Integrand size = 21, antiderivative size = 565 \[ \int \frac {x^2 \left (a+b \csc ^{-1}(c x)\right )}{d+e x^2} \, dx=\frac {x \left (a+b \csc ^{-1}(c x)\right )}{e}+\frac {b \text {arctanh}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{c e}-\frac {\sqrt {-d} \left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 e^{3/2}}+\frac {\sqrt {-d} \left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 e^{3/2}}-\frac {\sqrt {-d} \left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 e^{3/2}}+\frac {\sqrt {-d} \left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 e^{3/2}}-\frac {i b \sqrt {-d} \operatorname {PolyLog}\left (2,-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 e^{3/2}}+\frac {i b \sqrt {-d} \operatorname {PolyLog}\left (2,\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 e^{3/2}}-\frac {i b \sqrt {-d} \operatorname {PolyLog}\left (2,-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 e^{3/2}}+\frac {i b \sqrt {-d} \operatorname {PolyLog}\left (2,\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 e^{3/2}} \] Output:
x*(a+b*arccsc(c*x))/e+b*arctanh((1-1/c^2/x^2)^(1/2))/c/e-1/2*(-d)^(1/2)*(a +b*arccsc(c*x))*ln(1-I*c*(-d)^(1/2)*(I/c/x+(1-1/c^2/x^2)^(1/2))/(e^(1/2)-( c^2*d+e)^(1/2)))/e^(3/2)+1/2*(-d)^(1/2)*(a+b*arccsc(c*x))*ln(1+I*c*(-d)^(1 /2)*(I/c/x+(1-1/c^2/x^2)^(1/2))/(e^(1/2)-(c^2*d+e)^(1/2)))/e^(3/2)-1/2*(-d )^(1/2)*(a+b*arccsc(c*x))*ln(1-I*c*(-d)^(1/2)*(I/c/x+(1-1/c^2/x^2)^(1/2))/ (e^(1/2)+(c^2*d+e)^(1/2)))/e^(3/2)+1/2*(-d)^(1/2)*(a+b*arccsc(c*x))*ln(1+I *c*(-d)^(1/2)*(I/c/x+(1-1/c^2/x^2)^(1/2))/(e^(1/2)+(c^2*d+e)^(1/2)))/e^(3/ 2)-1/2*I*b*(-d)^(1/2)*polylog(2,-I*c*(-d)^(1/2)*(I/c/x+(1-1/c^2/x^2)^(1/2) )/(e^(1/2)-(c^2*d+e)^(1/2)))/e^(3/2)+1/2*I*b*(-d)^(1/2)*polylog(2,I*c*(-d) ^(1/2)*(I/c/x+(1-1/c^2/x^2)^(1/2))/(e^(1/2)-(c^2*d+e)^(1/2)))/e^(3/2)-1/2* I*b*(-d)^(1/2)*polylog(2,-I*c*(-d)^(1/2)*(I/c/x+(1-1/c^2/x^2)^(1/2))/(e^(1 /2)+(c^2*d+e)^(1/2)))/e^(3/2)+1/2*I*b*(-d)^(1/2)*polylog(2,I*c*(-d)^(1/2)* (I/c/x+(1-1/c^2/x^2)^(1/2))/(e^(1/2)+(c^2*d+e)^(1/2)))/e^(3/2)
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1260\) vs. \(2(565)=1130\).
Time = 1.78 (sec) , antiderivative size = 1260, normalized size of antiderivative = 2.23 \[ \int \frac {x^2 \left (a+b \csc ^{-1}(c x)\right )}{d+e x^2} \, dx =\text {Too large to display} \] Input:
Integrate[(x^2*(a + b*ArcCsc[c*x]))/(d + e*x^2),x]
Output:
((I/4)*((-4*I)*a*c*Sqrt[e]*x - (4*I)*b*c*Sqrt[e]*x*ArcCsc[c*x] + (4*I)*a*c *Sqrt[d]*ArcTan[(Sqrt[e]*x)/Sqrt[d]] + (8*I)*b*c*Sqrt[d]*ArcSin[Sqrt[1 - ( I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[(((-I)*c*Sqrt[d] + Sqrt[e])*Cot[(P i + 2*ArcCsc[c*x])/4])/Sqrt[c^2*d + e]] - (8*I)*b*c*Sqrt[d]*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[((I*c*Sqrt[d] + Sqrt[e])*Cot[(P i + 2*ArcCsc[c*x])/4])/Sqrt[c^2*d + e]] + b*c*Sqrt[d]*Pi*Log[1 + (Sqrt[e] - Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - 2*b*c*Sqrt[d]*ArcCsc[c *x]*Log[1 + (Sqrt[e] - Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + 4 *b*c*Sqrt[d]*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (Sq rt[e] - Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - b*c*Sqrt[d]*Pi*L og[1 + (-Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + 2*b*c *Sqrt[d]*ArcCsc[c*x]*Log[1 + (-Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I* ArcCsc[c*x]))] - 4*b*c*Sqrt[d]*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]))] - b*c*Sqrt[d]*Pi*Log[1 - (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcC sc[c*x]))] + 2*b*c*Sqrt[d]*ArcCsc[c*x]*Log[1 - (Sqrt[e] + Sqrt[c^2*d + e]) /(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + 4*b*c*Sqrt[d]*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]))] + b*c*Sqrt[d]*Pi*Log[1 + (Sqrt[e] + Sqrt[c^2*d + e])/(c*S qrt[d]*E^(I*ArcCsc[c*x]))] - 2*b*c*Sqrt[d]*ArcCsc[c*x]*Log[1 + (Sqrt[e]...
Time = 1.65 (sec) , antiderivative size = 617, normalized size of antiderivative = 1.09, 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.
| \(\displaystyle \int \frac {x^2 \left (a+b \csc ^{-1}(c x)\right )}{d+e x^2} \, dx\) |
| \(\Big \downarrow \) 5764 |
| \(\displaystyle -\int \frac {x^2 \left (a+b \arcsin \left (\frac {1}{c x}\right )\right )}{\frac {d}{x^2}+e}d\frac {1}{x}\) |
| \(\Big \downarrow \) 5232 |
| \(\displaystyle -\int \left (\frac {x^2 \left (a+b \arcsin \left (\frac {1}{c x}\right )\right )}{e}-\frac {d \left (a+b \arcsin \left (\frac {1}{c x}\right )\right )}{e \left (\frac {d}{x^2}+e\right )}\right )d\frac {1}{x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\sqrt {-d} \left (a+b \arcsin \left (\frac {1}{c x}\right )\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \arcsin \left (\frac {1}{c x}\right )}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 e^{3/2}}+\frac {\sqrt {-d} \left (a+b \arcsin \left (\frac {1}{c x}\right )\right ) \log \left (1+\frac {i c \sqrt {-d} e^{i \arcsin \left (\frac {1}{c x}\right )}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 e^{3/2}}-\frac {\sqrt {-d} \left (a+b \arcsin \left (\frac {1}{c x}\right )\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \arcsin \left (\frac {1}{c x}\right )}}{\sqrt {c^2 d+e}+\sqrt {e}}\right )}{2 e^{3/2}}+\frac {\sqrt {-d} \left (a+b \arcsin \left (\frac {1}{c x}\right )\right ) \log \left (1+\frac {i c \sqrt {-d} e^{i \arcsin \left (\frac {1}{c x}\right )}}{\sqrt {c^2 d+e}+\sqrt {e}}\right )}{2 e^{3/2}}+\frac {x \left (a+b \arcsin \left (\frac {1}{c x}\right )\right )}{e}-\frac {i b \sqrt {-d} \operatorname {PolyLog}\left (2,-\frac {i c \sqrt {-d} e^{i \arcsin \left (\frac {1}{c x}\right )}}{\sqrt {e}-\sqrt {d c^2+e}}\right )}{2 e^{3/2}}+\frac {i b \sqrt {-d} \operatorname {PolyLog}\left (2,\frac {i c \sqrt {-d} e^{i \arcsin \left (\frac {1}{c x}\right )}}{\sqrt {e}-\sqrt {d c^2+e}}\right )}{2 e^{3/2}}-\frac {i b \sqrt {-d} \operatorname {PolyLog}\left (2,-\frac {i c \sqrt {-d} e^{i \arcsin \left (\frac {1}{c x}\right )}}{\sqrt {e}+\sqrt {d c^2+e}}\right )}{2 e^{3/2}}+\frac {i b \sqrt {-d} \operatorname {PolyLog}\left (2,\frac {i c \sqrt {-d} e^{i \arcsin \left (\frac {1}{c x}\right )}}{\sqrt {e}+\sqrt {d c^2+e}}\right )}{2 e^{3/2}}+\frac {b \text {arctanh}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{c e}\) |
Input:
Int[(x^2*(a + b*ArcCsc[c*x]))/(d + e*x^2),x]
Output:
(x*(a + b*ArcSin[1/(c*x)]))/e + (b*ArcTanh[Sqrt[1 - 1/(c^2*x^2)]])/(c*e) - (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])])/(2*e^(3/2)) + (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])])/(2*e^(3/2)) - (Sqrt[-d]*(a + b*ArcSin[1/(c*x)])*Log[1 - (I*c*Sq rt[-d]*E^(I*ArcSin[1/(c*x)]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*e^(3/2)) + (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])])/(2*e^(3/2)) - ((I/2)*b*Sqrt[-d]*PolyLog [2, ((-I)*c*Sqrt[-d]*E^(I*ArcSin[1/(c*x)]))/(Sqrt[e] - Sqrt[c^2*d + e])])/ e^(3/2) + ((I/2)*b*Sqrt[-d]*PolyLog[2, (I*c*Sqrt[-d]*E^(I*ArcSin[1/(c*x)]) )/(Sqrt[e] - Sqrt[c^2*d + e])])/e^(3/2) - ((I/2)*b*Sqrt[-d]*PolyLog[2, ((- I)*c*Sqrt[-d]*E^(I*ArcSin[1/(c*x)]))/(Sqrt[e] + Sqrt[c^2*d + e])])/e^(3/2) + ((I/2)*b*Sqrt[-d]*PolyLog[2, (I*c*Sqrt[-d]*E^(I*ArcSin[1/(c*x)]))/(Sqrt [e] + Sqrt[c^2*d + e])])/e^(3/2)
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 = 66.77 (sec) , antiderivative size = 415, normalized size of antiderivative = 0.73
| method | result | size |
| parts | \(\frac {a x}{e}-\frac {a d \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{e \sqrt {d e}}+\frac {b \left (\frac {c^{3} \operatorname {arccsc}\left (c x \right ) x}{e}+\frac {c^{2} \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{e}-\frac {c^{2} \ln \left (-1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{e}-\frac {c^{4} d \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c^{2} d \,\textit {\_Z}^{4}+\left (-2 c^{2} d -4 e \right ) \textit {\_Z}^{2}+c^{2} d \right )}{\sum }\frac {\left (\textit {\_R1}^{2} c^{2} d -c^{2} d -4 e \right ) \left (i \operatorname {arccsc}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} c^{2} d -c^{2} d -2 e \right )}\right )}{8 e^{2}}+\frac {c^{4} d \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c^{2} d \,\textit {\_Z}^{4}+\left (-2 c^{2} d -4 e \right ) \textit {\_Z}^{2}+c^{2} d \right )}{\sum }\frac {\left (\textit {\_R1}^{2} c^{2} d +4 \textit {\_R1}^{2} e -c^{2} d \right ) \left (i \operatorname {arccsc}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} c^{2} d -c^{2} d -2 e \right )}\right )}{8 e^{2}}\right )}{c^{3}}\) | \(415\) |
| derivativedivides | \(\frac {\frac {a \,c^{3} x}{e}-\frac {a \,c^{3} d \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{e \sqrt {d e}}+b \,c^{2} \left (\frac {c x \,\operatorname {arccsc}\left (c x \right )}{e}+\frac {\ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{e}-\frac {\ln \left (-1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{e}-\frac {c^{2} d \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c^{2} d \,\textit {\_Z}^{4}+\left (-2 c^{2} d -4 e \right ) \textit {\_Z}^{2}+c^{2} d \right )}{\sum }\frac {\left (\textit {\_R1}^{2} c^{2} d -c^{2} d -4 e \right ) \left (i \operatorname {arccsc}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} c^{2} d -c^{2} d -2 e \right )}\right )}{8 e^{2}}+\frac {c^{2} d \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c^{2} d \,\textit {\_Z}^{4}+\left (-2 c^{2} d -4 e \right ) \textit {\_Z}^{2}+c^{2} d \right )}{\sum }\frac {\left (\textit {\_R1}^{2} c^{2} d +4 \textit {\_R1}^{2} e -c^{2} d \right ) \left (i \operatorname {arccsc}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} c^{2} d -c^{2} d -2 e \right )}\right )}{8 e^{2}}\right )}{c^{3}}\) | \(417\) |
| default | \(\frac {\frac {a \,c^{3} x}{e}-\frac {a \,c^{3} d \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{e \sqrt {d e}}+b \,c^{2} \left (\frac {c x \,\operatorname {arccsc}\left (c x \right )}{e}+\frac {\ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{e}-\frac {\ln \left (-1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{e}-\frac {c^{2} d \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c^{2} d \,\textit {\_Z}^{4}+\left (-2 c^{2} d -4 e \right ) \textit {\_Z}^{2}+c^{2} d \right )}{\sum }\frac {\left (\textit {\_R1}^{2} c^{2} d -c^{2} d -4 e \right ) \left (i \operatorname {arccsc}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} c^{2} d -c^{2} d -2 e \right )}\right )}{8 e^{2}}+\frac {c^{2} d \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c^{2} d \,\textit {\_Z}^{4}+\left (-2 c^{2} d -4 e \right ) \textit {\_Z}^{2}+c^{2} d \right )}{\sum }\frac {\left (\textit {\_R1}^{2} c^{2} d +4 \textit {\_R1}^{2} e -c^{2} d \right ) \left (i \operatorname {arccsc}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} c^{2} d -c^{2} d -2 e \right )}\right )}{8 e^{2}}\right )}{c^{3}}\) | \(417\) |
Input:
int(x^2*(a+b*arccsc(c*x))/(e*x^2+d),x,method=_RETURNVERBOSE)
Output:
a/e*x-a*d/e/(d*e)^(1/2)*arctan(e*x/(d*e)^(1/2))+b/c^3*(c^3*arccsc(c*x)/e*x +c^2/e*ln(1+I/c/x+(1-1/c^2/x^2)^(1/2))-c^2/e*ln(-1+I/c/x+(1-1/c^2/x^2)^(1/ 2))-1/8*c^4/e^2*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)) +1/8*c^4/e^2*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)+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)))
\[ \int \frac {x^2 \left (a+b \csc ^{-1}(c x)\right )}{d+e x^2} \, dx=\int { \frac {{\left (b \operatorname {arccsc}\left (c x\right ) + a\right )} x^{2}}{e x^{2} + d} \,d x } \] Input:
integrate(x^2*(a+b*arccsc(c*x))/(e*x^2+d),x, algorithm="fricas")
Output:
integral((b*x^2*arccsc(c*x) + a*x^2)/(e*x^2 + d), x)
\[ \int \frac {x^2 \left (a+b \csc ^{-1}(c x)\right )}{d+e x^2} \, dx=\int \frac {x^{2} \left (a + b \operatorname {acsc}{\left (c x \right )}\right )}{d + e x^{2}}\, dx \] Input:
integrate(x**2*(a+b*acsc(c*x))/(e*x**2+d),x)
Output:
Integral(x**2*(a + b*acsc(c*x))/(d + e*x**2), x)
Exception generated. \[ \int \frac {x^2 \left (a+b \csc ^{-1}(c x)\right )}{d+e x^2} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(x^2*(a+b*arccsc(c*x))/(e*x^2+d),x, algorithm="maxima")
Output:
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
Exception generated. \[ \int \frac {x^2 \left (a+b \csc ^{-1}(c x)\right )}{d+e x^2} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate(x^2*(a+b*arccsc(c*x))/(e*x^2+d),x, algorithm="giac")
Output:
Exception raised: RuntimeError >> an error occurred running a Giac command :INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve cteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {x^2 \left (a+b \csc ^{-1}(c x)\right )}{d+e x^2} \, dx=\int \frac {x^2\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}{e\,x^2+d} \,d x \] Input:
int((x^2*(a + b*asin(1/(c*x))))/(d + e*x^2),x)
Output:
int((x^2*(a + b*asin(1/(c*x))))/(d + e*x^2), x)
\[ \int \frac {x^2 \left (a+b \csc ^{-1}(c x)\right )}{d+e x^2} \, dx=\frac {-\sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) a +\left (\int \frac {\mathit {acsc} \left (c x \right ) x^{2}}{e \,x^{2}+d}d x \right ) b \,e^{2}+a e x}{e^{2}} \] Input:
int(x^2*(a+b*acsc(c*x))/(e*x^2+d),x)
Output:
( - sqrt(e)*sqrt(d)*atan((e*x)/(sqrt(e)*sqrt(d)))*a + int((acsc(c*x)*x**2) /(d + e*x**2),x)*b*e**2 + a*e*x)/e**2