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

Optimal. Leaf size=232 \[ \frac{1}{6} d x^6 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{1}{8} e x^8 \left (a+b \text{sech}^{-1}(c x)\right )-\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left (1-c^2 x^2\right )^{5/2} \left (4 c^2 d+9 e\right )}{120 c^8}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left (1-c^2 x^2\right )^{3/2} \left (8 c^2 d+9 e\right )}{72 c^8}-\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left (4 c^2 d+3 e\right )}{24 c^8}+\frac{b e \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left (1-c^2 x^2\right )^{7/2}}{56 c^8} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.16383, antiderivative size = 232, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {14, 6301, 12, 446, 77} \[ \frac{1}{6} d x^6 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{1}{8} e x^8 \left (a+b \text{sech}^{-1}(c x)\right )-\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left (1-c^2 x^2\right )^{5/2} \left (4 c^2 d+9 e\right )}{120 c^8}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left (1-c^2 x^2\right )^{3/2} \left (8 c^2 d+9 e\right )}{72 c^8}-\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left (4 c^2 d+3 e\right )}{24 c^8}+\frac{b e \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left (1-c^2 x^2\right )^{7/2}}{56 c^8} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 6301

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]}, Dist[a + b*ArcSech[c*x], u, x] + Dist[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, 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]))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((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]))))

Rubi steps

\begin{align*} \int x^5 \left (d+e x^2\right ) \left (a+b \text{sech}^{-1}(c x)\right ) \, 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 )+\left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{x^5 \left (4 d+3 e x^2\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 )+\frac{1}{24} \left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{x^5 \left (4 d+3 e x^2\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 )+\frac{1}{48} \left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{x^2 (4 d+3 e x)}{\sqrt{1-c^2 x}} \, dx,x,x^2\right )\\ &=\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} \left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \left (\frac{4 c^2 d+3 e}{c^6 \sqrt{1-c^2 x}}+\frac{\left (-8 c^2 d-9 e\right ) \sqrt{1-c^2 x}}{c^6}+\frac{\left (4 c^2 d+9 e\right ) \left (1-c^2 x\right )^{3/2}}{c^6}-\frac{3 e \left (1-c^2 x\right )^{5/2}}{c^6}\right ) \, dx,x,x^2\right )\\ &=-\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 )\\ \end{align*}

Mathematica [A]  time = 0.209894, size = 126, normalized size = 0.54 \[ \frac{1}{24} a x^6 \left (4 d+3 e x^2\right )-\frac{b \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left (c^6 \left (84 d x^4+45 e x^6\right )+2 c^4 \left (56 d x^2+27 e x^4\right )+8 c^2 \left (28 d+9 e x^2\right )+144 e\right )}{2520 c^8}+\frac{1}{24} b x^6 \text{sech}^{-1}(c x) \left (4 d+3 e x^2\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(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*(5
6*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

________________________________________________________________________________________

Maple [A]  time = 0.185, size = 150, normalized size = 0.7 \begin{align*}{\frac{1}{{c}^{6}} \left ({\frac{a}{{c}^{2}} \left ({\frac{e{c}^{8}{x}^{8}}{8}}+{\frac{{c}^{8}{x}^{6}d}{6}} \right ) }+{\frac{b}{{c}^{2}} \left ({\frac{{\rm arcsech} \left (cx\right )e{c}^{8}{x}^{8}}{8}}+{\frac{{\rm arcsech} \left (cx\right ){c}^{8}{x}^{6}d}{6}}-{\frac{cx \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\,{c}^{2}{x}^{2}e+224\,{c}^{2}d+144\,e \right ) }{2520}\sqrt{-{\frac{cx-1}{cx}}}\sqrt{{\frac{cx+1}{cx}}}} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c^6*(a/c^2*(1/8*e*c^8*x^8+1/6*c^8*x^6*d)+b/c^2*(1/8*arcsech(c*x)*e*c^8*x^8+1/6*arcsech(c*x)*c^8*x^6*d-1/2520
*(-(c*x-1)/c/x)^(1/2)*c*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)))

________________________________________________________________________________________

Maxima [A]  time = 1.00133, size = 239, normalized size = 1.03 \begin{align*} \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 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 2.05388, size = 381, normalized size = 1.64 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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]  time = 69.7732, size = 228, normalized size = 0.98 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a*d*x**6/6 + a*e*x**8/8 + b*d*x**6*asech(c*x)/6 + b*e*x**8*asech(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))

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (e x^{2} + d\right )}{\left (b \operatorname{arsech}\left (c x\right ) + a\right )} x^{5}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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