3.186 \(\int \frac{x^{11} (a+b \text{sech}^{-1}(c x))}{\sqrt{1-c^4 x^4}} \, dx\)

Optimal. Leaf size=473 \[ -\frac{\left (1-c^4 x^4\right )^{5/2} \left (a+b \text{sech}^{-1}(c x)\right )}{10 c^{12}}+\frac{\left (1-c^4 x^4\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{3 c^{12}}-\frac{\sqrt{1-c^4 x^4} \left (a+b \text{sech}^{-1}(c x)\right )}{2 c^{12}}-\frac{b \sqrt{1-c^2 x^2} \left (c^2 x^2+1\right )^{9/2}}{90 c^{13} x \sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}}+\frac{3 b \sqrt{1-c^2 x^2} \left (c^2 x^2+1\right )^{7/2}}{70 c^{13} x \sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}}-\frac{13 b \sqrt{1-c^2 x^2} \left (c^2 x^2+1\right )^{5/2}}{150 c^{13} x \sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}}+\frac{7 b \sqrt{1-c^2 x^2} \left (c^2 x^2+1\right )^{3/2}}{90 c^{13} x \sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}}-\frac{4 b \sqrt{1-c^2 x^2} \sqrt{c^2 x^2+1}}{15 c^{13} x \sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}}+\frac{4 b \sqrt{1-c^2 x^2} \tanh ^{-1}\left (\sqrt{c^2 x^2+1}\right )}{15 c^{13} x \sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}} \]

[Out]

(-4*b*Sqrt[1 - c^2*x^2]*Sqrt[1 + c^2*x^2])/(15*c^13*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]*x) + (7*b*Sqrt[1 - c^
2*x^2]*(1 + c^2*x^2)^(3/2))/(90*c^13*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]*x) - (13*b*Sqrt[1 - c^2*x^2]*(1 + c^
2*x^2)^(5/2))/(150*c^13*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]*x) + (3*b*Sqrt[1 - c^2*x^2]*(1 + c^2*x^2)^(7/2))/
(70*c^13*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]*x) - (b*Sqrt[1 - c^2*x^2]*(1 + c^2*x^2)^(9/2))/(90*c^13*Sqrt[-1
+ 1/(c*x)]*Sqrt[1 + 1/(c*x)]*x) - (Sqrt[1 - c^4*x^4]*(a + b*ArcSech[c*x]))/(2*c^12) + ((1 - c^4*x^4)^(3/2)*(a
+ b*ArcSech[c*x]))/(3*c^12) - ((1 - c^4*x^4)^(5/2)*(a + b*ArcSech[c*x]))/(10*c^12) + (4*b*Sqrt[1 - c^2*x^2]*Ar
cTanh[Sqrt[1 + c^2*x^2]])/(15*c^13*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]*x)

________________________________________________________________________________________

Rubi [A]  time = 1.57949, antiderivative size = 473, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 10, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385, Rules used = {266, 43, 6309, 12, 6742, 848, 50, 63, 208, 783} \[ -\frac{\left (1-c^4 x^4\right )^{5/2} \left (a+b \text{sech}^{-1}(c x)\right )}{10 c^{12}}+\frac{\left (1-c^4 x^4\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{3 c^{12}}-\frac{\sqrt{1-c^4 x^4} \left (a+b \text{sech}^{-1}(c x)\right )}{2 c^{12}}-\frac{b \sqrt{1-c^2 x^2} \left (c^2 x^2+1\right )^{9/2}}{90 c^{13} x \sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}}+\frac{3 b \sqrt{1-c^2 x^2} \left (c^2 x^2+1\right )^{7/2}}{70 c^{13} x \sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}}-\frac{13 b \sqrt{1-c^2 x^2} \left (c^2 x^2+1\right )^{5/2}}{150 c^{13} x \sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}}+\frac{7 b \sqrt{1-c^2 x^2} \left (c^2 x^2+1\right )^{3/2}}{90 c^{13} x \sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}}-\frac{4 b \sqrt{1-c^2 x^2} \sqrt{c^2 x^2+1}}{15 c^{13} x \sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}}+\frac{4 b \sqrt{1-c^2 x^2} \tanh ^{-1}\left (\sqrt{c^2 x^2+1}\right )}{15 c^{13} x \sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}} \]

Antiderivative was successfully verified.

[In]

Int[(x^11*(a + b*ArcSech[c*x]))/Sqrt[1 - c^4*x^4],x]

[Out]

(-4*b*Sqrt[1 - c^2*x^2]*Sqrt[1 + c^2*x^2])/(15*c^13*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]*x) + (7*b*Sqrt[1 - c^
2*x^2]*(1 + c^2*x^2)^(3/2))/(90*c^13*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]*x) - (13*b*Sqrt[1 - c^2*x^2]*(1 + c^
2*x^2)^(5/2))/(150*c^13*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]*x) + (3*b*Sqrt[1 - c^2*x^2]*(1 + c^2*x^2)^(7/2))/
(70*c^13*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]*x) - (b*Sqrt[1 - c^2*x^2]*(1 + c^2*x^2)^(9/2))/(90*c^13*Sqrt[-1
+ 1/(c*x)]*Sqrt[1 + 1/(c*x)]*x) - (Sqrt[1 - c^4*x^4]*(a + b*ArcSech[c*x]))/(2*c^12) + ((1 - c^4*x^4)^(3/2)*(a
+ b*ArcSech[c*x]))/(3*c^12) - ((1 - c^4*x^4)^(5/2)*(a + b*ArcSech[c*x]))/(10*c^12) + (4*b*Sqrt[1 - c^2*x^2]*Ar
cTanh[Sqrt[1 + c^2*x^2]])/(15*c^13*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]*x)

Rule 266

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 6309

Int[((a_.) + ArcSech[(c_.)*(x_)]*(b_.))*(u_), x_Symbol] :> With[{v = IntHide[u, x]}, Dist[a + b*ArcSech[c*x],
v, x] + Dist[(b*Sqrt[1 - c^2*x^2])/(c*x*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]), Int[SimplifyIntegrand[v/(x*Sqrt
[1 - c^2*x^2]), x], x], x] /; InverseFunctionFreeQ[v, x]] /; FreeQ[{a, b, c}, x]

Rule 12

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

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rule 848

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)
^(m + p)*(f + g*x)^n*(a/d + (c*x)/e)^p, x] /; FreeQ[{a, c, d, e, f, g, m, n}, x] && NeQ[e*f - d*g, 0] && EqQ[c
*d^2 + a*e^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[d, 0] && EqQ[m + p, 0]))

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rule 783

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m +
p)*(f + g*x)*(a/d + (c*x)/e)^p, x] /; FreeQ[{a, c, d, e, f, g, m}, x] && EqQ[c*d^2 + a*e^2, 0] && (IntegerQ[p
] || (GtQ[a, 0] && GtQ[d, 0] && EqQ[m + p, 0]))

Rubi steps

\begin{align*} \int \frac{x^{11} \left (a+b \text{sech}^{-1}(c x)\right )}{\sqrt{1-c^4 x^4}} \, dx &=-\frac{\sqrt{1-c^4 x^4} \left (a+b \text{sech}^{-1}(c x)\right )}{2 c^{12}}+\frac{\left (1-c^4 x^4\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{3 c^{12}}-\frac{\left (1-c^4 x^4\right )^{5/2} \left (a+b \text{sech}^{-1}(c x)\right )}{10 c^{12}}+\frac{\left (b \sqrt{1-c^2 x^2}\right ) \int \frac{\sqrt{1-c^4 x^4} \left (-8-4 c^4 x^4-3 c^8 x^8\right )}{30 c^{12} x \sqrt{1-c^2 x^2}} \, dx}{c \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}} x}\\ &=-\frac{\sqrt{1-c^4 x^4} \left (a+b \text{sech}^{-1}(c x)\right )}{2 c^{12}}+\frac{\left (1-c^4 x^4\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{3 c^{12}}-\frac{\left (1-c^4 x^4\right )^{5/2} \left (a+b \text{sech}^{-1}(c x)\right )}{10 c^{12}}+\frac{\left (b \sqrt{1-c^2 x^2}\right ) \int \frac{\sqrt{1-c^4 x^4} \left (-8-4 c^4 x^4-3 c^8 x^8\right )}{x \sqrt{1-c^2 x^2}} \, dx}{30 c^{13} \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}} x}\\ &=-\frac{\sqrt{1-c^4 x^4} \left (a+b \text{sech}^{-1}(c x)\right )}{2 c^{12}}+\frac{\left (1-c^4 x^4\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{3 c^{12}}-\frac{\left (1-c^4 x^4\right )^{5/2} \left (a+b \text{sech}^{-1}(c x)\right )}{10 c^{12}}-\frac{\left (b \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1-c^4 x^2} \left (8+4 c^4 x^2+3 c^8 x^4\right )}{x \sqrt{1-c^2 x}} \, dx,x,x^2\right )}{60 c^{13} \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}} x}\\ &=-\frac{\sqrt{1-c^4 x^4} \left (a+b \text{sech}^{-1}(c x)\right )}{2 c^{12}}+\frac{\left (1-c^4 x^4\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{3 c^{12}}-\frac{\left (1-c^4 x^4\right )^{5/2} \left (a+b \text{sech}^{-1}(c x)\right )}{10 c^{12}}-\frac{\left (b \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{8 \sqrt{1-c^4 x^2}}{x \sqrt{1-c^2 x}}+\frac{4 c^4 x \sqrt{1-c^4 x^2}}{\sqrt{1-c^2 x}}+\frac{3 c^8 x^3 \sqrt{1-c^4 x^2}}{\sqrt{1-c^2 x}}\right ) \, dx,x,x^2\right )}{60 c^{13} \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}} x}\\ &=-\frac{\sqrt{1-c^4 x^4} \left (a+b \text{sech}^{-1}(c x)\right )}{2 c^{12}}+\frac{\left (1-c^4 x^4\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{3 c^{12}}-\frac{\left (1-c^4 x^4\right )^{5/2} \left (a+b \text{sech}^{-1}(c x)\right )}{10 c^{12}}-\frac{\left (2 b \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1-c^4 x^2}}{x \sqrt{1-c^2 x}} \, dx,x,x^2\right )}{15 c^{13} \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}} x}-\frac{\left (b \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{x \sqrt{1-c^4 x^2}}{\sqrt{1-c^2 x}} \, dx,x,x^2\right )}{15 c^9 \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}} x}-\frac{\left (b \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{x^3 \sqrt{1-c^4 x^2}}{\sqrt{1-c^2 x}} \, dx,x,x^2\right )}{20 c^5 \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}} x}\\ &=-\frac{\sqrt{1-c^4 x^4} \left (a+b \text{sech}^{-1}(c x)\right )}{2 c^{12}}+\frac{\left (1-c^4 x^4\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{3 c^{12}}-\frac{\left (1-c^4 x^4\right )^{5/2} \left (a+b \text{sech}^{-1}(c x)\right )}{10 c^{12}}-\frac{\left (2 b \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+c^2 x}}{x} \, dx,x,x^2\right )}{15 c^{13} \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}} x}-\frac{\left (b \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int x \sqrt{1+c^2 x} \, dx,x,x^2\right )}{15 c^9 \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}} x}-\frac{\left (b \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int x^3 \sqrt{1+c^2 x} \, dx,x,x^2\right )}{20 c^5 \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}} x}\\ &=-\frac{4 b \sqrt{1-c^2 x^2} \sqrt{1+c^2 x^2}}{15 c^{13} \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}} x}-\frac{\sqrt{1-c^4 x^4} \left (a+b \text{sech}^{-1}(c x)\right )}{2 c^{12}}+\frac{\left (1-c^4 x^4\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{3 c^{12}}-\frac{\left (1-c^4 x^4\right )^{5/2} \left (a+b \text{sech}^{-1}(c x)\right )}{10 c^{12}}-\frac{\left (2 b \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1+c^2 x}} \, dx,x,x^2\right )}{15 c^{13} \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}} x}-\frac{\left (b \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \left (-\frac{\sqrt{1+c^2 x}}{c^2}+\frac{\left (1+c^2 x\right )^{3/2}}{c^2}\right ) \, dx,x,x^2\right )}{15 c^9 \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}} x}-\frac{\left (b \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \left (-\frac{\sqrt{1+c^2 x}}{c^6}+\frac{3 \left (1+c^2 x\right )^{3/2}}{c^6}-\frac{3 \left (1+c^2 x\right )^{5/2}}{c^6}+\frac{\left (1+c^2 x\right )^{7/2}}{c^6}\right ) \, dx,x,x^2\right )}{20 c^5 \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}} x}\\ &=-\frac{4 b \sqrt{1-c^2 x^2} \sqrt{1+c^2 x^2}}{15 c^{13} \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}} x}+\frac{7 b \sqrt{1-c^2 x^2} \left (1+c^2 x^2\right )^{3/2}}{90 c^{13} \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}} x}-\frac{13 b \sqrt{1-c^2 x^2} \left (1+c^2 x^2\right )^{5/2}}{150 c^{13} \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}} x}+\frac{3 b \sqrt{1-c^2 x^2} \left (1+c^2 x^2\right )^{7/2}}{70 c^{13} \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}} x}-\frac{b \sqrt{1-c^2 x^2} \left (1+c^2 x^2\right )^{9/2}}{90 c^{13} \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}} x}-\frac{\sqrt{1-c^4 x^4} \left (a+b \text{sech}^{-1}(c x)\right )}{2 c^{12}}+\frac{\left (1-c^4 x^4\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{3 c^{12}}-\frac{\left (1-c^4 x^4\right )^{5/2} \left (a+b \text{sech}^{-1}(c x)\right )}{10 c^{12}}-\frac{\left (4 b \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{1}{c^2}+\frac{x^2}{c^2}} \, dx,x,\sqrt{1+c^2 x^2}\right )}{15 c^{15} \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}} x}\\ &=-\frac{4 b \sqrt{1-c^2 x^2} \sqrt{1+c^2 x^2}}{15 c^{13} \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}} x}+\frac{7 b \sqrt{1-c^2 x^2} \left (1+c^2 x^2\right )^{3/2}}{90 c^{13} \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}} x}-\frac{13 b \sqrt{1-c^2 x^2} \left (1+c^2 x^2\right )^{5/2}}{150 c^{13} \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}} x}+\frac{3 b \sqrt{1-c^2 x^2} \left (1+c^2 x^2\right )^{7/2}}{70 c^{13} \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}} x}-\frac{b \sqrt{1-c^2 x^2} \left (1+c^2 x^2\right )^{9/2}}{90 c^{13} \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}} x}-\frac{\sqrt{1-c^4 x^4} \left (a+b \text{sech}^{-1}(c x)\right )}{2 c^{12}}+\frac{\left (1-c^4 x^4\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{3 c^{12}}-\frac{\left (1-c^4 x^4\right )^{5/2} \left (a+b \text{sech}^{-1}(c x)\right )}{10 c^{12}}+\frac{4 b \sqrt{1-c^2 x^2} \tanh ^{-1}\left (\sqrt{1+c^2 x^2}\right )}{15 c^{13} \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}} x}\\ \end{align*}

Mathematica [A]  time = 0.418814, size = 213, normalized size = 0.45 \[ \frac{-105 a \sqrt{1-c^4 x^4} \left (3 c^8 x^8+4 c^4 x^4+8\right )+\frac{b \sqrt{\frac{1-c x}{c x+1}} \sqrt{1-c^4 x^4} \left (35 c^8 x^8+5 c^6 x^6+78 c^4 x^4+36 c^2 x^2+768\right )}{c x-1}-840 b \log \left (-\sqrt{\frac{1-c x}{c x+1}} \sqrt{1-c^4 x^4}-c x+1\right )-105 b \sqrt{1-c^4 x^4} \left (3 c^8 x^8+4 c^4 x^4+8\right ) \text{sech}^{-1}(c x)+840 b \log (x (1-c x))}{3150 c^{12}} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^11*(a + b*ArcSech[c*x]))/Sqrt[1 - c^4*x^4],x]

[Out]

(-105*a*Sqrt[1 - c^4*x^4]*(8 + 4*c^4*x^4 + 3*c^8*x^8) + (b*Sqrt[(1 - c*x)/(1 + c*x)]*Sqrt[1 - c^4*x^4]*(768 +
36*c^2*x^2 + 78*c^4*x^4 + 5*c^6*x^6 + 35*c^8*x^8))/(-1 + c*x) - 105*b*Sqrt[1 - c^4*x^4]*(8 + 4*c^4*x^4 + 3*c^8
*x^8)*ArcSech[c*x] + 840*b*Log[x*(1 - c*x)] - 840*b*Log[1 - c*x - Sqrt[(1 - c*x)/(1 + c*x)]*Sqrt[1 - c^4*x^4]]
)/(3150*c^12)

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Maple [F]  time = 1.938, size = 0, normalized size = 0. \begin{align*} \int{{x}^{11} \left ( a+b{\rm arcsech} \left (cx\right ) \right ){\frac{1}{\sqrt{-{c}^{4}{x}^{4}+1}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^11*(a+b*arcsech(c*x))/(-c^4*x^4+1)^(1/2),x)

[Out]

int(x^11*(a+b*arcsech(c*x))/(-c^4*x^4+1)^(1/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{30} \, a{\left (\frac{3 \,{\left (-c^{4} x^{4} + 1\right )}^{\frac{5}{2}}}{c^{12}} - \frac{10 \,{\left (-c^{4} x^{4} + 1\right )}^{\frac{3}{2}}}{c^{12}} + \frac{15 \, \sqrt{-c^{4} x^{4} + 1}}{c^{12}}\right )} + \frac{1}{30} \, b{\left (\frac{{\left (3 \, c^{12} x^{12} + c^{8} x^{8} + 4 \, c^{4} x^{4} - 8\right )} \log \left (\sqrt{c x + 1} \sqrt{-c x + 1} + 1\right )}{\sqrt{c^{2} x^{2} + 1} \sqrt{c x + 1} \sqrt{-c x + 1} c^{12}} - 30 \, \int \frac{30 \, c^{10} x^{21} \log \left (c\right ) + 60 \, c^{10} x^{21} \log \left (\sqrt{x}\right ) +{\left (60 \, c^{10} x^{21} \log \left (\sqrt{x}\right ) +{\left (3 \, c^{10} x^{10}{\left (10 \, \log \left (c\right ) + 1\right )} + 3 \, c^{8} x^{8} + 4 \, c^{6} x^{6} + 4 \, c^{4} x^{4} + 8 \, c^{2} x^{2} + 8\right )} x^{11}\right )} e^{\left (\frac{1}{2} \, \log \left (c x + 1\right ) + \frac{1}{2} \, \log \left (-c x + 1\right )\right )}}{30 \,{\left (c^{10} x^{10} e^{\left (\log \left (c x + 1\right ) + \log \left (-c x + 1\right )\right )} + c^{10} x^{10} e^{\left (\frac{1}{2} \, \log \left (c x + 1\right ) + \frac{1}{2} \, \log \left (-c x + 1\right )\right )}\right )} \sqrt{c^{2} x^{2} + 1}}\,{d x}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11*(a+b*arcsech(c*x))/(-c^4*x^4+1)^(1/2),x, algorithm="maxima")

[Out]

-1/30*a*(3*(-c^4*x^4 + 1)^(5/2)/c^12 - 10*(-c^4*x^4 + 1)^(3/2)/c^12 + 15*sqrt(-c^4*x^4 + 1)/c^12) + 1/30*b*((3
*c^12*x^12 + c^8*x^8 + 4*c^4*x^4 - 8)*log(sqrt(c*x + 1)*sqrt(-c*x + 1) + 1)/(sqrt(c^2*x^2 + 1)*sqrt(c*x + 1)*s
qrt(-c*x + 1)*c^12) - 30*integrate(1/30*(30*c^10*x^21*log(c) + 60*c^10*x^21*log(sqrt(x)) + (60*c^10*x^21*log(s
qrt(x)) + (3*c^10*x^10*(10*log(c) + 1) + 3*c^8*x^8 + 4*c^6*x^6 + 4*c^4*x^4 + 8*c^2*x^2 + 8)*x^11)*e^(1/2*log(c
*x + 1) + 1/2*log(-c*x + 1)))/((c^10*x^10*e^(log(c*x + 1) + log(-c*x + 1)) + c^10*x^10*e^(1/2*log(c*x + 1) + 1
/2*log(-c*x + 1)))*sqrt(c^2*x^2 + 1)), x))

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Fricas [A]  time = 2.17649, size = 852, normalized size = 1.8 \begin{align*} -\frac{105 \,{\left (3 \, b c^{10} x^{10} - 3 \, b c^{8} x^{8} + 4 \, b c^{6} x^{6} - 4 \, b c^{4} x^{4} + 8 \, b c^{2} x^{2} - 8 \, b\right )} \sqrt{-c^{4} x^{4} + 1} \log \left (\frac{c x \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) -{\left (35 \, b c^{9} x^{9} + 5 \, b c^{7} x^{7} + 78 \, b c^{5} x^{5} + 36 \, b c^{3} x^{3} + 768 \, b c x\right )} \sqrt{-c^{4} x^{4} + 1} \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}} + 420 \,{\left (b c^{2} x^{2} - b\right )} \log \left (\frac{c^{2} x^{2} + \sqrt{-c^{4} x^{4} + 1} c x \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}} - 1}{c^{2} x^{2} - 1}\right ) - 420 \,{\left (b c^{2} x^{2} - b\right )} \log \left (-\frac{c^{2} x^{2} - \sqrt{-c^{4} x^{4} + 1} c x \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}} - 1}{c^{2} x^{2} - 1}\right ) + 105 \,{\left (3 \, a c^{10} x^{10} - 3 \, a c^{8} x^{8} + 4 \, a c^{6} x^{6} - 4 \, a c^{4} x^{4} + 8 \, a c^{2} x^{2} - 8 \, a\right )} \sqrt{-c^{4} x^{4} + 1}}{3150 \,{\left (c^{14} x^{2} - c^{12}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11*(a+b*arcsech(c*x))/(-c^4*x^4+1)^(1/2),x, algorithm="fricas")

[Out]

-1/3150*(105*(3*b*c^10*x^10 - 3*b*c^8*x^8 + 4*b*c^6*x^6 - 4*b*c^4*x^4 + 8*b*c^2*x^2 - 8*b)*sqrt(-c^4*x^4 + 1)*
log((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) + 1)/(c*x)) - (35*b*c^9*x^9 + 5*b*c^7*x^7 + 78*b*c^5*x^5 + 36*b*c^3*x^
3 + 768*b*c*x)*sqrt(-c^4*x^4 + 1)*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) + 420*(b*c^2*x^2 - b)*log((c^2*x^2 + sqrt(-c^
4*x^4 + 1)*c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) - 1)/(c^2*x^2 - 1)) - 420*(b*c^2*x^2 - b)*log(-(c^2*x^2 - sqrt(-
c^4*x^4 + 1)*c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) - 1)/(c^2*x^2 - 1)) + 105*(3*a*c^10*x^10 - 3*a*c^8*x^8 + 4*a*c
^6*x^6 - 4*a*c^4*x^4 + 8*a*c^2*x^2 - 8*a)*sqrt(-c^4*x^4 + 1))/(c^14*x^2 - c^12)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**11*(a+b*asech(c*x))/(-c**4*x**4+1)**(1/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \operatorname{arsech}\left (c x\right ) + a\right )} x^{11}}{\sqrt{-c^{4} x^{4} + 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11*(a+b*arcsech(c*x))/(-c^4*x^4+1)^(1/2),x, algorithm="giac")

[Out]

integrate((b*arcsech(c*x) + a)*x^11/sqrt(-c^4*x^4 + 1), x)