3.2.32 \(\int \frac {\tanh ^{-1}(a x)^3}{x^2 (c+a c x)} \, dx\) [132]

Optimal. Leaf size=191 \[ \frac {a \tanh ^{-1}(a x)^3}{c}-\frac {\tanh ^{-1}(a x)^3}{c x}+\frac {3 a \tanh ^{-1}(a x)^2 \log \left (2-\frac {2}{1+a x}\right )}{c}-\frac {a \tanh ^{-1}(a x)^3 \log \left (2-\frac {2}{1+a x}\right )}{c}-\frac {3 a \tanh ^{-1}(a x) \text {PolyLog}\left (2,-1+\frac {2}{1+a x}\right )}{c}+\frac {3 a \tanh ^{-1}(a x)^2 \text {PolyLog}\left (2,-1+\frac {2}{1+a x}\right )}{2 c}-\frac {3 a \text {PolyLog}\left (3,-1+\frac {2}{1+a x}\right )}{2 c}+\frac {3 a \tanh ^{-1}(a x) \text {PolyLog}\left (3,-1+\frac {2}{1+a x}\right )}{2 c}+\frac {3 a \text {PolyLog}\left (4,-1+\frac {2}{1+a x}\right )}{4 c} \]

[Out]

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

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Rubi [A]
time = 0.44, antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {6081, 6037, 6135, 6079, 6095, 6203, 6745, 6207} \begin {gather*} -\frac {3 a \text {Li}_3\left (\frac {2}{a x+1}-1\right )}{2 c}+\frac {3 a \text {Li}_4\left (\frac {2}{a x+1}-1\right )}{4 c}+\frac {3 a \text {Li}_2\left (\frac {2}{a x+1}-1\right ) \tanh ^{-1}(a x)^2}{2 c}-\frac {3 a \text {Li}_2\left (\frac {2}{a x+1}-1\right ) \tanh ^{-1}(a x)}{c}+\frac {3 a \text {Li}_3\left (\frac {2}{a x+1}-1\right ) \tanh ^{-1}(a x)}{2 c}+\frac {a \tanh ^{-1}(a x)^3}{c}-\frac {\tanh ^{-1}(a x)^3}{c x}-\frac {a \log \left (2-\frac {2}{a x+1}\right ) \tanh ^{-1}(a x)^3}{c}+\frac {3 a \log \left (2-\frac {2}{a x+1}\right ) \tanh ^{-1}(a x)^2}{c} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[ArcTanh[a*x]^3/(x^2*(c + a*c*x)),x]

[Out]

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

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 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 6081

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

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 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 6203

Int[(Log[u_]*((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(a + b*ArcTan
h[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 6207

Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*PolyLog[k_, u_])/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(-(a
+ b*ArcTanh[c*x])^p)*(PolyLog[k + 1, u]/(2*c*d)), x] + Dist[b*(p/2), Int[(a + b*ArcTanh[c*x])^(p - 1)*(PolyLog
[k + 1, u]/(d + e*x^2)), x], x] /; FreeQ[{a, b, c, d, e, k}, x] && IGtQ[p, 0] && EqQ[c^2*d + e, 0] && EqQ[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 {\tanh ^{-1}(a x)^3}{x^2 (c+a c x)} \, dx &=-\left (a \int \frac {\tanh ^{-1}(a x)^3}{x (c+a c x)} \, dx\right )+\frac {\int \frac {\tanh ^{-1}(a x)^3}{x^2} \, dx}{c}\\ &=-\frac {\tanh ^{-1}(a x)^3}{c x}-\frac {a \tanh ^{-1}(a x)^3 \log \left (2-\frac {2}{1+a x}\right )}{c}+\frac {(3 a) \int \frac {\tanh ^{-1}(a x)^2}{x \left (1-a^2 x^2\right )} \, dx}{c}+\frac {\left (3 a^2\right ) \int \frac {\tanh ^{-1}(a x)^2 \log \left (2-\frac {2}{1+a x}\right )}{1-a^2 x^2} \, dx}{c}\\ &=\frac {a \tanh ^{-1}(a x)^3}{c}-\frac {\tanh ^{-1}(a x)^3}{c x}-\frac {a \tanh ^{-1}(a x)^3 \log \left (2-\frac {2}{1+a x}\right )}{c}+\frac {3 a \tanh ^{-1}(a x)^2 \text {Li}_2\left (-1+\frac {2}{1+a x}\right )}{2 c}+\frac {(3 a) \int \frac {\tanh ^{-1}(a x)^2}{x (1+a x)} \, dx}{c}-\frac {\left (3 a^2\right ) \int \frac {\tanh ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1+a x}\right )}{1-a^2 x^2} \, dx}{c}\\ &=\frac {a \tanh ^{-1}(a x)^3}{c}-\frac {\tanh ^{-1}(a x)^3}{c x}+\frac {3 a \tanh ^{-1}(a x)^2 \log \left (2-\frac {2}{1+a x}\right )}{c}-\frac {a \tanh ^{-1}(a x)^3 \log \left (2-\frac {2}{1+a x}\right )}{c}+\frac {3 a \tanh ^{-1}(a x)^2 \text {Li}_2\left (-1+\frac {2}{1+a x}\right )}{2 c}+\frac {3 a \tanh ^{-1}(a x) \text {Li}_3\left (-1+\frac {2}{1+a x}\right )}{2 c}-\frac {\left (3 a^2\right ) \int \frac {\text {Li}_3\left (-1+\frac {2}{1+a x}\right )}{1-a^2 x^2} \, dx}{2 c}-\frac {\left (6 a^2\right ) \int \frac {\tanh ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )}{1-a^2 x^2} \, dx}{c}\\ &=\frac {a \tanh ^{-1}(a x)^3}{c}-\frac {\tanh ^{-1}(a x)^3}{c x}+\frac {3 a \tanh ^{-1}(a x)^2 \log \left (2-\frac {2}{1+a x}\right )}{c}-\frac {a \tanh ^{-1}(a x)^3 \log \left (2-\frac {2}{1+a x}\right )}{c}-\frac {3 a \tanh ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1+a x}\right )}{c}+\frac {3 a \tanh ^{-1}(a x)^2 \text {Li}_2\left (-1+\frac {2}{1+a x}\right )}{2 c}+\frac {3 a \tanh ^{-1}(a x) \text {Li}_3\left (-1+\frac {2}{1+a x}\right )}{2 c}+\frac {3 a \text {Li}_4\left (-1+\frac {2}{1+a x}\right )}{4 c}+\frac {\left (3 a^2\right ) \int \frac {\text {Li}_2\left (-1+\frac {2}{1+a x}\right )}{1-a^2 x^2} \, dx}{c}\\ &=\frac {a \tanh ^{-1}(a x)^3}{c}-\frac {\tanh ^{-1}(a x)^3}{c x}+\frac {3 a \tanh ^{-1}(a x)^2 \log \left (2-\frac {2}{1+a x}\right )}{c}-\frac {a \tanh ^{-1}(a x)^3 \log \left (2-\frac {2}{1+a x}\right )}{c}-\frac {3 a \tanh ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1+a x}\right )}{c}+\frac {3 a \tanh ^{-1}(a x)^2 \text {Li}_2\left (-1+\frac {2}{1+a x}\right )}{2 c}-\frac {3 a \text {Li}_3\left (-1+\frac {2}{1+a x}\right )}{2 c}+\frac {3 a \tanh ^{-1}(a x) \text {Li}_3\left (-1+\frac {2}{1+a x}\right )}{2 c}+\frac {3 a \text {Li}_4\left (-1+\frac {2}{1+a x}\right )}{4 c}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.17, size = 154, normalized size = 0.81 \begin {gather*} \frac {a \left (\frac {i \pi ^3}{8}-\frac {\pi ^4}{64}-\tanh ^{-1}(a x)^3-\frac {\tanh ^{-1}(a x)^3}{a x}+\frac {1}{2} \tanh ^{-1}(a x)^4+3 \tanh ^{-1}(a x)^2 \log \left (1-e^{2 \tanh ^{-1}(a x)}\right )-\tanh ^{-1}(a x)^3 \log \left (1-e^{2 \tanh ^{-1}(a x)}\right )-\frac {3}{2} \left (-2+\tanh ^{-1}(a x)\right ) \tanh ^{-1}(a x) \text {PolyLog}\left (2,e^{2 \tanh ^{-1}(a x)}\right )+\frac {3}{2} \left (-1+\tanh ^{-1}(a x)\right ) \text {PolyLog}\left (3,e^{2 \tanh ^{-1}(a x)}\right )-\frac {3}{4} \text {PolyLog}\left (4,e^{2 \tanh ^{-1}(a x)}\right )\right )}{c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a*((I/8)*Pi^3 - Pi^4/64 - ArcTanh[a*x]^3 - ArcTanh[a*x]^3/(a*x) + ArcTanh[a*x]^4/2 + 3*ArcTanh[a*x]^2*Log[1 -
E^(2*ArcTanh[a*x])] - ArcTanh[a*x]^3*Log[1 - E^(2*ArcTanh[a*x])] - (3*(-2 + ArcTanh[a*x])*ArcTanh[a*x]*PolyLo
g[2, E^(2*ArcTanh[a*x])])/2 + (3*(-1 + ArcTanh[a*x])*PolyLog[3, E^(2*ArcTanh[a*x])])/2 - (3*PolyLog[4, E^(2*Ar
cTanh[a*x])])/4))/c

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 26.78, size = 1339, normalized size = 7.01

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1339\)
default	\(\text {Expression too large to display}\)	\(1339\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*(-1/c*arctanh(a*x)^3/a/x-1/c*arctanh(a*x)^3*ln(a*x)+1/c*arctanh(a*x)^3*ln(a*x+1)-3/c*(-1/6*I*Pi*csgn(I/((a*x
+1)^2/(-a^2*x^2+1)+1))*csgn(I*(a*x+1)^2/(a^2*x^2-1))*csgn(I*(a*x+1)^2/(a^2*x^2-1)/((a*x+1)^2/(-a^2*x^2+1)+1))*
arctanh(a*x)^3+1/6*I*Pi*csgn(I*((a*x+1)^2/(-a^2*x^2+1)-1))*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))*csgn(I*((a*x+1)^
2/(-a^2*x^2+1)-1)/((a*x+1)^2/(-a^2*x^2+1)+1))*arctanh(a*x)^3+2*polylog(3,-(a*x+1)/(-a^2*x^2+1)^(1/2))+2*polylo
g(3,(a*x+1)/(-a^2*x^2+1)^(1/2))+2*polylog(4,-(a*x+1)/(-a^2*x^2+1)^(1/2))+2*polylog(4,(a*x+1)/(-a^2*x^2+1)^(1/2
))-2*arctanh(a*x)*polylog(2,-(a*x+1)/(-a^2*x^2+1)^(1/2))-2*arctanh(a*x)*polylog(2,(a*x+1)/(-a^2*x^2+1)^(1/2))+
arctanh(a*x)^2*polylog(2,-(a*x+1)/(-a^2*x^2+1)^(1/2))+arctanh(a*x)^2*polylog(2,(a*x+1)/(-a^2*x^2+1)^(1/2))-2*a
rctanh(a*x)*polylog(3,-(a*x+1)/(-a^2*x^2+1)^(1/2))-2*arctanh(a*x)*polylog(3,(a*x+1)/(-a^2*x^2+1)^(1/2))-1/6*ar
ctanh(a*x)^4+1/3*arctanh(a*x)^3-arctanh(a*x)^2*ln(1+(a*x+1)/(-a^2*x^2+1)^(1/2))-arctanh(a*x)^2*ln(1-(a*x+1)/(-
a^2*x^2+1)^(1/2))-1/3*arctanh(a*x)^3*ln((a*x+1)^2/(-a^2*x^2+1)-1)+1/3*arctanh(a*x)^3*ln(1-(a*x+1)/(-a^2*x^2+1)
^(1/2))+1/3*arctanh(a*x)^3*ln(1+(a*x+1)/(-a^2*x^2+1)^(1/2))+1/6*I*Pi*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))*csgn(I
*(a*x+1)^2/(a^2*x^2-1)/((a*x+1)^2/(-a^2*x^2+1)+1))^2*arctanh(a*x)^3+1/3*I*Pi*csgn(I*(a*x+1)/(-a^2*x^2+1)^(1/2)
)*csgn(I*(a*x+1)^2/(a^2*x^2-1))^2*arctanh(a*x)^3+1/6*I*Pi*csgn(I*(a*x+1)/(-a^2*x^2+1)^(1/2))^2*csgn(I*(a*x+1)^
2/(a^2*x^2-1))*arctanh(a*x)^3-1/6*I*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1))*csgn(I*(a*x+1)^2/(a^2*x^2-1)/((a*x+1)^2/(
-a^2*x^2+1)+1))^2*arctanh(a*x)^3-1/6*I*Pi*csgn(I*((a*x+1)^2/(-a^2*x^2+1)-1))*csgn(I*((a*x+1)^2/(-a^2*x^2+1)-1)
/((a*x+1)^2/(-a^2*x^2+1)+1))^2*arctanh(a*x)^3-1/6*I*Pi*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))*csgn(I*((a*x+1)^2/(-
a^2*x^2+1)-1)/((a*x+1)^2/(-a^2*x^2+1)+1))^2*arctanh(a*x)^3+1/6*I*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1)/((a*x+1)^2/(-
a^2*x^2+1)+1))^3*arctanh(a*x)^3+1/6*I*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1))^3*arctanh(a*x)^3+1/6*I*Pi*csgn(I*((a*x+
1)^2/(-a^2*x^2+1)-1)/((a*x+1)^2/(-a^2*x^2+1)+1))^3*arctanh(a*x)^3+1/3*ln(2)*arctanh(a*x)^3+2/3*arctanh(a*x)^3*
ln((a*x+1)/(-a^2*x^2+1)^(1/2))))

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

[Out]

-1/8*(a*x*log(a*x + 1) - 1)*log(-a*x + 1)^3/(c*x) + 1/8*integrate(((a*x - 1)*log(a*x + 1)^3 - 3*(a*x - 1)*log(
a*x + 1)^2*log(-a*x + 1) - 3*(a^2*x^2 + a*x - (a^3*x^3 + a^2*x^2 + a*x - 1)*log(a*x + 1))*log(-a*x + 1)^2)/(a^
2*c*x^4 - c*x^2), x)

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

[Out]

integral(arctanh(a*x)^3/(a*c*x^3 + c*x^2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(atanh(a*x)**3/(a*x**3 + x**2), x)/c

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

[Out]

integrate(arctanh(a*x)^3/((a*c*x + c)*x^2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\mathrm {atanh}\left (a\,x\right )}^3}{x^2\,\left (c+a\,c\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(atanh(a*x)^3/(x^2*(c + a*c*x)),x)

[Out]

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

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