Integrand size = 18, antiderivative size = 312 \[ \int \left (d+e x^2\right ) \text {arccosh}(a x) \log \left (c x^n\right ) \, dx=\frac {d n \sqrt {-1+a x} \sqrt {1+a x}}{a}+\frac {2 e n \sqrt {-1+a x} \sqrt {1+a x}}{27 a^3}+\frac {\left (9 a^2 d+2 e\right ) n \sqrt {-1+a x} \sqrt {1+a x}}{9 a^3}+\frac {e n x^2 \sqrt {-1+a x} \sqrt {1+a x}}{27 a}+\frac {e n (-1+a x)^{3/2} (1+a x)^{3/2}}{27 a^3}-d n x \text {arccosh}(a x)-\frac {1}{9} e n x^3 \text {arccosh}(a x)-\frac {\left (9 a^2 d+2 e\right ) n \arctan \left (\sqrt {-1+a x} \sqrt {1+a x}\right )}{9 a^3}-\frac {\left (9 a^2 d+2 e\right ) \sqrt {-1+a x} \sqrt {1+a x} \log \left (c x^n\right )}{9 a^3}-\frac {e x^2 \sqrt {-1+a x} \sqrt {1+a x} \log \left (c x^n\right )}{9 a}+d x \text {arccosh}(a x) \log \left (c x^n\right )+\frac {1}{3} e x^3 \text {arccosh}(a x) \log \left (c x^n\right ) \]
1/27*e*n*(a*x-1)^(3/2)*(a*x+1)^(3/2)/a^3-d*n*x*arccosh(a*x)-1/9*e*n*x^3*ar ccosh(a*x)-1/9*(9*a^2*d+2*e)*n*arctan((a*x-1)^(1/2)*(a*x+1)^(1/2))/a^3+d*x *arccosh(a*x)*ln(c*x^n)+1/3*e*x^3*arccosh(a*x)*ln(c*x^n)+d*n*(a*x-1)^(1/2) *(a*x+1)^(1/2)/a+2/27*e*n*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a^3+1/9*(9*a^2*d+2*e )*n*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a^3+1/27*e*n*x^2*(a*x-1)^(1/2)*(a*x+1)^(1/ 2)/a-1/9*(9*a^2*d+2*e)*ln(c*x^n)*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a^3-1/9*e*x^2 *ln(c*x^n)*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a
Time = 0.18 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.46 \[ \int \left (d+e x^2\right ) \text {arccosh}(a x) \log \left (c x^n\right ) \, dx=\frac {3 \left (9 a^2 d+2 e\right ) n \arctan \left (\frac {1}{\sqrt {-1+a x} \sqrt {1+a x}}\right )-3 a^3 x \text {arccosh}(a x) \left (n \left (9 d+e x^2\right )-3 \left (3 d+e x^2\right ) \log \left (c x^n\right )\right )+\sqrt {-1+a x} \sqrt {1+a x} \left (n \left (7 e+2 a^2 \left (27 d+e x^2\right )\right )-3 \left (2 e+a^2 \left (9 d+e x^2\right )\right ) \log \left (c x^n\right )\right )}{27 a^3} \]
(3*(9*a^2*d + 2*e)*n*ArcTan[1/(Sqrt[-1 + a*x]*Sqrt[1 + a*x])] - 3*a^3*x*Ar cCosh[a*x]*(n*(9*d + e*x^2) - 3*(3*d + e*x^2)*Log[c*x^n]) + Sqrt[-1 + a*x] *Sqrt[1 + a*x]*(n*(7*e + 2*a^2*(27*d + e*x^2)) - 3*(2*e + a^2*(9*d + e*x^2 ))*Log[c*x^n]))/(27*a^3)
Time = 0.45 (sec) , antiderivative size = 308, normalized size of antiderivative = 0.99, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2834, 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.
| \(\displaystyle \int \text {arccosh}(a x) \left (d+e x^2\right ) \log \left (c x^n\right ) \, dx\) |
| \(\Big \downarrow \) 2834 |
| \(\displaystyle -n \int \left (\frac {1}{3} e \text {arccosh}(a x) x^2-\frac {e \sqrt {a x-1} \sqrt {a x+1} x}{9 a}+d \text {arccosh}(a x)-\frac {\left (9 d a^2+2 e\right ) \sqrt {a x-1} \sqrt {a x+1}}{9 a^3 x}\right )dx-\frac {\sqrt {a x-1} \sqrt {a x+1} \left (9 a^2 d+2 e\right ) \log \left (c x^n\right )}{9 a^3}+d x \text {arccosh}(a x) \log \left (c x^n\right )+\frac {1}{3} e x^3 \text {arccosh}(a x) \log \left (c x^n\right )-\frac {e x^2 \sqrt {a x-1} \sqrt {a x+1} \log \left (c x^n\right )}{9 a}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -n \left (-\frac {e (a x-1)^{3/2} (a x+1)^{3/2}}{27 a^3}-\frac {2 e \sqrt {a x-1} \sqrt {a x+1}}{27 a^3}+\frac {\arctan \left (\sqrt {a x-1} \sqrt {a x+1}\right ) \left (9 a^2 d+2 e\right )}{9 a^3}-\frac {\sqrt {a x-1} \sqrt {a x+1} \left (9 a^2 d+2 e\right )}{9 a^3}+d x \text {arccosh}(a x)+\frac {1}{9} e x^3 \text {arccosh}(a x)-\frac {d \sqrt {a x-1} \sqrt {a x+1}}{a}-\frac {e x^2 \sqrt {a x-1} \sqrt {a x+1}}{27 a}\right )-\frac {\sqrt {a x-1} \sqrt {a x+1} \left (9 a^2 d+2 e\right ) \log \left (c x^n\right )}{9 a^3}+d x \text {arccosh}(a x) \log \left (c x^n\right )+\frac {1}{3} e x^3 \text {arccosh}(a x) \log \left (c x^n\right )-\frac {e x^2 \sqrt {a x-1} \sqrt {a x+1} \log \left (c x^n\right )}{9 a}\) |
-(n*(-((d*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/a) - (2*e*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(27*a^3) - ((9*a^2*d + 2*e)*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(9*a^3) - (e*x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(27*a) - (e*(-1 + a*x)^(3/2)*(1 + a*x )^(3/2))/(27*a^3) + d*x*ArcCosh[a*x] + (e*x^3*ArcCosh[a*x])/9 + ((9*a^2*d + 2*e)*ArcTan[Sqrt[-1 + a*x]*Sqrt[1 + a*x]])/(9*a^3))) - ((9*a^2*d + 2*e)* Sqrt[-1 + a*x]*Sqrt[1 + a*x]*Log[c*x^n])/(9*a^3) - (e*x^2*Sqrt[-1 + a*x]*S qrt[1 + a*x]*Log[c*x^n])/(9*a) + d*x*ArcCosh[a*x]*Log[c*x^n] + (e*x^3*ArcC osh[a*x]*Log[c*x^n])/3
3.2.91.3.1 Defintions of rubi rules used
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]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 5.01 (sec) , antiderivative size = 4757, normalized size of antiderivative = 15.25
-1/9/a^3*n*(3*arccosh(a*x)*x^3*a^3*e-(a*x+1)^(1/2)*(a*x-1)^(1/2)*x^2*a^2*e +9*arccosh(a*x)*x*a^3*d-9*(a*x+1)^(1/2)*(a*x-1)^(1/2)*a^2*d-2*(a*x+1)^(1/2 )*(a*x-1)^(1/2)*e)*ln(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))+1/54*I/a^3*(-6*Pi*c sgn(I/(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)))*(a*x+1)^(1/2)*(a*x-1)^(1/2)*csgn( I/(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))*(1+(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2 ))^2*e*n-6*Pi*csgn(I/a)*(a*x+1)^(1/2)*(a*x-1)^(1/2)*csgn(I/a*(1+(a*x+(a*x- 1)^(1/2)*(a*x+1)^(1/2))^2)/(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)))^2*e*n-6*Pi*( a*x+1)^(1/2)*(a*x-1)^(1/2)*csgn(I*(1+(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2)) *csgn(I/(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))*(1+(a*x+(a*x-1)^(1/2)*(a*x+1)^(1 /2))^2))^2*e*n-6*Pi*(a*x+1)^(1/2)*(a*x-1)^(1/2)*csgn(I/(a*x+(a*x-1)^(1/2)* (a*x+1)^(1/2))*(1+(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2))*csgn(I/a*(1+(a*x+( a*x-1)^(1/2)*(a*x+1)^(1/2))^2)/(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)))^2*e*n-54 *I*ln(a)*(a*x+1)^(1/2)*(a*x-1)^(1/2)*a^2*d*n+54*I*(a*x+1)^(1/2)*(a*x-1)^(1 /2)*ln(1+(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2)*a^2*d*n-9*Pi*arccosh(a*x)*cs gn(I/(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))*(1+(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2) )^2))^3*x^3*a^3*e*n-9*Pi*arccosh(a*x)*csgn(I/a*(1+(a*x+(a*x-1)^(1/2)*(a*x+ 1)^(1/2))^2)/(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)))^3*x^3*a^3*e*n+54*I*arccosh (a*x)*ln(a)*x*a^3*d*n-54*I*arccosh(a*x)*ln(1+(a*x+(a*x-1)^(1/2)*(a*x+1)^(1 /2))^2)*x*a^3*d*n+18*I*arccosh(a*x)*ln(a)*x^3*a^3*e*n-18*I*arccosh(a*x)*ln (1+(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2)*x^3*a^3*e*n+54*I*ln(2)*arccosh(...
Time = 0.39 (sec) , antiderivative size = 275, normalized size of antiderivative = 0.88 \[ \int \left (d+e x^2\right ) \text {arccosh}(a x) \log \left (c x^n\right ) \, dx=-\frac {6 \, {\left (9 \, a^{2} d + 2 \, e\right )} n \arctan \left (-a x + \sqrt {a^{2} x^{2} - 1}\right ) + 3 \, {\left (a^{3} e n x^{3} + 9 \, a^{3} d n x - {\left (9 \, a^{3} d + a^{3} e\right )} n - 3 \, {\left (a^{3} e x^{3} + 3 \, a^{3} d x - 3 \, a^{3} d - a^{3} e\right )} \log \left (c\right ) - 3 \, {\left (a^{3} e n x^{3} + 3 \, a^{3} d n x\right )} \log \left (x\right )\right )} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right ) - 3 \, {\left ({\left (9 \, a^{3} d + a^{3} e\right )} n - 3 \, {\left (3 \, a^{3} d + a^{3} e\right )} \log \left (c\right )\right )} \log \left (-a x + \sqrt {a^{2} x^{2} - 1}\right ) - {\left (2 \, a^{2} e n x^{2} + {\left (54 \, a^{2} d + 7 \, e\right )} n - 3 \, {\left (a^{2} e x^{2} + 9 \, a^{2} d + 2 \, e\right )} \log \left (c\right ) - 3 \, {\left (a^{2} e n x^{2} + {\left (9 \, a^{2} d + 2 \, e\right )} n\right )} \log \left (x\right )\right )} \sqrt {a^{2} x^{2} - 1}}{27 \, a^{3}} \]
-1/27*(6*(9*a^2*d + 2*e)*n*arctan(-a*x + sqrt(a^2*x^2 - 1)) + 3*(a^3*e*n*x ^3 + 9*a^3*d*n*x - (9*a^3*d + a^3*e)*n - 3*(a^3*e*x^3 + 3*a^3*d*x - 3*a^3* d - a^3*e)*log(c) - 3*(a^3*e*n*x^3 + 3*a^3*d*n*x)*log(x))*log(a*x + sqrt(a ^2*x^2 - 1)) - 3*((9*a^3*d + a^3*e)*n - 3*(3*a^3*d + a^3*e)*log(c))*log(-a *x + sqrt(a^2*x^2 - 1)) - (2*a^2*e*n*x^2 + (54*a^2*d + 7*e)*n - 3*(a^2*e*x ^2 + 9*a^2*d + 2*e)*log(c) - 3*(a^2*e*n*x^2 + (9*a^2*d + 2*e)*n)*log(x))*s qrt(a^2*x^2 - 1))/a^3
\[ \int \left (d+e x^2\right ) \text {arccosh}(a x) \log \left (c x^n\right ) \, dx=\int \left (d + e x^{2}\right ) \log {\left (c x^{n} \right )} \operatorname {acosh}{\left (a x \right )}\, dx \]
\[ \int \left (d+e x^2\right ) \text {arccosh}(a x) \log \left (c x^n\right ) \, dx=\int { {\left (e x^{2} + d\right )} \operatorname {arcosh}\left (a x\right ) \log \left (c x^{n}\right ) \,d x } \]
1/6*(3*a^2*d*n + e*n)*(log(a*x + 1)*log(x) + dilog(-a*x))/a^3 - 1/6*(3*a^2 *d*n + e*n)*(log(-a*x + 1)*log(x) + dilog(a*x))/a^3 - 1/18*(9*(d*n - d*log (c))*a^2 + e*n - 3*e*log(c))*log(a*x + 1)/a^3 + 1/18*(9*(d*n - d*log(c))*a ^2 + e*n - 3*e*log(c))*log(a*x - 1)/a^3 + 1/54*(2*(2*e*n - 3*e*log(c))*a^3 *x^3 - 9*(3*a^2*d*n + e*n)*log(a*x + 1)*log(x) + 9*(3*a^2*d*n + e*n)*log(a *x - 1)*log(x) + 6*(9*(2*d*n - d*log(c))*a^3 + (4*e*n - 3*e*log(c))*a)*x - 6*((e*n - 3*e*log(c))*a^3*x^3 + 9*(d*n - d*log(c))*a^3*x - 3*(a^3*e*x^3 + 3*a^3*d*x)*log(x^n))*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1)) - 3*(2*a^3*e* x^3 + 6*(3*a^3*d + a*e)*x - 3*(3*a^2*d + e)*log(a*x + 1) + 3*(3*a^2*d + e) *log(a*x - 1))*log(x^n))/a^3 + integrate(-1/9*((e*n - 3*e*log(c))*a*x^3 + 9*(d*n - d*log(c))*a*x - 3*(a*e*x^3 + 3*a*d*x)*log(x^n))/(a^3*x^3 + (a^2*x ^2 - 1)*sqrt(a*x + 1)*sqrt(a*x - 1) - a*x), x)
Exception generated. \[ \int \left (d+e x^2\right ) \text {arccosh}(a x) \log \left (c x^n\right ) \, dx=\text {Exception raised: TypeError} \]
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
Timed out. \[ \int \left (d+e x^2\right ) \text {arccosh}(a x) \log \left (c x^n\right ) \, dx=\int \ln \left (c\,x^n\right )\,\mathrm {acosh}\left (a\,x\right )\,\left (e\,x^2+d\right ) \,d x \]