3.2.44 \(\int \frac {(a+b \cot ^{-1}(c+d x))^3}{e+f x} \, dx\) [144]
Optimal. Leaf size=372 \[ -\frac {\left (a+b \cot ^{-1}(c+d x)\right )^3 \log \left (\frac {2}{1-i (c+d x)}\right )}{f}+\frac {\left (a+b \cot ^{-1}(c+d x)\right )^3 \log \left (\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{f}-\frac {3 i b \left (a+b \cot ^{-1}(c+d x)\right )^2 \text {PolyLog}\left (2,1-\frac {2}{1-i (c+d x)}\right )}{2 f}+\frac {3 i b \left (a+b \cot ^{-1}(c+d x)\right )^2 \text {PolyLog}\left (2,1-\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{2 f}-\frac {3 b^2 \left (a+b \cot ^{-1}(c+d x)\right ) \text {PolyLog}\left (3,1-\frac {2}{1-i (c+d x)}\right )}{2 f}+\frac {3 b^2 \left (a+b \cot ^{-1}(c+d x)\right ) \text {PolyLog}\left (3,1-\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{2 f}+\frac {3 i b^3 \text {PolyLog}\left (4,1-\frac {2}{1-i (c+d x)}\right )}{4 f}-\frac {3 i b^3 \text {PolyLog}\left (4,1-\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{4 f} \]
[Out]
-(a+b*arccot(d*x+c))^3*ln(2/(1-I*(d*x+c)))/f+(a+b*arccot(d*x+c))^3*ln(2*d*(f*x+e)/(d*e+I*f-c*f)/(1-I*(d*x+c)))
/f-3/2*I*b*(a+b*arccot(d*x+c))^2*polylog(2,1-2/(1-I*(d*x+c)))/f+3/2*I*b*(a+b*arccot(d*x+c))^2*polylog(2,1-2*d*
(f*x+e)/(d*e+I*f-c*f)/(1-I*(d*x+c)))/f-3/2*b^2*(a+b*arccot(d*x+c))*polylog(3,1-2/(1-I*(d*x+c)))/f+3/2*b^2*(a+b
*arccot(d*x+c))*polylog(3,1-2*d*(f*x+e)/(d*e+I*f-c*f)/(1-I*(d*x+c)))/f+3/4*I*b^3*polylog(4,1-2/(1-I*(d*x+c)))/
f-3/4*I*b^3*polylog(4,1-2*d*(f*x+e)/(d*e+I*f-c*f)/(1-I*(d*x+c)))/f
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Rubi [A]
time = 0.15, antiderivative size = 372, 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 = {5156, 4971}
\begin {gather*} \frac {3 b^2 \left (a+b \cot ^{-1}(c+d x)\right ) \text {Li}_3\left (1-\frac {2 d (e+f x)}{(d e-c f+i f) (1-i (c+d x))}\right )}{2 f}-\frac {3 b^2 \text {Li}_3\left (1-\frac {2}{1-i (c+d x)}\right ) \left (a+b \cot ^{-1}(c+d x)\right )}{2 f}+\frac {3 i b \left (a+b \cot ^{-1}(c+d x)\right )^2 \text {Li}_2\left (1-\frac {2 d (e+f x)}{(d e-c f+i f) (1-i (c+d x))}\right )}{2 f}+\frac {\left (a+b \cot ^{-1}(c+d x)\right )^3 \log \left (\frac {2 d (e+f x)}{(1-i (c+d x)) (-c f+d e+i f)}\right )}{f}-\frac {3 i b \text {Li}_2\left (1-\frac {2}{1-i (c+d x)}\right ) \left (a+b \cot ^{-1}(c+d x)\right )^2}{2 f}-\frac {\log \left (\frac {2}{1-i (c+d x)}\right ) \left (a+b \cot ^{-1}(c+d x)\right )^3}{f}-\frac {3 i b^3 \text {Li}_4\left (1-\frac {2 d (e+f x)}{(d e-c f+i f) (1-i (c+d x))}\right )}{4 f}+\frac {3 i b^3 \text {Li}_4\left (1-\frac {2}{1-i (c+d x)}\right )}{4 f} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(a + b*ArcCot[c + d*x])^3/(e + f*x),x]
[Out]
-(((a + b*ArcCot[c + d*x])^3*Log[2/(1 - I*(c + d*x))])/f) + ((a + b*ArcCot[c + d*x])^3*Log[(2*d*(e + f*x))/((d
*e + I*f - c*f)*(1 - I*(c + d*x)))])/f - (((3*I)/2)*b*(a + b*ArcCot[c + d*x])^2*PolyLog[2, 1 - 2/(1 - I*(c + d
*x))])/f + (((3*I)/2)*b*(a + b*ArcCot[c + d*x])^2*PolyLog[2, 1 - (2*d*(e + f*x))/((d*e + I*f - c*f)*(1 - I*(c
+ d*x)))])/f - (3*b^2*(a + b*ArcCot[c + d*x])*PolyLog[3, 1 - 2/(1 - I*(c + d*x))])/(2*f) + (3*b^2*(a + b*ArcCo
t[c + d*x])*PolyLog[3, 1 - (2*d*(e + f*x))/((d*e + I*f - c*f)*(1 - I*(c + d*x)))])/(2*f) + (((3*I)/4)*b^3*Poly
Log[4, 1 - 2/(1 - I*(c + d*x))])/f - (((3*I)/4)*b^3*PolyLog[4, 1 - (2*d*(e + f*x))/((d*e + I*f - c*f)*(1 - I*(
c + d*x)))])/f
Rule 4971
Int[((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^3/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcCot[c*x])^3)*(Log[
2/(1 - I*c*x)]/e), x] + (Simp[(a + b*ArcCot[c*x])^3*(Log[2*c*((d + e*x)/((c*d + I*e)*(1 - I*c*x)))]/e), x] - S
imp[3*I*b*(a + b*ArcCot[c*x])^2*(PolyLog[2, 1 - 2/(1 - I*c*x)]/(2*e)), x] + Simp[3*I*b*(a + b*ArcCot[c*x])^2*(
PolyLog[2, 1 - 2*c*((d + e*x)/((c*d + I*e)*(1 - I*c*x)))]/(2*e)), x] - Simp[3*b^2*(a + b*ArcCot[c*x])*(PolyLog
[3, 1 - 2/(1 - I*c*x)]/(2*e)), x] + Simp[3*b^2*(a + b*ArcCot[c*x])*(PolyLog[3, 1 - 2*c*((d + e*x)/((c*d + I*e)
*(1 - I*c*x)))]/(2*e)), x] + Simp[3*I*b^3*(PolyLog[4, 1 - 2/(1 - I*c*x)]/(4*e)), x] - Simp[3*I*b^3*(PolyLog[4,
1 - 2*c*((d + e*x)/((c*d + I*e)*(1 - I*c*x)))]/(4*e)), x]) /; FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d^2 + e^2,
0]
Rule 5156
Int[((a_.) + ArcCot[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Dist[1/d, Subst[I
nt[((d*e - c*f)/d + f*(x/d))^m*(a + b*ArcCot[x])^p, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x]
&& IGtQ[p, 0]
Rubi steps
\begin {align*} \int \frac {\left (a+b \cot ^{-1}(c+d x)\right )^3}{e+f x} \, dx &=\frac {\text {Subst}\left (\int \frac {\left (a+b \cot ^{-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 \cot ^{-1}(c+d x)\right )^3 \log \left (\frac {2}{1-i (c+d x)}\right )}{f}+\frac {\left (a+b \cot ^{-1}(c+d x)\right )^3 \log \left (\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{f}-\frac {3 i b \left (a+b \cot ^{-1}(c+d x)\right )^2 \text {Li}_2\left (1-\frac {2}{1-i (c+d x)}\right )}{2 f}+\frac {3 i b \left (a+b \cot ^{-1}(c+d x)\right )^2 \text {Li}_2\left (1-\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{2 f}-\frac {3 b^2 \left (a+b \cot ^{-1}(c+d x)\right ) \text {Li}_3\left (1-\frac {2}{1-i (c+d x)}\right )}{2 f}+\frac {3 b^2 \left (a+b \cot ^{-1}(c+d x)\right ) \text {Li}_3\left (1-\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{2 f}+\frac {3 i b^3 \text {Li}_4\left (1-\frac {2}{1-i (c+d x)}\right )}{4 f}-\frac {3 i b^3 \text {Li}_4\left (1-\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{4 f}\\ \end {align*}
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Mathematica [F]
time = 37.89, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a+b \cot ^{-1}(c+d x)\right )^3}{e+f x} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
Integrate[(a + b*ArcCot[c + d*x])^3/(e + f*x),x]
[Out]
Integrate[(a + b*ArcCot[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 = 11.31, size = 4291, normalized size = 11.53
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| method |
result |
size |
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| derivativedivides |
\(\text {Expression too large to display}\) |
\(4291\) |
| default |
\(\text {Expression too large to display}\) |
\(4291\) |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a+b*arccot(d*x+c))^3/(f*x+e),x,method=_RETURNVERBOSE)
[Out]
1/d*(1/2*I*b^3*d/f*Pi*arccot(d*x+c)^3*csgn(I/((d*x+c+I)^2/(1+(d*x+c)^2)-1))*csgn(I*(c*f*(d*x+c+I)^2/(1+(d*x+c)
^2)-d*e*(d*x+c+I)^2/(1+(d*x+c)^2)-c*f+d*e-I*f*(d*x+c+I)^2/(1+(d*x+c)^2)-I*f))*csgn(I*(c*f*(d*x+c+I)^2/(1+(d*x+
c)^2)-d*e*(d*x+c+I)^2/(1+(d*x+c)^2)-c*f+d*e-I*f*(d*x+c+I)^2/(1+(d*x+c)^2)-I*f)/((d*x+c+I)^2/(1+(d*x+c)^2)-1))+
6*I*a*b^2*d^2/f*e*arccot(d*x+c)*polylog(2,(d*e+I*f-c*f)/(-c*f+d*e-I*f)*(d*x+c+I)^2/(1+(d*x+c)^2))/(-2*I*f+2*c*
f-2*d*e)-3/2*I*a*b^2*d/f*Pi*arccot(d*x+c)^2*csgn(I*(c*f*(d*x+c+I)^2/(1+(d*x+c)^2)-d*e*(d*x+c+I)^2/(1+(d*x+c)^2
)-c*f+d*e-I*f*(d*x+c+I)^2/(1+(d*x+c)^2)-I*f))*csgn(I*(c*f*(d*x+c+I)^2/(1+(d*x+c)^2)-d*e*(d*x+c+I)^2/(1+(d*x+c)
^2)-c*f+d*e-I*f*(d*x+c+I)^2/(1+(d*x+c)^2)-I*f)/((d*x+c+I)^2/(1+(d*x+c)^2)-1))^2-3/2*I*a*b^2*d/f*Pi*arccot(d*x+
c)^2*csgn(I/((d*x+c+I)^2/(1+(d*x+c)^2)-1))*csgn(I*(c*f*(d*x+c+I)^2/(1+(d*x+c)^2)-d*e*(d*x+c+I)^2/(1+(d*x+c)^2)
-c*f+d*e-I*f*(d*x+c+I)^2/(1+(d*x+c)^2)-I*f)/((d*x+c+I)^2/(1+(d*x+c)^2)-1))^2-6*I*b^3*d/f*polylog(4,-(d*x+c+I)/
(1+(d*x+c)^2)^(1/2))+3*a*b^2*d*ln(c*f-d*e-f*(d*x+c))/f*arccot(d*x+c)^2-3*a*b^2*d/f*arccot(d*x+c)^2*ln(c*f*(d*x
+c+I)^2/(1+(d*x+c)^2)-d*e*(d*x+c+I)^2/(1+(d*x+c)^2)-c*f+d*e-I*f*(d*x+c+I)^2/(1+(d*x+c)^2)-I*f)+3*a*b^2*d/f*arc
cot(d*x+c)^2*ln((d*x+c+I)^2/(1+(d*x+c)^2)-1)-3*a*b^2*d/f*arccot(d*x+c)^2*ln(1+(d*x+c+I)/(1+(d*x+c)^2)^(1/2))-3
*a*b^2*d/f*arccot(d*x+c)^2*ln(1-(d*x+c+I)/(1+(d*x+c)^2)^(1/2))+3/2*a*b^2*d*c/(-I*f+c*f-d*e)*polylog(3,(d*e+I*f
-c*f)/(-c*f+d*e-I*f)*(d*x+c+I)^2/(1+(d*x+c)^2))-3*a*b^2*d/(-I*f+c*f-d*e)*arccot(d*x+c)*polylog(2,(d*e+I*f-c*f)
/(-c*f+d*e-I*f)*(d*x+c+I)^2/(1+(d*x+c)^2))-3/2*I*a*b^2*d/(-I*f+c*f-d*e)*polylog(3,(d*e+I*f-c*f)/(-c*f+d*e-I*f)
*(d*x+c+I)^2/(1+(d*x+c)^2))+3*a^2*b*d*ln(c*f-d*e-f*(d*x+c))/f*arccot(d*x+c)+3/2*I*a^2*b*d/f*dilog((I*f+f*(d*x+
c))/(c*f-d*e+I*f))-3/2*I*a^2*b*d/f*dilog((I*f-f*(d*x+c))/(d*e+I*f-c*f))+b^3*d*c/(-I*f+c*f-d*e)*arccot(d*x+c)^3
*ln(1-(d*e+I*f-c*f)/(-c*f+d*e-I*f)*(d*x+c+I)^2/(1+(d*x+c)^2))+3/2*b^3*d*c/(-I*f+c*f-d*e)*arccot(d*x+c)*polylog
(3,(d*e+I*f-c*f)/(-c*f+d*e-I*f)*(d*x+c+I)^2/(1+(d*x+c)^2))-I*b^3*d/(-I*f+c*f-d*e)*arccot(d*x+c)^3*ln(1-(d*e+I*
f-c*f)/(-c*f+d*e-I*f)*(d*x+c+I)^2/(1+(d*x+c)^2))+3*I*b^3*d/f*arccot(d*x+c)^2*polylog(2,-(d*x+c+I)/(1+(d*x+c)^2
)^(1/2))+3*I*b^3*d/f*arccot(d*x+c)^2*polylog(2,(d*x+c+I)/(1+(d*x+c)^2)^(1/2))-3/2*I*b^3*d/(-I*f+c*f-d*e)*arcco
t(d*x+c)*polylog(3,(d*e+I*f-c*f)/(-c*f+d*e-I*f)*(d*x+c+I)^2/(1+(d*x+c)^2))+3/4*I*b^3*d*c/(-I*f+c*f-d*e)*polylo
g(4,(d*e+I*f-c*f)/(-c*f+d*e-I*f)*(d*x+c+I)^2/(1+(d*x+c)^2))+a^3*d*ln(c*f-d*e-f*(d*x+c))/f+3/4*b^3*d/(-I*f+c*f-
d*e)*polylog(4,(d*e+I*f-c*f)/(-c*f+d*e-I*f)*(d*x+c+I)^2/(1+(d*x+c)^2))+3/2*I*a*b^2*d/f*Pi*arccot(d*x+c)^2*csgn
(I/((d*x+c+I)^2/(1+(d*x+c)^2)-1))*csgn(I*(c*f*(d*x+c+I)^2/(1+(d*x+c)^2)-d*e*(d*x+c+I)^2/(1+(d*x+c)^2)-c*f+d*e-
I*f*(d*x+c+I)^2/(1+(d*x+c)^2)-I*f))*csgn(I*(c*f*(d*x+c+I)^2/(1+(d*x+c)^2)-d*e*(d*x+c+I)^2/(1+(d*x+c)^2)-c*f+d*
e-I*f*(d*x+c+I)^2/(1+(d*x+c)^2)-I*f)/((d*x+c+I)^2/(1+(d*x+c)^2)-1))-1/2*I*b^3*d/f*Pi*arccot(d*x+c)^3*csgn(I/((
d*x+c+I)^2/(1+(d*x+c)^2)-1))*csgn(I*(c*f*(d*x+c+I)^2/(1+(d*x+c)^2)-d*e*(d*x+c+I)^2/(1+(d*x+c)^2)-c*f+d*e-I*f*(
d*x+c+I)^2/(1+(d*x+c)^2)-I*f)/((d*x+c+I)^2/(1+(d*x+c)^2)-1))^2+3*I*b^3*d^2/f*e*arccot(d*x+c)^2*polylog(2,(d*e+
I*f-c*f)/(-c*f+d*e-I*f)*(d*x+c+I)^2/(1+(d*x+c)^2))/(-2*I*f+2*c*f-2*d*e)-3*a*b^2*d^2/f*e/(-I*f+c*f-d*e)*arccot(
d*x+c)^2*ln(1-(d*e+I*f-c*f)/(-c*f+d*e-I*f)*(d*x+c+I)^2/(1+(d*x+c)^2))-3*I*a*b^2*d*c/(-I*f+c*f-d*e)*arccot(d*x+
c)*polylog(2,(d*e+I*f-c*f)/(-c*f+d*e-I*f)*(d*x+c+I)^2/(1+(d*x+c)^2))+3/2*I*a*b^2*d/f*Pi*arccot(d*x+c)^2*csgn(I
*(c*f*(d*x+c+I)^2/(1+(d*x+c)^2)-d*e*(d*x+c+I)^2/(1+(d*x+c)^2)-c*f+d*e-I*f*(d*x+c+I)^2/(1+(d*x+c)^2)-I*f)/((d*x
+c+I)^2/(1+(d*x+c)^2)-1))^3-6*a*b^2*d/f*polylog(3,-(d*x+c+I)/(1+(d*x+c)^2)^(1/2))-6*a*b^2*d/f*polylog(3,(d*x+c
+I)/(1+(d*x+c)^2)^(1/2))+b^3*d*ln(c*f-d*e-f*(d*x+c))/f*arccot(d*x+c)^3-3/2*b^3*d/(-I*f+c*f-d*e)*arccot(d*x+c)^
2*polylog(2,(d*e+I*f-c*f)/(-c*f+d*e-I*f)*(d*x+c+I)^2/(1+(d*x+c)^2))+b^3*d/f*arccot(d*x+c)^3*ln((d*x+c+I)^2/(1+
(d*x+c)^2)-1)-6*b^3*d/f*arccot(d*x+c)*polylog(3,(d*x+c+I)/(1+(d*x+c)^2)^(1/2))-b^3*d/f*arccot(d*x+c)^3*ln(c*f*
(d*x+c+I)^2/(1+(d*x+c)^2)-d*e*(d*x+c+I)^2/(1+(d*x+c)^2)-c*f+d*e-I*f*(d*x+c+I)^2/(1+(d*x+c)^2)-I*f)-b^3*d/f*arc
cot(d*x+c)^3*ln(1+(d*x+c+I)/(1+(d*x+c)^2)^(1/2))-6*b^3*d/f*arccot(d*x+c)*polylog(3,-(d*x+c+I)/(1+(d*x+c)^2)^(1
/2))-b^3*d/f*arccot(d*x+c)^3*ln(1-(d*x+c+I)/(1+(d*x+c)^2)^(1/2))-6*I*b^3*d/f*polylog(4,(d*x+c+I)/(1+(d*x+c)^2)
^(1/2))+1/2*I*b^3*d/f*Pi*arccot(d*x+c)^3*csgn(I*(c*f*(d*x+c+I)^2/(1+(d*x+c)^2)-d*e*(d*x+c+I)^2/(1+(d*x+c)^2)-c
*f+d*e-I*f*(d*x+c+I)^2/(1+(d*x+c)^2)-I*f)/((d*x+c+I)^2/(1+(d*x+c)^2)-1))^3+3/2*I*a^2*b*d*ln(c*f-d*e-f*(d*x+c))
/f*ln((I*f+f*(d*x+c))/(c*f-d*e+I*f))-3/2*I*a^2*b*d*ln(c*f-d*e-f*(d*x+c))/f*ln((I*f-f*(d*x+c))/(d*e+I*f-c*f))-1
/2*I*b^3*d/f*Pi*arccot(d*x+c)^3*csgn(I*(c*f*(d*x+c+I)^2/(1+(d*x+c)^2)-d*e*(d*x+c+I)^2/(1+(d*x+c)^2)-c*f+d*e-I*
f*(d*x+c+I)^2/(1+(d*x+c)^2)-I*f))*csgn(I*(c*f*(d*x+c+I)^2/(1+(d*x+c)^2)-d*e*(d*x+c+I)^2/(1+(d*x+c)^2)-c*f+d*e-
I*f*(d*x+c+I)^2/(1+(d*x+c)^2)-I*f)/((d*x+c+I)^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((a+b*arccot(d*x+c))^3/(f*x+e),x, algorithm="maxima")
[Out]
a^3*log(f*x + e)/f + integrate(1/32*(28*b^3*arctan2(1, d*x + c)^3 + 3*b^3*arctan2(1, d*x + c)*log(d^2*x^2 + 2*
c*d*x + c^2 + 1)^2 + 96*a*b^2*arctan2(1, d*x + c)^2 + 96*a^2*b*arctan2(1, d*x + c))/(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*arccot(d*x+c))^3/(f*x+e),x, algorithm="fricas")
[Out]
integral((b^3*arccot(d*x + c)^3 + 3*a*b^2*arccot(d*x + c)^2 + 3*a^2*b*arccot(d*x + c) + a^3)/(f*x + e), x)
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*acot(d*x+c))**3/(f*x+e),x)
[Out]
Timed out
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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*arccot(d*x+c))^3/(f*x+e),x, algorithm="giac")
[Out]
Timed out
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {acot}\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*acot(c + d*x))^3/(e + f*x),x)
[Out]
int((a + b*acot(c + d*x))^3/(e + f*x), x)
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