3.2.37 \(\int \frac {\tanh ^{-1}(a x)^4}{x (c-a c x)} \, dx\) [137]

Optimal. Leaf size=118 \[ \frac {\tanh ^{-1}(a x)^4 \log \left (2-\frac {2}{1-a x}\right )}{c}+\frac {2 \tanh ^{-1}(a x)^3 \text {PolyLog}\left (2,-1+\frac {2}{1-a x}\right )}{c}-\frac {3 \tanh ^{-1}(a x)^2 \text {PolyLog}\left (3,-1+\frac {2}{1-a x}\right )}{c}+\frac {3 \tanh ^{-1}(a x) \text {PolyLog}\left (4,-1+\frac {2}{1-a x}\right )}{c}-\frac {3 \text {PolyLog}\left (5,-1+\frac {2}{1-a x}\right )}{2 c} \]

[Out]

arctanh(a*x)^4*ln(2-2/(-a*x+1))/c+2*arctanh(a*x)^3*polylog(2,-1+2/(-a*x+1))/c-3*arctanh(a*x)^2*polylog(3,-1+2/
(-a*x+1))/c+3*arctanh(a*x)*polylog(4,-1+2/(-a*x+1))/c-3/2*polylog(5,-1+2/(-a*x+1))/c

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Rubi [A]
time = 0.16, antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {6079, 6095, 6205, 6209, 6745} \begin {gather*} -\frac {3 \text {Li}_5\left (\frac {2}{1-a x}-1\right )}{2 c}+\frac {2 \text {Li}_2\left (\frac {2}{1-a x}-1\right ) \tanh ^{-1}(a x)^3}{c}-\frac {3 \text {Li}_3\left (\frac {2}{1-a x}-1\right ) \tanh ^{-1}(a x)^2}{c}+\frac {3 \text {Li}_4\left (\frac {2}{1-a x}-1\right ) \tanh ^{-1}(a x)}{c}+\frac {\log \left (2-\frac {2}{1-a x}\right ) \tanh ^{-1}(a x)^4}{c} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[ArcTanh[a*x]^4/(x*(c - a*c*x)),x]

[Out]

(ArcTanh[a*x]^4*Log[2 - 2/(1 - a*x)])/c + (2*ArcTanh[a*x]^3*PolyLog[2, -1 + 2/(1 - a*x)])/c - (3*ArcTanh[a*x]^
2*PolyLog[3, -1 + 2/(1 - a*x)])/c + (3*ArcTanh[a*x]*PolyLog[4, -1 + 2/(1 - a*x)])/c - (3*PolyLog[5, -1 + 2/(1
- a*x)])/(2*c)

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 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 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 6209

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)^4}{x (c-a c x)} \, dx &=\frac {\tanh ^{-1}(a x)^4 \log \left (2-\frac {2}{1-a x}\right )}{c}-\frac {(4 a) \int \frac {\tanh ^{-1}(a x)^3 \log \left (2-\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx}{c}\\ &=\frac {\tanh ^{-1}(a x)^4 \log \left (2-\frac {2}{1-a x}\right )}{c}+\frac {2 \tanh ^{-1}(a x)^3 \text {Li}_2\left (-1+\frac {2}{1-a x}\right )}{c}-\frac {(6 a) \int \frac {\tanh ^{-1}(a x)^2 \text {Li}_2\left (-1+\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx}{c}\\ &=\frac {\tanh ^{-1}(a x)^4 \log \left (2-\frac {2}{1-a x}\right )}{c}+\frac {2 \tanh ^{-1}(a x)^3 \text {Li}_2\left (-1+\frac {2}{1-a x}\right )}{c}-\frac {3 \tanh ^{-1}(a x)^2 \text {Li}_3\left (-1+\frac {2}{1-a x}\right )}{c}+\frac {(6 a) \int \frac {\tanh ^{-1}(a x) \text {Li}_3\left (-1+\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx}{c}\\ &=\frac {\tanh ^{-1}(a x)^4 \log \left (2-\frac {2}{1-a x}\right )}{c}+\frac {2 \tanh ^{-1}(a x)^3 \text {Li}_2\left (-1+\frac {2}{1-a x}\right )}{c}-\frac {3 \tanh ^{-1}(a x)^2 \text {Li}_3\left (-1+\frac {2}{1-a x}\right )}{c}+\frac {3 \tanh ^{-1}(a x) \text {Li}_4\left (-1+\frac {2}{1-a x}\right )}{c}-\frac {(3 a) \int \frac {\text {Li}_4\left (-1+\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx}{c}\\ &=\frac {\tanh ^{-1}(a x)^4 \log \left (2-\frac {2}{1-a x}\right )}{c}+\frac {2 \tanh ^{-1}(a x)^3 \text {Li}_2\left (-1+\frac {2}{1-a x}\right )}{c}-\frac {3 \tanh ^{-1}(a x)^2 \text {Li}_3\left (-1+\frac {2}{1-a x}\right )}{c}+\frac {3 \tanh ^{-1}(a x) \text {Li}_4\left (-1+\frac {2}{1-a x}\right )}{c}-\frac {3 \text {Li}_5\left (-1+\frac {2}{1-a x}\right )}{2 c}\\ \end {align*}

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Mathematica [A]
time = 0.09, size = 102, normalized size = 0.86 \begin {gather*} \frac {\tanh ^{-1}(a x)^4 \log \left (1-e^{2 \tanh ^{-1}(a x)}\right )}{c}+\frac {2 \tanh ^{-1}(a x)^3 \text {PolyLog}\left (2,e^{2 \tanh ^{-1}(a x)}\right )}{c}-\frac {3 \tanh ^{-1}(a x)^2 \text {PolyLog}\left (3,e^{2 \tanh ^{-1}(a x)}\right )}{c}+\frac {3 \tanh ^{-1}(a x) \text {PolyLog}\left (4,e^{2 \tanh ^{-1}(a x)}\right )}{c}-\frac {3 \text {PolyLog}\left (5,e^{2 \tanh ^{-1}(a x)}\right )}{2 c} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[ArcTanh[a*x]^4/(x*(c - a*c*x)),x]

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 10.68, size = 754, normalized size = 6.39

method	result	size
derivativedivides	\(\frac {\arctanh \left (a x \right )^{4} \ln \left (a x \right )}{c}-\frac {\arctanh \left (a x \right )^{4} \ln \left (a x -1\right )}{c}+\frac {-\arctanh \left (a x \right )^{4} \ln \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}-1\right )+\frac {\left (-2 i \pi \mathrm {csgn}\left (\frac {i}{\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1}\right )^{2}+2 i \pi \mathrm {csgn}\left (\frac {i}{\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1}\right )^{3}+i \pi \,\mathrm {csgn}\left (i \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}-1\right )\right ) \mathrm {csgn}\left (\frac {i}{\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1}\right ) \mathrm {csgn}\left (\frac {i \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}-1\right )}{\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1}\right )-i \pi \,\mathrm {csgn}\left (i \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}-1\right )\right ) \mathrm {csgn}\left (\frac {i \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}-1\right )}{\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1}\right )^{2}-i \pi \,\mathrm {csgn}\left (\frac {i}{\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1}\right ) \mathrm {csgn}\left (\frac {i \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}-1\right )}{\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1}\right )^{2}+i \pi \mathrm {csgn}\left (\frac {i \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}-1\right )}{\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1}\right )^{3}+2 i \pi +2 \ln \left (2\right )\right ) \arctanh \left (a x \right )^{4}}{2}+\arctanh \left (a x \right )^{4} \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+4 \arctanh \left (a x \right )^{3} \polylog \left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-12 \arctanh \left (a x \right )^{2} \polylog \left (3, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+24 \arctanh \left (a x \right ) \polylog \left (4, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-24 \polylog \left (5, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+\arctanh \left (a x \right )^{4} \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+4 \arctanh \left (a x \right )^{3} \polylog \left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-12 \arctanh \left (a x \right )^{2} \polylog \left (3, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+24 \arctanh \left (a x \right ) \polylog \left (4, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-24 \polylog \left (5, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}\)	\(754\)
default	\(\frac {\arctanh \left (a x \right )^{4} \ln \left (a x \right )}{c}-\frac {\arctanh \left (a x \right )^{4} \ln \left (a x -1\right )}{c}+\frac {-\arctanh \left (a x \right )^{4} \ln \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}-1\right )+\frac {\left (-2 i \pi \mathrm {csgn}\left (\frac {i}{\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1}\right )^{2}+2 i \pi \mathrm {csgn}\left (\frac {i}{\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1}\right )^{3}+i \pi \,\mathrm {csgn}\left (i \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}-1\right )\right ) \mathrm {csgn}\left (\frac {i}{\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1}\right ) \mathrm {csgn}\left (\frac {i \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}-1\right )}{\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1}\right )-i \pi \,\mathrm {csgn}\left (i \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}-1\right )\right ) \mathrm {csgn}\left (\frac {i \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}-1\right )}{\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1}\right )^{2}-i \pi \,\mathrm {csgn}\left (\frac {i}{\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1}\right ) \mathrm {csgn}\left (\frac {i \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}-1\right )}{\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1}\right )^{2}+i \pi \mathrm {csgn}\left (\frac {i \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}-1\right )}{\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1}\right )^{3}+2 i \pi +2 \ln \left (2\right )\right ) \arctanh \left (a x \right )^{4}}{2}+\arctanh \left (a x \right )^{4} \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+4 \arctanh \left (a x \right )^{3} \polylog \left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-12 \arctanh \left (a x \right )^{2} \polylog \left (3, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+24 \arctanh \left (a x \right ) \polylog \left (4, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-24 \polylog \left (5, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+\arctanh \left (a x \right )^{4} \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+4 \arctanh \left (a x \right )^{3} \polylog \left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-12 \arctanh \left (a x \right )^{2} \polylog \left (3, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+24 \arctanh \left (a x \right ) \polylog \left (4, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-24 \polylog \left (5, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}\)	\(754\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c*arctanh(a*x)^4*ln(a*x)-1/c*arctanh(a*x)^4*ln(a*x-1)+4/c*(-1/4*arctanh(a*x)^4*ln((a*x+1)^2/(-a^2*x^2+1)-1)+
1/8*(-2*I*Pi*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))^2+2*I*Pi*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))^3+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))-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-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+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+2*I*Pi+2*ln(2))*arctanh(a*x)^4+1/4*arctanh
(a*x)^4*ln(1-(a*x+1)/(-a^2*x^2+1)^(1/2))+arctanh(a*x)^3*polylog(2,(a*x+1)/(-a^2*x^2+1)^(1/2))-3*arctanh(a*x)^2
*polylog(3,(a*x+1)/(-a^2*x^2+1)^(1/2))+6*arctanh(a*x)*polylog(4,(a*x+1)/(-a^2*x^2+1)^(1/2))-6*polylog(5,(a*x+1
)/(-a^2*x^2+1)^(1/2))+1/4*arctanh(a*x)^4*ln(1+(a*x+1)/(-a^2*x^2+1)^(1/2))+arctanh(a*x)^3*polylog(2,-(a*x+1)/(-
a^2*x^2+1)^(1/2))-3*arctanh(a*x)^2*polylog(3,-(a*x+1)/(-a^2*x^2+1)^(1/2))+6*arctanh(a*x)*polylog(4,-(a*x+1)/(-
a^2*x^2+1)^(1/2))-6*polylog(5,-(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)^4/x/(-a*c*x+c),x, algorithm="maxima")

[Out]

-1/80*log(-a*x + 1)^5/c + 1/16*integrate(-(log(a*x + 1)^4 - 4*log(a*x + 1)^3*log(-a*x + 1) + 6*log(a*x + 1)^2*
log(-a*x + 1)^2 - 4*log(a*x + 1)*log(-a*x + 1)^3)/(a*c*x^2 - c*x), x)

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Fricas [A]
time = 0.35, size = 155, normalized size = 1.31 \begin {gather*} \frac {\log \left (\frac {2 \, a x}{a x - 1}\right ) \log \left (-\frac {a x + 1}{a x - 1}\right )^{4} + 4 \, {\rm Li}_2\left (-\frac {2 \, a x}{a x - 1} + 1\right ) \log \left (-\frac {a x + 1}{a x - 1}\right )^{3} - 12 \, \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} {\rm polylog}\left (3, -\frac {a x + 1}{a x - 1}\right ) + 24 \, \log \left (-\frac {a x + 1}{a x - 1}\right ) {\rm polylog}\left (4, -\frac {a x + 1}{a x - 1}\right ) - 24 \, {\rm polylog}\left (5, -\frac {a x + 1}{a x - 1}\right )}{16 \, c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*(log(2*a*x/(a*x - 1))*log(-(a*x + 1)/(a*x - 1))^4 + 4*dilog(-2*a*x/(a*x - 1) + 1)*log(-(a*x + 1)/(a*x - 1
))^3 - 12*log(-(a*x + 1)/(a*x - 1))^2*polylog(3, -(a*x + 1)/(a*x - 1)) + 24*log(-(a*x + 1)/(a*x - 1))*polylog(
4, -(a*x + 1)/(a*x - 1)) - 24*polylog(5, -(a*x + 1)/(a*x - 1)))/c

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(atanh(a*x)**4/x/(-a*c*x+c),x)

[Out]

-Integral(atanh(a*x)**4/(a*x**2 - x), 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)^4/x/(-a*c*x+c),x, algorithm="giac")

[Out]

integrate(-arctanh(a*x)^4/((a*c*x - c)*x), 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 )}^4}{x\,\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)^4/(x*(c - a*c*x)),x)

[Out]

int(atanh(a*x)^4/(x*(c - a*c*x)), x)

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