3.2.19 \(\int \frac {\tanh ^{-1}(a x)^2}{c x-a c x^2} \, dx\) [119]

Optimal. Leaf size=67 \[ \frac {\tanh ^{-1}(a x)^2 \log \left (2-\frac {2}{1-a x}\right )}{c}+\frac {\tanh ^{-1}(a x) \text {PolyLog}\left (2,-1+\frac {2}{1-a x}\right )}{c}-\frac {\text {PolyLog}\left (3,-1+\frac {2}{1-a x}\right )}{2 c} \]

[Out]

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

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Rubi [A]
time = 0.10, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1607, 6079, 6095, 6205, 6745} \begin {gather*} -\frac {\text {Li}_3\left (\frac {2}{1-a x}-1\right )}{2 c}+\frac {\text {Li}_2\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)^2}{c} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[ArcTanh[a*x]^2/(c*x - a*c*x^2),x]

[Out]

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

Rule 1607

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

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

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Mathematica [A]
time = 0.08, size = 59, normalized size = 0.88 \begin {gather*} \frac {\tanh ^{-1}(a x)^2 \log \left (1-e^{2 \tanh ^{-1}(a x)}\right )}{c}+\frac {\tanh ^{-1}(a x) \text {PolyLog}\left (2,e^{2 \tanh ^{-1}(a x)}\right )}{c}-\frac {\text {PolyLog}\left (3,e^{2 \tanh ^{-1}(a x)}\right )}{2 c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(ArcTanh[a*x]^2*Log[1 - E^(2*ArcTanh[a*x])])/c + (ArcTanh[a*x]*PolyLog[2, E^(2*ArcTanh[a*x])])/c - PolyLog[3,
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 = 13.97, size = 647, normalized size = 9.66

method	result	size
derivativedivides	\(\frac {-\frac {a \arctanh \left (a x \right )^{2} \ln \left (a x -1\right )}{c}+\frac {a \arctanh \left (a x \right )^{2} \ln \left (a x \right )}{c}+\frac {2 a \left (-\frac {\arctanh \left (a x \right )^{2} \ln \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}-1\right )}{2}+\frac {\arctanh \left (a x \right )^{2} \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}+\arctanh \left (a x \right ) \polylog \left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-\polylog \left (3, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+\frac {\arctanh \left (a x \right )^{2} \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}+\arctanh \left (a x \right ) \polylog \left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-\polylog \left (3, \frac {a x +1}{\sqrt {-a^{2} x^{2}+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 )^{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 (\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 (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 \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 \mathrm {csgn}\left (\frac {i}{\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1}\right )^{2}+2 i \pi +2 \ln \left (2\right )\right ) \arctanh \left (a x \right )^{2}}{4}\right )}{c}}{a}\)	\(647\)
default	\(\frac {-\frac {a \arctanh \left (a x \right )^{2} \ln \left (a x -1\right )}{c}+\frac {a \arctanh \left (a x \right )^{2} \ln \left (a x \right )}{c}+\frac {2 a \left (-\frac {\arctanh \left (a x \right )^{2} \ln \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}-1\right )}{2}+\frac {\arctanh \left (a x \right )^{2} \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}+\arctanh \left (a x \right ) \polylog \left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-\polylog \left (3, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+\frac {\arctanh \left (a x \right )^{2} \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}+\arctanh \left (a x \right ) \polylog \left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-\polylog \left (3, \frac {a x +1}{\sqrt {-a^{2} x^{2}+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 )^{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 (\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 (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 \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 \mathrm {csgn}\left (\frac {i}{\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1}\right )^{2}+2 i \pi +2 \ln \left (2\right )\right ) \arctanh \left (a x \right )^{2}}{4}\right )}{c}}{a}\)	\(647\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

-1/12*log(-a*x + 1)^3/c + 1/4*integrate(-(log(a*x + 1)^2 - 2*log(a*x + 1)*log(-a*x + 1))/(a*c*x^2 - c*x), x)

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Fricas [A]
time = 0.35, size = 85, normalized size = 1.27 \begin {gather*} \frac {\log \left (\frac {2 \, a x}{a x - 1}\right ) \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} + 2 \, {\rm Li}_2\left (-\frac {2 \, a x}{a x - 1} + 1\right ) \log \left (-\frac {a x + 1}{a x - 1}\right ) - 2 \, {\rm polylog}\left (3, -\frac {a x + 1}{a x - 1}\right )}{4 \, c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(log(2*a*x/(a*x - 1))*log(-(a*x + 1)/(a*x - 1))^2 + 2*dilog(-2*a*x/(a*x - 1) + 1)*log(-(a*x + 1)/(a*x - 1)
) - 2*polylog(3, -(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}^{2}{\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)**2/(-a*c*x**2+c*x),x)

[Out]

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

[Out]

integrate(-arctanh(a*x)^2/(a*c*x^2 - 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 )}^2}{c\,x-a\,c\,x^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(atanh(a*x)^2/(c*x - a*c*x^2),x)

[Out]

int(atanh(a*x)^2/(c*x - a*c*x^2), x)

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