\(\int (d+e x^2) \arccos (a x)^2 \log (c x^n) \, dx\) [201]

Optimal result

Integrand size = 20, antiderivative size = 490 \[ \int \left (d+e x^2\right ) \arccos (a x)^2 \log \left (c x^n\right ) \, dx=2 d n x+\frac {2 e n x}{27 a^2}+\frac {4}{9} \left (9 d+\frac {2 e}{a^2}\right ) n x+\frac {2}{27} e n x^3+\frac {2 d n \sqrt {1-a^2 x^2} \arccos (a x)}{a}+\frac {4 e n \sqrt {1-a^2 x^2} \arccos (a x)}{27 a^3}+\frac {2 \left (9 a^2 d+2 e\right ) n \sqrt {1-a^2 x^2} \arccos (a x)}{9 a^3}+\frac {2 e n x^2 \sqrt {1-a^2 x^2} \arccos (a x)}{27 a}-\frac {2 e n \left (1-a^2 x^2\right )^{3/2} \arccos (a x)}{27 a^3}-d n x \arccos (a x)^2-\frac {1}{9} e n x^3 \arccos (a x)^2+\frac {4 i \left (9 a^2 d+2 e\right ) n \arccos (a x) \arctan \left (e^{i \arccos (a x)}\right )}{9 a^3}-2 d x \log \left (c x^n\right )-\frac {4 e x \log \left (c x^n\right )}{9 a^2}-\frac {2}{27} e x^3 \log \left (c x^n\right )-\frac {2 d \sqrt {1-a^2 x^2} \arccos (a x) \log \left (c x^n\right )}{a}-\frac {4 e \sqrt {1-a^2 x^2} \arccos (a x) \log \left (c x^n\right )}{9 a^3}-\frac {2 e x^2 \sqrt {1-a^2 x^2} \arccos (a x) \log \left (c x^n\right )}{9 a}+d x \arccos (a x)^2 \log \left (c x^n\right )+\frac {1}{3} e x^3 \arccos (a x)^2 \log \left (c x^n\right )-\frac {2 i \left (9 a^2 d+2 e\right ) n \operatorname {PolyLog}\left (2,-i e^{i \arccos (a x)}\right )}{9 a^3}+\frac {2 i \left (9 a^2 d+2 e\right ) n \operatorname {PolyLog}\left (2,i e^{i \arccos (a x)}\right )}{9 a^3} \] Output:

2*d*n*x+2/27*e*n*x/a^2+4/9*(9*d+2*e/a^2)*n*x+2/27*e*n*x^3+2*d*n*(-a^2*x^2+ 
1)^(1/2)*arccos(a*x)/a+4/27*e*n*(-a^2*x^2+1)^(1/2)*arccos(a*x)/a^3+2/9*(9* 
a^2*d+2*e)*n*(-a^2*x^2+1)^(1/2)*arccos(a*x)/a^3+2/27*e*n*x^2*(-a^2*x^2+1)^ 
(1/2)*arccos(a*x)/a-2/27*e*n*(-a^2*x^2+1)^(3/2)*arccos(a*x)/a^3-d*n*x*arcc 
os(a*x)^2-1/9*e*n*x^3*arccos(a*x)^2+4/9*I*(9*a^2*d+2*e)*n*arccos(a*x)*arct 
an(a*x+I*(-a^2*x^2+1)^(1/2))/a^3-2*d*x*ln(c*x^n)-4/9*e*x*ln(c*x^n)/a^2-2/2 
7*e*x^3*ln(c*x^n)-2*d*(-a^2*x^2+1)^(1/2)*arccos(a*x)*ln(c*x^n)/a-4/9*e*(-a 
^2*x^2+1)^(1/2)*arccos(a*x)*ln(c*x^n)/a^3-2/9*e*x^2*(-a^2*x^2+1)^(1/2)*arc 
cos(a*x)*ln(c*x^n)/a+d*x*arccos(a*x)^2*ln(c*x^n)+1/3*e*x^3*arccos(a*x)^2*l 
n(c*x^n)+2/9*I*(9*a^2*d+2*e)*n*polylog(2,I*(a*x+I*(-a^2*x^2+1)^(1/2)))/a^3 
-2/9*I*(9*a^2*d+2*e)*n*polylog(2,-I*(a*x+I*(-a^2*x^2+1)^(1/2)))/a^3
 

Mathematica [A] (verified)

Time = 1.20 (sec) , antiderivative size = 564, normalized size of antiderivative = 1.15 \[ \int \left (d+e x^2\right ) \arccos (a x)^2 \log \left (c x^n\right ) \, dx=2 d n x+\frac {4 e n x}{9 a^2}+\frac {2}{81} e n x^3+\frac {e n \left (-9 a x-12 \left (1-a^2 x^2\right )^{3/2} \arccos (a x)+\cos (3 \arccos (a x))\right )}{162 a^3}+\frac {d n \left (-2 a x-2 \sqrt {1-a^2 x^2} \arccos (a x)+a x \arccos (a x)^2\right ) \log (x)}{a}+\frac {e n \left (-12 a x-2 a^3 x^3-12 \sqrt {1-a^2 x^2} \arccos (a x)-6 a^2 x^2 \sqrt {1-a^2 x^2} \arccos (a x)+9 a^3 x^3 \arccos (a x)^2\right ) \log (x)}{27 a^3}+\frac {d \left (-2 \sqrt {1-a^2 x^2} \arccos (a x)+a x \left (-2+\arccos (a x)^2\right )\right ) \left (-n-n \log (x)+\log \left (c x^n\right )\right )}{a}+\frac {2 d n \left (a x+\sqrt {1-a^2 x^2} \arccos (a x)-\arccos (a x) \log \left (1-i e^{i \arccos (a x)}\right )+\arccos (a x) \log \left (1+i e^{i \arccos (a x)}\right )-i \operatorname {PolyLog}\left (2,-i e^{i \arccos (a x)}\right )+i \operatorname {PolyLog}\left (2,i e^{i \arccos (a x)}\right )\right )}{a}+\frac {4 e n \left (a x+\sqrt {1-a^2 x^2} \arccos (a x)-\arccos (a x) \log \left (1-i e^{i \arccos (a x)}\right )+\arccos (a x) \log \left (1+i e^{i \arccos (a x)}\right )-i \operatorname {PolyLog}\left (2,-i e^{i \arccos (a x)}\right )+i \operatorname {PolyLog}\left (2,i e^{i \arccos (a x)}\right )\right )}{9 a^3}+\frac {e \left (-n+3 \left (-n \log (x)+\log \left (c x^n\right )\right )\right ) \left (27 a x \left (-2+\arccos (a x)^2\right )-\left (2-9 \arccos (a x)^2\right ) \cos (3 \arccos (a x))-6 \arccos (a x) \left (9 \sqrt {1-a^2 x^2}+\sin (3 \arccos (a x))\right )\right )}{324 a^3} \] Input:

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

Output:

2*d*n*x + (4*e*n*x)/(9*a^2) + (2*e*n*x^3)/81 + (e*n*(-9*a*x - 12*(1 - a^2* 
x^2)^(3/2)*ArcCos[a*x] + Cos[3*ArcCos[a*x]]))/(162*a^3) + (d*n*(-2*a*x - 2 
*Sqrt[1 - a^2*x^2]*ArcCos[a*x] + a*x*ArcCos[a*x]^2)*Log[x])/a + (e*n*(-12* 
a*x - 2*a^3*x^3 - 12*Sqrt[1 - a^2*x^2]*ArcCos[a*x] - 6*a^2*x^2*Sqrt[1 - a^ 
2*x^2]*ArcCos[a*x] + 9*a^3*x^3*ArcCos[a*x]^2)*Log[x])/(27*a^3) + (d*(-2*Sq 
rt[1 - a^2*x^2]*ArcCos[a*x] + a*x*(-2 + ArcCos[a*x]^2))*(-n - n*Log[x] + L 
og[c*x^n]))/a + (2*d*n*(a*x + Sqrt[1 - a^2*x^2]*ArcCos[a*x] - ArcCos[a*x]* 
Log[1 - I*E^(I*ArcCos[a*x])] + ArcCos[a*x]*Log[1 + I*E^(I*ArcCos[a*x])] - 
I*PolyLog[2, (-I)*E^(I*ArcCos[a*x])] + I*PolyLog[2, I*E^(I*ArcCos[a*x])])) 
/a + (4*e*n*(a*x + Sqrt[1 - a^2*x^2]*ArcCos[a*x] - ArcCos[a*x]*Log[1 - I*E 
^(I*ArcCos[a*x])] + ArcCos[a*x]*Log[1 + I*E^(I*ArcCos[a*x])] - I*PolyLog[2 
, (-I)*E^(I*ArcCos[a*x])] + I*PolyLog[2, I*E^(I*ArcCos[a*x])]))/(9*a^3) + 
(e*(-n + 3*(-(n*Log[x]) + Log[c*x^n]))*(27*a*x*(-2 + ArcCos[a*x]^2) - (2 - 
 9*ArcCos[a*x]^2)*Cos[3*ArcCos[a*x]] - 6*ArcCos[a*x]*(9*Sqrt[1 - a^2*x^2] 
+ Sin[3*ArcCos[a*x]])))/(324*a^3)
 

Rubi [A] (verified)

Time = 1.14 (sec) , antiderivative size = 479, normalized size of antiderivative = 0.98, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {2834, 6, 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.

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

\(\Big \downarrow \) 2834

\(\displaystyle -n \int \left (\frac {1}{3} e \arccos (a x)^2 x^2-\frac {2 e x^2}{27}-\frac {2 e \sqrt {1-a^2 x^2} \arccos (a x) x}{9 a}+d \arccos (a x)^2-\frac {2}{9} \left (9 d+\frac {2 e}{a^2}\right )-\frac {2 d \sqrt {1-a^2 x^2} \arccos (a x)}{a x}-\frac {4 e \sqrt {1-a^2 x^2} \arccos (a x)}{9 a^3 x}\right )dx-\frac {2 d \sqrt {1-a^2 x^2} \arccos (a x) \log \left (c x^n\right )}{a}-\frac {2 e x^2 \sqrt {1-a^2 x^2} \arccos (a x) \log \left (c x^n\right )}{9 a}-\frac {4 e x \log \left (c x^n\right )}{9 a^2}-\frac {4 e \sqrt {1-a^2 x^2} \arccos (a x) \log \left (c x^n\right )}{9 a^3}+d x \arccos (a x)^2 \log \left (c x^n\right )+\frac {1}{3} e x^3 \arccos (a x)^2 \log \left (c x^n\right )-2 d x \log \left (c x^n\right )-\frac {2}{27} e x^3 \log \left (c x^n\right )\)

\(\Big \downarrow \) 6

\(\displaystyle -n \int \left (\frac {1}{3} e \arccos (a x)^2 x^2-\frac {2 e x^2}{27}-\frac {2 e \sqrt {1-a^2 x^2} \arccos (a x) x}{9 a}+d \arccos (a x)^2-\frac {2}{9} \left (9 d+\frac {2 e}{a^2}\right )+\frac {\left (-\frac {2 d}{a}-\frac {4 e}{9 a^3}\right ) \sqrt {1-a^2 x^2} \arccos (a x)}{x}\right )dx-\frac {2 d \sqrt {1-a^2 x^2} \arccos (a x) \log \left (c x^n\right )}{a}-\frac {2 e x^2 \sqrt {1-a^2 x^2} \arccos (a x) \log \left (c x^n\right )}{9 a}-\frac {4 e x \log \left (c x^n\right )}{9 a^2}-\frac {4 e \sqrt {1-a^2 x^2} \arccos (a x) \log \left (c x^n\right )}{9 a^3}+d x \arccos (a x)^2 \log \left (c x^n\right )+\frac {1}{3} e x^3 \arccos (a x)^2 \log \left (c x^n\right )-2 d x \log \left (c x^n\right )-\frac {2}{27} e x^3 \log \left (c x^n\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {2 d \sqrt {1-a^2 x^2} \arccos (a x) \log \left (c x^n\right )}{a}-\frac {2 e x^2 \sqrt {1-a^2 x^2} \arccos (a x) \log \left (c x^n\right )}{9 a}-\frac {4 e x \log \left (c x^n\right )}{9 a^2}-n \left (-\frac {2 d \sqrt {1-a^2 x^2} \arccos (a x)}{a}-\frac {2 e x^2 \sqrt {1-a^2 x^2} \arccos (a x)}{27 a}-\frac {4}{9} x \left (\frac {2 e}{a^2}+9 d\right )-\frac {2 e x}{27 a^2}-\frac {4 i \arccos (a x) \left (9 a^2 d+2 e\right ) \arctan \left (e^{i \arccos (a x)}\right )}{9 a^3}+\frac {2 i \left (9 a^2 d+2 e\right ) \operatorname {PolyLog}\left (2,-i e^{i \arccos (a x)}\right )}{9 a^3}-\frac {2 i \left (9 a^2 d+2 e\right ) \operatorname {PolyLog}\left (2,i e^{i \arccos (a x)}\right )}{9 a^3}-\frac {2 \sqrt {1-a^2 x^2} \arccos (a x) \left (9 a^2 d+2 e\right )}{9 a^3}+\frac {2 e \left (1-a^2 x^2\right )^{3/2} \arccos (a x)}{27 a^3}-\frac {4 e \sqrt {1-a^2 x^2} \arccos (a x)}{27 a^3}+d x \arccos (a x)^2+\frac {1}{9} e x^3 \arccos (a x)^2-2 d x-\frac {2 e x^3}{27}\right )-\frac {4 e \sqrt {1-a^2 x^2} \arccos (a x) \log \left (c x^n\right )}{9 a^3}+d x \arccos (a x)^2 \log \left (c x^n\right )+\frac {1}{3} e x^3 \arccos (a x)^2 \log \left (c x^n\right )-2 d x \log \left (c x^n\right )-\frac {2}{27} e x^3 \log \left (c x^n\right )\)

Input:

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

Output:

-2*d*x*Log[c*x^n] - (4*e*x*Log[c*x^n])/(9*a^2) - (2*e*x^3*Log[c*x^n])/27 - 
 (2*d*Sqrt[1 - a^2*x^2]*ArcCos[a*x]*Log[c*x^n])/a - (4*e*Sqrt[1 - a^2*x^2] 
*ArcCos[a*x]*Log[c*x^n])/(9*a^3) - (2*e*x^2*Sqrt[1 - a^2*x^2]*ArcCos[a*x]* 
Log[c*x^n])/(9*a) + d*x*ArcCos[a*x]^2*Log[c*x^n] + (e*x^3*ArcCos[a*x]^2*Lo 
g[c*x^n])/3 - n*(-2*d*x - (2*e*x)/(27*a^2) - (4*(9*d + (2*e)/a^2)*x)/9 - ( 
2*e*x^3)/27 - (2*d*Sqrt[1 - a^2*x^2]*ArcCos[a*x])/a - (4*e*Sqrt[1 - a^2*x^ 
2]*ArcCos[a*x])/(27*a^3) - (2*(9*a^2*d + 2*e)*Sqrt[1 - a^2*x^2]*ArcCos[a*x 
])/(9*a^3) - (2*e*x^2*Sqrt[1 - a^2*x^2]*ArcCos[a*x])/(27*a) + (2*e*(1 - a^ 
2*x^2)^(3/2)*ArcCos[a*x])/(27*a^3) + d*x*ArcCos[a*x]^2 + (e*x^3*ArcCos[a*x 
]^2)/9 - (((4*I)/9)*(9*a^2*d + 2*e)*ArcCos[a*x]*ArcTan[E^(I*ArcCos[a*x])]) 
/a^3 + (((2*I)/9)*(9*a^2*d + 2*e)*PolyLog[2, (-I)*E^(I*ArcCos[a*x])])/a^3 
- (((2*I)/9)*(9*a^2*d + 2*e)*PolyLog[2, I*E^(I*ArcCos[a*x])])/a^3)
 

Defintions of rubi rules used

rule 6

Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v 
+ (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] &&  !FreeQ[Fx, x]
 

rule 2009

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

rule 2834

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(Px_.)*(F_)[(d_.)*((e_.) + (f_.)* 
(x_))]^(m_.), x_Symbol] :> With[{u = IntHide[Px*F[d*(e + f*x)]^m, 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] && IGtQ[m, 0] && MemberQ[{ArcSi 
n, ArcCos, ArcSinh, ArcCosh}, F]
 

Maple [F]

\[\int \left (e \,x^{2}+d \right ) \arccos \left (a x \right )^{2} \ln \left (c \,x^{n}\right )d x\]

Input:

int((e*x^2+d)*arccos(a*x)^2*ln(c*x^n),x)
 

Output:

int((e*x^2+d)*arccos(a*x)^2*ln(c*x^n),x)
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

Integral((d + e*x**2)*log(c*x**n)*acos(a*x)**2, x)
 

Maxima [F]

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

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

Output:

1/3*(e*x^3 + 3*d*x)*arctan2(sqrt(a*x + 1)*sqrt(-a*x + 1), a*x)^2*log(x^n) 
- 1/9*((e*n - 3*e*log(c))*x^3 + 9*(d*n - d*log(c))*x)*arctan2(sqrt(a*x + 1 
)*sqrt(-a*x + 1), a*x)^2 - integrate(2/9*(3*(a*e*x^3 + 3*a*d*x)*arctan2(sq 
rt(a*x + 1)*sqrt(-a*x + 1), a*x)*log(x^n) - ((a*e*n - 3*a*e*log(c))*x^3 + 
9*(a*d*n - a*d*log(c))*x)*arctan2(sqrt(a*x + 1)*sqrt(-a*x + 1), a*x))*sqrt 
(a*x + 1)*sqrt(-a*x + 1)/(a^2*x^2 - 1), x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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