3.1.81 \(\int \frac {(d+c d x)^2 (a+b \tanh ^{-1}(c x))^2}{x^2} \, dx\) [81]

Optimal. Leaf size=283 \[ 2 c d^2 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {d^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{x}+c^2 d^2 x \left (a+b \tanh ^{-1}(c x)\right )^2+4 c d^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1-c x}\right )-2 b c d^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )+2 b c d^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac {2}{1+c x}\right )-b^2 c d^2 \text {PolyLog}\left (2,1-\frac {2}{1-c x}\right )-2 b c d^2 \left (a+b \tanh ^{-1}(c x)\right ) \text {PolyLog}\left (2,1-\frac {2}{1-c x}\right )+2 b c d^2 \left (a+b \tanh ^{-1}(c x)\right ) \text {PolyLog}\left (2,-1+\frac {2}{1-c x}\right )-b^2 c d^2 \text {PolyLog}\left (2,-1+\frac {2}{1+c x}\right )+b^2 c d^2 \text {PolyLog}\left (3,1-\frac {2}{1-c x}\right )-b^2 c d^2 \text {PolyLog}\left (3,-1+\frac {2}{1-c x}\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.46, antiderivative size = 283, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 15, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.682, Rules used = {6087, 6021, 6131, 6055, 2449, 2352, 6037, 6135, 6079, 2497, 6033, 6199, 6095, 6205, 6745} \begin {gather*} c^2 d^2 x \left (a+b \tanh ^{-1}(c x)\right )^2-2 b c d^2 \text {Li}_2\left (1-\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )+2 b c d^2 \text {Li}_2\left (\frac {2}{1-c x}-1\right ) \left (a+b \tanh ^{-1}(c x)\right )+2 c d^2 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {d^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{x}+4 c d^2 \tanh ^{-1}\left (1-\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2-2 b c d^2 \log \left (\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )+2 b c d^2 \log \left (2-\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )-b^2 c d^2 \text {Li}_2\left (1-\frac {2}{1-c x}\right )-b^2 c d^2 \text {Li}_2\left (\frac {2}{c x+1}-1\right )+b^2 c d^2 \text {Li}_3\left (1-\frac {2}{1-c x}\right )-b^2 c d^2 \text {Li}_3\left (\frac {2}{1-c x}-1\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

2*c*d^2*(a + b*ArcTanh[c*x])^2 - (d^2*(a + b*ArcTanh[c*x])^2)/x + c^2*d^2*x*(a + b*ArcTanh[c*x])^2 + 4*c*d^2*(
a + b*ArcTanh[c*x])^2*ArcTanh[1 - 2/(1 - c*x)] - 2*b*c*d^2*(a + b*ArcTanh[c*x])*Log[2/(1 - c*x)] + 2*b*c*d^2*(
a + b*ArcTanh[c*x])*Log[2 - 2/(1 + c*x)] - b^2*c*d^2*PolyLog[2, 1 - 2/(1 - c*x)] - 2*b*c*d^2*(a + b*ArcTanh[c*
x])*PolyLog[2, 1 - 2/(1 - c*x)] + 2*b*c*d^2*(a + b*ArcTanh[c*x])*PolyLog[2, -1 + 2/(1 - c*x)] - b^2*c*d^2*Poly
Log[2, -1 + 2/(1 + c*x)] + b^2*c*d^2*PolyLog[3, 1 - 2/(1 - c*x)] - b^2*c*d^2*PolyLog[3, -1 + 2/(1 - c*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 2497

Int[Log[u_]*(Pq_)^(m_.), x_Symbol] :> With[{C = FullSimplify[Pq^m*((1 - u)/D[u, x])]}, Simp[C*PolyLog[2, 1 - u
], x] /; FreeQ[C, x]] /; IntegerQ[m] && PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponen
ts[u, x][[2]], Expon[Pq, x]]

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 6033

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_)/(x_), x_Symbol] :> Simp[2*(a + b*ArcTanh[c*x])^p*ArcTanh[1 - 2/(1
- c*x)], x] - Dist[2*b*c*p, Int[(a + b*ArcTanh[c*x])^(p - 1)*(ArcTanh[1 - 2/(1 - c*x)]/(1 - c^2*x^2)), x], x]
/; FreeQ[{a, b, c}, x] && IGtQ[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 6079

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_))), x_Symbol] :> Simp[(a + b*ArcTanh[c*x
])^p*(Log[2 - 2/(1 + e*(x/d))]/d), x] - Dist[b*c*(p/d), Int[(a + b*ArcTanh[c*x])^(p - 1)*(Log[2 - 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 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 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 6135

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

Rule 6199

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

Rule 6205

Int[(Log[u_]*((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(-(a + b*ArcT
anh[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 {(d+c d x)^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{x^2} \, dx &=\int \left (c^2 d^2 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {d^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{x^2}+\frac {2 c d^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{x}\right ) \, dx\\ &=d^2 \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{x^2} \, dx+\left (2 c d^2\right ) \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{x} \, dx+\left (c^2 d^2\right ) \int \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx\\ &=-\frac {d^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{x}+c^2 d^2 x \left (a+b \tanh ^{-1}(c x)\right )^2+4 c d^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1-c x}\right )+\left (2 b c d^2\right ) \int \frac {a+b \tanh ^{-1}(c x)}{x \left (1-c^2 x^2\right )} \, dx-\left (8 b c^2 d^2\right ) \int \frac {\left (a+b \tanh ^{-1}(c x)\right ) \tanh ^{-1}\left (1-\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx-\left (2 b c^3 d^2\right ) \int \frac {x \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx\\ &=2 c d^2 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {d^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{x}+c^2 d^2 x \left (a+b \tanh ^{-1}(c x)\right )^2+4 c d^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1-c x}\right )+\left (2 b c d^2\right ) \int \frac {a+b \tanh ^{-1}(c x)}{x (1+c x)} \, dx-\left (2 b c^2 d^2\right ) \int \frac {a+b \tanh ^{-1}(c x)}{1-c x} \, dx+\left (4 b c^2 d^2\right ) \int \frac {\left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx-\left (4 b c^2 d^2\right ) \int \frac {\left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx\\ &=2 c d^2 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {d^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{x}+c^2 d^2 x \left (a+b \tanh ^{-1}(c x)\right )^2+4 c d^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1-c x}\right )-2 b c d^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )+2 b c d^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac {2}{1+c x}\right )-2 b c d^2 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-c x}\right )+2 b c d^2 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (-1+\frac {2}{1-c x}\right )+\left (2 b^2 c^2 d^2\right ) \int \frac {\log \left (\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx-\left (2 b^2 c^2 d^2\right ) \int \frac {\log \left (2-\frac {2}{1+c x}\right )}{1-c^2 x^2} \, dx+\left (2 b^2 c^2 d^2\right ) \int \frac {\text {Li}_2\left (1-\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx-\left (2 b^2 c^2 d^2\right ) \int \frac {\text {Li}_2\left (-1+\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx\\ &=2 c d^2 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {d^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{x}+c^2 d^2 x \left (a+b \tanh ^{-1}(c x)\right )^2+4 c d^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1-c x}\right )-2 b c d^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )+2 b c d^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac {2}{1+c x}\right )-2 b c d^2 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-c x}\right )+2 b c d^2 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (-1+\frac {2}{1-c x}\right )-b^2 c d^2 \text {Li}_2\left (-1+\frac {2}{1+c x}\right )+b^2 c d^2 \text {Li}_3\left (1-\frac {2}{1-c x}\right )-b^2 c d^2 \text {Li}_3\left (-1+\frac {2}{1-c x}\right )-\left (2 b^2 c d^2\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-c x}\right )\\ &=2 c d^2 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {d^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{x}+c^2 d^2 x \left (a+b \tanh ^{-1}(c x)\right )^2+4 c d^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1-c x}\right )-2 b c d^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )+2 b c d^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac {2}{1+c x}\right )-b^2 c d^2 \text {Li}_2\left (1-\frac {2}{1-c x}\right )-2 b c d^2 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-c x}\right )+2 b c d^2 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (-1+\frac {2}{1-c x}\right )-b^2 c d^2 \text {Li}_2\left (-1+\frac {2}{1+c x}\right )+b^2 c d^2 \text {Li}_3\left (1-\frac {2}{1-c x}\right )-b^2 c d^2 \text {Li}_3\left (-1+\frac {2}{1-c x}\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [C] Result contains complex when optimal does not.
time = 0.31, size = 341, normalized size = 1.20 \begin {gather*} \frac {d^2 \left (-12 a^2+i b^2 c \pi ^3 x+12 a^2 c^2 x^2-24 a b \tanh ^{-1}(c x)+24 a b c^2 x^2 \tanh ^{-1}(c x)-12 b^2 \tanh ^{-1}(c x)^2+12 b^2 c^2 x^2 \tanh ^{-1}(c x)^2-16 b^2 c x \tanh ^{-1}(c x)^3+24 b^2 c x \tanh ^{-1}(c x) \log \left (1-e^{-2 \tanh ^{-1}(c x)}\right )-24 b^2 c x \tanh ^{-1}(c x) \log \left (1+e^{-2 \tanh ^{-1}(c x)}\right )-24 b^2 c x \tanh ^{-1}(c x)^2 \log \left (1+e^{-2 \tanh ^{-1}(c x)}\right )+24 b^2 c x \tanh ^{-1}(c x)^2 \log \left (1-e^{2 \tanh ^{-1}(c x)}\right )+24 a^2 c x \log (x)+24 a b c x \log (c x)+12 b^2 c x \left (1+2 \tanh ^{-1}(c x)\right ) \text {PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c x)}\right )-12 b^2 c x \text {PolyLog}\left (2,e^{-2 \tanh ^{-1}(c x)}\right )+24 b^2 c x \tanh ^{-1}(c x) \text {PolyLog}\left (2,e^{2 \tanh ^{-1}(c x)}\right )-24 a b c x \text {PolyLog}(2,-c x)+24 a b c x \text {PolyLog}(2,c x)+12 b^2 c x \text {PolyLog}\left (3,-e^{-2 \tanh ^{-1}(c x)}\right )-12 b^2 c x \text {PolyLog}\left (3,e^{2 \tanh ^{-1}(c x)}\right )\right )}{12 x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(d^2*(-12*a^2 + I*b^2*c*Pi^3*x + 12*a^2*c^2*x^2 - 24*a*b*ArcTanh[c*x] + 24*a*b*c^2*x^2*ArcTanh[c*x] - 12*b^2*A
rcTanh[c*x]^2 + 12*b^2*c^2*x^2*ArcTanh[c*x]^2 - 16*b^2*c*x*ArcTanh[c*x]^3 + 24*b^2*c*x*ArcTanh[c*x]*Log[1 - E^
(-2*ArcTanh[c*x])] - 24*b^2*c*x*ArcTanh[c*x]*Log[1 + E^(-2*ArcTanh[c*x])] - 24*b^2*c*x*ArcTanh[c*x]^2*Log[1 +
E^(-2*ArcTanh[c*x])] + 24*b^2*c*x*ArcTanh[c*x]^2*Log[1 - E^(2*ArcTanh[c*x])] + 24*a^2*c*x*Log[x] + 24*a*b*c*x*
Log[c*x] + 12*b^2*c*x*(1 + 2*ArcTanh[c*x])*PolyLog[2, -E^(-2*ArcTanh[c*x])] - 12*b^2*c*x*PolyLog[2, E^(-2*ArcT
anh[c*x])] + 24*b^2*c*x*ArcTanh[c*x]*PolyLog[2, E^(2*ArcTanh[c*x])] - 24*a*b*c*x*PolyLog[2, -(c*x)] + 24*a*b*c
*x*PolyLog[2, c*x] + 12*b^2*c*x*PolyLog[3, -E^(-2*ArcTanh[c*x])] - 12*b^2*c*x*PolyLog[3, E^(2*ArcTanh[c*x])]))
/(12*x)

________________________________________________________________________________________

Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 6.00, size = 5974, normalized size = 21.11

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(5974\)
default	\(\text {Expression too large to display}\)	\(5974\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

________________________________________________________________________________________

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

[Out]

a^2*c^2*d^2*x - 1/2*b^2*c^2*d^2*integrate(log(c*x + 1)*log(-c*x + 1), x) + 1/4*b^2*c^2*d^2*integrate(log(c*x +
1)^2/(c^2*x^2), x) + (2*c*x*arctanh(c*x) + log(-c^2*x^2 + 1))*a*b*c*d^2 + 1/2*(c*x - (c*x - 1)*log(-c*x + 1)
- 1)*b^2*c*d^2 + 1/4*b^2*c*d^2*gamma(3, -log(c*x + 1)) + 1/2*b^2*c*d^2*integrate(log(c*x + 1)^2/x, x) - b^2*c*
d^2*integrate(log(c*x + 1)*log(-c*x + 1)/x, x) + 2*a*b*c*d^2*integrate(log(c*x + 1)/x, x) - 2*a*b*c*d^2*integr
ate(log(-c*x + 1)/x, x) - 1/2*b^2*c*d^2*integrate(log(-c*x + 1)/x, x) + 2*a^2*c*d^2*log(x) - (c*(log(c^2*x^2 -
1) - log(x^2)) + 2*arctanh(c*x)/x)*a*b*d^2 - 1/2*b^2*d^2*integrate(log(c*x + 1)*log(-c*x + 1)/x^2, x) - a^2*d
^2/x + 1/4*(b^2*c^2*d^2*x^2 - b^2*d^2)*log(-c*x + 1)^2/x

________________________________________________________________________________________

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^2,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^2, x)

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

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^2,x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

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^2} \,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^2,x)

[Out]

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

________________________________________________________________________________________