Integrand size = 19, antiderivative size = 180 \[ \int x^3 \left (d+e x^2\right ) \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=-\frac {b \left (3 c^2 d+2 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{12 c^6}+\frac {b \left (3 c^2 d+4 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \left (1-c^2 x^2\right )^{3/2}}{36 c^6}-\frac {b e \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \left (1-c^2 x^2\right )^{5/2}}{30 c^6}+\frac {1}{4} d x^4 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{6} e x^6 \left (a+b \text {sech}^{-1}(c x)\right ) \]
1/4*d*x^4*(a+b*arcsech(c*x))+1/6*e*x^6*(a+b*arcsech(c*x))+1/36*b*(3*c^2*d+ 4*e)*(-c^2*x^2+1)^(3/2)*(1/(c*x+1))^(1/2)*(c*x+1)^(1/2)/c^6-1/30*b*e*(-c^2 *x^2+1)^(5/2)*(1/(c*x+1))^(1/2)*(c*x+1)^(1/2)/c^6-1/12*b*(3*c^2*d+2*e)*(1/ (c*x+1))^(1/2)*(c*x+1)^(1/2)*(-c^2*x^2+1)^(1/2)/c^6
Time = 0.22 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.59 \[ \int x^3 \left (d+e x^2\right ) \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\frac {1}{180} \left (15 a x^4 \left (3 d+2 e x^2\right )-\frac {b \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (16 e+c^2 \left (30 d+8 e x^2\right )+3 c^4 \left (5 d x^2+2 e x^4\right )\right )}{c^6}+15 b x^4 \left (3 d+2 e x^2\right ) \text {sech}^{-1}(c x)\right ) \]
(15*a*x^4*(3*d + 2*e*x^2) - (b*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*(16*e + c^2*(30*d + 8*e*x^2) + 3*c^4*(5*d*x^2 + 2*e*x^4)))/c^6 + 15*b*x^4*(3*d + 2*e*x^2)*ArcSech[c*x])/180
Time = 0.35 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.78, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {6855, 27, 354, 86, 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.
| \(\displaystyle \int x^3 \left (d+e x^2\right ) \left (a+b \text {sech}^{-1}(c x)\right ) \, dx\) |
| \(\Big \downarrow \) 6855 |
| \(\displaystyle b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \int \frac {x^3 \left (2 e x^2+3 d\right )}{12 \sqrt {1-c^2 x^2}}dx+\frac {1}{4} d x^4 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{6} e x^6 \left (a+b \text {sech}^{-1}(c x)\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{12} b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \int \frac {x^3 \left (2 e x^2+3 d\right )}{\sqrt {1-c^2 x^2}}dx+\frac {1}{4} d x^4 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{6} e x^6 \left (a+b \text {sech}^{-1}(c x)\right )\) |
| \(\Big \downarrow \) 354 |
| \(\displaystyle \frac {1}{24} b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \int \frac {x^2 \left (2 e x^2+3 d\right )}{\sqrt {1-c^2 x^2}}dx^2+\frac {1}{4} d x^4 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{6} e x^6 \left (a+b \text {sech}^{-1}(c x)\right )\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle \frac {1}{24} b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \int \left (\frac {2 e \left (1-c^2 x^2\right )^{3/2}}{c^4}+\frac {\left (-3 d c^2-4 e\right ) \sqrt {1-c^2 x^2}}{c^4}+\frac {3 d c^2+2 e}{c^4 \sqrt {1-c^2 x^2}}\right )dx^2+\frac {1}{4} d x^4 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{6} e x^6 \left (a+b \text {sech}^{-1}(c x)\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{4} d x^4 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{6} e x^6 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{24} b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {2 \left (1-c^2 x^2\right )^{3/2} \left (3 c^2 d+4 e\right )}{3 c^6}-\frac {2 \sqrt {1-c^2 x^2} \left (3 c^2 d+2 e\right )}{c^6}-\frac {4 e \left (1-c^2 x^2\right )^{5/2}}{5 c^6}\right )\) |
(b*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*((-2*(3*c^2*d + 2*e)*Sqrt[1 - c^2*x^ 2])/c^6 + (2*(3*c^2*d + 4*e)*(1 - c^2*x^2)^(3/2))/(3*c^6) - (4*e*(1 - c^2* x^2)^(5/2))/(5*c^6)))/24 + (d*x^4*(a + b*ArcSech[c*x]))/4 + (e*x^6*(a + b* ArcSech[c*x]))/6
3.1.96.3.1 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[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ .), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 ] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S ymbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x , x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ [(m - 1)/2]
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 = 0.60 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.67
| method | result | size |
| parts | \(a \left (\frac {1}{6} e \,x^{6}+\frac {1}{4} d \,x^{4}\right )+\frac {b \left (\frac {c^{4} \operatorname {arcsech}\left (c x \right ) e \,x^{6}}{6}+\frac {\operatorname {arcsech}\left (c x \right ) x^{4} c^{4} d}{4}-\frac {\sqrt {-\frac {c x -1}{c x}}\, x \sqrt {\frac {c x +1}{c x}}\, \left (6 c^{4} e \,x^{4}+15 c^{4} d \,x^{2}+8 e \,c^{2} x^{2}+30 c^{2} d +16 e \right )}{180 c}\right )}{c^{4}}\) | \(121\) |
| derivativedivides | \(\frac {-\frac {a \left (\frac {c^{2} d \left (e \,c^{2} x^{2}+c^{2} d \right )^{2}}{2}-\frac {\left (e \,c^{2} x^{2}+c^{2} d \right )^{3}}{3}\right )}{2 c^{2} e^{2}}+\frac {b \left (-\frac {\operatorname {arcsech}\left (c x \right ) c^{6} d^{3}}{12 e^{2}}+\frac {\operatorname {arcsech}\left (c x \right ) c^{6} d \,x^{4}}{4}+\frac {e \,\operatorname {arcsech}\left (c x \right ) c^{6} x^{6}}{6}+\frac {\sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}\, \left (15 c^{6} d^{3} \operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )-15 c^{4} d \,e^{2} \sqrt {-c^{2} x^{2}+1}\, x^{2}-6 e^{3} \sqrt {-c^{2} x^{2}+1}\, c^{4} x^{4}-30 c^{2} d \,e^{2} \sqrt {-c^{2} x^{2}+1}-8 e^{3} c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}-16 e^{3} \sqrt {-c^{2} x^{2}+1}\right )}{180 e^{2} \sqrt {-c^{2} x^{2}+1}}\right )}{c^{2}}}{c^{4}}\) | \(281\) |
| default | \(\frac {-\frac {a \left (\frac {c^{2} d \left (e \,c^{2} x^{2}+c^{2} d \right )^{2}}{2}-\frac {\left (e \,c^{2} x^{2}+c^{2} d \right )^{3}}{3}\right )}{2 c^{2} e^{2}}+\frac {b \left (-\frac {\operatorname {arcsech}\left (c x \right ) c^{6} d^{3}}{12 e^{2}}+\frac {\operatorname {arcsech}\left (c x \right ) c^{6} d \,x^{4}}{4}+\frac {e \,\operatorname {arcsech}\left (c x \right ) c^{6} x^{6}}{6}+\frac {\sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}\, \left (15 c^{6} d^{3} \operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )-15 c^{4} d \,e^{2} \sqrt {-c^{2} x^{2}+1}\, x^{2}-6 e^{3} \sqrt {-c^{2} x^{2}+1}\, c^{4} x^{4}-30 c^{2} d \,e^{2} \sqrt {-c^{2} x^{2}+1}-8 e^{3} c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}-16 e^{3} \sqrt {-c^{2} x^{2}+1}\right )}{180 e^{2} \sqrt {-c^{2} x^{2}+1}}\right )}{c^{2}}}{c^{4}}\) | \(281\) |
a*(1/6*e*x^6+1/4*d*x^4)+b/c^4*(1/6*c^4*arcsech(c*x)*e*x^6+1/4*arcsech(c*x) *x^4*c^4*d-1/180/c*(-(c*x-1)/c/x)^(1/2)*x*((c*x+1)/c/x)^(1/2)*(6*c^4*e*x^4 +15*c^4*d*x^2+8*c^2*e*x^2+30*c^2*d+16*e))
Time = 0.29 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.82 \[ \int x^3 \left (d+e x^2\right ) \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\frac {30 \, a c^{5} e x^{6} + 45 \, a c^{5} d x^{4} + 15 \, {\left (2 \, b c^{5} e x^{6} + 3 \, b c^{5} d x^{4}\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) - {\left (6 \, b c^{4} e x^{5} + {\left (15 \, b c^{4} d + 8 \, b c^{2} e\right )} x^{3} + 2 \, {\left (15 \, b c^{2} d + 8 \, b e\right )} x\right )} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}}}{180 \, c^{5}} \]
1/180*(30*a*c^5*e*x^6 + 45*a*c^5*d*x^4 + 15*(2*b*c^5*e*x^6 + 3*b*c^5*d*x^4 )*log((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) + 1)/(c*x)) - (6*b*c^4*e*x^5 + ( 15*b*c^4*d + 8*b*c^2*e)*x^3 + 2*(15*b*c^2*d + 8*b*e)*x)*sqrt(-(c^2*x^2 - 1 )/(c^2*x^2)))/c^5
Time = 0.71 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.98 \[ \int x^3 \left (d+e x^2\right ) \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\begin {cases} \frac {a d x^{4}}{4} + \frac {a e x^{6}}{6} + \frac {b d x^{4} \operatorname {asech}{\left (c x \right )}}{4} + \frac {b e x^{6} \operatorname {asech}{\left (c x \right )}}{6} - \frac {b d x^{2} \sqrt {- c^{2} x^{2} + 1}}{12 c^{2}} - \frac {b e x^{4} \sqrt {- c^{2} x^{2} + 1}}{30 c^{2}} - \frac {b d \sqrt {- c^{2} x^{2} + 1}}{6 c^{4}} - \frac {2 b e x^{2} \sqrt {- c^{2} x^{2} + 1}}{45 c^{4}} - \frac {4 b e \sqrt {- c^{2} x^{2} + 1}}{45 c^{6}} & \text {for}\: c \neq 0 \\\left (a + \infty b\right ) \left (\frac {d x^{4}}{4} + \frac {e x^{6}}{6}\right ) & \text {otherwise} \end {cases} \]
Piecewise((a*d*x**4/4 + a*e*x**6/6 + b*d*x**4*asech(c*x)/4 + b*e*x**6*asec h(c*x)/6 - b*d*x**2*sqrt(-c**2*x**2 + 1)/(12*c**2) - b*e*x**4*sqrt(-c**2*x **2 + 1)/(30*c**2) - b*d*sqrt(-c**2*x**2 + 1)/(6*c**4) - 2*b*e*x**2*sqrt(- c**2*x**2 + 1)/(45*c**4) - 4*b*e*sqrt(-c**2*x**2 + 1)/(45*c**6), Ne(c, 0)) , ((a + oo*b)*(d*x**4/4 + e*x**6/6), True))
Time = 0.20 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.77 \[ \int x^3 \left (d+e x^2\right ) \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\frac {1}{6} \, a e x^{6} + \frac {1}{4} \, a d x^{4} + \frac {1}{12} \, {\left (3 \, x^{4} \operatorname {arsech}\left (c x\right ) + \frac {c^{2} x^{3} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {3}{2}} - 3 \, x \sqrt {\frac {1}{c^{2} x^{2}} - 1}}{c^{3}}\right )} b d + \frac {1}{90} \, {\left (15 \, x^{6} \operatorname {arsech}\left (c x\right ) - \frac {3 \, c^{4} x^{5} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {5}{2}} - 10 \, c^{2} x^{3} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {3}{2}} + 15 \, x \sqrt {\frac {1}{c^{2} x^{2}} - 1}}{c^{5}}\right )} b e \]
1/6*a*e*x^6 + 1/4*a*d*x^4 + 1/12*(3*x^4*arcsech(c*x) + (c^2*x^3*(1/(c^2*x^ 2) - 1)^(3/2) - 3*x*sqrt(1/(c^2*x^2) - 1))/c^3)*b*d + 1/90*(15*x^6*arcsech (c*x) - (3*c^4*x^5*(1/(c^2*x^2) - 1)^(5/2) - 10*c^2*x^3*(1/(c^2*x^2) - 1)^ (3/2) + 15*x*sqrt(1/(c^2*x^2) - 1))/c^5)*b*e
\[ \int x^3 \left (d+e x^2\right ) \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\int { {\left (e x^{2} + d\right )} {\left (b \operatorname {arsech}\left (c x\right ) + a\right )} x^{3} \,d x } \]
Timed out. \[ \int x^3 \left (d+e x^2\right ) \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\int x^3\,\left (e\,x^2+d\right )\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right ) \,d x \]