\(\int \frac {x^{11} (a+b \csc ^{-1}(c x))}{\sqrt {1-c^4 x^4}} \, dx\) [174]

Optimal result

Integrand size = 26, antiderivative size = 401 \[ \int \frac {x^{11} \left (a+b \csc ^{-1}(c x)\right )}{\sqrt {1-c^4 x^4}} \, dx=-\frac {4 b \sqrt {1-c^2 x^2} \sqrt {1+c^2 x^2}}{15 c^{13} \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {7 b \sqrt {1-c^2 x^2} \left (1+c^2 x^2\right )^{3/2}}{90 c^{13} \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {13 b \sqrt {1-c^2 x^2} \left (1+c^2 x^2\right )^{5/2}}{150 c^{13} \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {3 b \sqrt {1-c^2 x^2} \left (1+c^2 x^2\right )^{7/2}}{70 c^{13} \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {b \sqrt {1-c^2 x^2} \left (1+c^2 x^2\right )^{9/2}}{90 c^{13} \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {\sqrt {1-c^4 x^4} \left (a+b \csc ^{-1}(c x)\right )}{2 c^{12}}+\frac {\left (1-c^4 x^4\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 c^{12}}-\frac {\left (1-c^4 x^4\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{10 c^{12}}+\frac {4 b \sqrt {1-c^2 x^2} \text {arctanh}\left (\sqrt {1+c^2 x^2}\right )}{15 c^{13} \sqrt {1-\frac {1}{c^2 x^2}} x} \]

[Out]

1/3*(-c^4*x^4+1)^(3/2)*(a+b*arccsc(c*x))/c^12-1/10*(-c^4*x^4+1)^(5/2)*(a+b*arccsc(c*x))/c^12+7/90*b*(c^2*x^2+1
)^(3/2)*(-c^2*x^2+1)^(1/2)/c^13/x/(1-1/c^2/x^2)^(1/2)-13/150*b*(c^2*x^2+1)^(5/2)*(-c^2*x^2+1)^(1/2)/c^13/x/(1-
1/c^2/x^2)^(1/2)+3/70*b*(c^2*x^2+1)^(7/2)*(-c^2*x^2+1)^(1/2)/c^13/x/(1-1/c^2/x^2)^(1/2)-1/90*b*(c^2*x^2+1)^(9/
2)*(-c^2*x^2+1)^(1/2)/c^13/x/(1-1/c^2/x^2)^(1/2)+4/15*b*arctanh((c^2*x^2+1)^(1/2))*(-c^2*x^2+1)^(1/2)/c^13/x/(
1-1/c^2/x^2)^(1/2)-4/15*b*(-c^2*x^2+1)^(1/2)*(c^2*x^2+1)^(1/2)/c^13/x/(1-1/c^2/x^2)^(1/2)-1/2*(a+b*arccsc(c*x)
)*(-c^4*x^4+1)^(1/2)/c^12

Rubi [A] (verified)

Time = 1.72 (sec) , antiderivative size = 401, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.423, Rules used = {272, 45, 5355, 12, 6853, 6874, 862, 52, 65, 214, 797} \[ \int \frac {x^{11} \left (a+b \csc ^{-1}(c x)\right )}{\sqrt {1-c^4 x^4}} \, dx=-\frac {\left (1-c^4 x^4\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{10 c^{12}}+\frac {\left (1-c^4 x^4\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 c^{12}}-\frac {\sqrt {1-c^4 x^4} \left (a+b \csc ^{-1}(c x)\right )}{2 c^{12}}+\frac {4 b \sqrt {1-c^2 x^2} \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )}{15 c^{13} x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {b \sqrt {1-c^2 x^2} \left (c^2 x^2+1\right )^{9/2}}{90 c^{13} x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {3 b \sqrt {1-c^2 x^2} \left (c^2 x^2+1\right )^{7/2}}{70 c^{13} x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {13 b \sqrt {1-c^2 x^2} \left (c^2 x^2+1\right )^{5/2}}{150 c^{13} x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {7 b \sqrt {1-c^2 x^2} \left (c^2 x^2+1\right )^{3/2}}{90 c^{13} x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {4 b \sqrt {1-c^2 x^2} \sqrt {c^2 x^2+1}}{15 c^{13} x \sqrt {1-\frac {1}{c^2 x^2}}} \]

[In]

Int[(x^11*(a + b*ArcCsc[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^2*x^2)]*x) + (7*b*Sqrt[1 - c^2*x^2]*(1 + c^2
*x^2)^(3/2))/(90*c^13*Sqrt[1 - 1/(c^2*x^2)]*x) - (13*b*Sqrt[1 - c^2*x^2]*(1 + c^2*x^2)^(5/2))/(150*c^13*Sqrt[1
- 1/(c^2*x^2)]*x) + (3*b*Sqrt[1 - c^2*x^2]*(1 + c^2*x^2)^(7/2))/(70*c^13*Sqrt[1 - 1/(c^2*x^2)]*x) - (b*Sqrt[1
- c^2*x^2]*(1 + c^2*x^2)^(9/2))/(90*c^13*Sqrt[1 - 1/(c^2*x^2)]*x) - (Sqrt[1 - c^4*x^4]*(a + b*ArcCsc[c*x]))/(
2*c^12) + ((1 - c^4*x^4)^(3/2)*(a + b*ArcCsc[c*x]))/(3*c^12) - ((1 - c^4*x^4)^(5/2)*(a + b*ArcCsc[c*x]))/(10*c
^12) + (4*b*Sqrt[1 - c^2*x^2]*ArcTanh[Sqrt[1 + c^2*x^2]])/(15*c^13*Sqrt[1 - 1/(c^2*x^2)]*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 45

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 52

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 65

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 214

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

Rule 272

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 797

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/e)*x)^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]))

Rule 862

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/e)*x)^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 5355

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

Rule 6853

Int[(u_.)*((a_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[b^IntPart[p]*((a + b*x^n)^FracPart[p]/(x^(n*FracPa
rt[p])*(1 + a*(1/(x^n*b)))^FracPart[p])), Int[u*x^(n*p)*(1 + a*(1/(x^n*b)))^p, x], x] /; FreeQ[{a, b, p}, x] &
&  !IntegerQ[p] && ILtQ[n, 0] &&  !RationalFunctionQ[u, x] && IntegerQ[p + 1/2]

Rule 6874

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

Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {1-c^4 x^4} \left (a+b \csc ^{-1}(c x)\right )}{2 c^{12}}+\frac {\left (1-c^4 x^4\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 c^{12}}-\frac {\left (1-c^4 x^4\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{10 c^{12}}+\frac {b \int \frac {\sqrt {1-c^4 x^4} \left (-8-4 c^4 x^4-3 c^8 x^8\right )}{30 c^{12} \sqrt {1-\frac {1}{c^2 x^2}} x^2} \, dx}{c} \\ & = -\frac {\sqrt {1-c^4 x^4} \left (a+b \csc ^{-1}(c x)\right )}{2 c^{12}}+\frac {\left (1-c^4 x^4\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 c^{12}}-\frac {\left (1-c^4 x^4\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{10 c^{12}}+\frac {b \int \frac {\sqrt {1-c^4 x^4} \left (-8-4 c^4 x^4-3 c^8 x^8\right )}{\sqrt {1-\frac {1}{c^2 x^2}} x^2} \, dx}{30 c^{13}} \\ & = -\frac {\sqrt {1-c^4 x^4} \left (a+b \csc ^{-1}(c x)\right )}{2 c^{12}}+\frac {\left (1-c^4 x^4\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 c^{12}}-\frac {\left (1-c^4 x^4\right )^{5/2} \left (a+b \csc ^{-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^2 x^2}} x} \\ & = -\frac {\sqrt {1-c^4 x^4} \left (a+b \csc ^{-1}(c x)\right )}{2 c^{12}}+\frac {\left (1-c^4 x^4\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 c^{12}}-\frac {\left (1-c^4 x^4\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{10 c^{12}}-\frac {\left (b \sqrt {1-c^2 x^2}\right ) \text {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^2 x^2}} x} \\ & = -\frac {\sqrt {1-c^4 x^4} \left (a+b \csc ^{-1}(c x)\right )}{2 c^{12}}+\frac {\left (1-c^4 x^4\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 c^{12}}-\frac {\left (1-c^4 x^4\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{10 c^{12}}-\frac {\left (b \sqrt {1-c^2 x^2}\right ) \text {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^2 x^2}} x} \\ & = -\frac {\sqrt {1-c^4 x^4} \left (a+b \csc ^{-1}(c x)\right )}{2 c^{12}}+\frac {\left (1-c^4 x^4\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 c^{12}}-\frac {\left (1-c^4 x^4\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{10 c^{12}}-\frac {\left (2 b \sqrt {1-c^2 x^2}\right ) \text {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^2 x^2}} x}-\frac {\left (b \sqrt {1-c^2 x^2}\right ) \text {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^2 x^2}} x}-\frac {\left (b \sqrt {1-c^2 x^2}\right ) \text {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^2 x^2}} x} \\ & = -\frac {\sqrt {1-c^4 x^4} \left (a+b \csc ^{-1}(c x)\right )}{2 c^{12}}+\frac {\left (1-c^4 x^4\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 c^{12}}-\frac {\left (1-c^4 x^4\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{10 c^{12}}-\frac {\left (2 b \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\sqrt {1+c^2 x}}{x} \, dx,x,x^2\right )}{15 c^{13} \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {\left (b \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int x \sqrt {1+c^2 x} \, dx,x,x^2\right )}{15 c^9 \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {\left (b \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int x^3 \sqrt {1+c^2 x} \, dx,x,x^2\right )}{20 c^5 \sqrt {1-\frac {1}{c^2 x^2}} x} \\ & = -\frac {4 b \sqrt {1-c^2 x^2} \sqrt {1+c^2 x^2}}{15 c^{13} \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {\sqrt {1-c^4 x^4} \left (a+b \csc ^{-1}(c x)\right )}{2 c^{12}}+\frac {\left (1-c^4 x^4\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 c^{12}}-\frac {\left (1-c^4 x^4\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{10 c^{12}}-\frac {\left (2 b \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1+c^2 x}} \, dx,x,x^2\right )}{15 c^{13} \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {\left (b \sqrt {1-c^2 x^2}\right ) \text {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^2 x^2}} x}-\frac {\left (b \sqrt {1-c^2 x^2}\right ) \text {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^2 x^2}} x} \\ & = -\frac {4 b \sqrt {1-c^2 x^2} \sqrt {1+c^2 x^2}}{15 c^{13} \sqrt {1-\frac {1}{c^2 x^2}} 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^2 x^2}} 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^2 x^2}} 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^2 x^2}} 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^2 x^2}} x}-\frac {\sqrt {1-c^4 x^4} \left (a+b \csc ^{-1}(c x)\right )}{2 c^{12}}+\frac {\left (1-c^4 x^4\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 c^{12}}-\frac {\left (1-c^4 x^4\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{10 c^{12}}-\frac {\left (4 b \sqrt {1-c^2 x^2}\right ) \text {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^2 x^2}} x} \\ & = -\frac {4 b \sqrt {1-c^2 x^2} \sqrt {1+c^2 x^2}}{15 c^{13} \sqrt {1-\frac {1}{c^2 x^2}} 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^2 x^2}} 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^2 x^2}} 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^2 x^2}} 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^2 x^2}} x}-\frac {\sqrt {1-c^4 x^4} \left (a+b \csc ^{-1}(c x)\right )}{2 c^{12}}+\frac {\left (1-c^4 x^4\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 c^{12}}-\frac {\left (1-c^4 x^4\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{10 c^{12}}+\frac {4 b \sqrt {1-c^2 x^2} \text {arctanh}\left (\sqrt {1+c^2 x^2}\right )}{15 c^{13} \sqrt {1-\frac {1}{c^2 x^2}} x} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.48 \[ \int \frac {x^{11} \left (a+b \csc ^{-1}(c x)\right )}{\sqrt {1-c^4 x^4}} \, dx=-\frac {105 a \sqrt {1-c^4 x^4} \left (8+4 c^4 x^4+3 c^8 x^8\right )+\frac {b c \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {1-c^4 x^4} \left (768+36 c^2 x^2+78 c^4 x^4+5 c^6 x^6+35 c^8 x^8\right )}{-1+c^2 x^2}+105 b \sqrt {1-c^4 x^4} \left (8+4 c^4 x^4+3 c^8 x^8\right ) \csc ^{-1}(c x)+840 b \arctan \left (\frac {c \sqrt {1-\frac {1}{c^2 x^2}} x}{\sqrt {1-c^4 x^4}}\right )}{3150 c^{12}} \]

[In]

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

[Out]

-1/3150*(105*a*Sqrt[1 - c^4*x^4]*(8 + 4*c^4*x^4 + 3*c^8*x^8) + (b*c*Sqrt[1 - 1/(c^2*x^2)]*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^2*x^2) + 105*b*Sqrt[1 - c^4*x^4]*(8 + 4*c^4*
x^4 + 3*c^8*x^8)*ArcCsc[c*x] + 840*b*ArcTan[(c*Sqrt[1 - 1/(c^2*x^2)]*x)/Sqrt[1 - c^4*x^4]])/c^12

Maple [F]

\[\int \frac {x^{11} \left (a +b \,\operatorname {arccsc}\left (c x \right )\right )}{\sqrt {-c^{4} x^{4}+1}}d x\]

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 238, normalized size of antiderivative = 0.59 \[ \int \frac {x^{11} \left (a+b \csc ^{-1}(c x)\right )}{\sqrt {1-c^4 x^4}} \, dx=-\frac {{\left (35 \, b c^{8} x^{8} + 5 \, b c^{6} x^{6} + 78 \, b c^{4} x^{4} + 36 \, b c^{2} x^{2} + 768 \, b\right )} \sqrt {-c^{4} x^{4} + 1} \sqrt {c^{2} x^{2} - 1} - 840 \, {\left (b c^{2} x^{2} - b\right )} \arctan \left (\frac {\sqrt {-c^{4} x^{4} + 1}}{\sqrt {c^{2} x^{2} - 1}}\right ) + 105 \, {\left (3 \, a c^{10} x^{10} - 3 \, a c^{8} x^{8} + 4 \, a c^{6} x^{6} - 4 \, a c^{4} x^{4} + 8 \, a c^{2} x^{2} + {\left (3 \, b c^{10} x^{10} - 3 \, b c^{8} x^{8} + 4 \, b c^{6} x^{6} - 4 \, b c^{4} x^{4} + 8 \, b c^{2} x^{2} - 8 \, b\right )} \operatorname {arccsc}\left (c x\right ) - 8 \, a\right )} \sqrt {-c^{4} x^{4} + 1}}{3150 \, {\left (c^{14} x^{2} - c^{12}\right )}} \]

[In]

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

[Out]

-1/3150*((35*b*c^8*x^8 + 5*b*c^6*x^6 + 78*b*c^4*x^4 + 36*b*c^2*x^2 + 768*b)*sqrt(-c^4*x^4 + 1)*sqrt(c^2*x^2 -
1) - 840*(b*c^2*x^2 - b)*arctan(sqrt(-c^4*x^4 + 1)/sqrt(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 + (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)*arccsc(c*x) - 8*a)*sqrt(-c^4*x^4 + 1))/(c^14*x^2 - c^12)

Sympy [F(-1)]

Timed out. \[ \int \frac {x^{11} \left (a+b \csc ^{-1}(c x)\right )}{\sqrt {1-c^4 x^4}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int \frac {x^{11} \left (a+b \csc ^{-1}(c x)\right )}{\sqrt {1-c^4 x^4}} \, dx=\int { \frac {{\left (b \operatorname {arccsc}\left (c x\right ) + a\right )} x^{11}}{\sqrt {-c^{4} x^{4} + 1}} \,d x } \]

[In]

integrate(x^11*(a+b*arccsc(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*(30*c
^12*integrate(1/30*(3*c^10*x^11 + 3*c^8*x^9 + 4*c^6*x^7 + 4*c^4*x^5 + 8*c^2*x^3 + 8*x)*e^(-1/2*log(c^2*x^2 + 1
) + 1/2*log(c*x - 1))/(c^10*e^(log(c*x + 1) + log(c*x - 1) + 1/2*log(-c*x + 1)) + sqrt(-c*x + 1)*c^10), x) - (
3*c^8*x^8*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1)) + 4*c^4*x^4*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1)) + 8*arct
an2(1, sqrt(c*x + 1)*sqrt(c*x - 1)))*sqrt(c^2*x^2 + 1)*sqrt(c*x + 1)*sqrt(-c*x + 1))*b/c^12

Giac [F(-2)]

Exception generated. \[ \int \frac {x^{11} \left (a+b \csc ^{-1}(c x)\right )}{\sqrt {1-c^4 x^4}} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

Timed out. \[ \int \frac {x^{11} \left (a+b \csc ^{-1}(c x)\right )}{\sqrt {1-c^4 x^4}} \, dx=\int \frac {x^{11}\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}{\sqrt {1-c^4\,x^4}} \,d x \]

[In]

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

[Out]

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