Integrand size = 21, antiderivative size = 252 \[ \int x^2 \left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {b \left (280 c^4 d^2+252 c^2 d e+75 e^2\right ) x^2 \sqrt {-1+c^2 x^2}}{1680 c^5 \sqrt {c^2 x^2}}+\frac {b e \left (84 c^2 d+25 e\right ) x^4 \sqrt {-1+c^2 x^2}}{840 c^3 \sqrt {c^2 x^2}}+\frac {b e^2 x^6 \sqrt {-1+c^2 x^2}}{42 c \sqrt {c^2 x^2}}+\frac {1}{3} d^2 x^3 \left (a+b \csc ^{-1}(c x)\right )+\frac {2}{5} d e x^5 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{7} e^2 x^7 \left (a+b \csc ^{-1}(c x)\right )+\frac {b \left (280 c^4 d^2+252 c^2 d e+75 e^2\right ) x \text {arctanh}\left (\frac {c x}{\sqrt {-1+c^2 x^2}}\right )}{1680 c^6 \sqrt {c^2 x^2}} \]
1/3*d^2*x^3*(a+b*arccsc(c*x))+2/5*d*e*x^5*(a+b*arccsc(c*x))+1/7*e^2*x^7*(a +b*arccsc(c*x))+1/1680*b*(280*c^4*d^2+252*c^2*d*e+75*e^2)*x*arctanh(c*x/(c ^2*x^2-1)^(1/2))/c^6/(c^2*x^2)^(1/2)+1/1680*b*(280*c^4*d^2+252*c^2*d*e+75* e^2)*x^2*(c^2*x^2-1)^(1/2)/c^5/(c^2*x^2)^(1/2)+1/840*b*e*(84*c^2*d+25*e)*x ^4*(c^2*x^2-1)^(1/2)/c^3/(c^2*x^2)^(1/2)+1/42*b*e^2*x^6*(c^2*x^2-1)^(1/2)/ c/(c^2*x^2)^(1/2)
Time = 0.23 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.73 \[ \int x^2 \left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {c^2 x^2 \left (16 a c^5 x \left (35 d^2+42 d e x^2+15 e^2 x^4\right )+b \sqrt {1-\frac {1}{c^2 x^2}} \left (75 e^2+2 c^2 e \left (126 d+25 e x^2\right )+8 c^4 \left (35 d^2+21 d e x^2+5 e^2 x^4\right )\right )\right )+16 b c^7 x^3 \left (35 d^2+42 d e x^2+15 e^2 x^4\right ) \csc ^{-1}(c x)+b \left (280 c^4 d^2+252 c^2 d e+75 e^2\right ) \log \left (\left (1+\sqrt {1-\frac {1}{c^2 x^2}}\right ) x\right )}{1680 c^7} \]
(c^2*x^2*(16*a*c^5*x*(35*d^2 + 42*d*e*x^2 + 15*e^2*x^4) + b*Sqrt[1 - 1/(c^ 2*x^2)]*(75*e^2 + 2*c^2*e*(126*d + 25*e*x^2) + 8*c^4*(35*d^2 + 21*d*e*x^2 + 5*e^2*x^4))) + 16*b*c^7*x^3*(35*d^2 + 42*d*e*x^2 + 15*e^2*x^4)*ArcCsc[c* x] + b*(280*c^4*d^2 + 252*c^2*d*e + 75*e^2)*Log[(1 + Sqrt[1 - 1/(c^2*x^2)] )*x])/(1680*c^7)
Time = 0.45 (sec) , antiderivative size = 215, normalized size of antiderivative = 0.85, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {5762, 27, 1590, 27, 363, 262, 224, 219}
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.
| \(\displaystyle \int x^2 \left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right ) \, dx\) |
| \(\Big \downarrow \) 5762 |
| \(\displaystyle \frac {b c x \int \frac {x^2 \left (15 e^2 x^4+42 d e x^2+35 d^2\right )}{105 \sqrt {c^2 x^2-1}}dx}{\sqrt {c^2 x^2}}+\frac {1}{3} d^2 x^3 \left (a+b \csc ^{-1}(c x)\right )+\frac {2}{5} d e x^5 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{7} e^2 x^7 \left (a+b \csc ^{-1}(c x)\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b c x \int \frac {x^2 \left (15 e^2 x^4+42 d e x^2+35 d^2\right )}{\sqrt {c^2 x^2-1}}dx}{105 \sqrt {c^2 x^2}}+\frac {1}{3} d^2 x^3 \left (a+b \csc ^{-1}(c x)\right )+\frac {2}{5} d e x^5 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{7} e^2 x^7 \left (a+b \csc ^{-1}(c x)\right )\) |
| \(\Big \downarrow \) 1590 |
| \(\displaystyle \frac {b c x \left (\frac {\int \frac {3 x^2 \left (70 c^2 d^2+e \left (84 d c^2+25 e\right ) x^2\right )}{\sqrt {c^2 x^2-1}}dx}{6 c^2}+\frac {5 e^2 x^5 \sqrt {c^2 x^2-1}}{2 c^2}\right )}{105 \sqrt {c^2 x^2}}+\frac {1}{3} d^2 x^3 \left (a+b \csc ^{-1}(c x)\right )+\frac {2}{5} d e x^5 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{7} e^2 x^7 \left (a+b \csc ^{-1}(c x)\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b c x \left (\frac {\int \frac {x^2 \left (70 c^2 d^2+e \left (84 d c^2+25 e\right ) x^2\right )}{\sqrt {c^2 x^2-1}}dx}{2 c^2}+\frac {5 e^2 x^5 \sqrt {c^2 x^2-1}}{2 c^2}\right )}{105 \sqrt {c^2 x^2}}+\frac {1}{3} d^2 x^3 \left (a+b \csc ^{-1}(c x)\right )+\frac {2}{5} d e x^5 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{7} e^2 x^7 \left (a+b \csc ^{-1}(c x)\right )\) |
| \(\Big \downarrow \) 363 |
| \(\displaystyle \frac {b c x \left (\frac {\frac {\left (280 c^4 d^2+252 c^2 d e+75 e^2\right ) \int \frac {x^2}{\sqrt {c^2 x^2-1}}dx}{4 c^2}+\frac {e x^3 \sqrt {c^2 x^2-1} \left (84 c^2 d+25 e\right )}{4 c^2}}{2 c^2}+\frac {5 e^2 x^5 \sqrt {c^2 x^2-1}}{2 c^2}\right )}{105 \sqrt {c^2 x^2}}+\frac {1}{3} d^2 x^3 \left (a+b \csc ^{-1}(c x)\right )+\frac {2}{5} d e x^5 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{7} e^2 x^7 \left (a+b \csc ^{-1}(c x)\right )\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {b c x \left (\frac {\frac {\left (280 c^4 d^2+252 c^2 d e+75 e^2\right ) \left (\frac {\int \frac {1}{\sqrt {c^2 x^2-1}}dx}{2 c^2}+\frac {x \sqrt {c^2 x^2-1}}{2 c^2}\right )}{4 c^2}+\frac {e x^3 \sqrt {c^2 x^2-1} \left (84 c^2 d+25 e\right )}{4 c^2}}{2 c^2}+\frac {5 e^2 x^5 \sqrt {c^2 x^2-1}}{2 c^2}\right )}{105 \sqrt {c^2 x^2}}+\frac {1}{3} d^2 x^3 \left (a+b \csc ^{-1}(c x)\right )+\frac {2}{5} d e x^5 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{7} e^2 x^7 \left (a+b \csc ^{-1}(c x)\right )\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {b c x \left (\frac {\frac {\left (280 c^4 d^2+252 c^2 d e+75 e^2\right ) \left (\frac {\int \frac {1}{1-\frac {c^2 x^2}{c^2 x^2-1}}d\frac {x}{\sqrt {c^2 x^2-1}}}{2 c^2}+\frac {x \sqrt {c^2 x^2-1}}{2 c^2}\right )}{4 c^2}+\frac {e x^3 \sqrt {c^2 x^2-1} \left (84 c^2 d+25 e\right )}{4 c^2}}{2 c^2}+\frac {5 e^2 x^5 \sqrt {c^2 x^2-1}}{2 c^2}\right )}{105 \sqrt {c^2 x^2}}+\frac {1}{3} d^2 x^3 \left (a+b \csc ^{-1}(c x)\right )+\frac {2}{5} d e x^5 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{7} e^2 x^7 \left (a+b \csc ^{-1}(c x)\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{3} d^2 x^3 \left (a+b \csc ^{-1}(c x)\right )+\frac {2}{5} d e x^5 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{7} e^2 x^7 \left (a+b \csc ^{-1}(c x)\right )+\frac {b c x \left (\frac {\frac {\left (\frac {\text {arctanh}\left (\frac {c x}{\sqrt {c^2 x^2-1}}\right )}{2 c^3}+\frac {x \sqrt {c^2 x^2-1}}{2 c^2}\right ) \left (280 c^4 d^2+252 c^2 d e+75 e^2\right )}{4 c^2}+\frac {e x^3 \sqrt {c^2 x^2-1} \left (84 c^2 d+25 e\right )}{4 c^2}}{2 c^2}+\frac {5 e^2 x^5 \sqrt {c^2 x^2-1}}{2 c^2}\right )}{105 \sqrt {c^2 x^2}}\) |
(d^2*x^3*(a + b*ArcCsc[c*x]))/3 + (2*d*e*x^5*(a + b*ArcCsc[c*x]))/5 + (e^2 *x^7*(a + b*ArcCsc[c*x]))/7 + (b*c*x*((5*e^2*x^5*Sqrt[-1 + c^2*x^2])/(2*c^ 2) + ((e*(84*c^2*d + 25*e)*x^3*Sqrt[-1 + c^2*x^2])/(4*c^2) + ((280*c^4*d^2 + 252*c^2*d*e + 75*e^2)*((x*Sqrt[-1 + c^2*x^2])/(2*c^2) + ArcTanh[(c*x)/S qrt[-1 + c^2*x^2]]/(2*c^3)))/(4*c^2))/(2*c^2)))/(105*Sqrt[c^2*x^2])
3.1.88.3.1 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[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c , 2, m, p, x]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x _Symbol] :> Simp[d*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(b*e*(m + 2*p + 3))), x] - Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(b*(m + 2*p + 3)) Int[(e*x)^ m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b*c - a*d , 0] && NeQ[m + 2*p + 3, 0]
Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + ( c_.)*(x_)^4)^(p_.), x_Symbol] :> Simp[c^p*(f*x)^(m + 4*p - 1)*((d + e*x^2)^ (q + 1)/(e*f^(4*p - 1)*(m + 4*p + 2*q + 1))), x] + Simp[1/(e*(m + 4*p + 2*q + 1)) Int[(f*x)^m*(d + e*x^2)^q*ExpandToSum[e*(m + 4*p + 2*q + 1)*((a + b*x^2 + c*x^4)^p - c^p*x^(4*p)) - d*c^p*(m + 4*p - 1)*x^(4*p - 2), x], x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && !IntegerQ[q] && NeQ[m + 4*p + 2*q + 1, 0]
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]))
Leaf count of result is larger than twice the leaf count of optimal. \(458\) vs. \(2(222)=444\).
Time = 0.97 (sec) , antiderivative size = 459, normalized size of antiderivative = 1.82
| method | result | size |
| parts | \(a \left (\frac {1}{7} e^{2} x^{7}+\frac {2}{5} d e \,x^{5}+\frac {1}{3} x^{3} d^{2}\right )+\frac {b \,\operatorname {arccsc}\left (c x \right ) e^{2} x^{7}}{7}+\frac {2 b \,\operatorname {arccsc}\left (c x \right ) d e \,x^{5}}{5}+\frac {b \,\operatorname {arccsc}\left (c x \right ) d^{2} x^{3}}{3}+\frac {b \left (c^{2} x^{2}-1\right ) x^{4} e^{2}}{42 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {b \left (c^{2} x^{2}-1\right ) x^{2} d e}{10 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {5 b \left (c^{2} x^{2}-1\right ) x^{2} e^{2}}{168 c^{5} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {b \left (c^{2} x^{2}-1\right ) d^{2}}{6 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {3 b \left (c^{2} x^{2}-1\right ) d e}{20 c^{5} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {b \sqrt {c^{2} x^{2}-1}\, d^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{6 c^{4} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}+\frac {5 b \left (c^{2} x^{2}-1\right ) e^{2}}{112 c^{7} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {3 b \sqrt {c^{2} x^{2}-1}\, d e \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{20 c^{6} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}+\frac {5 b \sqrt {c^{2} x^{2}-1}\, e^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{112 c^{8} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}\) | \(459\) |
| derivativedivides | \(\frac {\frac {a \left (\frac {1}{3} d^{2} c^{7} x^{3}+\frac {2}{5} d \,c^{7} e \,x^{5}+\frac {1}{7} e^{2} c^{7} x^{7}\right )}{c^{4}}+\frac {b \,\operatorname {arccsc}\left (c x \right ) d^{2} c^{3} x^{3}}{3}+\frac {2 b \,c^{3} \operatorname {arccsc}\left (c x \right ) d e \,x^{5}}{5}+\frac {b \,c^{3} \operatorname {arccsc}\left (c x \right ) e^{2} x^{7}}{7}+\frac {b \left (c^{2} x^{2}-1\right ) d^{2}}{6 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {b \left (c^{2} x^{2}-1\right ) x^{2} d e}{10 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {b \left (c^{2} x^{2}-1\right ) x^{4} e^{2}}{42 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {b \sqrt {c^{2} x^{2}-1}\, d^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{6 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}+\frac {3 b \left (c^{2} x^{2}-1\right ) d e}{20 c^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {5 b \left (c^{2} x^{2}-1\right ) x^{2} e^{2}}{168 c^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {3 b \sqrt {c^{2} x^{2}-1}\, d e \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{20 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}+\frac {5 b \left (c^{2} x^{2}-1\right ) e^{2}}{112 c^{4} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {5 b \sqrt {c^{2} x^{2}-1}\, e^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{112 c^{5} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}}{c^{3}}\) | \(475\) |
| default | \(\frac {\frac {a \left (\frac {1}{3} d^{2} c^{7} x^{3}+\frac {2}{5} d \,c^{7} e \,x^{5}+\frac {1}{7} e^{2} c^{7} x^{7}\right )}{c^{4}}+\frac {b \,\operatorname {arccsc}\left (c x \right ) d^{2} c^{3} x^{3}}{3}+\frac {2 b \,c^{3} \operatorname {arccsc}\left (c x \right ) d e \,x^{5}}{5}+\frac {b \,c^{3} \operatorname {arccsc}\left (c x \right ) e^{2} x^{7}}{7}+\frac {b \left (c^{2} x^{2}-1\right ) d^{2}}{6 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {b \left (c^{2} x^{2}-1\right ) x^{2} d e}{10 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {b \left (c^{2} x^{2}-1\right ) x^{4} e^{2}}{42 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {b \sqrt {c^{2} x^{2}-1}\, d^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{6 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}+\frac {3 b \left (c^{2} x^{2}-1\right ) d e}{20 c^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {5 b \left (c^{2} x^{2}-1\right ) x^{2} e^{2}}{168 c^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {3 b \sqrt {c^{2} x^{2}-1}\, d e \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{20 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}+\frac {5 b \left (c^{2} x^{2}-1\right ) e^{2}}{112 c^{4} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {5 b \sqrt {c^{2} x^{2}-1}\, e^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{112 c^{5} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}}{c^{3}}\) | \(475\) |
a*(1/7*e^2*x^7+2/5*d*e*x^5+1/3*x^3*d^2)+1/7*b*arccsc(c*x)*e^2*x^7+2/5*b*ar ccsc(c*x)*d*e*x^5+1/3*b*arccsc(c*x)*d^2*x^3+1/42*b/c^3*(c^2*x^2-1)/((c^2*x ^2-1)/c^2/x^2)^(1/2)*x^4*e^2+1/10*b/c^3*(c^2*x^2-1)/((c^2*x^2-1)/c^2/x^2)^ (1/2)*x^2*d*e+5/168*b/c^5*(c^2*x^2-1)/((c^2*x^2-1)/c^2/x^2)^(1/2)*x^2*e^2+ 1/6*b/c^3*(c^2*x^2-1)/((c^2*x^2-1)/c^2/x^2)^(1/2)*d^2+3/20*b/c^5*(c^2*x^2- 1)/((c^2*x^2-1)/c^2/x^2)^(1/2)*d*e+1/6*b/c^4*(c^2*x^2-1)^(1/2)/((c^2*x^2-1 )/c^2/x^2)^(1/2)/x*d^2*ln(c*x+(c^2*x^2-1)^(1/2))+5/112*b/c^7*(c^2*x^2-1)/( (c^2*x^2-1)/c^2/x^2)^(1/2)*e^2+3/20*b/c^6*(c^2*x^2-1)^(1/2)/((c^2*x^2-1)/c ^2/x^2)^(1/2)/x*d*e*ln(c*x+(c^2*x^2-1)^(1/2))+5/112*b/c^8*(c^2*x^2-1)^(1/2 )/((c^2*x^2-1)/c^2/x^2)^(1/2)/x*e^2*ln(c*x+(c^2*x^2-1)^(1/2))
Time = 0.46 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.08 \[ \int x^2 \left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {240 \, a c^{7} e^{2} x^{7} + 672 \, a c^{7} d e x^{5} + 560 \, a c^{7} d^{2} x^{3} + 16 \, {\left (15 \, b c^{7} e^{2} x^{7} + 42 \, b c^{7} d e x^{5} + 35 \, b c^{7} d^{2} x^{3} - 35 \, b c^{7} d^{2} - 42 \, b c^{7} d e - 15 \, b c^{7} e^{2}\right )} \operatorname {arccsc}\left (c x\right ) - 32 \, {\left (35 \, b c^{7} d^{2} + 42 \, b c^{7} d e + 15 \, b c^{7} e^{2}\right )} \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (280 \, b c^{4} d^{2} + 252 \, b c^{2} d e + 75 \, b e^{2}\right )} \log \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + {\left (40 \, b c^{5} e^{2} x^{5} + 2 \, {\left (84 \, b c^{5} d e + 25 \, b c^{3} e^{2}\right )} x^{3} + {\left (280 \, b c^{5} d^{2} + 252 \, b c^{3} d e + 75 \, b c e^{2}\right )} x\right )} \sqrt {c^{2} x^{2} - 1}}{1680 \, c^{7}} \]
1/1680*(240*a*c^7*e^2*x^7 + 672*a*c^7*d*e*x^5 + 560*a*c^7*d^2*x^3 + 16*(15 *b*c^7*e^2*x^7 + 42*b*c^7*d*e*x^5 + 35*b*c^7*d^2*x^3 - 35*b*c^7*d^2 - 42*b *c^7*d*e - 15*b*c^7*e^2)*arccsc(c*x) - 32*(35*b*c^7*d^2 + 42*b*c^7*d*e + 1 5*b*c^7*e^2)*arctan(-c*x + sqrt(c^2*x^2 - 1)) - (280*b*c^4*d^2 + 252*b*c^2 *d*e + 75*b*e^2)*log(-c*x + sqrt(c^2*x^2 - 1)) + (40*b*c^5*e^2*x^5 + 2*(84 *b*c^5*d*e + 25*b*c^3*e^2)*x^3 + (280*b*c^5*d^2 + 252*b*c^3*d*e + 75*b*c*e ^2)*x)*sqrt(c^2*x^2 - 1))/c^7
Time = 11.86 (sec) , antiderivative size = 542, normalized size of antiderivative = 2.15 \[ \int x^2 \left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {a d^{2} x^{3}}{3} + \frac {2 a d e x^{5}}{5} + \frac {a e^{2} x^{7}}{7} + \frac {b d^{2} x^{3} \operatorname {acsc}{\left (c x \right )}}{3} + \frac {2 b d e x^{5} \operatorname {acsc}{\left (c x \right )}}{5} + \frac {b e^{2} x^{7} \operatorname {acsc}{\left (c x \right )}}{7} + \frac {b d^{2} \left (\begin {cases} \frac {x \sqrt {c^{2} x^{2} - 1}}{2 c} + \frac {\operatorname {acosh}{\left (c x \right )}}{2 c^{2}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\- \frac {i c x^{3}}{2 \sqrt {- c^{2} x^{2} + 1}} + \frac {i x}{2 c \sqrt {- c^{2} x^{2} + 1}} - \frac {i \operatorname {asin}{\left (c x \right )}}{2 c^{2}} & \text {otherwise} \end {cases}\right )}{3 c} + \frac {2 b d e \left (\begin {cases} \frac {c x^{5}}{4 \sqrt {c^{2} x^{2} - 1}} + \frac {x^{3}}{8 c \sqrt {c^{2} x^{2} - 1}} - \frac {3 x}{8 c^{3} \sqrt {c^{2} x^{2} - 1}} + \frac {3 \operatorname {acosh}{\left (c x \right )}}{8 c^{4}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\- \frac {i c x^{5}}{4 \sqrt {- c^{2} x^{2} + 1}} - \frac {i x^{3}}{8 c \sqrt {- c^{2} x^{2} + 1}} + \frac {3 i x}{8 c^{3} \sqrt {- c^{2} x^{2} + 1}} - \frac {3 i \operatorname {asin}{\left (c x \right )}}{8 c^{4}} & \text {otherwise} \end {cases}\right )}{5 c} + \frac {b e^{2} \left (\begin {cases} \frac {c x^{7}}{6 \sqrt {c^{2} x^{2} - 1}} + \frac {x^{5}}{24 c \sqrt {c^{2} x^{2} - 1}} + \frac {5 x^{3}}{48 c^{3} \sqrt {c^{2} x^{2} - 1}} - \frac {5 x}{16 c^{5} \sqrt {c^{2} x^{2} - 1}} + \frac {5 \operatorname {acosh}{\left (c x \right )}}{16 c^{6}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\- \frac {i c x^{7}}{6 \sqrt {- c^{2} x^{2} + 1}} - \frac {i x^{5}}{24 c \sqrt {- c^{2} x^{2} + 1}} - \frac {5 i x^{3}}{48 c^{3} \sqrt {- c^{2} x^{2} + 1}} + \frac {5 i x}{16 c^{5} \sqrt {- c^{2} x^{2} + 1}} - \frac {5 i \operatorname {asin}{\left (c x \right )}}{16 c^{6}} & \text {otherwise} \end {cases}\right )}{7 c} \]
a*d**2*x**3/3 + 2*a*d*e*x**5/5 + a*e**2*x**7/7 + b*d**2*x**3*acsc(c*x)/3 + 2*b*d*e*x**5*acsc(c*x)/5 + b*e**2*x**7*acsc(c*x)/7 + b*d**2*Piecewise((x* sqrt(c**2*x**2 - 1)/(2*c) + acosh(c*x)/(2*c**2), Abs(c**2*x**2) > 1), (-I* c*x**3/(2*sqrt(-c**2*x**2 + 1)) + I*x/(2*c*sqrt(-c**2*x**2 + 1)) - I*asin( c*x)/(2*c**2), True))/(3*c) + 2*b*d*e*Piecewise((c*x**5/(4*sqrt(c**2*x**2 - 1)) + x**3/(8*c*sqrt(c**2*x**2 - 1)) - 3*x/(8*c**3*sqrt(c**2*x**2 - 1)) + 3*acosh(c*x)/(8*c**4), Abs(c**2*x**2) > 1), (-I*c*x**5/(4*sqrt(-c**2*x** 2 + 1)) - I*x**3/(8*c*sqrt(-c**2*x**2 + 1)) + 3*I*x/(8*c**3*sqrt(-c**2*x** 2 + 1)) - 3*I*asin(c*x)/(8*c**4), True))/(5*c) + b*e**2*Piecewise((c*x**7/ (6*sqrt(c**2*x**2 - 1)) + x**5/(24*c*sqrt(c**2*x**2 - 1)) + 5*x**3/(48*c** 3*sqrt(c**2*x**2 - 1)) - 5*x/(16*c**5*sqrt(c**2*x**2 - 1)) + 5*acosh(c*x)/ (16*c**6), Abs(c**2*x**2) > 1), (-I*c*x**7/(6*sqrt(-c**2*x**2 + 1)) - I*x* *5/(24*c*sqrt(-c**2*x**2 + 1)) - 5*I*x**3/(48*c**3*sqrt(-c**2*x**2 + 1)) + 5*I*x/(16*c**5*sqrt(-c**2*x**2 + 1)) - 5*I*asin(c*x)/(16*c**6), True))/(7 *c)
Time = 0.23 (sec) , antiderivative size = 404, normalized size of antiderivative = 1.60 \[ \int x^2 \left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {1}{7} \, a e^{2} x^{7} + \frac {2}{5} \, a d e x^{5} + \frac {1}{3} \, a d^{2} x^{3} + \frac {1}{12} \, {\left (4 \, x^{3} \operatorname {arccsc}\left (c x\right ) + \frac {\frac {2 \, \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c^{2} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} + c^{2}} + \frac {\log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}{c^{2}} - \frac {\log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} - 1\right )}{c^{2}}}{c}\right )} b d^{2} + \frac {1}{40} \, {\left (16 \, x^{5} \operatorname {arccsc}\left (c x\right ) - \frac {\frac {2 \, {\left (3 \, {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} - 5 \, \sqrt {-\frac {1}{c^{2} x^{2}} + 1}\right )}}{c^{4} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2} + 2 \, c^{4} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} + c^{4}} - \frac {3 \, \log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}{c^{4}} + \frac {3 \, \log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} - 1\right )}{c^{4}}}{c}\right )} b d e + \frac {1}{672} \, {\left (96 \, x^{7} \operatorname {arccsc}\left (c x\right ) + \frac {\frac {2 \, {\left (15 \, {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {5}{2}} - 40 \, {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + 33 \, \sqrt {-\frac {1}{c^{2} x^{2}} + 1}\right )}}{c^{6} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{3} + 3 \, c^{6} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2} + 3 \, c^{6} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} + c^{6}} + \frac {15 \, \log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}{c^{6}} - \frac {15 \, \log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} - 1\right )}{c^{6}}}{c}\right )} b e^{2} \]
1/7*a*e^2*x^7 + 2/5*a*d*e*x^5 + 1/3*a*d^2*x^3 + 1/12*(4*x^3*arccsc(c*x) + (2*sqrt(-1/(c^2*x^2) + 1)/(c^2*(1/(c^2*x^2) - 1) + c^2) + log(sqrt(-1/(c^2 *x^2) + 1) + 1)/c^2 - log(sqrt(-1/(c^2*x^2) + 1) - 1)/c^2)/c)*b*d^2 + 1/40 *(16*x^5*arccsc(c*x) - (2*(3*(-1/(c^2*x^2) + 1)^(3/2) - 5*sqrt(-1/(c^2*x^2 ) + 1))/(c^4*(1/(c^2*x^2) - 1)^2 + 2*c^4*(1/(c^2*x^2) - 1) + c^4) - 3*log( sqrt(-1/(c^2*x^2) + 1) + 1)/c^4 + 3*log(sqrt(-1/(c^2*x^2) + 1) - 1)/c^4)/c )*b*d*e + 1/672*(96*x^7*arccsc(c*x) + (2*(15*(-1/(c^2*x^2) + 1)^(5/2) - 40 *(-1/(c^2*x^2) + 1)^(3/2) + 33*sqrt(-1/(c^2*x^2) + 1))/(c^6*(1/(c^2*x^2) - 1)^3 + 3*c^6*(1/(c^2*x^2) - 1)^2 + 3*c^6*(1/(c^2*x^2) - 1) + c^6) + 15*lo g(sqrt(-1/(c^2*x^2) + 1) + 1)/c^6 - 15*log(sqrt(-1/(c^2*x^2) + 1) - 1)/c^6 )/c)*b*e^2
Leaf count of result is larger than twice the leaf count of optimal. 1579 vs. \(2 (222) = 444\).
Time = 5.13 (sec) , antiderivative size = 1579, normalized size of antiderivative = 6.27 \[ \int x^2 \left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right ) \, dx=\text {Too large to display} \]
1/13440*(15*b*e^2*x^7*(sqrt(-1/(c^2*x^2) + 1) + 1)^7*arcsin(1/(c*x))/c + 1 5*a*e^2*x^7*(sqrt(-1/(c^2*x^2) + 1) + 1)^7/c + 5*b*e^2*x^6*(sqrt(-1/(c^2*x ^2) + 1) + 1)^6/c^2 + 168*b*d*e*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5*arcsin( 1/(c*x))/c + 168*a*d*e*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5/c + 105*b*e^2*x^ 5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5*arcsin(1/(c*x))/c^3 + 105*a*e^2*x^5*(sqrt (-1/(c^2*x^2) + 1) + 1)^5/c^3 + 84*b*d*e*x^4*(sqrt(-1/(c^2*x^2) + 1) + 1)^ 4/c^2 + 560*b*d^2*x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3*arcsin(1/(c*x))/c + 5 60*a*d^2*x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3/c + 45*b*e^2*x^4*(sqrt(-1/(c^2 *x^2) + 1) + 1)^4/c^4 + 840*b*d*e*x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3*arcsi n(1/(c*x))/c^3 + 840*a*d*e*x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3/c^3 + 560*b* d^2*x^2*(sqrt(-1/(c^2*x^2) + 1) + 1)^2/c^2 + 315*b*e^2*x^3*(sqrt(-1/(c^2*x ^2) + 1) + 1)^3*arcsin(1/(c*x))/c^5 + 315*a*e^2*x^3*(sqrt(-1/(c^2*x^2) + 1 ) + 1)^3/c^5 + 672*b*d*e*x^2*(sqrt(-1/(c^2*x^2) + 1) + 1)^2/c^4 + 1680*b*d ^2*x*(sqrt(-1/(c^2*x^2) + 1) + 1)*arcsin(1/(c*x))/c^3 + 1680*a*d^2*x*(sqrt (-1/(c^2*x^2) + 1) + 1)/c^3 + 225*b*e^2*x^2*(sqrt(-1/(c^2*x^2) + 1) + 1)^2 /c^6 + 1680*b*d*e*x*(sqrt(-1/(c^2*x^2) + 1) + 1)*arcsin(1/(c*x))/c^5 + 168 0*a*d*e*x*(sqrt(-1/(c^2*x^2) + 1) + 1)/c^5 + 2240*b*d^2*log(sqrt(-1/(c^2*x ^2) + 1) + 1)/c^4 - 2240*b*d^2*log(1/(abs(c)*abs(x)))/c^4 + 525*b*e^2*x*(s qrt(-1/(c^2*x^2) + 1) + 1)*arcsin(1/(c*x))/c^7 + 525*a*e^2*x*(sqrt(-1/(c^2 *x^2) + 1) + 1)/c^7 + 2016*b*d*e*log(sqrt(-1/(c^2*x^2) + 1) + 1)/c^6 - ...
Timed out. \[ \int x^2 \left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right ) \, dx=\int x^2\,{\left (e\,x^2+d\right )}^2\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right ) \,d x \]