\(\int \frac {\sqrt {d+e x^2} (a+b \csc ^{-1}(c x))}{x^6} \, dx\) [127]

Optimal result

Integrand size = 23, antiderivative size = 453 \[ \int \frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{x^6} \, dx=-\frac {b c \left (24 c^4 d^2+19 c^2 d e-31 e^2\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{225 d^2 \sqrt {c^2 x^2}}-\frac {b c \left (12 c^2 d-e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{225 d x^2 \sqrt {c^2 x^2}}-\frac {b c \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{25 d x^4 \sqrt {c^2 x^2}}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{5 d x^5}+\frac {2 e \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{15 d^2 x^3}+\frac {b c^2 \left (24 c^4 d^2+19 c^2 d e-31 e^2\right ) x \sqrt {1-c^2 x^2} \sqrt {d+e x^2} E\left (\arcsin (c x)\left |-\frac {e}{c^2 d}\right .\right )}{225 d^2 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {1+\frac {e x^2}{d}}}-\frac {b \left (c^2 d+e\right ) \left (24 c^4 d^2+7 c^2 d e-30 e^2\right ) x \sqrt {1-c^2 x^2} \sqrt {1+\frac {e x^2}{d}} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{225 d^2 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \]

[Out]

-1/5*(e*x^2+d)^(3/2)*(a+b*arccsc(c*x))/d/x^5+2/15*e*(e*x^2+d)^(3/2)*(a+b*arccsc(c*x))/d^2/x^3+2/15*b*c*e^2*(c^
2*x^2-1)^(1/2)*(e*x^2+d)^(1/2)/d^2/(c^2*x^2)^(1/2)-1/45*b*c*e*(2*c^2*d+e)*(c^2*x^2-1)^(1/2)*(e*x^2+d)^(1/2)/d^
2/(c^2*x^2)^(1/2)-1/75*b*c*(8*c^4*d^2+3*c^2*d*e-2*e^2)*(c^2*x^2-1)^(1/2)*(e*x^2+d)^(1/2)/d^2/(c^2*x^2)^(1/2)-1
/25*b*c*(c^2*x^2-1)^(1/2)*(e*x^2+d)^(1/2)/x^4/(c^2*x^2)^(1/2)-1/45*b*c*e*(c^2*x^2-1)^(1/2)*(e*x^2+d)^(1/2)/d/x
^2/(c^2*x^2)^(1/2)-1/75*b*c*(4*c^2*d+e)*(c^2*x^2-1)^(1/2)*(e*x^2+d)^(1/2)/d/x^2/(c^2*x^2)^(1/2)-2/15*b*c^2*e^2
*x*EllipticE(c*x,(-e/c^2/d)^(1/2))*(-c^2*x^2+1)^(1/2)*(e*x^2+d)^(1/2)/d^2/(c^2*x^2)^(1/2)/(c^2*x^2-1)^(1/2)/(1
+e*x^2/d)^(1/2)+1/45*b*c^2*e*(2*c^2*d+e)*x*EllipticE(c*x,(-e/c^2/d)^(1/2))*(-c^2*x^2+1)^(1/2)*(e*x^2+d)^(1/2)/
d^2/(c^2*x^2)^(1/2)/(c^2*x^2-1)^(1/2)/(1+e*x^2/d)^(1/2)+1/75*b*c^2*(8*c^4*d^2+3*c^2*d*e-2*e^2)*x*EllipticE(c*x
,(-e/c^2/d)^(1/2))*(-c^2*x^2+1)^(1/2)*(e*x^2+d)^(1/2)/d^2/(c^2*x^2)^(1/2)/(c^2*x^2-1)^(1/2)/(1+e*x^2/d)^(1/2)-
1/75*b*c^2*(8*c^2*d-e)*(c^2*d+e)*x*EllipticF(c*x,(-e/c^2/d)^(1/2))*(-c^2*x^2+1)^(1/2)*(1+e*x^2/d)^(1/2)/d/(c^2
*x^2)^(1/2)/(c^2*x^2-1)^(1/2)/(e*x^2+d)^(1/2)-2/45*b*c^2*e*(c^2*d+e)*x*EllipticF(c*x,(-e/c^2/d)^(1/2))*(-c^2*x
^2+1)^(1/2)*(1+e*x^2/d)^(1/2)/d/(c^2*x^2)^(1/2)/(c^2*x^2-1)^(1/2)/(e*x^2+d)^(1/2)+2/15*b*e^2*(c^2*d+e)*x*Ellip
ticF(c*x,(-e/c^2/d)^(1/2))*(-c^2*x^2+1)^(1/2)*(1+e*x^2/d)^(1/2)/d^2/(c^2*x^2)^(1/2)/(c^2*x^2-1)^(1/2)/(e*x^2+d
)^(1/2)

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 453, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {277, 270, 5347, 12, 594, 597, 538, 438, 437, 435, 432, 430} \[ \int \frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{x^6} \, dx=\frac {2 e \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{15 d^2 x^3}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{5 d x^5}-\frac {b x \sqrt {1-c^2 x^2} \left (c^2 d+e\right ) \left (24 c^4 d^2+7 c^2 d e-30 e^2\right ) \sqrt {\frac {e x^2}{d}+1} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{225 d^2 \sqrt {c^2 x^2} \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}+\frac {b c^2 x \sqrt {1-c^2 x^2} \left (24 c^4 d^2+19 c^2 d e-31 e^2\right ) \sqrt {d+e x^2} E\left (\arcsin (c x)\left |-\frac {e}{c^2 d}\right .\right )}{225 d^2 \sqrt {c^2 x^2} \sqrt {c^2 x^2-1} \sqrt {\frac {e x^2}{d}+1}}-\frac {b c \sqrt {c^2 x^2-1} \left (12 c^2 d-e\right ) \sqrt {d+e x^2}}{225 d x^2 \sqrt {c^2 x^2}}-\frac {b c \sqrt {c^2 x^2-1} \left (d+e x^2\right )^{3/2}}{25 d x^4 \sqrt {c^2 x^2}}-\frac {b c \sqrt {c^2 x^2-1} \left (24 c^4 d^2+19 c^2 d e-31 e^2\right ) \sqrt {d+e x^2}}{225 d^2 \sqrt {c^2 x^2}} \]

[In]

Int[(Sqrt[d + e*x^2]*(a + b*ArcCsc[c*x]))/x^6,x]

[Out]

-1/225*(b*c*(24*c^4*d^2 + 19*c^2*d*e - 31*e^2)*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])/(d^2*Sqrt[c^2*x^2]) - (b*c*
(12*c^2*d - e)*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])/(225*d*x^2*Sqrt[c^2*x^2]) - (b*c*Sqrt[-1 + c^2*x^2]*(d + e*
x^2)^(3/2))/(25*d*x^4*Sqrt[c^2*x^2]) - ((d + e*x^2)^(3/2)*(a + b*ArcCsc[c*x]))/(5*d*x^5) + (2*e*(d + e*x^2)^(3
/2)*(a + b*ArcCsc[c*x]))/(15*d^2*x^3) + (b*c^2*(24*c^4*d^2 + 19*c^2*d*e - 31*e^2)*x*Sqrt[1 - c^2*x^2]*Sqrt[d +
e*x^2]*EllipticE[ArcSin[c*x], -(e/(c^2*d))])/(225*d^2*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2]*Sqrt[1 + (e*x^2)/d]) -
(b*(c^2*d + e)*(24*c^4*d^2 + 7*c^2*d*e - 30*e^2)*x*Sqrt[1 - c^2*x^2]*Sqrt[1 + (e*x^2)/d]*EllipticF[ArcSin[c*x
], -(e/(c^2*d))])/(225*d^2*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])

Rule 12

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

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*
c*(m + 1))), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rule 277

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x^(m + 1)*((a + b*x^n)^(p + 1)/(a*(m + 1))), x]
- Dist[b*((m + n*(p + 1) + 1)/(a*(m + 1))), Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x]
&& ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]

Rule 430

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]
))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && Gt
Q[a, 0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])

Rule 432

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Dist[Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*
x^2], Int[1/(Sqrt[a + b*x^2]*Sqrt[1 + (d/c)*x^2]), x], x] /; FreeQ[{a, b, c, d}, x] &&  !GtQ[c, 0]

Rule 435

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*Ell
ipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0
]

Rule 437

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Dist[Sqrt[a + b*x^2]/Sqrt[1 + (b/a)*x^2]
, Int[Sqrt[1 + (b/a)*x^2]/Sqrt[c + d*x^2], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] &&  !GtQ
[a, 0]

Rule 438

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Dist[Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2]
, Int[Sqrt[a + b*x^2]/Sqrt[1 + (d/c)*x^2], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] &&  !GtQ[c, 0]

Rule 538

Int[((e_) + (f_.)*(x_)^(n_))/(Sqrt[(a_) + (b_.)*(x_)^(n_)]*Sqrt[(c_) + (d_.)*(x_)^(n_)]), x_Symbol] :> Dist[f/
b, Int[Sqrt[a + b*x^n]/Sqrt[c + d*x^n], x], x] + Dist[(b*e - a*f)/b, Int[1/(Sqrt[a + b*x^n]*Sqrt[c + d*x^n]),
x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] &&  !(EqQ[n, 2] && ((PosQ[b/a] && PosQ[d/c]) || (NegQ[b/a] && (PosQ[
d/c] || (GtQ[a, 0] && ( !GtQ[c, 0] || SimplerSqrtQ[-b/a, -d/c]))))))

Rule 594

Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^q/(a*g*(m + 1))), x] - Dist[1/(a*g^n*(m + 1
)), Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^(q - 1)*Simp[c*(b*e - a*f)*(m + 1) + e*n*(b*c*(p + 1) + a*d*q)
+ d*((b*e - a*f)*(m + 1) + b*e*n*(p + q + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && IGtQ[n
, 0] && GtQ[q, 0] && LtQ[m, -1] &&  !(EqQ[q, 1] && SimplerQ[e + f*x^n, c + d*x^n])

Rule 597

Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*c*g*(m + 1))), x] + Dist[1/(a*c*
g^n*(m + 1)), Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + n + 1) - e
*n*(b*c*p + a*d*q) - b*e*d*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] &&
IGtQ[n, 0] && LtQ[m, -1]

Rule 5347

Int[((a_.) + ArcCsc[(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*ArcCsc[c*x], u, x] + Dist[b*c*(x/Sqrt[c^2*x^2]), Int[SimplifyI
ntegrand[u/(x*Sqrt[c^2*x^2 - 1]), 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])) || (I
LtQ[(m + 2*p + 1)/2, 0] &&  !ILtQ[(m - 1)/2, 0]))

Rubi steps \begin{align*} \text {integral}& = -\frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{5 d x^5}+\frac {2 e \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{15 d^2 x^3}+\frac {(b c x) \int \frac {\left (d+e x^2\right )^{3/2} \left (-3 d+2 e x^2\right )}{15 d^2 x^6 \sqrt {-1+c^2 x^2}} \, dx}{\sqrt {c^2 x^2}} \\ & = -\frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{5 d x^5}+\frac {2 e \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{15 d^2 x^3}+\frac {(b c x) \int \frac {\left (d+e x^2\right )^{3/2} \left (-3 d+2 e x^2\right )}{x^6 \sqrt {-1+c^2 x^2}} \, dx}{15 d^2 \sqrt {c^2 x^2}} \\ & = -\frac {b c \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{25 d x^4 \sqrt {c^2 x^2}}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{5 d x^5}+\frac {2 e \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{15 d^2 x^3}-\frac {(b c x) \int \frac {\sqrt {d+e x^2} \left (d \left (12 c^2 d-e\right )+\left (3 c^2 d-10 e\right ) e x^2\right )}{x^4 \sqrt {-1+c^2 x^2}} \, dx}{75 d^2 \sqrt {c^2 x^2}} \\ & = -\frac {b c \left (12 c^2 d-e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{225 d x^2 \sqrt {c^2 x^2}}-\frac {b c \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{25 d x^4 \sqrt {c^2 x^2}}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{5 d x^5}+\frac {2 e \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{15 d^2 x^3}+\frac {(b c x) \int \frac {-d \left (24 c^4 d^2+19 c^2 d e-31 e^2\right )-2 e \left (6 c^4 d^2+4 c^2 d e-15 e^2\right ) x^2}{x^2 \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \, dx}{225 d^2 \sqrt {c^2 x^2}} \\ & = -\frac {b c \left (24 c^4 d^2+19 c^2 d e-31 e^2\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{225 d^2 \sqrt {c^2 x^2}}-\frac {b c \left (12 c^2 d-e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{225 d x^2 \sqrt {c^2 x^2}}-\frac {b c \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{25 d x^4 \sqrt {c^2 x^2}}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{5 d x^5}+\frac {2 e \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{15 d^2 x^3}+\frac {(b c x) \int \frac {-2 d e \left (6 c^4 d^2+4 c^2 d e-15 e^2\right )+c^2 d e \left (24 c^4 d^2+19 c^2 d e-31 e^2\right ) x^2}{\sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \, dx}{225 d^3 \sqrt {c^2 x^2}} \\ & = -\frac {b c \left (24 c^4 d^2+19 c^2 d e-31 e^2\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{225 d^2 \sqrt {c^2 x^2}}-\frac {b c \left (12 c^2 d-e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{225 d x^2 \sqrt {c^2 x^2}}-\frac {b c \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{25 d x^4 \sqrt {c^2 x^2}}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{5 d x^5}+\frac {2 e \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{15 d^2 x^3}+\frac {\left (b c^3 \left (24 c^4 d^2+19 c^2 d e-31 e^2\right ) x\right ) \int \frac {\sqrt {d+e x^2}}{\sqrt {-1+c^2 x^2}} \, dx}{225 d^2 \sqrt {c^2 x^2}}-\frac {\left (b c \left (c^2 d+e\right ) \left (24 c^4 d^2+7 c^2 d e-30 e^2\right ) x\right ) \int \frac {1}{\sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \, dx}{225 d^2 \sqrt {c^2 x^2}} \\ & = -\frac {b c \left (24 c^4 d^2+19 c^2 d e-31 e^2\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{225 d^2 \sqrt {c^2 x^2}}-\frac {b c \left (12 c^2 d-e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{225 d x^2 \sqrt {c^2 x^2}}-\frac {b c \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{25 d x^4 \sqrt {c^2 x^2}}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{5 d x^5}+\frac {2 e \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{15 d^2 x^3}+\frac {\left (b c^3 \left (24 c^4 d^2+19 c^2 d e-31 e^2\right ) x \sqrt {1-c^2 x^2}\right ) \int \frac {\sqrt {d+e x^2}}{\sqrt {1-c^2 x^2}} \, dx}{225 d^2 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2}}-\frac {\left (b c \left (c^2 d+e\right ) \left (24 c^4 d^2+7 c^2 d e-30 e^2\right ) x \sqrt {1+\frac {e x^2}{d}}\right ) \int \frac {1}{\sqrt {-1+c^2 x^2} \sqrt {1+\frac {e x^2}{d}}} \, dx}{225 d^2 \sqrt {c^2 x^2} \sqrt {d+e x^2}} \\ & = -\frac {b c \left (24 c^4 d^2+19 c^2 d e-31 e^2\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{225 d^2 \sqrt {c^2 x^2}}-\frac {b c \left (12 c^2 d-e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{225 d x^2 \sqrt {c^2 x^2}}-\frac {b c \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{25 d x^4 \sqrt {c^2 x^2}}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{5 d x^5}+\frac {2 e \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{15 d^2 x^3}+\frac {\left (b c^3 \left (24 c^4 d^2+19 c^2 d e-31 e^2\right ) x \sqrt {1-c^2 x^2} \sqrt {d+e x^2}\right ) \int \frac {\sqrt {1+\frac {e x^2}{d}}}{\sqrt {1-c^2 x^2}} \, dx}{225 d^2 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {1+\frac {e x^2}{d}}}-\frac {\left (b c \left (c^2 d+e\right ) \left (24 c^4 d^2+7 c^2 d e-30 e^2\right ) x \sqrt {1-c^2 x^2} \sqrt {1+\frac {e x^2}{d}}\right ) \int \frac {1}{\sqrt {1-c^2 x^2} \sqrt {1+\frac {e x^2}{d}}} \, dx}{225 d^2 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \\ & = -\frac {b c \left (24 c^4 d^2+19 c^2 d e-31 e^2\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{225 d^2 \sqrt {c^2 x^2}}-\frac {b c \left (12 c^2 d-e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{225 d x^2 \sqrt {c^2 x^2}}-\frac {b c \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{25 d x^4 \sqrt {c^2 x^2}}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{5 d x^5}+\frac {2 e \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{15 d^2 x^3}+\frac {b c^2 \left (24 c^4 d^2+19 c^2 d e-31 e^2\right ) x \sqrt {1-c^2 x^2} \sqrt {d+e x^2} E\left (\arcsin (c x)\left |-\frac {e}{c^2 d}\right .\right )}{225 d^2 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {1+\frac {e x^2}{d}}}-\frac {b \left (c^2 d+e\right ) \left (24 c^4 d^2+7 c^2 d e-30 e^2\right ) x \sqrt {1-c^2 x^2} \sqrt {1+\frac {e x^2}{d}} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{225 d^2 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 10.69 (sec) , antiderivative size = 325, normalized size of antiderivative = 0.72 \[ \int \frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{x^6} \, dx=-\frac {\sqrt {d+e x^2} \left (15 a \left (3 d^2+d e x^2-2 e^2 x^4\right )+b c \sqrt {1-\frac {1}{c^2 x^2}} x \left (-31 e^2 x^4+d e x^2 \left (8+19 c^2 x^2\right )+3 d^2 \left (3+4 c^2 x^2+8 c^4 x^4\right )\right )+15 b \left (3 d^2+d e x^2-2 e^2 x^4\right ) \csc ^{-1}(c x)\right )}{225 d^2 x^5}+\frac {i b c \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {1+\frac {e x^2}{d}} \left (c^2 d \left (24 c^4 d^2+19 c^2 d e-31 e^2\right ) E\left (i \text {arcsinh}\left (\sqrt {-c^2} x\right )|-\frac {e}{c^2 d}\right )+\left (-24 c^6 d^3-31 c^4 d^2 e+23 c^2 d e^2+30 e^3\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-c^2} x\right ),-\frac {e}{c^2 d}\right )\right )}{225 \sqrt {-c^2} d^2 \sqrt {1-c^2 x^2} \sqrt {d+e x^2}} \]

[In]

Integrate[(Sqrt[d + e*x^2]*(a + b*ArcCsc[c*x]))/x^6,x]

[Out]

-1/225*(Sqrt[d + e*x^2]*(15*a*(3*d^2 + d*e*x^2 - 2*e^2*x^4) + b*c*Sqrt[1 - 1/(c^2*x^2)]*x*(-31*e^2*x^4 + d*e*x
^2*(8 + 19*c^2*x^2) + 3*d^2*(3 + 4*c^2*x^2 + 8*c^4*x^4)) + 15*b*(3*d^2 + d*e*x^2 - 2*e^2*x^4)*ArcCsc[c*x]))/(d
^2*x^5) + ((I/225)*b*c*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[1 + (e*x^2)/d]*(c^2*d*(24*c^4*d^2 + 19*c^2*d*e - 31*e^2)*E
llipticE[I*ArcSinh[Sqrt[-c^2]*x], -(e/(c^2*d))] + (-24*c^6*d^3 - 31*c^4*d^2*e + 23*c^2*d*e^2 + 30*e^3)*Ellipti
cF[I*ArcSinh[Sqrt[-c^2]*x], -(e/(c^2*d))]))/(Sqrt[-c^2]*d^2*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2])

Maple [F]

\[\int \frac {\left (a +b \,\operatorname {arccsc}\left (c x \right )\right ) \sqrt {e \,x^{2}+d}}{x^{6}}d x\]

[In]

int((a+b*arccsc(c*x))*(e*x^2+d)^(1/2)/x^6,x)

[Out]

int((a+b*arccsc(c*x))*(e*x^2+d)^(1/2)/x^6,x)

Fricas [A] (verification not implemented)

none

Time = 0.11 (sec) , antiderivative size = 292, normalized size of antiderivative = 0.64 \[ \int \frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{x^6} \, dx=\frac {{\left (30 \, a c d e^{2} x^{4} - 15 \, a c d^{2} e x^{2} - 45 \, a c d^{3} + 15 \, {\left (2 \, b c d e^{2} x^{4} - b c d^{2} e x^{2} - 3 \, b c d^{3}\right )} \operatorname {arccsc}\left (c x\right ) - {\left (9 \, b c d^{3} + {\left (24 \, b c^{5} d^{3} + 19 \, b c^{3} d^{2} e - 31 \, b c d e^{2}\right )} x^{4} + 4 \, {\left (3 \, b c^{3} d^{3} + 2 \, b c d^{2} e\right )} x^{2}\right )} \sqrt {c^{2} x^{2} - 1}\right )} \sqrt {e x^{2} + d} - {\left ({\left (24 \, b c^{8} d^{3} + 19 \, b c^{6} d^{2} e - 31 \, b c^{4} d e^{2}\right )} x^{5} E(\arcsin \left (c x\right )\,|\,-\frac {e}{c^{2} d}) - {\left (24 \, b c^{8} d^{3} + {\left (19 \, b c^{6} + 12 \, b c^{4}\right )} d^{2} e - {\left (31 \, b c^{4} - 8 \, b c^{2}\right )} d e^{2} - 30 \, b e^{3}\right )} x^{5} F(\arcsin \left (c x\right )\,|\,-\frac {e}{c^{2} d})\right )} \sqrt {-d}}{225 \, c d^{3} x^{5}} \]

[In]

integrate((a+b*arccsc(c*x))*(e*x^2+d)^(1/2)/x^6,x, algorithm="fricas")

[Out]

1/225*((30*a*c*d*e^2*x^4 - 15*a*c*d^2*e*x^2 - 45*a*c*d^3 + 15*(2*b*c*d*e^2*x^4 - b*c*d^2*e*x^2 - 3*b*c*d^3)*ar
ccsc(c*x) - (9*b*c*d^3 + (24*b*c^5*d^3 + 19*b*c^3*d^2*e - 31*b*c*d*e^2)*x^4 + 4*(3*b*c^3*d^3 + 2*b*c*d^2*e)*x^
2)*sqrt(c^2*x^2 - 1))*sqrt(e*x^2 + d) - ((24*b*c^8*d^3 + 19*b*c^6*d^2*e - 31*b*c^4*d*e^2)*x^5*elliptic_e(arcsi
n(c*x), -e/(c^2*d)) - (24*b*c^8*d^3 + (19*b*c^6 + 12*b*c^4)*d^2*e - (31*b*c^4 - 8*b*c^2)*d*e^2 - 30*b*e^3)*x^5
*elliptic_f(arcsin(c*x), -e/(c^2*d)))*sqrt(-d))/(c*d^3*x^5)

Sympy [F]

\[ \int \frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{x^6} \, dx=\int \frac {\left (a + b \operatorname {acsc}{\left (c x \right )}\right ) \sqrt {d + e x^{2}}}{x^{6}}\, dx \]

[In]

integrate((a+b*acsc(c*x))*(e*x**2+d)**(1/2)/x**6,x)

[Out]

Integral((a + b*acsc(c*x))*sqrt(d + e*x**2)/x**6, x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{x^6} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((a+b*arccsc(c*x))*(e*x^2+d)^(1/2)/x^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more
details)Is e

Giac [F]

\[ \int \frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{x^6} \, dx=\int { \frac {\sqrt {e x^{2} + d} {\left (b \operatorname {arccsc}\left (c x\right ) + a\right )}}{x^{6}} \,d x } \]

[In]

integrate((a+b*arccsc(c*x))*(e*x^2+d)^(1/2)/x^6,x, algorithm="giac")

[Out]

integrate(sqrt(e*x^2 + d)*(b*arccsc(c*x) + a)/x^6, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{x^6} \, dx=\int \frac {\sqrt {e\,x^2+d}\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}{x^6} \,d x \]

[In]

int(((d + e*x^2)^(1/2)*(a + b*asin(1/(c*x))))/x^6,x)

[Out]

int(((d + e*x^2)^(1/2)*(a + b*asin(1/(c*x))))/x^6, x)