\(\int \frac {x^3 (a+b \csc ^{-1}(c x))}{(d+e x^2)^{5/2}} \, dx\) [157]

Optimal result

Integrand size = 23, antiderivative size = 163 \[ \int \frac {x^3 \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx=-\frac {b c x \sqrt {-1+c^2 x^2}}{3 e \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {d+e x^2}}+\frac {d \left (a+b \csc ^{-1}(c x)\right )}{3 e^2 \left (d+e x^2\right )^{3/2}}-\frac {a+b \csc ^{-1}(c x)}{e^2 \sqrt {d+e x^2}}+\frac {2 b c x \arctan \left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-1+c^2 x^2}}\right )}{3 \sqrt {d} e^2 \sqrt {c^2 x^2}} \] Output:

-1/3*b*c*x*(c^2*x^2-1)^(1/2)/e/(c^2*d+e)/(c^2*x^2)^(1/2)/(e*x^2+d)^(1/2)+1 
/3*d*(a+b*arccsc(c*x))/e^2/(e*x^2+d)^(3/2)-(a+b*arccsc(c*x))/e^2/(e*x^2+d) 
^(1/2)+2/3*b*c*x*arctan((e*x^2+d)^(1/2)/d^(1/2)/(c^2*x^2-1)^(1/2))/d^(1/2) 
/e^2/(c^2*x^2)^(1/2)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.

Time = 0.40 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.99 \[ \int \frac {x^3 \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx=\frac {b \left (c^2 d+e\right ) \sqrt {1+\frac {d}{e x^2}} \left (d+e x^2\right ) \operatorname {AppellF1}\left (1,\frac {1}{2},\frac {1}{2},2,\frac {1}{c^2 x^2},-\frac {d}{e x^2}\right )-c x \left (b c e \sqrt {1-\frac {1}{c^2 x^2}} x \left (d+e x^2\right )+a \left (c^2 d+e\right ) \left (2 d+3 e x^2\right )+b \left (c^2 d+e\right ) \left (2 d+3 e x^2\right ) \csc ^{-1}(c x)\right )}{3 c e^2 \left (c^2 d+e\right ) x \left (d+e x^2\right )^{3/2}} \] Input:

Integrate[(x^3*(a + b*ArcCsc[c*x]))/(d + e*x^2)^(5/2),x]
 

Output:

(b*(c^2*d + e)*Sqrt[1 + d/(e*x^2)]*(d + e*x^2)*AppellF1[1, 1/2, 1/2, 2, 1/ 
(c^2*x^2), -(d/(e*x^2))] - c*x*(b*c*e*Sqrt[1 - 1/(c^2*x^2)]*x*(d + e*x^2) 
+ a*(c^2*d + e)*(2*d + 3*e*x^2) + b*(c^2*d + e)*(2*d + 3*e*x^2)*ArcCsc[c*x 
]))/(3*c*e^2*(c^2*d + e)*x*(d + e*x^2)^(3/2))
 

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.91, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {5762, 27, 435, 169, 25, 27, 104, 217}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {x^3 \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 5762

\(\displaystyle \frac {b c x \int -\frac {3 e x^2+2 d}{3 e^2 x \sqrt {c^2 x^2-1} \left (e x^2+d\right )^{3/2}}dx}{\sqrt {c^2 x^2}}-\frac {a+b \csc ^{-1}(c x)}{e^2 \sqrt {d+e x^2}}+\frac {d \left (a+b \csc ^{-1}(c x)\right )}{3 e^2 \left (d+e x^2\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {b c x \int \frac {3 e x^2+2 d}{x \sqrt {c^2 x^2-1} \left (e x^2+d\right )^{3/2}}dx}{3 e^2 \sqrt {c^2 x^2}}-\frac {a+b \csc ^{-1}(c x)}{e^2 \sqrt {d+e x^2}}+\frac {d \left (a+b \csc ^{-1}(c x)\right )}{3 e^2 \left (d+e x^2\right )^{3/2}}\)

\(\Big \downarrow \) 435

\(\displaystyle -\frac {b c x \int \frac {3 e x^2+2 d}{x^2 \sqrt {c^2 x^2-1} \left (e x^2+d\right )^{3/2}}dx^2}{6 e^2 \sqrt {c^2 x^2}}-\frac {a+b \csc ^{-1}(c x)}{e^2 \sqrt {d+e x^2}}+\frac {d \left (a+b \csc ^{-1}(c x)\right )}{3 e^2 \left (d+e x^2\right )^{3/2}}\)

\(\Big \downarrow \) 169

\(\displaystyle -\frac {b c x \left (\frac {2 e \sqrt {c^2 x^2-1}}{\left (c^2 d+e\right ) \sqrt {d+e x^2}}-\frac {2 \int -\frac {d \left (d c^2+e\right )}{x^2 \sqrt {c^2 x^2-1} \sqrt {e x^2+d}}dx^2}{d \left (c^2 d+e\right )}\right )}{6 e^2 \sqrt {c^2 x^2}}-\frac {a+b \csc ^{-1}(c x)}{e^2 \sqrt {d+e x^2}}+\frac {d \left (a+b \csc ^{-1}(c x)\right )}{3 e^2 \left (d+e x^2\right )^{3/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {b c x \left (\frac {2 \int \frac {d \left (d c^2+e\right )}{x^2 \sqrt {c^2 x^2-1} \sqrt {e x^2+d}}dx^2}{d \left (c^2 d+e\right )}+\frac {2 e \sqrt {c^2 x^2-1}}{\left (c^2 d+e\right ) \sqrt {d+e x^2}}\right )}{6 e^2 \sqrt {c^2 x^2}}-\frac {a+b \csc ^{-1}(c x)}{e^2 \sqrt {d+e x^2}}+\frac {d \left (a+b \csc ^{-1}(c x)\right )}{3 e^2 \left (d+e x^2\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {b c x \left (2 \int \frac {1}{x^2 \sqrt {c^2 x^2-1} \sqrt {e x^2+d}}dx^2+\frac {2 e \sqrt {c^2 x^2-1}}{\left (c^2 d+e\right ) \sqrt {d+e x^2}}\right )}{6 e^2 \sqrt {c^2 x^2}}-\frac {a+b \csc ^{-1}(c x)}{e^2 \sqrt {d+e x^2}}+\frac {d \left (a+b \csc ^{-1}(c x)\right )}{3 e^2 \left (d+e x^2\right )^{3/2}}\)

\(\Big \downarrow \) 104

\(\displaystyle -\frac {b c x \left (4 \int \frac {1}{-x^4-d}d\frac {\sqrt {e x^2+d}}{\sqrt {c^2 x^2-1}}+\frac {2 e \sqrt {c^2 x^2-1}}{\left (c^2 d+e\right ) \sqrt {d+e x^2}}\right )}{6 e^2 \sqrt {c^2 x^2}}-\frac {a+b \csc ^{-1}(c x)}{e^2 \sqrt {d+e x^2}}+\frac {d \left (a+b \csc ^{-1}(c x)\right )}{3 e^2 \left (d+e x^2\right )^{3/2}}\)

\(\Big \downarrow \) 217

\(\displaystyle -\frac {a+b \csc ^{-1}(c x)}{e^2 \sqrt {d+e x^2}}+\frac {d \left (a+b \csc ^{-1}(c x)\right )}{3 e^2 \left (d+e x^2\right )^{3/2}}-\frac {b c x \left (\frac {2 e \sqrt {c^2 x^2-1}}{\left (c^2 d+e\right ) \sqrt {d+e x^2}}-\frac {4 \arctan \left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {c^2 x^2-1}}\right )}{\sqrt {d}}\right )}{6 e^2 \sqrt {c^2 x^2}}\)

Input:

Int[(x^3*(a + b*ArcCsc[c*x]))/(d + e*x^2)^(5/2),x]
 

Output:

(d*(a + b*ArcCsc[c*x]))/(3*e^2*(d + e*x^2)^(3/2)) - (a + b*ArcCsc[c*x])/(e 
^2*Sqrt[d + e*x^2]) - (b*c*x*((2*e*Sqrt[-1 + c^2*x^2])/((c^2*d + e)*Sqrt[d 
 + e*x^2]) - (4*ArcTan[Sqrt[d + e*x^2]/(Sqrt[d]*Sqrt[-1 + c^2*x^2])])/Sqrt 
[d]))/(6*e^2*Sqrt[c^2*x^2])
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 104

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x 
_)), x_] :> With[{q = Denominator[m]}, Simp[q   Subst[Int[x^(q*(m + 1) - 1) 
/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] 
] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L 
tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
 

rule 169

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + 
 d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S 
imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f))   Int[(a + b*x)^(m + 1)*(c + d*x)^n 
*(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* 
h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegersQ[ 
2*m, 2*n, 2*p]
 

rule 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[b, 0])
 

rule 435

Int[(x_)^(m_.)*((a_.) + (b_.)*(x_)^2)^(p_.)*((c_.) + (d_.)*(x_)^2)^(q_.)*(( 
e_.) + (f_.)*(x_)^2)^(r_.), x_Symbol] :> Simp[1/2   Subst[Int[x^((m - 1)/2) 
*(a + b*x)^p*(c + d*x)^q*(e + f*x)^r, x], x, x^2], x] /; FreeQ[{a, b, c, d, 
 e, f, p, q, r}, x] && IntegerQ[(m - 1)/2]
 

rule 5762

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]}, Sim 
p[(a + b*ArcCsc[c*x])   u, x] + Simp[b*c*(x/Sqrt[c^2*x^2])   Int[SimplifyIn 
tegrand[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])) || (ILtQ[(m 
 + 2*p + 1)/2, 0] &&  !ILtQ[(m - 1)/2, 0]))
 

Maple [F]

\[\int \frac {x^{3} \left (a +b \,\operatorname {arccsc}\left (c x \right )\right )}{\left (e \,x^{2}+d \right )^{\frac {5}{2}}}d x\]

Input:

int(x^3*(a+b*arccsc(c*x))/(e*x^2+d)^(5/2),x)
 

Output:

int(x^3*(a+b*arccsc(c*x))/(e*x^2+d)^(5/2),x)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 321 vs. \(2 (137) = 274\).

Time = 0.25 (sec) , antiderivative size = 663, normalized size of antiderivative = 4.07 \[ \int \frac {x^3 \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx=\left [-\frac {{\left (b c^{2} d^{3} + {\left (b c^{2} d e^{2} + b e^{3}\right )} x^{4} + b d^{2} e + 2 \, {\left (b c^{2} d^{2} e + b d e^{2}\right )} x^{2}\right )} \sqrt {-d} \log \left (\frac {{\left (c^{4} d^{2} - 6 \, c^{2} d e + e^{2}\right )} x^{4} - 8 \, {\left (c^{2} d^{2} - d e\right )} x^{2} + 4 \, \sqrt {c^{2} x^{2} - 1} {\left ({\left (c^{2} d - e\right )} x^{2} - 2 \, d\right )} \sqrt {e x^{2} + d} \sqrt {-d} + 8 \, d^{2}}{x^{4}}\right ) + 2 \, {\left (2 \, a c^{2} d^{3} + 2 \, a d^{2} e + 3 \, {\left (a c^{2} d^{2} e + a d e^{2}\right )} x^{2} + {\left (2 \, b c^{2} d^{3} + 2 \, b d^{2} e + 3 \, {\left (b c^{2} d^{2} e + b d e^{2}\right )} x^{2}\right )} \operatorname {arccsc}\left (c x\right ) + {\left (b d e^{2} x^{2} + b d^{2} e\right )} \sqrt {c^{2} x^{2} - 1}\right )} \sqrt {e x^{2} + d}}{6 \, {\left (c^{2} d^{4} e^{2} + d^{3} e^{3} + {\left (c^{2} d^{2} e^{4} + d e^{5}\right )} x^{4} + 2 \, {\left (c^{2} d^{3} e^{3} + d^{2} e^{4}\right )} x^{2}\right )}}, \frac {{\left (b c^{2} d^{3} + {\left (b c^{2} d e^{2} + b e^{3}\right )} x^{4} + b d^{2} e + 2 \, {\left (b c^{2} d^{2} e + b d e^{2}\right )} x^{2}\right )} \sqrt {d} \arctan \left (-\frac {\sqrt {c^{2} x^{2} - 1} {\left ({\left (c^{2} d - e\right )} x^{2} - 2 \, d\right )} \sqrt {e x^{2} + d} \sqrt {d}}{2 \, {\left (c^{2} d e x^{4} + {\left (c^{2} d^{2} - d e\right )} x^{2} - d^{2}\right )}}\right ) - {\left (2 \, a c^{2} d^{3} + 2 \, a d^{2} e + 3 \, {\left (a c^{2} d^{2} e + a d e^{2}\right )} x^{2} + {\left (2 \, b c^{2} d^{3} + 2 \, b d^{2} e + 3 \, {\left (b c^{2} d^{2} e + b d e^{2}\right )} x^{2}\right )} \operatorname {arccsc}\left (c x\right ) + {\left (b d e^{2} x^{2} + b d^{2} e\right )} \sqrt {c^{2} x^{2} - 1}\right )} \sqrt {e x^{2} + d}}{3 \, {\left (c^{2} d^{4} e^{2} + d^{3} e^{3} + {\left (c^{2} d^{2} e^{4} + d e^{5}\right )} x^{4} + 2 \, {\left (c^{2} d^{3} e^{3} + d^{2} e^{4}\right )} x^{2}\right )}}\right ] \] Input:

integrate(x^3*(a+b*arccsc(c*x))/(e*x^2+d)^(5/2),x, algorithm="fricas")
 

Output:

[-1/6*((b*c^2*d^3 + (b*c^2*d*e^2 + b*e^3)*x^4 + b*d^2*e + 2*(b*c^2*d^2*e + 
 b*d*e^2)*x^2)*sqrt(-d)*log(((c^4*d^2 - 6*c^2*d*e + e^2)*x^4 - 8*(c^2*d^2 
- d*e)*x^2 + 4*sqrt(c^2*x^2 - 1)*((c^2*d - e)*x^2 - 2*d)*sqrt(e*x^2 + d)*s 
qrt(-d) + 8*d^2)/x^4) + 2*(2*a*c^2*d^3 + 2*a*d^2*e + 3*(a*c^2*d^2*e + a*d* 
e^2)*x^2 + (2*b*c^2*d^3 + 2*b*d^2*e + 3*(b*c^2*d^2*e + b*d*e^2)*x^2)*arccs 
c(c*x) + (b*d*e^2*x^2 + b*d^2*e)*sqrt(c^2*x^2 - 1))*sqrt(e*x^2 + d))/(c^2* 
d^4*e^2 + d^3*e^3 + (c^2*d^2*e^4 + d*e^5)*x^4 + 2*(c^2*d^3*e^3 + d^2*e^4)* 
x^2), 1/3*((b*c^2*d^3 + (b*c^2*d*e^2 + b*e^3)*x^4 + b*d^2*e + 2*(b*c^2*d^2 
*e + b*d*e^2)*x^2)*sqrt(d)*arctan(-1/2*sqrt(c^2*x^2 - 1)*((c^2*d - e)*x^2 
- 2*d)*sqrt(e*x^2 + d)*sqrt(d)/(c^2*d*e*x^4 + (c^2*d^2 - d*e)*x^2 - d^2)) 
- (2*a*c^2*d^3 + 2*a*d^2*e + 3*(a*c^2*d^2*e + a*d*e^2)*x^2 + (2*b*c^2*d^3 
+ 2*b*d^2*e + 3*(b*c^2*d^2*e + b*d*e^2)*x^2)*arccsc(c*x) + (b*d*e^2*x^2 + 
b*d^2*e)*sqrt(c^2*x^2 - 1))*sqrt(e*x^2 + d))/(c^2*d^4*e^2 + d^3*e^3 + (c^2 
*d^2*e^4 + d*e^5)*x^4 + 2*(c^2*d^3*e^3 + d^2*e^4)*x^2)]
 

Sympy [F]

\[ \int \frac {x^3 \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx=\int \frac {x^{3} \left (a + b \operatorname {acsc}{\left (c x \right )}\right )}{\left (d + e x^{2}\right )^{\frac {5}{2}}}\, dx \] Input:

integrate(x**3*(a+b*acsc(c*x))/(e*x**2+d)**(5/2),x)
 

Output:

Integral(x**3*(a + b*acsc(c*x))/(d + e*x**2)**(5/2), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^3 \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(x^3*(a+b*arccsc(c*x))/(e*x^2+d)^(5/2),x, algorithm="maxima")
 

Output:

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

Giac [F]

\[ \int \frac {x^3 \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arccsc}\left (c x\right ) + a\right )} x^{3}}{{\left (e x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(x^3*(a+b*arccsc(c*x))/(e*x^2+d)^(5/2),x, algorithm="giac")
 

Output:

integrate((b*arccsc(c*x) + a)*x^3/(e*x^2 + d)^(5/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^3 \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx=\int \frac {x^3\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}{{\left (e\,x^2+d\right )}^{5/2}} \,d x \] Input:

int((x^3*(a + b*asin(1/(c*x))))/(d + e*x^2)^(5/2),x)
 

Output:

int((x^3*(a + b*asin(1/(c*x))))/(d + e*x^2)^(5/2), x)
 

Reduce [F]

\[ \int \frac {x^3 \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx=\frac {-2 \sqrt {e \,x^{2}+d}\, a d -3 \sqrt {e \,x^{2}+d}\, a e \,x^{2}+3 \left (\int \frac {\mathit {acsc} \left (c x \right ) x^{3}}{\sqrt {e \,x^{2}+d}\, d^{2}+2 \sqrt {e \,x^{2}+d}\, d e \,x^{2}+\sqrt {e \,x^{2}+d}\, e^{2} x^{4}}d x \right ) b \,d^{2} e^{2}+6 \left (\int \frac {\mathit {acsc} \left (c x \right ) x^{3}}{\sqrt {e \,x^{2}+d}\, d^{2}+2 \sqrt {e \,x^{2}+d}\, d e \,x^{2}+\sqrt {e \,x^{2}+d}\, e^{2} x^{4}}d x \right ) b d \,e^{3} x^{2}+3 \left (\int \frac {\mathit {acsc} \left (c x \right ) x^{3}}{\sqrt {e \,x^{2}+d}\, d^{2}+2 \sqrt {e \,x^{2}+d}\, d e \,x^{2}+\sqrt {e \,x^{2}+d}\, e^{2} x^{4}}d x \right ) b \,e^{4} x^{4}}{3 e^{2} \left (e^{2} x^{4}+2 d e \,x^{2}+d^{2}\right )} \] Input:

int(x^3*(a+b*acsc(c*x))/(e*x^2+d)^(5/2),x)
 

Output:

( - 2*sqrt(d + e*x**2)*a*d - 3*sqrt(d + e*x**2)*a*e*x**2 + 3*int((acsc(c*x 
)*x**3)/(sqrt(d + e*x**2)*d**2 + 2*sqrt(d + e*x**2)*d*e*x**2 + sqrt(d + e* 
x**2)*e**2*x**4),x)*b*d**2*e**2 + 6*int((acsc(c*x)*x**3)/(sqrt(d + e*x**2) 
*d**2 + 2*sqrt(d + e*x**2)*d*e*x**2 + sqrt(d + e*x**2)*e**2*x**4),x)*b*d*e 
**3*x**2 + 3*int((acsc(c*x)*x**3)/(sqrt(d + e*x**2)*d**2 + 2*sqrt(d + e*x* 
*2)*d*e*x**2 + sqrt(d + e*x**2)*e**2*x**4),x)*b*e**4*x**4)/(3*e**2*(d**2 + 
 2*d*e*x**2 + e**2*x**4))