Integrand size = 18, antiderivative size = 529 \[ \int \frac {a+b \csc ^{-1}(c x)}{d+e x^2} \, dx=-\frac {\left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {\left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {\left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {\left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {i b \operatorname {PolyLog}\left (2,-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {i b \operatorname {PolyLog}\left (2,\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {i b \operatorname {PolyLog}\left (2,-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {i b \operatorname {PolyLog}\left (2,\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}} \]
-1/2*(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^(1/2)+1/2*(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^(1/2)-1/2*(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^(1/2)+1/2*(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^(1/2)-1/2*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^(1/2)+1/2*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^(1/2)-1/2*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^(1/2)+1/2*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 ^(1/2)
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1068\) vs. \(2(529)=1058\).
Time = 0.45 (sec) , antiderivative size = 1068, normalized size of antiderivative = 2.02 \[ \int \frac {a+b \csc ^{-1}(c x)}{d+e x^2} \, dx=-\frac {i \left (4 i a \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )+8 i b \arcsin \left (\frac {\sqrt {1-\frac {i \sqrt {e}}{c \sqrt {d}}}}{\sqrt {2}}\right ) \arctan \left (\frac {\left (-i c \sqrt {d}+\sqrt {e}\right ) \cot \left (\frac {1}{4} \left (\pi +2 \csc ^{-1}(c x)\right )\right )}{\sqrt {c^2 d+e}}\right )-8 i b \arcsin \left (\frac {\sqrt {1+\frac {i \sqrt {e}}{c \sqrt {d}}}}{\sqrt {2}}\right ) \arctan \left (\frac {\left (i c \sqrt {d}+\sqrt {e}\right ) \cot \left (\frac {1}{4} \left (\pi +2 \csc ^{-1}(c x)\right )\right )}{\sqrt {c^2 d+e}}\right )+b \pi \log \left (1+\frac {\left (\sqrt {e}-\sqrt {c^2 d+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt {d}}\right )-2 b \csc ^{-1}(c x) \log \left (1+\frac {\left (\sqrt {e}-\sqrt {c^2 d+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt {d}}\right )+4 b \arcsin \left (\frac {\sqrt {1-\frac {i \sqrt {e}}{c \sqrt {d}}}}{\sqrt {2}}\right ) \log \left (1+\frac {\left (\sqrt {e}-\sqrt {c^2 d+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt {d}}\right )-b \pi \log \left (1+\frac {\left (-\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt {d}}\right )+2 b \csc ^{-1}(c x) \log \left (1+\frac {\left (-\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt {d}}\right )-4 b \arcsin \left (\frac {\sqrt {1+\frac {i \sqrt {e}}{c \sqrt {d}}}}{\sqrt {2}}\right ) \log \left (1+\frac {\left (-\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt {d}}\right )-b \pi \log \left (1-\frac {\left (\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt {d}}\right )+2 b \csc ^{-1}(c x) \log \left (1-\frac {\left (\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt {d}}\right )+4 b \arcsin \left (\frac {\sqrt {1+\frac {i \sqrt {e}}{c \sqrt {d}}}}{\sqrt {2}}\right ) \log \left (1-\frac {\left (\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt {d}}\right )+b \pi \log \left (1+\frac {\left (\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt {d}}\right )-2 b \csc ^{-1}(c x) \log \left (1+\frac {\left (\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt {d}}\right )-4 b \arcsin \left (\frac {\sqrt {1-\frac {i \sqrt {e}}{c \sqrt {d}}}}{\sqrt {2}}\right ) \log \left (1+\frac {\left (\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt {d}}\right )+b \pi \log \left (\sqrt {e}-\frac {i \sqrt {d}}{x}\right )-b \pi \log \left (\sqrt {e}+\frac {i \sqrt {d}}{x}\right )+2 i b \operatorname {PolyLog}\left (2,\frac {\left (\sqrt {e}-\sqrt {c^2 d+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt {d}}\right )-2 i b \operatorname {PolyLog}\left (2,\frac {\left (-\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt {d}}\right )-2 i b \operatorname {PolyLog}\left (2,-\frac {\left (\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt {d}}\right )+2 i b \operatorname {PolyLog}\left (2,\frac {\left (\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt {d}}\right )\right )}{4 \sqrt {d} \sqrt {e}} \]
((-1/4*I)*((4*I)*a*ArcTan[(Sqrt[e]*x)/Sqrt[d]] + (8*I)*b*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*ArcSin[Sqrt[1 + (I*Sqrt[ e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[((I*c*Sqrt[d] + Sqrt[e])*Cot[(Pi + 2*ArcC sc[c*x])/4])/Sqrt[c^2*d + e]] + b*Pi*Log[1 + (Sqrt[e] - Sqrt[c^2*d + e])/( c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - 2*b*ArcCsc[c*x]*Log[1 + (Sqrt[e] - Sqrt[c^ 2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + 4*b*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*Pi*Log[1 + (-Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^( I*ArcCsc[c*x]))] + 2*b*ArcCsc[c*x]*Log[1 + (-Sqrt[e] + Sqrt[c^2*d + e])/(c *Sqrt[d]*E^(I*ArcCsc[c*x]))] - 4*b*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*Pi*Log[1 - (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x ]))] + 2*b*ArcCsc[c*x]*Log[1 - (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I *ArcCsc[c*x]))] + 4*b*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Lo g[1 - (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + b*Pi*Lo g[1 + (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - 2*b*Arc Csc[c*x]*Log[1 + (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x])) ] - 4*b*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*Pi*Log[Sqrt[e] -...
Time = 1.36 (sec) , antiderivative size = 577, normalized size of antiderivative = 1.09, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {5754, 5172, 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.
| \(\displaystyle \int \frac {a+b \csc ^{-1}(c x)}{d+e x^2} \, dx\) |
| \(\Big \downarrow \) 5754 |
| \(\displaystyle -\int \frac {a+b \arcsin \left (\frac {1}{c x}\right )}{\frac {d}{x^2}+e}d\frac {1}{x}\) |
| \(\Big \downarrow \) 5172 |
| \(\displaystyle -\int \left (\frac {a+b \arcsin \left (\frac {1}{c x}\right )}{2 \sqrt {e} \left (\sqrt {e}-\frac {\sqrt {-d}}{x}\right )}+\frac {a+b \arcsin \left (\frac {1}{c x}\right )}{2 \sqrt {e} \left (\frac {\sqrt {-d}}{x}+\sqrt {e}\right )}\right )d\frac {1}{x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\left (a+b \arcsin \left (\frac {1}{c x}\right )\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \arcsin \left (\frac {1}{c x}\right )}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {\left (a+b \arcsin \left (\frac {1}{c x}\right )\right ) \log \left (1+\frac {i c \sqrt {-d} e^{i \arcsin \left (\frac {1}{c x}\right )}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {\left (a+b \arcsin \left (\frac {1}{c x}\right )\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \arcsin \left (\frac {1}{c x}\right )}}{\sqrt {c^2 d+e}+\sqrt {e}}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {\left (a+b \arcsin \left (\frac {1}{c x}\right )\right ) \log \left (1+\frac {i c \sqrt {-d} e^{i \arcsin \left (\frac {1}{c x}\right )}}{\sqrt {c^2 d+e}+\sqrt {e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {i b \operatorname {PolyLog}\left (2,-\frac {i c \sqrt {-d} e^{i \arcsin \left (\frac {1}{c x}\right )}}{\sqrt {e}-\sqrt {d c^2+e}}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {i b \operatorname {PolyLog}\left (2,\frac {i c \sqrt {-d} e^{i \arcsin \left (\frac {1}{c x}\right )}}{\sqrt {e}-\sqrt {d c^2+e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {i b \operatorname {PolyLog}\left (2,-\frac {i c \sqrt {-d} e^{i \arcsin \left (\frac {1}{c x}\right )}}{\sqrt {e}+\sqrt {d c^2+e}}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {i b \operatorname {PolyLog}\left (2,\frac {i c \sqrt {-d} e^{i \arcsin \left (\frac {1}{c x}\right )}}{\sqrt {e}+\sqrt {d c^2+e}}\right )}{2 \sqrt {-d} \sqrt {e}}\) |
-1/2*((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])])/(Sqrt[-d]*Sqrt[e]) + ((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*Sqrt[-d]*Sqrt[e]) - ((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*Sqrt[-d]*Sqrt[e]) + ((a + b*ArcSin[1/(c*x)])*Log[1 + (I*c*Sqrt[-d]*E^(I*ArcSin[1/(c*x)]))/(S qrt[e] + Sqrt[c^2*d + e])])/(2*Sqrt[-d]*Sqrt[e]) - ((I/2)*b*PolyLog[2, ((- I)*c*Sqrt[-d]*E^(I*ArcSin[1/(c*x)]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(Sqrt[- d]*Sqrt[e]) + ((I/2)*b*PolyLog[2, (I*c*Sqrt[-d]*E^(I*ArcSin[1/(c*x)]))/(Sq rt[e] - Sqrt[c^2*d + e])])/(Sqrt[-d]*Sqrt[e]) - ((I/2)*b*PolyLog[2, ((-I)* c*Sqrt[-d]*E^(I*ArcSin[1/(c*x)]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(Sqrt[-d]* Sqrt[e]) + ((I/2)*b*PolyLog[2, (I*c*Sqrt[-d]*E^(I*ArcSin[1/(c*x)]))/(Sqrt[ e] + Sqrt[c^2*d + e])])/(Sqrt[-d]*Sqrt[e])
3.1.100.3.1 Defintions of rubi rules used
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x _Symbol] :> Int[ExpandIntegrand[(a + b*ArcSin[c*x])^n, (d + e*x^2)^p, x], x ] /; FreeQ[{a, b, c, d, e, n}, x] && NeQ[c^2*d + e, 0] && IntegerQ[p] && (G tQ[p, 0] || IGtQ[n, 0])
Int[((a_.) + ArcCsc[(c_.)*(x_)]*(b_.))^(n_.)*((d_.) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> -Subst[Int[(e + d*x^2)^p*((a + b*ArcSin[x/c])^n/x^(2*(p + 1))) , x], x, 1/x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[n, 0] && IntegerQ[p]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 24.99 (sec) , antiderivative size = 272, normalized size of antiderivative = 0.51
| method | result | size |
| parts | \(\frac {a \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{\sqrt {d e}}-\frac {b c \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c^{2} d \,\textit {\_Z}^{4}+\left (-2 c^{2} d -4 e \right ) \textit {\_Z}^{2}+c^{2} d \right )}{\sum }\frac {i \operatorname {arccsc}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} c^{2} d -c^{2} d -2 e \right )}\right )}{2}-\frac {b c \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c^{2} d \,\textit {\_Z}^{4}+\left (-2 c^{2} d -4 e \right ) \textit {\_Z}^{2}+c^{2} d \right )}{\sum }\frac {\textit {\_R1} \left (i \operatorname {arccsc}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} c^{2} d -c^{2} d -2 e}\right )}{2}\) | \(272\) |
| derivativedivides | \(\frac {\frac {a c \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{\sqrt {d e}}+b \,c^{2} \left (-\frac {\left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c^{2} d \,\textit {\_Z}^{4}+\left (-2 c^{2} d -4 e \right ) \textit {\_Z}^{2}+c^{2} d \right )}{\sum }\frac {i \operatorname {arccsc}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} c^{2} d -c^{2} d -2 e \right )}\right )}{2}-\frac {\left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c^{2} d \,\textit {\_Z}^{4}+\left (-2 c^{2} d -4 e \right ) \textit {\_Z}^{2}+c^{2} d \right )}{\sum }\frac {\textit {\_R1} \left (i \operatorname {arccsc}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} c^{2} d -c^{2} d -2 e}\right )}{2}\right )}{c}\) | \(279\) |
| default | \(\frac {\frac {a c \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{\sqrt {d e}}+b \,c^{2} \left (-\frac {\left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c^{2} d \,\textit {\_Z}^{4}+\left (-2 c^{2} d -4 e \right ) \textit {\_Z}^{2}+c^{2} d \right )}{\sum }\frac {i \operatorname {arccsc}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} c^{2} d -c^{2} d -2 e \right )}\right )}{2}-\frac {\left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c^{2} d \,\textit {\_Z}^{4}+\left (-2 c^{2} d -4 e \right ) \textit {\_Z}^{2}+c^{2} d \right )}{\sum }\frac {\textit {\_R1} \left (i \operatorname {arccsc}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} c^{2} d -c^{2} d -2 e}\right )}{2}\right )}{c}\) | \(279\) |
a/(d*e)^(1/2)*arctan(e*x/(d*e)^(1/2))-1/2*b*c*sum(1/_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))-1/2*b*c*sum(_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)),_R 1=RootOf(c^2*d*_Z^4+(-2*c^2*d-4*e)*_Z^2+c^2*d))
\[ \int \frac {a+b \csc ^{-1}(c x)}{d+e x^2} \, dx=\int { \frac {b \operatorname {arccsc}\left (c x\right ) + a}{e x^{2} + d} \,d x } \]
\[ \int \frac {a+b \csc ^{-1}(c x)}{d+e x^2} \, dx=\int \frac {a + b \operatorname {acsc}{\left (c x \right )}}{d + e x^{2}}\, dx \]
Exception generated. \[ \int \frac {a+b \csc ^{-1}(c x)}{d+e x^2} \, dx=\text {Exception raised: ValueError} \]
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
Exception generated. \[ \int \frac {a+b \csc ^{-1}(c x)}{d+e x^2} \, dx=\text {Exception raised: RuntimeError} \]
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. \[ \int \frac {a+b \csc ^{-1}(c x)}{d+e x^2} \, dx=\int \frac {a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )}{e\,x^2+d} \,d x \]