Integrand size = 21, antiderivative size = 628 \[ \int \frac {x^5 \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x}{2 c e^2}+\frac {d \left (a+b \csc ^{-1}(c x)\right )}{2 e^2 \left (e+\frac {d}{x^2}\right )}+\frac {x^2 \left (a+b \csc ^{-1}(c x)\right )}{2 e^2}-\frac {b d \arctan \left (\frac {\sqrt {c^2 d+e}}{c \sqrt {e} \sqrt {1-\frac {1}{c^2 x^2}} x}\right )}{2 e^{5/2} \sqrt {c^2 d+e}}-\frac {d \left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{e^3}-\frac {d \left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{e^3}-\frac {d \left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{e^3}-\frac {d \left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{e^3}+\frac {2 d \left (a+b \csc ^{-1}(c x)\right ) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )}{e^3}+\frac {i b d \operatorname {PolyLog}\left (2,-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{e^3}+\frac {i b d \operatorname {PolyLog}\left (2,\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{e^3}+\frac {i b d \operatorname {PolyLog}\left (2,-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{e^3}+\frac {i b d \operatorname {PolyLog}\left (2,\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{e^3}-\frac {i b d \operatorname {PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right )}{e^3} \] Output:
1/2*b*(1-1/c^2/x^2)^(1/2)*x/c/e^2+1/2*d*(a+b*arccsc(c*x))/e^2/(e+d/x^2)+1/ 2*x^2*(a+b*arccsc(c*x))/e^2-1/2*b*d*arctan((c^2*d+e)^(1/2)/c/e^(1/2)/(1-1/ c^2/x^2)^(1/2)/x)/e^(5/2)/(c^2*d+e)^(1/2)-d*(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-d*(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-d*(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-d*(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*d*(a +b*arccsc(c*x))*ln(1-(I/c/x+(1-1/c^2/x^2)^(1/2))^2)/e^3+I*b*d*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+I *b*d*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+I*b*d*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+I*b*d*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-I*b*d*polylog(2,(I/c/x+(1- 1/c^2/x^2)^(1/2))^2)/e^3
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1480\) vs. \(2(628)=1256\).
Time = 3.82 (sec) , antiderivative size = 1480, normalized size of antiderivative = 2.36 \[ \int \frac {x^5 \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx =\text {Too large to display} \] Input:
Integrate[(x^5*(a + b*ArcCsc[c*x]))/(d + e*x^2)^2,x]
Output:
-1/4*(-2*a*e*x^2 + (2*a*d^2)/(d + e*x^2) + 4*a*d*Log[d + e*x^2] + b*(I*d*P i^2 - (2*e*Sqrt[1 - 1/(c^2*x^2)]*x)/c - (4*I)*d*Pi*ArcCsc[c*x] - 2*e*x^2*A rcCsc[c*x] + (d^(3/2)*ArcCsc[c*x])/(Sqrt[d] - I*Sqrt[e]*x) + (d^(3/2)*ArcC sc[c*x])/(Sqrt[d] + I*Sqrt[e]*x) + (8*I)*d*ArcCsc[c*x]^2 - 2*d*ArcSin[1/(c *x)] - (16*I)*d*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]] - (16*I)*d*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[((I*c*S qrt[d] + Sqrt[e])*Cot[(Pi + 2*ArcCsc[c*x])/4])/Sqrt[c^2*d + e]] - 2*d*Pi*L og[1 + (Sqrt[e] - Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + 4*d*Ar cCsc[c*x]*Log[1 + (Sqrt[e] - Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]) )] - 8*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]))] - 2*d*Pi*Log[1 + (-Sqr t[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + 4*d*ArcCsc[c*x]*L og[1 + (-Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - 8*d*A rcSin[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]))] - 2*d*Pi*Log[1 - (Sqrt[e] + Sqr t[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + 4*d*ArcCsc[c*x]*Log[1 - (Sq rt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + 8*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]))] - 2*d*Pi*Log[1 + (Sqrt[e] + Sqrt[c^2*d +...
Time = 1.78 (sec) , antiderivative size = 696, normalized size of antiderivative = 1.11, 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.
| \(\displaystyle \int \frac {x^5 \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 5764 |
| \(\displaystyle -\int \frac {x^3 \left (a+b \arcsin \left (\frac {1}{c x}\right )\right )}{\left (\frac {d}{x^2}+e\right )^2}d\frac {1}{x}\) |
| \(\Big \downarrow \) 5232 |
| \(\displaystyle -\int \left (\frac {\left (a+b \arcsin \left (\frac {1}{c x}\right )\right ) x^3}{e^2}-\frac {2 d \left (a+b \arcsin \left (\frac {1}{c x}\right )\right ) x}{e^3}+\frac {2 d^2 \left (a+b \arcsin \left (\frac {1}{c x}\right )\right )}{e^3 \left (\frac {d}{x^2}+e\right ) x}+\frac {d^2 \left (a+b \arcsin \left (\frac {1}{c x}\right )\right )}{e^2 \left (\frac {d}{x^2}+e\right )^2 x}\right )d\frac {1}{x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {d \left (a+b \arcsin \left (\frac {1}{c x}\right )\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \arcsin \left (\frac {1}{c x}\right )}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{e^3}-\frac {d \left (a+b \arcsin \left (\frac {1}{c x}\right )\right ) \log \left (1+\frac {i c \sqrt {-d} e^{i \arcsin \left (\frac {1}{c x}\right )}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{e^3}-\frac {d \left (a+b \arcsin \left (\frac {1}{c x}\right )\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \arcsin \left (\frac {1}{c x}\right )}}{\sqrt {c^2 d+e}+\sqrt {e}}\right )}{e^3}-\frac {d \left (a+b \arcsin \left (\frac {1}{c x}\right )\right ) \log \left (1+\frac {i c \sqrt {-d} e^{i \arcsin \left (\frac {1}{c x}\right )}}{\sqrt {c^2 d+e}+\sqrt {e}}\right )}{e^3}+\frac {2 d \log \left (1-e^{2 i \arcsin \left (\frac {1}{c x}\right )}\right ) \left (a+b \arcsin \left (\frac {1}{c x}\right )\right )}{e^3}+\frac {d \left (a+b \arcsin \left (\frac {1}{c x}\right )\right )}{2 e^2 \left (\frac {d}{x^2}+e\right )}+\frac {x^2 \left (a+b \arcsin \left (\frac {1}{c x}\right )\right )}{2 e^2}+\frac {i b d \operatorname {PolyLog}\left (2,-\frac {i c \sqrt {-d} e^{i \arcsin \left (\frac {1}{c x}\right )}}{\sqrt {e}-\sqrt {d c^2+e}}\right )}{e^3}+\frac {i b d \operatorname {PolyLog}\left (2,\frac {i c \sqrt {-d} e^{i \arcsin \left (\frac {1}{c x}\right )}}{\sqrt {e}-\sqrt {d c^2+e}}\right )}{e^3}+\frac {i b d \operatorname {PolyLog}\left (2,-\frac {i c \sqrt {-d} e^{i \arcsin \left (\frac {1}{c x}\right )}}{\sqrt {e}+\sqrt {d c^2+e}}\right )}{e^3}+\frac {i b d \operatorname {PolyLog}\left (2,\frac {i c \sqrt {-d} e^{i \arcsin \left (\frac {1}{c x}\right )}}{\sqrt {e}+\sqrt {d c^2+e}}\right )}{e^3}-\frac {i b d \operatorname {PolyLog}\left (2,e^{2 i \arcsin \left (\frac {1}{c x}\right )}\right )}{e^3}-\frac {b d \arctan \left (\frac {\sqrt {c^2 d+e}}{c \sqrt {e} x \sqrt {1-\frac {1}{c^2 x^2}}}\right )}{2 e^{5/2} \sqrt {c^2 d+e}}+\frac {b x \sqrt {1-\frac {1}{c^2 x^2}}}{2 c e^2}\) |
Input:
Int[(x^5*(a + b*ArcCsc[c*x]))/(d + e*x^2)^2,x]
Output:
(b*Sqrt[1 - 1/(c^2*x^2)]*x)/(2*c*e^2) + (d*(a + b*ArcSin[1/(c*x)]))/(2*e^2 *(e + d/x^2)) + (x^2*(a + b*ArcSin[1/(c*x)]))/(2*e^2) - (b*d*ArcTan[Sqrt[c ^2*d + e]/(c*Sqrt[e]*Sqrt[1 - 1/(c^2*x^2)]*x)])/(2*e^(5/2)*Sqrt[c^2*d + e] ) - (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])])/e^3 - (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])])/e^3 - (d *(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])])/e^3 - (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])])/e^3 + (2*d*(a + b*ArcSin[1/(c*x)])*Log[1 - E^((2*I)*ArcSin[1/(c*x)])])/e^3 + (I*b*d*Poly Log[2, ((-I)*c*Sqrt[-d]*E^(I*ArcSin[1/(c*x)]))/(Sqrt[e] - Sqrt[c^2*d + e]) ])/e^3 + (I*b*d*PolyLog[2, (I*c*Sqrt[-d]*E^(I*ArcSin[1/(c*x)]))/(Sqrt[e] - Sqrt[c^2*d + e])])/e^3 + (I*b*d*PolyLog[2, ((-I)*c*Sqrt[-d]*E^(I*ArcSin[1 /(c*x)]))/(Sqrt[e] + Sqrt[c^2*d + e])])/e^3 + (I*b*d*PolyLog[2, (I*c*Sqrt[ -d]*E^(I*ArcSin[1/(c*x)]))/(Sqrt[e] + Sqrt[c^2*d + e])])/e^3 - (I*b*d*Poly Log[2, E^((2*I)*ArcSin[1/(c*x)])])/e^3
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 = 3.61 (sec) , antiderivative size = 643, normalized size of antiderivative = 1.02
| method | result | size |
| parts | \(\frac {a \,x^{2}}{2 e^{2}}-\frac {a d \ln \left (e \,x^{2}+d \right )}{e^{3}}-\frac {a \,d^{2}}{2 e^{3} \left (e \,x^{2}+d \right )}+\frac {b \left (\frac {c^{4} \left (2 c^{4} d \,\operatorname {arccsc}\left (c x \right ) x^{2}+\operatorname {arccsc}\left (c x \right ) e \,c^{4} x^{4}+\sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{3} d x +\sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, e \,c^{3} x^{3}-i c^{2} d -i e \,c^{2} x^{2}\right )}{2 \left (e \,c^{2} x^{2}+c^{2} d \right ) e^{2}}+\frac {2 i d \,c^{6} \operatorname {dilog}\left (\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{e^{3}}+\frac {i d^{2} c^{8} \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}-1\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}^{2} c^{2} d -c^{2} d -2 e}\right )}{2 e^{3}}-\frac {2 i d \,c^{6} \operatorname {dilog}\left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{e^{3}}+\frac {i \sqrt {e \left (c^{2} d +e \right )}\, \operatorname {arctanh}\left (\frac {2 c^{2} d {\left (\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}^{2}-2 c^{2} d -4 e}{4 \sqrt {c^{2} d e +e^{2}}}\right ) d \,c^{6}}{2 \left (c^{2} d +e \right ) e^{3}}+\frac {i d \,c^{6} \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}^{2} c^{2} d -c^{2} d -2 e}\right )}{2 e^{3}}+\frac {2 d \,c^{6} \operatorname {arccsc}\left (c x \right ) \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{e^{3}}\right )}{c^{6}}\) | \(643\) |
| derivativedivides | \(\frac {\frac {a \,c^{6} x^{2}}{2 e^{2}}-\frac {a \,c^{8} d^{2}}{2 e^{3} \left (e \,c^{2} x^{2}+c^{2} d \right )}-\frac {a \,c^{6} d \ln \left (e \,c^{2} x^{2}+c^{2} d \right )}{e^{3}}+b \,c^{4} \left (\frac {2 c^{4} d \,\operatorname {arccsc}\left (c x \right ) x^{2}+\operatorname {arccsc}\left (c x \right ) e \,c^{4} x^{4}+\sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{3} d x +\sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, e \,c^{3} x^{3}-i c^{2} d -i e \,c^{2} x^{2}}{2 \left (e \,c^{2} x^{2}+c^{2} d \right ) e^{2}}+\frac {2 i c^{2} d \operatorname {dilog}\left (\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{e^{3}}+\frac {i c^{4} d^{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 {\left (\textit {\_R1}^{2}-1\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}^{2} c^{2} d -c^{2} d -2 e}\right )}{2 e^{3}}+\frac {i 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}^{2} c^{2} d -c^{2} d -2 e}\right )}{2 e^{3}}-\frac {2 i c^{2} d \operatorname {dilog}\left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{e^{3}}+\frac {i \sqrt {e \left (c^{2} d +e \right )}\, \operatorname {arctanh}\left (\frac {2 c^{2} d {\left (\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}^{2}-2 c^{2} d -4 e}{4 \sqrt {c^{2} d e +e^{2}}}\right ) d \,c^{2}}{2 e^{3} \left (c^{2} d +e \right )}+\frac {2 c^{2} d \,\operatorname {arccsc}\left (c x \right ) \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{e^{3}}\right )}{c^{6}}\) | \(667\) |
| default | \(\frac {\frac {a \,c^{6} x^{2}}{2 e^{2}}-\frac {a \,c^{8} d^{2}}{2 e^{3} \left (e \,c^{2} x^{2}+c^{2} d \right )}-\frac {a \,c^{6} d \ln \left (e \,c^{2} x^{2}+c^{2} d \right )}{e^{3}}+b \,c^{4} \left (\frac {2 c^{4} d \,\operatorname {arccsc}\left (c x \right ) x^{2}+\operatorname {arccsc}\left (c x \right ) e \,c^{4} x^{4}+\sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{3} d x +\sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, e \,c^{3} x^{3}-i c^{2} d -i e \,c^{2} x^{2}}{2 \left (e \,c^{2} x^{2}+c^{2} d \right ) e^{2}}+\frac {2 i c^{2} d \operatorname {dilog}\left (\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{e^{3}}+\frac {i c^{4} d^{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 {\left (\textit {\_R1}^{2}-1\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}^{2} c^{2} d -c^{2} d -2 e}\right )}{2 e^{3}}+\frac {i 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}^{2} c^{2} d -c^{2} d -2 e}\right )}{2 e^{3}}-\frac {2 i c^{2} d \operatorname {dilog}\left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{e^{3}}+\frac {i \sqrt {e \left (c^{2} d +e \right )}\, \operatorname {arctanh}\left (\frac {2 c^{2} d {\left (\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}^{2}-2 c^{2} d -4 e}{4 \sqrt {c^{2} d e +e^{2}}}\right ) d \,c^{2}}{2 e^{3} \left (c^{2} d +e \right )}+\frac {2 c^{2} d \,\operatorname {arccsc}\left (c x \right ) \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{e^{3}}\right )}{c^{6}}\) | \(667\) |
Input:
int(x^5*(a+b*arccsc(c*x))/(e*x^2+d)^2,x,method=_RETURNVERBOSE)
Output:
1/2*a*x^2/e^2-a*d/e^3*ln(e*x^2+d)-1/2*a*d^2/e^3/(e*x^2+d)+b/c^6*(1/2*c^4*( 2*c^4*d*arccsc(c*x)*x^2+arccsc(c*x)*e*c^4*x^4+((c^2*x^2-1)/c^2/x^2)^(1/2)* c^3*d*x+((c^2*x^2-1)/c^2/x^2)^(1/2)*e*c^3*x^3-I*c^2*d-I*e*c^2*x^2)/(c^2*e* x^2+c^2*d)/e^2+2*I/e^3*d*c^6*dilog(I/c/x+(1-1/c^2/x^2)^(1/2))+1/2*I/e^3*d^ 2*c^8*sum((_R1^2-1)/(_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=R ootOf(c^2*d*_Z^4+(-2*c^2*d-4*e)*_Z^2+c^2*d))-2*I/e^3*d*c^6*dilog(1+I/c/x+( 1-1/c^2/x^2)^(1/2))+1/2*I*(e*(c^2*d+e))^(1/2)/(c^2*d+e)/e^3*arctanh(1/4*(2 *c^2*d*(I/c/x+(1-1/c^2/x^2)^(1/2))^2-2*c^2*d-4*e)/(c^2*d*e+e^2)^(1/2))*d*c ^6+1/2*I/e^3*d*c^6*sum((_R1^2*c^2*d-c^2*d-4*e)/(_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))+2/ e^3*d*c^6*arccsc(c*x)*ln(1+I/c/x+(1-1/c^2/x^2)^(1/2)))
\[ \int \frac {x^5 \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=\int { \frac {{\left (b \operatorname {arccsc}\left (c x\right ) + a\right )} x^{5}}{{\left (e x^{2} + d\right )}^{2}} \,d x } \] Input:
integrate(x^5*(a+b*arccsc(c*x))/(e*x^2+d)^2,x, algorithm="fricas")
Output:
integral((b*x^5*arccsc(c*x) + a*x^5)/(e^2*x^4 + 2*d*e*x^2 + d^2), x)
Timed out. \[ \int \frac {x^5 \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=\text {Timed out} \] Input:
integrate(x**5*(a+b*acsc(c*x))/(e*x**2+d)**2,x)
Output:
Timed out
\[ \int \frac {x^5 \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=\int { \frac {{\left (b \operatorname {arccsc}\left (c x\right ) + a\right )} x^{5}}{{\left (e x^{2} + d\right )}^{2}} \,d x } \] Input:
integrate(x^5*(a+b*arccsc(c*x))/(e*x^2+d)^2,x, algorithm="maxima")
Output:
-1/2*a*(d^2/(e^4*x^2 + d*e^3) - x^2/e^2 + 2*d*log(e*x^2 + d)/e^3) + b*inte grate(x^5*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1))/(e^2*x^4 + 2*d*e*x^2 + d ^2), x)
Timed out. \[ \int \frac {x^5 \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=\text {Timed out} \] Input:
integrate(x^5*(a+b*arccsc(c*x))/(e*x^2+d)^2,x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int \frac {x^5 \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=\int \frac {x^5\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}{{\left (e\,x^2+d\right )}^2} \,d x \] Input:
int((x^5*(a + b*asin(1/(c*x))))/(d + e*x^2)^2,x)
Output:
int((x^5*(a + b*asin(1/(c*x))))/(d + e*x^2)^2, x)
\[ \int \frac {x^5 \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=\frac {2 \left (\int \frac {\mathit {acsc} \left (c x \right ) x^{5}}{e^{2} x^{4}+2 d e \,x^{2}+d^{2}}d x \right ) b d \,e^{3}+2 \left (\int \frac {\mathit {acsc} \left (c x \right ) x^{5}}{e^{2} x^{4}+2 d e \,x^{2}+d^{2}}d x \right ) b \,e^{4} x^{2}-2 \,\mathrm {log}\left (e \,x^{2}+d \right ) a \,d^{2}-2 \,\mathrm {log}\left (e \,x^{2}+d \right ) a d e \,x^{2}+2 a d e \,x^{2}+a \,e^{2} x^{4}}{2 e^{3} \left (e \,x^{2}+d \right )} \] Input:
int(x^5*(a+b*acsc(c*x))/(e*x^2+d)^2,x)
Output:
(2*int((acsc(c*x)*x**5)/(d**2 + 2*d*e*x**2 + e**2*x**4),x)*b*d*e**3 + 2*in t((acsc(c*x)*x**5)/(d**2 + 2*d*e*x**2 + e**2*x**4),x)*b*e**4*x**2 - 2*log( d + e*x**2)*a*d**2 - 2*log(d + e*x**2)*a*d*e*x**2 + 2*a*d*e*x**2 + a*e**2* x**4)/(2*e**3*(d + e*x**2))