Integrand size = 20, antiderivative size = 458 \[ \int \left (d+e x^2\right ) \text {arcsinh}(a x)^2 \log \left (c x^n\right ) \, dx=-2 d n x+\frac {2 e n x}{27 a^2}-\frac {4}{9} \left (9 d-\frac {2 e}{a^2}\right ) n x-\frac {2}{27} e n x^3+\frac {2 d n \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{a}+\frac {2 \left (9 a^2 d-2 e\right ) n \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{9 a^3}-\frac {4 e n \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{27 a^3}+\frac {2 e n x^2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{27 a}+\frac {2 e n \left (1+a^2 x^2\right )^{3/2} \text {arcsinh}(a x)}{27 a^3}-d n x \text {arcsinh}(a x)^2-\frac {1}{9} e n x^3 \text {arcsinh}(a x)^2-\frac {4 \left (9 a^2 d-2 e\right ) n \text {arcsinh}(a x) \text {arctanh}\left (e^{\text {arcsinh}(a x)}\right )}{9 a^3}+2 d x \log \left (c x^n\right )-\frac {4 e x \log \left (c x^n\right )}{9 a^2}+\frac {2}{27} e x^3 \log \left (c x^n\right )-\frac {2 d \sqrt {1+a^2 x^2} \text {arcsinh}(a x) \log \left (c x^n\right )}{a}+\frac {4 e \sqrt {1+a^2 x^2} \text {arcsinh}(a x) \log \left (c x^n\right )}{9 a^3}-\frac {2 e x^2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x) \log \left (c x^n\right )}{9 a}+d x \text {arcsinh}(a x)^2 \log \left (c x^n\right )+\frac {1}{3} e x^3 \text {arcsinh}(a x)^2 \log \left (c x^n\right )-\frac {2 \left (9 a^2 d-2 e\right ) n \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(a x)}\right )}{9 a^3}+\frac {2 \left (9 a^2 d-2 e\right ) n \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(a x)}\right )}{9 a^3} \]
-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+2/27*e*n*(a^2*x ^2+1)^(3/2)*arcsinh(a*x)/a^3-d*n*x*arcsinh(a*x)^2-1/9*e*n*x^3*arcsinh(a*x) ^2-4/9*(9*a^2*d-2*e)*n*arcsinh(a*x)*arctanh(a*x+(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+2/27*e*x^3*ln(c*x^n)+d*x*arcsinh(a*x)^2 *ln(c*x^n)+1/3*e*x^3*arcsinh(a*x)^2*ln(c*x^n)-2/9*(9*a^2*d-2*e)*n*polylog( 2,-a*x-(a^2*x^2+1)^(1/2))/a^3+2/9*(9*a^2*d-2*e)*n*polylog(2,a*x+(a^2*x^2+1 )^(1/2))/a^3+2*d*n*arcsinh(a*x)*(a^2*x^2+1)^(1/2)/a+2/9*(9*a^2*d-2*e)*n*ar csinh(a*x)*(a^2*x^2+1)^(1/2)/a^3-4/27*e*n*arcsinh(a*x)*(a^2*x^2+1)^(1/2)/a ^3+2/27*e*n*x^2*arcsinh(a*x)*(a^2*x^2+1)^(1/2)/a-2*d*arcsinh(a*x)*ln(c*x^n )*(a^2*x^2+1)^(1/2)/a+4/9*e*arcsinh(a*x)*ln(c*x^n)*(a^2*x^2+1)^(1/2)/a^3-2 /9*e*x^2*arcsinh(a*x)*ln(c*x^n)*(a^2*x^2+1)^(1/2)/a
Time = 0.54 (sec) , antiderivative size = 516, normalized size of antiderivative = 1.13 \[ \int \left (d+e x^2\right ) \text {arcsinh}(a x)^2 \log \left (c x^n\right ) \, dx=-2 d n x+\frac {4 e n x}{9 a^2}-\frac {2}{81} e n x^3+\frac {2 e n \left (-\frac {a x}{3}-\frac {a^3 x^3}{9}+\frac {1}{3} \left (1+a^2 x^2\right )^{3/2} \text {arcsinh}(a x)\right )}{9 a^3}+\frac {d n \left (2 a x-2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)+a x \text {arcsinh}(a x)^2\right ) \log (x)}{a}+\frac {e n \left (-12 a x+2 a^3 x^3+12 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)-6 a^2 x^2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)+9 a^3 x^3 \text {arcsinh}(a x)^2\right ) \log (x)}{27 a^3}+\frac {d \left (-2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)+a x \left (2+\text {arcsinh}(a x)^2\right )\right ) \left (-n-n \log (x)+\log \left (c x^n\right )\right )}{a}+\frac {e \left (27 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)+a x \left (-26-9 \text {arcsinh}(a x)^2+\left (2+9 \text {arcsinh}(a x)^2\right ) \cosh (2 \text {arcsinh}(a x))\right )-3 \text {arcsinh}(a x) \cosh (3 \text {arcsinh}(a x))\right ) \left (-n+3 \left (-n \log (x)+\log \left (c x^n\right )\right )\right )}{162 a^3}+\frac {2 d n \left (-a x+\sqrt {1+a^2 x^2} \text {arcsinh}(a x)+\text {arcsinh}(a x) \log \left (1-e^{-\text {arcsinh}(a x)}\right )-\text {arcsinh}(a x) \log \left (1+e^{-\text {arcsinh}(a x)}\right )+\operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(a x)}\right )-\operatorname {PolyLog}\left (2,e^{-\text {arcsinh}(a x)}\right )\right )}{a}-\frac {4 e n \left (-a x+\sqrt {1+a^2 x^2} \text {arcsinh}(a x)+\text {arcsinh}(a x) \log \left (1-e^{-\text {arcsinh}(a x)}\right )-\text {arcsinh}(a x) \log \left (1+e^{-\text {arcsinh}(a x)}\right )+\operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(a x)}\right )-\operatorname {PolyLog}\left (2,e^{-\text {arcsinh}(a x)}\right )\right )}{9 a^3} \]
-2*d*n*x + (4*e*n*x)/(9*a^2) - (2*e*n*x^3)/81 + (2*e*n*(-1/3*(a*x) - (a^3* x^3)/9 + ((1 + a^2*x^2)^(3/2)*ArcSinh[a*x])/3))/(9*a^3) + (d*n*(2*a*x - 2* Sqrt[1 + a^2*x^2]*ArcSinh[a*x] + a*x*ArcSinh[a*x]^2)*Log[x])/a + (e*n*(-12 *a*x + 2*a^3*x^3 + 12*Sqrt[1 + a^2*x^2]*ArcSinh[a*x] - 6*a^2*x^2*Sqrt[1 + a^2*x^2]*ArcSinh[a*x] + 9*a^3*x^3*ArcSinh[a*x]^2)*Log[x])/(27*a^3) + (d*(- 2*Sqrt[1 + a^2*x^2]*ArcSinh[a*x] + a*x*(2 + ArcSinh[a*x]^2))*(-n - n*Log[x ] + Log[c*x^n]))/a + (e*(27*Sqrt[1 + a^2*x^2]*ArcSinh[a*x] + a*x*(-26 - 9* ArcSinh[a*x]^2 + (2 + 9*ArcSinh[a*x]^2)*Cosh[2*ArcSinh[a*x]]) - 3*ArcSinh[ a*x]*Cosh[3*ArcSinh[a*x]])*(-n + 3*(-(n*Log[x]) + Log[c*x^n])))/(162*a^3) + (2*d*n*(-(a*x) + Sqrt[1 + a^2*x^2]*ArcSinh[a*x] + ArcSinh[a*x]*Log[1 - E ^(-ArcSinh[a*x])] - ArcSinh[a*x]*Log[1 + E^(-ArcSinh[a*x])] + PolyLog[2, - E^(-ArcSinh[a*x])] - PolyLog[2, E^(-ArcSinh[a*x])]))/a - (4*e*n*(-(a*x) + Sqrt[1 + a^2*x^2]*ArcSinh[a*x] + ArcSinh[a*x]*Log[1 - E^(-ArcSinh[a*x])] - ArcSinh[a*x]*Log[1 + E^(-ArcSinh[a*x])] + PolyLog[2, -E^(-ArcSinh[a*x])] - PolyLog[2, E^(-ArcSinh[a*x])]))/(9*a^3)
Time = 0.97 (sec) , antiderivative size = 447, 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.
| \(\displaystyle \int \text {arcsinh}(a x)^2 \left (d+e x^2\right ) \log \left (c x^n\right ) \, dx\) |
| \(\Big \downarrow \) 2834 |
| \(\displaystyle -n \int \left (\frac {1}{3} e \text {arcsinh}(a x)^2 x^2+\frac {2 e x^2}{27}-\frac {2 e \sqrt {a^2 x^2+1} \text {arcsinh}(a x) x}{9 a}+d \text {arcsinh}(a x)^2+\frac {2}{9} \left (9 d-\frac {2 e}{a^2}\right )-\frac {2 d \sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{a x}+\frac {4 e \sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{9 a^3 x}\right )dx-\frac {2 d \sqrt {a^2 x^2+1} \text {arcsinh}(a x) \log \left (c x^n\right )}{a}-\frac {2 e x^2 \sqrt {a^2 x^2+1} \text {arcsinh}(a x) \log \left (c x^n\right )}{9 a}-\frac {4 e x \log \left (c x^n\right )}{9 a^2}+\frac {4 e \sqrt {a^2 x^2+1} \text {arcsinh}(a x) \log \left (c x^n\right )}{9 a^3}+d x \text {arcsinh}(a x)^2 \log \left (c x^n\right )+\frac {1}{3} e x^3 \text {arcsinh}(a x)^2 \log \left (c x^n\right )+2 d x \log \left (c x^n\right )+\frac {2}{27} e x^3 \log \left (c x^n\right )\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle -n \int \left (\frac {1}{3} e \text {arcsinh}(a x)^2 x^2+\frac {2 e x^2}{27}-\frac {2 e \sqrt {a^2 x^2+1} \text {arcsinh}(a x) x}{9 a}+d \text {arcsinh}(a x)^2+\frac {2}{9} \left (9 d-\frac {2 e}{a^2}\right )+\frac {\left (\frac {4 e}{9 a^3}-\frac {2 d}{a}\right ) \sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{x}\right )dx-\frac {2 d \sqrt {a^2 x^2+1} \text {arcsinh}(a x) \log \left (c x^n\right )}{a}-\frac {2 e x^2 \sqrt {a^2 x^2+1} \text {arcsinh}(a x) \log \left (c x^n\right )}{9 a}-\frac {4 e x \log \left (c x^n\right )}{9 a^2}+\frac {4 e \sqrt {a^2 x^2+1} \text {arcsinh}(a x) \log \left (c x^n\right )}{9 a^3}+d x \text {arcsinh}(a x)^2 \log \left (c x^n\right )+\frac {1}{3} e x^3 \text {arcsinh}(a x)^2 \log \left (c x^n\right )+2 d x \log \left (c x^n\right )+\frac {2}{27} e x^3 \log \left (c x^n\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 d \sqrt {a^2 x^2+1} \text {arcsinh}(a x) \log \left (c x^n\right )}{a}-\frac {2 e x^2 \sqrt {a^2 x^2+1} \text {arcsinh}(a x) \log \left (c x^n\right )}{9 a}-\frac {4 e x \log \left (c x^n\right )}{9 a^2}-n \left (\frac {4 \text {arcsinh}(a x) \left (9 d-\frac {2 e}{a^2}\right ) \text {arctanh}\left (e^{\text {arcsinh}(a x)}\right )}{9 a}-\frac {2 \left (9 d-\frac {2 e}{a^2}\right ) \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(a x)}\right )}{9 a}-\frac {2 \sqrt {a^2 x^2+1} \text {arcsinh}(a x) \left (9 d-\frac {2 e}{a^2}\right )}{9 a}-\frac {2 d \sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{a}-\frac {2 e x^2 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{27 a}+\frac {4}{9} x \left (9 d-\frac {2 e}{a^2}\right )-\frac {2 e x}{27 a^2}+\frac {2 \left (9 a^2 d-2 e\right ) \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(a x)}\right )}{9 a^3}-\frac {2 e \left (a^2 x^2+1\right )^{3/2} \text {arcsinh}(a x)}{27 a^3}+\frac {4 e \sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{27 a^3}+d x \text {arcsinh}(a x)^2+\frac {1}{9} e x^3 \text {arcsinh}(a x)^2+2 d x+\frac {2 e x^3}{27}\right )+\frac {4 e \sqrt {a^2 x^2+1} \text {arcsinh}(a x) \log \left (c x^n\right )}{9 a^3}+d x \text {arcsinh}(a x)^2 \log \left (c x^n\right )+\frac {1}{3} e x^3 \text {arcsinh}(a x)^2 \log \left (c x^n\right )+2 d x \log \left (c x^n\right )+\frac {2}{27} e x^3 \log \left (c x^n\right )\) |
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]*ArcSinh[a*x]*Log[c*x^n])/a + (4*e*Sqrt[1 + a^2*x^2] *ArcSinh[a*x]*Log[c*x^n])/(9*a^3) - (2*e*x^2*Sqrt[1 + a^2*x^2]*ArcSinh[a*x ]*Log[c*x^n])/(9*a) + d*x*ArcSinh[a*x]^2*Log[c*x^n] + (e*x^3*ArcSinh[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^2*x^2]*ArcSinh[a*x])/a + (4*e*Sqrt[1 + a^ 2*x^2]*ArcSinh[a*x])/(27*a^3) - (2*(9*d - (2*e)/a^2)*Sqrt[1 + a^2*x^2]*Arc Sinh[a*x])/(9*a) - (2*e*x^2*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/(27*a) - (2*e* (1 + a^2*x^2)^(3/2)*ArcSinh[a*x])/(27*a^3) + d*x*ArcSinh[a*x]^2 + (e*x^3*A rcSinh[a*x]^2)/9 + (4*(9*d - (2*e)/a^2)*ArcSinh[a*x]*ArcTanh[E^ArcSinh[a*x ]])/(9*a) + (2*(9*a^2*d - 2*e)*PolyLog[2, -E^ArcSinh[a*x]])/(9*a^3) - (2*( 9*d - (2*e)/a^2)*PolyLog[2, E^ArcSinh[a*x]])/(9*a))
3.2.96.3.1 Defintions of rubi rules used
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]
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]
\[\int \left (e \,x^{2}+d \right ) \operatorname {arcsinh}\left (a x \right )^{2} \ln \left (c \,x^{n}\right )d x\]
\[ \int \left (d+e x^2\right ) \text {arcsinh}(a x)^2 \log \left (c x^n\right ) \, dx=\int { {\left (e x^{2} + d\right )} \operatorname {arsinh}\left (a x\right )^{2} \log \left (c x^{n}\right ) \,d x } \]
\[ \int \left (d+e x^2\right ) \text {arcsinh}(a x)^2 \log \left (c x^n\right ) \, dx=\int \left (d + e x^{2}\right ) \log {\left (c x^{n} \right )} \operatorname {asinh}^{2}{\left (a x \right )}\, dx \]
\[ \int \left (d+e x^2\right ) \text {arcsinh}(a x)^2 \log \left (c x^n\right ) \, dx=\int { {\left (e x^{2} + d\right )} \operatorname {arsinh}\left (a x\right )^{2} \log \left (c x^{n}\right ) \,d x } \]
-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^2*x^2 + 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 - 3*(a^3*e*x^5 + (3*a^3*d + a*e)*x^3 + 3*a*d*x)*log(x^n) + ((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^2*x^2 + 1))*log(a*x + sqrt(a^2*x^2 + 1))/(a^ 3*x^3 + a*x + (a^2*x^2 + 1)^(3/2)), x)
Exception generated. \[ \int \left (d+e x^2\right ) \text {arcsinh}(a x)^2 \log \left (c x^n\right ) \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \left (d+e x^2\right ) \text {arcsinh}(a x)^2 \log \left (c x^n\right ) \, dx=\int \ln \left (c\,x^n\right )\,{\mathrm {asinh}\left (a\,x\right )}^2\,\left (e\,x^2+d\right ) \,d x \]