3.1.83 \(\int \frac {(d+c d x)^2 (a+b \tanh ^{-1}(c x))^2}{x^4} \, dx\) [83]

Optimal. Leaf size=244 \[ -\frac {b^2 c^2 d^2}{3 x}+\frac {1}{3} b^2 c^3 d^2 \tanh ^{-1}(c x)-\frac {b c d^2 \left (a+b \tanh ^{-1}(c x)\right )}{3 x^2}-\frac {2 b c^2 d^2 \left (a+b \tanh ^{-1}(c x)\right )}{x}-\frac {d^2 (1+c x)^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{3 x^3}+\frac {8}{3} a b c^3 d^2 \log (x)+2 b^2 c^3 d^2 \log (x)+\frac {8}{3} b c^3 d^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )-b^2 c^3 d^2 \log \left (1-c^2 x^2\right )-\frac {4}{3} b^2 c^3 d^2 \text {PolyLog}(2,-c x)+\frac {4}{3} b^2 c^3 d^2 \text {PolyLog}(2,c x)+\frac {4}{3} b^2 c^3 d^2 \text {PolyLog}\left (2,1-\frac {2}{1-c x}\right ) \]

[Out]

-1/3*b^2*c^2*d^2/x+1/3*b^2*c^3*d^2*arctanh(c*x)-1/3*b*c*d^2*(a+b*arctanh(c*x))/x^2-2*b*c^2*d^2*(a+b*arctanh(c*
x))/x-1/3*d^2*(c*x+1)^3*(a+b*arctanh(c*x))^2/x^3+8/3*a*b*c^3*d^2*ln(x)+2*b^2*c^3*d^2*ln(x)+8/3*b*c^3*d^2*(a+b*
arctanh(c*x))*ln(2/(-c*x+1))-b^2*c^3*d^2*ln(-c^2*x^2+1)-4/3*b^2*c^3*d^2*polylog(2,-c*x)+4/3*b^2*c^3*d^2*polylo
g(2,c*x)+4/3*b^2*c^3*d^2*polylog(2,1-2/(-c*x+1))

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Rubi [A]
time = 0.19, antiderivative size = 244, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 13, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.591, Rules used = {37, 6085, 6037, 331, 212, 272, 36, 29, 31, 6031, 6055, 2449, 2352} \begin {gather*} \frac {8}{3} a b c^3 d^2 \log (x)+\frac {8}{3} b c^3 d^2 \log \left (\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )-\frac {2 b c^2 d^2 \left (a+b \tanh ^{-1}(c x)\right )}{x}-\frac {d^2 (c x+1)^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{3 x^3}-\frac {b c d^2 \left (a+b \tanh ^{-1}(c x)\right )}{3 x^2}-\frac {4}{3} b^2 c^3 d^2 \text {Li}_2(-c x)+\frac {4}{3} b^2 c^3 d^2 \text {Li}_2(c x)+\frac {4}{3} b^2 c^3 d^2 \text {Li}_2\left (1-\frac {2}{1-c x}\right )+2 b^2 c^3 d^2 \log (x)+\frac {1}{3} b^2 c^3 d^2 \tanh ^{-1}(c x)-\frac {b^2 c^2 d^2}{3 x}-b^2 c^3 d^2 \log \left (1-c^2 x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*(b^2*c^2*d^2)/x + (b^2*c^3*d^2*ArcTanh[c*x])/3 - (b*c*d^2*(a + b*ArcTanh[c*x]))/(3*x^2) - (2*b*c^2*d^2*(a
+ b*ArcTanh[c*x]))/x - (d^2*(1 + c*x)^3*(a + b*ArcTanh[c*x])^2)/(3*x^3) + (8*a*b*c^3*d^2*Log[x])/3 + 2*b^2*c^
3*d^2*Log[x] + (8*b*c^3*d^2*(a + b*ArcTanh[c*x])*Log[2/(1 - c*x)])/3 - b^2*c^3*d^2*Log[1 - c^2*x^2] - (4*b^2*c
^3*d^2*PolyLog[2, -(c*x)])/3 + (4*b^2*c^3*d^2*PolyLog[2, c*x])/3 + (4*b^2*c^3*d^2*PolyLog[2, 1 - 2/(1 - c*x)])
/3

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

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Mathematica [A]
time = 0.39, size = 270, normalized size = 1.11 \begin {gather*} -\frac {d^2 \left (a^2+3 a^2 c x+a b c x+3 a^2 c^2 x^2+6 a b c^2 x^2+b^2 c^2 x^2+b^2 \left (1+3 c x+3 c^2 x^2-7 c^3 x^3\right ) \tanh ^{-1}(c x)^2+b \tanh ^{-1}(c x) \left (b c x \left (1+6 c x-c^2 x^2\right )+a \left (2+6 c x+6 c^2 x^2\right )-8 b c^3 x^3 \log \left (1-e^{-2 \tanh ^{-1}(c x)}\right )\right )-8 a b c^3 x^3 \log (c x)+3 a b c^3 x^3 \log (1-c x)-3 a b c^3 x^3 \log (1+c x)-6 b^2 c^3 x^3 \log \left (\frac {c x}{\sqrt {1-c^2 x^2}}\right )+4 a b c^3 x^3 \log \left (1-c^2 x^2\right )+4 b^2 c^3 x^3 \text {PolyLog}\left (2,e^{-2 \tanh ^{-1}(c x)}\right )\right )}{3 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*(d^2*(a^2 + 3*a^2*c*x + a*b*c*x + 3*a^2*c^2*x^2 + 6*a*b*c^2*x^2 + b^2*c^2*x^2 + b^2*(1 + 3*c*x + 3*c^2*x^
2 - 7*c^3*x^3)*ArcTanh[c*x]^2 + b*ArcTanh[c*x]*(b*c*x*(1 + 6*c*x - c^2*x^2) + a*(2 + 6*c*x + 6*c^2*x^2) - 8*b*
c^3*x^3*Log[1 - E^(-2*ArcTanh[c*x])]) - 8*a*b*c^3*x^3*Log[c*x] + 3*a*b*c^3*x^3*Log[1 - c*x] - 3*a*b*c^3*x^3*Lo
g[1 + c*x] - 6*b^2*c^3*x^3*Log[(c*x)/Sqrt[1 - c^2*x^2]] + 4*a*b*c^3*x^3*Log[1 - c^2*x^2] + 4*b^2*c^3*x^3*PolyL
og[2, E^(-2*ArcTanh[c*x])]))/x^3

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(508\) vs. \(2(226)=452\).
time = 0.65, size = 509, normalized size = 2.09

method	result	size
derivativedivides	\(c^{3} \left (d^{2} a^{2} \left (-\frac {1}{c^{2} x^{2}}-\frac {1}{3 c^{3} x^{3}}-\frac {1}{c x}\right )+\frac {b^{2} \ln \left (c x +1\right )^{2} d^{2}}{12}-\frac {2 d^{2} a b \arctanh \left (c x \right )}{3 c^{3} x^{3}}+\frac {8 d^{2} b^{2} \arctanh \left (c x \right ) \ln \left (c x \right )}{3}-\frac {b^{2} \arctanh \left (c x \right ) \ln \left (c x +1\right ) d^{2}}{3}-\frac {7 b^{2} \arctanh \left (c x \right ) \ln \left (c x -1\right ) d^{2}}{3}-\frac {d^{2} b^{2} \arctanh \left (c x \right )^{2}}{3 c^{3} x^{3}}-\frac {2 d^{2} a b \arctanh \left (c x \right )}{c x}-\frac {4 d^{2} b^{2} \dilog \left (c x +1\right )}{3}-\frac {4 d^{2} b^{2} \dilog \left (c x \right )}{3}-\frac {4 d^{2} b^{2} \ln \left (c x \right ) \ln \left (c x +1\right )}{3}-\frac {d^{2} b^{2}}{3 c x}-\frac {7 b^{2} \ln \left (c x -1\right )^{2} d^{2}}{12}-\frac {7 a b \ln \left (c x -1\right ) d^{2}}{3}-\frac {a b \ln \left (c x +1\right ) d^{2}}{3}+\frac {7 b^{2} \ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right ) d^{2}}{6}-\frac {b^{2} \ln \left (c x +1\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) d^{2}}{6}+\frac {b^{2} \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right ) d^{2}}{6}-\frac {7 d^{2} b^{2} \ln \left (c x -1\right )}{6}-\frac {5 d^{2} b^{2} \ln \left (c x +1\right )}{6}+2 d^{2} b^{2} \ln \left (c x \right )-\frac {d^{2} a b}{3 c^{2} x^{2}}+\frac {4 b^{2} \dilog \left (\frac {c x}{2}+\frac {1}{2}\right ) d^{2}}{3}-\frac {d^{2} b^{2} \arctanh \left (c x \right )}{3 c^{2} x^{2}}-\frac {2 d^{2} a b \arctanh \left (c x \right )}{c^{2} x^{2}}+\frac {8 d^{2} a b \ln \left (c x \right )}{3}-\frac {d^{2} b^{2} \arctanh \left (c x \right )^{2}}{c x}-\frac {2 d^{2} a b}{c x}-\frac {d^{2} b^{2} \arctanh \left (c x \right )^{2}}{c^{2} x^{2}}-\frac {2 d^{2} b^{2} \arctanh \left (c x \right )}{c x}\right )\)	\(509\)
default	\(c^{3} \left (d^{2} a^{2} \left (-\frac {1}{c^{2} x^{2}}-\frac {1}{3 c^{3} x^{3}}-\frac {1}{c x}\right )+\frac {b^{2} \ln \left (c x +1\right )^{2} d^{2}}{12}-\frac {2 d^{2} a b \arctanh \left (c x \right )}{3 c^{3} x^{3}}+\frac {8 d^{2} b^{2} \arctanh \left (c x \right ) \ln \left (c x \right )}{3}-\frac {b^{2} \arctanh \left (c x \right ) \ln \left (c x +1\right ) d^{2}}{3}-\frac {7 b^{2} \arctanh \left (c x \right ) \ln \left (c x -1\right ) d^{2}}{3}-\frac {d^{2} b^{2} \arctanh \left (c x \right )^{2}}{3 c^{3} x^{3}}-\frac {2 d^{2} a b \arctanh \left (c x \right )}{c x}-\frac {4 d^{2} b^{2} \dilog \left (c x +1\right )}{3}-\frac {4 d^{2} b^{2} \dilog \left (c x \right )}{3}-\frac {4 d^{2} b^{2} \ln \left (c x \right ) \ln \left (c x +1\right )}{3}-\frac {d^{2} b^{2}}{3 c x}-\frac {7 b^{2} \ln \left (c x -1\right )^{2} d^{2}}{12}-\frac {7 a b \ln \left (c x -1\right ) d^{2}}{3}-\frac {a b \ln \left (c x +1\right ) d^{2}}{3}+\frac {7 b^{2} \ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right ) d^{2}}{6}-\frac {b^{2} \ln \left (c x +1\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) d^{2}}{6}+\frac {b^{2} \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right ) d^{2}}{6}-\frac {7 d^{2} b^{2} \ln \left (c x -1\right )}{6}-\frac {5 d^{2} b^{2} \ln \left (c x +1\right )}{6}+2 d^{2} b^{2} \ln \left (c x \right )-\frac {d^{2} a b}{3 c^{2} x^{2}}+\frac {4 b^{2} \dilog \left (\frac {c x}{2}+\frac {1}{2}\right ) d^{2}}{3}-\frac {d^{2} b^{2} \arctanh \left (c x \right )}{3 c^{2} x^{2}}-\frac {2 d^{2} a b \arctanh \left (c x \right )}{c^{2} x^{2}}+\frac {8 d^{2} a b \ln \left (c x \right )}{3}-\frac {d^{2} b^{2} \arctanh \left (c x \right )^{2}}{c x}-\frac {2 d^{2} a b}{c x}-\frac {d^{2} b^{2} \arctanh \left (c x \right )^{2}}{c^{2} x^{2}}-\frac {2 d^{2} b^{2} \arctanh \left (c x \right )}{c x}\right )\)	\(509\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c^3*(d^2*a^2*(-1/c^2/x^2-1/3/c^3/x^3-1/c/x)+1/12*b^2*ln(c*x+1)^2*d^2-2/3*d^2*a*b*arctanh(c*x)/c^3/x^3-2*d^2*a*
b*arctanh(c*x)/c/x-2*d^2*a*b*arctanh(c*x)/c^2/x^2-1/3*d^2*b^2*arctanh(c*x)*ln(c*x+1)+8/3*d^2*b^2*arctanh(c*x)*
ln(c*x)-4/3*d^2*b^2*ln(c*x)*ln(c*x+1)-1/3*d^2*b^2/c/x-7/12*b^2*ln(c*x-1)^2*d^2-7/3*d^2*b^2*arctanh(c*x)*ln(c*x
-1)-7/3*a*b*ln(c*x-1)*d^2-1/3*a*b*ln(c*x+1)*d^2+7/6*b^2*ln(c*x-1)*ln(1/2*c*x+1/2)*d^2-1/6*b^2*ln(c*x+1)*ln(-1/
2*c*x+1/2)*d^2+1/6*b^2*ln(-1/2*c*x+1/2)*ln(1/2*c*x+1/2)*d^2+4/3*d^2*b^2*dilog(1/2*c*x+1/2)-7/6*d^2*b^2*ln(c*x-
1)-5/6*d^2*b^2*ln(c*x+1)-4/3*d^2*b^2*dilog(c*x+1)+2*d^2*b^2*ln(c*x)-4/3*d^2*b^2*dilog(c*x)-1/3*d^2*a*b/c^2/x^2
-1/3*d^2*b^2*arctanh(c*x)^2/c^3/x^3-1/3*d^2*b^2*arctanh(c*x)/c^2/x^2-d^2*b^2*arctanh(c*x)^2/c/x+8/3*d^2*a*b*ln
(c*x)-d^2*b^2*arctanh(c*x)^2/c^2/x^2-2*d^2*b^2/c/x*arctanh(c*x)-2*d^2*a*b/c/x)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 555 vs. \(2 (221) = 442\).
time = 0.65, size = 555, normalized size = 2.27 \begin {gather*} -\frac {4}{3} \, {\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^{3} d^{2} - \frac {4}{3} \, {\left (\log \left (c x\right ) \log \left (-c x + 1\right ) + {\rm Li}_2\left (-c x + 1\right )\right )} b^{2} c^{3} d^{2} + \frac {4}{3} \, {\left (\log \left (c x + 1\right ) \log \left (-c x\right ) + {\rm Li}_2\left (c x + 1\right )\right )} b^{2} c^{3} d^{2} - \frac {5}{6} \, b^{2} c^{3} d^{2} \log \left (c x + 1\right ) - \frac {7}{6} \, b^{2} c^{3} d^{2} \log \left (c x - 1\right ) + 2 \, b^{2} c^{3} d^{2} \log \left (x\right ) - {\left (c {\left (\log \left (c^{2} x^{2} - 1\right ) - \log \left (x^{2}\right )\right )} + \frac {2 \, \operatorname {artanh}\left (c x\right )}{x}\right )} a b c^{2} d^{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 d^{2} - \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 d^{2} - \frac {a^{2} c^{2} d^{2}}{x} - \frac {a^{2} c d^{2}}{x^{2}} - \frac {a^{2} d^{2}}{3 \, x^{3}} - \frac {4 \, b^{2} c^{2} d^{2} x^{2} + {\left (b^{2} c^{3} d^{2} x^{3} + 3 \, b^{2} c^{2} d^{2} x^{2} + 3 \, b^{2} c d^{2} x + b^{2} d^{2}\right )} \log \left (c x + 1\right )^{2} - {\left (7 \, b^{2} c^{3} d^{2} x^{3} - 3 \, b^{2} c^{2} d^{2} x^{2} - 3 \, b^{2} c d^{2} x - b^{2} d^{2}\right )} \log \left (-c x + 1\right )^{2} + 2 \, {\left (6 \, b^{2} c^{2} d^{2} x^{2} + b^{2} c d^{2} x\right )} \log \left (c x + 1\right ) - 2 \, {\left (6 \, b^{2} c^{2} d^{2} x^{2} + b^{2} c d^{2} x + {\left (b^{2} c^{3} d^{2} x^{3} + 3 \, b^{2} c^{2} d^{2} x^{2} + 3 \, b^{2} c d^{2} x + b^{2} d^{2}\right )} \log \left (c x + 1\right )\right )} \log \left (-c x + 1\right )}{12 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/3*(log(c*x + 1)*log(-1/2*c*x + 1/2) + dilog(1/2*c*x + 1/2))*b^2*c^3*d^2 - 4/3*(log(c*x)*log(-c*x + 1) + dil
og(-c*x + 1))*b^2*c^3*d^2 + 4/3*(log(c*x + 1)*log(-c*x) + dilog(c*x + 1))*b^2*c^3*d^2 - 5/6*b^2*c^3*d^2*log(c*
x + 1) - 7/6*b^2*c^3*d^2*log(c*x - 1) + 2*b^2*c^3*d^2*log(x) - (c*(log(c^2*x^2 - 1) - log(x^2)) + 2*arctanh(c*
x)/x)*a*b*c^2*d^2 + ((c*log(c*x + 1) - c*log(c*x - 1) - 2/x)*c - 2*arctanh(c*x)/x^2)*a*b*c*d^2 - 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*d^2 - a^2*c^2*d^2/x - a^2*c*d^2/x^2 - 1/3*a^
2*d^2/x^3 - 1/12*(4*b^2*c^2*d^2*x^2 + (b^2*c^3*d^2*x^3 + 3*b^2*c^2*d^2*x^2 + 3*b^2*c*d^2*x + b^2*d^2)*log(c*x
+ 1)^2 - (7*b^2*c^3*d^2*x^3 - 3*b^2*c^2*d^2*x^2 - 3*b^2*c*d^2*x - b^2*d^2)*log(-c*x + 1)^2 + 2*(6*b^2*c^2*d^2*
x^2 + b^2*c*d^2*x)*log(c*x + 1) - 2*(6*b^2*c^2*d^2*x^2 + b^2*c*d^2*x + (b^2*c^3*d^2*x^3 + 3*b^2*c^2*d^2*x^2 +
3*b^2*c*d^2*x + b^2*d^2)*log(c*x + 1))*log(-c*x + 1))/x^3

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

[Out]

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

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