3.1.42 \(\int \frac {(a+b \tanh ^{-1}(c+d x))^2}{e+f x} \, dx\) [42]
Optimal. Leaf size=214 \[ -\frac {\left (a+b \tanh ^{-1}(c+d x)\right )^2 \log \left (\frac {2}{1+c+d x}\right )}{f}+\frac {\left (a+b \tanh ^{-1}(c+d x)\right )^2 \log \left (\frac {2 d (e+f x)}{(d e+f-c f) (1+c+d x)}\right )}{f}+\frac {b \left (a+b \tanh ^{-1}(c+d x)\right ) \text {PolyLog}\left (2,1-\frac {2}{1+c+d x}\right )}{f}-\frac {b \left (a+b \tanh ^{-1}(c+d x)\right ) \text {PolyLog}\left (2,1-\frac {2 d (e+f x)}{(d e+f-c f) (1+c+d x)}\right )}{f}+\frac {b^2 \text {PolyLog}\left (3,1-\frac {2}{1+c+d x}\right )}{2 f}-\frac {b^2 \text {PolyLog}\left (3,1-\frac {2 d (e+f x)}{(d e+f-c f) (1+c+d x)}\right )}{2 f} \]
[Out]
-(a+b*arctanh(d*x+c))^2*ln(2/(d*x+c+1))/f+(a+b*arctanh(d*x+c))^2*ln(2*d*(f*x+e)/(-c*f+d*e+f)/(d*x+c+1))/f+b*(a
+b*arctanh(d*x+c))*polylog(2,1-2/(d*x+c+1))/f-b*(a+b*arctanh(d*x+c))*polylog(2,1-2*d*(f*x+e)/(-c*f+d*e+f)/(d*x
+c+1))/f+1/2*b^2*polylog(3,1-2/(d*x+c+1))/f-1/2*b^2*polylog(3,1-2*d*(f*x+e)/(-c*f+d*e+f)/(d*x+c+1))/f
________________________________________________________________________________________
Rubi [A]
time = 0.11, antiderivative size = 214, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {6246, 6059}
\begin {gather*} -\frac {b \left (a+b \tanh ^{-1}(c+d x)\right ) \text {Li}_2\left (1-\frac {2 d (e+f x)}{(d e-c f+f) (c+d x+1)}\right )}{f}+\frac {\left (a+b \tanh ^{-1}(c+d x)\right )^2 \log \left (\frac {2 d (e+f x)}{(c+d x+1) (-c f+d e+f)}\right )}{f}+\frac {b \text {Li}_2\left (1-\frac {2}{c+d x+1}\right ) \left (a+b \tanh ^{-1}(c+d x)\right )}{f}-\frac {\log \left (\frac {2}{c+d x+1}\right ) \left (a+b \tanh ^{-1}(c+d x)\right )^2}{f}-\frac {b^2 \text {Li}_3\left (1-\frac {2 d (e+f x)}{(d e-c f+f) (c+d x+1)}\right )}{2 f}+\frac {b^2 \text {Li}_3\left (1-\frac {2}{c+d x+1}\right )}{2 f} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(a + b*ArcTanh[c + d*x])^2/(e + f*x),x]
[Out]
-(((a + b*ArcTanh[c + d*x])^2*Log[2/(1 + c + d*x)])/f) + ((a + b*ArcTanh[c + d*x])^2*Log[(2*d*(e + f*x))/((d*e
+ f - c*f)*(1 + c + d*x))])/f + (b*(a + b*ArcTanh[c + d*x])*PolyLog[2, 1 - 2/(1 + c + d*x)])/f - (b*(a + b*Ar
cTanh[c + d*x])*PolyLog[2, 1 - (2*d*(e + f*x))/((d*e + f - c*f)*(1 + c + d*x))])/f + (b^2*PolyLog[3, 1 - 2/(1
+ c + d*x)])/(2*f) - (b^2*PolyLog[3, 1 - (2*d*(e + f*x))/((d*e + f - c*f)*(1 + c + d*x))])/(2*f)
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]
Rule 6246
Int[((a_.) + ArcTanh[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Dist[1/d, Subst[
Int[((d*e - c*f)/d + f*(x/d))^m*(a + b*ArcTanh[x])^p, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] &
& IGtQ[p, 0]
Rubi steps
\begin {align*} \int \frac {\left (a+b \tanh ^{-1}(c+d x)\right )^2}{e+f x} \, dx &=\frac {\text {Subst}\left (\int \frac {\left (a+b \tanh ^{-1}(x)\right )^2}{\frac {d e-c f}{d}+\frac {f x}{d}} \, dx,x,c+d x\right )}{d}\\ &=-\frac {\left (a+b \tanh ^{-1}(c+d x)\right )^2 \log \left (\frac {2}{1+c+d x}\right )}{f}+\frac {\left (a+b \tanh ^{-1}(c+d x)\right )^2 \log \left (\frac {2 d (e+f x)}{(d e+f-c f) (1+c+d x)}\right )}{f}+\frac {b \left (a+b \tanh ^{-1}(c+d x)\right ) \text {Li}_2\left (1-\frac {2}{1+c+d x}\right )}{f}-\frac {b \left (a+b \tanh ^{-1}(c+d x)\right ) \text {Li}_2\left (1-\frac {2 d (e+f x)}{(d e+f-c f) (1+c+d x)}\right )}{f}+\frac {b^2 \text {Li}_3\left (1-\frac {2}{1+c+d x}\right )}{2 f}-\frac {b^2 \text {Li}_3\left (1-\frac {2 d (e+f x)}{(d e+f-c f) (1+c+d x)}\right )}{2 f}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains complex when optimal does not.
time = 20.62, size = 2404, normalized size = 11.23 \begin {gather*} \text {Result too large to show} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
Integrate[(a + b*ArcTanh[c + d*x])^2/(e + f*x),x]
[Out]
(a^2*Log[e + f*x])/f - ((2*I)*a*b*(I*ArcTanh[c + d*x]*(-Log[1/Sqrt[1 - (c + d*x)^2]] + Log[I*Sinh[ArcTanh[(d*e
- c*f)/f] + ArcTanh[c + d*x]]]) + ((-I)*(I*ArcTanh[(d*e - c*f)/f] + I*ArcTanh[c + d*x])^2 - (I/4)*(Pi - (2*I)
*ArcTanh[c + d*x])^2 + 2*(I*ArcTanh[(d*e - c*f)/f] + I*ArcTanh[c + d*x])*Log[1 - E^((2*I)*(I*ArcTanh[(d*e - c*
f)/f] + I*ArcTanh[c + d*x]))] + (Pi - (2*I)*ArcTanh[c + d*x])*Log[1 - E^(I*(Pi - (2*I)*ArcTanh[c + d*x]))] - (
Pi - (2*I)*ArcTanh[c + d*x])*Log[2*Sin[(Pi - (2*I)*ArcTanh[c + d*x])/2]] - 2*(I*ArcTanh[(d*e - c*f)/f] + I*Arc
Tanh[c + d*x])*Log[(2*I)*Sinh[ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x]]] - I*PolyLog[2, E^((2*I)*(I*ArcTanh[(
d*e - c*f)/f] + I*ArcTanh[c + d*x]))] - I*PolyLog[2, E^(I*(Pi - (2*I)*ArcTanh[c + d*x]))])/2))/f + (b^2*(d*e -
c*f + f*(c + d*x))*((2*ArcTanh[c + d*x]^2*(d*e*ArcTanh[c + d*x] - (1 + c)*f*ArcTanh[c + d*x] + 3*(d*e - c*f)*
Log[1 + E^(-2*ArcTanh[c + d*x])]) + (-6*d*e*ArcTanh[c + d*x] + 6*c*f*ArcTanh[c + d*x])*PolyLog[2, -E^(-2*ArcTa
nh[c + d*x])] + (-3*d*e + 3*c*f)*PolyLog[3, -E^(-2*ArcTanh[c + d*x])])/(6*f*(-(d*e) + c*f)) + ((-(d*e) - f + c
*f)*(-(d*e) + f + c*f)*((ArcTanh[c + d*x]^2*(f*ArcTanh[c + d*x] + (-(d*e) + c*f)*Log[(d*e)/Sqrt[1 - (c + d*x)^
2] - (c*f)/Sqrt[1 - (c + d*x)^2] + (f*(c + d*x))/Sqrt[1 - (c + d*x)^2]]))/((d*e + f - c*f)*(d*e - (1 + c)*f))
- (ArcTanh[c + d*x]*(I*d*e*Pi*ArcTanh[c + d*x] - I*c*f*Pi*ArcTanh[c + d*x] + 2*f*ArcTanh[c + d*x]^2 - (Sqrt[1
- c^2 - (d^2*e^2)/f^2 + (2*c*d*e)/f]*f*ArcTanh[c + d*x]^2)/E^ArcTanh[(d*e - c*f)/f] - I*d*e*Pi*Log[1 + E^(2*Ar
cTanh[c + d*x])] + I*c*f*Pi*Log[1 + E^(2*ArcTanh[c + d*x])] + 2*d*e*ArcTanh[c + d*x]*Log[1 - E^(-2*(ArcTanh[(d
*e - c*f)/f] + ArcTanh[c + d*x]))] - 2*c*f*ArcTanh[c + d*x]*Log[1 - E^(-2*(ArcTanh[(d*e - c*f)/f] + ArcTanh[c
+ d*x]))] + I*d*e*Pi*Log[1/Sqrt[1 - (c + d*x)^2]] - I*c*f*Pi*Log[1/Sqrt[1 - (c + d*x)^2]] - 2*d*e*ArcTanh[c +
d*x]*Log[(d*e)/Sqrt[1 - (c + d*x)^2] - (c*f)/Sqrt[1 - (c + d*x)^2] + (f*(c + d*x))/Sqrt[1 - (c + d*x)^2]] + 2*
c*f*ArcTanh[c + d*x]*Log[(d*e)/Sqrt[1 - (c + d*x)^2] - (c*f)/Sqrt[1 - (c + d*x)^2] + (f*(c + d*x))/Sqrt[1 - (c
+ d*x)^2]] + 2*(d*e - c*f)*ArcTanh[(d*e - c*f)/f]*(ArcTanh[c + d*x] + Log[1 - E^(-2*(ArcTanh[(d*e - c*f)/f] +
ArcTanh[c + d*x]))] - Log[I*Sinh[ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x]]]) + (-(d*e) + c*f)*PolyLog[2, E^(
-2*(ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x]))]))/((d*e + f - c*f)*(d*e - (1 + c)*f)) + (((-2*d*e + (2 + 2*c
- Sqrt[1 - c^2 - (d^2*e^2)/f^2 + (2*c*d*e)/f]/E^ArcTanh[(d*e - c*f)/f])*f)*ArcTanh[c + d*x]^3)/3 + (d*e - c*f)
*ArcTanh[c + d*x]^2*Log[-1 + E^(2*(ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x]))] + (d*e - c*f)*ArcTanh[c + d*x]
*(I*Pi*(ArcTanh[c + d*x] - Log[1 + E^(2*ArcTanh[c + d*x])] + Log[(1 + E^(2*ArcTanh[c + d*x]))/(2*E^ArcTanh[c +
d*x])]) + 2*ArcTanh[(d*e - c*f)/f]*(ArcTanh[c + d*x] + Log[1 - E^(-2*(ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*
x]))] - Log[(I/2)*E^(-ArcTanh[(d*e - c*f)/f] - ArcTanh[c + d*x])*(-1 + E^(2*(ArcTanh[(d*e - c*f)/f] + ArcTanh[
c + d*x])))])) - (d*e - c*f)*ArcTanh[c + d*x]^2*Log[d*e*(1 + E^(2*ArcTanh[c + d*x])) - (1 + c - E^(2*ArcTanh[c
+ d*x]) + c*E^(2*ArcTanh[c + d*x]))*f] + (d*e - c*f)*ArcTanh[c + d*x]^2*(ArcTanh[c + d*x] + Log[1 - E^(-2*(Ar
cTanh[(d*e - c*f)/f] + ArcTanh[c + d*x]))] - Log[-1 + E^(2*(ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x]))] + Log
[d*e*(1 + E^(2*ArcTanh[c + d*x])) - (1 + c - E^(2*ArcTanh[c + d*x]) + c*E^(2*ArcTanh[c + d*x]))*f] - Log[(d*e*
(1 + E^(2*ArcTanh[c + d*x])) - (1 + c - E^(2*ArcTanh[c + d*x]) + c*E^(2*ArcTanh[c + d*x]))*f)/(2*E^ArcTanh[c +
d*x])]) + ((d*e - c*f)*(2*ArcTanh[c + d*x]^3 + 3*ArcTanh[c + d*x]^2*Log[1 - E^(-2*(ArcTanh[(d*e - c*f)/f] + A
rcTanh[c + d*x]))] - 3*ArcTanh[c + d*x]^2*Log[1 - E^(ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x])] - 3*ArcTanh[c
+ d*x]^2*Log[1 + E^(ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x])] - 3*ArcTanh[c + d*x]*PolyLog[2, E^(-2*(ArcTan
h[(d*e - c*f)/f] + ArcTanh[c + d*x]))] - 6*ArcTanh[c + d*x]*PolyLog[2, -E^(ArcTanh[(d*e - c*f)/f] + ArcTanh[c
+ d*x])] - 6*ArcTanh[c + d*x]*PolyLog[2, E^(ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x])] + 6*PolyLog[3, -E^(Arc
Tanh[(d*e - c*f)/f] + ArcTanh[c + d*x])] + 6*PolyLog[3, E^(ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x])]))/3 + (
(d*e - c*f)*(4*ArcTanh[c + d*x]^3 - 6*ArcTanh[c + d*x]^2*Log[1 - E^(2*(ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*
x]))] - 6*ArcTanh[c + d*x]*PolyLog[2, E^(2*(ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x]))] + 3*PolyLog[3, E^(2*(
ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x]))]))/6 - ((d*e - c*f)*(4*ArcTanh[c + d*x]^3 - 6*ArcTanh[c + d*x]^2*L
og[1 + (E^(2*ArcTanh[c + d*x])*(d*e + f - c*f))/(d*e - (1 + c)*f)] - 6*ArcTanh[c + d*x]*PolyLog[2, -((E^(2*Arc
Tanh[c + d*x])*(d*e + f - c*f))/(d*e - (1 + c)*f))] + 3*PolyLog[3, -((E^(2*ArcTanh[c + d*x])*(d*e + f - c*f))/
(d*e - (1 + c)*f))]))/6)/(d^2*e^2 - 2*c*d*e*f + (-1 + c^2)*f^2)))/(f*(-(d*e) + c*f))))/(d*(e + f*x))
________________________________________________________________________________________
Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 30.40, size = 1872, normalized size = 8.75
| | |
| method |
result |
size |
| | |
| derivativedivides |
\(\text {Expression too large to display}\) |
\(1872\) |
| default |
\(\text {Expression too large to display}\) |
\(1872\) |
| | |
|
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a+b*arctanh(d*x+c))^2/(f*x+e),x,method=_RETURNVERBOSE)
[Out]
1/d*(a^2*d*ln(c*f-d*e-f*(d*x+c))/f+b^2*d*ln(c*f-d*e-f*(d*x+c))/f*arctanh(d*x+c)^2-b^2*d/f*arctanh(d*x+c)^2*ln(
f*c*(1+(d*x+c+1)^2/(1-(d*x+c)^2))+(-(d*x+c+1)^2/(1-(d*x+c)^2)-1)*e*d+(-(d*x+c+1)^2/(1-(d*x+c)^2)+1)*f)+1/2*I*b
^2*d/f*Pi*arctanh(d*x+c)^2*csgn(I*(f*c*(1+(d*x+c+1)^2/(1-(d*x+c)^2))+(-(d*x+c+1)^2/(1-(d*x+c)^2)-1)*e*d+(-(d*x
+c+1)^2/(1-(d*x+c)^2)+1)*f)/(1+(d*x+c+1)^2/(1-(d*x+c)^2)))^3-1/2*I*b^2*d/f*Pi*arctanh(d*x+c)^2*csgn(I/(1+(d*x+
c+1)^2/(1-(d*x+c)^2)))*csgn(I*(f*c*(1+(d*x+c+1)^2/(1-(d*x+c)^2))+(-(d*x+c+1)^2/(1-(d*x+c)^2)-1)*e*d+(-(d*x+c+1
)^2/(1-(d*x+c)^2)+1)*f)/(1+(d*x+c+1)^2/(1-(d*x+c)^2)))^2-1/2*I*b^2*d/f*Pi*arctanh(d*x+c)^2*csgn(I*(f*c*(1+(d*x
+c+1)^2/(1-(d*x+c)^2))+(-(d*x+c+1)^2/(1-(d*x+c)^2)-1)*e*d+(-(d*x+c+1)^2/(1-(d*x+c)^2)+1)*f)/(1+(d*x+c+1)^2/(1-
(d*x+c)^2)))^2*csgn(I*(f*c*(1+(d*x+c+1)^2/(1-(d*x+c)^2))+(-(d*x+c+1)^2/(1-(d*x+c)^2)-1)*e*d+(-(d*x+c+1)^2/(1-(
d*x+c)^2)+1)*f))+1/2*I*b^2*d/f*Pi*arctanh(d*x+c)^2*csgn(I/(1+(d*x+c+1)^2/(1-(d*x+c)^2)))*csgn(I*(f*c*(1+(d*x+c
+1)^2/(1-(d*x+c)^2))+(-(d*x+c+1)^2/(1-(d*x+c)^2)-1)*e*d+(-(d*x+c+1)^2/(1-(d*x+c)^2)+1)*f)/(1+(d*x+c+1)^2/(1-(d
*x+c)^2)))*csgn(I*(f*c*(1+(d*x+c+1)^2/(1-(d*x+c)^2))+(-(d*x+c+1)^2/(1-(d*x+c)^2)-1)*e*d+(-(d*x+c+1)^2/(1-(d*x+
c)^2)+1)*f))-b^2*d/f*arctanh(d*x+c)*polylog(2,-(d*x+c+1)^2/(1-(d*x+c)^2))+1/2*b^2*d/f*polylog(3,-(d*x+c+1)^2/(
1-(d*x+c)^2))+b^2*d*c/(c*f-d*e-f)*arctanh(d*x+c)^2*ln(1-(c*f-d*e-f)*(d*x+c+1)^2/(1-(d*x+c)^2)/(-c*f+d*e-f))+b^
2*d*c/(c*f-d*e-f)*arctanh(d*x+c)*polylog(2,(c*f-d*e-f)*(d*x+c+1)^2/(1-(d*x+c)^2)/(-c*f+d*e-f))-1/2*b^2*d*c/(c*
f-d*e-f)*polylog(3,(c*f-d*e-f)*(d*x+c+1)^2/(1-(d*x+c)^2)/(-c*f+d*e-f))-b^2*d/(c*f-d*e-f)*arctanh(d*x+c)^2*ln(1
-(c*f-d*e-f)*(d*x+c+1)^2/(1-(d*x+c)^2)/(-c*f+d*e-f))-b^2*d/(c*f-d*e-f)*arctanh(d*x+c)*polylog(2,(c*f-d*e-f)*(d
*x+c+1)^2/(1-(d*x+c)^2)/(-c*f+d*e-f))+1/2*b^2*d/(c*f-d*e-f)*polylog(3,(c*f-d*e-f)*(d*x+c+1)^2/(1-(d*x+c)^2)/(-
c*f+d*e-f))-b^2*d^2/f*e/(c*f-d*e-f)*arctanh(d*x+c)^2*ln(1-(c*f-d*e-f)*(d*x+c+1)^2/(1-(d*x+c)^2)/(-c*f+d*e-f))-
b^2*d^2/f*e/(c*f-d*e-f)*arctanh(d*x+c)*polylog(2,(c*f-d*e-f)*(d*x+c+1)^2/(1-(d*x+c)^2)/(-c*f+d*e-f))+1/2*b^2*d
^2/f*e/(c*f-d*e-f)*polylog(3,(c*f-d*e-f)*(d*x+c+1)^2/(1-(d*x+c)^2)/(-c*f+d*e-f))+2*a*b*d*ln(c*f-d*e-f*(d*x+c))
/f*arctanh(d*x+c)-a*b*d/f*ln(c*f-d*e-f*(d*x+c))*ln((-f*(d*x+c)-f)/(-c*f+d*e-f))-a*b*d/f*dilog((-f*(d*x+c)-f)/(
-c*f+d*e-f))+a*b*d/f*ln(c*f-d*e-f*(d*x+c))*ln((-f*(d*x+c)+f)/(-c*f+d*e+f))+a*b*d/f*dilog((-f*(d*x+c)+f)/(-c*f+
d*e+f)))
________________________________________________________________________________________
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(d*x+c))^2/(f*x+e),x, algorithm="maxima")
[Out]
a^2*log(f*x + e)/f + integrate(1/4*b^2*(log(d*x + c + 1) - log(-d*x - c + 1))^2/(f*x + e) + a*b*(log(d*x + c +
1) - log(-d*x - c + 1))/(f*x + e), x)
________________________________________________________________________________________
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(d*x+c))^2/(f*x+e),x, algorithm="fricas")
[Out]
integral((b^2*arctanh(d*x + c)^2 + 2*a*b*arctanh(d*x + c) + a^2)/(f*x + e), x)
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \operatorname {atanh}{\left (c + d x \right )}\right )^{2}}{e + f x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*atanh(d*x+c))**2/(f*x+e),x)
[Out]
Integral((a + b*atanh(c + d*x))**2/(e + f*x), x)
________________________________________________________________________________________
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(d*x+c))^2/(f*x+e),x, algorithm="giac")
[Out]
integrate((b*arctanh(d*x + c) + a)^2/(f*x + e), x)
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {atanh}\left (c+d\,x\right )\right )}^2}{e+f\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a + b*atanh(c + d*x))^2/(e + f*x),x)
[Out]
int((a + b*atanh(c + d*x))^2/(e + f*x), x)
________________________________________________________________________________________