3.1.48 \(\int \frac {(a+b \tanh ^{-1}(c+d x))^3}{e+f x} \, dx\) [48]
Optimal. Leaf size=308 \[ -\frac {\left (a+b \tanh ^{-1}(c+d x)\right )^3 \log \left (\frac {2}{1+c+d x}\right )}{f}+\frac {\left (a+b \tanh ^{-1}(c+d x)\right )^3 \log \left (\frac {2 d (e+f x)}{(d e+f-c f) (1+c+d x)}\right )}{f}+\frac {3 b \left (a+b \tanh ^{-1}(c+d x)\right )^2 \text {PolyLog}\left (2,1-\frac {2}{1+c+d x}\right )}{2 f}-\frac {3 b \left (a+b \tanh ^{-1}(c+d x)\right )^2 \text {PolyLog}\left (2,1-\frac {2 d (e+f x)}{(d e+f-c f) (1+c+d x)}\right )}{2 f}+\frac {3 b^2 \left (a+b \tanh ^{-1}(c+d x)\right ) \text {PolyLog}\left (3,1-\frac {2}{1+c+d x}\right )}{2 f}-\frac {3 b^2 \left (a+b \tanh ^{-1}(c+d x)\right ) \text {PolyLog}\left (3,1-\frac {2 d (e+f x)}{(d e+f-c f) (1+c+d x)}\right )}{2 f}+\frac {3 b^3 \text {PolyLog}\left (4,1-\frac {2}{1+c+d x}\right )}{4 f}-\frac {3 b^3 \text {PolyLog}\left (4,1-\frac {2 d (e+f x)}{(d e+f-c f) (1+c+d x)}\right )}{4 f} \]
[Out]
-(a+b*arctanh(d*x+c))^3*ln(2/(d*x+c+1))/f+(a+b*arctanh(d*x+c))^3*ln(2*d*(f*x+e)/(-c*f+d*e+f)/(d*x+c+1))/f+3/2*
b*(a+b*arctanh(d*x+c))^2*polylog(2,1-2/(d*x+c+1))/f-3/2*b*(a+b*arctanh(d*x+c))^2*polylog(2,1-2*d*(f*x+e)/(-c*f
+d*e+f)/(d*x+c+1))/f+3/2*b^2*(a+b*arctanh(d*x+c))*polylog(3,1-2/(d*x+c+1))/f-3/2*b^2*(a+b*arctanh(d*x+c))*poly
log(3,1-2*d*(f*x+e)/(-c*f+d*e+f)/(d*x+c+1))/f+3/4*b^3*polylog(4,1-2/(d*x+c+1))/f-3/4*b^3*polylog(4,1-2*d*(f*x+
e)/(-c*f+d*e+f)/(d*x+c+1))/f
________________________________________________________________________________________
Rubi [A]
time = 0.13, antiderivative size = 308, 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, 6061}
\begin {gather*} -\frac {3 b^2 \left (a+b \tanh ^{-1}(c+d x)\right ) \text {Li}_3\left (1-\frac {2 d (e+f x)}{(d e-c f+f) (c+d x+1)}\right )}{2 f}+\frac {3 b^2 \text {Li}_3\left (1-\frac {2}{c+d x+1}\right ) \left (a+b \tanh ^{-1}(c+d x)\right )}{2 f}-\frac {3 b \left (a+b \tanh ^{-1}(c+d x)\right )^2 \text {Li}_2\left (1-\frac {2 d (e+f x)}{(d e-c f+f) (c+d x+1)}\right )}{2 f}+\frac {\left (a+b \tanh ^{-1}(c+d x)\right )^3 \log \left (\frac {2 d (e+f x)}{(c+d x+1) (-c f+d e+f)}\right )}{f}+\frac {3 b \text {Li}_2\left (1-\frac {2}{c+d x+1}\right ) \left (a+b \tanh ^{-1}(c+d x)\right )^2}{2 f}-\frac {\log \left (\frac {2}{c+d x+1}\right ) \left (a+b \tanh ^{-1}(c+d x)\right )^3}{f}-\frac {3 b^3 \text {Li}_4\left (1-\frac {2 d (e+f x)}{(d e-c f+f) (c+d x+1)}\right )}{4 f}+\frac {3 b^3 \text {Li}_4\left (1-\frac {2}{c+d x+1}\right )}{4 f} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(a + b*ArcTanh[c + d*x])^3/(e + f*x),x]
[Out]
-(((a + b*ArcTanh[c + d*x])^3*Log[2/(1 + c + d*x)])/f) + ((a + b*ArcTanh[c + d*x])^3*Log[(2*d*(e + f*x))/((d*e
+ f - c*f)*(1 + c + d*x))])/f + (3*b*(a + b*ArcTanh[c + d*x])^2*PolyLog[2, 1 - 2/(1 + c + d*x)])/(2*f) - (3*b
*(a + b*ArcTanh[c + d*x])^2*PolyLog[2, 1 - (2*d*(e + f*x))/((d*e + f - c*f)*(1 + c + d*x))])/(2*f) + (3*b^2*(a
+ b*ArcTanh[c + d*x])*PolyLog[3, 1 - 2/(1 + c + d*x)])/(2*f) - (3*b^2*(a + b*ArcTanh[c + d*x])*PolyLog[3, 1 -
(2*d*(e + f*x))/((d*e + f - c*f)*(1 + c + d*x))])/(2*f) + (3*b^3*PolyLog[4, 1 - 2/(1 + c + d*x)])/(4*f) - (3*
b^3*PolyLog[4, 1 - (2*d*(e + f*x))/((d*e + f - c*f)*(1 + c + d*x))])/(4*f)
Rule 6061
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^3/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcTanh[c*x])^3)*(Lo
g[2/(1 + c*x)]/e), x] + (Simp[(a + b*ArcTanh[c*x])^3*(Log[2*c*((d + e*x)/((c*d + e)*(1 + c*x)))]/e), x] + Simp
[3*b*(a + b*ArcTanh[c*x])^2*(PolyLog[2, 1 - 2/(1 + c*x)]/(2*e)), x] - Simp[3*b*(a + b*ArcTanh[c*x])^2*(PolyLog
[2, 1 - 2*c*((d + e*x)/((c*d + e)*(1 + c*x)))]/(2*e)), x] + Simp[3*b^2*(a + b*ArcTanh[c*x])*(PolyLog[3, 1 - 2/
(1 + c*x)]/(2*e)), x] - Simp[3*b^2*(a + b*ArcTanh[c*x])*(PolyLog[3, 1 - 2*c*((d + e*x)/((c*d + e)*(1 + c*x)))]
/(2*e)), x] + Simp[3*b^3*(PolyLog[4, 1 - 2/(1 + c*x)]/(4*e)), x] - Simp[3*b^3*(PolyLog[4, 1 - 2*c*((d + e*x)/(
(c*d + e)*(1 + c*x)))]/(4*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 )^3}{e+f x} \, dx &=\frac {\text {Subst}\left (\int \frac {\left (a+b \tanh ^{-1}(x)\right )^3}{\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 )^3 \log \left (\frac {2}{1+c+d x}\right )}{f}+\frac {\left (a+b \tanh ^{-1}(c+d x)\right )^3 \log \left (\frac {2 d (e+f x)}{(d e+f-c f) (1+c+d x)}\right )}{f}+\frac {3 b \left (a+b \tanh ^{-1}(c+d x)\right )^2 \text {Li}_2\left (1-\frac {2}{1+c+d x}\right )}{2 f}-\frac {3 b \left (a+b \tanh ^{-1}(c+d x)\right )^2 \text {Li}_2\left (1-\frac {2 d (e+f x)}{(d e+f-c f) (1+c+d x)}\right )}{2 f}+\frac {3 b^2 \left (a+b \tanh ^{-1}(c+d x)\right ) \text {Li}_3\left (1-\frac {2}{1+c+d x}\right )}{2 f}-\frac {3 b^2 \left (a+b \tanh ^{-1}(c+d x)\right ) \text {Li}_3\left (1-\frac {2 d (e+f x)}{(d e+f-c f) (1+c+d x)}\right )}{2 f}+\frac {3 b^3 \text {Li}_4\left (1-\frac {2}{1+c+d x}\right )}{4 f}-\frac {3 b^3 \text {Li}_4\left (1-\frac {2 d (e+f x)}{(d e+f-c f) (1+c+d x)}\right )}{4 f}\\ \end {align*}
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Mathematica [F]
time = 46.23, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a+b \tanh ^{-1}(c+d x)\right )^3}{e+f x} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
Integrate[(a + b*ArcTanh[c + d*x])^3/(e + f*x),x]
[Out]
Integrate[(a + b*ArcTanh[c + d*x])^3/(e + f*x), x]
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 7.70, size = 3825, normalized size = 12.42
| | |
| method |
result |
size |
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| derivativedivides |
\(\text {Expression too large to display}\) |
\(3825\) |
| default |
\(\text {Expression too large to display}\) |
\(3825\) |
| | |
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|
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a+b*arctanh(d*x+c))^3/(f*x+e),x,method=_RETURNVERBOSE)
[Out]
1/d*(-3/2*I*a*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))
)^2-3/2*I*a*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^3*d/f*Pi*arctanh(d*x+c)^3*
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))+3/2*I*a*b^2*d/f*Pi*arctanh(d*x+c)^2*cs
gn(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^3*d*c/(c*f-d*e-f)*arctanh(d*x+c)^3*ln(
1-(c*f-d*e-f)*(d*x+c+1)^2/(1-(d*x+c)^2)/(-c*f+d*e-f))+3*a*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))+3/2*a*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))+3*a*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^3*d^2/f*e/(c*f-d*e-f)*arctanh(d*x+c)^3*ln(1-(c*f-d*e-f)*(d*x+c+1)^2/(1-(d*x+c)^2)/(-c*f+d*e-f
))-3/2*b^3*d^2/f*e/(c*f-d*e-f)*arctanh(d*x+c)^2*polylog(2,(c*f-d*e-f)*(d*x+c+1)^2/(1-(d*x+c)^2)/(-c*f+d*e-f))+
3/2*b^3*d^2/f*e/(c*f-d*e-f)*arctanh(d*x+c)*polylog(3,(c*f-d*e-f)*(d*x+c+1)^2/(1-(d*x+c)^2)/(-c*f+d*e-f))+1/2*I
*b^3*d/f*Pi*arctanh(d*x+c)^3*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+a^3*d*ln(c*f-d*e-f*(d*x+c))/f-3/4*b^3*d/f*polyl
og(4,-(d*x+c+1)^2/(1-(d*x+c)^2))-3/4*b^3*d/(c*f-d*e-f)*polylog(4,(c*f-d*e-f)*(d*x+c+1)^2/(1-(d*x+c)^2)/(-c*f+d
*e-f))-3/2*a^2*b*d/f*dilog((-f*(d*x+c)-f)/(-c*f+d*e-f))+3/2*a^2*b*d/f*dilog((-f*(d*x+c)+f)/(-c*f+d*e+f))+3/2*a
*b^2*d/f*polylog(3,-(d*x+c+1)^2/(1-(d*x+c)^2))+3/2*a*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^3*d*ln(c*f-d*e-f*(d*x+c))/f*arctanh(d*x+c)^3-b^3*d/f*arctanh(d*x+c)^3*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)-3/2*b^3*d/f*arctanh
(d*x+c)^2*polylog(2,-(d*x+c+1)^2/(1-(d*x+c)^2))+3/2*b^3*d/f*arctanh(d*x+c)*polylog(3,-(d*x+c+1)^2/(1-(d*x+c)^2
))+3/4*b^3*d*c/(c*f-d*e-f)*polylog(4,(c*f-d*e-f)*(d*x+c+1)^2/(1-(d*x+c)^2)/(-c*f+d*e-f))-b^3*d/(c*f-d*e-f)*arc
tanh(d*x+c)^3*ln(1-(c*f-d*e-f)*(d*x+c+1)^2/(1-(d*x+c)^2)/(-c*f+d*e-f))-3/2*b^3*d/(c*f-d*e-f)*arctanh(d*x+c)^2*
polylog(2,(c*f-d*e-f)*(d*x+c+1)^2/(1-(d*x+c)^2)/(-c*f+d*e-f))+3/2*b^3*d/(c*f-d*e-f)*arctanh(d*x+c)*polylog(3,(
c*f-d*e-f)*(d*x+c+1)^2/(1-(d*x+c)^2)/(-c*f+d*e-f))-1/2*I*b^3*d/f*Pi*arctanh(d*x+c)^3*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^3*d/f*Pi*arctanh(d*x+c)^3*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)))^2-3*a*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))-3
*a*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))+3/2*I*
a*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+3/2*b^3*d*c/(c*f-d*e-f)*arctanh(d*x+c)^2*polyl
og(2,(c*f-d*e-f)*(d*x+c+1)^2/(1-(d*x+c)^2)/(-c*f+d*e-f))-3/2*b^3*d*c/(c*f-d*e-f)*arctanh(d*x+c)*polylog(3,(c*f
-d*e-f)*(d*x+c+1)^2/(1-(d*x+c)^2)/(-c*f+d*e-f))-3/4*b^3*d^2/f*e/(c*f-d*e-f)*polylog(4,(c*f-d*e-f)*(d*x+c+1)^2/
(1-(d*x+c)^2)/(-c*f+d*e-f))+3*a^2*b*d*ln(c*f-d*e-f*(d*x+c))/f*arctanh(d*x+c)-3/2*a^2*b*d/f*ln(c*f-d*e-f*(d*x+c
))*ln((-f*(d*x+c)-f)/(-c*f+d*e-f))+3/2*a^2*b*d/f*ln(c*f-d*e-f*(d*x+c))*ln((-f*(d*x+c)+f)/(-c*f+d*e+f))+3*a*b^2
*d*ln(c*f-d*e-f*(d*x+c))/f*arctanh(d*x+c)^2-3*a*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)-3*a*b^2*d/f*arctanh(d*x+c)*polylog(2,-(d*
x+c+1)^2/(1-(d*x+c)^2))-3/2*a*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)
)-3*a*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))-3*a*b^2*d/(c
*f-d*e-f)*arctanh(d*x+c)*polylog(2,(c*f-d*e-f)*...
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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(d*x+c))^3/(f*x+e),x, algorithm="maxima")
[Out]
a^3*log(f*x + e)/f + integrate(1/8*b^3*(log(d*x + c + 1) - log(-d*x - c + 1))^3/(f*x + e) + 3/4*a*b^2*(log(d*x
+ c + 1) - log(-d*x - c + 1))^2/(f*x + e) + 3/2*a^2*b*(log(d*x + c + 1) - log(-d*x - c + 1))/(f*x + e), 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(d*x+c))^3/(f*x+e),x, algorithm="fricas")
[Out]
integral((b^3*arctanh(d*x + c)^3 + 3*a*b^2*arctanh(d*x + c)^2 + 3*a^2*b*arctanh(d*x + c) + a^3)/(f*x + e), 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 + d x \right )}\right )^{3}}{e + f x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*atanh(d*x+c))**3/(f*x+e),x)
[Out]
Integral((a + b*atanh(c + d*x))**3/(e + f*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(d*x+c))^3/(f*x+e),x, algorithm="giac")
[Out]
integrate((b*arctanh(d*x + c) + a)^3/(f*x + e), x)
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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 )}^3}{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))^3/(e + f*x),x)
[Out]
int((a + b*atanh(c + d*x))^3/(e + f*x), x)
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