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

Optimal. Leaf size=188 \[ -\frac {45 x}{256 a \left (1-a^2 x^2\right )}-\frac {3 x}{128 a \left (1-a^2 x^2\right )^2}+\frac {\tanh ^{-1}(a x)^3}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac {9 x \tanh ^{-1}(a x)^2}{32 a \left (1-a^2 x^2\right )}-\frac {3 x \tanh ^{-1}(a x)^2}{16 a \left (1-a^2 x^2\right )^2}+\frac {9 \tanh ^{-1}(a x)}{32 a^2 \left (1-a^2 x^2\right )}+\frac {3 \tanh ^{-1}(a x)}{32 a^2 \left (1-a^2 x^2\right )^2}-\frac {3 \tanh ^{-1}(a x)^3}{32 a^2}-\frac {45 \tanh ^{-1}(a x)}{256 a^2} \]

[Out]

-3/128*x/a/(-a^2*x^2+1)^2-45/256*x/a/(-a^2*x^2+1)-45/256*arctanh(a*x)/a^2+3/32*arctanh(a*x)/a^2/(-a^2*x^2+1)^2
+9/32*arctanh(a*x)/a^2/(-a^2*x^2+1)-3/16*x*arctanh(a*x)^2/a/(-a^2*x^2+1)^2-9/32*x*arctanh(a*x)^2/a/(-a^2*x^2+1
)-3/32*arctanh(a*x)^3/a^2+1/4*arctanh(a*x)^3/a^2/(-a^2*x^2+1)^2

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Rubi [A]  time = 0.16, antiderivative size = 188, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5994, 5964, 5956, 199, 206} \[ -\frac {45 x}{256 a \left (1-a^2 x^2\right )}-\frac {3 x}{128 a \left (1-a^2 x^2\right )^2}+\frac {\tanh ^{-1}(a x)^3}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac {9 x \tanh ^{-1}(a x)^2}{32 a \left (1-a^2 x^2\right )}-\frac {3 x \tanh ^{-1}(a x)^2}{16 a \left (1-a^2 x^2\right )^2}+\frac {9 \tanh ^{-1}(a x)}{32 a^2 \left (1-a^2 x^2\right )}+\frac {3 \tanh ^{-1}(a x)}{32 a^2 \left (1-a^2 x^2\right )^2}-\frac {3 \tanh ^{-1}(a x)^3}{32 a^2}-\frac {45 \tanh ^{-1}(a x)}{256 a^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*x)/(128*a*(1 - a^2*x^2)^2) - (45*x)/(256*a*(1 - a^2*x^2)) - (45*ArcTanh[a*x])/(256*a^2) + (3*ArcTanh[a*x])
/(32*a^2*(1 - a^2*x^2)^2) + (9*ArcTanh[a*x])/(32*a^2*(1 - a^2*x^2)) - (3*x*ArcTanh[a*x]^2)/(16*a*(1 - a^2*x^2)
^2) - (9*x*ArcTanh[a*x]^2)/(32*a*(1 - a^2*x^2)) - (3*ArcTanh[a*x]^3)/(32*a^2) + ArcTanh[a*x]^3/(4*a^2*(1 - a^2
*x^2)^2)

Rule 199

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

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 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 {x \tanh ^{-1}(a x)^3}{\left (1-a^2 x^2\right )^3} \, dx &=\frac {\tanh ^{-1}(a x)^3}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac {3 \int \frac {\tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^3} \, dx}{4 a}\\ &=\frac {3 \tanh ^{-1}(a x)}{32 a^2 \left (1-a^2 x^2\right )^2}-\frac {3 x \tanh ^{-1}(a x)^2}{16 a \left (1-a^2 x^2\right )^2}+\frac {\tanh ^{-1}(a x)^3}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac {3 \int \frac {1}{\left (1-a^2 x^2\right )^3} \, dx}{32 a}-\frac {9 \int \frac {\tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^2} \, dx}{16 a}\\ &=-\frac {3 x}{128 a \left (1-a^2 x^2\right )^2}+\frac {3 \tanh ^{-1}(a x)}{32 a^2 \left (1-a^2 x^2\right )^2}-\frac {3 x \tanh ^{-1}(a x)^2}{16 a \left (1-a^2 x^2\right )^2}-\frac {9 x \tanh ^{-1}(a x)^2}{32 a \left (1-a^2 x^2\right )}-\frac {3 \tanh ^{-1}(a x)^3}{32 a^2}+\frac {\tanh ^{-1}(a x)^3}{4 a^2 \left (1-a^2 x^2\right )^2}+\frac {9}{16} \int \frac {x \tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^2} \, dx-\frac {9 \int \frac {1}{\left (1-a^2 x^2\right )^2} \, dx}{128 a}\\ &=-\frac {3 x}{128 a \left (1-a^2 x^2\right )^2}-\frac {9 x}{256 a \left (1-a^2 x^2\right )}+\frac {3 \tanh ^{-1}(a x)}{32 a^2 \left (1-a^2 x^2\right )^2}+\frac {9 \tanh ^{-1}(a x)}{32 a^2 \left (1-a^2 x^2\right )}-\frac {3 x \tanh ^{-1}(a x)^2}{16 a \left (1-a^2 x^2\right )^2}-\frac {9 x \tanh ^{-1}(a x)^2}{32 a \left (1-a^2 x^2\right )}-\frac {3 \tanh ^{-1}(a x)^3}{32 a^2}+\frac {\tanh ^{-1}(a x)^3}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac {9 \int \frac {1}{1-a^2 x^2} \, dx}{256 a}-\frac {9 \int \frac {1}{\left (1-a^2 x^2\right )^2} \, dx}{32 a}\\ &=-\frac {3 x}{128 a \left (1-a^2 x^2\right )^2}-\frac {45 x}{256 a \left (1-a^2 x^2\right )}-\frac {9 \tanh ^{-1}(a x)}{256 a^2}+\frac {3 \tanh ^{-1}(a x)}{32 a^2 \left (1-a^2 x^2\right )^2}+\frac {9 \tanh ^{-1}(a x)}{32 a^2 \left (1-a^2 x^2\right )}-\frac {3 x \tanh ^{-1}(a x)^2}{16 a \left (1-a^2 x^2\right )^2}-\frac {9 x \tanh ^{-1}(a x)^2}{32 a \left (1-a^2 x^2\right )}-\frac {3 \tanh ^{-1}(a x)^3}{32 a^2}+\frac {\tanh ^{-1}(a x)^3}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac {9 \int \frac {1}{1-a^2 x^2} \, dx}{64 a}\\ &=-\frac {3 x}{128 a \left (1-a^2 x^2\right )^2}-\frac {45 x}{256 a \left (1-a^2 x^2\right )}-\frac {45 \tanh ^{-1}(a x)}{256 a^2}+\frac {3 \tanh ^{-1}(a x)}{32 a^2 \left (1-a^2 x^2\right )^2}+\frac {9 \tanh ^{-1}(a x)}{32 a^2 \left (1-a^2 x^2\right )}-\frac {3 x \tanh ^{-1}(a x)^2}{16 a \left (1-a^2 x^2\right )^2}-\frac {9 x \tanh ^{-1}(a x)^2}{32 a \left (1-a^2 x^2\right )}-\frac {3 \tanh ^{-1}(a x)^3}{32 a^2}+\frac {\tanh ^{-1}(a x)^3}{4 a^2 \left (1-a^2 x^2\right )^2}\\ \end {align*}

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Mathematica [A]  time = 0.09, size = 148, normalized size = 0.79 \[ \frac {-45 a^4 x^4 \log (a x+1)+90 a^3 x^3+90 a^2 x^2 \log (a x+1)+45 \left (a^2 x^2-1\right )^2 \log (1-a x)+48 a x \left (3 a^2 x^2-5\right ) \tanh ^{-1}(a x)^2-48 \left (3 a^2 x^2-4\right ) \tanh ^{-1}(a x)+\left (-48 a^4 x^4+96 a^2 x^2+80\right ) \tanh ^{-1}(a x)^3-102 a x-45 \log (a x+1)}{512 a^2 \left (a^2 x^2-1\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-102*a*x + 90*a^3*x^3 - 48*(-4 + 3*a^2*x^2)*ArcTanh[a*x] + 48*a*x*(-5 + 3*a^2*x^2)*ArcTanh[a*x]^2 + (80 + 96*
a^2*x^2 - 48*a^4*x^4)*ArcTanh[a*x]^3 + 45*(-1 + a^2*x^2)^2*Log[1 - a*x] - 45*Log[1 + a*x] + 90*a^2*x^2*Log[1 +
a*x] - 45*a^4*x^4*Log[1 + a*x])/(512*a^2*(-1 + a^2*x^2)^2)

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fricas [A]  time = 0.46, size = 140, normalized size = 0.74 \[ \frac {90 \, a^{3} x^{3} - 2 \, {\left (3 \, a^{4} x^{4} - 6 \, a^{2} x^{2} - 5\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{3} + 12 \, {\left (3 \, a^{3} x^{3} - 5 \, a x\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} - 102 \, a x - 3 \, {\left (15 \, a^{4} x^{4} - 6 \, a^{2} x^{2} - 17\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )}{512 \, {\left (a^{6} x^{4} - 2 \, a^{4} x^{2} + a^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/512*(90*a^3*x^3 - 2*(3*a^4*x^4 - 6*a^2*x^2 - 5)*log(-(a*x + 1)/(a*x - 1))^3 + 12*(3*a^3*x^3 - 5*a*x)*log(-(a
*x + 1)/(a*x - 1))^2 - 102*a*x - 3*(15*a^4*x^4 - 6*a^2*x^2 - 17)*log(-(a*x + 1)/(a*x - 1)))/(a^6*x^4 - 2*a^4*x
^2 + a^2)

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giac [B]  time = 0.29, size = 342, normalized size = 1.82 \[ -\frac {1}{2048} \, {\left (4 \, {\left (\frac {{\left (a x - 1\right )}^{2} {\left (\frac {4 \, {\left (a x + 1\right )}}{a x - 1} - 1\right )}}{{\left (a x + 1\right )}^{2} a^{3}} - \frac {{\left (a x + 1\right )}^{2}}{{\left (a x - 1\right )}^{2} a^{3}} + \frac {4 \, {\left (a x + 1\right )}}{{\left (a x - 1\right )} a^{3}}\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{3} + 6 \, {\left (\frac {{\left (a x - 1\right )}^{2} {\left (\frac {8 \, {\left (a x + 1\right )}}{a x - 1} - 1\right )}}{{\left (a x + 1\right )}^{2} a^{3}} + \frac {{\left (a x + 1\right )}^{2}}{{\left (a x - 1\right )}^{2} a^{3}} - \frac {8 \, {\left (a x + 1\right )}}{{\left (a x - 1\right )} a^{3}}\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} + 6 \, {\left (\frac {{\left (a x - 1\right )}^{2} {\left (\frac {16 \, {\left (a x + 1\right )}}{a x - 1} - 1\right )}}{{\left (a x + 1\right )}^{2} a^{3}} - \frac {{\left (a x + 1\right )}^{2}}{{\left (a x - 1\right )}^{2} a^{3}} + \frac {16 \, {\left (a x + 1\right )}}{{\left (a x - 1\right )} a^{3}}\right )} \log \left (-\frac {a x + 1}{a x - 1}\right ) + \frac {3 \, {\left (a x - 1\right )}^{2} {\left (\frac {32 \, {\left (a x + 1\right )}}{a x - 1} - 1\right )}}{{\left (a x + 1\right )}^{2} a^{3}} + \frac {3 \, {\left (a x + 1\right )}^{2}}{{\left (a x - 1\right )}^{2} a^{3}} - \frac {96 \, {\left (a x + 1\right )}}{{\left (a x - 1\right )} a^{3}}\right )} a \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2048*(4*((a*x - 1)^2*(4*(a*x + 1)/(a*x - 1) - 1)/((a*x + 1)^2*a^3) - (a*x + 1)^2/((a*x - 1)^2*a^3) + 4*(a*x
+ 1)/((a*x - 1)*a^3))*log(-(a*x + 1)/(a*x - 1))^3 + 6*((a*x - 1)^2*(8*(a*x + 1)/(a*x - 1) - 1)/((a*x + 1)^2*a
^3) + (a*x + 1)^2/((a*x - 1)^2*a^3) - 8*(a*x + 1)/((a*x - 1)*a^3))*log(-(a*x + 1)/(a*x - 1))^2 + 6*((a*x - 1)^
2*(16*(a*x + 1)/(a*x - 1) - 1)/((a*x + 1)^2*a^3) - (a*x + 1)^2/((a*x - 1)^2*a^3) + 16*(a*x + 1)/((a*x - 1)*a^3
))*log(-(a*x + 1)/(a*x - 1)) + 3*(a*x - 1)^2*(32*(a*x + 1)/(a*x - 1) - 1)/((a*x + 1)^2*a^3) + 3*(a*x + 1)^2/((
a*x - 1)^2*a^3) - 96*(a*x + 1)/((a*x - 1)*a^3))*a

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maple [C]  time = 0.72, size = 2582, normalized size = 13.73 \[ \text {Expression too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-9/32*I/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)^2*csgn(I*(a*x+1)^2/(a^2*x^2-1))^2*csgn(I*(a*x+1)/(-a^2*x^2+1)^(1/2))*
Pi*x^2-9/64*I/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)^2*csgn(I*(a*x+1)^2/(a^2*x^2-1))*csgn(I*(a*x+1)/(-a^2*x^2+1)^(1/
2))^2*Pi*x^2-9/64*I/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)^2*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*Pi*x^2-9/64*I*a^2/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)^2*csgn(I/(1+(a*x+1)^
2/(-a^2*x^2+1)))^3*Pi*x^4+9/128*I*a^2/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)^2*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(1+(a*x+
1)^2/(-a^2*x^2+1)))^3*Pi*x^4+9/128*I*a^2/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)^2*csgn(I*(a*x+1)^2/(a^2*x^2-1))^3*Pi
*x^4+9/64*I*a^2/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)^2*csgn(I/(1+(a*x+1)^2/(-a^2*x^2+1)))^2*Pi*x^4+9/128*I/a^2/(a*
x-1)^2/(a*x+1)^2*Pi*arctanh(a*x)^2*csgn(I*(a*x+1)/(-a^2*x^2+1)^(1/2))^2*csgn(I*(a*x+1)^2/(a^2*x^2-1))+9/128*I/
a^2/(a*x-1)^2/(a*x+1)^2*Pi*arctanh(a*x)^2*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)))^2-9/128*I/a^2/(a*x-1)^2/(a*x+1)^2*Pi*arctanh(a*x)^2*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+9/64*I/a^2/(a*x-1)^2/(a*x+1)^2*Pi*arctanh(a*x)^2*csgn(I
*(a*x+1)/(-a^2*x^2+1)^(1/2))*csgn(I*(a*x+1)^2/(a^2*x^2-1))^2+9/64*I/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)^2*csgn(I*
(a*x+1)^2/(a^2*x^2-1)/(1+(a*x+1)^2/(-a^2*x^2+1)))^2*csgn(I*(a*x+1)^2/(a^2*x^2-1))*Pi*x^2-9/128*I*a^2/(a*x-1)^2
/(a*x+1)^2*arctanh(a*x)^2*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(1+(a*x+1)^2/(-a^2*x^2+1)))^2*csgn(I*(a*x+1)^2/(a^2*x^2
-1))*Pi*x^4+9/64*I*a^2/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)^2*csgn(I*(a*x+1)^2/(a^2*x^2-1))^2*csgn(I*(a*x+1)/(-a^2
*x^2+1)^(1/2))*Pi*x^4+9/128*I*a^2/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)^2*csgn(I*(a*x+1)^2/(a^2*x^2-1))*csgn(I*(a*x
+1)/(-a^2*x^2+1)^(1/2))^2*Pi*x^4+9/64*I/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)^2*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)))*csgn(I*(a*x+1)^2/(a^2*x^2-1))*Pi*x^2-9/128*I/a^2/(a*x
-1)^2/(a*x+1)^2*Pi*arctanh(a*x)^2*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)))+9/128*I*a^2/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)^2*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*Pi*x^4-9/128*I*a^2/(a*x-1)^2/(a*x
+1)^2*arctanh(a*x)^2*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)
))*csgn(I*(a*x+1)^2/(a^2*x^2-1))*Pi*x^4-3/32*a^2/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)^3*x^4-45/256*a^2/(a*x-1)^2/(
a*x+1)^2*arctanh(a*x)*x^4+1/4/a^2/(a^2*x^2-1)^2*arctanh(a*x)^3-3/64/a^2*arctanh(a*x)^2/(a*x-1)^2+9/64/a^2*arct
anh(a*x)^2/(a*x-1)+9/64/a^2*arctanh(a*x)^2*ln(a*x-1)+3/64/a^2*arctanh(a*x)^2/(a*x+1)^2+9/64/a^2*arctanh(a*x)^2
/(a*x+1)-9/64/a^2*arctanh(a*x)^2*ln(a*x+1)+9/32/a^2*arctanh(a*x)^2*ln((a*x+1)/(-a^2*x^2+1)^(1/2))-9/64*I/a^2/(
a*x-1)^2/(a*x+1)^2*Pi*arctanh(a*x)^2+9/32*I/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)^2*Pi*x^2+9/128*I/a^2/(a*x-1)^2/(a
*x+1)^2*Pi*arctanh(a*x)^2*csgn(I*(a*x+1)^2/(a^2*x^2-1))^3+9/64*I/a^2/(a*x-1)^2/(a*x+1)^2*Pi*arctanh(a*x)^2*csg
n(I/(1+(a*x+1)^2/(-a^2*x^2+1)))^2-9/64*I*a^2/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)^2*Pi*x^4-9/64*I/(a*x-1)^2/(a*x+1
)^2*arctanh(a*x)^2*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(1+(a*x+1)^2/(-a^2*x^2+1)))^3*Pi*x^2-9/64*I/(a*x-1)^2/(a*x+1)^
2*arctanh(a*x)^2*csgn(I*(a*x+1)^2/(a^2*x^2-1))^3*Pi*x^2-9/32*I/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)^2*csgn(I/(1+(a
*x+1)^2/(-a^2*x^2+1)))^2*Pi*x^2+9/32*I/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)^2*csgn(I/(1+(a*x+1)^2/(-a^2*x^2+1)))^3
*Pi*x^2-9/64*I/a^2/(a*x-1)^2/(a*x+1)^2*Pi*arctanh(a*x)^2*csgn(I/(1+(a*x+1)^2/(-a^2*x^2+1)))^3+9/128*I/a^2/(a*x
-1)^2/(a*x+1)^2*Pi*arctanh(a*x)^2*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(1+(a*x+1)^2/(-a^2*x^2+1)))^3+3/16/(a*x-1)^2/(a
*x+1)^2*arctanh(a*x)^3*x^2+9/128/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)*x^2-3/32/a^2/(a*x-1)^2/(a*x+1)^2*arctanh(a*x
)^3+51/256/a^2/(a*x-1)^2/(a*x+1)^2*arctanh(a*x)+45/256*a/(a*x-1)^2/(a*x+1)^2*x^3-51/256/a/(a*x-1)^2/(a*x+1)^2*
x

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maxima [B]  time = 0.34, size = 422, normalized size = 2.24 \[ \frac {3 \, {\left (\frac {2 \, {\left (3 \, a^{2} x^{3} - 5 \, x\right )}}{a^{4} x^{4} - 2 \, a^{2} x^{2} + 1} - \frac {3 \, \log \left (a x + 1\right )}{a} + \frac {3 \, \log \left (a x - 1\right )}{a}\right )} \operatorname {artanh}\left (a x\right )^{2}}{64 \, a} + \frac {3 \, {\left (\frac {{\left (30 \, a^{3} x^{3} - 2 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right )^{3} + 6 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right )^{2} \log \left (a x - 1\right ) + 2 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x - 1\right )^{3} - 34 \, a x - 3 \, {\left (5 \, a^{4} x^{4} - 10 \, a^{2} x^{2} + 2 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x - 1\right )^{2} + 5\right )} \log \left (a x + 1\right ) + 15 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x - 1\right )\right )} a^{2}}{a^{7} x^{4} - 2 \, a^{5} x^{2} + a^{3}} - \frac {4 \, {\left (12 \, a^{2} x^{2} - 3 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right )^{2} + 6 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right ) \log \left (a x - 1\right ) - 3 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x - 1\right )^{2} - 16\right )} a \operatorname {artanh}\left (a x\right )}{a^{6} x^{4} - 2 \, a^{4} x^{2} + a^{2}}\right )}}{512 \, a} + \frac {\operatorname {artanh}\left (a x\right )^{3}}{4 \, {\left (a^{2} x^{2} - 1\right )}^{2} a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/64*(2*(3*a^2*x^3 - 5*x)/(a^4*x^4 - 2*a^2*x^2 + 1) - 3*log(a*x + 1)/a + 3*log(a*x - 1)/a)*arctanh(a*x)^2/a +
3/512*((30*a^3*x^3 - 2*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x + 1)^3 + 6*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x + 1)^2*l
og(a*x - 1) + 2*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x - 1)^3 - 34*a*x - 3*(5*a^4*x^4 - 10*a^2*x^2 + 2*(a^4*x^4 - 2
*a^2*x^2 + 1)*log(a*x - 1)^2 + 5)*log(a*x + 1) + 15*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x - 1))*a^2/(a^7*x^4 - 2*a
^5*x^2 + a^3) - 4*(12*a^2*x^2 - 3*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x + 1)^2 + 6*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a
*x + 1)*log(a*x - 1) - 3*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x - 1)^2 - 16)*a*arctanh(a*x)/(a^6*x^4 - 2*a^4*x^2 +
a^2))/a + 1/4*arctanh(a*x)^3/((a^2*x^2 - 1)^2*a^2)

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mupad [B]  time = 2.69, size = 414, normalized size = 2.20 \[ -\frac {48\,\ln \left (1-a\,x\right )-48\,\ln \left (a\,x+1\right )+45\,\mathrm {atanh}\left (a\,x\right )+51\,a\,x-5\,{\ln \left (a\,x+1\right )}^3+5\,{\ln \left (1-a\,x\right )}^3-15\,\ln \left (a\,x+1\right )\,{\ln \left (1-a\,x\right )}^2+15\,{\ln \left (a\,x+1\right )}^2\,\ln \left (1-a\,x\right )-45\,a^3\,x^3-6\,a^2\,x^2\,{\ln \left (a\,x+1\right )}^3+6\,a^2\,x^2\,{\ln \left (1-a\,x\right )}^3-18\,a^3\,x^3\,{\ln \left (a\,x+1\right )}^2-18\,a^3\,x^3\,{\ln \left (1-a\,x\right )}^2+3\,a^4\,x^4\,{\ln \left (a\,x+1\right )}^3-3\,a^4\,x^4\,{\ln \left (1-a\,x\right )}^3-90\,a^2\,x^2\,\mathrm {atanh}\left (a\,x\right )+45\,a^4\,x^4\,\mathrm {atanh}\left (a\,x\right )+30\,a\,x\,{\ln \left (a\,x+1\right )}^2+30\,a\,x\,{\ln \left (1-a\,x\right )}^2+36\,a^2\,x^2\,\ln \left (a\,x+1\right )-36\,a^2\,x^2\,\ln \left (1-a\,x\right )-60\,a\,x\,\ln \left (a\,x+1\right )\,\ln \left (1-a\,x\right )-18\,a^2\,x^2\,\ln \left (a\,x+1\right )\,{\ln \left (1-a\,x\right )}^2+18\,a^2\,x^2\,{\ln \left (a\,x+1\right )}^2\,\ln \left (1-a\,x\right )+9\,a^4\,x^4\,\ln \left (a\,x+1\right )\,{\ln \left (1-a\,x\right )}^2-9\,a^4\,x^4\,{\ln \left (a\,x+1\right )}^2\,\ln \left (1-a\,x\right )+36\,a^3\,x^3\,\ln \left (a\,x+1\right )\,\ln \left (1-a\,x\right )}{256\,a^2\,{\left (a^2\,x^2-1\right )}^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x*atanh(a*x)^3)/(a^2*x^2 - 1)^3,x)

[Out]

-(48*log(1 - a*x) - 48*log(a*x + 1) + 45*atanh(a*x) + 51*a*x - 5*log(a*x + 1)^3 + 5*log(1 - a*x)^3 - 15*log(a*
x + 1)*log(1 - a*x)^2 + 15*log(a*x + 1)^2*log(1 - a*x) - 45*a^3*x^3 - 6*a^2*x^2*log(a*x + 1)^3 + 6*a^2*x^2*log
(1 - a*x)^3 - 18*a^3*x^3*log(a*x + 1)^2 - 18*a^3*x^3*log(1 - a*x)^2 + 3*a^4*x^4*log(a*x + 1)^3 - 3*a^4*x^4*log
(1 - a*x)^3 - 90*a^2*x^2*atanh(a*x) + 45*a^4*x^4*atanh(a*x) + 30*a*x*log(a*x + 1)^2 + 30*a*x*log(1 - a*x)^2 +
36*a^2*x^2*log(a*x + 1) - 36*a^2*x^2*log(1 - a*x) - 60*a*x*log(a*x + 1)*log(1 - a*x) - 18*a^2*x^2*log(a*x + 1)
*log(1 - a*x)^2 + 18*a^2*x^2*log(a*x + 1)^2*log(1 - a*x) + 9*a^4*x^4*log(a*x + 1)*log(1 - a*x)^2 - 9*a^4*x^4*l
og(a*x + 1)^2*log(1 - a*x) + 36*a^3*x^3*log(a*x + 1)*log(1 - a*x))/(256*a^2*(a^2*x^2 - 1)^2)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - \int \frac {x \operatorname {atanh}^{3}{\left (a x \right )}}{a^{6} x^{6} - 3 a^{4} x^{4} + 3 a^{2} x^{2} - 1}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(x*atanh(a*x)**3/(a**6*x**6 - 3*a**4*x**4 + 3*a**2*x**2 - 1), x)

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