Integrand size = 21, antiderivative size = 551 \[ \int \frac {a+b \sec ^{-1}(c x)}{x^2 \left (d+e x^2\right )} \, dx=\frac {b c \sqrt {1-\frac {1}{c^2 x^2}}}{d}-\frac {a}{d x}-\frac {b \sec ^{-1}(c x)}{d x}+\frac {\sqrt {e} \left (a+b \sec ^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{i \sec ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 (-d)^{3/2}}-\frac {\sqrt {e} \left (a+b \sec ^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{i \sec ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 (-d)^{3/2}}+\frac {\sqrt {e} \left (a+b \sec ^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{i \sec ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 (-d)^{3/2}}-\frac {\sqrt {e} \left (a+b \sec ^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{i \sec ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 (-d)^{3/2}}+\frac {i b \sqrt {e} \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{i \sec ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 (-d)^{3/2}}-\frac {i b \sqrt {e} \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{i \sec ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 (-d)^{3/2}}+\frac {i b \sqrt {e} \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{i \sec ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 (-d)^{3/2}}-\frac {i b \sqrt {e} \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{i \sec ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 (-d)^{3/2}} \]
-a/d/x-b*arcsec(c*x)/d/x+1/2*(a+b*arcsec(c*x))*ln(1-c*(1/c/x+I*(1-1/c^2/x^ 2)^(1/2))*(-d)^(1/2)/(e^(1/2)-(c^2*d+e)^(1/2)))*e^(1/2)/(-d)^(3/2)-1/2*(a+ b*arcsec(c*x))*ln(1+c*(1/c/x+I*(1-1/c^2/x^2)^(1/2))*(-d)^(1/2)/(e^(1/2)-(c ^2*d+e)^(1/2)))*e^(1/2)/(-d)^(3/2)+1/2*(a+b*arcsec(c*x))*ln(1-c*(1/c/x+I*( 1-1/c^2/x^2)^(1/2))*(-d)^(1/2)/(e^(1/2)+(c^2*d+e)^(1/2)))*e^(1/2)/(-d)^(3/ 2)-1/2*(a+b*arcsec(c*x))*ln(1+c*(1/c/x+I*(1-1/c^2/x^2)^(1/2))*(-d)^(1/2)/( e^(1/2)+(c^2*d+e)^(1/2)))*e^(1/2)/(-d)^(3/2)+1/2*I*b*polylog(2,-c*(1/c/x+I *(1-1/c^2/x^2)^(1/2))*(-d)^(1/2)/(e^(1/2)-(c^2*d+e)^(1/2)))*e^(1/2)/(-d)^( 3/2)-1/2*I*b*polylog(2,c*(1/c/x+I*(1-1/c^2/x^2)^(1/2))*(-d)^(1/2)/(e^(1/2) -(c^2*d+e)^(1/2)))*e^(1/2)/(-d)^(3/2)+1/2*I*b*polylog(2,-c*(1/c/x+I*(1-1/c ^2/x^2)^(1/2))*(-d)^(1/2)/(e^(1/2)+(c^2*d+e)^(1/2)))*e^(1/2)/(-d)^(3/2)-1/ 2*I*b*polylog(2,c*(1/c/x+I*(1-1/c^2/x^2)^(1/2))*(-d)^(1/2)/(e^(1/2)+(c^2*d +e)^(1/2)))*e^(1/2)/(-d)^(3/2)+b*c*(1-1/c^2/x^2)^(1/2)/d
Time = 1.79 (sec) , antiderivative size = 997, normalized size of antiderivative = 1.81 \[ \int \frac {a+b \sec ^{-1}(c x)}{x^2 \left (d+e x^2\right )} \, dx=-\frac {a}{d x}-\frac {a \sqrt {e} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d^{3/2}}+b \left (\frac {c \sqrt {1-\frac {1}{c^2 x^2}}-\frac {\sec ^{-1}(c x)}{x}}{d}-\frac {\sqrt {e} \left (8 \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 ) \tan \left (\frac {1}{2} \sec ^{-1}(c x)\right )}{\sqrt {c^2 d+e}}\right )-2 i \sec ^{-1}(c x) \log \left (1+\frac {i \left (\sqrt {e}-\sqrt {c^2 d+e}\right ) e^{i \sec ^{-1}(c x)}}{c \sqrt {d}}\right )-4 i \arcsin \left (\frac {\sqrt {1+\frac {i \sqrt {e}}{c \sqrt {d}}}}{\sqrt {2}}\right ) \log \left (1+\frac {i \left (\sqrt {e}-\sqrt {c^2 d+e}\right ) e^{i \sec ^{-1}(c x)}}{c \sqrt {d}}\right )-2 i \sec ^{-1}(c x) \log \left (1+\frac {i \left (\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{i \sec ^{-1}(c x)}}{c \sqrt {d}}\right )+4 i \arcsin \left (\frac {\sqrt {1+\frac {i \sqrt {e}}{c \sqrt {d}}}}{\sqrt {2}}\right ) \log \left (1+\frac {i \left (\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{i \sec ^{-1}(c x)}}{c \sqrt {d}}\right )+2 i \sec ^{-1}(c x) \log \left (1+e^{2 i \sec ^{-1}(c x)}\right )-2 \operatorname {PolyLog}\left (2,\frac {i \left (-\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{i \sec ^{-1}(c x)}}{c \sqrt {d}}\right )-2 \operatorname {PolyLog}\left (2,-\frac {i \left (\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{i \sec ^{-1}(c x)}}{c \sqrt {d}}\right )+\operatorname {PolyLog}\left (2,-e^{2 i \sec ^{-1}(c x)}\right )\right )}{4 d^{3/2}}+\frac {\sqrt {e} \left (8 \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 ) \tan \left (\frac {1}{2} \sec ^{-1}(c x)\right )}{\sqrt {c^2 d+e}}\right )-2 i \sec ^{-1}(c x) \log \left (1+\frac {i \left (-\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{i \sec ^{-1}(c x)}}{c \sqrt {d}}\right )-4 i \arcsin \left (\frac {\sqrt {1-\frac {i \sqrt {e}}{c \sqrt {d}}}}{\sqrt {2}}\right ) \log \left (1+\frac {i \left (-\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{i \sec ^{-1}(c x)}}{c \sqrt {d}}\right )-2 i \sec ^{-1}(c x) \log \left (1-\frac {i \left (\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{i \sec ^{-1}(c x)}}{c \sqrt {d}}\right )+4 i \arcsin \left (\frac {\sqrt {1-\frac {i \sqrt {e}}{c \sqrt {d}}}}{\sqrt {2}}\right ) \log \left (1-\frac {i \left (\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{i \sec ^{-1}(c x)}}{c \sqrt {d}}\right )+2 i \sec ^{-1}(c x) \log \left (1+e^{2 i \sec ^{-1}(c x)}\right )-2 \operatorname {PolyLog}\left (2,-\frac {i \left (-\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{i \sec ^{-1}(c x)}}{c \sqrt {d}}\right )-2 \operatorname {PolyLog}\left (2,\frac {i \left (\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{i \sec ^{-1}(c x)}}{c \sqrt {d}}\right )+\operatorname {PolyLog}\left (2,-e^{2 i \sec ^{-1}(c x)}\right )\right )}{4 d^{3/2}}\right ) \]
-(a/(d*x)) - (a*Sqrt[e]*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/d^(3/2) + b*((c*Sqrt[ 1 - 1/(c^2*x^2)] - ArcSec[c*x]/x)/d - (Sqrt[e]*(8*ArcSin[Sqrt[1 + (I*Sqrt[ e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[((I*c*Sqrt[d] + Sqrt[e])*Tan[ArcSec[c*x]/ 2])/Sqrt[c^2*d + e]] - (2*I)*ArcSec[c*x]*Log[1 + (I*(Sqrt[e] - Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] - (4*I)*ArcSin[Sqrt[1 + (I*Sqrt[e])/ (c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(Sqrt[e] - Sqrt[c^2*d + e])*E^(I*ArcSec[c *x]))/(c*Sqrt[d])] - (2*I)*ArcSec[c*x]*Log[1 + (I*(Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] + (4*I)*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c *Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x ]))/(c*Sqrt[d])] + (2*I)*ArcSec[c*x]*Log[1 + E^((2*I)*ArcSec[c*x])] - 2*Po lyLog[2, (I*(-Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] - 2*PolyLog[2, ((-I)*(Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt [d])] + PolyLog[2, -E^((2*I)*ArcSec[c*x])]))/(4*d^(3/2)) + (Sqrt[e]*(8*Arc Sin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[(((-I)*c*Sqrt[d] + S qrt[e])*Tan[ArcSec[c*x]/2])/Sqrt[c^2*d + e]] - (2*I)*ArcSec[c*x]*Log[1 + ( I*(-Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] - (4*I)*Arc Sin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(-Sqrt[e] + Sqrt [c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] - (2*I)*ArcSec[c*x]*Log[1 - ( I*(Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] + (4*I)*ArcS in[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 - (I*(Sqrt[e] + Sqr...
Time = 1.52 (sec) , antiderivative size = 603, 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 = {5763, 5233, 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.
| \(\displaystyle \int \frac {a+b \sec ^{-1}(c x)}{x^2 \left (d+e x^2\right )} \, dx\) |
| \(\Big \downarrow \) 5763 |
| \(\displaystyle -\int \frac {a+b \arccos \left (\frac {1}{c x}\right )}{\left (\frac {d}{x^2}+e\right ) x^2}d\frac {1}{x}\) |
| \(\Big \downarrow \) 5233 |
| \(\displaystyle -\int \left (\frac {a+b \arccos \left (\frac {1}{c x}\right )}{d}-\frac {e \left (a+b \arccos \left (\frac {1}{c x}\right )\right )}{d \left (\frac {d}{x^2}+e\right )}\right )d\frac {1}{x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {e} \left (a+b \arccos \left (\frac {1}{c x}\right )\right ) \log \left (1-\frac {c \sqrt {-d} e^{i \arccos \left (\frac {1}{c x}\right )}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 (-d)^{3/2}}-\frac {\sqrt {e} \left (a+b \arccos \left (\frac {1}{c x}\right )\right ) \log \left (1+\frac {c \sqrt {-d} e^{i \arccos \left (\frac {1}{c x}\right )}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 (-d)^{3/2}}+\frac {\sqrt {e} \left (a+b \arccos \left (\frac {1}{c x}\right )\right ) \log \left (1-\frac {c \sqrt {-d} e^{i \arccos \left (\frac {1}{c x}\right )}}{\sqrt {c^2 d+e}+\sqrt {e}}\right )}{2 (-d)^{3/2}}-\frac {\sqrt {e} \left (a+b \arccos \left (\frac {1}{c x}\right )\right ) \log \left (1+\frac {c \sqrt {-d} e^{i \arccos \left (\frac {1}{c x}\right )}}{\sqrt {c^2 d+e}+\sqrt {e}}\right )}{2 (-d)^{3/2}}-\frac {a}{d x}+\frac {i b \sqrt {e} \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{i \arccos \left (\frac {1}{c x}\right )}}{\sqrt {e}-\sqrt {d c^2+e}}\right )}{2 (-d)^{3/2}}-\frac {i b \sqrt {e} \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{i \arccos \left (\frac {1}{c x}\right )}}{\sqrt {e}-\sqrt {d c^2+e}}\right )}{2 (-d)^{3/2}}+\frac {i b \sqrt {e} \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{i \arccos \left (\frac {1}{c x}\right )}}{\sqrt {e}+\sqrt {d c^2+e}}\right )}{2 (-d)^{3/2}}-\frac {i b \sqrt {e} \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{i \arccos \left (\frac {1}{c x}\right )}}{\sqrt {e}+\sqrt {d c^2+e}}\right )}{2 (-d)^{3/2}}-\frac {b \arccos \left (\frac {1}{c x}\right )}{d x}+\frac {b c \sqrt {1-\frac {1}{c^2 x^2}}}{d}\) |
(b*c*Sqrt[1 - 1/(c^2*x^2)])/d - a/(d*x) - (b*ArcCos[1/(c*x)])/(d*x) + (Sqr t[e]*(a + b*ArcCos[1/(c*x)])*Log[1 - (c*Sqrt[-d]*E^(I*ArcCos[1/(c*x)]))/(S qrt[e] - Sqrt[c^2*d + e])])/(2*(-d)^(3/2)) - (Sqrt[e]*(a + b*ArcCos[1/(c*x )])*Log[1 + (c*Sqrt[-d]*E^(I*ArcCos[1/(c*x)]))/(Sqrt[e] - Sqrt[c^2*d + e]) ])/(2*(-d)^(3/2)) + (Sqrt[e]*(a + b*ArcCos[1/(c*x)])*Log[1 - (c*Sqrt[-d]*E ^(I*ArcCos[1/(c*x)]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*(-d)^(3/2)) - (Sqrt [e]*(a + b*ArcCos[1/(c*x)])*Log[1 + (c*Sqrt[-d]*E^(I*ArcCos[1/(c*x)]))/(Sq rt[e] + Sqrt[c^2*d + e])])/(2*(-d)^(3/2)) + ((I/2)*b*Sqrt[e]*PolyLog[2, -( (c*Sqrt[-d]*E^(I*ArcCos[1/(c*x)]))/(Sqrt[e] - Sqrt[c^2*d + e]))])/(-d)^(3/ 2) - ((I/2)*b*Sqrt[e]*PolyLog[2, (c*Sqrt[-d]*E^(I*ArcCos[1/(c*x)]))/(Sqrt[ e] - Sqrt[c^2*d + e])])/(-d)^(3/2) + ((I/2)*b*Sqrt[e]*PolyLog[2, -((c*Sqrt [-d]*E^(I*ArcCos[1/(c*x)]))/(Sqrt[e] + Sqrt[c^2*d + e]))])/(-d)^(3/2) - (( I/2)*b*Sqrt[e]*PolyLog[2, (c*Sqrt[-d]*E^(I*ArcCos[1/(c*x)]))/(Sqrt[e] + Sq rt[c^2*d + e])])/(-d)^(3/2)
3.1.95.3.1 Defintions of rubi rules used
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_.)*((d_) + (e_ .)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*ArcCos[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_.) + ArcSec[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_.) + (e_.)*(x_) ^2)^(p_.), x_Symbol] :> -Subst[Int[(e + d*x^2)^p*((a + b*ArcCos[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 = 27.44 (sec) , antiderivative size = 331, normalized size of antiderivative = 0.60
| method | result | size |
| parts | \(-\frac {a}{d x}-\frac {a e \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{d \sqrt {d e}}+\frac {b c \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{d}-\frac {b \,\operatorname {arcsec}\left (c x \right )}{d x}-\frac {i b c e \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c^{2} d \,\textit {\_Z}^{4}+\left (2 c^{2} d +4 e \right ) \textit {\_Z}^{2}+c^{2} d \right )}{\sum }\frac {\textit {\_R1} \left (i \operatorname {arcsec}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -\frac {1}{c x}-i \sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {1}{c x}-i \sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} c^{2} d +c^{2} d +2 e}\right )}{2 d}+\frac {i b c e \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c^{2} d \,\textit {\_Z}^{4}+\left (2 c^{2} d +4 e \right ) \textit {\_Z}^{2}+c^{2} d \right )}{\sum }\frac {i \operatorname {arcsec}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -\frac {1}{c x}-i \sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {1}{c x}-i \sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} c^{2} d +c^{2} d +2 e \right )}\right )}{2 d}\) | \(331\) |
| derivativedivides | \(c \left (-\frac {a}{d c x}-\frac {a e \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{c d \sqrt {d e}}+\frac {b \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{d}-\frac {b \,\operatorname {arcsec}\left (c x \right )}{d c x}-\frac {i b e \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c^{2} d \,\textit {\_Z}^{4}+\left (2 c^{2} d +4 e \right ) \textit {\_Z}^{2}+c^{2} d \right )}{\sum }\frac {\textit {\_R1} \left (i \operatorname {arcsec}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -\frac {1}{c x}-i \sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {1}{c x}-i \sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} c^{2} d +c^{2} d +2 e}\right )}{2 d}+\frac {i b e \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c^{2} d \,\textit {\_Z}^{4}+\left (2 c^{2} d +4 e \right ) \textit {\_Z}^{2}+c^{2} d \right )}{\sum }\frac {i \operatorname {arcsec}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -\frac {1}{c x}-i \sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {1}{c x}-i \sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} c^{2} d +c^{2} d +2 e \right )}\right )}{2 d}\right )\) | \(339\) |
| default | \(c \left (-\frac {a}{d c x}-\frac {a e \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{c d \sqrt {d e}}+\frac {b \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{d}-\frac {b \,\operatorname {arcsec}\left (c x \right )}{d c x}-\frac {i b e \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c^{2} d \,\textit {\_Z}^{4}+\left (2 c^{2} d +4 e \right ) \textit {\_Z}^{2}+c^{2} d \right )}{\sum }\frac {\textit {\_R1} \left (i \operatorname {arcsec}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -\frac {1}{c x}-i \sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {1}{c x}-i \sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} c^{2} d +c^{2} d +2 e}\right )}{2 d}+\frac {i b e \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c^{2} d \,\textit {\_Z}^{4}+\left (2 c^{2} d +4 e \right ) \textit {\_Z}^{2}+c^{2} d \right )}{\sum }\frac {i \operatorname {arcsec}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -\frac {1}{c x}-i \sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {1}{c x}-i \sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} c^{2} d +c^{2} d +2 e \right )}\right )}{2 d}\right )\) | \(339\) |
-a/d/x-a*e/d/(d*e)^(1/2)*arctan(e*x/(d*e)^(1/2))+b*c/d*((c^2*x^2-1)/c^2/x^ 2)^(1/2)-b*arcsec(c*x)/d/x-1/2*I*b*c*e/d*sum(_R1/(_R1^2*c^2*d+c^2*d+2*e)*( I*arcsec(c*x)*ln((_R1-1/c/x-I*(1-1/c^2/x^2)^(1/2))/_R1)+dilog((_R1-1/c/x-I *(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*I*b*c*e/d*sum(1/_R1/(_R1^2*c^2*d+c^2*d+2*e)*(I*arcsec(c*x)*ln((_R1- 1/c/x-I*(1-1/c^2/x^2)^(1/2))/_R1)+dilog((_R1-1/c/x-I*(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))
\[ \int \frac {a+b \sec ^{-1}(c x)}{x^2 \left (d+e x^2\right )} \, dx=\int { \frac {b \operatorname {arcsec}\left (c x\right ) + a}{{\left (e x^{2} + d\right )} x^{2}} \,d x } \]
\[ \int \frac {a+b \sec ^{-1}(c x)}{x^2 \left (d+e x^2\right )} \, dx=\int \frac {a + b \operatorname {asec}{\left (c x \right )}}{x^{2} \left (d + e x^{2}\right )}\, dx \]
Exception generated. \[ \int \frac {a+b \sec ^{-1}(c x)}{x^2 \left (d+e x^2\right )} \, dx=\text {Exception raised: ValueError} \]
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. \[ \int \frac {a+b \sec ^{-1}(c x)}{x^2 \left (d+e x^2\right )} \, dx=\text {Exception raised: RuntimeError} \]
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. \[ \int \frac {a+b \sec ^{-1}(c x)}{x^2 \left (d+e x^2\right )} \, dx=\int \frac {a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )}{x^2\,\left (e\,x^2+d\right )} \,d x \]