Integrand size = 18, antiderivative size = 762 \[ \int \frac {a+b \csc ^{-1}(c x)}{\left (d+e x^2\right )^2} \, dx=\frac {-a-b \csc ^{-1}(c x)}{4 d \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}+\frac {a+b \csc ^{-1}(c x)}{4 d \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}+\frac {b \text {arctanh}\left (\frac {c^2 d-\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {c^2 d+e} \sqrt {1-\frac {1}{c^2 x^2}}}\right )}{4 d^{3/2} \sqrt {c^2 d+e}}+\frac {b \text {arctanh}\left (\frac {c^2 d+\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {c^2 d+e} \sqrt {1-\frac {1}{c^2 x^2}}}\right )}{4 d^{3/2} \sqrt {c^2 d+e}}+\frac {\left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}-\frac {\left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {\left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}-\frac {\left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {i b \operatorname {PolyLog}\left (2,-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}-\frac {i b \operatorname {PolyLog}\left (2,\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {i b \operatorname {PolyLog}\left (2,-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}-\frac {i b \operatorname {PolyLog}\left (2,\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}} \] Output:
1/4*(-a-b*arccsc(c*x))/d/((-d)^(1/2)*e^(1/2)-d/x)+1/4*(a+b*arccsc(c*x))/d/ ((-d)^(1/2)*e^(1/2)+d/x)+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^(3/2)/(c^2*d+e)^(1/2)+1/4*b*arc tanh((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^(3/2)/(c^2*d+e)^(1/2)+1/4*(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)))/(-d)^(3/2)/e^(1/2)-1 /4*(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)))/(-d)^(3/2)/e^(1/2)+1/4*(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)))/(-d)^(3/2) /e^(1/2)-1/4*(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)))/(-d)^(3/2)/e^(1/2)+1/4*I*b*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)))/(-d)^(3/ 2)/e^(1/2)-1/4*I*b*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)))/(-d)^(3/2)/e^(1/2)+1/4*I*b*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)))/(-d)^(3/2)/e^(1 /2)-1/4*I*b*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)))/(-d)^(3/2)/e^(1/2)
Time = 1.80 (sec) , antiderivative size = 1477, normalized size of antiderivative = 1.94 \[ \int \frac {a+b \csc ^{-1}(c x)}{\left (d+e x^2\right )^2} \, dx =\text {Too large to display} \] Input:
Integrate[(a + b*ArcCsc[c*x])/(d + e*x^2)^2,x]
Output:
((a*x)/(d^2 + d*e*x^2) + (a*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(d^(3/2)*Sqrt[e]) + (b*((2*Sqrt[d]*ArcCsc[c*x])/((-I)*Sqrt[d]*Sqrt[e] + e*x) + (2*Sqrt[d]*A rcCsc[c*x])/(I*Sqrt[d]*Sqrt[e] + e*x) + (8*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]])/Sqrt[e] - (8*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqr t[d])]/Sqrt[2]]*ArcTan[((I*c*Sqrt[d] + Sqrt[e])*Cot[(Pi + 2*ArcCsc[c*x])/4 ])/Sqrt[c^2*d + e]])/Sqrt[e] - (I*Pi*Log[1 + (Sqrt[e] - Sqrt[c^2*d + e])/( c*Sqrt[d]*E^(I*ArcCsc[c*x]))])/Sqrt[e] + ((2*I)*ArcCsc[c*x]*Log[1 + (Sqrt[ e] - Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))])/Sqrt[e] - ((4*I)*Arc Sin[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]))])/Sqrt[e] + (I*Pi*Log[1 + (-Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))])/Sqrt[e] - ((2*I)*ArcCs c[c*x]*Log[1 + (-Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] )/Sqrt[e] + ((4*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]))])/Sqrt[e] + (I*Pi*Log[1 - (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))])/ Sqrt[e] - ((2*I)*ArcCsc[c*x]*Log[1 - (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d ]*E^(I*ArcCsc[c*x]))])/Sqrt[e] - ((4*I)*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqr t[d])]/Sqrt[2]]*Log[1 - (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc [c*x]))])/Sqrt[e] - (I*Pi*Log[1 + (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d...
Time = 2.69 (sec) , antiderivative size = 815, normalized size of antiderivative = 1.07, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {5754, 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.
| \(\displaystyle \int \frac {a+b \csc ^{-1}(c x)}{\left (d+e x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 5754 |
| \(\displaystyle -\int \frac {a+b \arcsin \left (\frac {1}{c x}\right )}{\left (\frac {d}{x^2}+e\right )^2 x^2}d\frac {1}{x}\) |
| \(\Big \downarrow \) 5232 |
| \(\displaystyle -\int \left (\frac {a+b \arcsin \left (\frac {1}{c x}\right )}{d \left (\frac {d}{x^2}+e\right )}-\frac {e \left (a+b \arcsin \left (\frac {1}{c x}\right )\right )}{d \left (\frac {d}{x^2}+e\right )^2}\right )d\frac {1}{x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\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 (-d)^{3/2} \sqrt {e}}-\frac {\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 (-d)^{3/2} \sqrt {e}}+\frac {\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 (-d)^{3/2} \sqrt {e}}-\frac {\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 (-d)^{3/2} \sqrt {e}}-\frac {a+b \arcsin \left (\frac {1}{c x}\right )}{4 d \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}+\frac {a+b \arcsin \left (\frac {1}{c x}\right )}{4 d \left (\frac {d}{x}+\sqrt {-d} \sqrt {e}\right )}+\frac {b \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 d^{3/2} \sqrt {d c^2+e}}+\frac {b \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 d^{3/2} \sqrt {d c^2+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 )}{4 (-d)^{3/2} \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 )}{4 (-d)^{3/2} \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 )}{4 (-d)^{3/2} \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 )}{4 (-d)^{3/2} \sqrt {e}}\) |
Input:
Int[(a + b*ArcCsc[c*x])/(d + e*x^2)^2,x]
Output:
-1/4*(a + b*ArcSin[1/(c*x)])/(d*(Sqrt[-d]*Sqrt[e] - d/x)) + (a + b*ArcSin[ 1/(c*x)])/(4*d*(Sqrt[-d]*Sqrt[e] + d/x)) + (b*ArcTanh[(c^2*d - (Sqrt[-d]*S qrt[e])/x)/(c*Sqrt[d]*Sqrt[c^2*d + e]*Sqrt[1 - 1/(c^2*x^2)])])/(4*d^(3/2)* Sqrt[c^2*d + e]) + (b*ArcTanh[(c^2*d + (Sqrt[-d]*Sqrt[e])/x)/(c*Sqrt[d]*Sq rt[c^2*d + e]*Sqrt[1 - 1/(c^2*x^2)])])/(4*d^(3/2)*Sqrt[c^2*d + 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])])/(4*(-d)^(3/2)*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])])/(4 *(-d)^(3/2)*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])])/(4*(-d)^(3/2)*Sqrt[e]) - ((a + b*ArcSin[1/(c*x)])*Log[1 + (I*c*Sqrt[-d]*E^(I*ArcSin[1/(c*x)]))/(Sqr t[e] + Sqrt[c^2*d + e])])/(4*(-d)^(3/2)*Sqrt[e]) + ((I/4)*b*PolyLog[2, ((- I)*c*Sqrt[-d]*E^(I*ArcSin[1/(c*x)]))/(Sqrt[e] - Sqrt[c^2*d + e])])/((-d)^( 3/2)*Sqrt[e]) - ((I/4)*b*PolyLog[2, (I*c*Sqrt[-d]*E^(I*ArcSin[1/(c*x)]))/( Sqrt[e] - Sqrt[c^2*d + e])])/((-d)^(3/2)*Sqrt[e]) + ((I/4)*b*PolyLog[2, (( -I)*c*Sqrt[-d]*E^(I*ArcSin[1/(c*x)]))/(Sqrt[e] + Sqrt[c^2*d + e])])/((-d)^ (3/2)*Sqrt[e]) - ((I/4)*b*PolyLog[2, (I*c*Sqrt[-d]*E^(I*ArcSin[1/(c*x)]))/ (Sqrt[e] + Sqrt[c^2*d + e])])/((-d)^(3/2)*Sqrt[e])
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_.)*((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 = 55.82 (sec) , antiderivative size = 832, normalized size of antiderivative = 1.09
| method | result | size |
| parts | \(\frac {a x}{2 d \left (e \,x^{2}+d \right )}+\frac {a \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{2 d \sqrt {d e}}+\frac {b \left (\frac {c^{3} \operatorname {arccsc}\left (c x \right ) x}{2 d \left (e \,c^{2} x^{2}+c^{2} d \right )}-\frac {c^{2} \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 )}{4 d}-\frac {c^{2} \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 )}{4 d}+\frac {\sqrt {-\left (c^{2} d -2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) d}\, \left (c^{2} d +2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) \arctan \left (\frac {c d \left (\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{\sqrt {\left (-c^{2} d +2 \sqrt {e \left (c^{2} d +e \right )}-2 e \right ) d}}\right )}{2 d^{4} c^{3}}-\frac {\sqrt {-\left (c^{2} d -2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) d}\, \left (\sqrt {e \left (c^{2} d +e \right )}\, c^{2} d +2 c^{2} d e +2 \sqrt {e \left (c^{2} d +e \right )}\, e +2 e^{2}\right ) \arctan \left (\frac {c d \left (\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{\sqrt {\left (-c^{2} d +2 \sqrt {e \left (c^{2} d +e \right )}-2 e \right ) d}}\right )}{2 d^{4} \left (c^{2} d +e \right ) c^{3}}+\frac {\sqrt {\left (c^{2} d +2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) d}\, \left (c^{2} d -2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) \operatorname {arctanh}\left (\frac {c d \left (\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{\sqrt {\left (c^{2} d +2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) d}}\right )}{2 d^{4} c^{3}}-\frac {\sqrt {\left (c^{2} d +2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) d}\, \left (-\sqrt {e \left (c^{2} d +e \right )}\, c^{2} d +2 c^{2} d e -2 \sqrt {e \left (c^{2} d +e \right )}\, e +2 e^{2}\right ) \operatorname {arctanh}\left (\frac {c d \left (\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{\sqrt {\left (c^{2} d +2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) d}}\right )}{2 d^{4} \left (c^{2} d +e \right ) c^{3}}\right )}{c}\) | \(832\) |
| derivativedivides | \(\frac {\frac {a \,c^{3} x}{2 d \left (e \,c^{2} x^{2}+c^{2} d \right )}+\frac {a c \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{2 d \sqrt {d e}}+b \,c^{4} \left (\frac {\operatorname {arccsc}\left (c x \right ) x}{2 c d \left (e \,c^{2} x^{2}+c^{2} d \right )}-\frac {\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 )}}{4 d \,c^{2}}-\frac {\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}}{4 d \,c^{2}}+\frac {\sqrt {-\left (c^{2} d -2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) d}\, \left (c^{2} d +2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) \arctan \left (\frac {c d \left (\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{\sqrt {\left (-c^{2} d +2 \sqrt {e \left (c^{2} d +e \right )}-2 e \right ) d}}\right )}{2 c^{7} d^{4}}-\frac {\sqrt {-\left (c^{2} d -2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) d}\, \left (\sqrt {e \left (c^{2} d +e \right )}\, c^{2} d +2 c^{2} d e +2 \sqrt {e \left (c^{2} d +e \right )}\, e +2 e^{2}\right ) \arctan \left (\frac {c d \left (\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{\sqrt {\left (-c^{2} d +2 \sqrt {e \left (c^{2} d +e \right )}-2 e \right ) d}}\right )}{2 c^{7} d^{4} \left (c^{2} d +e \right )}+\frac {\sqrt {\left (c^{2} d +2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) d}\, \left (c^{2} d -2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) \operatorname {arctanh}\left (\frac {c d \left (\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{\sqrt {\left (c^{2} d +2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) d}}\right )}{2 c^{7} d^{4}}-\frac {\sqrt {\left (c^{2} d +2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) d}\, \left (-\sqrt {e \left (c^{2} d +e \right )}\, c^{2} d +2 c^{2} d e -2 \sqrt {e \left (c^{2} d +e \right )}\, e +2 e^{2}\right ) \operatorname {arctanh}\left (\frac {c d \left (\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{\sqrt {\left (c^{2} d +2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) d}}\right )}{2 c^{7} d^{4} \left (c^{2} d +e \right )}\right )}{c}\) | \(847\) |
| default | \(\frac {\frac {a \,c^{3} x}{2 d \left (e \,c^{2} x^{2}+c^{2} d \right )}+\frac {a c \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{2 d \sqrt {d e}}+b \,c^{4} \left (\frac {\operatorname {arccsc}\left (c x \right ) x}{2 c d \left (e \,c^{2} x^{2}+c^{2} d \right )}-\frac {\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 )}}{4 d \,c^{2}}-\frac {\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}}{4 d \,c^{2}}+\frac {\sqrt {-\left (c^{2} d -2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) d}\, \left (c^{2} d +2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) \arctan \left (\frac {c d \left (\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{\sqrt {\left (-c^{2} d +2 \sqrt {e \left (c^{2} d +e \right )}-2 e \right ) d}}\right )}{2 c^{7} d^{4}}-\frac {\sqrt {-\left (c^{2} d -2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) d}\, \left (\sqrt {e \left (c^{2} d +e \right )}\, c^{2} d +2 c^{2} d e +2 \sqrt {e \left (c^{2} d +e \right )}\, e +2 e^{2}\right ) \arctan \left (\frac {c d \left (\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{\sqrt {\left (-c^{2} d +2 \sqrt {e \left (c^{2} d +e \right )}-2 e \right ) d}}\right )}{2 c^{7} d^{4} \left (c^{2} d +e \right )}+\frac {\sqrt {\left (c^{2} d +2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) d}\, \left (c^{2} d -2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) \operatorname {arctanh}\left (\frac {c d \left (\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{\sqrt {\left (c^{2} d +2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) d}}\right )}{2 c^{7} d^{4}}-\frac {\sqrt {\left (c^{2} d +2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) d}\, \left (-\sqrt {e \left (c^{2} d +e \right )}\, c^{2} d +2 c^{2} d e -2 \sqrt {e \left (c^{2} d +e \right )}\, e +2 e^{2}\right ) \operatorname {arctanh}\left (\frac {c d \left (\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{\sqrt {\left (c^{2} d +2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) d}}\right )}{2 c^{7} d^{4} \left (c^{2} d +e \right )}\right )}{c}\) | \(847\) |
Input:
int((a+b*arccsc(c*x))/(e*x^2+d)^2,x,method=_RETURNVERBOSE)
Output:
1/2*a*x/d/(e*x^2+d)+1/2*a/d/(d*e)^(1/2)*arctan(e*x/(d*e)^(1/2))+b/c*(1/2*c ^3*arccsc(c*x)*x/d/(c^2*e*x^2+c^2*d)-1/4/d*c^2*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))-1/4/d*c^2*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)) ,_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)*(c^2*d+2*(e*(c^2*d+e))^(1/2)+2*e)*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^4 /c^3-1/2*(-(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)*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^4/(c^2*d+e)/c ^3+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)*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^4/c^3-1/2*((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)*a rctanh(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^4/(c^2*d+e)/c^3)
\[ \int \frac {a+b \csc ^{-1}(c x)}{\left (d+e x^2\right )^2} \, dx=\int { \frac {b \operatorname {arccsc}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{2}} \,d x } \] Input:
integrate((a+b*arccsc(c*x))/(e*x^2+d)^2,x, algorithm="fricas")
Output:
integral((b*arccsc(c*x) + a)/(e^2*x^4 + 2*d*e*x^2 + d^2), x)
\[ \int \frac {a+b \csc ^{-1}(c x)}{\left (d+e x^2\right )^2} \, dx=\int \frac {a + b \operatorname {acsc}{\left (c x \right )}}{\left (d + e x^{2}\right )^{2}}\, dx \] Input:
integrate((a+b*acsc(c*x))/(e*x**2+d)**2,x)
Output:
Integral((a + b*acsc(c*x))/(d + e*x**2)**2, x)
Exception generated. \[ \int \frac {a+b \csc ^{-1}(c x)}{\left (d+e x^2\right )^2} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((a+b*arccsc(c*x))/(e*x^2+d)^2,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. \[ \int \frac {a+b \csc ^{-1}(c x)}{\left (d+e x^2\right )^2} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((a+b*arccsc(c*x))/(e*x^2+d)^2,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. \[ \int \frac {a+b \csc ^{-1}(c x)}{\left (d+e x^2\right )^2} \, dx=\int \frac {a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )}{{\left (e\,x^2+d\right )}^2} \,d x \] Input:
int((a + b*asin(1/(c*x)))/(d + e*x^2)^2,x)
Output:
int((a + b*asin(1/(c*x)))/(d + e*x^2)^2, x)
\[ \int \frac {a+b \csc ^{-1}(c x)}{\left (d+e x^2\right )^2} \, dx=\frac {\sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) a d +\sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) a e \,x^{2}+2 \left (\int \frac {\mathit {acsc} \left (c x \right )}{e^{2} x^{4}+2 d e \,x^{2}+d^{2}}d x \right ) b \,d^{3} e +2 \left (\int \frac {\mathit {acsc} \left (c x \right )}{e^{2} x^{4}+2 d e \,x^{2}+d^{2}}d x \right ) b \,d^{2} e^{2} x^{2}+a d e x}{2 d^{2} e \left (e \,x^{2}+d \right )} \] Input:
int((a+b*acsc(c*x))/(e*x^2+d)^2,x)
Output:
(sqrt(e)*sqrt(d)*atan((e*x)/(sqrt(e)*sqrt(d)))*a*d + sqrt(e)*sqrt(d)*atan( (e*x)/(sqrt(e)*sqrt(d)))*a*e*x**2 + 2*int(acsc(c*x)/(d**2 + 2*d*e*x**2 + e **2*x**4),x)*b*d**3*e + 2*int(acsc(c*x)/(d**2 + 2*d*e*x**2 + e**2*x**4),x) *b*d**2*e**2*x**2 + a*d*e*x)/(2*d**2*e*(d + e*x**2))