b^2 \log \left (1-c^2 x^2\right )}{c^5 d^2}-\frac {10 b^2 \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{3 c^5 d^2}-\frac {4 b \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{c^5 d^2}-\frac {2 b^2 \text {Li}_3\left (1-\frac {2}{1+c x}\right )}{c^5 d^2}-\frac {b^2 \int \frac {1}{-1+c^2 x^2} \, dx}{2 c^4 d^2}\\ &=-\frac {2 a b x}{c^4 d^2}+\frac {b^2 x}{3 c^4 d^2}-\frac {b^2}{2 c^5 d^2 (1+c x)}+\frac {b^2 \tanh ^{-1}(c x)}{6 c^5 d^2}-\frac {2 b^2 x \tanh ^{-1}(c x)}{c^4 d^2}+\frac {b x^2 \left (a+b \tanh ^{-1}(c x)\right )}{3 c^3 d^2}-\frac {b \left (a+b \tanh ^{-1}(c x)\right )}{c^5 d^2 (1+c x)}+\frac {29 \left (a+b \tanh ^{-1}(c x)\right )^2}{6 c^5 d^2}+\frac {3 x \left (a+b \tanh ^{-1}(c x)\right )^2}{c^4 d^2}-\frac {x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{c^3 d^2}+\frac {x^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{3 c^2 d^2}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{c^5 d^2 (1+c x)}-\frac {20 b \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{3 c^5 d^2}+\frac {4 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+c x}\right )}{c^5 d^2}-\frac {b^2 \log \left (1-c^2 x^2\right )}{c^5 d^2}-\frac {10 b^2 \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{3 c^5 d^2}-\frac {4 b \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{c^5 d^2}-\frac {2 b^2 \text {Li}_3\left (1-\frac {2}{1+c x}\right )}{c^5 d^2}\\ \end {align*}

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Mathematica [A]
time = 1.15, size = 425, normalized size = 1.08 \begin {gather*} \frac {36 a^2 c x-12 a^2 c^2 x^2+4 a^2 c^3 x^3-\frac {12 a^2}{1+c x}-48 a^2 \log (1+c x)+b^2 \left (4 c x-4 \tanh ^{-1}(c x)-24 c x \tanh ^{-1}(c x)+4 c^2 x^2 \tanh ^{-1}(c x)-28 \tanh ^{-1}(c x)^2+36 c x \tanh ^{-1}(c x)^2-12 c^2 x^2 \tanh ^{-1}(c x)^2+4 c^3 x^3 \tanh ^{-1}(c x)^2-3 \cosh \left (2 \tanh ^{-1}(c x)\right )-6 \tanh ^{-1}(c x) \cosh \left (2 \tanh ^{-1}(c x)\right )-6 \tanh ^{-1}(c x)^2 \cosh \left (2 \tanh ^{-1}(c x)\right )-80 \tanh ^{-1}(c x) \log \left (1+e^{-2 \tanh ^{-1}(c x)}\right )+48 \tanh ^{-1}(c x)^2 \log \left (1+e^{-2 \tanh ^{-1}(c x)}\right )-12 \log \left (1-c^2 x^2\right )-8 \left (-5+6 \tanh ^{-1}(c x)\right ) \text {PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c x)}\right )-24 \text {PolyLog}\left (3,-e^{-2 \tanh ^{-1}(c x)}\right )+3 \sinh \left (2 \tanh ^{-1}(c x)\right )+6 \tanh ^{-1}(c x) \sinh \left (2 \tanh ^{-1}(c x)\right )+6 \tanh ^{-1}(c x)^2 \sinh \left (2 \tanh ^{-1}(c x)\right )\right )+2 a b \left (-2-12 c x+2 c^2 x^2-3 \cosh \left (2 \tanh ^{-1}(c x)\right )+20 \log \left (1-c^2 x^2\right )-24 \text {PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c x)}\right )+3 \sinh \left (2 \tanh ^{-1}(c x)\right )+2 \tanh ^{-1}(c x) \left (6+18 c x-6 c^2 x^2+2 c^3 x^3-3 \cosh \left (2 \tanh ^{-1}(c x)\right )+24 \log \left (1+e^{-2 \tanh ^{-1}(c x)}\right )+3 \sinh \left (2 \tanh ^{-1}(c x)\right )\right )\right )}{12 c^5 d^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^4*(a + b*ArcTanh[c*x])^2)/(d + c*d*x)^2,x]

[Out]

(36*a^2*c*x - 12*a^2*c^2*x^2 + 4*a^2*c^3*x^3 - (12*a^2)/(1 + c*x) - 48*a^2*Log[1 + c*x] + b^2*(4*c*x - 4*ArcTa
nh[c*x] - 24*c*x*ArcTanh[c*x] + 4*c^2*x^2*ArcTanh[c*x] - 28*ArcTanh[c*x]^2 + 36*c*x*ArcTanh[c*x]^2 - 12*c^2*x^
2*ArcTanh[c*x]^2 + 4*c^3*x^3*ArcTanh[c*x]^2 - 3*Cosh[2*ArcTanh[c*x]] - 6*ArcTanh[c*x]*Cosh[2*ArcTanh[c*x]] - 6
*ArcTanh[c*x]^2*Cosh[2*ArcTanh[c*x]] - 80*ArcTanh[c*x]*Log[1 + E^(-2*ArcTanh[c*x])] + 48*ArcTanh[c*x]^2*Log[1
+ E^(-2*ArcTanh[c*x])] - 12*Log[1 - c^2*x^2] - 8*(-5 + 6*ArcTanh[c*x])*PolyLog[2, -E^(-2*ArcTanh[c*x])] - 24*P
olyLog[3, -E^(-2*ArcTanh[c*x])] + 3*Sinh[2*ArcTanh[c*x]] + 6*ArcTanh[c*x]*Sinh[2*ArcTanh[c*x]] + 6*ArcTanh[c*x
]^2*Sinh[2*ArcTanh[c*x]]) + 2*a*b*(-2 - 12*c*x + 2*c^2*x^2 - 3*Cosh[2*ArcTanh[c*x]] + 20*Log[1 - c^2*x^2] - 24
*PolyLog[2, -E^(-2*ArcTanh[c*x])] + 3*Sinh[2*ArcTanh[c*x]] + 2*ArcTanh[c*x]*(6 + 18*c*x - 6*c^2*x^2 + 2*c^3*x^
3 - 3*Cosh[2*ArcTanh[c*x]] + 24*Log[1 + E^(-2*ArcTanh[c*x])] + 3*Sinh[2*ArcTanh[c*x]])))/(12*c^5*d^2)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 12.34, size = 1347, normalized size = 3.42

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1347\)
default	\(\text {Expression too large to display}\)	\(1347\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(a+b*arctanh(c*x))^2/(c*d*x+d)^2,x,method=_RETURNVERBOSE)

[Out]

1/c^5*(2*I*b^2/d^2*arctanh(c*x)^2*Pi*csgn(I*(c*x+1)^2/(c^2*x^2-1))^3-a^2/d^2/(c*x+1)-4*a^2/d^2*ln(c*x+1)+29/6*
b^2/d^2*arctanh(c*x)^2-8/3*b^2/d^2*arctanh(c*x)^3+2*b^2/d^2*ln(1+(c*x+1)^2/(-c^2*x^2+1))-2*b^2/d^2*polylog(3,-
(c*x+1)^2/(-c^2*x^2+1))-20/3*b^2/d^2*dilog(1-I*(c*x+1)/(-c^2*x^2+1)^(1/2))-20/3*b^2/d^2*dilog(1+I*(c*x+1)/(-c^
2*x^2+1)^(1/2))-1/3*b^2/d^2-7/3*a*b/d^2+1/3*a^2/d^2*c^3*x^3-a^2/d^2*c^2*x^2+3*a^2/d^2*c*x+1/3*b^2/d^2*c*x-2*I*
b^2/d^2*arctanh(c*x)^2*Pi*csgn(I/(1+(c*x+1)^2/(-c^2*x^2+1)))*csgn(I*(c*x+1)^2/(c^2*x^2-1))*csgn(I*(c*x+1)^2/(c
^2*x^2-1)/(1+(c*x+1)^2/(-c^2*x^2+1)))-1/4*b^2/d^2/(c*x+1)-7/3*b^2*arctanh(c*x)/d^2+2*I*b^2/d^2*arctanh(c*x)^2*
Pi*csgn(I*(c*x+1)^2/(c^2*x^2-1)/(1+(c*x+1)^2/(-c^2*x^2+1)))^3+1/2*b^2/d^2*arctanh(c*x)/(c*x+1)*c*x+2/3*a*b/d^2
*arctanh(c*x)*c^3*x^3-2*a*b/d^2*arctanh(c*x)*c^2*x^2+6*a*b/d^2*arctanh(c*x)*c*x-b^2/d^2*arctanh(c*x)^2/(c*x+1)
-4*b^2/d^2*arctanh(c*x)^2*ln(c*x+1)-1/2*b^2/d^2*arctanh(c*x)/(c*x+1)-20/3*b^2/d^2*arctanh(c*x)*ln(1-I*(c*x+1)/
(-c^2*x^2+1)^(1/2))-20/3*b^2/d^2*arctanh(c*x)*ln(1+I*(c*x+1)/(-c^2*x^2+1)^(1/2))+4*b^2/d^2*arctanh(c*x)*polylo
g(2,-(c*x+1)^2/(-c^2*x^2+1))-a*b/d^2/(c*x+1)+4*a*b/d^2*dilog(1/2*c*x+1/2)+2*a*b/d^2*ln(c*x+1)^2+11/6*a*b/d^2*l
n(c*x-1)+29/6*a*b/d^2*ln(c*x+1)+4*b^2/d^2*arctanh(c*x)^2*ln(2)+8*b^2/d^2*arctanh(c*x)^2*ln((c*x+1)/(-c^2*x^2+1
)^(1/2))+1/3*a*b/d^2*c^2*x^2-2*a*b/d^2*c*x+1/3*b^2/d^2*arctanh(c*x)^2*c^3*x^3-b^2/d^2*arctanh(c*x)^2*c^2*x^2+3
*b^2/d^2*arctanh(c*x)^2*c*x-2*b^2/d^2*arctanh(c*x)*c*x+1/3*b^2/d^2*arctanh(c*x)*c^2*x^2+1/4*b^2/d^2/(c*x+1)*c*
x-2*a*b/d^2*arctanh(c*x)/(c*x+1)-8*a*b/d^2*arctanh(c*x)*ln(c*x+1)+4*a*b/d^2*ln(-1/2*c*x+1/2)*ln(1/2*c*x+1/2)-4
*a*b/d^2*ln(-1/2*c*x+1/2)*ln(c*x+1)+2*I*b^2/d^2*arctanh(c*x)^2*Pi*csgn(I*(c*x+1)/(-c^2*x^2+1)^(1/2))^2*csgn(I*
(c*x+1)^2/(c^2*x^2-1))+2*I*b^2/d^2*arctanh(c*x)^2*Pi*csgn(I/(1+(c*x+1)^2/(-c^2*x^2+1)))*csgn(I*(c*x+1)^2/(c^2*
x^2-1)/(1+(c*x+1)^2/(-c^2*x^2+1)))^2+4*I*b^2/d^2*arctanh(c*x)^2*Pi*csgn(I*(c*x+1)/(-c^2*x^2+1)^(1/2))*csgn(I*(
c*x+1)^2/(c^2*x^2-1))^2-2*I*b^2/d^2*arctanh(c*x)^2*Pi*csgn(I*(c*x+1)^2/(c^2*x^2-1))*csgn(I*(c*x+1)^2/(c^2*x^2-
1)/(1+(c*x+1)^2/(-c^2*x^2+1)))^2)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(a+b*arctanh(c*x))^2/(c*d*x+d)^2,x, algorithm="maxima")

[Out]

-1/3*a^2*(3/(c^6*d^2*x + c^5*d^2) - (c^2*x^3 - 3*c*x^2 + 9*x)/(c^4*d^2) + 12*log(c*x + 1)/(c^5*d^2)) + 1/12*(b
^2*c^4*x^4 - 2*b^2*c^3*x^3 + 6*b^2*c^2*x^2 + 9*b^2*c*x - 3*b^2 - 12*(b^2*c*x + b^2)*log(c*x + 1))*log(-c*x + 1
)^2/(c^6*d^2*x + c^5*d^2) - integrate(-1/12*(3*(b^2*c^5*x^5 - b^2*c^4*x^4)*log(c*x + 1)^2 + 12*(a*b*c^5*x^5 -
a*b*c^4*x^4)*log(c*x + 1) - 2*(4*b^2*c^3*x^3 + 15*b^2*c^2*x^2 + (6*a*b*c^5 + b^2*c^5)*x^5 - (6*a*b*c^4 + b^2*c
^4)*x^4 + 6*b^2*c*x - 3*b^2 + 3*(b^2*c^5*x^5 - b^2*c^4*x^4 - 4*b^2*c^2*x^2 - 8*b^2*c*x - 4*b^2)*log(c*x + 1))*
log(-c*x + 1))/(c^7*d^2*x^3 + c^6*d^2*x^2 - c^5*d^2*x - c^4*d^2), x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(a+b*arctanh(c*x))^2/(c*d*x+d)^2,x, algorithm="fricas")

[Out]

integral((b^2*x^4*arctanh(c*x)^2 + 2*a*b*x^4*arctanh(c*x) + a^2*x^4)/(c^2*d^2*x^2 + 2*c*d^2*x + d^2), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {a^{2} x^{4}}{c^{2} x^{2} + 2 c x + 1}\, dx + \int \frac {b^{2} x^{4} \operatorname {atanh}^{2}{\left (c x \right )}}{c^{2} x^{2} + 2 c x + 1}\, dx + \int \frac {2 a b x^{4} \operatorname {atanh}{\left (c x \right )}}{c^{2} x^{2} + 2 c x + 1}\, dx}{d^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(a+b*atanh(c*x))**2/(c*d*x+d)**2,x)

[Out]

(Integral(a**2*x**4/(c**2*x**2 + 2*c*x + 1), x) + Integral(b**2*x**4*atanh(c*x)**2/(c**2*x**2 + 2*c*x + 1), x)
+ Integral(2*a*b*x**4*atanh(c*x)/(c**2*x**2 + 2*c*x + 1), x))/d**2

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(a+b*arctanh(c*x))^2/(c*d*x+d)^2,x, algorithm="giac")

[Out]

integrate((b*arctanh(c*x) + a)^2*x^4/(c*d*x + d)^2, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^4\,{\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^2}{{\left (d+c\,d\,x\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4*(a + b*atanh(c*x))^2)/(d + c*d*x)^2,x)

[Out]

int((x^4*(a + b*atanh(c*x))^2)/(d + c*d*x)^2, x)

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