\(\int (d+e x^2) \cot ^{-1}(a x) \log (c x^n) \, dx\) [195]

Optimal result

Integrand size = 18, antiderivative size = 182 \[ \int \left (d+e x^2\right ) \cot ^{-1}(a x) \log \left (c x^n\right ) \, dx=-\frac {5 e n x^2}{36 a}-d n x \cot ^{-1}(a x)-\frac {1}{9} e n x^3 \cot ^{-1}(a x)+\frac {e x^2 \log \left (c x^n\right )}{6 a}+d x \cot ^{-1}(a x) \log \left (c x^n\right )+\frac {1}{3} e x^3 \cot ^{-1}(a x) \log \left (c x^n\right )-\frac {d n \log \left (1+a^2 x^2\right )}{2 a}+\frac {e n \log \left (1+a^2 x^2\right )}{18 a^3}+\frac {\left (3 a^2 d-e\right ) \log \left (c x^n\right ) \log \left (1+a^2 x^2\right )}{6 a^3}+\frac {\left (3 a^2 d-e\right ) n \operatorname {PolyLog}\left (2,-a^2 x^2\right )}{12 a^3} \] Output:

-5/36*e*n*x^2/a-d*n*x*arccot(a*x)-1/9*e*n*x^3*arccot(a*x)+1/6*e*x^2*ln(c*x 
^n)/a+d*x*arccot(a*x)*ln(c*x^n)+1/3*e*x^3*arccot(a*x)*ln(c*x^n)-1/2*d*n*ln 
(a^2*x^2+1)/a+1/18*e*n*ln(a^2*x^2+1)/a^3+1/6*(3*a^2*d-e)*ln(c*x^n)*ln(a^2* 
x^2+1)/a^3+1/12*(3*a^2*d-e)*n*polylog(2,-a^2*x^2)/a^3
 

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.98 \[ \int \left (d+e x^2\right ) \cot ^{-1}(a x) \log \left (c x^n\right ) \, dx=\frac {-5 a^2 e n x^2+36 a^2 d n \log \left (\frac {1}{a \sqrt {1+\frac {1}{a^2 x^2}} x}\right )+6 a^2 e x^2 \log \left (c x^n\right )-4 a^3 x \cot ^{-1}(a x) \left (n \left (9 d+e x^2\right )-3 \left (3 d+e x^2\right ) \log \left (c x^n\right )\right )+2 e n \log \left (1+a^2 x^2\right )+18 a^2 d \log \left (c x^n\right ) \log \left (1+a^2 x^2\right )-6 e \log \left (c x^n\right ) \log \left (1+a^2 x^2\right )+\left (9 a^2 d n-3 e n\right ) \operatorname {PolyLog}\left (2,-a^2 x^2\right )}{36 a^3} \] Input:

Integrate[(d + e*x^2)*ArcCot[a*x]*Log[c*x^n],x]
 

Output:

(-5*a^2*e*n*x^2 + 36*a^2*d*n*Log[1/(a*Sqrt[1 + 1/(a^2*x^2)]*x)] + 6*a^2*e* 
x^2*Log[c*x^n] - 4*a^3*x*ArcCot[a*x]*(n*(9*d + e*x^2) - 3*(3*d + e*x^2)*Lo 
g[c*x^n]) + 2*e*n*Log[1 + a^2*x^2] + 18*a^2*d*Log[c*x^n]*Log[1 + a^2*x^2] 
- 6*e*Log[c*x^n]*Log[1 + a^2*x^2] + (9*a^2*d*n - 3*e*n)*PolyLog[2, -(a^2*x 
^2)])/(36*a^3)
 

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.98, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2835, 2009}

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.

\(\displaystyle \int \cot ^{-1}(a x) \left (d+e x^2\right ) \log \left (c x^n\right ) \, dx\)

\(\Big \downarrow \) 2835

\(\displaystyle -n \int \left (\frac {1}{3} e \cot ^{-1}(a x) x^2+\frac {e x}{6 a}+d \cot ^{-1}(a x)+\frac {\left (3 a^2 d-e\right ) \log \left (a^2 x^2+1\right )}{6 a^3 x}\right )dx+\frac {\left (3 a^2 d-e\right ) \log \left (a^2 x^2+1\right ) \log \left (c x^n\right )}{6 a^3}+d x \cot ^{-1}(a x) \log \left (c x^n\right )+\frac {1}{3} e x^3 \cot ^{-1}(a x) \log \left (c x^n\right )+\frac {e x^2 \log \left (c x^n\right )}{6 a}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\left (3 a^2 d-e\right ) \log \left (a^2 x^2+1\right ) \log \left (c x^n\right )}{6 a^3}-n \left (\frac {d \log \left (a^2 x^2+1\right )}{2 a}-\frac {\left (3 a^2 d-e\right ) \operatorname {PolyLog}\left (2,-a^2 x^2\right )}{12 a^3}-\frac {e \log \left (a^2 x^2+1\right )}{18 a^3}+d x \cot ^{-1}(a x)+\frac {1}{9} e x^3 \cot ^{-1}(a x)+\frac {5 e x^2}{36 a}\right )+d x \cot ^{-1}(a x) \log \left (c x^n\right )+\frac {1}{3} e x^3 \cot ^{-1}(a x) \log \left (c x^n\right )+\frac {e x^2 \log \left (c x^n\right )}{6 a}\)

Input:

Int[(d + e*x^2)*ArcCot[a*x]*Log[c*x^n],x]
 

Output:

(e*x^2*Log[c*x^n])/(6*a) + d*x*ArcCot[a*x]*Log[c*x^n] + (e*x^3*ArcCot[a*x] 
*Log[c*x^n])/3 + ((3*a^2*d - e)*Log[c*x^n]*Log[1 + a^2*x^2])/(6*a^3) - n*( 
(5*e*x^2)/(36*a) + d*x*ArcCot[a*x] + (e*x^3*ArcCot[a*x])/9 + (d*Log[1 + a^ 
2*x^2])/(2*a) - (e*Log[1 + a^2*x^2])/(18*a^3) - ((3*a^2*d - e)*PolyLog[2, 
-(a^2*x^2)])/(12*a^3))
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2835

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(Px_.)*(F_)[(d_.)*((e_.) + (f_.)* 
(x_))], x_Symbol] :> With[{u = IntHide[Px*F[d*(e + f*x)], x]}, Simp[(a + b* 
Log[c*x^n])   u, x] - Simp[b*n   Int[1/x   u, x], x]] /; FreeQ[{a, b, c, d, 
 e, f, n}, x] && PolynomialQ[Px, x] && MemberQ[{ArcTan, ArcCot, ArcTanh, Ar 
cCoth}, F]
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 63.14 (sec) , antiderivative size = 1633, normalized size of antiderivative = 8.97

method	result	size
risch	\(\text {Expression too large to display}\)	\(1633\)
default	\(\text {Expression too large to display}\)	\(147949\)

Input:

int((e*x^2+d)*arccot(a*x)*ln(c*x^n),x,method=_RETURNVERBOSE)
 

Output:

1/2*d*n/a*dilog(-I*(I+a*x))-1/6*e*n/a^3*dilog(-I*(I+a*x))-1/2*d*n/a*dilog( 
-I*a*x)+1/6*e*n/a^3*dilog(-I*a*x)-1/4*d*n*Pi*ln(x)*csgn(a*x-I)^3*x+3/4*d*n 
*Pi*ln(x)*csgn(a*x-I)^2*x-1/4*d*n*Pi*csgn(a*x-I)^2*csgn(I*(a*x-I))*x-1/4*d 
*n*Pi*csgn(a*x-I)*csgn(I*(a*x-I))*x-1/4*d*n*csgn(I+a*x)^3*Pi*ln(x)*x-1/4*d 
*n*csgn(I+a*x)^2*csgn(I*(I+a*x))*Pi*x-3/4*d*n*csgn(I+a*x)^2*Pi*ln(x)*x+1/4 
*d*n*csgn(I+a*x)*csgn(I*(I+a*x))*Pi*x-1/12*e*n*Pi*ln(x)*csgn(a*x-I)^3*x^3+ 
1/4*e*n*Pi*ln(x)*csgn(a*x-I)^2*x^3-1/36*e*n*Pi*csgn(a*x-I)^2*csgn(I*(a*x-I 
))*x^3-1/36*e*n*Pi*csgn(a*x-I)*csgn(I*(a*x-I))*x^3-1/12*e*n*Pi*csgn(I+a*x) 
^3*ln(x)*x^3-1/4*e*n*Pi*csgn(I+a*x)^2*ln(x)*x^3-1/36*e*n*Pi*csgn(I+a*x)^2* 
csgn(I*(I+a*x))*x^3+1/36*e*n*Pi*csgn(I+a*x)*csgn(I*(I+a*x))*x^3-11/18*e/a^ 
3*n*ln(x)+1/18*I*e*n*x^3*ln(I+a*x)+1/2*I*d*n*x*ln(I+a*x)-1/2*I*d*ln(1-I*a* 
x)*ln(x^n)*x-1/6*I*ln(1-I*a*x)*ln(x^n)*e*x^3-1/2*d*n/a*ln(-I*(I-a*x))*ln(- 
I*a*x)+1/4*d*n*Pi*csgn(a*x-I)^3*x-3/4*d*n*Pi*csgn(a*x-I)^2*x+1/4*d*n*Pi*ln 
(x)*csgn(a*x-I)^2*csgn(I*(a*x-I))*x+1/4*d*n*Pi*ln(x)*csgn(a*x-I)*csgn(I*(a 
*x-I))*x+1/12*e*n*Pi*ln(x)*csgn(a*x-I)^2*csgn(I*(a*x-I))*x^3+1/12*e*n*Pi*l 
n(x)*csgn(a*x-I)*csgn(I*(a*x-I))*x^3+1/4*d*n*csgn(I+a*x)^2*csgn(I*(I+a*x)) 
*Pi*ln(x)*x-1/4*d*n*csgn(I+a*x)*csgn(I*(I+a*x))*Pi*ln(x)*x-1/12*e*n*Pi*csg 
n(I+a*x)*csgn(I*(I+a*x))*ln(x)*x^3+1/12*e*n*Pi*csgn(I+a*x)^2*csgn(I*(I+a*x 
))*ln(x)*x^3+1/2*d*n/a*ln(x)*ln(-I*(I+a*x))-1/6*e*n/a^3*ln(x)*ln(-I*(I+a*x 
))+1/36*e*n*Pi*csgn(I+a*x)^3*x^3+1/12*e*n*Pi*csgn(I+a*x)^2*x^3+1/4*d*n*...
 

Fricas [F]

\[ \int \left (d+e x^2\right ) \cot ^{-1}(a x) \log \left (c x^n\right ) \, dx=\int { {\left (e x^{2} + d\right )} \operatorname {arccot}\left (a x\right ) \log \left (c x^{n}\right ) \,d x } \] Input:

integrate((e*x^2+d)*arccot(a*x)*log(c*x^n),x, algorithm="fricas")
 

Output:

integral((e*x^2 + d)*arccot(a*x)*log(c*x^n), x)
 

Sympy [A] (verification not implemented)

Time = 39.59 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.27 \[ \int \left (d+e x^2\right ) \cot ^{-1}(a x) \log \left (c x^n\right ) \, dx=- d n \left (\begin {cases} \frac {\pi x}{2} & \text {for}\: a = 0 \\\begin {cases} x \operatorname {acot}{\left (a x \right )} + \frac {\log {\left (a^{2} x^{2} + 1 \right )}}{2 a} & \text {for}\: a \neq 0 \\\frac {\pi x}{2} & \text {otherwise} \end {cases} - \frac {\operatorname {Li}_{2}\left (a^{2} x^{2} e^{i \pi }\right )}{4 a} & \text {otherwise} \end {cases}\right ) + d \left (\begin {cases} \frac {\pi x}{2} & \text {for}\: a = 0 \\x \operatorname {acot}{\left (a x \right )} + \frac {\log {\left (a^{2} x^{2} + 1 \right )}}{2 a} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )} - \frac {e n x^{3} \operatorname {acot}{\left (a x \right )}}{9} + \frac {e x^{3} \log {\left (c x^{n} \right )} \operatorname {acot}{\left (a x \right )}}{3} - \frac {5 e n x^{2}}{36 a} + \frac {e n \left (\begin {cases} \frac {x^{2}}{2} & \text {for}\: a = 0 \\- \frac {\operatorname {Li}_{2}\left (a^{2} x^{2} e^{i \pi }\right )}{2 a^{2}} & \text {otherwise} \end {cases}\right )}{6 a} + \frac {e n \left (\begin {cases} x^{2} & \text {for}\: a^{2} = 0 \\\frac {\log {\left (a^{2} x^{2} + 1 \right )}}{a^{2}} & \text {otherwise} \end {cases}\right )}{18 a} + \frac {e x^{2} \log {\left (c x^{n} \right )}}{6 a} - \frac {e \left (\begin {cases} x^{2} & \text {for}\: a^{2} = 0 \\\frac {\log {\left (a^{2} x^{2} + 1 \right )}}{a^{2}} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )}}{6 a} \] Input:

integrate((e*x**2+d)*acot(a*x)*ln(c*x**n),x)
 

Output:

-d*n*Piecewise((pi*x/2, Eq(a, 0)), (Piecewise((x*acot(a*x) + log(a**2*x**2 
 + 1)/(2*a), Ne(a, 0)), (pi*x/2, True)) - polylog(2, a**2*x**2*exp_polar(I 
*pi))/(4*a), True)) + d*Piecewise((pi*x/2, Eq(a, 0)), (x*acot(a*x) + log(a 
**2*x**2 + 1)/(2*a), True))*log(c*x**n) - e*n*x**3*acot(a*x)/9 + e*x**3*lo 
g(c*x**n)*acot(a*x)/3 - 5*e*n*x**2/(36*a) + e*n*Piecewise((x**2/2, Eq(a, 0 
)), (-polylog(2, a**2*x**2*exp_polar(I*pi))/(2*a**2), True))/(6*a) + e*n*P 
iecewise((x**2, Eq(a**2, 0)), (log(a**2*x**2 + 1)/a**2, True))/(18*a) + e* 
x**2*log(c*x**n)/(6*a) - e*Piecewise((x**2, Eq(a**2, 0)), (log(a**2*x**2 + 
 1)/a**2, True))*log(c*x**n)/(6*a)
 

Maxima [F]

\[ \int \left (d+e x^2\right ) \cot ^{-1}(a x) \log \left (c x^n\right ) \, dx=\int { {\left (e x^{2} + d\right )} \operatorname {arccot}\left (a x\right ) \log \left (c x^{n}\right ) \,d x } \] Input:

integrate((e*x^2+d)*arccot(a*x)*log(c*x^n),x, algorithm="maxima")
 

Output:

1/36*(69984*a^4*e*n*integrate(1/11664*x^4*log(x)/(a^2*x^3 + x), x) + 20995 
2*a^4*d*n*integrate(1/11664*x^2*log(x)/(a^2*x^3 + x), x) + 1944*a^4*e*inte 
grate(1/216*(2*a*x^4*arctan2(1, a*x) + x^3*log(a^2*x^2 + 1))/(a^2*x^2 + 1) 
, x)*log(c) + 1944*a^4*d*integrate(1/216*(2*a*x^2*arctan2(1, a*x) + x*log( 
a^2*x^2 + 1))/(a^2*x^2 + 1), x)*log(c) + 1944*a^4*e*integrate(1/216*(2*a*x 
^4*arctan2(1, a*x) + x^3*log(a^2*x^2 + 1))*log(x^n)/(a^2*x^2 + 1), x) + 19 
44*a^4*d*integrate(1/216*(2*a*x^2*arctan2(1, a*x) + x*log(a^2*x^2 + 1))*lo 
g(x^n)/(a^2*x^2 + 1), x) - 9*(216*a*integrate(1/216*x*log(a^2*x^2 + 1)/(a^ 
2*x^2 + 1), x) - arctan(a*x)^2/a - 2*arctan(a*x)*arctan(1/(a*x))/a)*a^3*d* 
log(c) - 1944*a^3*e*integrate(1/216*(a*x^3*log(a^2*x^2 + 1) - 2*x^2*arctan 
2(1, a*x))/(a^2*x^2 + 1), x)*log(c) - 1944*a^3*e*integrate(1/216*(a*x^3*lo 
g(a^2*x^2 + 1) - 2*x^2*arctan2(1, a*x))*log(x^n)/(a^2*x^2 + 1), x) - 1944* 
a^3*d*integrate(1/216*(a*x*log(a^2*x^2 + 1) - 2*arctan2(1, a*x))*log(x^n)/ 
(a^2*x^2 + 1), x) - 2*(a^3*e*n*arctan2(1, a*x) - 3*a^3*e*arctan2(1, a*x)*l 
og(c))*x^3 - (a^2*e*n - 3*a^2*e*log(c))*x^2 - 18*(a^3*d*n*arctan2(1, a*x) 
- a^3*d*arctan2(1, a*x)*log(c))*x + (9*a^2*d*log(c) - (9*a^2*d - e)*n - 3* 
e*log(c))*log(a^2*x^2 + 1) + 6*(a^3*e*x^3*arctan2(1, a*x) + 3*a^3*d*x*arct 
an2(1, a*x))*log(x^n))/a^3
 

Giac [F]

\[ \int \left (d+e x^2\right ) \cot ^{-1}(a x) \log \left (c x^n\right ) \, dx=\int { {\left (e x^{2} + d\right )} \operatorname {arccot}\left (a x\right ) \log \left (c x^{n}\right ) \,d x } \] Input:

integrate((e*x^2+d)*arccot(a*x)*log(c*x^n),x, algorithm="giac")
 

Output:

integrate((e*x^2 + d)*arccot(a*x)*log(c*x^n), x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int \left (d+e x^2\right ) \cot ^{-1}(a x) \log \left (c x^n\right ) \, dx=\int \ln \left (c\,x^n\right )\,\mathrm {acot}\left (a\,x\right )\,\left (e\,x^2+d\right ) \,d x \] Input:

int(log(c*x^n)*acot(a*x)*(d + e*x^2),x)
 

Output:

int(log(c*x^n)*acot(a*x)*(d + e*x^2), x)
 

Reduce [F]

\[ \int \left (d+e x^2\right ) \cot ^{-1}(a x) \log \left (c x^n\right ) \, dx=\frac {36 \mathit {acot} \left (a x \right ) \mathrm {log}\left (x^{n} c \right ) a^{3} d n x +12 \mathit {acot} \left (a x \right ) \mathrm {log}\left (x^{n} c \right ) a^{3} e n \,x^{3}-36 \mathit {acot} \left (a x \right ) a^{3} d \,n^{2} x -4 \mathit {acot} \left (a x \right ) a^{3} e \,n^{2} x^{3}-36 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{a^{2} x^{3}+x}d x \right ) a^{2} d n +12 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{a^{2} x^{3}+x}d x \right ) e n -18 \,\mathrm {log}\left (a^{2} x^{2}+1\right ) a^{2} d \,n^{2}+2 \,\mathrm {log}\left (a^{2} x^{2}+1\right ) e \,n^{2}+18 \mathrm {log}\left (x^{n} c \right )^{2} a^{2} d -6 \mathrm {log}\left (x^{n} c \right )^{2} e +6 \,\mathrm {log}\left (x^{n} c \right ) a^{2} e n \,x^{2}-5 a^{2} e \,n^{2} x^{2}}{36 a^{3} n} \] Input:

int((e*x^2+d)*acot(a*x)*log(c*x^n),x)
 

Output:

(36*acot(a*x)*log(x**n*c)*a**3*d*n*x + 12*acot(a*x)*log(x**n*c)*a**3*e*n*x 
**3 - 36*acot(a*x)*a**3*d*n**2*x - 4*acot(a*x)*a**3*e*n**2*x**3 - 36*int(l 
og(x**n*c)/(a**2*x**3 + x),x)*a**2*d*n + 12*int(log(x**n*c)/(a**2*x**3 + x 
),x)*e*n - 18*log(a**2*x**2 + 1)*a**2*d*n**2 + 2*log(a**2*x**2 + 1)*e*n**2 
 + 18*log(x**n*c)**2*a**2*d - 6*log(x**n*c)**2*e + 6*log(x**n*c)*a**2*e*n* 
x**2 - 5*a**2*e*n**2*x**2)/(36*a**3*n)