3.2.56 \(\int \frac {(a+b \tanh ^{-1}(c x))^2}{d+e x} \, dx\) [156]

Optimal. Leaf size=188 \[ -\frac {\left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+c x}\right )}{e}+\frac {\left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{e}+\frac {b \left (a+b \tanh ^{-1}(c x)\right ) \text {PolyLog}\left (2,1-\frac {2}{1+c x}\right )}{e}-\frac {b \left (a+b \tanh ^{-1}(c x)\right ) \text {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{e}+\frac {b^2 \text {PolyLog}\left (3,1-\frac {2}{1+c x}\right )}{2 e}-\frac {b^2 \text {PolyLog}\left (3,1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{2 e} \]

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 188, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {6059} \begin {gather*} -\frac {b \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+e) (c x+1)}\right )}{e}+\frac {\left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2 c (d+e x)}{(c x+1) (c d+e)}\right )}{e}+\frac {b \text {Li}_2\left (1-\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{e}-\frac {\log \left (\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{e}-\frac {b^2 \text {Li}_3\left (1-\frac {2 c (d+e x)}{(c d+e) (c x+1)}\right )}{2 e}+\frac {b^2 \text {Li}_3\left (1-\frac {2}{c x+1}\right )}{2 e} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcTanh[c*x])^2/(d + e*x),x]

[Out]

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

Rule 6059

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^2/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcTanh[c*x])^2)*(Lo
g[2/(1 + c*x)]/e), x] + (Simp[(a + b*ArcTanh[c*x])^2*(Log[2*c*((d + e*x)/((c*d + e)*(1 + c*x)))]/e), x] + Simp
[b*(a + b*ArcTanh[c*x])*(PolyLog[2, 1 - 2/(1 + c*x)]/e), x] - Simp[b*(a + b*ArcTanh[c*x])*(PolyLog[2, 1 - 2*c*
((d + e*x)/((c*d + e)*(1 + c*x)))]/e), x] + Simp[b^2*(PolyLog[3, 1 - 2/(1 + c*x)]/(2*e)), x] - Simp[b^2*(PolyL
og[3, 1 - 2*c*((d + e*x)/((c*d + e)*(1 + c*x)))]/(2*e)), x]) /; FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d^2 - e^2
, 0]

Rubi steps

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

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Mathematica [C] Result contains complex when optimal does not.
time = 8.85, size = 938, normalized size = 4.99 \begin {gather*} \frac {6 a^2 \log (d+e x)+6 a b \tanh ^{-1}(c x) \left (\log \left (1-c^2 x^2\right )+2 \log \left (i \sinh \left (\tanh ^{-1}\left (\frac {c d}{e}\right )+\tanh ^{-1}(c x)\right )\right )\right )-6 i a b \left (-\frac {1}{4} i \left (\pi -2 i \tanh ^{-1}(c x)\right )^2+i \left (\tanh ^{-1}\left (\frac {c d}{e}\right )+\tanh ^{-1}(c x)\right )^2+\left (\pi -2 i \tanh ^{-1}(c x)\right ) \log \left (1+e^{2 \tanh ^{-1}(c x)}\right )+2 i \left (\tanh ^{-1}\left (\frac {c d}{e}\right )+\tanh ^{-1}(c x)\right ) \log \left (1-e^{-2 \left (\tanh ^{-1}\left (\frac {c d}{e}\right )+\tanh ^{-1}(c x)\right )}\right )-\left (\pi -2 i \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{\sqrt {1-c^2 x^2}}\right )-2 i \left (\tanh ^{-1}\left (\frac {c d}{e}\right )+\tanh ^{-1}(c x)\right ) \log \left (2 i \sinh \left (\tanh ^{-1}\left (\frac {c d}{e}\right )+\tanh ^{-1}(c x)\right )\right )-i \text {PolyLog}\left (2,-e^{2 \tanh ^{-1}(c x)}\right )-i \text {PolyLog}\left (2,e^{-2 \left (\tanh ^{-1}\left (\frac {c d}{e}\right )+\tanh ^{-1}(c x)\right )}\right )\right )+\frac {b^2 \left (-8 c d \tanh ^{-1}(c x)^3+4 e \tanh ^{-1}(c x)^3-4 \sqrt {1-\frac {c^2 d^2}{e^2}} e e^{-\tanh ^{-1}\left (\frac {c d}{e}\right )} \tanh ^{-1}(c x)^3-6 c d \tanh ^{-1}(c x)^2 \log \left (1+e^{-2 \tanh ^{-1}(c x)}\right )-6 i c d \pi \tanh ^{-1}(c x) \log \left (\frac {1}{2} \left (e^{-\tanh ^{-1}(c x)}+e^{\tanh ^{-1}(c x)}\right )\right )-6 c d \tanh ^{-1}(c x)^2 \log \left (1+\frac {(c d+e) e^{2 \tanh ^{-1}(c x)}}{c d-e}\right )+6 c d \tanh ^{-1}(c x)^2 \log \left (1-e^{\tanh ^{-1}\left (\frac {c d}{e}\right )+\tanh ^{-1}(c x)}\right )+6 c d \tanh ^{-1}(c x)^2 \log \left (1+e^{\tanh ^{-1}\left (\frac {c d}{e}\right )+\tanh ^{-1}(c x)}\right )+6 c d \tanh ^{-1}(c x)^2 \log \left (1-e^{2 \left (\tanh ^{-1}\left (\frac {c d}{e}\right )+\tanh ^{-1}(c x)\right )}\right )+12 c d \tanh ^{-1}\left (\frac {c d}{e}\right ) \tanh ^{-1}(c x) \log \left (\frac {1}{2} i e^{-\tanh ^{-1}\left (\frac {c d}{e}\right )-\tanh ^{-1}(c x)} \left (-1+e^{2 \left (\tanh ^{-1}\left (\frac {c d}{e}\right )+\tanh ^{-1}(c x)\right )}\right )\right )+6 c d \tanh ^{-1}(c x)^2 \log \left (\frac {1}{2} e^{-\tanh ^{-1}(c x)} \left (e \left (-1+e^{2 \tanh ^{-1}(c x)}\right )+c d \left (1+e^{2 \tanh ^{-1}(c x)}\right )\right )\right )-6 c d \tanh ^{-1}(c x)^2 \log \left (\frac {c (d+e x)}{\sqrt {1-c^2 x^2}}\right )-3 i c d \pi \tanh ^{-1}(c x) \log \left (1-c^2 x^2\right )-12 c d \tanh ^{-1}\left (\frac {c d}{e}\right ) \tanh ^{-1}(c x) \log \left (i \sinh \left (\tanh ^{-1}\left (\frac {c d}{e}\right )+\tanh ^{-1}(c x)\right )\right )+6 c d \tanh ^{-1}(c x) \text {PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c x)}\right )-6 c d \tanh ^{-1}(c x) \text {PolyLog}\left (2,-\frac {(c d+e) e^{2 \tanh ^{-1}(c x)}}{c d-e}\right )+12 c d \tanh ^{-1}(c x) \text {PolyLog}\left (2,-e^{\tanh ^{-1}\left (\frac {c d}{e}\right )+\tanh ^{-1}(c x)}\right )+12 c d \tanh ^{-1}(c x) \text {PolyLog}\left (2,e^{\tanh ^{-1}\left (\frac {c d}{e}\right )+\tanh ^{-1}(c x)}\right )+6 c d \tanh ^{-1}(c x) \text {PolyLog}\left (2,e^{2 \left (\tanh ^{-1}\left (\frac {c d}{e}\right )+\tanh ^{-1}(c x)\right )}\right )+3 c d \text {PolyLog}\left (3,-e^{-2 \tanh ^{-1}(c x)}\right )+3 c d \text {PolyLog}\left (3,-\frac {(c d+e) e^{2 \tanh ^{-1}(c x)}}{c d-e}\right )-12 c d \text {PolyLog}\left (3,-e^{\tanh ^{-1}\left (\frac {c d}{e}\right )+\tanh ^{-1}(c x)}\right )-12 c d \text {PolyLog}\left (3,e^{\tanh ^{-1}\left (\frac {c d}{e}\right )+\tanh ^{-1}(c x)}\right )-3 c d \text {PolyLog}\left (3,e^{2 \left (\tanh ^{-1}\left (\frac {c d}{e}\right )+\tanh ^{-1}(c x)\right )}\right )\right )}{c d}}{6 e} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcTanh[c*x])^2/(d + e*x),x]

[Out]

(6*a^2*Log[d + e*x] + 6*a*b*ArcTanh[c*x]*(Log[1 - c^2*x^2] + 2*Log[I*Sinh[ArcTanh[(c*d)/e] + ArcTanh[c*x]]]) -
(6*I)*a*b*((-1/4*I)*(Pi - (2*I)*ArcTanh[c*x])^2 + I*(ArcTanh[(c*d)/e] + ArcTanh[c*x])^2 + (Pi - (2*I)*ArcTanh
[c*x])*Log[1 + E^(2*ArcTanh[c*x])] + (2*I)*(ArcTanh[(c*d)/e] + ArcTanh[c*x])*Log[1 - E^(-2*(ArcTanh[(c*d)/e] +
ArcTanh[c*x]))] - (Pi - (2*I)*ArcTanh[c*x])*Log[2/Sqrt[1 - c^2*x^2]] - (2*I)*(ArcTanh[(c*d)/e] + ArcTanh[c*x]
)*Log[(2*I)*Sinh[ArcTanh[(c*d)/e] + ArcTanh[c*x]]] - I*PolyLog[2, -E^(2*ArcTanh[c*x])] - I*PolyLog[2, E^(-2*(A
rcTanh[(c*d)/e] + ArcTanh[c*x]))]) + (b^2*(-8*c*d*ArcTanh[c*x]^3 + 4*e*ArcTanh[c*x]^3 - (4*Sqrt[1 - (c^2*d^2)/
e^2]*e*ArcTanh[c*x]^3)/E^ArcTanh[(c*d)/e] - 6*c*d*ArcTanh[c*x]^2*Log[1 + E^(-2*ArcTanh[c*x])] - (6*I)*c*d*Pi*A
rcTanh[c*x]*Log[(E^(-ArcTanh[c*x]) + E^ArcTanh[c*x])/2] - 6*c*d*ArcTanh[c*x]^2*Log[1 + ((c*d + e)*E^(2*ArcTanh
[c*x]))/(c*d - e)] + 6*c*d*ArcTanh[c*x]^2*Log[1 - E^(ArcTanh[(c*d)/e] + ArcTanh[c*x])] + 6*c*d*ArcTanh[c*x]^2*
Log[1 + E^(ArcTanh[(c*d)/e] + ArcTanh[c*x])] + 6*c*d*ArcTanh[c*x]^2*Log[1 - E^(2*(ArcTanh[(c*d)/e] + ArcTanh[c
*x]))] + 12*c*d*ArcTanh[(c*d)/e]*ArcTanh[c*x]*Log[(I/2)*E^(-ArcTanh[(c*d)/e] - ArcTanh[c*x])*(-1 + E^(2*(ArcTa
nh[(c*d)/e] + ArcTanh[c*x])))] + 6*c*d*ArcTanh[c*x]^2*Log[(e*(-1 + E^(2*ArcTanh[c*x])) + c*d*(1 + E^(2*ArcTanh
[c*x])))/(2*E^ArcTanh[c*x])] - 6*c*d*ArcTanh[c*x]^2*Log[(c*(d + e*x))/Sqrt[1 - c^2*x^2]] - (3*I)*c*d*Pi*ArcTan
h[c*x]*Log[1 - c^2*x^2] - 12*c*d*ArcTanh[(c*d)/e]*ArcTanh[c*x]*Log[I*Sinh[ArcTanh[(c*d)/e] + ArcTanh[c*x]]] +
6*c*d*ArcTanh[c*x]*PolyLog[2, -E^(-2*ArcTanh[c*x])] - 6*c*d*ArcTanh[c*x]*PolyLog[2, -(((c*d + e)*E^(2*ArcTanh[
c*x]))/(c*d - e))] + 12*c*d*ArcTanh[c*x]*PolyLog[2, -E^(ArcTanh[(c*d)/e] + ArcTanh[c*x])] + 12*c*d*ArcTanh[c*x
]*PolyLog[2, E^(ArcTanh[(c*d)/e] + ArcTanh[c*x])] + 6*c*d*ArcTanh[c*x]*PolyLog[2, E^(2*(ArcTanh[(c*d)/e] + Arc
Tanh[c*x]))] + 3*c*d*PolyLog[3, -E^(-2*ArcTanh[c*x])] + 3*c*d*PolyLog[3, -(((c*d + e)*E^(2*ArcTanh[c*x]))/(c*d
- e))] - 12*c*d*PolyLog[3, -E^(ArcTanh[(c*d)/e] + ArcTanh[c*x])] - 12*c*d*PolyLog[3, E^(ArcTanh[(c*d)/e] + Ar
cTanh[c*x])] - 3*c*d*PolyLog[3, E^(2*(ArcTanh[(c*d)/e] + ArcTanh[c*x]))]))/(c*d))/(6*e)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.24, size = 1197, normalized size = 6.37

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1197\)
default	\(\text {Expression too large to display}\)	\(1197\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c*(a^2*c*ln(c*e*x+c*d)/e+b^2*c*ln(c*e*x+c*d)/e*arctanh(c*x)^2-b^2*c/e*arctanh(c*x)^2*ln(d*c*(1+(c*x+1)^2/(-c
^2*x^2+1))+e*((c*x+1)^2/(-c^2*x^2+1)-1))-1/2*I*b^2*c/e*arctanh(c*x)^2*Pi*csgn(I*(d*c*(1+(c*x+1)^2/(-c^2*x^2+1)
)+e*((c*x+1)^2/(-c^2*x^2+1)-1))/(1+(c*x+1)^2/(-c^2*x^2+1)))^2*csgn(I/(1+(c*x+1)^2/(-c^2*x^2+1)))+1/2*I*b^2*c/e
*arctanh(c*x)^2*Pi*csgn(I*(d*c*(1+(c*x+1)^2/(-c^2*x^2+1))+e*((c*x+1)^2/(-c^2*x^2+1)-1))/(1+(c*x+1)^2/(-c^2*x^2
+1)))^3+1/2*I*b^2*c/e*arctanh(c*x)^2*Pi*csgn(I*(d*c*(1+(c*x+1)^2/(-c^2*x^2+1))+e*((c*x+1)^2/(-c^2*x^2+1)-1))/(
1+(c*x+1)^2/(-c^2*x^2+1)))*csgn(I*(d*c*(1+(c*x+1)^2/(-c^2*x^2+1))+e*((c*x+1)^2/(-c^2*x^2+1)-1)))*csgn(I/(1+(c*
x+1)^2/(-c^2*x^2+1)))-1/2*I*b^2*c/e*arctanh(c*x)^2*Pi*csgn(I*(d*c*(1+(c*x+1)^2/(-c^2*x^2+1))+e*((c*x+1)^2/(-c^
2*x^2+1)-1))/(1+(c*x+1)^2/(-c^2*x^2+1)))^2*csgn(I*(d*c*(1+(c*x+1)^2/(-c^2*x^2+1))+e*((c*x+1)^2/(-c^2*x^2+1)-1)
))-b^2*c/e*arctanh(c*x)*polylog(2,-(c*x+1)^2/(-c^2*x^2+1))+1/2*b^2*c/e*polylog(3,-(c*x+1)^2/(-c^2*x^2+1))+b^2*
c/(c*d+e)*arctanh(c*x)^2*ln(1-(c*d+e)*(c*x+1)^2/(-c^2*x^2+1)/(-c*d+e))+b^2*c/(c*d+e)*arctanh(c*x)*polylog(2,(c
*d+e)*(c*x+1)^2/(-c^2*x^2+1)/(-c*d+e))-1/2*b^2*c/(c*d+e)*polylog(3,(c*d+e)*(c*x+1)^2/(-c^2*x^2+1)/(-c*d+e))+b^
2*c^2/e*d/(c*d+e)*arctanh(c*x)^2*ln(1-(c*d+e)*(c*x+1)^2/(-c^2*x^2+1)/(-c*d+e))+b^2*c^2/e*d/(c*d+e)*arctanh(c*x
)*polylog(2,(c*d+e)*(c*x+1)^2/(-c^2*x^2+1)/(-c*d+e))-1/2*b^2*c^2/e*d/(c*d+e)*polylog(3,(c*d+e)*(c*x+1)^2/(-c^2
*x^2+1)/(-c*d+e))+2*a*b*c*ln(c*e*x+c*d)/e*arctanh(c*x)-a*b*c/e*ln(c*e*x+c*d)*ln((c*e*x+e)/(-c*d+e))-a*b*c/e*di
log((c*e*x+e)/(-c*d+e))+a*b*c/e*ln((c*e*x-e)/(-c*d-e))*ln(c*e*x+c*d)+a*b*c/e*dilog((c*e*x-e)/(-c*d-e)))

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

[Out]

a^2*e^(-1)*log(x*e + d) + integrate(1/4*b^2*(log(c*x + 1) - log(-c*x + 1))^2/(x*e + d) + a*b*(log(c*x + 1) - l
og(-c*x + 1))/(x*e + d), 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((a+b*arctanh(c*x))^2/(e*x+d),x, algorithm="fricas")

[Out]

integral((b^2*arctanh(c*x)^2 + 2*a*b*arctanh(c*x) + a^2)/(x*e + d), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*atanh(c*x))**2/(e*x+d),x)

[Out]

Integral((a + b*atanh(c*x))**2/(d + e*x), x)

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

[Out]

integrate((b*arctanh(c*x) + a)^2/(e*x + d), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*atanh(c*x))^2/(d + e*x),x)

[Out]

int((a + b*atanh(c*x))^2/(d + e*x), x)

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