Integrand size = 19, antiderivative size = 195 \[ \int x \left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {b \left (3 c^4 d^2+3 c^2 d e+e^2\right ) x \sqrt {-1+c^2 x^2}}{6 c^5 \sqrt {c^2 x^2}}+\frac {b e \left (3 c^2 d+2 e\right ) x \left (-1+c^2 x^2\right )^{3/2}}{18 c^5 \sqrt {c^2 x^2}}+\frac {b e^2 x \left (-1+c^2 x^2\right )^{5/2}}{30 c^5 \sqrt {c^2 x^2}}+\frac {\left (d+e x^2\right )^3 \left (a+b \csc ^{-1}(c x)\right )}{6 e}+\frac {b c d^3 x \arctan \left (\sqrt {-1+c^2 x^2}\right )}{6 e \sqrt {c^2 x^2}} \]
1/6*(e*x^2+d)^3*(a+b*arccsc(c*x))/e+1/18*b*e*(3*c^2*d+2*e)*x*(c^2*x^2-1)^( 3/2)/c^5/(c^2*x^2)^(1/2)+1/30*b*e^2*x*(c^2*x^2-1)^(5/2)/c^5/(c^2*x^2)^(1/2 )+1/6*b*c*d^3*x*arctan((c^2*x^2-1)^(1/2))/e/(c^2*x^2)^(1/2)+1/6*b*(3*c^4*d ^2+3*c^2*d*e+e^2)*x*(c^2*x^2-1)^(1/2)/c^5/(c^2*x^2)^(1/2)
Time = 0.17 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.64 \[ \int x \left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {1}{90} x \left (15 a x \left (3 d^2+3 d e x^2+e^2 x^4\right )+\frac {b \sqrt {1-\frac {1}{c^2 x^2}} \left (8 e^2+2 c^2 e \left (15 d+2 e x^2\right )+3 c^4 \left (15 d^2+5 d e x^2+e^2 x^4\right )\right )}{c^5}+15 b x \left (3 d^2+3 d e x^2+e^2 x^4\right ) \csc ^{-1}(c x)\right ) \]
(x*(15*a*x*(3*d^2 + 3*d*e*x^2 + e^2*x^4) + (b*Sqrt[1 - 1/(c^2*x^2)]*(8*e^2 + 2*c^2*e*(15*d + 2*e*x^2) + 3*c^4*(15*d^2 + 5*d*e*x^2 + e^2*x^4)))/c^5 + 15*b*x*(3*d^2 + 3*d*e*x^2 + e^2*x^4)*ArcCsc[c*x]))/90
Time = 0.38 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.82, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {5760, 354, 99, 2009}
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.
| \(\displaystyle \int x \left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right ) \, dx\) |
| \(\Big \downarrow \) 5760 |
| \(\displaystyle \frac {b c x \int \frac {\left (e x^2+d\right )^3}{x \sqrt {c^2 x^2-1}}dx}{6 e \sqrt {c^2 x^2}}+\frac {\left (d+e x^2\right )^3 \left (a+b \csc ^{-1}(c x)\right )}{6 e}\) |
| \(\Big \downarrow \) 354 |
| \(\displaystyle \frac {b c x \int \frac {\left (e x^2+d\right )^3}{x^2 \sqrt {c^2 x^2-1}}dx^2}{12 e \sqrt {c^2 x^2}}+\frac {\left (d+e x^2\right )^3 \left (a+b \csc ^{-1}(c x)\right )}{6 e}\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \frac {b c x \int \left (\frac {d^3}{x^2 \sqrt {c^2 x^2-1}}+\frac {e^3 \left (c^2 x^2-1\right )^{3/2}}{c^4}+\frac {e^2 \left (3 d c^2+2 e\right ) \sqrt {c^2 x^2-1}}{c^4}+\frac {e \left (3 d^2 c^4+3 d e c^2+e^2\right )}{c^4 \sqrt {c^2 x^2-1}}\right )dx^2}{12 e \sqrt {c^2 x^2}}+\frac {\left (d+e x^2\right )^3 \left (a+b \csc ^{-1}(c x)\right )}{6 e}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\left (d+e x^2\right )^3 \left (a+b \csc ^{-1}(c x)\right )}{6 e}+\frac {b c x \left (2 d^3 \arctan \left (\sqrt {c^2 x^2-1}\right )+\frac {2 e^2 \left (c^2 x^2-1\right )^{3/2} \left (3 c^2 d+2 e\right )}{3 c^6}+\frac {2 e^3 \left (c^2 x^2-1\right )^{5/2}}{5 c^6}+\frac {2 e \sqrt {c^2 x^2-1} \left (3 c^4 d^2+3 c^2 d e+e^2\right )}{c^6}\right )}{12 e \sqrt {c^2 x^2}}\) |
((d + e*x^2)^3*(a + b*ArcCsc[c*x]))/(6*e) + (b*c*x*((2*e*(3*c^4*d^2 + 3*c^ 2*d*e + e^2)*Sqrt[-1 + c^2*x^2])/c^6 + (2*e^2*(3*c^2*d + 2*e)*(-1 + c^2*x^ 2)^(3/2))/(3*c^6) + (2*e^3*(-1 + c^2*x^2)^(5/2))/(5*c^6) + 2*d^3*ArcTan[Sq rt[-1 + c^2*x^2]]))/(12*e*Sqrt[c^2*x^2])
3.1.95.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | | (GtQ[m, 0] && GeQ[n, -1]))
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S ymbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x , x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ [(m - 1)/2]
Int[((a_.) + ArcCsc[(c_.)*(x_)]*(b_.))*(x_)*((d_.) + (e_.)*(x_)^2)^(p_.), x _Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcCsc[c*x])/(2*e*(p + 1))), x ] + Simp[b*c*(x/(2*e*(p + 1)*Sqrt[c^2*x^2])) Int[(d + e*x^2)^(p + 1)/(x*S qrt[c^2*x^2 - 1]), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[p, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(351\) vs. \(2(169)=338\).
Time = 0.97 (sec) , antiderivative size = 352, normalized size of antiderivative = 1.81
| method | result | size |
| parts | \(\frac {a \left (e \,x^{2}+d \right )^{3}}{6 e}+\frac {b \,\operatorname {arccsc}\left (c x \right ) e^{2} x^{6}}{6}+\frac {b \,\operatorname {arccsc}\left (c x \right ) d e \,x^{4}}{2}+\frac {b \,\operatorname {arccsc}\left (c x \right ) d^{2} x^{2}}{2}+\frac {b \,d^{3} \operatorname {arccsc}\left (c x \right )}{6 e}+\frac {b \left (c^{2} x^{2}-1\right ) x^{3} e^{2}}{30 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {b \left (c^{2} x^{2}-1\right ) x d e}{6 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {b \sqrt {c^{2} x^{2}-1}\, d^{3} \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{6 c e \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}+\frac {2 b \left (c^{2} x^{2}-1\right ) x \,e^{2}}{45 c^{5} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {b \left (c^{2} x^{2}-1\right ) d^{2}}{2 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}+\frac {b e \left (c^{2} x^{2}-1\right ) d}{3 c^{5} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}+\frac {4 b \,e^{2} \left (c^{2} x^{2}-1\right )}{45 c^{7} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}\) | \(352\) |
| derivativedivides | \(\frac {\frac {a \left (c^{2} e \,x^{2}+c^{2} d \right )^{3}}{6 c^{4} e}+\frac {b \,c^{2} \operatorname {arccsc}\left (c x \right ) d^{3}}{6 e}+\frac {b \,\operatorname {arccsc}\left (c x \right ) d^{2} c^{2} x^{2}}{2}+\frac {b \,c^{2} e \,\operatorname {arccsc}\left (c x \right ) d \,x^{4}}{2}+\frac {b \,c^{2} e^{2} \operatorname {arccsc}\left (c x \right ) x^{6}}{6}-\frac {b c \sqrt {c^{2} x^{2}-1}\, d^{3} \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{6 e \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}+\frac {b \left (c^{2} x^{2}-1\right ) d^{2}}{2 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}+\frac {b e \left (c^{2} x^{2}-1\right ) x d}{6 c \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {b \,e^{2} \left (c^{2} x^{2}-1\right ) x^{3}}{30 c \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {b e \left (c^{2} x^{2}-1\right ) d}{3 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}+\frac {2 b \left (c^{2} x^{2}-1\right ) x \,e^{2}}{45 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {4 b \,e^{2} \left (c^{2} x^{2}-1\right )}{45 c^{5} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}}{c^{2}}\) | \(376\) |
| default | \(\frac {\frac {a \left (c^{2} e \,x^{2}+c^{2} d \right )^{3}}{6 c^{4} e}+\frac {b \,c^{2} \operatorname {arccsc}\left (c x \right ) d^{3}}{6 e}+\frac {b \,\operatorname {arccsc}\left (c x \right ) d^{2} c^{2} x^{2}}{2}+\frac {b \,c^{2} e \,\operatorname {arccsc}\left (c x \right ) d \,x^{4}}{2}+\frac {b \,c^{2} e^{2} \operatorname {arccsc}\left (c x \right ) x^{6}}{6}-\frac {b c \sqrt {c^{2} x^{2}-1}\, d^{3} \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{6 e \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}+\frac {b \left (c^{2} x^{2}-1\right ) d^{2}}{2 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}+\frac {b e \left (c^{2} x^{2}-1\right ) x d}{6 c \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {b \,e^{2} \left (c^{2} x^{2}-1\right ) x^{3}}{30 c \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {b e \left (c^{2} x^{2}-1\right ) d}{3 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}+\frac {2 b \left (c^{2} x^{2}-1\right ) x \,e^{2}}{45 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {4 b \,e^{2} \left (c^{2} x^{2}-1\right )}{45 c^{5} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}}{c^{2}}\) | \(376\) |
1/6*a*(e*x^2+d)^3/e+1/6*b*arccsc(c*x)*e^2*x^6+1/2*b*arccsc(c*x)*d*e*x^4+1/ 2*b*arccsc(c*x)*d^2*x^2+1/6*b*d^3*arccsc(c*x)/e+1/30*b/c^3*(c^2*x^2-1)/((c ^2*x^2-1)/c^2/x^2)^(1/2)*x^3*e^2+1/6*b/c^3*(c^2*x^2-1)/((c^2*x^2-1)/c^2/x^ 2)^(1/2)*x*d*e-1/6*b/c/e*(c^2*x^2-1)^(1/2)/((c^2*x^2-1)/c^2/x^2)^(1/2)/x*d ^3*arctan(1/(c^2*x^2-1)^(1/2))+2/45*b/c^5*(c^2*x^2-1)/((c^2*x^2-1)/c^2/x^2 )^(1/2)*x*e^2+1/2*b/c^3*(c^2*x^2-1)/((c^2*x^2-1)/c^2/x^2)^(1/2)/x*d^2+1/3* b/c^5*e*(c^2*x^2-1)/((c^2*x^2-1)/c^2/x^2)^(1/2)/x*d+4/45*b/c^7*e^2*(c^2*x^ 2-1)/((c^2*x^2-1)/c^2/x^2)^(1/2)/x
Time = 0.31 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.78 \[ \int x \left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {15 \, a c^{6} e^{2} x^{6} + 45 \, a c^{6} d e x^{4} + 45 \, a c^{6} d^{2} x^{2} + 15 \, {\left (b c^{6} e^{2} x^{6} + 3 \, b c^{6} d e x^{4} + 3 \, b c^{6} d^{2} x^{2}\right )} \operatorname {arccsc}\left (c x\right ) + {\left (3 \, b c^{4} e^{2} x^{4} + 45 \, b c^{4} d^{2} + 30 \, b c^{2} d e + 8 \, b e^{2} + {\left (15 \, b c^{4} d e + 4 \, b c^{2} e^{2}\right )} x^{2}\right )} \sqrt {c^{2} x^{2} - 1}}{90 \, c^{6}} \]
1/90*(15*a*c^6*e^2*x^6 + 45*a*c^6*d*e*x^4 + 45*a*c^6*d^2*x^2 + 15*(b*c^6*e ^2*x^6 + 3*b*c^6*d*e*x^4 + 3*b*c^6*d^2*x^2)*arccsc(c*x) + (3*b*c^4*e^2*x^4 + 45*b*c^4*d^2 + 30*b*c^2*d*e + 8*b*e^2 + (15*b*c^4*d*e + 4*b*c^2*e^2)*x^ 2)*sqrt(c^2*x^2 - 1))/c^6
Time = 2.78 (sec) , antiderivative size = 352, normalized size of antiderivative = 1.81 \[ \int x \left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {a d^{2} x^{2}}{2} + \frac {a d e x^{4}}{2} + \frac {a e^{2} x^{6}}{6} + \frac {b d^{2} x^{2} \operatorname {acsc}{\left (c x \right )}}{2} + \frac {b d e x^{4} \operatorname {acsc}{\left (c x \right )}}{2} + \frac {b e^{2} x^{6} \operatorname {acsc}{\left (c x \right )}}{6} + \frac {b d^{2} \left (\begin {cases} \frac {\sqrt {c^{2} x^{2} - 1}}{c} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\\frac {i \sqrt {- c^{2} x^{2} + 1}}{c} & \text {otherwise} \end {cases}\right )}{2 c} + \frac {b d e \left (\begin {cases} \frac {x^{2} \sqrt {c^{2} x^{2} - 1}}{3 c} + \frac {2 \sqrt {c^{2} x^{2} - 1}}{3 c^{3}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\\frac {i x^{2} \sqrt {- c^{2} x^{2} + 1}}{3 c} + \frac {2 i \sqrt {- c^{2} x^{2} + 1}}{3 c^{3}} & \text {otherwise} \end {cases}\right )}{2 c} + \frac {b e^{2} \left (\begin {cases} \frac {x^{4} \sqrt {c^{2} x^{2} - 1}}{5 c} + \frac {4 x^{2} \sqrt {c^{2} x^{2} - 1}}{15 c^{3}} + \frac {8 \sqrt {c^{2} x^{2} - 1}}{15 c^{5}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\\frac {i x^{4} \sqrt {- c^{2} x^{2} + 1}}{5 c} + \frac {4 i x^{2} \sqrt {- c^{2} x^{2} + 1}}{15 c^{3}} + \frac {8 i \sqrt {- c^{2} x^{2} + 1}}{15 c^{5}} & \text {otherwise} \end {cases}\right )}{6 c} \]
a*d**2*x**2/2 + a*d*e*x**4/2 + a*e**2*x**6/6 + b*d**2*x**2*acsc(c*x)/2 + b *d*e*x**4*acsc(c*x)/2 + b*e**2*x**6*acsc(c*x)/6 + b*d**2*Piecewise((sqrt(c **2*x**2 - 1)/c, Abs(c**2*x**2) > 1), (I*sqrt(-c**2*x**2 + 1)/c, True))/(2 *c) + b*d*e*Piecewise((x**2*sqrt(c**2*x**2 - 1)/(3*c) + 2*sqrt(c**2*x**2 - 1)/(3*c**3), Abs(c**2*x**2) > 1), (I*x**2*sqrt(-c**2*x**2 + 1)/(3*c) + 2* I*sqrt(-c**2*x**2 + 1)/(3*c**3), True))/(2*c) + b*e**2*Piecewise((x**4*sqr t(c**2*x**2 - 1)/(5*c) + 4*x**2*sqrt(c**2*x**2 - 1)/(15*c**3) + 8*sqrt(c** 2*x**2 - 1)/(15*c**5), Abs(c**2*x**2) > 1), (I*x**4*sqrt(-c**2*x**2 + 1)/( 5*c) + 4*I*x**2*sqrt(-c**2*x**2 + 1)/(15*c**3) + 8*I*sqrt(-c**2*x**2 + 1)/ (15*c**5), True))/(6*c)
Time = 0.21 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.97 \[ \int x \left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {1}{6} \, a e^{2} x^{6} + \frac {1}{2} \, a d e x^{4} + \frac {1}{2} \, a d^{2} x^{2} + \frac {1}{2} \, {\left (x^{2} \operatorname {arccsc}\left (c x\right ) + \frac {x \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c}\right )} b d^{2} + \frac {1}{6} \, {\left (3 \, x^{4} \operatorname {arccsc}\left (c x\right ) + \frac {c^{2} x^{3} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + 3 \, x \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c^{3}}\right )} b d e + \frac {1}{90} \, {\left (15 \, x^{6} \operatorname {arccsc}\left (c x\right ) + \frac {3 \, c^{4} x^{5} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {5}{2}} + 10 \, c^{2} x^{3} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + 15 \, x \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c^{5}}\right )} b e^{2} \]
1/6*a*e^2*x^6 + 1/2*a*d*e*x^4 + 1/2*a*d^2*x^2 + 1/2*(x^2*arccsc(c*x) + x*s qrt(-1/(c^2*x^2) + 1)/c)*b*d^2 + 1/6*(3*x^4*arccsc(c*x) + (c^2*x^3*(-1/(c^ 2*x^2) + 1)^(3/2) + 3*x*sqrt(-1/(c^2*x^2) + 1))/c^3)*b*d*e + 1/90*(15*x^6* arccsc(c*x) + (3*c^4*x^5*(-1/(c^2*x^2) + 1)^(5/2) + 10*c^2*x^3*(-1/(c^2*x^ 2) + 1)^(3/2) + 15*x*sqrt(-1/(c^2*x^2) + 1))/c^5)*b*e^2
Leaf count of result is larger than twice the leaf count of optimal. 1160 vs. \(2 (169) = 338\).
Time = 0.39 (sec) , antiderivative size = 1160, normalized size of antiderivative = 5.95 \[ \int x \left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right ) \, dx=\text {Too large to display} \]
1/5760*(15*b*e^2*x^6*(sqrt(-1/(c^2*x^2) + 1) + 1)^6*arcsin(1/(c*x))/c + 15 *a*e^2*x^6*(sqrt(-1/(c^2*x^2) + 1) + 1)^6/c + 6*b*e^2*x^5*(sqrt(-1/(c^2*x^ 2) + 1) + 1)^5/c^2 + 180*b*d*e*x^4*(sqrt(-1/(c^2*x^2) + 1) + 1)^4*arcsin(1 /(c*x))/c + 180*a*d*e*x^4*(sqrt(-1/(c^2*x^2) + 1) + 1)^4/c + 90*b*e^2*x^4* (sqrt(-1/(c^2*x^2) + 1) + 1)^4*arcsin(1/(c*x))/c^3 + 90*a*e^2*x^4*(sqrt(-1 /(c^2*x^2) + 1) + 1)^4/c^3 + 120*b*d*e*x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3/ c^2 + 720*b*d^2*x^2*(sqrt(-1/(c^2*x^2) + 1) + 1)^2*arcsin(1/(c*x))/c + 720 *a*d^2*x^2*(sqrt(-1/(c^2*x^2) + 1) + 1)^2/c + 50*b*e^2*x^3*(sqrt(-1/(c^2*x ^2) + 1) + 1)^3/c^4 + 720*b*d*e*x^2*(sqrt(-1/(c^2*x^2) + 1) + 1)^2*arcsin( 1/(c*x))/c^3 + 720*a*d*e*x^2*(sqrt(-1/(c^2*x^2) + 1) + 1)^2/c^3 + 1440*b*d ^2*x*(sqrt(-1/(c^2*x^2) + 1) + 1)/c^2 + 225*b*e^2*x^2*(sqrt(-1/(c^2*x^2) + 1) + 1)^2*arcsin(1/(c*x))/c^5 + 225*a*e^2*x^2*(sqrt(-1/(c^2*x^2) + 1) + 1 )^2/c^5 + 1080*b*d*e*x*(sqrt(-1/(c^2*x^2) + 1) + 1)/c^4 + 1440*b*d^2*arcsi n(1/(c*x))/c^3 + 1440*a*d^2/c^3 + 300*b*e^2*x*(sqrt(-1/(c^2*x^2) + 1) + 1) /c^6 + 1080*b*d*e*arcsin(1/(c*x))/c^5 + 1080*a*d*e/c^5 - 1440*b*d^2/(c^4*x *(sqrt(-1/(c^2*x^2) + 1) + 1)) + 300*b*e^2*arcsin(1/(c*x))/c^7 + 300*a*e^2 /c^7 - 1080*b*d*e/(c^6*x*(sqrt(-1/(c^2*x^2) + 1) + 1)) + 720*b*d^2*arcsin( 1/(c*x))/(c^5*x^2*(sqrt(-1/(c^2*x^2) + 1) + 1)^2) + 720*a*d^2/(c^5*x^2*(sq rt(-1/(c^2*x^2) + 1) + 1)^2) - 300*b*e^2/(c^8*x*(sqrt(-1/(c^2*x^2) + 1) + 1)) + 720*b*d*e*arcsin(1/(c*x))/(c^7*x^2*(sqrt(-1/(c^2*x^2) + 1) + 1)^2...
Timed out. \[ \int x \left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right ) \, dx=\int x\,{\left (e\,x^2+d\right )}^2\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right ) \,d x \]