Integrand size = 19, antiderivative size = 174 \[ \int x^2 \left (d+e x^2\right ) \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=-\frac {b \left (20 c^2 d+9 e\right ) x \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{120 c^4}-\frac {b e x^3 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{20 c^2}+\frac {1}{3} d x^3 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{5} e x^5 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {b \left (20 c^2 d+9 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \arcsin (c x)}{120 c^5} \]
1/3*d*x^3*(a+b*arcsech(c*x))+1/5*e*x^5*(a+b*arcsech(c*x))+1/120*b*(20*c^2* d+9*e)*arcsin(c*x)*(1/(c*x+1))^(1/2)*(c*x+1)^(1/2)/c^5-1/120*b*(20*c^2*d+9 *e)*x*(1/(c*x+1))^(1/2)*(c*x+1)^(1/2)*(-c^2*x^2+1)^(1/2)/c^4-1/20*b*e*x^3* (1/(c*x+1))^(1/2)*(c*x+1)^(1/2)*(-c^2*x^2+1)^(1/2)/c^2
Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.83 \[ \int x^2 \left (d+e x^2\right ) \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\frac {8 a c^5 x^3 \left (5 d+3 e x^2\right )-b c x \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (9 e+c^2 \left (20 d+6 e x^2\right )\right )+8 b c^5 x^3 \left (5 d+3 e x^2\right ) \text {sech}^{-1}(c x)+i b \left (20 c^2 d+9 e\right ) \log \left (-2 i c x+2 \sqrt {\frac {1-c x}{1+c x}} (1+c x)\right )}{120 c^5} \]
(8*a*c^5*x^3*(5*d + 3*e*x^2) - b*c*x*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*( 9*e + c^2*(20*d + 6*e*x^2)) + 8*b*c^5*x^3*(5*d + 3*e*x^2)*ArcSech[c*x] + I *b*(20*c^2*d + 9*e)*Log[(-2*I)*c*x + 2*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x) ])/(120*c^5)
Time = 0.31 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.76, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {6855, 27, 363, 262, 223}
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.
| \(\displaystyle \int x^2 \left (d+e x^2\right ) \left (a+b \text {sech}^{-1}(c x)\right ) \, dx\) |
| \(\Big \downarrow \) 6855 |
| \(\displaystyle b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \int \frac {x^2 \left (3 e x^2+5 d\right )}{15 \sqrt {1-c^2 x^2}}dx+\frac {1}{3} d x^3 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{5} e x^5 \left (a+b \text {sech}^{-1}(c x)\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{15} b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \int \frac {x^2 \left (3 e x^2+5 d\right )}{\sqrt {1-c^2 x^2}}dx+\frac {1}{3} d x^3 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{5} e x^5 \left (a+b \text {sech}^{-1}(c x)\right )\) |
| \(\Big \downarrow \) 363 |
| \(\displaystyle \frac {1}{15} b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {1}{4} \left (\frac {9 e}{c^2}+20 d\right ) \int \frac {x^2}{\sqrt {1-c^2 x^2}}dx-\frac {3 e x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {1}{3} d x^3 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{5} e x^5 \left (a+b \text {sech}^{-1}(c x)\right )\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {1}{15} b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {1}{4} \left (\frac {9 e}{c^2}+20 d\right ) \left (\frac {\int \frac {1}{\sqrt {1-c^2 x^2}}dx}{2 c^2}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )-\frac {3 e x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {1}{3} d x^3 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{5} e x^5 \left (a+b \text {sech}^{-1}(c x)\right )\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \frac {1}{3} d x^3 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{5} e x^5 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{15} b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {1}{4} \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right ) \left (\frac {9 e}{c^2}+20 d\right )-\frac {3 e x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )\) |
(d*x^3*(a + b*ArcSech[c*x]))/3 + (e*x^5*(a + b*ArcSech[c*x]))/5 + (b*Sqrt[ (1 + c*x)^(-1)]*Sqrt[1 + c*x]*((-3*e*x^3*Sqrt[1 - c^2*x^2])/(4*c^2) + ((20 *d + (9*e)/c^2)*(-1/2*(x*Sqrt[1 - c^2*x^2])/c^2 + ArcSin[c*x]/(2*c^3)))/4) )/15
3.1.89.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt [a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c , 2, m, p, x]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x _Symbol] :> Simp[d*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(b*e*(m + 2*p + 3))), x] - Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(b*(m + 2*p + 3)) Int[(e*x)^ m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b*c - a*d , 0] && NeQ[m + 2*p + 3, 0]
Int[((a_.) + ArcSech[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_.)*((d_.) + (e_.)*( x_)^2)^(p_.), x_Symbol] :> With[{u = IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Si mp[(a + b*ArcSech[c*x]) u, x] + Simp[b*Sqrt[1 + c*x]*Sqrt[1/(1 + c*x)] Int[SimplifyIntegrand[u/(x*Sqrt[1 - c*x]*Sqrt[1 + c*x]), x], x], x]] /; Fre eQ[{a, b, c, d, e, f, m, p}, x] && ((IGtQ[p, 0] && !(ILtQ[(m - 1)/2, 0] && GtQ[m + 2*p + 3, 0])) || (IGtQ[(m + 1)/2, 0] && !(ILtQ[p, 0] && GtQ[m + 2 *p + 3, 0])) || (ILtQ[(m + 2*p + 1)/2, 0] && !ILtQ[(m - 1)/2, 0]))
Time = 0.59 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.98
| method | result | size |
| parts | \(a \left (\frac {1}{5} e \,x^{5}+\frac {1}{3} d \,x^{3}\right )+\frac {b \left (\frac {c^{3} \operatorname {arcsech}\left (c x \right ) e \,x^{5}}{5}+\frac {\operatorname {arcsech}\left (c x \right ) x^{3} c^{3} d}{3}+\frac {\sqrt {-\frac {c x -1}{c x}}\, x \sqrt {\frac {c x +1}{c x}}\, \left (-6 e \,c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}-20 d \,c^{3} x \sqrt {-c^{2} x^{2}+1}+20 d \,c^{2} \arcsin \left (c x \right )-9 e c x \sqrt {-c^{2} x^{2}+1}+9 e \arcsin \left (c x \right )\right )}{120 c \sqrt {-c^{2} x^{2}+1}}\right )}{c^{3}}\) | \(171\) |
| derivativedivides | \(\frac {\frac {a \left (\frac {1}{3} d \,c^{5} x^{3}+\frac {1}{5} e \,c^{5} x^{5}\right )}{c^{2}}+\frac {b \left (\frac {\operatorname {arcsech}\left (c x \right ) d \,c^{5} x^{3}}{3}+\frac {\operatorname {arcsech}\left (c x \right ) e \,c^{5} x^{5}}{5}-\frac {\sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}\, \left (20 d \,c^{3} x \sqrt {-c^{2} x^{2}+1}+6 e \,c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}-20 d \,c^{2} \arcsin \left (c x \right )+9 e c x \sqrt {-c^{2} x^{2}+1}-9 e \arcsin \left (c x \right )\right )}{120 \sqrt {-c^{2} x^{2}+1}}\right )}{c^{2}}}{c^{3}}\) | \(182\) |
| default | \(\frac {\frac {a \left (\frac {1}{3} d \,c^{5} x^{3}+\frac {1}{5} e \,c^{5} x^{5}\right )}{c^{2}}+\frac {b \left (\frac {\operatorname {arcsech}\left (c x \right ) d \,c^{5} x^{3}}{3}+\frac {\operatorname {arcsech}\left (c x \right ) e \,c^{5} x^{5}}{5}-\frac {\sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}\, \left (20 d \,c^{3} x \sqrt {-c^{2} x^{2}+1}+6 e \,c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}-20 d \,c^{2} \arcsin \left (c x \right )+9 e c x \sqrt {-c^{2} x^{2}+1}-9 e \arcsin \left (c x \right )\right )}{120 \sqrt {-c^{2} x^{2}+1}}\right )}{c^{2}}}{c^{3}}\) | \(182\) |
a*(1/5*e*x^5+1/3*d*x^3)+b/c^3*(1/5*c^3*arcsech(c*x)*e*x^5+1/3*arcsech(c*x) *x^3*c^3*d+1/120/c*(-(c*x-1)/c/x)^(1/2)*x*((c*x+1)/c/x)^(1/2)*(-6*e*c^3*x^ 3*(-c^2*x^2+1)^(1/2)-20*d*c^3*x*(-c^2*x^2+1)^(1/2)+20*d*c^2*arcsin(c*x)-9* e*c*x*(-c^2*x^2+1)^(1/2)+9*e*arcsin(c*x))/(-c^2*x^2+1)^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 238 vs. \(2 (100) = 200\).
Time = 0.32 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.37 \[ \int x^2 \left (d+e x^2\right ) \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\frac {24 \, a c^{5} e x^{5} + 40 \, a c^{5} d x^{3} - 2 \, {\left (20 \, b c^{2} d + 9 \, b e\right )} \arctan \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - 1}{c x}\right ) - 8 \, {\left (5 \, b c^{5} d + 3 \, b c^{5} e\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - 1}{x}\right ) + 8 \, {\left (3 \, b c^{5} e x^{5} + 5 \, b c^{5} d x^{3} - 5 \, b c^{5} d - 3 \, b c^{5} e\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) - {\left (6 \, b c^{4} e x^{4} + {\left (20 \, b c^{4} d + 9 \, b c^{2} e\right )} x^{2}\right )} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}}}{120 \, c^{5}} \]
1/120*(24*a*c^5*e*x^5 + 40*a*c^5*d*x^3 - 2*(20*b*c^2*d + 9*b*e)*arctan((c* x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) - 1)/(c*x)) - 8*(5*b*c^5*d + 3*b*c^5*e)*l og((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) - 1)/x) + 8*(3*b*c^5*e*x^5 + 5*b*c^ 5*d*x^3 - 5*b*c^5*d - 3*b*c^5*e)*log((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) + 1)/(c*x)) - (6*b*c^4*e*x^4 + (20*b*c^4*d + 9*b*c^2*e)*x^2)*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)))/c^5
\[ \int x^2 \left (d+e x^2\right ) \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\int x^{2} \left (a + b \operatorname {asech}{\left (c x \right )}\right ) \left (d + e x^{2}\right )\, dx \]
Time = 0.28 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.05 \[ \int x^2 \left (d+e x^2\right ) \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\frac {1}{5} \, a e x^{5} + \frac {1}{3} \, a d x^{3} + \frac {1}{6} \, {\left (2 \, x^{3} \operatorname {arsech}\left (c x\right ) - \frac {\frac {\sqrt {\frac {1}{c^{2} x^{2}} - 1}}{c^{2} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} + c^{2}} + \frac {\arctan \left (\sqrt {\frac {1}{c^{2} x^{2}} - 1}\right )}{c^{2}}}{c}\right )} b d + \frac {1}{40} \, {\left (8 \, x^{5} \operatorname {arsech}\left (c x\right ) - \frac {\frac {3 \, {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {3}{2}} + 5 \, \sqrt {\frac {1}{c^{2} x^{2}} - 1}}{c^{4} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2} + 2 \, c^{4} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} + c^{4}} + \frac {3 \, \arctan \left (\sqrt {\frac {1}{c^{2} x^{2}} - 1}\right )}{c^{4}}}{c}\right )} b e \]
1/5*a*e*x^5 + 1/3*a*d*x^3 + 1/6*(2*x^3*arcsech(c*x) - (sqrt(1/(c^2*x^2) - 1)/(c^2*(1/(c^2*x^2) - 1) + c^2) + arctan(sqrt(1/(c^2*x^2) - 1))/c^2)/c)*b *d + 1/40*(8*x^5*arcsech(c*x) - ((3*(1/(c^2*x^2) - 1)^(3/2) + 5*sqrt(1/(c^ 2*x^2) - 1))/(c^4*(1/(c^2*x^2) - 1)^2 + 2*c^4*(1/(c^2*x^2) - 1) + c^4) + 3 *arctan(sqrt(1/(c^2*x^2) - 1))/c^4)/c)*b*e
\[ \int x^2 \left (d+e x^2\right ) \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\int { {\left (e x^{2} + d\right )} {\left (b \operatorname {arsech}\left (c x\right ) + a\right )} x^{2} \,d x } \]
Timed out. \[ \int x^2 \left (d+e x^2\right ) \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\int x^2\,\left (e\,x^2+d\right )\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right ) \,d x \]