3.1.93 \(\int \frac {(d+c d x)^3 (a+b \tanh ^{-1}(c x))^2}{x^6} \, dx\) [93]

Optimal. Leaf size=352 \[ -\frac {b^2 c^2 d^3}{30 x^3}-\frac {b^2 c^3 d^3}{4 x^2}-\frac {13 b^2 c^4 d^3}{10 x}+\frac {13}{10} b^2 c^5 d^3 \tanh ^{-1}(c x)-\frac {b c d^3 \left (a+b \tanh ^{-1}(c x)\right )}{10 x^4}-\frac {b c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{2 x^3}-\frac {6 b c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{5 x^2}-\frac {5 b c^4 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{2 x}-\frac {d^3 (1+c x)^4 \left (a+b \tanh ^{-1}(c x)\right )^2}{5 x^5}+\frac {c d^3 (1+c x)^4 \left (a+b \tanh ^{-1}(c x)\right )^2}{20 x^4}+\frac {12}{5} a b c^5 d^3 \log (x)+3 b^2 c^5 d^3 \log (x)+\frac {12}{5} b c^5 d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )-\frac {3}{2} b^2 c^5 d^3 \log \left (1-c^2 x^2\right )-\frac {6}{5} b^2 c^5 d^3 \text {PolyLog}(2,-c x)+\frac {6}{5} b^2 c^5 d^3 \text {PolyLog}(2,c x)+\frac {6}{5} b^2 c^5 d^3 \text {PolyLog}\left (2,1-\frac {2}{1-c x}\right ) \]

[Out]

-1/30*b^2*c^2*d^3/x^3-1/4*b^2*c^3*d^3/x^2-13/10*b^2*c^4*d^3/x+13/10*b^2*c^5*d^3*arctanh(c*x)-1/10*b*c*d^3*(a+b
*arctanh(c*x))/x^4-1/2*b*c^2*d^3*(a+b*arctanh(c*x))/x^3-6/5*b*c^3*d^3*(a+b*arctanh(c*x))/x^2-5/2*b*c^4*d^3*(a+
b*arctanh(c*x))/x-1/5*d^3*(c*x+1)^4*(a+b*arctanh(c*x))^2/x^5+1/20*c*d^3*(c*x+1)^4*(a+b*arctanh(c*x))^2/x^4+12/
5*a*b*c^5*d^3*ln(x)+3*b^2*c^5*d^3*ln(x)+12/5*b*c^5*d^3*(a+b*arctanh(c*x))*ln(2/(-c*x+1))-3/2*b^2*c^5*d^3*ln(-c
^2*x^2+1)-6/5*b^2*c^5*d^3*polylog(2,-c*x)+6/5*b^2*c^5*d^3*polylog(2,c*x)+6/5*b^2*c^5*d^3*polylog(2,1-2/(-c*x+1
))

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Rubi [A]
time = 0.27, antiderivative size = 352, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 15, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.682, Rules used = {47, 37, 6085, 6037, 331, 212, 272, 46, 36, 29, 31, 6031, 6055, 2449, 2352} \begin {gather*} \frac {12}{5} a b c^5 d^3 \log (x)+\frac {12}{5} b c^5 d^3 \log \left (\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )-\frac {5 b c^4 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{2 x}-\frac {6 b c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{5 x^2}-\frac {b c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{2 x^3}-\frac {d^3 (c x+1)^4 \left (a+b \tanh ^{-1}(c x)\right )^2}{5 x^5}+\frac {c d^3 (c x+1)^4 \left (a+b \tanh ^{-1}(c x)\right )^2}{20 x^4}-\frac {b c d^3 \left (a+b \tanh ^{-1}(c x)\right )}{10 x^4}-\frac {6}{5} b^2 c^5 d^3 \text {Li}_2(-c x)+\frac {6}{5} b^2 c^5 d^3 \text {Li}_2(c x)+\frac {6}{5} b^2 c^5 d^3 \text {Li}_2\left (1-\frac {2}{1-c x}\right )+3 b^2 c^5 d^3 \log (x)+\frac {13}{10} b^2 c^5 d^3 \tanh ^{-1}(c x)-\frac {13 b^2 c^4 d^3}{10 x}-\frac {b^2 c^3 d^3}{4 x^2}-\frac {b^2 c^2 d^3}{30 x^3}-\frac {3}{2} b^2 c^5 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^6,x]

[Out]

-1/30*(b^2*c^2*d^3)/x^3 - (b^2*c^3*d^3)/(4*x^2) - (13*b^2*c^4*d^3)/(10*x) + (13*b^2*c^5*d^3*ArcTanh[c*x])/10 -
(b*c*d^3*(a + b*ArcTanh[c*x]))/(10*x^4) - (b*c^2*d^3*(a + b*ArcTanh[c*x]))/(2*x^3) - (6*b*c^3*d^3*(a + b*ArcT
anh[c*x]))/(5*x^2) - (5*b*c^4*d^3*(a + b*ArcTanh[c*x]))/(2*x) - (d^3*(1 + c*x)^4*(a + b*ArcTanh[c*x])^2)/(5*x^
5) + (c*d^3*(1 + c*x)^4*(a + b*ArcTanh[c*x])^2)/(20*x^4) + (12*a*b*c^5*d^3*Log[x])/5 + 3*b^2*c^5*d^3*Log[x] +
(12*b*c^5*d^3*(a + b*ArcTanh[c*x])*Log[2/(1 - c*x)])/5 - (3*b^2*c^5*d^3*Log[1 - c^2*x^2])/2 - (6*b^2*c^5*d^3*P
olyLog[2, -(c*x)])/5 + (6*b^2*c^5*d^3*PolyLog[2, c*x])/5 + (6*b^2*c^5*d^3*PolyLog[2, 1 - 2/(1 - c*x)])/5

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 37

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

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 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n + 1
)/((b*c - a*d)*(m + 1))), x] - Dist[d*(Simplify[m + n + 2]/((b*c - a*d)*(m + 1))), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

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^6} \, dx &=-\frac {d^3 (1+c x)^4 \left (a+b \tanh ^{-1}(c x)\right )^2}{5 x^5}+\frac {c d^3 (1+c x)^4 \left (a+b \tanh ^{-1}(c x)\right )^2}{20 x^4}-(2 b c) \int \left (-\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right )}{5 x^5}-\frac {3 c d^3 \left (a+b \tanh ^{-1}(c x)\right )}{4 x^4}-\frac {6 c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{5 x^3}-\frac {5 c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{4 x^2}-\frac {6 c^4 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{5 x}+\frac {6 c^5 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{5 (-1+c x)}\right ) \, dx\\ &=-\frac {d^3 (1+c x)^4 \left (a+b \tanh ^{-1}(c x)\right )^2}{5 x^5}+\frac {c d^3 (1+c x)^4 \left (a+b \tanh ^{-1}(c x)\right )^2}{20 x^4}+\frac {1}{5} \left (2 b c d^3\right ) \int \frac {a+b \tanh ^{-1}(c x)}{x^5} \, dx+\frac {1}{2} \left (3 b c^2 d^3\right ) \int \frac {a+b \tanh ^{-1}(c x)}{x^4} \, dx+\frac {1}{5} \left (12 b c^3 d^3\right ) \int \frac {a+b \tanh ^{-1}(c x)}{x^3} \, dx+\frac {1}{2} \left (5 b c^4 d^3\right ) \int \frac {a+b \tanh ^{-1}(c x)}{x^2} \, dx+\frac {1}{5} \left (12 b c^5 d^3\right ) \int \frac {a+b \tanh ^{-1}(c x)}{x} \, dx-\frac {1}{5} \left (12 b c^6 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 )}{10 x^4}-\frac {b c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{2 x^3}-\frac {6 b c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{5 x^2}-\frac {5 b c^4 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{2 x}-\frac {d^3 (1+c x)^4 \left (a+b \tanh ^{-1}(c x)\right )^2}{5 x^5}+\frac {c d^3 (1+c x)^4 \left (a+b \tanh ^{-1}(c x)\right )^2}{20 x^4}+\frac {12}{5} a b c^5 d^3 \log (x)+\frac {12}{5} b c^5 d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )-\frac {6}{5} b^2 c^5 d^3 \text {Li}_2(-c x)+\frac {6}{5} b^2 c^5 d^3 \text {Li}_2(c x)+\frac {1}{10} \left (b^2 c^2 d^3\right ) \int \frac {1}{x^4 \left (1-c^2 x^2\right )} \, dx+\frac {1}{2} \left (b^2 c^3 d^3\right ) \int \frac {1}{x^3 \left (1-c^2 x^2\right )} \, dx+\frac {1}{5} \left (6 b^2 c^4 d^3\right ) \int \frac {1}{x^2 \left (1-c^2 x^2\right )} \, dx+\frac {1}{2} \left (5 b^2 c^5 d^3\right ) \int \frac {1}{x \left (1-c^2 x^2\right )} \, dx-\frac {1}{5} \left (12 b^2 c^6 d^3\right ) \int \frac {\log \left (\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx\\ &=-\frac {b^2 c^2 d^3}{30 x^3}-\frac {6 b^2 c^4 d^3}{5 x}-\frac {b c d^3 \left (a+b \tanh ^{-1}(c x)\right )}{10 x^4}-\frac {b c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{2 x^3}-\frac {6 b c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{5 x^2}-\frac {5 b c^4 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{2 x}-\frac {d^3 (1+c x)^4 \left (a+b \tanh ^{-1}(c x)\right )^2}{5 x^5}+\frac {c d^3 (1+c x)^4 \left (a+b \tanh ^{-1}(c x)\right )^2}{20 x^4}+\frac {12}{5} a b c^5 d^3 \log (x)+\frac {12}{5} b c^5 d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )-\frac {6}{5} b^2 c^5 d^3 \text {Li}_2(-c x)+\frac {6}{5} b^2 c^5 d^3 \text {Li}_2(c x)+\frac {1}{4} \left (b^2 c^3 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 (b^2 c^4 d^3\right ) \int \frac {1}{x^2 \left (1-c^2 x^2\right )} \, dx+\frac {1}{4} \left (5 b^2 c^5 d^3\right ) \text {Subst}\left (\int \frac {1}{x \left (1-c^2 x\right )} \, dx,x,x^2\right )+\frac {1}{5} \left (12 b^2 c^5 d^3\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-c x}\right )+\frac {1}{5} \left (6 b^2 c^6 d^3\right ) \int \frac {1}{1-c^2 x^2} \, dx\\ &=-\frac {b^2 c^2 d^3}{30 x^3}-\frac {13 b^2 c^4 d^3}{10 x}+\frac {6}{5} b^2 c^5 d^3 \tanh ^{-1}(c x)-\frac {b c d^3 \left (a+b \tanh ^{-1}(c x)\right )}{10 x^4}-\frac {b c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{2 x^3}-\frac {6 b c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{5 x^2}-\frac {5 b c^4 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{2 x}-\frac {d^3 (1+c x)^4 \left (a+b \tanh ^{-1}(c x)\right )^2}{5 x^5}+\frac {c d^3 (1+c x)^4 \left (a+b \tanh ^{-1}(c x)\right )^2}{20 x^4}+\frac {12}{5} a b c^5 d^3 \log (x)+\frac {12}{5} b c^5 d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )-\frac {6}{5} b^2 c^5 d^3 \text {Li}_2(-c x)+\frac {6}{5} b^2 c^5 d^3 \text {Li}_2(c x)+\frac {6}{5} b^2 c^5 d^3 \text {Li}_2\left (1-\frac {2}{1-c x}\right )+\frac {1}{4} \left (b^2 c^3 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}{4} \left (5 b^2 c^5 d^3\right ) \text {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )+\frac {1}{10} \left (b^2 c^6 d^3\right ) \int \frac {1}{1-c^2 x^2} \, dx+\frac {1}{4} \left (5 b^2 c^7 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}{30 x^3}-\frac {b^2 c^3 d^3}{4 x^2}-\frac {13 b^2 c^4 d^3}{10 x}+\frac {13}{10} b^2 c^5 d^3 \tanh ^{-1}(c x)-\frac {b c d^3 \left (a+b \tanh ^{-1}(c x)\right )}{10 x^4}-\frac {b c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{2 x^3}-\frac {6 b c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{5 x^2}-\frac {5 b c^4 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{2 x}-\frac {d^3 (1+c x)^4 \left (a+b \tanh ^{-1}(c x)\right )^2}{5 x^5}+\frac {c d^3 (1+c x)^4 \left (a+b \tanh ^{-1}(c x)\right )^2}{20 x^4}+\frac {12}{5} a b c^5 d^3 \log (x)+3 b^2 c^5 d^3 \log (x)+\frac {12}{5} b c^5 d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )-\frac {3}{2} b^2 c^5 d^3 \log \left (1-c^2 x^2\right )-\frac {6}{5} b^2 c^5 d^3 \text {Li}_2(-c x)+\frac {6}{5} b^2 c^5 d^3 \text {Li}_2(c x)+\frac {6}{5} b^2 c^5 d^3 \text {Li}_2\left (1-\frac {2}{1-c x}\right )\\ \end {align*}

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Mathematica [A]
time = 0.84, size = 372, normalized size = 1.06 \begin {gather*} -\frac {d^3 \left (12 a^2+45 a^2 c x+6 a b c x+60 a^2 c^2 x^2+30 a b c^2 x^2+2 b^2 c^2 x^2+30 a^2 c^3 x^3+72 a b c^3 x^3+15 b^2 c^3 x^3+150 a b c^4 x^4+78 b^2 c^4 x^4-15 b^2 c^5 x^5+3 b^2 \left (4+15 c x+20 c^2 x^2+10 c^3 x^3-49 c^5 x^5\right ) \tanh ^{-1}(c x)^2+6 b \tanh ^{-1}(c x) \left (a \left (4+15 c x+20 c^2 x^2+10 c^3 x^3\right )+b c x \left (1+5 c x+12 c^2 x^2+25 c^3 x^3-13 c^4 x^4\right )-24 b c^5 x^5 \log \left (1-e^{-2 \tanh ^{-1}(c x)}\right )\right )-144 a b c^5 x^5 \log (c x)+75 a b c^5 x^5 \log (1-c x)-75 a b c^5 x^5 \log (1+c x)-180 b^2 c^5 x^5 \log \left (\frac {c x}{\sqrt {1-c^2 x^2}}\right )+72 a b c^5 x^5 \log \left (1-c^2 x^2\right )+72 b^2 c^5 x^5 \text {PolyLog}\left (2,e^{-2 \tanh ^{-1}(c x)}\right )\right )}{60 x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((d + c*d*x)^3*(a + b*ArcTanh[c*x])^2)/x^6,x]

[Out]

-1/60*(d^3*(12*a^2 + 45*a^2*c*x + 6*a*b*c*x + 60*a^2*c^2*x^2 + 30*a*b*c^2*x^2 + 2*b^2*c^2*x^2 + 30*a^2*c^3*x^3
+ 72*a*b*c^3*x^3 + 15*b^2*c^3*x^3 + 150*a*b*c^4*x^4 + 78*b^2*c^4*x^4 - 15*b^2*c^5*x^5 + 3*b^2*(4 + 15*c*x + 2
0*c^2*x^2 + 10*c^3*x^3 - 49*c^5*x^5)*ArcTanh[c*x]^2 + 6*b*ArcTanh[c*x]*(a*(4 + 15*c*x + 20*c^2*x^2 + 10*c^3*x^
3) + b*c*x*(1 + 5*c*x + 12*c^2*x^2 + 25*c^3*x^3 - 13*c^4*x^4) - 24*b*c^5*x^5*Log[1 - E^(-2*ArcTanh[c*x])]) - 1
44*a*b*c^5*x^5*Log[c*x] + 75*a*b*c^5*x^5*Log[1 - c*x] - 75*a*b*c^5*x^5*Log[1 + c*x] - 180*b^2*c^5*x^5*Log[(c*x
)/Sqrt[1 - c^2*x^2]] + 72*a*b*c^5*x^5*Log[1 - c^2*x^2] + 72*b^2*c^5*x^5*PolyLog[2, E^(-2*ArcTanh[c*x])]))/x^5

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(643\) vs. \(2(320)=640\).
time = 0.73, size = 644, normalized size = 1.83

method	result	size
derivativedivides	\(c^{5} \left (d^{3} a^{2} \left (-\frac {1}{2 c^{2} x^{2}}-\frac {1}{5 c^{5} x^{5}}-\frac {3}{4 c^{4} x^{4}}-\frac {1}{c^{3} x^{3}}\right )-\frac {3 d^{3} a b \arctanh \left (c x \right )}{2 c^{4} x^{4}}-\frac {d^{3} b^{2} \arctanh \left (c x \right )}{10 c^{4} x^{4}}-\frac {49 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 )}{20}-\frac {3 d^{3} b^{2} \arctanh \left (c x \right )^{2}}{4 c^{4} x^{4}}-\frac {d^{3} a b}{10 c^{4} x^{4}}-\frac {17 d^{3} b^{2} \ln \left (c x +1\right )}{20}-\frac {d^{3} b^{2} \arctanh \left (c x \right )^{2}}{5 c^{5} x^{5}}-\frac {6 d^{3} b^{2} \dilog \left (c x \right )}{5}-\frac {6 d^{3} b^{2} \dilog \left (c x +1\right )}{5}-\frac {49 d^{3} b^{2} \ln \left (c x -1\right )^{2}}{80}-\frac {d^{3} b^{2} \ln \left (c x +1\right )^{2}}{80}-\frac {43 d^{3} b^{2} \ln \left (c x -1\right )}{20}+\frac {49 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 )}{40}-\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 )}{40}-\frac {49 d^{3} a b \ln \left (c x -1\right )}{20}+\frac {d^{3} a b \ln \left (c x +1\right )}{20}-\frac {d^{3} b^{2} \arctanh \left (c x \right )}{2 c^{3} x^{3}}+3 d^{3} b^{2} \ln \left (c x \right )-\frac {d^{3} a b}{2 c^{3} x^{3}}-\frac {2 d^{3} a b \arctanh \left (c x \right )}{5 c^{5} x^{5}}+\frac {6 d^{3} b^{2} \dilog \left (\frac {c x}{2}+\frac {1}{2}\right )}{5}-\frac {d^{3} b^{2}}{30 c^{3} x^{3}}-\frac {d^{3} b^{2}}{4 c^{2} x^{2}}-\frac {13 d^{3} b^{2}}{10 c x}-\frac {6 d^{3} b^{2} \ln \left (c x \right ) \ln \left (c x +1\right )}{5}-\frac {6 d^{3} a b}{5 c^{2} x^{2}}-\frac {2 d^{3} a b \arctanh \left (c x \right )}{c^{3} x^{3}}+\frac {12 d^{3} b^{2} \arctanh \left (c x \right ) \ln \left (c x \right )}{5}-\frac {6 d^{3} b^{2} \arctanh \left (c x \right )}{5 c^{2} x^{2}}-\frac {d^{3} b^{2} \arctanh \left (c x \right )^{2}}{c^{3} x^{3}}-\frac {d^{3} b^{2} \arctanh \left (c x \right )^{2}}{2 c^{2} x^{2}}-\frac {5 d^{3} b^{2} \arctanh \left (c x \right )}{2 c x}-\frac {5 d^{3} a b}{2 c x}+\frac {12 d^{3} a b \ln \left (c x \right )}{5}-\frac {d^{3} a b \arctanh \left (c x \right )}{c^{2} x^{2}}\right )\)	\(644\)
default	\(c^{5} \left (d^{3} a^{2} \left (-\frac {1}{2 c^{2} x^{2}}-\frac {1}{5 c^{5} x^{5}}-\frac {3}{4 c^{4} x^{4}}-\frac {1}{c^{3} x^{3}}\right )-\frac {3 d^{3} a b \arctanh \left (c x \right )}{2 c^{4} x^{4}}-\frac {d^{3} b^{2} \arctanh \left (c x \right )}{10 c^{4} x^{4}}-\frac {49 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 )}{20}-\frac {3 d^{3} b^{2} \arctanh \left (c x \right )^{2}}{4 c^{4} x^{4}}-\frac {d^{3} a b}{10 c^{4} x^{4}}-\frac {17 d^{3} b^{2} \ln \left (c x +1\right )}{20}-\frac {d^{3} b^{2} \arctanh \left (c x \right )^{2}}{5 c^{5} x^{5}}-\frac {6 d^{3} b^{2} \dilog \left (c x \right )}{5}-\frac {6 d^{3} b^{2} \dilog \left (c x +1\right )}{5}-\frac {49 d^{3} b^{2} \ln \left (c x -1\right )^{2}}{80}-\frac {d^{3} b^{2} \ln \left (c x +1\right )^{2}}{80}-\frac {43 d^{3} b^{2} \ln \left (c x -1\right )}{20}+\frac {49 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 )}{40}-\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 )}{40}-\frac {49 d^{3} a b \ln \left (c x -1\right )}{20}+\frac {d^{3} a b \ln \left (c x +1\right )}{20}-\frac {d^{3} b^{2} \arctanh \left (c x \right )}{2 c^{3} x^{3}}+3 d^{3} b^{2} \ln \left (c x \right )-\frac {d^{3} a b}{2 c^{3} x^{3}}-\frac {2 d^{3} a b \arctanh \left (c x \right )}{5 c^{5} x^{5}}+\frac {6 d^{3} b^{2} \dilog \left (\frac {c x}{2}+\frac {1}{2}\right )}{5}-\frac {d^{3} b^{2}}{30 c^{3} x^{3}}-\frac {d^{3} b^{2}}{4 c^{2} x^{2}}-\frac {13 d^{3} b^{2}}{10 c x}-\frac {6 d^{3} b^{2} \ln \left (c x \right ) \ln \left (c x +1\right )}{5}-\frac {6 d^{3} a b}{5 c^{2} x^{2}}-\frac {2 d^{3} a b \arctanh \left (c x \right )}{c^{3} x^{3}}+\frac {12 d^{3} b^{2} \arctanh \left (c x \right ) \ln \left (c x \right )}{5}-\frac {6 d^{3} b^{2} \arctanh \left (c x \right )}{5 c^{2} x^{2}}-\frac {d^{3} b^{2} \arctanh \left (c x \right )^{2}}{c^{3} x^{3}}-\frac {d^{3} b^{2} \arctanh \left (c x \right )^{2}}{2 c^{2} x^{2}}-\frac {5 d^{3} b^{2} \arctanh \left (c x \right )}{2 c x}-\frac {5 d^{3} a b}{2 c x}+\frac {12 d^{3} a b \ln \left (c x \right )}{5}-\frac {d^{3} a b \arctanh \left (c x \right )}{c^{2} x^{2}}\right )\)	\(644\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*d*x+d)^3*(a+b*arctanh(c*x))^2/x^6,x,method=_RETURNVERBOSE)

[Out]

c^5*(d^3*a^2*(-1/2/c^2/x^2-1/5/c^5/x^5-3/4/c^4/x^4-1/c^3/x^3)-2/5*d^3*a*b*arctanh(c*x)/c^5/x^5-1/10*d^3*b^2*ar
ctanh(c*x)/c^4/x^4-1/5*d^3*b^2*arctanh(c*x)^2/c^5/x^5-1/10*d^3*a*b/c^4/x^4-d^3*a*b*arctanh(c*x)/c^2/x^2-3/2*d^
3*a*b*arctanh(c*x)/c^4/x^4-17/20*d^3*b^2*ln(c*x+1)+6/5*d^3*b^2*dilog(1/2*c*x+1/2)-49/80*d^3*b^2*ln(c*x-1)^2-1/
80*d^3*b^2*ln(c*x+1)^2-43/20*d^3*b^2*ln(c*x-1)-49/20*d^3*b^2*arctanh(c*x)*ln(c*x-1)+1/20*d^3*b^2*arctanh(c*x)*
ln(c*x+1)+49/40*d^3*b^2*ln(c*x-1)*ln(1/2*c*x+1/2)+1/40*d^3*b^2*ln(-1/2*c*x+1/2)*ln(c*x+1)-1/40*d^3*b^2*ln(-1/2
*c*x+1/2)*ln(1/2*c*x+1/2)-49/20*d^3*a*b*ln(c*x-1)+1/20*d^3*a*b*ln(c*x+1)-6/5*d^3*b^2*dilog(c*x)-6/5*d^3*b^2*di
log(c*x+1)+3*d^3*b^2*ln(c*x)-1/2*d^3*a*b/c^3/x^3-3/4*d^3*b^2*arctanh(c*x)^2/c^4/x^4-1/2*d^3*b^2*arctanh(c*x)/c
^3/x^3-1/30*d^3*b^2/c^3/x^3-1/4*d^3*b^2/c^2/x^2-13/10*d^3*b^2/c/x+12/5*d^3*b^2*arctanh(c*x)*ln(c*x)-6/5*d^3*b^
2*ln(c*x)*ln(c*x+1)-6/5*d^3*b^2*arctanh(c*x)/c^2/x^2-d^3*b^2*arctanh(c*x)^2/c^3/x^3-6/5*d^3*a*b/c^2/x^2-2*d^3*
a*b*arctanh(c*x)/c^3/x^3-1/2*d^3*b^2*arctanh(c*x)^2/c^2/x^2-5/2*d^3*b^2*arctanh(c*x)/c/x-5/2*d^3*a*b/c/x+12/5*
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. 783 vs. \(2 (315) = 630\).
time = 0.66, size = 783, normalized size = 2.22 \begin {gather*} -\frac {6}{5} \, {\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^{5} d^{3} - \frac {6}{5} \, {\left (\log \left (c x\right ) \log \left (-c x + 1\right ) + {\rm Li}_2\left (-c x + 1\right )\right )} b^{2} c^{5} d^{3} + \frac {6}{5} \, {\left (\log \left (c x + 1\right ) \log \left (-c x\right ) + {\rm Li}_2\left (c x + 1\right )\right )} b^{2} c^{5} d^{3} - \frac {17}{20} \, b^{2} c^{5} d^{3} \log \left (c x + 1\right ) - \frac {43}{20} \, b^{2} c^{5} d^{3} \log \left (c x - 1\right ) + 3 \, b^{2} c^{5} d^{3} \log \left (x\right ) + \frac {1}{2} \, {\left ({\left (c \log \left (c x + 1\right ) - c \log \left (c x - 1\right ) - \frac {2}{x}\right )} c - \frac {2 \, \operatorname {artanh}\left (c x\right )}{x^{2}}\right )} a b c^{3} d^{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^{2} 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 d^{3} - \frac {1}{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 d^{3} - \frac {a^{2} c^{3} d^{3}}{2 \, x^{2}} - \frac {a^{2} c^{2} d^{3}}{x^{3}} - \frac {3 \, a^{2} c d^{3}}{4 \, x^{4}} - \frac {a^{2} d^{3}}{5 \, x^{5}} - \frac {312 \, b^{2} c^{4} d^{3} x^{4} + 60 \, b^{2} c^{3} d^{3} x^{3} + 8 \, b^{2} c^{2} d^{3} x^{2} - 3 \, {\left (b^{2} c^{5} d^{3} x^{5} - 10 \, b^{2} c^{3} d^{3} x^{3} - 20 \, b^{2} c^{2} d^{3} x^{2} - 15 \, b^{2} c d^{3} x - 4 \, b^{2} d^{3}\right )} \log \left (c x + 1\right )^{2} - 3 \, {\left (49 \, b^{2} c^{5} d^{3} x^{5} - 10 \, b^{2} c^{3} d^{3} x^{3} - 20 \, b^{2} c^{2} d^{3} x^{2} - 15 \, b^{2} c d^{3} x - 4 \, b^{2} d^{3}\right )} \log \left (-c x + 1\right )^{2} + 12 \, {\left (25 \, b^{2} c^{4} d^{3} x^{4} + 12 \, b^{2} c^{3} d^{3} x^{3} + 5 \, b^{2} c^{2} d^{3} x^{2} + b^{2} c d^{3} x\right )} \log \left (c x + 1\right ) - 6 \, {\left (50 \, b^{2} c^{4} d^{3} x^{4} + 24 \, b^{2} c^{3} d^{3} x^{3} + 10 \, b^{2} c^{2} d^{3} x^{2} + 2 \, b^{2} c d^{3} x - {\left (b^{2} c^{5} d^{3} x^{5} - 10 \, b^{2} c^{3} d^{3} x^{3} - 20 \, b^{2} c^{2} d^{3} x^{2} - 15 \, b^{2} c d^{3} x - 4 \, b^{2} 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^6,x, algorithm="maxima")

[Out]

-6/5*(log(c*x + 1)*log(-1/2*c*x + 1/2) + dilog(1/2*c*x + 1/2))*b^2*c^5*d^3 - 6/5*(log(c*x)*log(-c*x + 1) + dil
og(-c*x + 1))*b^2*c^5*d^3 + 6/5*(log(c*x + 1)*log(-c*x) + dilog(c*x + 1))*b^2*c^5*d^3 - 17/20*b^2*c^5*d^3*log(
c*x + 1) - 43/20*b^2*c^5*d^3*log(c*x - 1) + 3*b^2*c^5*d^3*log(x) + 1/2*((c*log(c*x + 1) - c*log(c*x - 1) - 2/x
)*c - 2*arctanh(c*x)/x^2)*a*b*c^3*d^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^2*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*d^3 - 1/10*((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*
d^3 - 1/2*a^2*c^3*d^3/x^2 - a^2*c^2*d^3/x^3 - 3/4*a^2*c*d^3/x^4 - 1/5*a^2*d^3/x^5 - 1/240*(312*b^2*c^4*d^3*x^4
+ 60*b^2*c^3*d^3*x^3 + 8*b^2*c^2*d^3*x^2 - 3*(b^2*c^5*d^3*x^5 - 10*b^2*c^3*d^3*x^3 - 20*b^2*c^2*d^3*x^2 - 15*
b^2*c*d^3*x - 4*b^2*d^3)*log(c*x + 1)^2 - 3*(49*b^2*c^5*d^3*x^5 - 10*b^2*c^3*d^3*x^3 - 20*b^2*c^2*d^3*x^2 - 15
*b^2*c*d^3*x - 4*b^2*d^3)*log(-c*x + 1)^2 + 12*(25*b^2*c^4*d^3*x^4 + 12*b^2*c^3*d^3*x^3 + 5*b^2*c^2*d^3*x^2 +
b^2*c*d^3*x)*log(c*x + 1) - 6*(50*b^2*c^4*d^3*x^4 + 24*b^2*c^3*d^3*x^3 + 10*b^2*c^2*d^3*x^2 + 2*b^2*c*d^3*x -
(b^2*c^5*d^3*x^5 - 10*b^2*c^3*d^3*x^3 - 20*b^2*c^2*d^3*x^2 - 15*b^2*c*d^3*x - 4*b^2*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^6,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^6, 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^{6}}\, dx + \int \frac {3 a^{2} c}{x^{5}}\, dx + \int \frac {3 a^{2} c^{2}}{x^{4}}\, dx + \int \frac {a^{2} c^{3}}{x^{3}}\, dx + \int \frac {b^{2} \operatorname {atanh}^{2}{\left (c x \right )}}{x^{6}}\, dx + \int \frac {2 a b \operatorname {atanh}{\left (c x \right )}}{x^{6}}\, dx + \int \frac {3 b^{2} c \operatorname {atanh}^{2}{\left (c x \right )}}{x^{5}}\, dx + \int \frac {3 b^{2} c^{2} \operatorname {atanh}^{2}{\left (c x \right )}}{x^{4}}\, dx + \int \frac {b^{2} c^{3} \operatorname {atanh}^{2}{\left (c x \right )}}{x^{3}}\, dx + \int \frac {6 a b c \operatorname {atanh}{\left (c x \right )}}{x^{5}}\, dx + \int \frac {6 a b c^{2} \operatorname {atanh}{\left (c x \right )}}{x^{4}}\, dx + \int \frac {2 a b c^{3} \operatorname {atanh}{\left (c x \right )}}{x^{3}}\, 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**6,x)

[Out]

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

[Out]

integrate((c*d*x + d)^3*(b*arctanh(c*x) + a)^2/x^6, 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^6} \,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^6,x)

[Out]

int(((a + b*atanh(c*x))^2*(d + c*d*x)^3)/x^6, x)

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