3.1.94 \(\int \frac {(d+c d x)^3 (a+b \tanh ^{-1}(c x))^2}{x^7} \, dx\) [94]

Optimal. Leaf size=479 \[ -\frac {b^2 c^2 d^3}{60 x^4}-\frac {b^2 c^3 d^3}{10 x^3}-\frac {61 b^2 c^4 d^3}{180 x^2}-\frac {37 b^2 c^5 d^3}{30 x}+\frac {37}{30} b^2 c^6 d^3 \tanh ^{-1}(c x)-\frac {b c d^3 \left (a+b \tanh ^{-1}(c x)\right )}{15 x^5}-\frac {3 b c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{10 x^4}-\frac {11 b c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{18 x^3}-\frac {14 b c^4 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{15 x^2}-\frac {11 b c^5 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{6 x}-\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{6 x^6}-\frac {3 c d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{5 x^5}-\frac {3 c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{4 x^4}-\frac {c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{3 x^3}+\frac {28}{15} a b c^6 d^3 \log (x)+\frac {113}{45} b^2 c^6 d^3 \log (x)+\frac {37}{20} b c^6 d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )+\frac {1}{60} b c^6 d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1+c x}\right )-\frac {113}{90} b^2 c^6 d^3 \log \left (1-c^2 x^2\right )-\frac {14}{15} b^2 c^6 d^3 \text {PolyLog}(2,-c x)+\frac {14}{15} b^2 c^6 d^3 \text {PolyLog}(2,c x)+\frac {37}{40} b^2 c^6 d^3 \text {PolyLog}\left (2,1-\frac {2}{1-c x}\right )-\frac {1}{120} b^2 c^6 d^3 \text {PolyLog}\left (2,1-\frac {2}{1+c x}\right ) \]

[Out]

-1/60*b^2*c^2*d^3/x^4-1/10*b^2*c^3*d^3/x^3-61/180*b^2*c^4*d^3/x^2-37/30*b^2*c^5*d^3/x+37/30*b^2*c^6*d^3*arctan
h(c*x)-1/15*b*c*d^3*(a+b*arctanh(c*x))/x^5-3/10*b*c^2*d^3*(a+b*arctanh(c*x))/x^4-11/18*b*c^3*d^3*(a+b*arctanh(
c*x))/x^3-14/15*b*c^4*d^3*(a+b*arctanh(c*x))/x^2-11/6*b*c^5*d^3*(a+b*arctanh(c*x))/x-1/6*d^3*(a+b*arctanh(c*x)
)^2/x^6-3/5*c*d^3*(a+b*arctanh(c*x))^2/x^5-3/4*c^2*d^3*(a+b*arctanh(c*x))^2/x^4-1/3*c^3*d^3*(a+b*arctanh(c*x))
^2/x^3+28/15*a*b*c^6*d^3*ln(x)+113/45*b^2*c^6*d^3*ln(x)+37/20*b*c^6*d^3*(a+b*arctanh(c*x))*ln(2/(-c*x+1))+1/60
*b*c^6*d^3*(a+b*arctanh(c*x))*ln(2/(c*x+1))-113/90*b^2*c^6*d^3*ln(-c^2*x^2+1)-14/15*b^2*c^6*d^3*polylog(2,-c*x
)+14/15*b^2*c^6*d^3*polylog(2,c*x)+37/40*b^2*c^6*d^3*polylog(2,1-2/(-c*x+1))-1/120*b^2*c^6*d^3*polylog(2,1-2/(
c*x+1))

________________________________________________________________________________________

Rubi [A]
time = 0.38, antiderivative size = 479, normalized size of antiderivative = 1.00, number of steps used = 29, number of rules used = 14, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {45, 6085, 6037, 272, 46, 331, 212, 36, 29, 31, 6031, 6055, 2449, 2352} \begin {gather*} \frac {28}{15} a b c^6 d^3 \log (x)+\frac {37}{20} b c^6 d^3 \log \left (\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{60} b c^6 d^3 \log \left (\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )-\frac {11 b c^5 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{6 x}-\frac {14 b c^4 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{15 x^2}-\frac {c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{3 x^3}-\frac {11 b c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{18 x^3}-\frac {3 c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{4 x^4}-\frac {3 b c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{10 x^4}-\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{6 x^6}-\frac {3 c d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{5 x^5}-\frac {b c d^3 \left (a+b \tanh ^{-1}(c x)\right )}{15 x^5}-\frac {14}{15} b^2 c^6 d^3 \text {Li}_2(-c x)+\frac {14}{15} b^2 c^6 d^3 \text {Li}_2(c x)+\frac {37}{40} b^2 c^6 d^3 \text {Li}_2\left (1-\frac {2}{1-c x}\right )-\frac {1}{120} b^2 c^6 d^3 \text {Li}_2\left (1-\frac {2}{c x+1}\right )+\frac {113}{45} b^2 c^6 d^3 \log (x)+\frac {37}{30} b^2 c^6 d^3 \tanh ^{-1}(c x)-\frac {37 b^2 c^5 d^3}{30 x}-\frac {61 b^2 c^4 d^3}{180 x^2}-\frac {b^2 c^3 d^3}{10 x^3}-\frac {b^2 c^2 d^3}{60 x^4}-\frac {113}{90} b^2 c^6 d^3 \log \left (1-c^2 x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((d + c*d*x)^3*(a + b*ArcTanh[c*x])^2)/x^7,x]

[Out]

-1/60*(b^2*c^2*d^3)/x^4 - (b^2*c^3*d^3)/(10*x^3) - (61*b^2*c^4*d^3)/(180*x^2) - (37*b^2*c^5*d^3)/(30*x) + (37*
b^2*c^6*d^3*ArcTanh[c*x])/30 - (b*c*d^3*(a + b*ArcTanh[c*x]))/(15*x^5) - (3*b*c^2*d^3*(a + b*ArcTanh[c*x]))/(1
0*x^4) - (11*b*c^3*d^3*(a + b*ArcTanh[c*x]))/(18*x^3) - (14*b*c^4*d^3*(a + b*ArcTanh[c*x]))/(15*x^2) - (11*b*c
^5*d^3*(a + b*ArcTanh[c*x]))/(6*x) - (d^3*(a + b*ArcTanh[c*x])^2)/(6*x^6) - (3*c*d^3*(a + b*ArcTanh[c*x])^2)/(
5*x^5) - (3*c^2*d^3*(a + b*ArcTanh[c*x])^2)/(4*x^4) - (c^3*d^3*(a + b*ArcTanh[c*x])^2)/(3*x^3) + (28*a*b*c^6*d
^3*Log[x])/15 + (113*b^2*c^6*d^3*Log[x])/45 + (37*b*c^6*d^3*(a + b*ArcTanh[c*x])*Log[2/(1 - c*x)])/20 + (b*c^6
*d^3*(a + b*ArcTanh[c*x])*Log[2/(1 + c*x)])/60 - (113*b^2*c^6*d^3*Log[1 - c^2*x^2])/90 - (14*b^2*c^6*d^3*PolyL
og[2, -(c*x)])/15 + (14*b^2*c^6*d^3*PolyLog[2, c*x])/15 + (37*b^2*c^6*d^3*PolyLog[2, 1 - 2/(1 - c*x)])/40 - (b
^2*c^6*d^3*PolyLog[2, 1 - 2/(1 + c*x)])/120

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

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 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 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 331

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c
*(m + 1))), x] - Dist[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && 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 6031

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))/(x_), x_Symbol] :> Simp[a*Log[x], x] + (-Simp[(b/2)*PolyLog[2, (-c)*x]
, x] + Simp[(b/2)*PolyLog[2, c*x], x]) /; FreeQ[{a, b, c}, x]

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 6085

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_)*((f_.)*(x_))^(m_.)*((d_.) + (e_.)*(x_))^(q_), x_Symbol] :> With[{
u = IntHide[(f*x)^m*(d + e*x)^q, x]}, Dist[(a + b*ArcTanh[c*x])^p, u, x] - Dist[b*c*p, Int[ExpandIntegrand[(a
+ b*ArcTanh[c*x])^(p - 1), u/(1 - c^2*x^2), x], x], x]] /; FreeQ[{a, b, c, d, e, f, q}, x] && IGtQ[p, 1] && Eq
Q[c^2*d^2 - e^2, 0] && IntegersQ[m, q] && NeQ[m, -1] && NeQ[q, -1] && ILtQ[m + q + 1, 0] && LtQ[m*q, 0]

Rubi steps

\begin {align*} \int \frac {(d+c d x)^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{x^7} \, dx &=-\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{6 x^6}-\frac {3 c d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{5 x^5}-\frac {3 c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{4 x^4}-\frac {c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{3 x^3}-(2 b c) \int \left (-\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right )}{6 x^6}-\frac {3 c d^3 \left (a+b \tanh ^{-1}(c x)\right )}{5 x^5}-\frac {11 c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{12 x^4}-\frac {14 c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{15 x^3}-\frac {11 c^4 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{12 x^2}-\frac {14 c^5 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{15 x}+\frac {37 c^6 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{40 (-1+c x)}+\frac {c^6 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{120 (1+c x)}\right ) \, dx\\ &=-\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{6 x^6}-\frac {3 c d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{5 x^5}-\frac {3 c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{4 x^4}-\frac {c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{3 x^3}+\frac {1}{3} \left (b c d^3\right ) \int \frac {a+b \tanh ^{-1}(c x)}{x^6} \, dx+\frac {1}{5} \left (6 b c^2 d^3\right ) \int \frac {a+b \tanh ^{-1}(c x)}{x^5} \, dx+\frac {1}{6} \left (11 b c^3 d^3\right ) \int \frac {a+b \tanh ^{-1}(c x)}{x^4} \, dx+\frac {1}{15} \left (28 b c^4 d^3\right ) \int \frac {a+b \tanh ^{-1}(c x)}{x^3} \, dx+\frac {1}{6} \left (11 b c^5 d^3\right ) \int \frac {a+b \tanh ^{-1}(c x)}{x^2} \, dx+\frac {1}{15} \left (28 b c^6 d^3\right ) \int \frac {a+b \tanh ^{-1}(c x)}{x} \, dx-\frac {1}{60} \left (b c^7 d^3\right ) \int \frac {a+b \tanh ^{-1}(c x)}{1+c x} \, dx-\frac {1}{20} \left (37 b c^7 d^3\right ) \int \frac {a+b \tanh ^{-1}(c x)}{-1+c x} \, dx\\ &=-\frac {b c d^3 \left (a+b \tanh ^{-1}(c x)\right )}{15 x^5}-\frac {3 b c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{10 x^4}-\frac {11 b c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{18 x^3}-\frac {14 b c^4 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{15 x^2}-\frac {11 b c^5 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{6 x}-\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{6 x^6}-\frac {3 c d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{5 x^5}-\frac {3 c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{4 x^4}-\frac {c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{3 x^3}+\frac {28}{15} a b c^6 d^3 \log (x)+\frac {37}{20} b c^6 d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )+\frac {1}{60} b c^6 d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1+c x}\right )-\frac {14}{15} b^2 c^6 d^3 \text {Li}_2(-c x)+\frac {14}{15} b^2 c^6 d^3 \text {Li}_2(c x)+\frac {1}{15} \left (b^2 c^2 d^3\right ) \int \frac {1}{x^5 \left (1-c^2 x^2\right )} \, dx+\frac {1}{10} \left (3 b^2 c^3 d^3\right ) \int \frac {1}{x^4 \left (1-c^2 x^2\right )} \, dx+\frac {1}{18} \left (11 b^2 c^4 d^3\right ) \int \frac {1}{x^3 \left (1-c^2 x^2\right )} \, dx+\frac {1}{15} \left (14 b^2 c^5 d^3\right ) \int \frac {1}{x^2 \left (1-c^2 x^2\right )} \, dx+\frac {1}{6} \left (11 b^2 c^6 d^3\right ) \int \frac {1}{x \left (1-c^2 x^2\right )} \, dx-\frac {1}{60} \left (b^2 c^7 d^3\right ) \int \frac {\log \left (\frac {2}{1+c x}\right )}{1-c^2 x^2} \, dx-\frac {1}{20} \left (37 b^2 c^7 d^3\right ) \int \frac {\log \left (\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx\\ &=-\frac {b^2 c^3 d^3}{10 x^3}-\frac {14 b^2 c^5 d^3}{15 x}-\frac {b c d^3 \left (a+b \tanh ^{-1}(c x)\right )}{15 x^5}-\frac {3 b c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{10 x^4}-\frac {11 b c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{18 x^3}-\frac {14 b c^4 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{15 x^2}-\frac {11 b c^5 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{6 x}-\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{6 x^6}-\frac {3 c d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{5 x^5}-\frac {3 c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{4 x^4}-\frac {c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{3 x^3}+\frac {28}{15} a b c^6 d^3 \log (x)+\frac {37}{20} b c^6 d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )+\frac {1}{60} b c^6 d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1+c x}\right )-\frac {14}{15} b^2 c^6 d^3 \text {Li}_2(-c x)+\frac {14}{15} b^2 c^6 d^3 \text {Li}_2(c x)+\frac {1}{30} \left (b^2 c^2 d^3\right ) \text {Subst}\left (\int \frac {1}{x^3 \left (1-c^2 x\right )} \, dx,x,x^2\right )+\frac {1}{36} \left (11 b^2 c^4 d^3\right ) \text {Subst}\left (\int \frac {1}{x^2 \left (1-c^2 x\right )} \, dx,x,x^2\right )+\frac {1}{10} \left (3 b^2 c^5 d^3\right ) \int \frac {1}{x^2 \left (1-c^2 x^2\right )} \, dx-\frac {1}{60} \left (b^2 c^6 d^3\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+c x}\right )+\frac {1}{12} \left (11 b^2 c^6 d^3\right ) \text {Subst}\left (\int \frac {1}{x \left (1-c^2 x\right )} \, dx,x,x^2\right )+\frac {1}{20} \left (37 b^2 c^6 d^3\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-c x}\right )+\frac {1}{15} \left (14 b^2 c^7 d^3\right ) \int \frac {1}{1-c^2 x^2} \, dx\\ &=-\frac {b^2 c^3 d^3}{10 x^3}-\frac {37 b^2 c^5 d^3}{30 x}+\frac {14}{15} b^2 c^6 d^3 \tanh ^{-1}(c x)-\frac {b c d^3 \left (a+b \tanh ^{-1}(c x)\right )}{15 x^5}-\frac {3 b c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{10 x^4}-\frac {11 b c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{18 x^3}-\frac {14 b c^4 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{15 x^2}-\frac {11 b c^5 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{6 x}-\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{6 x^6}-\frac {3 c d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{5 x^5}-\frac {3 c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{4 x^4}-\frac {c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{3 x^3}+\frac {28}{15} a b c^6 d^3 \log (x)+\frac {37}{20} b c^6 d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )+\frac {1}{60} b c^6 d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1+c x}\right )-\frac {14}{15} b^2 c^6 d^3 \text {Li}_2(-c x)+\frac {14}{15} b^2 c^6 d^3 \text {Li}_2(c x)+\frac {37}{40} b^2 c^6 d^3 \text {Li}_2\left (1-\frac {2}{1-c x}\right )-\frac {1}{120} b^2 c^6 d^3 \text {Li}_2\left (1-\frac {2}{1+c x}\right )+\frac {1}{30} \left (b^2 c^2 d^3\right ) \text {Subst}\left (\int \left (\frac {1}{x^3}+\frac {c^2}{x^2}+\frac {c^4}{x}-\frac {c^6}{-1+c^2 x}\right ) \, dx,x,x^2\right )+\frac {1}{36} \left (11 b^2 c^4 d^3\right ) \text {Subst}\left (\int \left (\frac {1}{x^2}+\frac {c^2}{x}-\frac {c^4}{-1+c^2 x}\right ) \, dx,x,x^2\right )+\frac {1}{12} \left (11 b^2 c^6 d^3\right ) \text {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )+\frac {1}{10} \left (3 b^2 c^7 d^3\right ) \int \frac {1}{1-c^2 x^2} \, dx+\frac {1}{12} \left (11 b^2 c^8 d^3\right ) \text {Subst}\left (\int \frac {1}{1-c^2 x} \, dx,x,x^2\right )\\ &=-\frac {b^2 c^2 d^3}{60 x^4}-\frac {b^2 c^3 d^3}{10 x^3}-\frac {61 b^2 c^4 d^3}{180 x^2}-\frac {37 b^2 c^5 d^3}{30 x}+\frac {37}{30} b^2 c^6 d^3 \tanh ^{-1}(c x)-\frac {b c d^3 \left (a+b \tanh ^{-1}(c x)\right )}{15 x^5}-\frac {3 b c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{10 x^4}-\frac {11 b c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{18 x^3}-\frac {14 b c^4 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{15 x^2}-\frac {11 b c^5 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{6 x}-\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{6 x^6}-\frac {3 c d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{5 x^5}-\frac {3 c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{4 x^4}-\frac {c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{3 x^3}+\frac {28}{15} a b c^6 d^3 \log (x)+\frac {113}{45} b^2 c^6 d^3 \log (x)+\frac {37}{20} b c^6 d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )+\frac {1}{60} b c^6 d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1+c x}\right )-\frac {113}{90} b^2 c^6 d^3 \log \left (1-c^2 x^2\right )-\frac {14}{15} b^2 c^6 d^3 \text {Li}_2(-c x)+\frac {14}{15} b^2 c^6 d^3 \text {Li}_2(c x)+\frac {37}{40} b^2 c^6 d^3 \text {Li}_2\left (1-\frac {2}{1-c x}\right )-\frac {1}{120} b^2 c^6 d^3 \text {Li}_2\left (1-\frac {2}{1+c x}\right )\\ \end {align*}

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Mathematica [A]
time = 1.19, size = 402, normalized size = 0.84 \begin {gather*} -\frac {d^3 \left (30 a^2+108 a^2 c x+12 a b c x+135 a^2 c^2 x^2+54 a b c^2 x^2+3 b^2 c^2 x^2+60 a^2 c^3 x^3+110 a b c^3 x^3+18 b^2 c^3 x^3+168 a b c^4 x^4+61 b^2 c^4 x^4+330 a b c^5 x^5+222 b^2 c^5 x^5-64 b^2 c^6 x^6+3 b^2 \left (10+36 c x+45 c^2 x^2+20 c^3 x^3-111 c^6 x^6\right ) \tanh ^{-1}(c x)^2+2 b \tanh ^{-1}(c x) \left (3 a \left (10+36 c x+45 c^2 x^2+20 c^3 x^3\right )+b c x \left (6+27 c x+55 c^2 x^2+84 c^3 x^3+165 c^4 x^4-111 c^5 x^5\right )-168 b c^6 x^6 \log \left (1-e^{-2 \tanh ^{-1}(c x)}\right )\right )-336 a b c^6 x^6 \log (c x)+165 a b c^6 x^6 \log (1-c x)-165 a b c^6 x^6 \log (1+c x)-452 b^2 c^6 x^6 \log \left (\frac {c x}{\sqrt {1-c^2 x^2}}\right )+168 a b c^6 x^6 \log \left (1-c^2 x^2\right )+168 b^2 c^6 x^6 \text {PolyLog}\left (2,e^{-2 \tanh ^{-1}(c x)}\right )\right )}{180 x^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((d + c*d*x)^3*(a + b*ArcTanh[c*x])^2)/x^7,x]

[Out]

-1/180*(d^3*(30*a^2 + 108*a^2*c*x + 12*a*b*c*x + 135*a^2*c^2*x^2 + 54*a*b*c^2*x^2 + 3*b^2*c^2*x^2 + 60*a^2*c^3
*x^3 + 110*a*b*c^3*x^3 + 18*b^2*c^3*x^3 + 168*a*b*c^4*x^4 + 61*b^2*c^4*x^4 + 330*a*b*c^5*x^5 + 222*b^2*c^5*x^5
- 64*b^2*c^6*x^6 + 3*b^2*(10 + 36*c*x + 45*c^2*x^2 + 20*c^3*x^3 - 111*c^6*x^6)*ArcTanh[c*x]^2 + 2*b*ArcTanh[c
*x]*(3*a*(10 + 36*c*x + 45*c^2*x^2 + 20*c^3*x^3) + b*c*x*(6 + 27*c*x + 55*c^2*x^2 + 84*c^3*x^3 + 165*c^4*x^4 -
111*c^5*x^5) - 168*b*c^6*x^6*Log[1 - E^(-2*ArcTanh[c*x])]) - 336*a*b*c^6*x^6*Log[c*x] + 165*a*b*c^6*x^6*Log[1
- c*x] - 165*a*b*c^6*x^6*Log[1 + c*x] - 452*b^2*c^6*x^6*Log[(c*x)/Sqrt[1 - c^2*x^2]] + 168*a*b*c^6*x^6*Log[1
- c^2*x^2] + 168*b^2*c^6*x^6*PolyLog[2, E^(-2*ArcTanh[c*x])]))/x^6

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Maple [A]
time = 0.74, size = 689, normalized size = 1.44

method	result	size
derivativedivides	\(c^{6} \left (d^{3} a^{2} \left (-\frac {1}{6 c^{6} x^{6}}-\frac {3}{5 c^{5} x^{5}}-\frac {1}{3 c^{3} x^{3}}-\frac {3}{4 c^{4} x^{4}}\right )-\frac {d^{3} b^{2} \arctanh \left (c x \right )^{2}}{6 c^{6} x^{6}}-\frac {d^{3} b^{2} \arctanh \left (c x \right )}{15 c^{5} x^{5}}-\frac {3 d^{3} a b \arctanh \left (c x \right )}{2 c^{4} x^{4}}-\frac {3 d^{3} b^{2} \arctanh \left (c x \right )}{10 c^{4} x^{4}}-\frac {37 d^{3} b^{2} \arctanh \left (c x \right ) \ln \left (c x -1\right )}{20}-\frac {d^{3} b^{2} \arctanh \left (c x \right ) \ln \left (c x +1\right )}{60}-\frac {3 d^{3} b^{2} \arctanh \left (c x \right )^{2}}{4 c^{4} x^{4}}-\frac {3 d^{3} a b}{10 c^{4} x^{4}}-\frac {d^{3} a b}{15 c^{5} x^{5}}-\frac {23 d^{3} b^{2} \ln \left (c x +1\right )}{36}-\frac {3 d^{3} b^{2} \arctanh \left (c x \right )^{2}}{5 c^{5} x^{5}}-\frac {14 d^{3} b^{2} \dilog \left (c x \right )}{15}-\frac {14 d^{3} b^{2} \dilog \left (c x +1\right )}{15}-\frac {37 d^{3} b^{2} \ln \left (c x -1\right )^{2}}{80}+\frac {d^{3} b^{2} \ln \left (c x +1\right )^{2}}{240}-\frac {337 d^{3} b^{2} \ln \left (c x -1\right )}{180}+\frac {37 d^{3} b^{2} \ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{40}-\frac {d^{3} b^{2} \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (c x +1\right )}{120}+\frac {d^{3} b^{2} \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{120}-\frac {37 d^{3} a b \ln \left (c x -1\right )}{20}-\frac {d^{3} a b \ln \left (c x +1\right )}{60}-\frac {11 d^{3} b^{2} \arctanh \left (c x \right )}{18 c^{3} x^{3}}+\frac {113 d^{3} b^{2} \ln \left (c x \right )}{45}-\frac {d^{3} a b \arctanh \left (c x \right )}{3 c^{6} x^{6}}-\frac {11 d^{3} a b}{18 c^{3} x^{3}}-\frac {6 d^{3} a b \arctanh \left (c x \right )}{5 c^{5} x^{5}}+\frac {14 d^{3} b^{2} \dilog \left (\frac {c x}{2}+\frac {1}{2}\right )}{15}-\frac {d^{3} b^{2}}{60 c^{4} x^{4}}-\frac {d^{3} b^{2}}{10 c^{3} x^{3}}-\frac {61 d^{3} b^{2}}{180 c^{2} x^{2}}-\frac {37 d^{3} b^{2}}{30 c x}-\frac {14 d^{3} b^{2} \ln \left (c x \right ) \ln \left (c x +1\right )}{15}-\frac {14 d^{3} a b}{15 c^{2} x^{2}}-\frac {2 d^{3} a b \arctanh \left (c x \right )}{3 c^{3} x^{3}}+\frac {28 d^{3} b^{2} \arctanh \left (c x \right ) \ln \left (c x \right )}{15}-\frac {14 d^{3} b^{2} \arctanh \left (c x \right )}{15 c^{2} x^{2}}-\frac {d^{3} b^{2} \arctanh \left (c x \right )^{2}}{3 c^{3} x^{3}}-\frac {11 d^{3} b^{2} \arctanh \left (c x \right )}{6 c x}-\frac {11 d^{3} a b}{6 c x}+\frac {28 d^{3} a b \ln \left (c x \right )}{15}\right )\)	\(689\)
default	\(c^{6} \left (d^{3} a^{2} \left (-\frac {1}{6 c^{6} x^{6}}-\frac {3}{5 c^{5} x^{5}}-\frac {1}{3 c^{3} x^{3}}-\frac {3}{4 c^{4} x^{4}}\right )-\frac {d^{3} b^{2} \arctanh \left (c x \right )^{2}}{6 c^{6} x^{6}}-\frac {d^{3} b^{2} \arctanh \left (c x \right )}{15 c^{5} x^{5}}-\frac {3 d^{3} a b \arctanh \left (c x \right )}{2 c^{4} x^{4}}-\frac {3 d^{3} b^{2} \arctanh \left (c x \right )}{10 c^{4} x^{4}}-\frac {37 d^{3} b^{2} \arctanh \left (c x \right ) \ln \left (c x -1\right )}{20}-\frac {d^{3} b^{2} \arctanh \left (c x \right ) \ln \left (c x +1\right )}{60}-\frac {3 d^{3} b^{2} \arctanh \left (c x \right )^{2}}{4 c^{4} x^{4}}-\frac {3 d^{3} a b}{10 c^{4} x^{4}}-\frac {d^{3} a b}{15 c^{5} x^{5}}-\frac {23 d^{3} b^{2} \ln \left (c x +1\right )}{36}-\frac {3 d^{3} b^{2} \arctanh \left (c x \right )^{2}}{5 c^{5} x^{5}}-\frac {14 d^{3} b^{2} \dilog \left (c x \right )}{15}-\frac {14 d^{3} b^{2} \dilog \left (c x +1\right )}{15}-\frac {37 d^{3} b^{2} \ln \left (c x -1\right )^{2}}{80}+\frac {d^{3} b^{2} \ln \left (c x +1\right )^{2}}{240}-\frac {337 d^{3} b^{2} \ln \left (c x -1\right )}{180}+\frac {37 d^{3} b^{2} \ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{40}-\frac {d^{3} b^{2} \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (c x +1\right )}{120}+\frac {d^{3} b^{2} \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{120}-\frac {37 d^{3} a b \ln \left (c x -1\right )}{20}-\frac {d^{3} a b \ln \left (c x +1\right )}{60}-\frac {11 d^{3} b^{2} \arctanh \left (c x \right )}{18 c^{3} x^{3}}+\frac {113 d^{3} b^{2} \ln \left (c x \right )}{45}-\frac {d^{3} a b \arctanh \left (c x \right )}{3 c^{6} x^{6}}-\frac {11 d^{3} a b}{18 c^{3} x^{3}}-\frac {6 d^{3} a b \arctanh \left (c x \right )}{5 c^{5} x^{5}}+\frac {14 d^{3} b^{2} \dilog \left (\frac {c x}{2}+\frac {1}{2}\right )}{15}-\frac {d^{3} b^{2}}{60 c^{4} x^{4}}-\frac {d^{3} b^{2}}{10 c^{3} x^{3}}-\frac {61 d^{3} b^{2}}{180 c^{2} x^{2}}-\frac {37 d^{3} b^{2}}{30 c x}-\frac {14 d^{3} b^{2} \ln \left (c x \right ) \ln \left (c x +1\right )}{15}-\frac {14 d^{3} a b}{15 c^{2} x^{2}}-\frac {2 d^{3} a b \arctanh \left (c x \right )}{3 c^{3} x^{3}}+\frac {28 d^{3} b^{2} \arctanh \left (c x \right ) \ln \left (c x \right )}{15}-\frac {14 d^{3} b^{2} \arctanh \left (c x \right )}{15 c^{2} x^{2}}-\frac {d^{3} b^{2} \arctanh \left (c x \right )^{2}}{3 c^{3} x^{3}}-\frac {11 d^{3} b^{2} \arctanh \left (c x \right )}{6 c x}-\frac {11 d^{3} a b}{6 c x}+\frac {28 d^{3} a b \ln \left (c x \right )}{15}\right )\)	\(689\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*d*x+d)^3*(a+b*arctanh(c*x))^2/x^7,x,method=_RETURNVERBOSE)

[Out]

c^6*(d^3*a^2*(-1/6/c^6/x^6-3/5/c^5/x^5-1/3/c^3/x^3-3/4/c^4/x^4)-6/5*d^3*a*b*arctanh(c*x)/c^5/x^5-3/10*d^3*b^2*
arctanh(c*x)/c^4/x^4-3/5*d^3*b^2*arctanh(c*x)^2/c^5/x^5-3/10*d^3*a*b/c^4/x^4-1/6*d^3*b^2*arctanh(c*x)^2/c^6/x^
6-1/15*d^3*a*b/c^5/x^5-1/15*d^3*b^2*arctanh(c*x)/c^5/x^5-3/2*d^3*a*b*arctanh(c*x)/c^4/x^4-1/3*d^3*a*b*arctanh(
c*x)/c^6/x^6-23/36*d^3*b^2*ln(c*x+1)+14/15*d^3*b^2*dilog(1/2*c*x+1/2)-37/80*d^3*b^2*ln(c*x-1)^2+1/240*d^3*b^2*
ln(c*x+1)^2-337/180*d^3*b^2*ln(c*x-1)-37/20*d^3*b^2*arctanh(c*x)*ln(c*x-1)-1/60*d^3*b^2*arctanh(c*x)*ln(c*x+1)
+37/40*d^3*b^2*ln(c*x-1)*ln(1/2*c*x+1/2)-1/120*d^3*b^2*ln(-1/2*c*x+1/2)*ln(c*x+1)+1/120*d^3*b^2*ln(-1/2*c*x+1/
2)*ln(1/2*c*x+1/2)-37/20*d^3*a*b*ln(c*x-1)-1/60*d^3*a*b*ln(c*x+1)-14/15*d^3*b^2*dilog(c*x)-14/15*d^3*b^2*dilog
(c*x+1)+113/45*d^3*b^2*ln(c*x)-11/18*d^3*a*b/c^3/x^3-3/4*d^3*b^2*arctanh(c*x)^2/c^4/x^4-11/18*d^3*b^2*arctanh(
c*x)/c^3/x^3-1/60*d^3*b^2/c^4/x^4-1/10*d^3*b^2/c^3/x^3-61/180*d^3*b^2/c^2/x^2-37/30*d^3*b^2/c/x+28/15*d^3*b^2*
arctanh(c*x)*ln(c*x)-14/15*d^3*b^2*ln(c*x)*ln(c*x+1)-14/15*d^3*b^2*arctanh(c*x)/c^2/x^2-1/3*d^3*b^2*arctanh(c*
x)^2/c^3/x^3-14/15*d^3*a*b/c^2/x^2-2/3*d^3*a*b*arctanh(c*x)/c^3/x^3-11/6*d^3*b^2*arctanh(c*x)/c/x-11/6*d^3*a*b
/c/x+28/15*d^3*a*b*ln(c*x))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 961 vs. \(2 (427) = 854\).
time = 0.66, size = 961, normalized size = 2.01 \begin {gather*} -\frac {14}{15} \, {\left (\log \left (c x + 1\right ) \log \left (-\frac {1}{2} \, c x + \frac {1}{2}\right ) + {\rm Li}_2\left (\frac {1}{2} \, c x + \frac {1}{2}\right )\right )} b^{2} c^{6} d^{3} - \frac {14}{15} \, {\left (\log \left (c x\right ) \log \left (-c x + 1\right ) + {\rm Li}_2\left (-c x + 1\right )\right )} b^{2} c^{6} d^{3} + \frac {14}{15} \, {\left (\log \left (c x + 1\right ) \log \left (-c x\right ) + {\rm Li}_2\left (c x + 1\right )\right )} b^{2} c^{6} d^{3} - \frac {23}{60} \, b^{2} c^{6} d^{3} \log \left (c x + 1\right ) - \frac {97}{60} \, b^{2} c^{6} d^{3} \log \left (c x - 1\right ) + 2 \, b^{2} c^{6} d^{3} \log \left (x\right ) - \frac {1}{3} \, {\left ({\left (c^{2} \log \left (c^{2} x^{2} - 1\right ) - c^{2} \log \left (x^{2}\right ) + \frac {1}{x^{2}}\right )} c + \frac {2 \, \operatorname {artanh}\left (c x\right )}{x^{3}}\right )} a b c^{3} d^{3} + \frac {1}{4} \, {\left ({\left (3 \, c^{3} \log \left (c x + 1\right ) - 3 \, c^{3} \log \left (c x - 1\right ) - \frac {2 \, {\left (3 \, c^{2} x^{2} + 1\right )}}{x^{3}}\right )} c - \frac {6 \, \operatorname {artanh}\left (c x\right )}{x^{4}}\right )} a b c^{2} d^{3} - \frac {3}{10} \, {\left ({\left (2 \, c^{4} \log \left (c^{2} x^{2} - 1\right ) - 2 \, c^{4} \log \left (x^{2}\right ) + \frac {2 \, c^{2} x^{2} + 1}{x^{4}}\right )} c + \frac {4 \, \operatorname {artanh}\left (c x\right )}{x^{5}}\right )} a b c d^{3} + \frac {1}{90} \, {\left ({\left (15 \, c^{5} \log \left (c x + 1\right ) - 15 \, c^{5} \log \left (c x - 1\right ) - \frac {2 \, {\left (15 \, c^{4} x^{4} + 5 \, c^{2} x^{2} + 3\right )}}{x^{5}}\right )} c - \frac {30 \, \operatorname {artanh}\left (c x\right )}{x^{6}}\right )} a b d^{3} + \frac {1}{360} \, {\left ({\left (184 \, c^{4} \log \left (x\right ) - \frac {15 \, c^{4} x^{4} \log \left (c x + 1\right )^{2} + 15 \, c^{4} x^{4} \log \left (c x - 1\right )^{2} + 92 \, c^{4} x^{4} \log \left (c x - 1\right ) + 32 \, c^{2} x^{2} - 2 \, {\left (15 \, c^{4} x^{4} \log \left (c x - 1\right ) - 46 \, c^{4} x^{4}\right )} \log \left (c x + 1\right ) + 6}{x^{4}}\right )} c^{2} + 4 \, {\left (15 \, c^{5} \log \left (c x + 1\right ) - 15 \, c^{5} \log \left (c x - 1\right ) - \frac {2 \, {\left (15 \, c^{4} x^{4} + 5 \, c^{2} x^{2} + 3\right )}}{x^{5}}\right )} c \operatorname {artanh}\left (c x\right )\right )} b^{2} d^{3} - \frac {a^{2} c^{3} d^{3}}{3 \, x^{3}} - \frac {3 \, a^{2} c^{2} d^{3}}{4 \, x^{4}} - \frac {3 \, a^{2} c d^{3}}{5 \, x^{5}} - \frac {b^{2} d^{3} \operatorname {artanh}\left (c x\right )^{2}}{6 \, x^{6}} - \frac {a^{2} d^{3}}{6 \, x^{6}} - \frac {296 \, b^{2} c^{5} d^{3} x^{4} + 60 \, b^{2} c^{4} d^{3} x^{3} + 24 \, b^{2} c^{3} d^{3} x^{2} + {\left (11 \, b^{2} c^{6} d^{3} x^{5} + 20 \, b^{2} c^{3} d^{3} x^{2} + 45 \, b^{2} c^{2} d^{3} x + 36 \, b^{2} c d^{3}\right )} \log \left (c x + 1\right )^{2} - {\left (101 \, b^{2} c^{6} d^{3} x^{5} - 20 \, b^{2} c^{3} d^{3} x^{2} - 45 \, b^{2} c^{2} d^{3} x - 36 \, b^{2} c d^{3}\right )} \log \left (-c x + 1\right )^{2} + 4 \, {\left (45 \, b^{2} c^{5} d^{3} x^{4} + 28 \, b^{2} c^{4} d^{3} x^{3} + 15 \, b^{2} c^{3} d^{3} x^{2} + 9 \, b^{2} c^{2} d^{3} x\right )} \log \left (c x + 1\right ) - 2 \, {\left (90 \, b^{2} c^{5} d^{3} x^{4} + 56 \, b^{2} c^{4} d^{3} x^{3} + 30 \, b^{2} c^{3} d^{3} x^{2} + 18 \, b^{2} c^{2} d^{3} x + {\left (11 \, b^{2} c^{6} d^{3} x^{5} + 20 \, b^{2} c^{3} d^{3} x^{2} + 45 \, b^{2} c^{2} d^{3} x + 36 \, b^{2} c d^{3}\right )} \log \left (c x + 1\right )\right )} \log \left (-c x + 1\right )}{240 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*d*x+d)^3*(a+b*arctanh(c*x))^2/x^7,x, algorithm="maxima")

[Out]

-14/15*(log(c*x + 1)*log(-1/2*c*x + 1/2) + dilog(1/2*c*x + 1/2))*b^2*c^6*d^3 - 14/15*(log(c*x)*log(-c*x + 1) +
dilog(-c*x + 1))*b^2*c^6*d^3 + 14/15*(log(c*x + 1)*log(-c*x) + dilog(c*x + 1))*b^2*c^6*d^3 - 23/60*b^2*c^6*d^
3*log(c*x + 1) - 97/60*b^2*c^6*d^3*log(c*x - 1) + 2*b^2*c^6*d^3*log(x) - 1/3*((c^2*log(c^2*x^2 - 1) - c^2*log(
x^2) + 1/x^2)*c + 2*arctanh(c*x)/x^3)*a*b*c^3*d^3 + 1/4*((3*c^3*log(c*x + 1) - 3*c^3*log(c*x - 1) - 2*(3*c^2*x
^2 + 1)/x^3)*c - 6*arctanh(c*x)/x^4)*a*b*c^2*d^3 - 3/10*((2*c^4*log(c^2*x^2 - 1) - 2*c^4*log(x^2) + (2*c^2*x^2
+ 1)/x^4)*c + 4*arctanh(c*x)/x^5)*a*b*c*d^3 + 1/90*((15*c^5*log(c*x + 1) - 15*c^5*log(c*x - 1) - 2*(15*c^4*x^
4 + 5*c^2*x^2 + 3)/x^5)*c - 30*arctanh(c*x)/x^6)*a*b*d^3 + 1/360*((184*c^4*log(x) - (15*c^4*x^4*log(c*x + 1)^2
+ 15*c^4*x^4*log(c*x - 1)^2 + 92*c^4*x^4*log(c*x - 1) + 32*c^2*x^2 - 2*(15*c^4*x^4*log(c*x - 1) - 46*c^4*x^4)
*log(c*x + 1) + 6)/x^4)*c^2 + 4*(15*c^5*log(c*x + 1) - 15*c^5*log(c*x - 1) - 2*(15*c^4*x^4 + 5*c^2*x^2 + 3)/x^
5)*c*arctanh(c*x))*b^2*d^3 - 1/3*a^2*c^3*d^3/x^3 - 3/4*a^2*c^2*d^3/x^4 - 3/5*a^2*c*d^3/x^5 - 1/6*b^2*d^3*arcta
nh(c*x)^2/x^6 - 1/6*a^2*d^3/x^6 - 1/240*(296*b^2*c^5*d^3*x^4 + 60*b^2*c^4*d^3*x^3 + 24*b^2*c^3*d^3*x^2 + (11*b
^2*c^6*d^3*x^5 + 20*b^2*c^3*d^3*x^2 + 45*b^2*c^2*d^3*x + 36*b^2*c*d^3)*log(c*x + 1)^2 - (101*b^2*c^6*d^3*x^5 -
20*b^2*c^3*d^3*x^2 - 45*b^2*c^2*d^3*x - 36*b^2*c*d^3)*log(-c*x + 1)^2 + 4*(45*b^2*c^5*d^3*x^4 + 28*b^2*c^4*d^
3*x^3 + 15*b^2*c^3*d^3*x^2 + 9*b^2*c^2*d^3*x)*log(c*x + 1) - 2*(90*b^2*c^5*d^3*x^4 + 56*b^2*c^4*d^3*x^3 + 30*b
^2*c^3*d^3*x^2 + 18*b^2*c^2*d^3*x + (11*b^2*c^6*d^3*x^5 + 20*b^2*c^3*d^3*x^2 + 45*b^2*c^2*d^3*x + 36*b^2*c*d^3
)*log(c*x + 1))*log(-c*x + 1))/x^5

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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((c*d*x+d)^3*(a+b*arctanh(c*x))^2/x^7,x, algorithm="fricas")

[Out]

integral((a^2*c^3*d^3*x^3 + 3*a^2*c^2*d^3*x^2 + 3*a^2*c*d^3*x + a^2*d^3 + (b^2*c^3*d^3*x^3 + 3*b^2*c^2*d^3*x^2
+ 3*b^2*c*d^3*x + b^2*d^3)*arctanh(c*x)^2 + 2*(a*b*c^3*d^3*x^3 + 3*a*b*c^2*d^3*x^2 + 3*a*b*c*d^3*x + a*b*d^3)
*arctanh(c*x))/x^7, x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} d^{3} \left (\int \frac {a^{2}}{x^{7}}\, dx + \int \frac {3 a^{2} c}{x^{6}}\, dx + \int \frac {3 a^{2} c^{2}}{x^{5}}\, dx + \int \frac {a^{2} c^{3}}{x^{4}}\, dx + \int \frac {b^{2} \operatorname {atanh}^{2}{\left (c x \right )}}{x^{7}}\, dx + \int \frac {2 a b \operatorname {atanh}{\left (c x \right )}}{x^{7}}\, dx + \int \frac {3 b^{2} c \operatorname {atanh}^{2}{\left (c x \right )}}{x^{6}}\, dx + \int \frac {3 b^{2} c^{2} \operatorname {atanh}^{2}{\left (c x \right )}}{x^{5}}\, dx + \int \frac {b^{2} c^{3} \operatorname {atanh}^{2}{\left (c x \right )}}{x^{4}}\, dx + \int \frac {6 a b c \operatorname {atanh}{\left (c x \right )}}{x^{6}}\, dx + \int \frac {6 a b c^{2} \operatorname {atanh}{\left (c x \right )}}{x^{5}}\, dx + \int \frac {2 a b c^{3} \operatorname {atanh}{\left (c x \right )}}{x^{4}}\, dx\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*d*x+d)**3*(a+b*atanh(c*x))**2/x**7,x)

[Out]

d**3*(Integral(a**2/x**7, x) + Integral(3*a**2*c/x**6, x) + Integral(3*a**2*c**2/x**5, x) + Integral(a**2*c**3
/x**4, x) + Integral(b**2*atanh(c*x)**2/x**7, x) + Integral(2*a*b*atanh(c*x)/x**7, x) + Integral(3*b**2*c*atan
h(c*x)**2/x**6, x) + Integral(3*b**2*c**2*atanh(c*x)**2/x**5, x) + Integral(b**2*c**3*atanh(c*x)**2/x**4, x) +
Integral(6*a*b*c*atanh(c*x)/x**6, x) + Integral(6*a*b*c**2*atanh(c*x)/x**5, x) + Integral(2*a*b*c**3*atanh(c*
x)/x**4, 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((c*d*x+d)^3*(a+b*arctanh(c*x))^2/x^7,x, algorithm="giac")

[Out]

integrate((c*d*x + d)^3*(b*arctanh(c*x) + a)^2/x^7, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^2\,{\left (d+c\,d\,x\right )}^3}{x^7} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*atanh(c*x))^2*(d + c*d*x)^3)/x^7,x)

[Out]

int(((a + b*atanh(c*x))^2*(d + c*d*x)^3)/x^7, x)

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