Integrand size = 21, antiderivative size = 242 \[ \int x^3 \left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {b \left (6 c^4 d^2+8 c^2 d e+3 e^2\right ) x \sqrt {-1+c^2 x^2}}{24 c^7 \sqrt {c^2 x^2}}+\frac {b \left (6 c^4 d^2+16 c^2 d e+9 e^2\right ) x \left (-1+c^2 x^2\right )^{3/2}}{72 c^7 \sqrt {c^2 x^2}}+\frac {b e \left (8 c^2 d+9 e\right ) x \left (-1+c^2 x^2\right )^{5/2}}{120 c^7 \sqrt {c^2 x^2}}+\frac {b e^2 x \left (-1+c^2 x^2\right )^{7/2}}{56 c^7 \sqrt {c^2 x^2}}+\frac {1}{4} d^2 x^4 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{3} d e x^6 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{8} e^2 x^8 \left (a+b \csc ^{-1}(c x)\right ) \]
1/4*d^2*x^4*(a+b*arccsc(c*x))+1/3*d*e*x^6*(a+b*arccsc(c*x))+1/8*e^2*x^8*(a +b*arccsc(c*x))+1/72*b*(6*c^4*d^2+16*c^2*d*e+9*e^2)*x*(c^2*x^2-1)^(3/2)/c^ 7/(c^2*x^2)^(1/2)+1/120*b*e*(8*c^2*d+9*e)*x*(c^2*x^2-1)^(5/2)/c^7/(c^2*x^2 )^(1/2)+1/56*b*e^2*x*(c^2*x^2-1)^(7/2)/c^7/(c^2*x^2)^(1/2)+1/24*b*(6*c^4*d ^2+8*c^2*d*e+3*e^2)*x*(c^2*x^2-1)^(1/2)/c^7/(c^2*x^2)^(1/2)
Time = 0.20 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.66 \[ \int x^3 \left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {x \left (105 a x^3 \left (6 d^2+8 d e x^2+3 e^2 x^4\right )+\frac {b \sqrt {1-\frac {1}{c^2 x^2}} \left (144 e^2+8 c^2 e \left (56 d+9 e x^2\right )+c^4 \left (420 d^2+224 d e x^2+54 e^2 x^4\right )+3 c^6 \left (70 d^2 x^2+56 d e x^4+15 e^2 x^6\right )\right )}{c^7}+105 b x^3 \left (6 d^2+8 d e x^2+3 e^2 x^4\right ) \csc ^{-1}(c x)\right )}{2520} \]
(x*(105*a*x^3*(6*d^2 + 8*d*e*x^2 + 3*e^2*x^4) + (b*Sqrt[1 - 1/(c^2*x^2)]*( 144*e^2 + 8*c^2*e*(56*d + 9*e*x^2) + c^4*(420*d^2 + 224*d*e*x^2 + 54*e^2*x ^4) + 3*c^6*(70*d^2*x^2 + 56*d*e*x^4 + 15*e^2*x^6)))/c^7 + 105*b*x^3*(6*d^ 2 + 8*d*e*x^2 + 3*e^2*x^4)*ArcCsc[c*x]))/2520
Time = 0.49 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.86, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {5762, 27, 1578, 1195, 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.
| \(\displaystyle \int x^3 \left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right ) \, dx\) |
| \(\Big \downarrow \) 5762 |
| \(\displaystyle \frac {b c x \int \frac {x^3 \left (3 e^2 x^4+8 d e x^2+6 d^2\right )}{24 \sqrt {c^2 x^2-1}}dx}{\sqrt {c^2 x^2}}+\frac {1}{4} d^2 x^4 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{3} d e x^6 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{8} e^2 x^8 \left (a+b \csc ^{-1}(c x)\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b c x \int \frac {x^3 \left (3 e^2 x^4+8 d e x^2+6 d^2\right )}{\sqrt {c^2 x^2-1}}dx}{24 \sqrt {c^2 x^2}}+\frac {1}{4} d^2 x^4 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{3} d e x^6 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{8} e^2 x^8 \left (a+b \csc ^{-1}(c x)\right )\) |
| \(\Big \downarrow \) 1578 |
| \(\displaystyle \frac {b c x \int \frac {x^2 \left (3 e^2 x^4+8 d e x^2+6 d^2\right )}{\sqrt {c^2 x^2-1}}dx^2}{48 \sqrt {c^2 x^2}}+\frac {1}{4} d^2 x^4 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{3} d e x^6 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{8} e^2 x^8 \left (a+b \csc ^{-1}(c x)\right )\) |
| \(\Big \downarrow \) 1195 |
| \(\displaystyle \frac {b c x \int \left (\frac {3 e^2 \left (c^2 x^2-1\right )^{5/2}}{c^6}+\frac {e \left (8 d c^2+9 e\right ) \left (c^2 x^2-1\right )^{3/2}}{c^6}+\frac {\left (6 d^2 c^4+16 d e c^2+9 e^2\right ) \sqrt {c^2 x^2-1}}{c^6}+\frac {6 d^2 c^4+8 d e c^2+3 e^2}{c^6 \sqrt {c^2 x^2-1}}\right )dx^2}{48 \sqrt {c^2 x^2}}+\frac {1}{4} d^2 x^4 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{3} d e x^6 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{8} e^2 x^8 \left (a+b \csc ^{-1}(c x)\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{4} d^2 x^4 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{3} d e x^6 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{8} e^2 x^8 \left (a+b \csc ^{-1}(c x)\right )+\frac {b c x \left (\frac {2 e \left (c^2 x^2-1\right )^{5/2} \left (8 c^2 d+9 e\right )}{5 c^8}+\frac {6 e^2 \left (c^2 x^2-1\right )^{7/2}}{7 c^8}+\frac {2 \left (c^2 x^2-1\right )^{3/2} \left (6 c^4 d^2+16 c^2 d e+9 e^2\right )}{3 c^8}+\frac {2 \sqrt {c^2 x^2-1} \left (6 c^4 d^2+8 c^2 d e+3 e^2\right )}{c^8}\right )}{48 \sqrt {c^2 x^2}}\) |
(b*c*x*((2*(6*c^4*d^2 + 8*c^2*d*e + 3*e^2)*Sqrt[-1 + c^2*x^2])/c^8 + (2*(6 *c^4*d^2 + 16*c^2*d*e + 9*e^2)*(-1 + c^2*x^2)^(3/2))/(3*c^8) + (2*e*(8*c^2 *d + 9*e)*(-1 + c^2*x^2)^(5/2))/(5*c^8) + (6*e^2*(-1 + c^2*x^2)^(7/2))/(7* c^8)))/(48*Sqrt[c^2*x^2]) + (d^2*x^4*(a + b*ArcCsc[c*x]))/4 + (d*e*x^6*(a + b*ArcCsc[c*x]))/3 + (e^2*x^8*(a + b*ArcCsc[c*x]))/8
3.1.94.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x _) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x ] && IGtQ[p, 0]
Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_ )^4)^(p_.), x_Symbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && Int egerQ[(m - 1)/2]
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]))
Time = 1.05 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.82
| method | result | size |
| parts | \(a \left (\frac {1}{8} e^{2} x^{8}+\frac {1}{3} d e \,x^{6}+\frac {1}{4} x^{4} d^{2}\right )+\frac {b \left (\frac {c^{4} \operatorname {arccsc}\left (c x \right ) e^{2} x^{8}}{8}+\frac {c^{4} \operatorname {arccsc}\left (c x \right ) d e \,x^{6}}{3}+\frac {\operatorname {arccsc}\left (c x \right ) d^{2} x^{4} c^{4}}{4}+\frac {\left (c^{2} x^{2}-1\right ) \left (45 c^{6} e^{2} x^{6}+168 c^{6} d e \,x^{4}+210 c^{6} d^{2} x^{2}+54 c^{4} e^{2} x^{4}+224 c^{4} d e \,x^{2}+420 c^{4} d^{2}+72 c^{2} e^{2} x^{2}+448 c^{2} d e +144 e^{2}\right )}{2520 c^{5} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}\right )}{c^{4}}\) | \(198\) |
| derivativedivides | \(\frac {-\frac {a \left (\frac {c^{2} d \left (c^{2} e \,x^{2}+c^{2} d \right )^{3}}{3}-\frac {\left (c^{2} e \,x^{2}+c^{2} d \right )^{4}}{4}\right )}{2 c^{4} e^{2}}-\frac {b \,c^{4} \operatorname {arccsc}\left (c x \right ) d^{4}}{24 e^{2}}+\frac {b \,\operatorname {arccsc}\left (c x \right ) d^{2} c^{4} x^{4}}{4}+\frac {b \,c^{4} e \,\operatorname {arccsc}\left (c x \right ) d \,x^{6}}{3}+\frac {b \,c^{4} e^{2} \operatorname {arccsc}\left (c x \right ) x^{8}}{8}+\frac {b \,c^{3} \sqrt {c^{2} x^{2}-1}\, d^{4} \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{24 e^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}+\frac {b \left (c^{2} x^{2}-1\right ) c x \,d^{2}}{12 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {b c e \left (c^{2} x^{2}-1\right ) x^{3} d}{15 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {b c \,e^{2} \left (c^{2} x^{2}-1\right ) x^{5}}{56 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {b \left (c^{2} x^{2}-1\right ) d^{2}}{6 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}+\frac {4 b e \left (c^{2} x^{2}-1\right ) x d}{45 c \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {3 b \,e^{2} \left (c^{2} x^{2}-1\right ) x^{3}}{140 c \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {8 b e \left (c^{2} x^{2}-1\right ) d}{45 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}+\frac {b \left (c^{2} x^{2}-1\right ) x \,e^{2}}{35 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {2 b \,e^{2} \left (c^{2} x^{2}-1\right )}{35 c^{5} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}}{c^{4}}\) | \(511\) |
| default | \(\frac {-\frac {a \left (\frac {c^{2} d \left (c^{2} e \,x^{2}+c^{2} d \right )^{3}}{3}-\frac {\left (c^{2} e \,x^{2}+c^{2} d \right )^{4}}{4}\right )}{2 c^{4} e^{2}}-\frac {b \,c^{4} \operatorname {arccsc}\left (c x \right ) d^{4}}{24 e^{2}}+\frac {b \,\operatorname {arccsc}\left (c x \right ) d^{2} c^{4} x^{4}}{4}+\frac {b \,c^{4} e \,\operatorname {arccsc}\left (c x \right ) d \,x^{6}}{3}+\frac {b \,c^{4} e^{2} \operatorname {arccsc}\left (c x \right ) x^{8}}{8}+\frac {b \,c^{3} \sqrt {c^{2} x^{2}-1}\, d^{4} \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{24 e^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}+\frac {b \left (c^{2} x^{2}-1\right ) c x \,d^{2}}{12 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {b c e \left (c^{2} x^{2}-1\right ) x^{3} d}{15 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {b c \,e^{2} \left (c^{2} x^{2}-1\right ) x^{5}}{56 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {b \left (c^{2} x^{2}-1\right ) d^{2}}{6 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}+\frac {4 b e \left (c^{2} x^{2}-1\right ) x d}{45 c \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {3 b \,e^{2} \left (c^{2} x^{2}-1\right ) x^{3}}{140 c \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {8 b e \left (c^{2} x^{2}-1\right ) d}{45 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}+\frac {b \left (c^{2} x^{2}-1\right ) x \,e^{2}}{35 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {2 b \,e^{2} \left (c^{2} x^{2}-1\right )}{35 c^{5} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}}{c^{4}}\) | \(511\) |
a*(1/8*e^2*x^8+1/3*d*e*x^6+1/4*x^4*d^2)+b/c^4*(1/8*c^4*arccsc(c*x)*e^2*x^8 +1/3*c^4*arccsc(c*x)*d*e*x^6+1/4*arccsc(c*x)*d^2*x^4*c^4+1/2520/c^5*(c^2*x ^2-1)*(45*c^6*e^2*x^6+168*c^6*d*e*x^4+210*c^6*d^2*x^2+54*c^4*e^2*x^4+224*c ^4*d*e*x^2+420*c^4*d^2+72*c^2*e^2*x^2+448*c^2*d*e+144*e^2)/((c^2*x^2-1)/c^ 2/x^2)^(1/2)/x)
Time = 0.34 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.77 \[ \int x^3 \left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {315 \, a c^{8} e^{2} x^{8} + 840 \, a c^{8} d e x^{6} + 630 \, a c^{8} d^{2} x^{4} + 105 \, {\left (3 \, b c^{8} e^{2} x^{8} + 8 \, b c^{8} d e x^{6} + 6 \, b c^{8} d^{2} x^{4}\right )} \operatorname {arccsc}\left (c x\right ) + {\left (45 \, b c^{6} e^{2} x^{6} + 420 \, b c^{4} d^{2} + 448 \, b c^{2} d e + 6 \, {\left (28 \, b c^{6} d e + 9 \, b c^{4} e^{2}\right )} x^{4} + 144 \, b e^{2} + 2 \, {\left (105 \, b c^{6} d^{2} + 112 \, b c^{4} d e + 36 \, b c^{2} e^{2}\right )} x^{2}\right )} \sqrt {c^{2} x^{2} - 1}}{2520 \, c^{8}} \]
1/2520*(315*a*c^8*e^2*x^8 + 840*a*c^8*d*e*x^6 + 630*a*c^8*d^2*x^4 + 105*(3 *b*c^8*e^2*x^8 + 8*b*c^8*d*e*x^6 + 6*b*c^8*d^2*x^4)*arccsc(c*x) + (45*b*c^ 6*e^2*x^6 + 420*b*c^4*d^2 + 448*b*c^2*d*e + 6*(28*b*c^6*d*e + 9*b*c^4*e^2) *x^4 + 144*b*e^2 + 2*(105*b*c^6*d^2 + 112*b*c^4*d*e + 36*b*c^2*e^2)*x^2)*s qrt(c^2*x^2 - 1))/c^8
Time = 4.39 (sec) , antiderivative size = 493, normalized size of antiderivative = 2.04 \[ \int x^3 \left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {a d^{2} x^{4}}{4} + \frac {a d e x^{6}}{3} + \frac {a e^{2} x^{8}}{8} + \frac {b d^{2} x^{4} \operatorname {acsc}{\left (c x \right )}}{4} + \frac {b d e x^{6} \operatorname {acsc}{\left (c x \right )}}{3} + \frac {b e^{2} x^{8} \operatorname {acsc}{\left (c x \right )}}{8} + \frac {b d^{2} \left (\begin {cases} \frac {x^{2} \sqrt {c^{2} x^{2} - 1}}{3 c} + \frac {2 \sqrt {c^{2} x^{2} - 1}}{3 c^{3}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\\frac {i x^{2} \sqrt {- c^{2} x^{2} + 1}}{3 c} + \frac {2 i \sqrt {- c^{2} x^{2} + 1}}{3 c^{3}} & \text {otherwise} \end {cases}\right )}{4 c} + \frac {b d e \left (\begin {cases} \frac {x^{4} \sqrt {c^{2} x^{2} - 1}}{5 c} + \frac {4 x^{2} \sqrt {c^{2} x^{2} - 1}}{15 c^{3}} + \frac {8 \sqrt {c^{2} x^{2} - 1}}{15 c^{5}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\\frac {i x^{4} \sqrt {- c^{2} x^{2} + 1}}{5 c} + \frac {4 i x^{2} \sqrt {- c^{2} x^{2} + 1}}{15 c^{3}} + \frac {8 i \sqrt {- c^{2} x^{2} + 1}}{15 c^{5}} & \text {otherwise} \end {cases}\right )}{3 c} + \frac {b e^{2} \left (\begin {cases} \frac {x^{6} \sqrt {c^{2} x^{2} - 1}}{7 c} + \frac {6 x^{4} \sqrt {c^{2} x^{2} - 1}}{35 c^{3}} + \frac {8 x^{2} \sqrt {c^{2} x^{2} - 1}}{35 c^{5}} + \frac {16 \sqrt {c^{2} x^{2} - 1}}{35 c^{7}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\\frac {i x^{6} \sqrt {- c^{2} x^{2} + 1}}{7 c} + \frac {6 i x^{4} \sqrt {- c^{2} x^{2} + 1}}{35 c^{3}} + \frac {8 i x^{2} \sqrt {- c^{2} x^{2} + 1}}{35 c^{5}} + \frac {16 i \sqrt {- c^{2} x^{2} + 1}}{35 c^{7}} & \text {otherwise} \end {cases}\right )}{8 c} \]
a*d**2*x**4/4 + a*d*e*x**6/3 + a*e**2*x**8/8 + b*d**2*x**4*acsc(c*x)/4 + b *d*e*x**6*acsc(c*x)/3 + b*e**2*x**8*acsc(c*x)/8 + b*d**2*Piecewise((x**2*s qrt(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))/(4*c) + b*d*e*Piecewise((x**4*sqrt(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))/(3*c) + b*e* *2*Piecewise((x**6*sqrt(c**2*x**2 - 1)/(7*c) + 6*x**4*sqrt(c**2*x**2 - 1)/ (35*c**3) + 8*x**2*sqrt(c**2*x**2 - 1)/(35*c**5) + 16*sqrt(c**2*x**2 - 1)/ (35*c**7), Abs(c**2*x**2) > 1), (I*x**6*sqrt(-c**2*x**2 + 1)/(7*c) + 6*I*x **4*sqrt(-c**2*x**2 + 1)/(35*c**3) + 8*I*x**2*sqrt(-c**2*x**2 + 1)/(35*c** 5) + 16*I*sqrt(-c**2*x**2 + 1)/(35*c**7), True))/(8*c)
Time = 0.21 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.05 \[ \int x^3 \left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {1}{8} \, a e^{2} x^{8} + \frac {1}{3} \, a d e x^{6} + \frac {1}{4} \, a d^{2} x^{4} + \frac {1}{12} \, {\left (3 \, x^{4} \operatorname {arccsc}\left (c x\right ) + \frac {c^{2} x^{3} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + 3 \, x \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c^{3}}\right )} b d^{2} + \frac {1}{45} \, {\left (15 \, x^{6} \operatorname {arccsc}\left (c x\right ) + \frac {3 \, c^{4} x^{5} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {5}{2}} + 10 \, c^{2} x^{3} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + 15 \, x \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c^{5}}\right )} b d e + \frac {1}{280} \, {\left (35 \, x^{8} \operatorname {arccsc}\left (c x\right ) + \frac {5 \, c^{6} x^{7} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {7}{2}} + 21 \, c^{4} x^{5} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {5}{2}} + 35 \, c^{2} x^{3} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + 35 \, x \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c^{7}}\right )} b e^{2} \]
1/8*a*e^2*x^8 + 1/3*a*d*e*x^6 + 1/4*a*d^2*x^4 + 1/12*(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^2 + 1/45*(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*d*e + 1/280*(35*x^8*arccsc(c*x) + (5*c^6*x^7*(-1/(c^2*x^2) + 1)^(7/2) + 21*c^4*x ^5*(-1/(c^2*x^2) + 1)^(5/2) + 35*c^2*x^3*(-1/(c^2*x^2) + 1)^(3/2) + 35*x*s qrt(-1/(c^2*x^2) + 1))/c^7)*b*e^2
Leaf count of result is larger than twice the leaf count of optimal. 1706 vs. \(2 (212) = 424\).
Time = 0.45 (sec) , antiderivative size = 1706, normalized size of antiderivative = 7.05 \[ \int x^3 \left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right ) \, dx=\text {Too large to display} \]
1/645120*(315*b*e^2*x^8*(sqrt(-1/(c^2*x^2) + 1) + 1)^8*arcsin(1/(c*x))/c + 315*a*e^2*x^8*(sqrt(-1/(c^2*x^2) + 1) + 1)^8/c + 90*b*e^2*x^7*(sqrt(-1/(c ^2*x^2) + 1) + 1)^7/c^2 + 3360*b*d*e*x^6*(sqrt(-1/(c^2*x^2) + 1) + 1)^6*ar csin(1/(c*x))/c + 3360*a*d*e*x^6*(sqrt(-1/(c^2*x^2) + 1) + 1)^6/c + 2520*b *e^2*x^6*(sqrt(-1/(c^2*x^2) + 1) + 1)^6*arcsin(1/(c*x))/c^3 + 2520*a*e^2*x ^6*(sqrt(-1/(c^2*x^2) + 1) + 1)^6/c^3 + 1344*b*d*e*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5/c^2 + 10080*b*d^2*x^4*(sqrt(-1/(c^2*x^2) + 1) + 1)^4*arcsin(1/ (c*x))/c + 10080*a*d^2*x^4*(sqrt(-1/(c^2*x^2) + 1) + 1)^4/c + 882*b*e^2*x^ 5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5/c^4 + 20160*b*d*e*x^4*(sqrt(-1/(c^2*x^2) + 1) + 1)^4*arcsin(1/(c*x))/c^3 + 20160*a*d*e*x^4*(sqrt(-1/(c^2*x^2) + 1) + 1)^4/c^3 + 6720*b*d^2*x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3/c^2 + 8820*b*e^ 2*x^4*(sqrt(-1/(c^2*x^2) + 1) + 1)^4*arcsin(1/(c*x))/c^5 + 8820*a*e^2*x^4* (sqrt(-1/(c^2*x^2) + 1) + 1)^4/c^5 + 11200*b*d*e*x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3/c^4 + 40320*b*d^2*x^2*(sqrt(-1/(c^2*x^2) + 1) + 1)^2*arcsin(1/(c *x))/c^3 + 40320*a*d^2*x^2*(sqrt(-1/(c^2*x^2) + 1) + 1)^2/c^3 + 4410*b*e^2 *x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3/c^6 + 50400*b*d*e*x^2*(sqrt(-1/(c^2*x^ 2) + 1) + 1)^2*arcsin(1/(c*x))/c^5 + 50400*a*d*e*x^2*(sqrt(-1/(c^2*x^2) + 1) + 1)^2/c^5 + 60480*b*d^2*x*(sqrt(-1/(c^2*x^2) + 1) + 1)/c^4 + 17640*b*e ^2*x^2*(sqrt(-1/(c^2*x^2) + 1) + 1)^2*arcsin(1/(c*x))/c^7 + 17640*a*e^2*x^ 2*(sqrt(-1/(c^2*x^2) + 1) + 1)^2/c^7 + 67200*b*d*e*x*(sqrt(-1/(c^2*x^2)...
Timed out. \[ \int x^3 \left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right ) \, dx=\int x^3\,{\left (e\,x^2+d\right )}^2\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right ) \,d x \]