\(\int x^5 (d+e x^2) (a+b \text {sech}^{-1}(c x)) \, dx\) [94]

Optimal result

Integrand size = 19, antiderivative size = 232 \[ \int x^5 \left (d+e x^2\right ) \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=-\frac {b \left (4 c^2 d+3 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{24 c^8}+\frac {b \left (8 c^2 d+9 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \left (1-c^2 x^2\right )^{3/2}}{72 c^8}-\frac {b \left (4 c^2 d+9 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \left (1-c^2 x^2\right )^{5/2}}{120 c^8}+\frac {b e \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \left (1-c^2 x^2\right )^{7/2}}{56 c^8}+\frac {1}{6} d x^6 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{8} e x^8 \left (a+b \text {sech}^{-1}(c x)\right ) \] Output:

-1/24*b*(4*c^2*d+3*e)*(1/(c*x+1))^(1/2)*(c*x+1)^(1/2)*(-c^2*x^2+1)^(1/2)/c 
^8+1/72*b*(8*c^2*d+9*e)*(1/(c*x+1))^(1/2)*(c*x+1)^(1/2)*(-c^2*x^2+1)^(3/2) 
/c^8-1/120*b*(4*c^2*d+9*e)*(1/(c*x+1))^(1/2)*(c*x+1)^(1/2)*(-c^2*x^2+1)^(5 
/2)/c^8+1/56*b*e*(1/(c*x+1))^(1/2)*(c*x+1)^(1/2)*(-c^2*x^2+1)^(7/2)/c^8+1/ 
6*d*x^6*(a+b*arcsech(c*x))+1/8*e*x^8*(a+b*arcsech(c*x))
 

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.54 \[ \int x^5 \left (d+e x^2\right ) \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\frac {1}{24} a x^6 \left (4 d+3 e x^2\right )-\frac {b \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (144 e+8 c^2 \left (28 d+9 e x^2\right )+2 c^4 \left (56 d x^2+27 e x^4\right )+c^6 \left (84 d x^4+45 e x^6\right )\right )}{2520 c^8}+\frac {1}{24} b x^6 \left (4 d+3 e x^2\right ) \text {sech}^{-1}(c x) \] Input:

Integrate[x^5*(d + e*x^2)*(a + b*ArcSech[c*x]),x]
 

Output:

(a*x^6*(4*d + 3*e*x^2))/24 - (b*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*(144*e 
 + 8*c^2*(28*d + 9*e*x^2) + 2*c^4*(56*d*x^2 + 27*e*x^4) + c^6*(84*d*x^4 + 
45*e*x^6)))/(2520*c^8) + (b*x^6*(4*d + 3*e*x^2)*ArcSech[c*x])/24
 

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.74, 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.

\(\displaystyle \int x^5 \left (d+e x^2\right ) \left (a+b \text {sech}^{-1}(c x)\right ) \, dx\)

\(\Big \downarrow \) 6855

\(\displaystyle b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \int \frac {x^5 \left (3 e x^2+4 d\right )}{24 \sqrt {1-c^2 x^2}}dx+\frac {1}{6} d x^6 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{8} e x^8 \left (a+b \text {sech}^{-1}(c x)\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{24} b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \int \frac {x^5 \left (3 e x^2+4 d\right )}{\sqrt {1-c^2 x^2}}dx+\frac {1}{6} d x^6 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{8} e x^8 \left (a+b \text {sech}^{-1}(c x)\right )\)

\(\Big \downarrow \) 354

\(\displaystyle \frac {1}{48} b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \int \frac {x^4 \left (3 e x^2+4 d\right )}{\sqrt {1-c^2 x^2}}dx^2+\frac {1}{6} d x^6 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{8} e x^8 \left (a+b \text {sech}^{-1}(c x)\right )\)

\(\Big \downarrow \) 86

\(\displaystyle \frac {1}{48} b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \int \left (-\frac {3 e \left (1-c^2 x^2\right )^{5/2}}{c^6}+\frac {\left (4 d c^2+9 e\right ) \left (1-c^2 x^2\right )^{3/2}}{c^6}+\frac {\left (-8 d c^2-9 e\right ) \sqrt {1-c^2 x^2}}{c^6}+\frac {4 d c^2+3 e}{c^6 \sqrt {1-c^2 x^2}}\right )dx^2+\frac {1}{6} d x^6 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{8} e x^8 \left (a+b \text {sech}^{-1}(c x)\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{6} d x^6 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{8} e x^8 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{48} b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (-\frac {2 \left (1-c^2 x^2\right )^{5/2} \left (4 c^2 d+9 e\right )}{5 c^8}+\frac {2 \left (1-c^2 x^2\right )^{3/2} \left (8 c^2 d+9 e\right )}{3 c^8}-\frac {2 \sqrt {1-c^2 x^2} \left (4 c^2 d+3 e\right )}{c^8}+\frac {6 e \left (1-c^2 x^2\right )^{7/2}}{7 c^8}\right )\)

Input:

Int[x^5*(d + e*x^2)*(a + b*ArcSech[c*x]),x]
 

Output:

(b*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*((-2*(4*c^2*d + 3*e)*Sqrt[1 - c^2*x^ 
2])/c^8 + (2*(8*c^2*d + 9*e)*(1 - c^2*x^2)^(3/2))/(3*c^8) - (2*(4*c^2*d + 
9*e)*(1 - c^2*x^2)^(5/2))/(5*c^8) + (6*e*(1 - c^2*x^2)^(7/2))/(7*c^8)))/48 
 + (d*x^6*(a + b*ArcSech[c*x]))/6 + (e*x^8*(a + b*ArcSech[c*x]))/8
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 86

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]))))
 

rule 354

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]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 6855

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]))
 

Maple [A] (verified)

Time = 0.50 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.60

method	result	size
parts	\(a \left (\frac {1}{8} e \,x^{8}+\frac {1}{6} d \,x^{6}\right )+\frac {b \left (\frac {c^{6} \operatorname {arcsech}\left (c x \right ) e \,x^{8}}{8}+\frac {\operatorname {arcsech}\left (c x \right ) d \,c^{6} x^{6}}{6}-\frac {\sqrt {-\frac {c x -1}{c x}}\, x \sqrt {\frac {c x +1}{c x}}\, \left (45 c^{6} e \,x^{6}+84 c^{6} d \,x^{4}+54 c^{4} e \,x^{4}+112 c^{4} d \,x^{2}+72 e \,c^{2} x^{2}+224 c^{2} d +144 e \right )}{2520 c}\right )}{c^{6}}\)	\(139\)
derivativedivides	\(\frac {\frac {a \left (\frac {1}{6} c^{8} d \,x^{6}+\frac {1}{8} e \,c^{8} x^{8}\right )}{c^{2}}+\frac {b \left (\frac {\operatorname {arcsech}\left (c x \right ) d \,c^{8} x^{6}}{6}+\frac {\operatorname {arcsech}\left (c x \right ) e \,c^{8} x^{8}}{8}-\frac {\sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}\, \left (45 c^{6} e \,x^{6}+84 c^{6} d \,x^{4}+54 c^{4} e \,x^{4}+112 c^{4} d \,x^{2}+72 e \,c^{2} x^{2}+224 c^{2} d +144 e \right )}{2520}\right )}{c^{2}}}{c^{6}}\)	\(150\)
default	\(\frac {\frac {a \left (\frac {1}{6} c^{8} d \,x^{6}+\frac {1}{8} e \,c^{8} x^{8}\right )}{c^{2}}+\frac {b \left (\frac {\operatorname {arcsech}\left (c x \right ) d \,c^{8} x^{6}}{6}+\frac {\operatorname {arcsech}\left (c x \right ) e \,c^{8} x^{8}}{8}-\frac {\sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}\, \left (45 c^{6} e \,x^{6}+84 c^{6} d \,x^{4}+54 c^{4} e \,x^{4}+112 c^{4} d \,x^{2}+72 e \,c^{2} x^{2}+224 c^{2} d +144 e \right )}{2520}\right )}{c^{2}}}{c^{6}}\)	\(150\)

Input:

int(x^5*(e*x^2+d)*(a+b*arcsech(c*x)),x,method=_RETURNVERBOSE)
 

Output:

a*(1/8*e*x^8+1/6*d*x^6)+b/c^6*(1/8*c^6*arcsech(c*x)*e*x^8+1/6*arcsech(c*x) 
*d*c^6*x^6-1/2520/c*(-(c*x-1)/c/x)^(1/2)*x*((c*x+1)/c/x)^(1/2)*(45*c^6*e*x 
^6+84*c^6*d*x^4+54*c^4*e*x^4+112*c^4*d*x^2+72*c^2*e*x^2+224*c^2*d+144*e))
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.72 \[ \int x^5 \left (d+e x^2\right ) \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\frac {315 \, a c^{7} e x^{8} + 420 \, a c^{7} d x^{6} + 105 \, {\left (3 \, b c^{7} e x^{8} + 4 \, b c^{7} d x^{6}\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) - {\left (45 \, b c^{6} e x^{7} + 6 \, {\left (14 \, b c^{6} d + 9 \, b c^{4} e\right )} x^{5} + 8 \, {\left (14 \, b c^{4} d + 9 \, b c^{2} e\right )} x^{3} + 16 \, {\left (14 \, b c^{2} d + 9 \, b e\right )} x\right )} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}}}{2520 \, c^{7}} \] Input:

integrate(x^5*(e*x^2+d)*(a+b*arcsech(c*x)),x, algorithm="fricas")
 

Output:

1/2520*(315*a*c^7*e*x^8 + 420*a*c^7*d*x^6 + 105*(3*b*c^7*e*x^8 + 4*b*c^7*d 
*x^6)*log((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) + 1)/(c*x)) - (45*b*c^6*e*x^ 
7 + 6*(14*b*c^6*d + 9*b*c^4*e)*x^5 + 8*(14*b*c^4*d + 9*b*c^2*e)*x^3 + 16*( 
14*b*c^2*d + 9*b*e)*x)*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)))/c^7
                                                                                    
                                                                                    
 

Sympy [A] (verification not implemented)

Time = 1.62 (sec) , antiderivative size = 228, normalized size of antiderivative = 0.98 \[ \int x^5 \left (d+e x^2\right ) \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\begin {cases} \frac {a d x^{6}}{6} + \frac {a e x^{8}}{8} + \frac {b d x^{6} \operatorname {asech}{\left (c x \right )}}{6} + \frac {b e x^{8} \operatorname {asech}{\left (c x \right )}}{8} - \frac {b d x^{4} \sqrt {- c^{2} x^{2} + 1}}{30 c^{2}} - \frac {b e x^{6} \sqrt {- c^{2} x^{2} + 1}}{56 c^{2}} - \frac {2 b d x^{2} \sqrt {- c^{2} x^{2} + 1}}{45 c^{4}} - \frac {3 b e x^{4} \sqrt {- c^{2} x^{2} + 1}}{140 c^{4}} - \frac {4 b d \sqrt {- c^{2} x^{2} + 1}}{45 c^{6}} - \frac {b e x^{2} \sqrt {- c^{2} x^{2} + 1}}{35 c^{6}} - \frac {2 b e \sqrt {- c^{2} x^{2} + 1}}{35 c^{8}} & \text {for}\: c \neq 0 \\\left (a + \infty b\right ) \left (\frac {d x^{6}}{6} + \frac {e x^{8}}{8}\right ) & \text {otherwise} \end {cases} \] Input:

integrate(x**5*(e*x**2+d)*(a+b*asech(c*x)),x)
 

Output:

Piecewise((a*d*x**6/6 + a*e*x**8/8 + b*d*x**6*asech(c*x)/6 + b*e*x**8*asec 
h(c*x)/8 - b*d*x**4*sqrt(-c**2*x**2 + 1)/(30*c**2) - b*e*x**6*sqrt(-c**2*x 
**2 + 1)/(56*c**2) - 2*b*d*x**2*sqrt(-c**2*x**2 + 1)/(45*c**4) - 3*b*e*x** 
4*sqrt(-c**2*x**2 + 1)/(140*c**4) - 4*b*d*sqrt(-c**2*x**2 + 1)/(45*c**6) - 
 b*e*x**2*sqrt(-c**2*x**2 + 1)/(35*c**6) - 2*b*e*sqrt(-c**2*x**2 + 1)/(35* 
c**8), Ne(c, 0)), ((a + oo*b)*(d*x**6/6 + e*x**8/8), True))
 

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.76 \[ \int x^5 \left (d+e x^2\right ) \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\frac {1}{8} \, a e x^{8} + \frac {1}{6} \, a d x^{6} + \frac {1}{90} \, {\left (15 \, x^{6} \operatorname {arsech}\left (c x\right ) - \frac {3 \, c^{4} x^{5} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {5}{2}} - 10 \, c^{2} x^{3} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {3}{2}} + 15 \, x \sqrt {\frac {1}{c^{2} x^{2}} - 1}}{c^{5}}\right )} b d + \frac {1}{280} \, {\left (35 \, x^{8} \operatorname {arsech}\left (c x\right ) + \frac {5 \, c^{6} x^{7} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {7}{2}} - 21 \, c^{4} x^{5} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {5}{2}} + 35 \, c^{2} x^{3} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {3}{2}} - 35 \, x \sqrt {\frac {1}{c^{2} x^{2}} - 1}}{c^{7}}\right )} b e \] Input:

integrate(x^5*(e*x^2+d)*(a+b*arcsech(c*x)),x, algorithm="maxima")
 

Output:

1/8*a*e*x^8 + 1/6*a*d*x^6 + 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*d + 1/280*(35*x^8*arcsech(c*x) + (5*c^6*x^7*(1/(c^2*x^2) 
- 1)^(7/2) - 21*c^4*x^5*(1/(c^2*x^2) - 1)^(5/2) + 35*c^2*x^3*(1/(c^2*x^2) 
- 1)^(3/2) - 35*x*sqrt(1/(c^2*x^2) - 1))/c^7)*b*e
 

Giac [F]

\[ \int x^5 \left (d+e x^2\right ) \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\int { {\left (e x^{2} + d\right )} {\left (b \operatorname {arsech}\left (c x\right ) + a\right )} x^{5} \,d x } \] Input:

integrate(x^5*(e*x^2+d)*(a+b*arcsech(c*x)),x, algorithm="giac")
 

Output:

integrate((e*x^2 + d)*(b*arcsech(c*x) + a)*x^5, x)
 

Mupad [F(-1)]

Timed out. \[ \int x^5 \left (d+e x^2\right ) \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\int x^5\,\left (e\,x^2+d\right )\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right ) \,d x \] Input:

int(x^5*(d + e*x^2)*(a + b*acosh(1/(c*x))),x)
 

Output:

int(x^5*(d + e*x^2)*(a + b*acosh(1/(c*x))), x)
 

Reduce [F]

\[ \int x^5 \left (d+e x^2\right ) \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\left (\int \mathit {asech} \left (c x \right ) x^{7}d x \right ) b e +\left (\int \mathit {asech} \left (c x \right ) x^{5}d x \right ) b d +\frac {a d \,x^{6}}{6}+\frac {a e \,x^{8}}{8} \] Input:

int(x^5*(e*x^2+d)*(a+b*asech(c*x)),x)
 

Output:

(24*int(asech(c*x)*x**7,x)*b*e + 24*int(asech(c*x)*x**5,x)*b*d + 4*a*d*x** 
6 + 3*a*e*x**8)/24