\(\int (d+e x^2) \text {arccosh}(a x)^2 \log (c x^n) \, dx\) [203]

Optimal result

Integrand size = 20, antiderivative size = 508 \[ \int \left (d+e x^2\right ) \text {arccosh}(a x)^2 \log \left (c x^n\right ) \, dx=-2 d n x-\frac {2 e n x}{27 a^2}-\frac {4}{9} \left (9 d+\frac {2 e}{a^2}\right ) n x-\frac {2}{27} e n x^3+\frac {2 d n \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)}{a}+\frac {4 e n \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)}{27 a^3}+\frac {2 \left (9 a^2 d+2 e\right ) n \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)}{9 a^3}+\frac {2 e n x^2 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)}{27 a}+\frac {2 e n (-1+a x)^{3/2} (1+a x)^{3/2} \text {arccosh}(a x)}{27 a^3}-d n x \text {arccosh}(a x)^2-\frac {1}{9} e n x^3 \text {arccosh}(a x)^2-\frac {4 \left (9 a^2 d+2 e\right ) n \text {arccosh}(a x) \arctan \left (e^{\text {arccosh}(a x)}\right )}{9 a^3}+2 d x \log \left (c x^n\right )+\frac {4 e x \log \left (c x^n\right )}{9 a^2}+\frac {2}{27} e x^3 \log \left (c x^n\right )-\frac {2 d \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x) \log \left (c x^n\right )}{a}-\frac {4 e \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x) \log \left (c x^n\right )}{9 a^3}-\frac {2 e x^2 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x) \log \left (c x^n\right )}{9 a}+d x \text {arccosh}(a x)^2 \log \left (c x^n\right )+\frac {1}{3} e x^3 \text {arccosh}(a x)^2 \log \left (c x^n\right )+\frac {2 i \left (9 a^2 d+2 e\right ) n \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )}{9 a^3}-\frac {2 i \left (9 a^2 d+2 e\right ) n \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )}{9 a^3} \] 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*x-1)^( 
1/2)*(a*x+1)^(1/2)*arccosh(a*x)/a+4/27*e*n*(a*x-1)^(1/2)*(a*x+1)^(1/2)*arc 
cosh(a*x)/a^3+2/9*(9*a^2*d+2*e)*n*(a*x-1)^(1/2)*(a*x+1)^(1/2)*arccosh(a*x) 
/a^3+2/27*e*n*x^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)*arccosh(a*x)/a+2/27*e*n*(a*x 
-1)^(3/2)*(a*x+1)^(3/2)*arccosh(a*x)/a^3-d*n*x*arccosh(a*x)^2-1/9*e*n*x^3* 
arccosh(a*x)^2-4/9*(9*a^2*d+2*e)*n*arccosh(a*x)*arctan(a*x+(a*x-1)^(1/2)*( 
a*x+1)^(1/2))/a^3+2*d*x*ln(c*x^n)+4/9*e*x*ln(c*x^n)/a^2+2/27*e*x^3*ln(c*x^ 
n)-2*d*(a*x-1)^(1/2)*(a*x+1)^(1/2)*arccosh(a*x)*ln(c*x^n)/a-4/9*e*(a*x-1)^ 
(1/2)*(a*x+1)^(1/2)*arccosh(a*x)*ln(c*x^n)/a^3-2/9*e*x^2*(a*x-1)^(1/2)*(a* 
x+1)^(1/2)*arccosh(a*x)*ln(c*x^n)/a+d*x*arccosh(a*x)^2*ln(c*x^n)+1/3*e*x^3 
*arccosh(a*x)^2*ln(c*x^n)+2/9*I*(9*a^2*d+2*e)*n*polylog(2,-I*(a*x+(a*x-1)^ 
(1/2)*(a*x+1)^(1/2)))/a^3-2/9*I*(9*a^2*d+2*e)*n*polylog(2,I*(a*x+(a*x-1)^( 
1/2)*(a*x+1)^(1/2)))/a^3
 

Mathematica [A] (warning: unable to verify)

Time = 4.88 (sec) , antiderivative size = 619, normalized size of antiderivative = 1.22 \[ \int \left (d+e x^2\right ) \text {arccosh}(a x)^2 \log \left (c x^n\right ) \, dx=\frac {-648 a^3 d n x-144 a e n x-8 a^3 e n x^3+2 e n \left (9 a x+12 \left (\frac {-1+a x}{1+a x}\right )^{3/2} (1+a x)^3 \text {arccosh}(a x)-\cosh (3 \text {arccosh}(a x))\right )+324 a^2 d n \left (2 a x-2 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)+a x \text {arccosh}(a x)^2\right ) \log (x)+12 e n \left (2 a x \left (6+a^2 x^2\right )-6 \sqrt {-1+a x} \sqrt {1+a x} \left (2+a^2 x^2\right ) \text {arccosh}(a x)+9 a^3 x^3 \text {arccosh}(a x)^2\right ) \log (x)+324 a^2 d \left (2 \sqrt {\frac {-1+a x}{1+a x}} (1+a x) \text {arccosh}(a x)-a x \left (2+\text {arccosh}(a x)^2\right )\right ) \left (n+n \log (x)-\log \left (c x^n\right )\right )+648 a^2 d n \left (-a x+\sqrt {\frac {-1+a x}{1+a x}} \text {arccosh}(a x)+a x \sqrt {\frac {-1+a x}{1+a x}} \text {arccosh}(a x)+i \text {arccosh}(a x) \log \left (1-i e^{-\text {arccosh}(a x)}\right )-i \text {arccosh}(a x) \log \left (1+i e^{-\text {arccosh}(a x)}\right )+i \operatorname {PolyLog}\left (2,-i e^{-\text {arccosh}(a x)}\right )-i \operatorname {PolyLog}\left (2,i e^{-\text {arccosh}(a x)}\right )\right )+144 e n \left (-a x+\sqrt {\frac {-1+a x}{1+a x}} \text {arccosh}(a x)+a x \sqrt {\frac {-1+a x}{1+a x}} \text {arccosh}(a x)+i \text {arccosh}(a x) \log \left (1-i e^{-\text {arccosh}(a x)}\right )-i \text {arccosh}(a x) \log \left (1+i e^{-\text {arccosh}(a x)}\right )+i \operatorname {PolyLog}\left (2,-i e^{-\text {arccosh}(a x)}\right )-i \operatorname {PolyLog}\left (2,i e^{-\text {arccosh}(a x)}\right )\right )-e \left (n+3 n \log (x)-3 \log \left (c x^n\right )\right ) \left (27 a x \left (2+\text {arccosh}(a x)^2\right )+\left (2+9 \text {arccosh}(a x)^2\right ) \cosh (3 \text {arccosh}(a x))-6 \text {arccosh}(a x) \left (9 \sqrt {\frac {-1+a x}{1+a x}} (1+a x)+\sinh (3 \text {arccosh}(a x))\right )\right )}{324 a^3} \] Input:

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

Output:

(-648*a^3*d*n*x - 144*a*e*n*x - 8*a^3*e*n*x^3 + 2*e*n*(9*a*x + 12*((-1 + a 
*x)/(1 + a*x))^(3/2)*(1 + a*x)^3*ArcCosh[a*x] - Cosh[3*ArcCosh[a*x]]) + 32 
4*a^2*d*n*(2*a*x - 2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x] + a*x*ArcCo 
sh[a*x]^2)*Log[x] + 12*e*n*(2*a*x*(6 + a^2*x^2) - 6*Sqrt[-1 + a*x]*Sqrt[1 
+ a*x]*(2 + a^2*x^2)*ArcCosh[a*x] + 9*a^3*x^3*ArcCosh[a*x]^2)*Log[x] + 324 
*a^2*d*(2*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)*ArcCosh[a*x] - a*x*(2 + Arc 
Cosh[a*x]^2))*(n + n*Log[x] - Log[c*x^n]) + 648*a^2*d*n*(-(a*x) + Sqrt[(-1 
 + a*x)/(1 + a*x)]*ArcCosh[a*x] + a*x*Sqrt[(-1 + a*x)/(1 + a*x)]*ArcCosh[a 
*x] + I*ArcCosh[a*x]*Log[1 - I/E^ArcCosh[a*x]] - I*ArcCosh[a*x]*Log[1 + I/ 
E^ArcCosh[a*x]] + I*PolyLog[2, (-I)/E^ArcCosh[a*x]] - I*PolyLog[2, I/E^Arc 
Cosh[a*x]]) + 144*e*n*(-(a*x) + Sqrt[(-1 + a*x)/(1 + a*x)]*ArcCosh[a*x] + 
a*x*Sqrt[(-1 + a*x)/(1 + a*x)]*ArcCosh[a*x] + I*ArcCosh[a*x]*Log[1 - I/E^A 
rcCosh[a*x]] - I*ArcCosh[a*x]*Log[1 + I/E^ArcCosh[a*x]] + I*PolyLog[2, (-I 
)/E^ArcCosh[a*x]] - I*PolyLog[2, I/E^ArcCosh[a*x]]) - e*(n + 3*n*Log[x] - 
3*Log[c*x^n])*(27*a*x*(2 + ArcCosh[a*x]^2) + (2 + 9*ArcCosh[a*x]^2)*Cosh[3 
*ArcCosh[a*x]] - 6*ArcCosh[a*x]*(9*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x) + 
Sinh[3*ArcCosh[a*x]])))/(324*a^3)
 

Rubi [A] (verified)

Time = 1.99 (sec) , antiderivative size = 497, 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.

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

\(\Big \downarrow \) 2834

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

\(\Big \downarrow \) 6

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

\(\Big \downarrow \) 2009

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

Input:

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

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

-1/9*((e*n - 3*e*log(c))*x^3 + 9*(d*n - d*log(c))*x - 3*(e*x^3 + 3*d*x)*lo 
g(x^n))*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))^2 - integrate(-2/9*((e*n - 
3*e*log(c))*a^3*x^5 + (9*(d*n - d*log(c))*a^3 - (e*n - 3*e*log(c))*a)*x^3 
- 9*(d*n - d*log(c))*a*x + ((e*n - 3*e*log(c))*a^2*x^4 + 9*(d*n - d*log(c) 
)*a^2*x^2 - 3*(a^2*e*x^4 + 3*a^2*d*x^2)*log(x^n))*sqrt(a*x + 1)*sqrt(a*x - 
 1) - 3*(a^3*e*x^5 + (3*a^3*d - a*e)*x^3 - 3*a*d*x)*log(x^n))*log(a*x + sq 
rt(a*x + 1)*sqrt(a*x - 1))/(a^3*x^3 + (a^2*x^2 - 1)*sqrt(a*x + 1)*sqrt(a*x 
 - 1) - a*x), x)
 

Giac [F(-2)]

Exception generated. \[ \int \left (d+e x^2\right ) \text {arccosh}(a x)^2 \log \left (c x^n\right ) \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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