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

Optimal. Leaf size=453 \[ \frac {2 e \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{15 d^2 x^3}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{5 d x^5}-\frac {b c \sqrt {c^2 x^2-1} \left (12 c^2 d-e\right ) \sqrt {d+e x^2}}{225 d x^2 \sqrt {c^2 x^2}}-\frac {b c \sqrt {c^2 x^2-1} \left (d+e x^2\right )^{3/2}}{25 d x^4 \sqrt {c^2 x^2}}-\frac {b c \sqrt {c^2 x^2-1} \left (24 c^4 d^2+19 c^2 d e-31 e^2\right ) \sqrt {d+e x^2}}{225 d^2 \sqrt {c^2 x^2}}-\frac {b x \sqrt {1-c^2 x^2} \left (c^2 d+e\right ) \left (24 c^4 d^2+7 c^2 d e-30 e^2\right ) \sqrt {\frac {e x^2}{d}+1} F\left (\sin ^{-1}(c x)|-\frac {e}{c^2 d}\right )}{225 d^2 \sqrt {c^2 x^2} \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}+\frac {b c^2 x \sqrt {1-c^2 x^2} \left (24 c^4 d^2+19 c^2 d e-31 e^2\right ) \sqrt {d+e x^2} E\left (\sin ^{-1}(c x)|-\frac {e}{c^2 d}\right )}{225 d^2 \sqrt {c^2 x^2} \sqrt {c^2 x^2-1} \sqrt {\frac {e x^2}{d}+1}} \]

[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]  time = 0.62, antiderivative size = 453, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 12, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {271, 264, 5239, 12, 580, 583, 524, 427, 426, 424, 421, 419} \[ \frac {2 e \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{15 d^2 x^3}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{5 d x^5}-\frac {b c \sqrt {c^2 x^2-1} \left (24 c^4 d^2+19 c^2 d e-31 e^2\right ) \sqrt {d+e x^2}}{225 d^2 \sqrt {c^2 x^2}}-\frac {b x \sqrt {1-c^2 x^2} \left (c^2 d+e\right ) \left (24 c^4 d^2+7 c^2 d e-30 e^2\right ) \sqrt {\frac {e x^2}{d}+1} F\left (\sin ^{-1}(c x)|-\frac {e}{c^2 d}\right )}{225 d^2 \sqrt {c^2 x^2} \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}+\frac {b c^2 x \sqrt {1-c^2 x^2} \left (24 c^4 d^2+19 c^2 d e-31 e^2\right ) \sqrt {d+e x^2} E\left (\sin ^{-1}(c x)|-\frac {e}{c^2 d}\right )}{225 d^2 \sqrt {c^2 x^2} \sqrt {c^2 x^2-1} \sqrt {\frac {e x^2}{d}+1}}-\frac {b c \sqrt {c^2 x^2-1} \left (d+e x^2\right )^{3/2}}{25 d x^4 \sqrt {c^2 x^2}}-\frac {b c \sqrt {c^2 x^2-1} \left (12 c^2 d-e\right ) \sqrt {d+e x^2}}{225 d x^2 \sqrt {c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(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])/(225*d^2*Sqrt[c^2*x^2]) - (b*c*(1
2*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 264

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 271

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 419

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

Rule 421

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

Rule 424

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

Rule 426

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

Rule 427

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

Rule 524

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 580

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 583

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 5239

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*} \int \frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{x^6} \, dx &=-\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 (\sin ^{-1}(c x)|-\frac {e}{c^2 d}\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}} F\left (\sin ^{-1}(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*}

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Mathematica [C]  time = 0.72, size = 325, normalized size = 0.72 \[ -\frac {\sqrt {d+e x^2} \left (15 a \left (3 d^2+d e x^2-2 e^2 x^4\right )+b c x \sqrt {1-\frac {1}{c^2 x^2}} \left (d e x^2 \left (19 c^2 x^2+8\right )+3 d^2 \left (8 c^4 x^4+4 c^2 x^2+3\right )-31 e^2 x^4\right )+15 b \csc ^{-1}(c x) \left (3 d^2+d e x^2-2 e^2 x^4\right )\right )}{225 d^2 x^5}+\frac {i b c x \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {\frac {e x^2}{d}+1} \left (c^2 d \left (24 c^4 d^2+19 c^2 d e-31 e^2\right ) E\left (i \sinh ^{-1}\left (\sqrt {-c^2} x\right )|-\frac {e}{c^2 d}\right )+\left (-24 c^6 d^3-31 c^4 d^2 e+23 c^2 d e^2+30 e^3\right ) F\left (i \sinh ^{-1}\left (\sqrt {-c^2} x\right )|-\frac {e}{c^2 d}\right )\right )}{225 \sqrt {-c^2} d^2 \sqrt {1-c^2 x^2} \sqrt {d+e x^2}} \]

Antiderivative was successfully verified.

[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])

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fricas [F]  time = 0.73, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {e x^{2} + d} {\left (b \operatorname {arccsc}\left (c x\right ) + a\right )}}{x^{6}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {e x^{2} + d} {\left (b \operatorname {arccsc}\left (c x\right ) + a\right )}}{x^{6}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [F]  time = 8.95, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \,\mathrm {arccsc}\left (c x \right )\right ) \sqrt {e \,x^{2}+d}}{x^{6}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{15} \, a {\left (\frac {2 \, {\left (e x^{2} + d\right )}^{\frac {3}{2}} e}{d^{2} x^{3}} - \frac {3 \, {\left (e x^{2} + d\right )}^{\frac {3}{2}}}{d x^{5}}\right )} + \frac {{\left (d^{2} x^{5} \int \frac {{\left (2 \, c^{2} e^{2} x^{4} - c^{2} d e x^{2} - 3 \, c^{2} d^{2}\right )} e^{\left (\frac {1}{2} \, \log \left (e x^{2} + d\right ) + \frac {1}{2} \, \log \left (c x + 1\right ) + \frac {1}{2} \, \log \left (c x - 1\right )\right )}}{c^{2} d^{2} x^{6} - d^{2} x^{4} + {\left (c^{2} d^{2} x^{6} - d^{2} x^{4}\right )} {\left (c x + 1\right )} {\left (c x - 1\right )}}\,{d x} + {\left (2 \, e^{2} x^{4} \arctan \left (1, \sqrt {c x + 1} \sqrt {c x - 1}\right ) - d e x^{2} \arctan \left (1, \sqrt {c x + 1} \sqrt {c x - 1}\right ) - 3 \, d^{2} \arctan \left (1, \sqrt {c x + 1} \sqrt {c x - 1}\right )\right )} \sqrt {e x^{2} + d}\right )} b}{15 \, d^{2} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/15*a*(2*(e*x^2 + d)^(3/2)*e/(d^2*x^3) - 3*(e*x^2 + d)^(3/2)/(d*x^5)) + 1/15*(15*d^2*x^5*integrate(1/15*(2*c^
2*e^2*x^4 - c^2*d*e*x^2 - 3*c^2*d^2)*e^(1/2*log(e*x^2 + d) + 1/2*log(c*x + 1) + 1/2*log(c*x - 1))/(c^2*d^2*x^6
- d^2*x^4 + (c^2*d^2*x^6 - d^2*x^4)*e^(log(c*x + 1) + log(c*x - 1))), x) + (2*e^2*x^4*arctan2(1, sqrt(c*x + 1
)*sqrt(c*x - 1)) - d*e*x^2*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1)) - 3*d^2*arctan2(1, sqrt(c*x + 1)*sqrt(c*x -
1)))*sqrt(e*x^2 + d))*b/(d^2*x^5)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\sqrt {e\,x^2+d}\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}{x^6} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \operatorname {acsc}{\left (c x \right )}\right ) \sqrt {d + e x^{2}}}{x^{6}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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