\(\int (d+e x^2) \arctan (a x) \log (c x^n) \, dx\) [194]

Optimal result

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

5/36*e*n*x^2/a-d*n*x*arctan(a*x)-1/9*e*n*x^3*arctan(a*x)-1/6*e*x^2*ln(c*x^ 
n)/a+d*x*arctan(a*x)*ln(c*x^n)+1/3*e*x^3*arctan(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 = 0.20 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.91 \[ \int \left (d+e x^2\right ) \arctan (a x) \log \left (c x^n\right ) \, dx=\frac {5 a^2 e n x^2-6 a^2 e x^2 \log \left (c x^n\right )-4 a^3 x \arctan (a x) \left (n \left (9 d+e x^2\right )-3 \left (3 d+e x^2\right ) \log \left (c x^n\right )\right )+18 a^2 d n \log \left (1+a^2 x^2\right )-2 e n \log \left (1+a^2 x^2\right )-18 a^2 d \log \left (c x^n\right ) \log \left (1+a^2 x^2\right )+6 e \log \left (c x^n\right ) \log \left (1+a^2 x^2\right )+3 \left (-3 a^2 d+e\right ) n \operatorname {PolyLog}\left (2,-a^2 x^2\right )}{36 a^3} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.43 (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.

\(\displaystyle \int \arctan (a x) \left (d+e x^2\right ) \log \left (c x^n\right ) \, dx\)

\(\Big \downarrow \) 2835

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

-1/6*(e*x^2*Log[c*x^n])/a + d*x*ArcTan[a*x]*Log[c*x^n] + (e*x^3*ArcTan[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*ArcTan[a*x] + (e*x^3*ArcTan[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 = 66.76 (sec) , antiderivative size = 1490, normalized size of antiderivative = 8.19

method	result	size
risch	\(\text {Expression too large to display}\)	\(1490\)
default	\(\text {Expression too large to display}\)	\(76733\)

Input:

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

Fricas [F]

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

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

Output:

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

Sympy [A] (verification not implemented)

Time = 44.26 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.21 \[ \int \left (d+e x^2\right ) \arctan (a x) \log \left (c x^n\right ) \, dx=- d n \left (\begin {cases} 0 & \text {for}\: a = 0 \\\begin {cases} x \operatorname {atan}{\left (a x \right )} - \frac {\log {\left (a^{2} x^{2} + 1 \right )}}{2 a} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} + \frac {\operatorname {Li}_{2}\left (a^{2} x^{2} e^{i \pi }\right )}{4 a} & \text {otherwise} \end {cases}\right ) + d \left (\begin {cases} 0 & \text {for}\: a = 0 \\x \operatorname {atan}{\left (a x \right )} - \frac {\log {\left (a^{2} x^{2} + 1 \right )}}{2 a} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )} - \frac {e n x^{3} \operatorname {atan}{\left (a x \right )}}{9} + \frac {e x^{3} \log {\left (c x^{n} \right )} \operatorname {atan}{\left (a x \right )}}{3} + \frac {5 e n x^{2}}{36 a} - \frac {e n \left (\begin {cases} \frac {x^{2}}{2} & \text {for}\: a = 0 \\- \frac {\operatorname {Li}_{2}\left (a^{2} x^{2} e^{i \pi }\right )}{2 a^{2}} & \text {otherwise} \end {cases}\right )}{6 a} - \frac {e n \left (\begin {cases} x^{2} & \text {for}\: a^{2} = 0 \\\frac {\log {\left (a^{2} x^{2} + 1 \right )}}{a^{2}} & \text {otherwise} \end {cases}\right )}{18 a} - \frac {e x^{2} \log {\left (c x^{n} \right )}}{6 a} + \frac {e \left (\begin {cases} x^{2} & \text {for}\: a^{2} = 0 \\\frac {\log {\left (a^{2} x^{2} + 1 \right )}}{a^{2}} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )}}{6 a} \] Input:

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

Output:

-d*n*Piecewise((0, Eq(a, 0)), (Piecewise((x*atan(a*x) - log(a**2*x**2 + 1) 
/(2*a), Ne(a, 0)), (0, True)) + polylog(2, a**2*x**2*exp_polar(I*pi))/(4*a 
), True)) + d*Piecewise((0, Eq(a, 0)), (x*atan(a*x) - log(a**2*x**2 + 1)/( 
2*a), True))*log(c*x**n) - e*n*x**3*atan(a*x)/9 + e*x**3*log(c*x**n)*atan( 
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*Piecewise((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]

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

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

Output:

-1/6*(a^2*e*x^2*log(c) - 3*a^3*integrate(2*(e*x^2 + d)*arctan(a*x)*log(x^n 
), x) - 2*(a^3*e*x^3*log(c) + 3*a^3*d*x*log(c))*arctan(a*x) + (3*a^2*d*log 
(c) - e*log(c))*log(a^2*x^2 + 1))/a^3
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(36*atan(a*x)*log(x**n*c)*a**3*d*n*x + 12*atan(a*x)*log(x**n*c)*a**3*e*n*x 
**3 - 36*atan(a*x)*a**3*d*n**2*x - 4*atan(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)