3.347 \(\int \frac {\tanh ^{-1}(a x)^3}{(1-a^2 x^2)^4} \, dx\)

Optimal. Leaf size=291 \[ -\frac {245}{768 a \left (1-a^2 x^2\right )}-\frac {65}{2304 a \left (1-a^2 x^2\right )^2}-\frac {1}{216 a \left (1-a^2 x^2\right )^3}+\frac {5 x \tanh ^{-1}(a x)^3}{16 \left (1-a^2 x^2\right )}+\frac {5 x \tanh ^{-1}(a x)^3}{24 \left (1-a^2 x^2\right )^2}+\frac {x \tanh ^{-1}(a x)^3}{6 \left (1-a^2 x^2\right )^3}-\frac {15 \tanh ^{-1}(a x)^2}{32 a \left (1-a^2 x^2\right )}-\frac {5 \tanh ^{-1}(a x)^2}{32 a \left (1-a^2 x^2\right )^2}-\frac {\tanh ^{-1}(a x)^2}{12 a \left (1-a^2 x^2\right )^3}+\frac {245 x \tanh ^{-1}(a x)}{384 \left (1-a^2 x^2\right )}+\frac {65 x \tanh ^{-1}(a x)}{576 \left (1-a^2 x^2\right )^2}+\frac {x \tanh ^{-1}(a x)}{36 \left (1-a^2 x^2\right )^3}+\frac {5 \tanh ^{-1}(a x)^4}{64 a}+\frac {245 \tanh ^{-1}(a x)^2}{768 a} \]

[Out]

-1/216/a/(-a^2*x^2+1)^3-65/2304/a/(-a^2*x^2+1)^2-245/768/a/(-a^2*x^2+1)+1/36*x*arctanh(a*x)/(-a^2*x^2+1)^3+65/
576*x*arctanh(a*x)/(-a^2*x^2+1)^2+245/384*x*arctanh(a*x)/(-a^2*x^2+1)+245/768*arctanh(a*x)^2/a-1/12*arctanh(a*
x)^2/a/(-a^2*x^2+1)^3-5/32*arctanh(a*x)^2/a/(-a^2*x^2+1)^2-15/32*arctanh(a*x)^2/a/(-a^2*x^2+1)+1/6*x*arctanh(a
*x)^3/(-a^2*x^2+1)^3+5/24*x*arctanh(a*x)^3/(-a^2*x^2+1)^2+5/16*x*arctanh(a*x)^3/(-a^2*x^2+1)+5/64*arctanh(a*x)
^4/a

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Rubi [A]  time = 0.33, antiderivative size = 291, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {5964, 5956, 5994, 261, 5960} \[ -\frac {245}{768 a \left (1-a^2 x^2\right )}-\frac {65}{2304 a \left (1-a^2 x^2\right )^2}-\frac {1}{216 a \left (1-a^2 x^2\right )^3}+\frac {5 x \tanh ^{-1}(a x)^3}{16 \left (1-a^2 x^2\right )}+\frac {5 x \tanh ^{-1}(a x)^3}{24 \left (1-a^2 x^2\right )^2}+\frac {x \tanh ^{-1}(a x)^3}{6 \left (1-a^2 x^2\right )^3}-\frac {15 \tanh ^{-1}(a x)^2}{32 a \left (1-a^2 x^2\right )}-\frac {5 \tanh ^{-1}(a x)^2}{32 a \left (1-a^2 x^2\right )^2}-\frac {\tanh ^{-1}(a x)^2}{12 a \left (1-a^2 x^2\right )^3}+\frac {245 x \tanh ^{-1}(a x)}{384 \left (1-a^2 x^2\right )}+\frac {65 x \tanh ^{-1}(a x)}{576 \left (1-a^2 x^2\right )^2}+\frac {x \tanh ^{-1}(a x)}{36 \left (1-a^2 x^2\right )^3}+\frac {5 \tanh ^{-1}(a x)^4}{64 a}+\frac {245 \tanh ^{-1}(a x)^2}{768 a} \]

Antiderivative was successfully verified.

[In]

Int[ArcTanh[a*x]^3/(1 - a^2*x^2)^4,x]

[Out]

-1/(216*a*(1 - a^2*x^2)^3) - 65/(2304*a*(1 - a^2*x^2)^2) - 245/(768*a*(1 - a^2*x^2)) + (x*ArcTanh[a*x])/(36*(1
- a^2*x^2)^3) + (65*x*ArcTanh[a*x])/(576*(1 - a^2*x^2)^2) + (245*x*ArcTanh[a*x])/(384*(1 - a^2*x^2)) + (245*A
rcTanh[a*x]^2)/(768*a) - ArcTanh[a*x]^2/(12*a*(1 - a^2*x^2)^3) - (5*ArcTanh[a*x]^2)/(32*a*(1 - a^2*x^2)^2) - (
15*ArcTanh[a*x]^2)/(32*a*(1 - a^2*x^2)) + (x*ArcTanh[a*x]^3)/(6*(1 - a^2*x^2)^3) + (5*x*ArcTanh[a*x]^3)/(24*(1
- a^2*x^2)^2) + (5*x*ArcTanh[a*x]^3)/(16*(1 - a^2*x^2)) + (5*ArcTanh[a*x]^4)/(64*a)

Rule 261

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

Rule 5956

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

Rule 5960

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> -Simp[(b*(d + e*x^2)^(q + 1))
/(4*c*d*(q + 1)^2), x] + (Dist[(2*q + 3)/(2*d*(q + 1)), Int[(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x]), x], x] -
Simp[(x*(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x]))/(2*d*(q + 1)), x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*
d + e, 0] && LtQ[q, -1] && NeQ[q, -3/2]

Rule 5964

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_)*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> -Simp[(b*p*(d + e*x^2)^(
q + 1)*(a + b*ArcTanh[c*x])^(p - 1))/(4*c*d*(q + 1)^2), x] + (Dist[(2*q + 3)/(2*d*(q + 1)), Int[(d + e*x^2)^(q
+ 1)*(a + b*ArcTanh[c*x])^p, x], x] + Dist[(b^2*p*(p - 1))/(4*(q + 1)^2), Int[(d + e*x^2)^q*(a + b*ArcTanh[c*
x])^(p - 2), x], x] - Simp[(x*(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x])^p)/(2*d*(q + 1)), x]) /; FreeQ[{a, b, c
, d, e}, x] && EqQ[c^2*d + e, 0] && LtQ[q, -1] && GtQ[p, 1] && NeQ[q, -3/2]

Rule 5994

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

Rubi steps

\begin {align*} \int \frac {\tanh ^{-1}(a x)^3}{\left (1-a^2 x^2\right )^4} \, dx &=-\frac {\tanh ^{-1}(a x)^2}{12 a \left (1-a^2 x^2\right )^3}+\frac {x \tanh ^{-1}(a x)^3}{6 \left (1-a^2 x^2\right )^3}+\frac {1}{6} \int \frac {\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^4} \, dx+\frac {5}{6} \int \frac {\tanh ^{-1}(a x)^3}{\left (1-a^2 x^2\right )^3} \, dx\\ &=-\frac {1}{216 a \left (1-a^2 x^2\right )^3}+\frac {x \tanh ^{-1}(a x)}{36 \left (1-a^2 x^2\right )^3}-\frac {\tanh ^{-1}(a x)^2}{12 a \left (1-a^2 x^2\right )^3}-\frac {5 \tanh ^{-1}(a x)^2}{32 a \left (1-a^2 x^2\right )^2}+\frac {x \tanh ^{-1}(a x)^3}{6 \left (1-a^2 x^2\right )^3}+\frac {5 x \tanh ^{-1}(a x)^3}{24 \left (1-a^2 x^2\right )^2}+\frac {5}{36} \int \frac {\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^3} \, dx+\frac {5}{16} \int \frac {\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^3} \, dx+\frac {5}{8} \int \frac {\tanh ^{-1}(a x)^3}{\left (1-a^2 x^2\right )^2} \, dx\\ &=-\frac {1}{216 a \left (1-a^2 x^2\right )^3}-\frac {65}{2304 a \left (1-a^2 x^2\right )^2}+\frac {x \tanh ^{-1}(a x)}{36 \left (1-a^2 x^2\right )^3}+\frac {65 x \tanh ^{-1}(a x)}{576 \left (1-a^2 x^2\right )^2}-\frac {\tanh ^{-1}(a x)^2}{12 a \left (1-a^2 x^2\right )^3}-\frac {5 \tanh ^{-1}(a x)^2}{32 a \left (1-a^2 x^2\right )^2}+\frac {x \tanh ^{-1}(a x)^3}{6 \left (1-a^2 x^2\right )^3}+\frac {5 x \tanh ^{-1}(a x)^3}{24 \left (1-a^2 x^2\right )^2}+\frac {5 x \tanh ^{-1}(a x)^3}{16 \left (1-a^2 x^2\right )}+\frac {5 \tanh ^{-1}(a x)^4}{64 a}+\frac {5}{48} \int \frac {\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^2} \, dx+\frac {15}{64} \int \frac {\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^2} \, dx-\frac {1}{16} (15 a) \int \frac {x \tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^2} \, dx\\ &=-\frac {1}{216 a \left (1-a^2 x^2\right )^3}-\frac {65}{2304 a \left (1-a^2 x^2\right )^2}+\frac {x \tanh ^{-1}(a x)}{36 \left (1-a^2 x^2\right )^3}+\frac {65 x \tanh ^{-1}(a x)}{576 \left (1-a^2 x^2\right )^2}+\frac {65 x \tanh ^{-1}(a x)}{384 \left (1-a^2 x^2\right )}+\frac {65 \tanh ^{-1}(a x)^2}{768 a}-\frac {\tanh ^{-1}(a x)^2}{12 a \left (1-a^2 x^2\right )^3}-\frac {5 \tanh ^{-1}(a x)^2}{32 a \left (1-a^2 x^2\right )^2}-\frac {15 \tanh ^{-1}(a x)^2}{32 a \left (1-a^2 x^2\right )}+\frac {x \tanh ^{-1}(a x)^3}{6 \left (1-a^2 x^2\right )^3}+\frac {5 x \tanh ^{-1}(a x)^3}{24 \left (1-a^2 x^2\right )^2}+\frac {5 x \tanh ^{-1}(a x)^3}{16 \left (1-a^2 x^2\right )}+\frac {5 \tanh ^{-1}(a x)^4}{64 a}+\frac {15}{16} \int \frac {\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^2} \, dx-\frac {1}{96} (5 a) \int \frac {x}{\left (1-a^2 x^2\right )^2} \, dx-\frac {1}{128} (15 a) \int \frac {x}{\left (1-a^2 x^2\right )^2} \, dx\\ &=-\frac {1}{216 a \left (1-a^2 x^2\right )^3}-\frac {65}{2304 a \left (1-a^2 x^2\right )^2}-\frac {65}{768 a \left (1-a^2 x^2\right )}+\frac {x \tanh ^{-1}(a x)}{36 \left (1-a^2 x^2\right )^3}+\frac {65 x \tanh ^{-1}(a x)}{576 \left (1-a^2 x^2\right )^2}+\frac {245 x \tanh ^{-1}(a x)}{384 \left (1-a^2 x^2\right )}+\frac {245 \tanh ^{-1}(a x)^2}{768 a}-\frac {\tanh ^{-1}(a x)^2}{12 a \left (1-a^2 x^2\right )^3}-\frac {5 \tanh ^{-1}(a x)^2}{32 a \left (1-a^2 x^2\right )^2}-\frac {15 \tanh ^{-1}(a x)^2}{32 a \left (1-a^2 x^2\right )}+\frac {x \tanh ^{-1}(a x)^3}{6 \left (1-a^2 x^2\right )^3}+\frac {5 x \tanh ^{-1}(a x)^3}{24 \left (1-a^2 x^2\right )^2}+\frac {5 x \tanh ^{-1}(a x)^3}{16 \left (1-a^2 x^2\right )}+\frac {5 \tanh ^{-1}(a x)^4}{64 a}-\frac {1}{32} (15 a) \int \frac {x}{\left (1-a^2 x^2\right )^2} \, dx\\ &=-\frac {1}{216 a \left (1-a^2 x^2\right )^3}-\frac {65}{2304 a \left (1-a^2 x^2\right )^2}-\frac {245}{768 a \left (1-a^2 x^2\right )}+\frac {x \tanh ^{-1}(a x)}{36 \left (1-a^2 x^2\right )^3}+\frac {65 x \tanh ^{-1}(a x)}{576 \left (1-a^2 x^2\right )^2}+\frac {245 x \tanh ^{-1}(a x)}{384 \left (1-a^2 x^2\right )}+\frac {245 \tanh ^{-1}(a x)^2}{768 a}-\frac {\tanh ^{-1}(a x)^2}{12 a \left (1-a^2 x^2\right )^3}-\frac {5 \tanh ^{-1}(a x)^2}{32 a \left (1-a^2 x^2\right )^2}-\frac {15 \tanh ^{-1}(a x)^2}{32 a \left (1-a^2 x^2\right )}+\frac {x \tanh ^{-1}(a x)^3}{6 \left (1-a^2 x^2\right )^3}+\frac {5 x \tanh ^{-1}(a x)^3}{24 \left (1-a^2 x^2\right )^2}+\frac {5 x \tanh ^{-1}(a x)^3}{16 \left (1-a^2 x^2\right )}+\frac {5 \tanh ^{-1}(a x)^4}{64 a}\\ \end {align*}

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Mathematica [A]  time = 0.15, size = 143, normalized size = 0.49 \[ \frac {2205 a^4 x^4-4605 a^2 x^2+540 \left (a^2 x^2-1\right )^3 \tanh ^{-1}(a x)^4-144 a x \left (15 a^4 x^4-40 a^2 x^2+33\right ) \tanh ^{-1}(a x)^3-6 a x \left (735 a^4 x^4-1600 a^2 x^2+897\right ) \tanh ^{-1}(a x)+9 \left (245 a^6 x^6-375 a^4 x^4-105 a^2 x^2+299\right ) \tanh ^{-1}(a x)^2+2432}{6912 a \left (a^2 x^2-1\right )^3} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcTanh[a*x]^3/(1 - a^2*x^2)^4,x]

[Out]

(2432 - 4605*a^2*x^2 + 2205*a^4*x^4 - 6*a*x*(897 - 1600*a^2*x^2 + 735*a^4*x^4)*ArcTanh[a*x] + 9*(299 - 105*a^2
*x^2 - 375*a^4*x^4 + 245*a^6*x^6)*ArcTanh[a*x]^2 - 144*a*x*(33 - 40*a^2*x^2 + 15*a^4*x^4)*ArcTanh[a*x]^3 + 540
*(-1 + a^2*x^2)^3*ArcTanh[a*x]^4)/(6912*a*(-1 + a^2*x^2)^3)

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fricas [A]  time = 0.71, size = 216, normalized size = 0.74 \[ \frac {8820 \, a^{4} x^{4} + 135 \, {\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{4} - 18420 \, a^{2} x^{2} - 72 \, {\left (15 \, a^{5} x^{5} - 40 \, a^{3} x^{3} + 33 \, a x\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{3} + 9 \, {\left (245 \, a^{6} x^{6} - 375 \, a^{4} x^{4} - 105 \, a^{2} x^{2} + 299\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} - 12 \, {\left (735 \, a^{5} x^{5} - 1600 \, a^{3} x^{3} + 897 \, a x\right )} \log \left (-\frac {a x + 1}{a x - 1}\right ) + 9728}{27648 \, {\left (a^{7} x^{6} - 3 \, a^{5} x^{4} + 3 \, a^{3} x^{2} - a\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(a*x)^3/(-a^2*x^2+1)^4,x, algorithm="fricas")

[Out]

1/27648*(8820*a^4*x^4 + 135*(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)*log(-(a*x + 1)/(a*x - 1))^4 - 18420*a^2*x^2
- 72*(15*a^5*x^5 - 40*a^3*x^3 + 33*a*x)*log(-(a*x + 1)/(a*x - 1))^3 + 9*(245*a^6*x^6 - 375*a^4*x^4 - 105*a^2*x
^2 + 299)*log(-(a*x + 1)/(a*x - 1))^2 - 12*(735*a^5*x^5 - 1600*a^3*x^3 + 897*a*x)*log(-(a*x + 1)/(a*x - 1)) +
9728)/(a^7*x^6 - 3*a^5*x^4 + 3*a^3*x^2 - a)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {artanh}\left (a x\right )^{3}}{{\left (a^{2} x^{2} - 1\right )}^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(a*x)^3/(-a^2*x^2+1)^4,x, algorithm="giac")

[Out]

integrate(arctanh(a*x)^3/(a^2*x^2 - 1)^4, x)

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maple [C]  time = 0.97, size = 3550, normalized size = 12.20 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arctanh(a*x)^3/(-a^2*x^2+1)^4,x)

[Out]

5/32*I*a^5/(a*x-1)^3/(a*x+1)^3*arctanh(a*x)^3*Pi*csgn(I/(1+(a*x+1)^2/(-a^2*x^2+1)))^3*x^6-5/64*I*a^5/(a*x-1)^3
/(a*x+1)^3*arctanh(a*x)^3*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(1+(a*x+1)^2/(-a^2*x^2+1)))^3*x^6-5/32*I*a^5/(a*x-1)
^3/(a*x+1)^3*arctanh(a*x)^3*Pi*csgn(I/(1+(a*x+1)^2/(-a^2*x^2+1)))^2*x^6+15/64*I*a^3/(a*x-1)^3/(a*x+1)^3*arctan
h(a*x)^3*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1))^3*x^4-15/32*I*a^3/(a*x-1)^3/(a*x+1)^3*arctanh(a*x)^3*Pi*csgn(I/(1+(a
*x+1)^2/(-a^2*x^2+1)))^3*x^4+15/64*I*a^3/(a*x-1)^3/(a*x+1)^3*arctanh(a*x)^3*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(1
+(a*x+1)^2/(-a^2*x^2+1)))^3*x^4+15/32*I*a^3/(a*x-1)^3/(a*x+1)^3*arctanh(a*x)^3*Pi*csgn(I/(1+(a*x+1)^2/(-a^2*x^
2+1)))^2*x^4-15/64*I*a/(a*x-1)^3/(a*x+1)^3*arctanh(a*x)^3*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1))^3*x^2+15/32*I*a/(a*
x-1)^3/(a*x+1)^3*arctanh(a*x)^3*Pi*csgn(I/(1+(a*x+1)^2/(-a^2*x^2+1)))^3*x^2-15/64*I*a/(a*x-1)^3/(a*x+1)^3*arct
anh(a*x)^3*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(1+(a*x+1)^2/(-a^2*x^2+1)))^3*x^2+5/64*I/a/(a*x-1)^3/(a*x+1)^3*Pi*a
rctanh(a*x)^3*csgn(I*(a*x+1)/(-a^2*x^2+1)^(1/2))^2*csgn(I*(a*x+1)^2/(a^2*x^2-1))+5/64*I/a/(a*x-1)^3/(a*x+1)^3*
Pi*arctanh(a*x)^3*csgn(I/(1+(a*x+1)^2/(-a^2*x^2+1)))*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(1+(a*x+1)^2/(-a^2*x^2+1)))^
2+5/32*I/a/(a*x-1)^3/(a*x+1)^3*Pi*arctanh(a*x)^3*csgn(I*(a*x+1)/(-a^2*x^2+1)^(1/2))*csgn(I*(a*x+1)^2/(a^2*x^2-
1))^2-5/64*I/a/(a*x-1)^3/(a*x+1)^3*Pi*arctanh(a*x)^3*csgn(I*(a*x+1)^2/(a^2*x^2-1))*csgn(I*(a*x+1)^2/(a^2*x^2-1
)/(1+(a*x+1)^2/(-a^2*x^2+1)))^2-15/32*I*a/(a*x-1)^3/(a*x+1)^3*arctanh(a*x)^3*Pi*csgn(I/(1+(a*x+1)^2/(-a^2*x^2+
1)))^2*x^2-5/64*I*a^5/(a*x-1)^3/(a*x+1)^3*arctanh(a*x)^3*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1))^3*x^6+5/64*I*a^5/(a*
x-1)^3/(a*x+1)^3*arctanh(a*x)^3*Pi*csgn(I/(1+(a*x+1)^2/(-a^2*x^2+1)))*csgn(I*(a*x+1)^2/(a^2*x^2-1))*csgn(I*(a*
x+1)^2/(a^2*x^2-1)/(1+(a*x+1)^2/(-a^2*x^2+1)))*x^6+15/64*I*a/(a*x-1)^3/(a*x+1)^3*arctanh(a*x)^3*Pi*csgn(I/(1+(
a*x+1)^2/(-a^2*x^2+1)))*csgn(I*(a*x+1)^2/(a^2*x^2-1))*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(1+(a*x+1)^2/(-a^2*x^2+1)))
*x^2-15/64*I*a^3/(a*x-1)^3/(a*x+1)^3*arctanh(a*x)^3*Pi*csgn(I/(1+(a*x+1)^2/(-a^2*x^2+1)))*csgn(I*(a*x+1)^2/(a^
2*x^2-1))*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(1+(a*x+1)^2/(-a^2*x^2+1)))*x^4+9485/55296*a^5/(a*x-1)^3/(a*x+1)^3*x^6-
3605/18432*a^3/(a*x-1)^3/(a*x+1)^3*x^4-2795/18432*a/(a*x-1)^3/(a*x+1)^3*x^2-299/384/(a*x-1)^3/(a*x+1)^3*arctan
h(a*x)*x-5/64/a/(a*x-1)^3/(a*x+1)^3*arctanh(a*x)^4+299/768/a/(a*x-1)^3/(a*x+1)^3*arctanh(a*x)^2+15/32*I*a^3/(a
*x-1)^3/(a*x+1)^3*arctanh(a*x)^3*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1))^2*csgn(I*(a*x+1)/(-a^2*x^2+1)^(1/2))*x^4-5/6
4*I/a/(a*x-1)^3/(a*x+1)^3*Pi*arctanh(a*x)^3*csgn(I/(1+(a*x+1)^2/(-a^2*x^2+1)))*csgn(I*(a*x+1)^2/(a^2*x^2-1))*c
sgn(I*(a*x+1)^2/(a^2*x^2-1)/(1+(a*x+1)^2/(-a^2*x^2+1)))-15/64*I*a/(a*x-1)^3/(a*x+1)^3*arctanh(a*x)^3*Pi*csgn(I
/(1+(a*x+1)^2/(-a^2*x^2+1)))*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(1+(a*x+1)^2/(-a^2*x^2+1)))^2*x^2-15/64*I*a^3/(a*x-1
)^3/(a*x+1)^3*arctanh(a*x)^3*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1))*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(1+(a*x+1)^2/(-a^2*
x^2+1)))^2*x^4+15/64*I*a^3/(a*x-1)^3/(a*x+1)^3*arctanh(a*x)^3*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1))*csgn(I*(a*x+1)/
(-a^2*x^2+1)^(1/2))^2*x^4+15/64*I*a^3/(a*x-1)^3/(a*x+1)^3*arctanh(a*x)^3*Pi*csgn(I/(1+(a*x+1)^2/(-a^2*x^2+1)))
*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(1+(a*x+1)^2/(-a^2*x^2+1)))^2*x^4-15/32*I*a/(a*x-1)^3/(a*x+1)^3*arctanh(a*x)^3*P
i*csgn(I*(a*x+1)^2/(a^2*x^2-1))^2*csgn(I*(a*x+1)/(-a^2*x^2+1)^(1/2))*x^2+15/64*I*a/(a*x-1)^3/(a*x+1)^3*arctanh
(a*x)^3*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1))*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(1+(a*x+1)^2/(-a^2*x^2+1)))^2*x^2-15/64*
I*a/(a*x-1)^3/(a*x+1)^3*arctanh(a*x)^3*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1))*csgn(I*(a*x+1)/(-a^2*x^2+1)^(1/2))^2*x
^2-5/32*I*a^5/(a*x-1)^3/(a*x+1)^3*arctanh(a*x)^3*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1))^2*csgn(I*(a*x+1)/(-a^2*x^2+1
)^(1/2))*x^6+5/64*I*a^5/(a*x-1)^3/(a*x+1)^3*arctanh(a*x)^3*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1))*csgn(I*(a*x+1)^2/(
a^2*x^2-1)/(1+(a*x+1)^2/(-a^2*x^2+1)))^2*x^6-5/64*I*a^5/(a*x-1)^3/(a*x+1)^3*arctanh(a*x)^3*Pi*csgn(I*(a*x+1)^2
/(a^2*x^2-1))*csgn(I*(a*x+1)/(-a^2*x^2+1)^(1/2))^2*x^6-5/64*I*a^5/(a*x-1)^3/(a*x+1)^3*arctanh(a*x)^3*Pi*csgn(I
/(1+(a*x+1)^2/(-a^2*x^2+1)))*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(1+(a*x+1)^2/(-a^2*x^2+1)))^2*x^6-5/32*I/a/(a*x-1)^3
/(a*x+1)^3*Pi*arctanh(a*x)^3+5/32/a*arctanh(a*x)^3*ln(a*x+1)-5/16/a*arctanh(a*x)^3*ln((a*x+1)/(-a^2*x^2+1)^(1/
2))+15/64*a/(a*x-1)^3/(a*x+1)^3*arctanh(a*x)^4*x^2+25/18*a^2/(a*x-1)^3/(a*x+1)^3*arctanh(a*x)*x^3+9971/55296/a
/(a*x-1)^3/(a*x+1)^3-1/48/a*arctanh(a*x)^3/(a*x-1)^3+1/16/a*arctanh(a*x)^3/(a*x-1)^2-5/32/a*arctanh(a*x)^3/(a*
x-1)-5/32/a*arctanh(a*x)^3*ln(a*x-1)-1/48/a*arctanh(a*x)^3/(a*x+1)^3-1/16/a*arctanh(a*x)^3/(a*x+1)^2-35/256*a/
(a*x-1)^3/(a*x+1)^3*arctanh(a*x)^2*x^2+5/64*a^5/(a*x-1)^3/(a*x+1)^3*arctanh(a*x)^4*x^6+245/768*a^5/(a*x-1)^3/(
a*x+1)^3*arctanh(a*x)^2*x^6-15/64*a^3/(a*x-1)^3/(a*x+1)^3*arctanh(a*x)^4*x^4-245/384*a^4/(a*x-1)^3/(a*x+1)^3*a
rctanh(a*x)*x^5-125/256*a^3/(a*x-1)^3/(a*x+1)^3*arctanh(a*x)^2*x^4-5/32/a*arctanh(a*x)^3/(a*x+1)-15/32*I*a^3/(
a*x-1)^3/(a*x+1)^3*arctanh(a*x)^3*Pi*x^4+15/32*I*a/(a*x-1)^3/(a*x+1)^3*arctanh(a*x)^3*Pi*x^2+5/64*I/a/(a*x-1)^
3/(a*x+1)^3*Pi*arctanh(a*x)^3*csgn(I*(a*x+1)^2/(a^2*x^2-1))^3-5/32*I/a/(a*x-1)^3/(a*x+1)^3*Pi*arctanh(a*x)^3*c
sgn(I/(1+(a*x+1)^2/(-a^2*x^2+1)))^3+5/64*I/a/(a*x-1)^3/(a*x+1)^3*Pi*arctanh(a*x)^3*csgn(I*(a*x+1)^2/(a^2*x^2-1
)/(1+(a*x+1)^2/(-a^2*x^2+1)))^3+5/32*I/a/(a*x-1)^3/(a*x+1)^3*Pi*arctanh(a*x)^3*csgn(I/(1+(a*x+1)^2/(-a^2*x^2+1
)))^2+5/32*I*a^5/(a*x-1)^3/(a*x+1)^3*arctanh(a*x)^3*Pi*x^6

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maxima [B]  time = 0.37, size = 871, normalized size = 2.99 \[ -\frac {1}{96} \, {\left (\frac {2 \, {\left (15 \, a^{4} x^{5} - 40 \, a^{2} x^{3} + 33 \, x\right )}}{a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1} - \frac {15 \, \log \left (a x + 1\right )}{a} + \frac {15 \, \log \left (a x - 1\right )}{a}\right )} \operatorname {artanh}\left (a x\right )^{3} + \frac {{\left (180 \, a^{4} x^{4} - 420 \, a^{2} x^{2} - 45 \, {\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \log \left (a x + 1\right )^{2} + 90 \, {\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \log \left (a x + 1\right ) \log \left (a x - 1\right ) - 45 \, {\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )^{2} + 272\right )} a \operatorname {artanh}\left (a x\right )^{2}}{384 \, {\left (a^{8} x^{6} - 3 \, a^{6} x^{4} + 3 \, a^{4} x^{2} - a^{2}\right )}} + \frac {1}{27648} \, {\left (\frac {{\left (8820 \, a^{4} x^{4} - 135 \, {\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \log \left (a x + 1\right )^{4} + 540 \, {\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \log \left (a x + 1\right )^{3} \log \left (a x - 1\right ) - 135 \, {\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )^{4} - 18420 \, a^{2} x^{2} - 45 \, {\left (49 \, a^{6} x^{6} - 147 \, a^{4} x^{4} + 147 \, a^{2} x^{2} + 18 \, {\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )^{2} - 49\right )} \log \left (a x + 1\right )^{2} - 2205 \, {\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )^{2} + 90 \, {\left (6 \, {\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )^{3} + 49 \, {\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )\right )} \log \left (a x + 1\right ) + 9728\right )} a^{2}}{a^{10} x^{6} - 3 \, a^{8} x^{4} + 3 \, a^{6} x^{2} - a^{4}} - \frac {12 \, {\left (1470 \, a^{5} x^{5} - 3200 \, a^{3} x^{3} - 90 \, {\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \log \left (a x + 1\right )^{3} + 270 \, {\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \log \left (a x + 1\right )^{2} \log \left (a x - 1\right ) + 90 \, {\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )^{3} + 1794 \, a x - 15 \, {\left (49 \, a^{6} x^{6} - 147 \, a^{4} x^{4} + 147 \, a^{2} x^{2} + 18 \, {\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )^{2} - 49\right )} \log \left (a x + 1\right ) + 735 \, {\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )\right )} a \operatorname {artanh}\left (a x\right )}{a^{9} x^{6} - 3 \, a^{7} x^{4} + 3 \, a^{5} x^{2} - a^{3}}\right )} a \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(a*x)^3/(-a^2*x^2+1)^4,x, algorithm="maxima")

[Out]

-1/96*(2*(15*a^4*x^5 - 40*a^2*x^3 + 33*x)/(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1) - 15*log(a*x + 1)/a + 15*log(a
*x - 1)/a)*arctanh(a*x)^3 + 1/384*(180*a^4*x^4 - 420*a^2*x^2 - 45*(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)*log(a*
x + 1)^2 + 90*(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)*log(a*x + 1)*log(a*x - 1) - 45*(a^6*x^6 - 3*a^4*x^4 + 3*a^
2*x^2 - 1)*log(a*x - 1)^2 + 272)*a*arctanh(a*x)^2/(a^8*x^6 - 3*a^6*x^4 + 3*a^4*x^2 - a^2) + 1/27648*((8820*a^4
*x^4 - 135*(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)*log(a*x + 1)^4 + 540*(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)*lo
g(a*x + 1)^3*log(a*x - 1) - 135*(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)*log(a*x - 1)^4 - 18420*a^2*x^2 - 45*(49*
a^6*x^6 - 147*a^4*x^4 + 147*a^2*x^2 + 18*(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)*log(a*x - 1)^2 - 49)*log(a*x +
1)^2 - 2205*(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)*log(a*x - 1)^2 + 90*(6*(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)
*log(a*x - 1)^3 + 49*(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)*log(a*x - 1))*log(a*x + 1) + 9728)*a^2/(a^10*x^6 -
3*a^8*x^4 + 3*a^6*x^2 - a^4) - 12*(1470*a^5*x^5 - 3200*a^3*x^3 - 90*(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)*log(
a*x + 1)^3 + 270*(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)*log(a*x + 1)^2*log(a*x - 1) + 90*(a^6*x^6 - 3*a^4*x^4 +
3*a^2*x^2 - 1)*log(a*x - 1)^3 + 1794*a*x - 15*(49*a^6*x^6 - 147*a^4*x^4 + 147*a^2*x^2 + 18*(a^6*x^6 - 3*a^4*x
^4 + 3*a^2*x^2 - 1)*log(a*x - 1)^2 - 49)*log(a*x + 1) + 735*(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)*log(a*x - 1)
)*a*arctanh(a*x)/(a^9*x^6 - 3*a^7*x^4 + 3*a^5*x^2 - a^3))*a

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mupad [B]  time = 2.97, size = 1041, normalized size = 3.58 \[ \frac {\frac {1216}{3\,a}-\frac {1535\,a\,x^2}{2}+\frac {735\,a^3\,x^4}{2}}{1152\,a^6\,x^6-3456\,a^4\,x^4+3456\,a^2\,x^2-1152}-{\ln \left (1-a\,x\right )}^3\,\left (\frac {5\,\ln \left (a\,x+1\right )}{256\,a}-\frac {\frac {5\,a^4\,x^5}{16}-\frac {5\,a^2\,x^3}{6}+\frac {11\,x}{16}}{8\,a^6\,x^6-24\,a^4\,x^4+24\,a^2\,x^2-8}\right )+\frac {5\,{\ln \left (a\,x+1\right )}^4}{1024\,a}+\frac {5\,{\ln \left (1-a\,x\right )}^4}{1024\,a}+{\ln \left (1-a\,x\right )}^2\,\left (\frac {15\,{\ln \left (a\,x+1\right )}^2}{512\,a}+\frac {245}{3072\,a}+\frac {\frac {37\,x}{2}-35\,a\,x^2+\frac {68}{3\,a}-\frac {82\,a^2\,x^3}{3}+15\,a^3\,x^4+\frac {23\,a^4\,x^5}{2}}{256\,a^6\,x^6-768\,a^4\,x^4+768\,a^2\,x^2-256}-\frac {\frac {37\,x}{2}+35\,a\,x^2-\frac {68}{3\,a}-\frac {82\,a^2\,x^3}{3}-15\,a^3\,x^4+\frac {23\,a^4\,x^5}{2}}{256\,a^6\,x^6-768\,a^4\,x^4+768\,a^2\,x^2-256}-\frac {\ln \left (a\,x+1\right )\,\left (30\,a^4\,x^5-80\,a^2\,x^3+66\,x\right )}{256\,a^6\,x^6-768\,a^4\,x^4+768\,a^2\,x^2-256}\right )+{\ln \left (a\,x+1\right )}^2\,\left (\frac {\frac {17}{96\,a^2}-\frac {35\,x^2}{128}+\frac {15\,a^2\,x^4}{128}}{3\,a\,x^2-\frac {1}{a}-3\,a^3\,x^4+a^5\,x^6}+\frac {245}{3072\,a}\right )+\ln \left (1-a\,x\right )\,\left (\frac {36\,x+22\,a\,x^2-\frac {23}{2\,a}-67\,a^2\,x^3-\frac {21\,a^3\,x^4}{2}+31\,a^4\,x^5}{768\,a^6\,x^6-2304\,a^4\,x^4+2304\,a^2\,x^2-768}-\frac {5\,{\ln \left (a\,x+1\right )}^3}{256\,a}-\ln \left (a\,x+1\right )\,\left (\frac {\frac {37\,x}{2}-35\,a\,x^2+\frac {68}{3\,a}-\frac {82\,a^2\,x^3}{3}+15\,a^3\,x^4+\frac {23\,a^4\,x^5}{2}}{128\,a^6\,x^6-384\,a^4\,x^4+384\,a^2\,x^2-128}-\frac {\frac {37\,x}{2}+35\,a\,x^2-\frac {68}{3\,a}-\frac {82\,a^2\,x^3}{3}-15\,a^3\,x^4+\frac {23\,a^4\,x^5}{2}}{128\,a^6\,x^6-384\,a^4\,x^4+384\,a^2\,x^2-128}+\frac {245\,\left (a^6\,x^6-3\,a^4\,x^4+3\,a^2\,x^2-1\right )}{12\,a\,\left (128\,a^6\,x^6-384\,a^4\,x^4+384\,a^2\,x^2-128\right )}\right )+\frac {\frac {227\,x}{2}+173\,a\,x^2-\frac {593}{6\,a}-\frac {599\,a^2\,x^3}{3}-\frac {159\,a^3\,x^4}{2}+\frac {183\,a^4\,x^5}{2}}{768\,a^6\,x^6-2304\,a^4\,x^4+2304\,a^2\,x^2-768}+\frac {\frac {299\,x}{2}-195\,a\,x^2+\frac {331}{3\,a}-\frac {800\,a^2\,x^3}{3}+90\,a^3\,x^4+\frac {245\,a^4\,x^5}{2}}{768\,a^6\,x^6-2304\,a^4\,x^4+2304\,a^2\,x^2-768}+\frac {{\ln \left (a\,x+1\right )}^2\,\left (30\,a^4\,x^5-80\,a^2\,x^3+66\,x\right )}{256\,a^6\,x^6-768\,a^4\,x^4+768\,a^2\,x^2-256}\right )-\frac {\ln \left (a\,x+1\right )\,\left (\frac {299\,x}{768\,a}-\frac {25\,a\,x^3}{36}+\frac {245\,a^3\,x^5}{768}\right )}{3\,a\,x^2-\frac {1}{a}-3\,a^3\,x^4+a^5\,x^6}-\frac {{\ln \left (a\,x+1\right )}^3\,\left (\frac {11\,x}{128\,a}-\frac {5\,a\,x^3}{48}+\frac {5\,a^3\,x^5}{128}\right )}{3\,a\,x^2-\frac {1}{a}-3\,a^3\,x^4+a^5\,x^6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(atanh(a*x)^3/(a^2*x^2 - 1)^4,x)

[Out]

(1216/(3*a) - (1535*a*x^2)/2 + (735*a^3*x^4)/2)/(3456*a^2*x^2 - 3456*a^4*x^4 + 1152*a^6*x^6 - 1152) - log(1 -
a*x)^3*((5*log(a*x + 1))/(256*a) - ((11*x)/16 - (5*a^2*x^3)/6 + (5*a^4*x^5)/16)/(24*a^2*x^2 - 24*a^4*x^4 + 8*a
^6*x^6 - 8)) + (5*log(a*x + 1)^4)/(1024*a) + (5*log(1 - a*x)^4)/(1024*a) + log(1 - a*x)^2*((15*log(a*x + 1)^2)
/(512*a) + 245/(3072*a) + ((37*x)/2 - 35*a*x^2 + 68/(3*a) - (82*a^2*x^3)/3 + 15*a^3*x^4 + (23*a^4*x^5)/2)/(768
*a^2*x^2 - 768*a^4*x^4 + 256*a^6*x^6 - 256) - ((37*x)/2 + 35*a*x^2 - 68/(3*a) - (82*a^2*x^3)/3 - 15*a^3*x^4 +
(23*a^4*x^5)/2)/(768*a^2*x^2 - 768*a^4*x^4 + 256*a^6*x^6 - 256) - (log(a*x + 1)*(66*x - 80*a^2*x^3 + 30*a^4*x^
5))/(768*a^2*x^2 - 768*a^4*x^4 + 256*a^6*x^6 - 256)) + log(a*x + 1)^2*((17/(96*a^2) - (35*x^2)/128 + (15*a^2*x
^4)/128)/(3*a*x^2 - 1/a - 3*a^3*x^4 + a^5*x^6) + 245/(3072*a)) + log(1 - a*x)*((36*x + 22*a*x^2 - 23/(2*a) - 6
7*a^2*x^3 - (21*a^3*x^4)/2 + 31*a^4*x^5)/(2304*a^2*x^2 - 2304*a^4*x^4 + 768*a^6*x^6 - 768) - (5*log(a*x + 1)^3
)/(256*a) - log(a*x + 1)*(((37*x)/2 - 35*a*x^2 + 68/(3*a) - (82*a^2*x^3)/3 + 15*a^3*x^4 + (23*a^4*x^5)/2)/(384
*a^2*x^2 - 384*a^4*x^4 + 128*a^6*x^6 - 128) - ((37*x)/2 + 35*a*x^2 - 68/(3*a) - (82*a^2*x^3)/3 - 15*a^3*x^4 +
(23*a^4*x^5)/2)/(384*a^2*x^2 - 384*a^4*x^4 + 128*a^6*x^6 - 128) + (245*(3*a^2*x^2 - 3*a^4*x^4 + a^6*x^6 - 1))/
(12*a*(384*a^2*x^2 - 384*a^4*x^4 + 128*a^6*x^6 - 128))) + ((227*x)/2 + 173*a*x^2 - 593/(6*a) - (599*a^2*x^3)/3
- (159*a^3*x^4)/2 + (183*a^4*x^5)/2)/(2304*a^2*x^2 - 2304*a^4*x^4 + 768*a^6*x^6 - 768) + ((299*x)/2 - 195*a*x
^2 + 331/(3*a) - (800*a^2*x^3)/3 + 90*a^3*x^4 + (245*a^4*x^5)/2)/(2304*a^2*x^2 - 2304*a^4*x^4 + 768*a^6*x^6 -
768) + (log(a*x + 1)^2*(66*x - 80*a^2*x^3 + 30*a^4*x^5))/(768*a^2*x^2 - 768*a^4*x^4 + 256*a^6*x^6 - 256)) - (l
og(a*x + 1)*((299*x)/(768*a) - (25*a*x^3)/36 + (245*a^3*x^5)/768))/(3*a*x^2 - 1/a - 3*a^3*x^4 + a^5*x^6) - (lo
g(a*x + 1)^3*((11*x)/(128*a) - (5*a*x^3)/48 + (5*a^3*x^5)/128))/(3*a*x^2 - 1/a - 3*a^3*x^4 + a^5*x^6)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {atanh}^{3}{\left (a x \right )}}{\left (a x - 1\right )^{4} \left (a x + 1\right )^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(atanh(a*x)**3/(-a**2*x**2+1)**4,x)

[Out]

Integral(atanh(a*x)**3/((a*x - 1)**4*(a*x + 1)**4), x)

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