Integrand size = 18, antiderivative size = 244 \[ \int \left (d+e x^2\right ) \text {arcsinh}(a x) \log \left (c x^n\right ) \, dx=\frac {d n \sqrt {1+a^2 x^2}}{a}+\frac {\left (3 a^2 d-e\right ) n \sqrt {1+a^2 x^2}}{3 a^3}+\frac {2 e n \left (1+a^2 x^2\right )^{3/2}}{27 a^3}-d n x \text {arcsinh}(a x)-\frac {1}{9} e n x^3 \text {arcsinh}(a x)-\frac {\left (3 a^2 d-e\right ) n \text {arctanh}\left (\sqrt {1+a^2 x^2}\right )}{3 a^3}-\frac {e n \text {arctanh}\left (\sqrt {1+a^2 x^2}\right )}{9 a^3}-\frac {\left (3 a^2 d-e\right ) \sqrt {1+a^2 x^2} \log \left (c x^n\right )}{3 a^3}-\frac {e \left (1+a^2 x^2\right )^{3/2} \log \left (c x^n\right )}{9 a^3}+d x \text {arcsinh}(a x) \log \left (c x^n\right )+\frac {1}{3} e x^3 \text {arcsinh}(a x) \log \left (c x^n\right ) \]
2/27*e*n*(a^2*x^2+1)^(3/2)/a^3-d*n*x*arcsinh(a*x)-1/9*e*n*x^3*arcsinh(a*x) -1/3*(3*a^2*d-e)*n*arctanh((a^2*x^2+1)^(1/2))/a^3-1/9*e*n*arctanh((a^2*x^2 +1)^(1/2))/a^3-1/9*e*(a^2*x^2+1)^(3/2)*ln(c*x^n)/a^3+d*x*arcsinh(a*x)*ln(c *x^n)+1/3*e*x^3*arcsinh(a*x)*ln(c*x^n)+d*n*(a^2*x^2+1)^(1/2)/a+1/3*(3*a^2* d-e)*n*(a^2*x^2+1)^(1/2)/a^3-1/3*(3*a^2*d-e)*ln(c*x^n)*(a^2*x^2+1)^(1/2)/a ^3
Time = 0.12 (sec) , antiderivative size = 240, normalized size of antiderivative = 0.98 \[ \int \left (d+e x^2\right ) \text {arcsinh}(a x) \log \left (c x^n\right ) \, dx=\frac {54 a^2 d n \sqrt {1+a^2 x^2}-7 e n \sqrt {1+a^2 x^2}+2 a^2 e n x^2 \sqrt {1+a^2 x^2}+3 \left (9 a^2 d-2 e\right ) n \log (x)-27 a^2 d \sqrt {1+a^2 x^2} \log \left (c x^n\right )+6 e \sqrt {1+a^2 x^2} \log \left (c x^n\right )-3 a^2 e x^2 \sqrt {1+a^2 x^2} \log \left (c x^n\right )-3 a^3 x \text {arcsinh}(a x) \left (n \left (9 d+e x^2\right )-3 \left (3 d+e x^2\right ) \log \left (c x^n\right )\right )-27 a^2 d n \log \left (1+\sqrt {1+a^2 x^2}\right )+6 e n \log \left (1+\sqrt {1+a^2 x^2}\right )}{27 a^3} \]
(54*a^2*d*n*Sqrt[1 + a^2*x^2] - 7*e*n*Sqrt[1 + a^2*x^2] + 2*a^2*e*n*x^2*Sq rt[1 + a^2*x^2] + 3*(9*a^2*d - 2*e)*n*Log[x] - 27*a^2*d*Sqrt[1 + a^2*x^2]* Log[c*x^n] + 6*e*Sqrt[1 + a^2*x^2]*Log[c*x^n] - 3*a^2*e*x^2*Sqrt[1 + a^2*x ^2]*Log[c*x^n] - 3*a^3*x*ArcSinh[a*x]*(n*(9*d + e*x^2) - 3*(3*d + e*x^2)*L og[c*x^n]) - 27*a^2*d*n*Log[1 + Sqrt[1 + a^2*x^2]] + 6*e*n*Log[1 + Sqrt[1 + a^2*x^2]])/(27*a^3)
Time = 0.47 (sec) , antiderivative size = 241, 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.
| \(\displaystyle \int \text {arcsinh}(a x) \left (d+e x^2\right ) \log \left (c x^n\right ) \, dx\) |
| \(\Big \downarrow \) 2834 |
| \(\displaystyle -n \int \left (\frac {1}{3} e \text {arcsinh}(a x) x^2+d \text {arcsinh}(a x)-\frac {e \left (a^2 x^2+1\right )^{3/2}}{9 a^3 x}-\frac {\left (3 a^2 d-e\right ) \sqrt {a^2 x^2+1}}{3 a^3 x}\right )dx-\frac {\sqrt {a^2 x^2+1} \left (3 a^2 d-e\right ) \log \left (c x^n\right )}{3 a^3}-\frac {e \left (a^2 x^2+1\right )^{3/2} \log \left (c x^n\right )}{9 a^3}+d x \text {arcsinh}(a x) \log \left (c x^n\right )+\frac {1}{3} e x^3 \text {arcsinh}(a x) \log \left (c x^n\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -n \left (-\frac {d \sqrt {a^2 x^2+1}}{a}+\frac {\text {arctanh}\left (\sqrt {a^2 x^2+1}\right ) \left (3 a^2 d-e\right )}{3 a^3}+\frac {e \text {arctanh}\left (\sqrt {a^2 x^2+1}\right )}{9 a^3}-\frac {\sqrt {a^2 x^2+1} \left (3 a^2 d-e\right )}{3 a^3}-\frac {2 e \left (a^2 x^2+1\right )^{3/2}}{27 a^3}+d x \text {arcsinh}(a x)+\frac {1}{9} e x^3 \text {arcsinh}(a x)\right )-\frac {\sqrt {a^2 x^2+1} \left (3 a^2 d-e\right ) \log \left (c x^n\right )}{3 a^3}-\frac {e \left (a^2 x^2+1\right )^{3/2} \log \left (c x^n\right )}{9 a^3}+d x \text {arcsinh}(a x) \log \left (c x^n\right )+\frac {1}{3} e x^3 \text {arcsinh}(a x) \log \left (c x^n\right )\) |
-(n*(-((d*Sqrt[1 + a^2*x^2])/a) - ((3*a^2*d - e)*Sqrt[1 + a^2*x^2])/(3*a^3 ) - (2*e*(1 + a^2*x^2)^(3/2))/(27*a^3) + d*x*ArcSinh[a*x] + (e*x^3*ArcSinh [a*x])/9 + ((3*a^2*d - e)*ArcTanh[Sqrt[1 + a^2*x^2]])/(3*a^3) + (e*ArcTanh [Sqrt[1 + a^2*x^2]])/(9*a^3))) - ((3*a^2*d - e)*Sqrt[1 + a^2*x^2]*Log[c*x^ n])/(3*a^3) - (e*(1 + a^2*x^2)^(3/2)*Log[c*x^n])/(9*a^3) + d*x*ArcSinh[a*x ]*Log[c*x^n] + (e*x^3*ArcSinh[a*x]*Log[c*x^n])/3
3.2.90.3.1 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 = 3.88 (sec) , antiderivative size = 4121, normalized size of antiderivative = 16.89
-1/9/a^3*n*(3*arcsinh(a*x)*x^3*a^3*e+9*arcsinh(a*x)*x*a^3*d-(a^2*x^2+1)^(1 /2)*x^2*a^2*e-9*(a^2*x^2+1)^(1/2)*a^2*d+2*(a^2*x^2+1)^(1/2)*e)*ln(a*x+(a^2 *x^2+1)^(1/2))-1/54/a^3*(-27*I*Pi*arcsinh(a*x)*csgn(I*(-1+(a*x+(a^2*x^2+1) ^(1/2))^2))*csgn(I/(a*x+(a^2*x^2+1)^(1/2))*(-1+(a*x+(a^2*x^2+1)^(1/2))^2)) ^2*x*a^3*d*n-27*I*Pi*arcsinh(a*x)*csgn(I/(a*x+(a^2*x^2+1)^(1/2))*(-1+(a*x+ (a^2*x^2+1)^(1/2))^2))^2*csgn(I/(a*x+(a^2*x^2+1)^(1/2)))*x*a^3*d*n-9*I*Pi* arcsinh(a*x)*csgn(I/a*(-1+(a*x+(a^2*x^2+1)^(1/2))^2)/(a*x+(a^2*x^2+1)^(1/2 )))^2*csgn(I/(a*x+(a^2*x^2+1)^(1/2))*(-1+(a*x+(a^2*x^2+1)^(1/2))^2))*x^3*a ^3*e*n-9*I*Pi*arcsinh(a*x)*csgn(I/a*(-1+(a*x+(a^2*x^2+1)^(1/2))^2)/(a*x+(a ^2*x^2+1)^(1/2)))^2*csgn(I/a)*x^3*a^3*e*n-3*I*Pi*csgn(I*(-1+(a*x+(a^2*x^2+ 1)^(1/2))^2))*csgn(I/(a*x+(a^2*x^2+1)^(1/2))*(-1+(a*x+(a^2*x^2+1)^(1/2))^2 ))*csgn(I/(a*x+(a^2*x^2+1)^(1/2)))*(a^2*x^2+1)^(1/2)*x^2*a^2*e*n+9*I*Pi*ar csinh(a*x)*csgn(I/a*(-1+(a*x+(a^2*x^2+1)^(1/2))^2)/(a*x+(a^2*x^2+1)^(1/2)) )*csgn(I/(a*x+(a^2*x^2+1)^(1/2))*(-1+(a*x+(a^2*x^2+1)^(1/2))^2))*csgn(I/a) *x^3*a^3*e*n+9*I*Pi*arcsinh(a*x)*csgn(I*(-1+(a*x+(a^2*x^2+1)^(1/2))^2))*cs gn(I/(a*x+(a^2*x^2+1)^(1/2))*(-1+(a*x+(a^2*x^2+1)^(1/2))^2))*csgn(I/(a*x+( a^2*x^2+1)^(1/2)))*x^3*a^3*e*n+27*I*Pi*arcsinh(a*x)*csgn(I/a*(-1+(a*x+(a^2 *x^2+1)^(1/2))^2)/(a*x+(a^2*x^2+1)^(1/2)))*csgn(I/(a*x+(a^2*x^2+1)^(1/2))* (-1+(a*x+(a^2*x^2+1)^(1/2))^2))*csgn(I/a)*x*a^3*d*n+27*I*Pi*arcsinh(a*x)*c sgn(I*(-1+(a*x+(a^2*x^2+1)^(1/2))^2))*csgn(I/(a*x+(a^2*x^2+1)^(1/2))*(-...
Time = 0.39 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.26 \[ \int \left (d+e x^2\right ) \text {arcsinh}(a x) \log \left (c x^n\right ) \, dx=-\frac {3 \, {\left (9 \, a^{2} d - 2 \, e\right )} n \log \left (-a x + \sqrt {a^{2} x^{2} + 1} + 1\right ) - 3 \, {\left (9 \, a^{2} d - 2 \, e\right )} n \log \left (-a x + \sqrt {a^{2} x^{2} + 1} - 1\right ) + 3 \, {\left (a^{3} e n x^{3} + 9 \, a^{3} d n x - {\left (9 \, a^{3} d + a^{3} e\right )} n - 3 \, {\left (a^{3} e x^{3} + 3 \, a^{3} d x - 3 \, a^{3} d - a^{3} e\right )} \log \left (c\right ) - 3 \, {\left (a^{3} e n x^{3} + 3 \, a^{3} d n x\right )} \log \left (x\right )\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right ) - 3 \, {\left ({\left (9 \, a^{3} d + a^{3} e\right )} n - 3 \, {\left (3 \, a^{3} d + a^{3} e\right )} \log \left (c\right )\right )} \log \left (-a x + \sqrt {a^{2} x^{2} + 1}\right ) - {\left (2 \, a^{2} e n x^{2} + {\left (54 \, a^{2} d - 7 \, e\right )} n - 3 \, {\left (a^{2} e x^{2} + 9 \, a^{2} d - 2 \, e\right )} \log \left (c\right ) - 3 \, {\left (a^{2} e n x^{2} + {\left (9 \, a^{2} d - 2 \, e\right )} n\right )} \log \left (x\right )\right )} \sqrt {a^{2} x^{2} + 1}}{27 \, a^{3}} \]
-1/27*(3*(9*a^2*d - 2*e)*n*log(-a*x + sqrt(a^2*x^2 + 1) + 1) - 3*(9*a^2*d - 2*e)*n*log(-a*x + sqrt(a^2*x^2 + 1) - 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))*sqrt(a^2*x^2 + 1)) /a^3
\[ \int \left (d+e x^2\right ) \text {arcsinh}(a x) \log \left (c x^n\right ) \, dx=\int \left (d + e x^{2}\right ) \log {\left (c x^{n} \right )} \operatorname {asinh}{\left (a x \right )}\, dx \]
\[ \int \left (d+e x^2\right ) \text {arcsinh}(a x) \log \left (c x^n\right ) \, dx=\int { {\left (e x^{2} + d\right )} \operatorname {arsinh}\left (a x\right ) \log \left (c x^{n}\right ) \,d x } \]
1/2*a^2*d*n*(2*x/a^2 + I*(log(I*a*x + 1) - log(-I*a*x + 1))/a^3) + 1/54*a^ 2*e*n*(2*(a^2*x^3 - 3*x)/a^4 - 3*I*(log(I*a*x + 1) - log(-I*a*x + 1))/a^5) - 3*a^2*e*n*integrate(1/9*x^4*log(x)/(a^2*x^2 + 1), x) - 9*a^2*d*n*integr ate(1/9*x^2*log(x)/(a^2*x^2 + 1), x) - 1/2*a^2*d*(2*x/a^2 + I*(log(I*a*x + 1) - log(-I*a*x + 1))/a^3)*log(c) - 1/18*a^2*e*(2*(a^2*x^3 - 3*x)/a^4 - 3 *I*(log(I*a*x + 1) - log(-I*a*x + 1))/a^5)*log(c) - 1/9*((e*n - 3*e*log(c) )*x^3 + 9*(d*n - d*log(c))*x - 3*(e*x^3 + 3*d*x)*log(x^n))*log(a*x + sqrt( a^2*x^2 + 1)) - 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*x + (a^2*x^2 + 1)^( 3/2)), x)
Exception generated. \[ \int \left (d+e x^2\right ) \text {arcsinh}(a x) \log \left (c x^n\right ) \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \left (d+e x^2\right ) \text {arcsinh}(a x) \log \left (c x^n\right ) \, dx=\int \ln \left (c\,x^n\right )\,\mathrm {asinh}\left (a\,x\right )\,\left (e\,x^2+d\right ) \,d x \]