Integrand size = 21, antiderivative size = 157 \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right )}{x^4} \, dx=-\frac {2 b c d \left (c^2 d+9 e\right ) \sqrt {-1+c^2 x^2}}{9 \sqrt {c^2 x^2}}-\frac {b c d^2 \sqrt {-1+c^2 x^2}}{9 x^2 \sqrt {c^2 x^2}}-\frac {d^2 \left (a+b \csc ^{-1}(c x)\right )}{3 x^3}-\frac {2 d e \left (a+b \csc ^{-1}(c x)\right )}{x}+e^2 x \left (a+b \csc ^{-1}(c x)\right )+\frac {b e^2 x \text {arctanh}\left (\frac {c x}{\sqrt {-1+c^2 x^2}}\right )}{\sqrt {c^2 x^2}} \]
-1/3*d^2*(a+b*arccsc(c*x))/x^3-2*d*e*(a+b*arccsc(c*x))/x+e^2*x*(a+b*arccsc (c*x))+b*e^2*x*arctanh(c*x/(c^2*x^2-1)^(1/2))/(c^2*x^2)^(1/2)-2/9*b*c*d*(c ^2*d+9*e)*(c^2*x^2-1)^(1/2)/(c^2*x^2)^(1/2)-1/9*b*c*d^2*(c^2*x^2-1)^(1/2)/ x^2/(c^2*x^2)^(1/2)
Time = 0.17 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.80 \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right )}{x^4} \, dx=-\frac {b c d \sqrt {1-\frac {1}{c^2 x^2}} x \left (d+2 c^2 d x^2+18 e x^2\right )+3 a \left (d^2+6 d e x^2-3 e^2 x^4\right )}{9 x^3}-\frac {b \left (d^2+6 d e x^2-3 e^2 x^4\right ) \csc ^{-1}(c x)}{3 x^3}+\frac {b e^2 \log \left (\left (1+\sqrt {1-\frac {1}{c^2 x^2}}\right ) x\right )}{c} \]
-1/9*(b*c*d*Sqrt[1 - 1/(c^2*x^2)]*x*(d + 2*c^2*d*x^2 + 18*e*x^2) + 3*a*(d^ 2 + 6*d*e*x^2 - 3*e^2*x^4))/x^3 - (b*(d^2 + 6*d*e*x^2 - 3*e^2*x^4)*ArcCsc[ c*x])/(3*x^3) + (b*e^2*Log[(1 + Sqrt[1 - 1/(c^2*x^2)])*x])/c
Time = 0.39 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.94, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {5762, 27, 1588, 358, 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.
| \(\displaystyle \int \frac {\left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right )}{x^4} \, dx\) |
| \(\Big \downarrow \) 5762 |
| \(\displaystyle \frac {b c x \int -\frac {-3 e^2 x^4+6 d e x^2+d^2}{3 x^4 \sqrt {c^2 x^2-1}}dx}{\sqrt {c^2 x^2}}-\frac {d^2 \left (a+b \csc ^{-1}(c x)\right )}{3 x^3}-\frac {2 d e \left (a+b \csc ^{-1}(c x)\right )}{x}+e^2 x \left (a+b \csc ^{-1}(c x)\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {b c x \int \frac {-3 e^2 x^4+6 d e x^2+d^2}{x^4 \sqrt {c^2 x^2-1}}dx}{3 \sqrt {c^2 x^2}}-\frac {d^2 \left (a+b \csc ^{-1}(c x)\right )}{3 x^3}-\frac {2 d e \left (a+b \csc ^{-1}(c x)\right )}{x}+e^2 x \left (a+b \csc ^{-1}(c x)\right )\) |
| \(\Big \downarrow \) 1588 |
| \(\displaystyle -\frac {b c x \left (\frac {1}{3} \int \frac {2 d \left (d c^2+9 e\right )-9 e^2 x^2}{x^2 \sqrt {c^2 x^2-1}}dx+\frac {d^2 \sqrt {c^2 x^2-1}}{3 x^3}\right )}{3 \sqrt {c^2 x^2}}-\frac {d^2 \left (a+b \csc ^{-1}(c x)\right )}{3 x^3}-\frac {2 d e \left (a+b \csc ^{-1}(c x)\right )}{x}+e^2 x \left (a+b \csc ^{-1}(c x)\right )\) |
| \(\Big \downarrow \) 358 |
| \(\displaystyle -\frac {b c x \left (\frac {1}{3} \left (\frac {2 d \sqrt {c^2 x^2-1} \left (c^2 d+9 e\right )}{x}-9 e^2 \int \frac {1}{\sqrt {c^2 x^2-1}}dx\right )+\frac {d^2 \sqrt {c^2 x^2-1}}{3 x^3}\right )}{3 \sqrt {c^2 x^2}}-\frac {d^2 \left (a+b \csc ^{-1}(c x)\right )}{3 x^3}-\frac {2 d e \left (a+b \csc ^{-1}(c x)\right )}{x}+e^2 x \left (a+b \csc ^{-1}(c x)\right )\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle -\frac {b c x \left (\frac {1}{3} \left (\frac {2 d \sqrt {c^2 x^2-1} \left (c^2 d+9 e\right )}{x}-9 e^2 \int \frac {1}{1-\frac {c^2 x^2}{c^2 x^2-1}}d\frac {x}{\sqrt {c^2 x^2-1}}\right )+\frac {d^2 \sqrt {c^2 x^2-1}}{3 x^3}\right )}{3 \sqrt {c^2 x^2}}-\frac {d^2 \left (a+b \csc ^{-1}(c x)\right )}{3 x^3}-\frac {2 d e \left (a+b \csc ^{-1}(c x)\right )}{x}+e^2 x \left (a+b \csc ^{-1}(c x)\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {d^2 \left (a+b \csc ^{-1}(c x)\right )}{3 x^3}-\frac {2 d e \left (a+b \csc ^{-1}(c x)\right )}{x}+e^2 x \left (a+b \csc ^{-1}(c x)\right )-\frac {b c x \left (\frac {1}{3} \left (\frac {2 d \sqrt {c^2 x^2-1} \left (c^2 d+9 e\right )}{x}-\frac {9 e^2 \text {arctanh}\left (\frac {c x}{\sqrt {c^2 x^2-1}}\right )}{c}\right )+\frac {d^2 \sqrt {c^2 x^2-1}}{3 x^3}\right )}{3 \sqrt {c^2 x^2}}\) |
-1/3*(d^2*(a + b*ArcCsc[c*x]))/x^3 - (2*d*e*(a + b*ArcCsc[c*x]))/x + e^2*x *(a + b*ArcCsc[c*x]) - (b*c*x*((d^2*Sqrt[-1 + c^2*x^2])/(3*x^3) + ((2*d*(c ^2*d + 9*e)*Sqrt[-1 + c^2*x^2])/x - (9*e^2*ArcTanh[(c*x)/Sqrt[-1 + c^2*x^2 ]])/c)/3))/(3*Sqrt[c^2*x^2])
3.1.91.3.1 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[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x_ Symbol] :> Simp[c*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*e*(m + 1))), x] + S imp[d/e^2 Int[(e*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e , m, p}, x] && NeQ[b*c - a*d, 0] && EqQ[Simplify[m + 2*p + 3], 0] && NeQ[m, -1]
Int[((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c _.)*(x_)^4)^(p_.), x_Symbol] :> With[{Qx = PolynomialQuotient[(a + b*x^2 + c*x^4)^p, f*x, x], R = PolynomialRemainder[(a + b*x^2 + c*x^4)^p, f*x, x]}, Simp[R*(f*x)^(m + 1)*((d + e*x^2)^(q + 1)/(d*f*(m + 1))), x] + Simp[1/(d*f ^2*(m + 1)) Int[(f*x)^(m + 2)*(d + e*x^2)^q*ExpandToSum[d*f*(m + 1)*(Qx/x ) - e*R*(m + 2*q + 3), x], x], x]] /; FreeQ[{a, b, c, d, e, f, q}, x] && Ne Q[b^2 - 4*a*c, 0] && IGtQ[p, 0] && LtQ[m, -1]
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 = 0.47 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.46
| method | result | size |
| parts | \(a \left (e^{2} x -\frac {2 d e}{x}-\frac {d^{2}}{3 x^{3}}\right )+b \,\operatorname {arccsc}\left (c x \right ) e^{2} x -\frac {2 b \,\operatorname {arccsc}\left (c x \right ) d e}{x}-\frac {b \,\operatorname {arccsc}\left (c x \right ) d^{2}}{3 x^{3}}-\frac {2 b c \left (c^{2} x^{2}-1\right ) d^{2}}{9 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x^{2}}-\frac {2 b \left (c^{2} x^{2}-1\right ) d e}{c \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x^{2}}+\frac {b \sqrt {c^{2} x^{2}-1}\, e^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{c^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}-\frac {b \left (c^{2} x^{2}-1\right ) d^{2}}{9 c \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x^{4}}\) | \(230\) |
| derivativedivides | \(c^{3} \left (\frac {a \left (e^{2} c x -\frac {c \,d^{2}}{3 x^{3}}-\frac {2 c d e}{x}\right )}{c^{4}}+\frac {b \,\operatorname {arccsc}\left (c x \right ) e^{2} x}{c^{3}}-\frac {b \,\operatorname {arccsc}\left (c x \right ) d^{2}}{3 c^{3} x^{3}}-\frac {2 b \,\operatorname {arccsc}\left (c x \right ) d e}{c^{3} x}-\frac {2 b \left (c^{2} x^{2}-1\right ) d^{2}}{9 c^{2} x^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {b \left (c^{2} x^{2}-1\right ) d^{2}}{9 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{4} x^{4}}-\frac {2 b \left (c^{2} x^{2}-1\right ) d e}{c^{4} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x^{2}}+\frac {b \sqrt {c^{2} x^{2}-1}\, e^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{c^{5} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}\right )\) | \(251\) |
| default | \(c^{3} \left (\frac {a \left (e^{2} c x -\frac {c \,d^{2}}{3 x^{3}}-\frac {2 c d e}{x}\right )}{c^{4}}+\frac {b \,\operatorname {arccsc}\left (c x \right ) e^{2} x}{c^{3}}-\frac {b \,\operatorname {arccsc}\left (c x \right ) d^{2}}{3 c^{3} x^{3}}-\frac {2 b \,\operatorname {arccsc}\left (c x \right ) d e}{c^{3} x}-\frac {2 b \left (c^{2} x^{2}-1\right ) d^{2}}{9 c^{2} x^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {b \left (c^{2} x^{2}-1\right ) d^{2}}{9 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{4} x^{4}}-\frac {2 b \left (c^{2} x^{2}-1\right ) d e}{c^{4} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x^{2}}+\frac {b \sqrt {c^{2} x^{2}-1}\, e^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{c^{5} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}\right )\) | \(251\) |
a*(e^2*x-2*d*e/x-1/3*d^2/x^3)+b*arccsc(c*x)*e^2*x-2*b*arccsc(c*x)*d*e/x-1/ 3*b*arccsc(c*x)*d^2/x^3-2/9*b*c*(c^2*x^2-1)/((c^2*x^2-1)/c^2/x^2)^(1/2)/x^ 2*d^2-2*b/c*(c^2*x^2-1)/((c^2*x^2-1)/c^2/x^2)^(1/2)/x^2*d*e+b/c^2*(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))-1/9* b/c*(c^2*x^2-1)/((c^2*x^2-1)/c^2/x^2)^(1/2)/x^4*d^2
Time = 0.32 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.41 \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right )}{x^4} \, dx=\frac {9 \, a c e^{2} x^{4} - 9 \, b e^{2} x^{3} \log \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) - 18 \, a c d e x^{2} + 6 \, {\left (b c d^{2} + 6 \, b c d e - 3 \, b c e^{2}\right )} x^{3} \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) - 3 \, a c d^{2} - 2 \, {\left (b c^{4} d^{2} + 9 \, b c^{2} d e\right )} x^{3} + 3 \, {\left (3 \, b c e^{2} x^{4} - 6 \, b c d e x^{2} - b c d^{2} + {\left (b c d^{2} + 6 \, b c d e - 3 \, b c e^{2}\right )} x^{3}\right )} \operatorname {arccsc}\left (c x\right ) - {\left (b c d^{2} + 2 \, {\left (b c^{3} d^{2} + 9 \, b c d e\right )} x^{2}\right )} \sqrt {c^{2} x^{2} - 1}}{9 \, c x^{3}} \]
1/9*(9*a*c*e^2*x^4 - 9*b*e^2*x^3*log(-c*x + sqrt(c^2*x^2 - 1)) - 18*a*c*d* e*x^2 + 6*(b*c*d^2 + 6*b*c*d*e - 3*b*c*e^2)*x^3*arctan(-c*x + sqrt(c^2*x^2 - 1)) - 3*a*c*d^2 - 2*(b*c^4*d^2 + 9*b*c^2*d*e)*x^3 + 3*(3*b*c*e^2*x^4 - 6*b*c*d*e*x^2 - b*c*d^2 + (b*c*d^2 + 6*b*c*d*e - 3*b*c*e^2)*x^3)*arccsc(c* x) - (b*c*d^2 + 2*(b*c^3*d^2 + 9*b*c*d*e)*x^2)*sqrt(c^2*x^2 - 1))/(c*x^3)
Time = 4.02 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.34 \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right )}{x^4} \, dx=- \frac {a d^{2}}{3 x^{3}} - \frac {2 a d e}{x} + a e^{2} x - 2 b c d e \sqrt {1 - \frac {1}{c^{2} x^{2}}} - \frac {b d^{2} \operatorname {acsc}{\left (c x \right )}}{3 x^{3}} - \frac {2 b d e \operatorname {acsc}{\left (c x \right )}}{x} + b e^{2} x \operatorname {acsc}{\left (c x \right )} - \frac {b d^{2} \left (\begin {cases} \frac {2 c^{3} \sqrt {c^{2} x^{2} - 1}}{3 x} + \frac {c \sqrt {c^{2} x^{2} - 1}}{3 x^{3}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\\frac {2 i c^{3} \sqrt {- c^{2} x^{2} + 1}}{3 x} + \frac {i c \sqrt {- c^{2} x^{2} + 1}}{3 x^{3}} & \text {otherwise} \end {cases}\right )}{3 c} + \frac {b e^{2} \left (\begin {cases} \operatorname {acosh}{\left (c x \right )} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\- i \operatorname {asin}{\left (c x \right )} & \text {otherwise} \end {cases}\right )}{c} \]
-a*d**2/(3*x**3) - 2*a*d*e/x + a*e**2*x - 2*b*c*d*e*sqrt(1 - 1/(c**2*x**2) ) - b*d**2*acsc(c*x)/(3*x**3) - 2*b*d*e*acsc(c*x)/x + b*e**2*x*acsc(c*x) - b*d**2*Piecewise((2*c**3*sqrt(c**2*x**2 - 1)/(3*x) + c*sqrt(c**2*x**2 - 1 )/(3*x**3), Abs(c**2*x**2) > 1), (2*I*c**3*sqrt(-c**2*x**2 + 1)/(3*x) + I* c*sqrt(-c**2*x**2 + 1)/(3*x**3), True))/(3*c) + b*e**2*Piecewise((acosh(c* x), Abs(c**2*x**2) > 1), (-I*asin(c*x), True))/c
Time = 0.22 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.01 \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right )}{x^4} \, dx=-2 \, {\left (c \sqrt {-\frac {1}{c^{2} x^{2}} + 1} + \frac {\operatorname {arccsc}\left (c x\right )}{x}\right )} b d e + a e^{2} x + \frac {1}{9} \, b d^{2} {\left (\frac {c^{4} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} - 3 \, c^{4} \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c} - \frac {3 \, \operatorname {arccsc}\left (c x\right )}{x^{3}}\right )} + \frac {{\left (2 \, c x \operatorname {arccsc}\left (c x\right ) + \log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right ) - \log \left (-\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )\right )} b e^{2}}{2 \, c} - \frac {2 \, a d e}{x} - \frac {a d^{2}}{3 \, x^{3}} \]
-2*(c*sqrt(-1/(c^2*x^2) + 1) + arccsc(c*x)/x)*b*d*e + a*e^2*x + 1/9*b*d^2* ((c^4*(-1/(c^2*x^2) + 1)^(3/2) - 3*c^4*sqrt(-1/(c^2*x^2) + 1))/c - 3*arccs c(c*x)/x^3) + 1/2*(2*c*x*arccsc(c*x) + log(sqrt(-1/(c^2*x^2) + 1) + 1) - l og(-sqrt(-1/(c^2*x^2) + 1) + 1))*b*e^2/c - 2*a*d*e/x - 1/3*a*d^2/x^3
Leaf count of result is larger than twice the leaf count of optimal. 4288 vs. \(2 (139) = 278\).
Time = 98.89 (sec) , antiderivative size = 4288, normalized size of antiderivative = 27.31 \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right )}{x^4} \, dx=\text {Too large to display} \]
-1/18*(4*b*c^3*d^2/(x*(sqrt(-1/(c^2*x^2) + 1) + 1)*(c/(x*(sqrt(-1/(c^2*x^2 ) + 1) + 1)) + 3/(c*x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3) + 3/(c^3*x^5*(sqrt (-1/(c^2*x^2) + 1) + 1)^5) + 1/(c^5*x^7*(sqrt(-1/(c^2*x^2) + 1) + 1)^7))) - 9*b*e^2*arcsin(1/(c*x))/(c/(x*(sqrt(-1/(c^2*x^2) + 1) + 1)) + 3/(c*x^3*( sqrt(-1/(c^2*x^2) + 1) + 1)^3) + 3/(c^3*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5 ) + 1/(c^5*x^7*(sqrt(-1/(c^2*x^2) + 1) + 1)^7)) - 9*a*e^2/(c/(x*(sqrt(-1/( c^2*x^2) + 1) + 1)) + 3/(c*x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3) + 3/(c^3*x^ 5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5) + 1/(c^5*x^7*(sqrt(-1/(c^2*x^2) + 1) + 1 )^7)) + 36*b*c*d*e/(x*(sqrt(-1/(c^2*x^2) + 1) + 1)*(c/(x*(sqrt(-1/(c^2*x^2 ) + 1) + 1)) + 3/(c*x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3) + 3/(c^3*x^5*(sqrt (-1/(c^2*x^2) + 1) + 1)^5) + 1/(c^5*x^7*(sqrt(-1/(c^2*x^2) + 1) + 1)^7))) - 18*b*e^2*log(sqrt(-1/(c^2*x^2) + 1) + 1)/(c*x*(sqrt(-1/(c^2*x^2) + 1) + 1)*(c/(x*(sqrt(-1/(c^2*x^2) + 1) + 1)) + 3/(c*x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3) + 3/(c^3*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5) + 1/(c^5*x^7*(sqrt(-1 /(c^2*x^2) + 1) + 1)^7))) + 18*b*e^2*log(1/(abs(c)*abs(x)))/(c*x*(sqrt(-1/ (c^2*x^2) + 1) + 1)*(c/(x*(sqrt(-1/(c^2*x^2) + 1) + 1)) + 3/(c*x^3*(sqrt(- 1/(c^2*x^2) + 1) + 1)^3) + 3/(c^3*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5) + 1/ (c^5*x^7*(sqrt(-1/(c^2*x^2) + 1) + 1)^7))) + 72*b*d*e*arcsin(1/(c*x))/(x^2 *(sqrt(-1/(c^2*x^2) + 1) + 1)^2*(c/(x*(sqrt(-1/(c^2*x^2) + 1) + 1)) + 3/(c *x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3) + 3/(c^3*x^5*(sqrt(-1/(c^2*x^2) + ...
Timed out. \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right )}{x^4} \, dx=\int \frac {{\left (e\,x^2+d\right )}^2\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}{x^4} \,d x \]