Integrand size = 18, antiderivative size = 204 \[ \int \left (d+e x^2\right )^2 \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=-\frac {b e \left (40 c^2 d+9 e\right ) x \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{120 c^4}-\frac {b e^2 x^3 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{20 c^2}+d^2 x \left (a+b \text {sech}^{-1}(c x)\right )+\frac {2}{3} d e x^3 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{5} e^2 x^5 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {b \left (120 c^4 d^2+40 c^2 d e+9 e^2\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \arcsin (c x)}{120 c^5} \]
d^2*x*(a+b*arcsech(c*x))+2/3*d*e*x^3*(a+b*arcsech(c*x))+1/5*e^2*x^5*(a+b*a rcsech(c*x))+1/120*b*(120*c^4*d^2+40*c^2*d*e+9*e^2)*arcsin(c*x)*(1/(c*x+1) )^(1/2)*(c*x+1)^(1/2)/c^5-1/120*b*e*(40*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^2*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 = 0.28 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.85 \[ \int \left (d+e x^2\right )^2 \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\frac {8 a c^5 x \left (15 d^2+10 d e x^2+3 e^2 x^4\right )-b c e x \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (9 e+c^2 \left (40 d+6 e x^2\right )\right )+8 b c^5 x \left (15 d^2+10 d e x^2+3 e^2 x^4\right ) \text {sech}^{-1}(c x)+i b \left (120 c^4 d^2+40 c^2 d e+9 e^2\right ) \log \left (-2 i c x+2 \sqrt {\frac {1-c x}{1+c x}} (1+c x)\right )}{120 c^5} \]
(8*a*c^5*x*(15*d^2 + 10*d*e*x^2 + 3*e^2*x^4) - b*c*e*x*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*(9*e + c^2*(40*d + 6*e*x^2)) + 8*b*c^5*x*(15*d^2 + 10*d*e *x^2 + 3*e^2*x^4)*ArcSech[c*x] + I*b*(120*c^4*d^2 + 40*c^2*d*e + 9*e^2)*Lo g[(-2*I)*c*x + 2*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)])/(120*c^5)
Time = 0.37 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.86, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6845, 27, 1473, 25, 299, 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.
| \(\displaystyle \int \left (d+e x^2\right )^2 \left (a+b \text {sech}^{-1}(c x)\right ) \, dx\) |
| \(\Big \downarrow \) 6845 |
| \(\displaystyle b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \int \frac {3 e^2 x^4+10 d e x^2+15 d^2}{15 \sqrt {1-c^2 x^2}}dx+d^2 x \left (a+b \text {sech}^{-1}(c x)\right )+\frac {2}{3} d e x^3 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{5} e^2 x^5 \left (a+b \text {sech}^{-1}(c x)\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{15} b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \int \frac {3 e^2 x^4+10 d e x^2+15 d^2}{\sqrt {1-c^2 x^2}}dx+d^2 x \left (a+b \text {sech}^{-1}(c x)\right )+\frac {2}{3} d e x^3 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{5} e^2 x^5 \left (a+b \text {sech}^{-1}(c x)\right )\) |
| \(\Big \downarrow \) 1473 |
| \(\displaystyle \frac {1}{15} b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (-\frac {\int -\frac {60 c^2 d^2+e \left (40 d c^2+9 e\right ) x^2}{\sqrt {1-c^2 x^2}}dx}{4 c^2}-\frac {3 e^2 x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+d^2 x \left (a+b \text {sech}^{-1}(c x)\right )+\frac {2}{3} d e x^3 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{5} e^2 x^5 \left (a+b \text {sech}^{-1}(c x)\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{15} b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {\int \frac {60 c^2 d^2+e \left (40 d c^2+9 e\right ) x^2}{\sqrt {1-c^2 x^2}}dx}{4 c^2}-\frac {3 e^2 x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+d^2 x \left (a+b \text {sech}^{-1}(c x)\right )+\frac {2}{3} d e x^3 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{5} e^2 x^5 \left (a+b \text {sech}^{-1}(c x)\right )\) |
| \(\Big \downarrow \) 299 |
| \(\displaystyle \frac {1}{15} b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {\frac {\left (120 c^4 d^2+40 c^2 d e+9 e^2\right ) \int \frac {1}{\sqrt {1-c^2 x^2}}dx}{2 c^2}-\frac {e x \sqrt {1-c^2 x^2} \left (40 c^2 d+9 e\right )}{2 c^2}}{4 c^2}-\frac {3 e^2 x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+d^2 x \left (a+b \text {sech}^{-1}(c x)\right )+\frac {2}{3} d e x^3 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{5} e^2 x^5 \left (a+b \text {sech}^{-1}(c x)\right )\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle d^2 x \left (a+b \text {sech}^{-1}(c x)\right )+\frac {2}{3} d e x^3 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{5} e^2 x^5 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{15} b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {\frac {\arcsin (c x) \left (120 c^4 d^2+40 c^2 d e+9 e^2\right )}{2 c^3}-\frac {e x \sqrt {1-c^2 x^2} \left (40 c^2 d+9 e\right )}{2 c^2}}{4 c^2}-\frac {3 e^2 x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )\) |
d^2*x*(a + b*ArcSech[c*x]) + (2*d*e*x^3*(a + b*ArcSech[c*x]))/3 + (e^2*x^5 *(a + b*ArcSech[c*x]))/5 + (b*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*((-3*e^2* x^3*Sqrt[1 - c^2*x^2])/(4*c^2) + (-1/2*(e*(40*c^2*d + 9*e)*x*Sqrt[1 - c^2* x^2])/c^2 + ((120*c^4*d^2 + 40*c^2*d*e + 9*e^2)*ArcSin[c*x])/(2*c^3))/(4*c ^2)))/15
3.2.1.3.1 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[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[d*x *((a + b*x^2)^(p + 1)/(b*(2*p + 3))), x] - Simp[(a*d - b*c*(2*p + 3))/(b*(2 *p + 3)) Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && NeQ[2*p + 3, 0]
Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Simp[c^p*x^(4*p - 1)*((d + e*x^2)^(q + 1)/(e*(4*p + 2*q + 1))) , x] + Simp[1/(e*(4*p + 2*q + 1)) Int[(d + e*x^2)^q*ExpandToSum[e*(4*p + 2*q + 1)*(a + b*x^2 + c*x^4)^p - d*c^p*(4*p - 1)*x^(4*p - 2) - e*c^p*(4*p + 2*q + 1)*x^(4*p), x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && !LtQ[q, -1]
Int[((a_.) + ArcSech[(c_.)*(x_)]*(b_.))*((d_.) + (e_.)*(x_)^2)^(p_.), x_Sym bol] :> With[{u = IntHide[(d + e*x^2)^p, x]}, Simp[(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]] /; FreeQ[{a, b, c, d, e}, x] && ( IGtQ[p, 0] || ILtQ[p + 1/2, 0])
Time = 0.62 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.02
| method | result | size |
| parts | \(a \left (\frac {1}{5} e^{2} x^{5}+\frac {2}{3} d e \,x^{3}+x \,d^{2}\right )+\frac {b \left (\frac {c \,\operatorname {arcsech}\left (c x \right ) e^{2} x^{5}}{5}+\frac {2 c \,\operatorname {arcsech}\left (c x \right ) d e \,x^{3}}{3}+\operatorname {arcsech}\left (c x \right ) x c \,d^{2}+\frac {\sqrt {-\frac {c x -1}{c x}}\, x \sqrt {\frac {c x +1}{c x}}\, \left (120 d^{2} c^{4} \arcsin \left (c x \right )-40 d \,c^{3} e x \sqrt {-c^{2} x^{2}+1}-6 e^{2} c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}+40 d \,c^{2} e \arcsin \left (c x \right )-9 e^{2} c x \sqrt {-c^{2} x^{2}+1}+9 e^{2} \arcsin \left (c x \right )\right )}{120 c^{3} \sqrt {-c^{2} x^{2}+1}}\right )}{c}\) | \(208\) |
| derivativedivides | \(\frac {\frac {a \left (d^{2} c^{5} x +\frac {2}{3} d \,c^{5} e \,x^{3}+\frac {1}{5} e^{2} c^{5} x^{5}\right )}{c^{4}}+\frac {b \left (\operatorname {arcsech}\left (c x \right ) d^{2} c^{5} x +\frac {2 \,\operatorname {arcsech}\left (c x \right ) d \,c^{5} e \,x^{3}}{3}+\frac {\operatorname {arcsech}\left (c x \right ) e^{2} c^{5} x^{5}}{5}+\frac {\sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}\, \left (120 d^{2} c^{4} \arcsin \left (c x \right )-40 d \,c^{3} e x \sqrt {-c^{2} x^{2}+1}-6 e^{2} c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}+40 d \,c^{2} e \arcsin \left (c x \right )-9 e^{2} c x \sqrt {-c^{2} x^{2}+1}+9 e^{2} \arcsin \left (c x \right )\right )}{120 \sqrt {-c^{2} x^{2}+1}}\right )}{c^{4}}}{c}\) | \(228\) |
| default | \(\frac {\frac {a \left (d^{2} c^{5} x +\frac {2}{3} d \,c^{5} e \,x^{3}+\frac {1}{5} e^{2} c^{5} x^{5}\right )}{c^{4}}+\frac {b \left (\operatorname {arcsech}\left (c x \right ) d^{2} c^{5} x +\frac {2 \,\operatorname {arcsech}\left (c x \right ) d \,c^{5} e \,x^{3}}{3}+\frac {\operatorname {arcsech}\left (c x \right ) e^{2} c^{5} x^{5}}{5}+\frac {\sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}\, \left (120 d^{2} c^{4} \arcsin \left (c x \right )-40 d \,c^{3} e x \sqrt {-c^{2} x^{2}+1}-6 e^{2} c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}+40 d \,c^{2} e \arcsin \left (c x \right )-9 e^{2} c x \sqrt {-c^{2} x^{2}+1}+9 e^{2} \arcsin \left (c x \right )\right )}{120 \sqrt {-c^{2} x^{2}+1}}\right )}{c^{4}}}{c}\) | \(228\) |
a*(1/5*e^2*x^5+2/3*d*e*x^3+x*d^2)+b/c*(1/5*c*arcsech(c*x)*e^2*x^5+2/3*c*ar csech(c*x)*d*e*x^3+arcsech(c*x)*x*c*d^2+1/120/c^3*(-(c*x-1)/c/x)^(1/2)*x*( (c*x+1)/c/x)^(1/2)*(120*d^2*c^4*arcsin(c*x)-40*d*c^3*e*x*(-c^2*x^2+1)^(1/2 )-6*e^2*c^3*x^3*(-c^2*x^2+1)^(1/2)+40*d*c^2*e*arcsin(c*x)-9*e^2*c*x*(-c^2* x^2+1)^(1/2)+9*e^2*arcsin(c*x))/(-c^2*x^2+1)^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 305 vs. \(2 (130) = 260\).
Time = 0.37 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.50 \[ \int \left (d+e x^2\right )^2 \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\frac {24 \, a c^{5} e^{2} x^{5} + 80 \, a c^{5} d e x^{3} + 120 \, a c^{5} d^{2} x - 2 \, {\left (120 \, b c^{4} d^{2} + 40 \, b c^{2} d e + 9 \, b e^{2}\right )} \arctan \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - 1}{c x}\right ) - 8 \, {\left (15 \, b c^{5} d^{2} + 10 \, b c^{5} d e + 3 \, b c^{5} e^{2}\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - 1}{x}\right ) + 8 \, {\left (3 \, b c^{5} e^{2} x^{5} + 10 \, b c^{5} d e x^{3} + 15 \, b c^{5} d^{2} x - 15 \, b c^{5} d^{2} - 10 \, b c^{5} d e - 3 \, b c^{5} e^{2}\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) - {\left (6 \, b c^{4} e^{2} x^{4} + {\left (40 \, b c^{4} d e + 9 \, b c^{2} e^{2}\right )} x^{2}\right )} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}}}{120 \, c^{5}} \]
1/120*(24*a*c^5*e^2*x^5 + 80*a*c^5*d*e*x^3 + 120*a*c^5*d^2*x - 2*(120*b*c^ 4*d^2 + 40*b*c^2*d*e + 9*b*e^2)*arctan((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) - 1)/(c*x)) - 8*(15*b*c^5*d^2 + 10*b*c^5*d*e + 3*b*c^5*e^2)*log((c*x*sqrt (-(c^2*x^2 - 1)/(c^2*x^2)) - 1)/x) + 8*(3*b*c^5*e^2*x^5 + 10*b*c^5*d*e*x^3 + 15*b*c^5*d^2*x - 15*b*c^5*d^2 - 10*b*c^5*d*e - 3*b*c^5*e^2)*log((c*x*sq rt(-(c^2*x^2 - 1)/(c^2*x^2)) + 1)/(c*x)) - (6*b*c^4*e^2*x^4 + (40*b*c^4*d* e + 9*b*c^2*e^2)*x^2)*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)))/c^5
\[ \int \left (d+e x^2\right )^2 \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\int \left (a + b \operatorname {asech}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{2}\, dx \]
Time = 0.29 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.10 \[ \int \left (d+e x^2\right )^2 \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\frac {1}{5} \, a e^{2} x^{5} + \frac {2}{3} \, a d e x^{3} + \frac {1}{3} \, {\left (2 \, x^{3} \operatorname {arsech}\left (c x\right ) - \frac {\frac {\sqrt {\frac {1}{c^{2} x^{2}} - 1}}{c^{2} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} + c^{2}} + \frac {\arctan \left (\sqrt {\frac {1}{c^{2} x^{2}} - 1}\right )}{c^{2}}}{c}\right )} b d e + \frac {1}{40} \, {\left (8 \, x^{5} \operatorname {arsech}\left (c x\right ) - \frac {\frac {3 \, {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {3}{2}} + 5 \, \sqrt {\frac {1}{c^{2} x^{2}} - 1}}{c^{4} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2} + 2 \, c^{4} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} + c^{4}} + \frac {3 \, \arctan \left (\sqrt {\frac {1}{c^{2} x^{2}} - 1}\right )}{c^{4}}}{c}\right )} b e^{2} + a d^{2} x + \frac {{\left (c x \operatorname {arsech}\left (c x\right ) - \arctan \left (\sqrt {\frac {1}{c^{2} x^{2}} - 1}\right )\right )} b d^{2}}{c} \]
1/5*a*e^2*x^5 + 2/3*a*d*e*x^3 + 1/3*(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*e + 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^2 + a*d^2*x + (c*x*arcsec h(c*x) - arctan(sqrt(1/(c^2*x^2) - 1)))*b*d^2/c
\[ \int \left (d+e x^2\right )^2 \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\int { {\left (e x^{2} + d\right )}^{2} {\left (b \operatorname {arsech}\left (c x\right ) + a\right )} \,d x } \]
Timed out. \[ \int \left (d+e x^2\right )^2 \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\int {\left (e\,x^2+d\right )}^2\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right ) \,d x \]