Integrand size = 21, antiderivative size = 803 \[ \int \frac {x^4 \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=-\frac {d \left (a+b \csc ^{-1}(c x)\right )}{4 e^2 \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}+\frac {d \left (a+b \csc ^{-1}(c x)\right )}{4 e^2 \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}+\frac {x \left (a+b \csc ^{-1}(c x)\right )}{e^2}+\frac {b \text {arctanh}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{c e^2}+\frac {b \sqrt {d} \text {arctanh}\left (\frac {c^2 d-\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {c^2 d+e} \sqrt {1-\frac {1}{c^2 x^2}}}\right )}{4 e^2 \sqrt {c^2 d+e}}+\frac {b \sqrt {d} \text {arctanh}\left (\frac {c^2 d+\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {c^2 d+e} \sqrt {1-\frac {1}{c^2 x^2}}}\right )}{4 e^2 \sqrt {c^2 d+e}}-\frac {3 \sqrt {-d} \left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{4 e^{5/2}}+\frac {3 \sqrt {-d} \left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{4 e^{5/2}}-\frac {3 \sqrt {-d} \left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{4 e^{5/2}}+\frac {3 \sqrt {-d} \left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{4 e^{5/2}}-\frac {3 i b \sqrt {-d} \operatorname {PolyLog}\left (2,-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{4 e^{5/2}}+\frac {3 i b \sqrt {-d} \operatorname {PolyLog}\left (2,\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{4 e^{5/2}}-\frac {3 i b \sqrt {-d} \operatorname {PolyLog}\left (2,-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{4 e^{5/2}}+\frac {3 i b \sqrt {-d} \operatorname {PolyLog}\left (2,\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{4 e^{5/2}} \]
x*(a+b*arccsc(c*x))/e^2+b*arctanh((1-1/c^2/x^2)^(1/2))/c/e^2-3/4*(a+b*arcc sc(c*x))*ln(1-I*c*(I/c/x+(1-1/c^2/x^2)^(1/2))*(-d)^(1/2)/(e^(1/2)-(c^2*d+e )^(1/2)))*(-d)^(1/2)/e^(5/2)+3/4*(a+b*arccsc(c*x))*ln(1+I*c*(I/c/x+(1-1/c^ 2/x^2)^(1/2))*(-d)^(1/2)/(e^(1/2)-(c^2*d+e)^(1/2)))*(-d)^(1/2)/e^(5/2)-3/4 *(a+b*arccsc(c*x))*ln(1-I*c*(I/c/x+(1-1/c^2/x^2)^(1/2))*(-d)^(1/2)/(e^(1/2 )+(c^2*d+e)^(1/2)))*(-d)^(1/2)/e^(5/2)+3/4*(a+b*arccsc(c*x))*ln(1+I*c*(I/c /x+(1-1/c^2/x^2)^(1/2))*(-d)^(1/2)/(e^(1/2)+(c^2*d+e)^(1/2)))*(-d)^(1/2)/e ^(5/2)-3/4*I*b*polylog(2,-I*c*(I/c/x+(1-1/c^2/x^2)^(1/2))*(-d)^(1/2)/(e^(1 /2)-(c^2*d+e)^(1/2)))*(-d)^(1/2)/e^(5/2)+3/4*I*b*polylog(2,I*c*(I/c/x+(1-1 /c^2/x^2)^(1/2))*(-d)^(1/2)/(e^(1/2)-(c^2*d+e)^(1/2)))*(-d)^(1/2)/e^(5/2)- 3/4*I*b*polylog(2,-I*c*(I/c/x+(1-1/c^2/x^2)^(1/2))*(-d)^(1/2)/(e^(1/2)+(c^ 2*d+e)^(1/2)))*(-d)^(1/2)/e^(5/2)+3/4*I*b*polylog(2,I*c*(I/c/x+(1-1/c^2/x^ 2)^(1/2))*(-d)^(1/2)/(e^(1/2)+(c^2*d+e)^(1/2)))*(-d)^(1/2)/e^(5/2)-1/4*d*( a+b*arccsc(c*x))/e^2/(-d/x+(-d)^(1/2)*e^(1/2))+1/4*d*(a+b*arccsc(c*x))/e^2 /(d/x+(-d)^(1/2)*e^(1/2))+1/4*b*arctanh((c^2*d-(-d)^(1/2)*e^(1/2)/x)/c/d^( 1/2)/(c^2*d+e)^(1/2)/(1-1/c^2/x^2)^(1/2))*d^(1/2)/e^2/(c^2*d+e)^(1/2)+1/4* b*arctanh((c^2*d+(-d)^(1/2)*e^(1/2)/x)/c/d^(1/2)/(c^2*d+e)^(1/2)/(1-1/c^2/ x^2)^(1/2))*d^(1/2)/e^2/(c^2*d+e)^(1/2)
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1634\) vs. \(2(803)=1606\).
Time = 6.05 (sec) , antiderivative size = 1634, normalized size of antiderivative = 2.03 \[ \int \frac {x^4 \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx =\text {Too large to display} \]
(a*x)/e^2 + (a*d*x)/(2*e^2*(d + e*x^2)) - (3*a*Sqrt[d]*ArcTan[(Sqrt[e]*x)/ Sqrt[d]])/(2*e^(5/2)) + b*(-1/4*(d*(-(ArcCsc[c*x]/((-I)*Sqrt[d]*Sqrt[e] + e*x)) + (I*(ArcSin[1/(c*x)]/Sqrt[e] - Log[(2*Sqrt[d]*Sqrt[e]*(Sqrt[e] + c* ((-I)*c*Sqrt[d] - Sqrt[-(c^2*d) - e]*Sqrt[1 - 1/(c^2*x^2)])*x))/(Sqrt[-(c^ 2*d) - e]*(Sqrt[d] + I*Sqrt[e]*x))]/Sqrt[-(c^2*d) - e]))/Sqrt[d]))/e^2 - ( d*(-(ArcCsc[c*x]/(I*Sqrt[d]*Sqrt[e] + e*x)) - (I*(ArcSin[1/(c*x)]/Sqrt[e] - Log[(2*Sqrt[d]*Sqrt[e]*(-Sqrt[e] + c*((-I)*c*Sqrt[d] + Sqrt[-(c^2*d) - e ]*Sqrt[1 - 1/(c^2*x^2)])*x))/(Sqrt[-(c^2*d) - e]*(Sqrt[d] - I*Sqrt[e]*x))] /Sqrt[-(c^2*d) - e]))/Sqrt[d]))/(4*e^2) + (3*Sqrt[d]*(Pi^2 - 4*Pi*ArcCsc[c *x] + 8*ArcCsc[c*x]^2 - 32*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2 ]]*ArcTan[(((-I)*c*Sqrt[d] + Sqrt[e])*Cot[(Pi + 2*ArcCsc[c*x])/4])/Sqrt[c^ 2*d + e]] + (4*I)*Pi*Log[1 + (Sqrt[e] - Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*A rcCsc[c*x]))] - (8*I)*ArcCsc[c*x]*Log[1 + (Sqrt[e] - Sqrt[c^2*d + e])/(c*S qrt[d]*E^(I*ArcCsc[c*x]))] + (16*I)*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d] )]/Sqrt[2]]*Log[1 + (Sqrt[e] - Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x ]))] + (4*I)*Pi*Log[1 + (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc [c*x]))] - (8*I)*ArcCsc[c*x]*Log[1 + (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d ]*E^(I*ArcCsc[c*x]))] - (16*I)*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sq rt[2]]*Log[1 + (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + (8*I)*ArcCsc[c*x]*Log[1 - E^((2*I)*ArcCsc[c*x])] - (4*I)*Pi*Log[Sqrt[...
Time = 2.94 (sec) , antiderivative size = 863, normalized size of antiderivative = 1.07, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {5764, 5232, 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.
\(\displaystyle \int \frac {x^4 \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx\) |
\(\Big \downarrow \) 5764 |
\(\displaystyle -\int \frac {x^2 \left (a+b \arcsin \left (\frac {1}{c x}\right )\right )}{\left (\frac {d}{x^2}+e\right )^2}d\frac {1}{x}\) |
\(\Big \downarrow \) 5232 |
\(\displaystyle -\int \left (\frac {\left (a+b \arcsin \left (\frac {1}{c x}\right )\right ) x^2}{e^2}-\frac {d \left (a+b \arcsin \left (\frac {1}{c x}\right )\right )}{e^2 \left (\frac {d}{x^2}+e\right )}-\frac {d \left (a+b \arcsin \left (\frac {1}{c x}\right )\right )}{e \left (\frac {d}{x^2}+e\right )^2}\right )d\frac {1}{x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {x \left (a+b \arcsin \left (\frac {1}{c x}\right )\right )}{e^2}-\frac {3 \sqrt {-d} \log \left (1-\frac {i c \sqrt {-d} e^{i \arcsin \left (\frac {1}{c x}\right )}}{\sqrt {e}-\sqrt {d c^2+e}}\right ) \left (a+b \arcsin \left (\frac {1}{c x}\right )\right )}{4 e^{5/2}}+\frac {3 \sqrt {-d} \log \left (\frac {i \sqrt {-d} e^{i \arcsin \left (\frac {1}{c x}\right )} c}{\sqrt {e}-\sqrt {d c^2+e}}+1\right ) \left (a+b \arcsin \left (\frac {1}{c x}\right )\right )}{4 e^{5/2}}-\frac {3 \sqrt {-d} \log \left (1-\frac {i c \sqrt {-d} e^{i \arcsin \left (\frac {1}{c x}\right )}}{\sqrt {e}+\sqrt {d c^2+e}}\right ) \left (a+b \arcsin \left (\frac {1}{c x}\right )\right )}{4 e^{5/2}}+\frac {3 \sqrt {-d} \log \left (\frac {i \sqrt {-d} e^{i \arcsin \left (\frac {1}{c x}\right )} c}{\sqrt {e}+\sqrt {d c^2+e}}+1\right ) \left (a+b \arcsin \left (\frac {1}{c x}\right )\right )}{4 e^{5/2}}-\frac {d \left (a+b \arcsin \left (\frac {1}{c x}\right )\right )}{4 e^2 \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}+\frac {d \left (a+b \arcsin \left (\frac {1}{c x}\right )\right )}{4 e^2 \left (\frac {d}{x}+\sqrt {-d} \sqrt {e}\right )}+\frac {b \text {arctanh}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{c e^2}+\frac {b \sqrt {d} \text {arctanh}\left (\frac {c^2 d-\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {d c^2+e} \sqrt {1-\frac {1}{c^2 x^2}}}\right )}{4 e^2 \sqrt {d c^2+e}}+\frac {b \sqrt {d} \text {arctanh}\left (\frac {d c^2+\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {d c^2+e} \sqrt {1-\frac {1}{c^2 x^2}}}\right )}{4 e^2 \sqrt {d c^2+e}}-\frac {3 i b \sqrt {-d} \operatorname {PolyLog}\left (2,-\frac {i c \sqrt {-d} e^{i \arcsin \left (\frac {1}{c x}\right )}}{\sqrt {e}-\sqrt {d c^2+e}}\right )}{4 e^{5/2}}+\frac {3 i b \sqrt {-d} \operatorname {PolyLog}\left (2,\frac {i c \sqrt {-d} e^{i \arcsin \left (\frac {1}{c x}\right )}}{\sqrt {e}-\sqrt {d c^2+e}}\right )}{4 e^{5/2}}-\frac {3 i b \sqrt {-d} \operatorname {PolyLog}\left (2,-\frac {i c \sqrt {-d} e^{i \arcsin \left (\frac {1}{c x}\right )}}{\sqrt {e}+\sqrt {d c^2+e}}\right )}{4 e^{5/2}}+\frac {3 i b \sqrt {-d} \operatorname {PolyLog}\left (2,\frac {i c \sqrt {-d} e^{i \arcsin \left (\frac {1}{c x}\right )}}{\sqrt {e}+\sqrt {d c^2+e}}\right )}{4 e^{5/2}}\) |
-1/4*(d*(a + b*ArcSin[1/(c*x)]))/(e^2*(Sqrt[-d]*Sqrt[e] - d/x)) + (d*(a + b*ArcSin[1/(c*x)]))/(4*e^2*(Sqrt[-d]*Sqrt[e] + d/x)) + (x*(a + b*ArcSin[1/ (c*x)]))/e^2 + (b*ArcTanh[Sqrt[1 - 1/(c^2*x^2)]])/(c*e^2) + (b*Sqrt[d]*Arc Tanh[(c^2*d - (Sqrt[-d]*Sqrt[e])/x)/(c*Sqrt[d]*Sqrt[c^2*d + e]*Sqrt[1 - 1/ (c^2*x^2)])])/(4*e^2*Sqrt[c^2*d + e]) + (b*Sqrt[d]*ArcTanh[(c^2*d + (Sqrt[ -d]*Sqrt[e])/x)/(c*Sqrt[d]*Sqrt[c^2*d + e]*Sqrt[1 - 1/(c^2*x^2)])])/(4*e^2 *Sqrt[c^2*d + e]) - (3*Sqrt[-d]*(a + b*ArcSin[1/(c*x)])*Log[1 - (I*c*Sqrt[ -d]*E^(I*ArcSin[1/(c*x)]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(4*e^(5/2)) + (3* Sqrt[-d]*(a + b*ArcSin[1/(c*x)])*Log[1 + (I*c*Sqrt[-d]*E^(I*ArcSin[1/(c*x) ]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(4*e^(5/2)) - (3*Sqrt[-d]*(a + b*ArcSin[ 1/(c*x)])*Log[1 - (I*c*Sqrt[-d]*E^(I*ArcSin[1/(c*x)]))/(Sqrt[e] + Sqrt[c^2 *d + e])])/(4*e^(5/2)) + (3*Sqrt[-d]*(a + b*ArcSin[1/(c*x)])*Log[1 + (I*c* Sqrt[-d]*E^(I*ArcSin[1/(c*x)]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(4*e^(5/2)) - (((3*I)/4)*b*Sqrt[-d]*PolyLog[2, ((-I)*c*Sqrt[-d]*E^(I*ArcSin[1/(c*x)])) /(Sqrt[e] - Sqrt[c^2*d + e])])/e^(5/2) + (((3*I)/4)*b*Sqrt[-d]*PolyLog[2, (I*c*Sqrt[-d]*E^(I*ArcSin[1/(c*x)]))/(Sqrt[e] - Sqrt[c^2*d + e])])/e^(5/2) - (((3*I)/4)*b*Sqrt[-d]*PolyLog[2, ((-I)*c*Sqrt[-d]*E^(I*ArcSin[1/(c*x)]) )/(Sqrt[e] + Sqrt[c^2*d + e])])/e^(5/2) + (((3*I)/4)*b*Sqrt[-d]*PolyLog[2, (I*c*Sqrt[-d]*E^(I*ArcSin[1/(c*x)]))/(Sqrt[e] + Sqrt[c^2*d + e])])/e^(5/2 )
3.2.7.3.1 Defintions of rubi rules used
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_.)*((d_) + (e_ .)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*ArcSin[c*x])^n, ( f*x)^m*(d + e*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[c^2*d + e, 0] && IGtQ[n, 0] && IntegerQ[p] && IntegerQ[m]
Int[((a_.) + ArcCsc[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_.) + (e_.)*(x_) ^2)^(p_.), x_Symbol] :> -Subst[Int[(e + d*x^2)^p*((a + b*ArcSin[x/c])^n/x^( m + 2*(p + 1))), x], x, 1/x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[n, 0] && IntegerQ[m] && IntegerQ[p]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 44.66 (sec) , antiderivative size = 965, normalized size of antiderivative = 1.20
method | result | size |
parts | \(\text {Expression too large to display}\) | \(965\) |
derivativedivides | \(\text {Expression too large to display}\) | \(987\) |
default | \(\text {Expression too large to display}\) | \(987\) |
a*(1/e^2*x-1/e^2*d*(-1/2*x/(e*x^2+d)+3/2/(d*e)^(1/2)*arctan(e*x/(d*e)^(1/2 ))))+b/c^5*(1/2*x*c^5*arccsc(c*x)*(2*c^2*e*x^2+3*c^2*d)/e^2/(c^2*e*x^2+c^2 *d)-1/e^2*c^4*ln(-1+I/c/x+(1-1/c^2/x^2)^(1/2))+1/e^2*c^4*ln(1+I/c/x+(1-1/c ^2/x^2)^(1/2))-3/16/e^3*c^6*d*sum((_R1^2*c^2*d-c^2*d-4*e)/_R1/(_R1^2*c^2*d -c^2*d-2*e)*(I*arccsc(c*x)*ln((_R1-I/c/x-(1-1/c^2/x^2)^(1/2))/_R1)+dilog(( _R1-I/c/x-(1-1/c^2/x^2)^(1/2))/_R1)),_R1=RootOf(c^2*d*_Z^4+(-2*c^2*d-4*e)* _Z^2+c^2*d))+3/16/e^3*c^6*d*sum((_R1^2*c^2*d+4*_R1^2*e-c^2*d)/_R1/(_R1^2*c ^2*d-c^2*d-2*e)*(I*arccsc(c*x)*ln((_R1-I/c/x-(1-1/c^2/x^2)^(1/2))/_R1)+dil og((_R1-I/c/x-(1-1/c^2/x^2)^(1/2))/_R1)),_R1=RootOf(c^2*d*_Z^4+(-2*c^2*d-4 *e)*_Z^2+c^2*d))-1/2*(-(c^2*d-2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2)*((e*(c^2 *d+e))^(1/2)*c^2*d+2*c^2*d*e+2*(e*(c^2*d+e))^(1/2)*e+2*e^2)*c*arctan(c*d*( I/c/x+(1-1/c^2/x^2)^(1/2))/((-c^2*d+2*(e*(c^2*d+e))^(1/2)-2*e)*d)^(1/2))/e ^2/(c^2*d+e)/d^2-1/2*((c^2*d+2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2)*(-(e*(c^2 *d+e))^(1/2)*c^2*d+2*c^2*d*e-2*(e*(c^2*d+e))^(1/2)*e+2*e^2)*c*arctanh(c*d* (I/c/x+(1-1/c^2/x^2)^(1/2))/((c^2*d+2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2))/e ^2/(c^2*d+e)/d^2+1/2*(-(c^2*d-2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2)*(c^2*d+2 *(e*(c^2*d+e))^(1/2)+2*e)*c*arctan(c*d*(I/c/x+(1-1/c^2/x^2)^(1/2))/((-c^2* d+2*(e*(c^2*d+e))^(1/2)-2*e)*d)^(1/2))/d^2/e^2+1/2*((c^2*d+2*(e*(c^2*d+e)) ^(1/2)+2*e)*d)^(1/2)*(c^2*d-2*(e*(c^2*d+e))^(1/2)+2*e)*c*arctanh(c*d*(I/c/ x+(1-1/c^2/x^2)^(1/2))/((c^2*d+2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2))/d^2...
\[ \int \frac {x^4 \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=\int { \frac {{\left (b \operatorname {arccsc}\left (c x\right ) + a\right )} x^{4}}{{\left (e x^{2} + d\right )}^{2}} \,d x } \]
\[ \int \frac {x^4 \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=\int \frac {x^{4} \left (a + b \operatorname {acsc}{\left (c x \right )}\right )}{\left (d + e x^{2}\right )^{2}}\, dx \]
Exception generated. \[ \int \frac {x^4 \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=\text {Exception raised: ValueError} \]
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
Timed out. \[ \int \frac {x^4 \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {x^4 \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=\int \frac {x^4\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}{{\left (e\,x^2+d\right )}^2} \,d x \]