\(\int x \cot ^{-1}(a x)^3 \, dx\) [27]

Optimal result

Integrand size = 8, antiderivative size = 103 \[ \int x \cot ^{-1}(a x)^3 \, dx=\frac {3 i \cot ^{-1}(a x)^2}{2 a^2}+\frac {3 x \cot ^{-1}(a x)^2}{2 a}+\frac {\cot ^{-1}(a x)^3}{2 a^2}+\frac {1}{2} x^2 \cot ^{-1}(a x)^3-\frac {3 \cot ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{a^2}+\frac {3 i \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{2 a^2} \] Output:

3/2*I*arccot(a*x)^2/a^2+3/2*x*arccot(a*x)^2/a+1/2*arccot(a*x)^3/a^2+1/2*x^ 
2*arccot(a*x)^3-3*arccot(a*x)*ln(2/(1+I*a*x))/a^2+3/2*I*polylog(2,1-2/(1+I 
*a*x))/a^2
 

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.74 \[ \int x \cot ^{-1}(a x)^3 \, dx=\frac {\cot ^{-1}(a x) \left (3 (i+a x) \cot ^{-1}(a x)+\left (1+a^2 x^2\right ) \cot ^{-1}(a x)^2-6 \log \left (1-e^{2 i \cot ^{-1}(a x)}\right )\right )+3 i \operatorname {PolyLog}\left (2,e^{2 i \cot ^{-1}(a x)}\right )}{2 a^2} \] Input:

Integrate[x*ArcCot[a*x]^3,x]
 

Output:

(ArcCot[a*x]*(3*(I + a*x)*ArcCot[a*x] + (1 + a^2*x^2)*ArcCot[a*x]^2 - 6*Lo 
g[1 - E^((2*I)*ArcCot[a*x])]) + (3*I)*PolyLog[2, E^((2*I)*ArcCot[a*x])])/( 
2*a^2)
 

Rubi [A] (verified)

Time = 0.73 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.14, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {5362, 5452, 5346, 5420, 5456, 5380, 2849, 2752}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int x \cot ^{-1}(a x)^3 \, dx\)

\(\Big \downarrow \) 5362

\(\displaystyle \frac {3}{2} a \int \frac {x^2 \cot ^{-1}(a x)^2}{a^2 x^2+1}dx+\frac {1}{2} x^2 \cot ^{-1}(a x)^3\)

\(\Big \downarrow \) 5452

\(\displaystyle \frac {3}{2} a \left (\frac {\int \cot ^{-1}(a x)^2dx}{a^2}-\frac {\int \frac {\cot ^{-1}(a x)^2}{a^2 x^2+1}dx}{a^2}\right )+\frac {1}{2} x^2 \cot ^{-1}(a x)^3\)

\(\Big \downarrow \) 5346

\(\displaystyle \frac {3}{2} a \left (\frac {2 a \int \frac {x \cot ^{-1}(a x)}{a^2 x^2+1}dx+x \cot ^{-1}(a x)^2}{a^2}-\frac {\int \frac {\cot ^{-1}(a x)^2}{a^2 x^2+1}dx}{a^2}\right )+\frac {1}{2} x^2 \cot ^{-1}(a x)^3\)

\(\Big \downarrow \) 5420

\(\displaystyle \frac {3}{2} a \left (\frac {2 a \int \frac {x \cot ^{-1}(a x)}{a^2 x^2+1}dx+x \cot ^{-1}(a x)^2}{a^2}+\frac {\cot ^{-1}(a x)^3}{3 a^3}\right )+\frac {1}{2} x^2 \cot ^{-1}(a x)^3\)

\(\Big \downarrow \) 5456

\(\displaystyle \frac {1}{2} x^2 \cot ^{-1}(a x)^3+\frac {3}{2} a \left (\frac {\cot ^{-1}(a x)^3}{3 a^3}+\frac {x \cot ^{-1}(a x)^2+2 a \left (\frac {i \cot ^{-1}(a x)^2}{2 a^2}-\frac {\int \frac {\cot ^{-1}(a x)}{i-a x}dx}{a}\right )}{a^2}\right )\)

\(\Big \downarrow \) 5380

\(\displaystyle \frac {1}{2} x^2 \cot ^{-1}(a x)^3+\frac {3}{2} a \left (\frac {\cot ^{-1}(a x)^3}{3 a^3}+\frac {x \cot ^{-1}(a x)^2+2 a \left (\frac {i \cot ^{-1}(a x)^2}{2 a^2}-\frac {\int \frac {\log \left (\frac {2}{i a x+1}\right )}{a^2 x^2+1}dx+\frac {\log \left (\frac {2}{1+i a x}\right ) \cot ^{-1}(a x)}{a}}{a}\right )}{a^2}\right )\)

\(\Big \downarrow \) 2849

\(\displaystyle \frac {1}{2} x^2 \cot ^{-1}(a x)^3+\frac {3}{2} a \left (\frac {\cot ^{-1}(a x)^3}{3 a^3}+\frac {x \cot ^{-1}(a x)^2+2 a \left (\frac {i \cot ^{-1}(a x)^2}{2 a^2}-\frac {\frac {\log \left (\frac {2}{1+i a x}\right ) \cot ^{-1}(a x)}{a}-\frac {i \int \frac {\log \left (\frac {2}{i a x+1}\right )}{1-\frac {2}{i a x+1}}d\frac {1}{i a x+1}}{a}}{a}\right )}{a^2}\right )\)

\(\Big \downarrow \) 2752

\(\displaystyle \frac {1}{2} x^2 \cot ^{-1}(a x)^3+\frac {3}{2} a \left (\frac {\cot ^{-1}(a x)^3}{3 a^3}+\frac {x \cot ^{-1}(a x)^2+2 a \left (\frac {i \cot ^{-1}(a x)^2}{2 a^2}-\frac {\frac {\log \left (\frac {2}{1+i a x}\right ) \cot ^{-1}(a x)}{a}-\frac {i \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )}{2 a}}{a}\right )}{a^2}\right )\)

Input:

Int[x*ArcCot[a*x]^3,x]
 

Output:

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

Defintions of rubi rules used

rule 2752

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLo 
g[2, 1 - c*x], x] /; FreeQ[{c, d, e}, x] && EqQ[e + c*d, 0]
 

rule 2849

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Simp 
[-e/g   Subst[Int[Log[2*d*x]/(1 - 2*d*x), x], x, 1/(d + e*x)], x] /; FreeQ[ 
{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]
 

rule 5346

Int[((a_.) + ArcCot[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a 
+ b*ArcCot[c*x^n])^p, x] + Simp[b*c*n*p   Int[x^n*((a + b*ArcCot[c*x^n])^(p 
 - 1)/(1 + c^2*x^(2*n))), x], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[p, 0] && 
 (EqQ[n, 1] || EqQ[p, 1])
 

rule 5362

Int[((a_.) + ArcCot[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> 
 Simp[x^(m + 1)*((a + b*ArcCot[c*x^n])^p/(m + 1)), x] + Simp[b*c*n*(p/(m + 
1))   Int[x^(m + n)*((a + b*ArcCot[c*x^n])^(p - 1)/(1 + c^2*x^(2*n))), x], 
x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0] && (EqQ[p, 1] || (EqQ[n, 1] & 
& IntegerQ[m])) && NeQ[m, -1]
 

rule 5380

Int[((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] 
 :> Simp[(-(a + b*ArcCot[c*x])^p)*(Log[2/(1 + e*(x/d))]/e), x] - Simp[b*c*( 
p/e)   Int[(a + b*ArcCot[c*x])^(p - 1)*(Log[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 5420

Int[((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symbo 
l] :> Simp[-(a + b*ArcCot[c*x])^(p + 1)/(b*c*d*(p + 1)), x] /; FreeQ[{a, b, 
 c, d, e, p}, x] && EqQ[e, c^2*d] && NeQ[p, -1]
 

rule 5452

Int[(((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e 
_.)*(x_)^2), x_Symbol] :> Simp[f^2/e   Int[(f*x)^(m - 2)*(a + b*ArcCot[c*x] 
)^p, x], x] - Simp[d*(f^2/e)   Int[(f*x)^(m - 2)*((a + b*ArcCot[c*x])^p/(d 
+ e*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && GtQ[m, 1]
 

rule 5456

Int[(((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), 
x_Symbol] :> Simp[I*((a + b*ArcCot[c*x])^(p + 1)/(b*e*(p + 1))), x] - Simp[ 
1/(c*d)   Int[(a + b*ArcCot[c*x])^p/(I - c*x), x], x] /; FreeQ[{a, b, c, d, 
 e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0]
 

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 591 vs. \(2 (89 ) = 178\).

Time = 5.10 (sec) , antiderivative size = 592, normalized size of antiderivative = 5.75

method	result	size
risch	\(\frac {\pi ^{3} x^{2}}{16}+\frac {3 \pi ^{2} x}{8 a}+\frac {21 \arctan \left (a x \right )}{32 a^{2}}-\frac {3 \pi ^{2} \arctan \left (a x \right )}{8 a^{2}}+\frac {21 \pi \ln \left (a^{2} x^{2}+1\right )}{32 a^{2}}+\frac {\pi ^{3}}{16 a^{2}}-\frac {3 i \pi \ln \left (-i a x +1\right ) x}{4 a}-\frac {3 i \ln \left (-i a x +1\right )^{2} x^{2}}{32}+\frac {3 i \ln \left (-i a x +1\right ) x^{2}}{32}+\frac {i \ln \left (-i a x +1\right )^{3}}{16 a^{2}}-\frac {9 i \ln \left (-i a x +1\right )^{2}}{32 a^{2}}+\frac {i \ln \left (-i a x +1\right )^{3} x^{2}}{16}-\frac {3 i \ln \left (-i a x +1\right )}{4 a^{2}}+\frac {3 i \operatorname {dilog}\left (\frac {1}{2}-\frac {i a x}{2}\right )}{2 a^{2}}+\frac {21 i \ln \left (a^{2} x^{2}+1\right )}{64 a^{2}}+\frac {3 i \pi ^{2}}{8 a^{2}}-\frac {3 \left (-i x^{2} \ln \left (-i a x +1\right ) a^{2}+\pi \,a^{2} x^{2}-i \ln \left (-i a x +1\right )+2 a x +\pi -2 i\right ) \ln \left (i a x +1\right )^{2}}{16 a^{2}}-\frac {3 \pi \ln \left (-i a x +1\right )^{2}}{16 a^{2}}+\frac {3 \pi \ln \left (-i a x +1\right )}{16 a^{2}}-\frac {3 \pi \ln \left (-i a x +1\right )^{2} x^{2}}{16}-\frac {3 \ln \left (-i a x +1\right )^{2} x}{16 a}-\frac {3 \ln \left (-i a x +1\right ) x}{16 a}+\left (-\frac {3 i \left (a^{2} x^{2}+1\right ) \ln \left (-i a x +1\right )^{2}}{16 a^{2}}+\frac {3 x \left (\pi a x +2\right ) \ln \left (-i a x +1\right )}{8 a}+\frac {\frac {3 i \pi ^{2} a^{2} x^{2}}{16}+\frac {3 i \pi a x}{4}+\frac {3 \ln \left (-i a x +1\right ) \pi }{8}+\frac {3 i \ln \left (-i a x +1\right )}{4}}{a^{2}}\right ) \ln \left (i a x +1\right )+\frac {3 i \left (-i a x +1\right )^{2} \ln \left (-i a x +1\right )}{32 a^{2}}-\frac {3 i \left (-i a x +1\right )^{2} \ln \left (-i a x +1\right )^{2}}{32 a^{2}}+\frac {3 i \ln \left (\frac {1}{2}-\frac {i a x}{2}\right ) \ln \left (\frac {1}{2}+\frac {i a x}{2}\right )}{2 a^{2}}-\frac {3 i \ln \left (\frac {1}{2}+\frac {i a x}{2}\right ) \ln \left (-i a x +1\right )}{2 a^{2}}+\frac {3 i \pi \arctan \left (a x \right )}{16 a^{2}}-\frac {i \left (a^{2} x^{2}+1\right ) \ln \left (i a x +1\right )^{3}}{16 a^{2}}-\frac {3 i \pi ^{2} \ln \left (-i a x +1\right ) x^{2}}{16}\)	\(592\)
parts	\(\text {Expression too large to display}\)	\(4391\)
derivativedivides	\(\text {Expression too large to display}\)	\(4393\)
default	\(\text {Expression too large to display}\)	\(4393\)

Input:

int(x*arccot(a*x)^3,x,method=_RETURNVERBOSE)
 

Output:

3/32*I/a^2*(1-I*a*x)^2*ln(1-I*a*x)-3/32*I/a^2*(1-I*a*x)^2*ln(1-I*a*x)^2-3/ 
32*I*ln(1-I*a*x)^2*x^2+3/32*I*ln(1-I*a*x)*x^2+1/16*I/a^2*ln(1-I*a*x)^3-9/3 
2*I/a^2*ln(1-I*a*x)^2+1/16*I*ln(1-I*a*x)^3*x^2+3/2*I/a^2*ln(1/2-1/2*I*a*x) 
*ln(1/2+1/2*I*a*x)-3/2*I/a^2*ln(1/2+1/2*I*a*x)*ln(1-I*a*x)+3/16*I/a^2*Pi*a 
rctan(a*x)+1/16*Pi^3*x^2+3/8*Pi^2*x/a+21/32*arctan(a*x)/a^2-3/4*I/a^2*ln(1 
-I*a*x)+3/2*I/a^2*dilog(1/2-1/2*I*a*x)-1/16*I*(a^2*x^2+1)/a^2*ln(1+I*a*x)^ 
3-3/16*I*Pi^2*ln(1-I*a*x)*x^2-3/4*I/a*Pi*ln(1-I*a*x)*x+21/64*I/a^2*ln(a^2* 
x^2+1)-3/8/a^2*Pi^2*arctan(a*x)+21/32/a^2*Pi*ln(a^2*x^2+1)+3/8*I*Pi^2/a^2- 
3/16*(-I*x^2*ln(1-I*a*x)*a^2+Pi*a^2*x^2-I*ln(1-I*a*x)+2*a*x+Pi-2*I)/a^2*ln 
(1+I*a*x)^2-3/16/a^2*Pi*ln(1-I*a*x)^2+3/16/a^2*Pi*ln(1-I*a*x)-3/16*Pi*ln(1 
-I*a*x)^2*x^2-3/16/a*ln(1-I*a*x)^2*x-3/16/a*ln(1-I*a*x)*x+1/16/a^2*Pi^3+(- 
3/16*I*(a^2*x^2+1)/a^2*ln(1-I*a*x)^2+3/8*x*(Pi*a*x+2)/a*ln(1-I*a*x)+3/16*( 
I*Pi^2*a^2*x^2+4*I*Pi*a*x+2*ln(1-I*a*x)*Pi+4*I*ln(1-I*a*x))/a^2)*ln(1+I*a* 
x)
                                                                                    
                                                                                    
 

Fricas [F]

\[ \int x \cot ^{-1}(a x)^3 \, dx=\int { x \operatorname {arccot}\left (a x\right )^{3} \,d x } \] Input:

integrate(x*arccot(a*x)^3,x, algorithm="fricas")
 

Output:

integral(x*arccot(a*x)^3, x)
 

Sympy [F]

\[ \int x \cot ^{-1}(a x)^3 \, dx=\int x \operatorname {acot}^{3}{\left (a x \right )}\, dx \] Input:

integrate(x*acot(a*x)**3,x)
 

Output:

Integral(x*acot(a*x)**3, x)
 

Maxima [F]

\[ \int x \cot ^{-1}(a x)^3 \, dx=\int { x \operatorname {arccot}\left (a x\right )^{3} \,d x } \] Input:

integrate(x*arccot(a*x)^3,x, algorithm="maxima")
 

Output:

1/32*(8*a^2*x^2*arctan2(1, a*x)^3 + 12*a*x*arctan2(1, a*x)^2 - 3*a*x*log(a 
^2*x^2 + 1)^2 + 4*(128*a^3*integrate(1/32*x^3*arctan(1/(a*x))^3/(a^3*x^2 + 
 a), x) + 96*a^2*integrate(1/32*x^2*arctan(1/(a*x))^2/(a^3*x^2 + a), x) + 
24*a^2*integrate(1/32*x^2*log(a^2*x^2 + 1)^2/(a^3*x^2 + a), x) + 96*a^2*in 
tegrate(1/32*x^2*log(a^2*x^2 + 1)/(a^3*x^2 + a), x) + 128*a*integrate(1/32 
*x*arctan(1/(a*x))^3/(a^3*x^2 + a), x) + 192*a*integrate(1/32*x*arctan(1/( 
a*x))/(a^3*x^2 + a), x) + arctan(a*x)^3/a^2 + 3*arctan(a*x)^2*arctan(1/(a* 
x))/a^2 + 3*arctan(a*x)*arctan(1/(a*x))^2/a^2 + 24*integrate(1/32*log(a^2* 
x^2 + 1)^2/(a^3*x^2 + a), x))*a^2 + 8*arctan2(1, a*x)^3)/a^2
 

Giac [F]

\[ \int x \cot ^{-1}(a x)^3 \, dx=\int { x \operatorname {arccot}\left (a x\right )^{3} \,d x } \] Input:

integrate(x*arccot(a*x)^3,x, algorithm="giac")
 

Output:

integrate(x*arccot(a*x)^3, x)
 

Mupad [F(-1)]

Timed out. \[ \int x \cot ^{-1}(a x)^3 \, dx=\int x\,{\mathrm {acot}\left (a\,x\right )}^3 \,d x \] Input:

int(x*acot(a*x)^3,x)
 

Output:

int(x*acot(a*x)^3, x)
 

Reduce [F]

\[ \int x \cot ^{-1}(a x)^3 \, dx=\frac {\mathit {acot} \left (a x \right )^{3} a^{2} x^{2}+\mathit {acot} \left (a x \right )^{3}+3 \mathit {acot} \left (a x \right )^{2} a x +3 \mathit {acot} \left (a x \right ) a^{2} x^{2}+3 \mathit {acot} \left (a x \right )-6 \left (\int \frac {\mathit {acot} \left (a x \right ) x^{3}}{a^{2} x^{2}+1}d x \right ) a^{4}+3 a x}{2 a^{2}} \] Input:

int(x*acot(a*x)^3,x)
 

Output:

(acot(a*x)**3*a**2*x**2 + acot(a*x)**3 + 3*acot(a*x)**2*a*x + 3*acot(a*x)* 
a**2*x**2 + 3*acot(a*x) - 6*int((acot(a*x)*x**3)/(a**2*x**2 + 1),x)*a**4 + 
 3*a*x)/(2*a**2)