Integrand size = 21, antiderivative size = 1124 \[ \int \frac {x^2 \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\frac {b c \sqrt {1-\frac {1}{c^2 x^2}}}{16 \sqrt {-d} \sqrt {e} \left (c^2 d+e\right ) \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}+\frac {b c \sqrt {1-\frac {1}{c^2 x^2}}}{16 \sqrt {-d} \sqrt {e} \left (c^2 d+e\right ) \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}+\frac {a+b \sec ^{-1}(c x)}{16 \sqrt {-d} \sqrt {e} \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )^2}+\frac {a+b \sec ^{-1}(c x)}{16 d e \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {a+b \sec ^{-1}(c x)}{16 \sqrt {-d} \sqrt {e} \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )^2}-\frac {a+b \sec ^{-1}(c x)}{16 d e \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 )}{16 d^{3/2} \left (c^2 d+e\right )^{3/2}}+\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 )}{16 d^{3/2} e \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 )}{16 d^{3/2} \left (c^2 d+e\right )^{3/2}}+\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 )}{16 d^{3/2} e \sqrt {c^2 d+e}}-\frac {\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 )}{16 (-d)^{3/2} e^{3/2}}+\frac {\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 )}{16 (-d)^{3/2} e^{3/2}}-\frac {\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 )}{16 (-d)^{3/2} e^{3/2}}+\frac {\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 )}{16 (-d)^{3/2} e^{3/2}}-\frac {i b \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{i \sec ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{16 (-d)^{3/2} e^{3/2}}+\frac {i b \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{i \sec ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{16 (-d)^{3/2} e^{3/2}}-\frac {i b \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{i \sec ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{16 (-d)^{3/2} e^{3/2}}+\frac {i b \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{i \sec ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{16 (-d)^{3/2} e^{3/2}} \]
-1/16*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)^(3/2)-1/16*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)^(3/2)-1/16*(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)))/(-d)^(3/2)/e^(3/2)+1/16*(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)) )/(-d)^(3/2)/e^(3/2)-1/16*(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)))/(-d)^(3/2)/e^(3/2)+1/16*(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)))/(-d)^(3/2)/e^(3/2)+1/16*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)))/(-d)^(3/2)/e^(3/2)-1/16*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)))/(-d)^(3/2)/e^(3/2)+1/16*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)))/(-d)^(3/2)/e^(3/2)-1/16*I*b*pol ylog(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 )))/(-d)^(3/2)/e^(3/2)+1/16*(a+b*arcsec(c*x))/(-d)^(1/2)/e^(1/2)/(-d/x+(-d )^(1/2)*e^(1/2))^2+1/16*(a+b*arcsec(c*x))/d/e/(-d/x+(-d)^(1/2)*e^(1/2))+1/ 16*(-a-b*arcsec(c*x))/(-d)^(1/2)/e^(1/2)/(d/x+(-d)^(1/2)*e^(1/2))^2+1/16*( -a-b*arcsec(c*x))/d/e/(d/x+(-d)^(1/2)*e^(1/2))+1/16*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)...
Time = 6.07 (sec) , antiderivative size = 1827, normalized size of antiderivative = 1.63 \[ \int \frac {x^2 \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx =\text {Too large to display} \]
-1/4*(a*x)/(e*(d + e*x^2)^2) + (a*x)/(8*d*e*(d + e*x^2)) + (a*ArcTan[(Sqrt [e]*x)/Sqrt[d]])/(8*d^(3/2)*e^(3/2)) + b*(-1/16*(-(ArcSec[c*x]/(I*Sqrt[d]* Sqrt[e] + e*x)) + (I*(ArcSin[1/(c*x)]/Sqrt[e] - Log[(2*Sqrt[d]*Sqrt[e]*(Sq rt[e] + c*(I*c*Sqrt[d] - Sqrt[-(c^2*d) - e]*Sqrt[1 - 1/(c^2*x^2)])*x))/(Sq rt[-(c^2*d) - e]*(Sqrt[d] - I*Sqrt[e]*x))]/Sqrt[-(c^2*d) - e]))/Sqrt[d])/( d*e) - (-(ArcSec[c*x]/((-I)*Sqrt[d]*Sqrt[e] + e*x)) - (I*(ArcSin[1/(c*x)]/ Sqrt[e] - Log[(2*Sqrt[d]*Sqrt[e]*(-Sqrt[e] + c*(I*c*Sqrt[d] + Sqrt[-(c^2*d ) - e]*Sqrt[1 - 1/(c^2*x^2)])*x))/(Sqrt[-(c^2*d) - e]*(Sqrt[d] + I*Sqrt[e] *x))]/Sqrt[-(c^2*d) - e]))/Sqrt[d])/(16*d*e) - ((I/16)*(-(ArcSec[c*x]/(Sqr t[e]*((-I)*Sqrt[d] + Sqrt[e]*x)^2)) + (ArcSin[1/(c*x)]/Sqrt[e] - I*((c*Sqr t[d]*Sqrt[e]*Sqrt[1 - 1/(c^2*x^2)]*x)/((c^2*d + e)*((-I)*Sqrt[d] + Sqrt[e] *x)) + ((2*c^2*d + e)*Log[(-4*d*Sqrt[e]*Sqrt[c^2*d + e]*(I*Sqrt[e] + c*(c* Sqrt[d] - Sqrt[c^2*d + e]*Sqrt[1 - 1/(c^2*x^2)])*x))/((2*c^2*d + e)*((-I)* Sqrt[d] + Sqrt[e]*x))])/(c^2*d + e)^(3/2)))/d))/(Sqrt[d]*e) + ((I/16)*((I* c*Sqrt[e]*Sqrt[1 - 1/(c^2*x^2)]*x)/(Sqrt[d]*(c^2*d + e)*(I*Sqrt[d] + Sqrt[ e]*x)) - ArcSec[c*x]/(Sqrt[e]*(I*Sqrt[d] + Sqrt[e]*x)^2) + ArcSin[1/(c*x)] /(d*Sqrt[e]) - (I*(2*c^2*d + e)*Log[(4*d*Sqrt[e]*Sqrt[c^2*d + e]*((-I)*Sqr t[e] + c*(c*Sqrt[d] + Sqrt[c^2*d + e]*Sqrt[1 - 1/(c^2*x^2)])*x))/((2*c^2*d + e)*(I*Sqrt[d] + Sqrt[e]*x))])/(d*(c^2*d + e)^(3/2))))/(Sqrt[d]*e) + (8* ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[((I*c*Sqrt[d] ...
Time = 3.43 (sec) , antiderivative size = 1188, normalized size of antiderivative = 1.06, 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.
| \(\displaystyle \int \frac {x^2 \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 5763 |
| \(\displaystyle -\int \frac {a+b \arccos \left (\frac {1}{c x}\right )}{\left (\frac {d}{x^2}+e\right )^3 x^2}d\frac {1}{x}\) |
| \(\Big \downarrow \) 5233 |
| \(\displaystyle -\int \left (\frac {a+b \arccos \left (\frac {1}{c x}\right )}{d \left (\frac {d}{x^2}+e\right )^2}-\frac {e \left (a+b \arccos \left (\frac {1}{c x}\right )\right )}{d \left (\frac {d}{x^2}+e\right )^3}\right )d\frac {1}{x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {b \sqrt {1-\frac {1}{c^2 x^2}} c}{16 \sqrt {-d} \sqrt {e} \left (d c^2+e\right ) \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}+\frac {b \sqrt {1-\frac {1}{c^2 x^2}} c}{16 \sqrt {-d} \sqrt {e} \left (d c^2+e\right ) \left (\frac {d}{x}+\sqrt {-d} \sqrt {e}\right )}+\frac {a+b \arccos \left (\frac {1}{c x}\right )}{16 d e \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {a+b \arccos \left (\frac {1}{c x}\right )}{16 d e \left (\frac {d}{x}+\sqrt {-d} \sqrt {e}\right )}+\frac {a+b \arccos \left (\frac {1}{c x}\right )}{16 \sqrt {-d} \sqrt {e} \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )^2}-\frac {a+b \arccos \left (\frac {1}{c x}\right )}{16 \sqrt {-d} \sqrt {e} \left (\frac {d}{x}+\sqrt {-d} \sqrt {e}\right )^2}+\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 )}{16 d^{3/2} e \sqrt {d c^2+e}}-\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 )}{16 d^{3/2} \left (d c^2+e\right )^{3/2}}+\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 )}{16 d^{3/2} e \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 )}{16 d^{3/2} \left (d c^2+e\right )^{3/2}}-\frac {\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 {d c^2+e}}\right )}{16 (-d)^{3/2} e^{3/2}}+\frac {\left (a+b \arccos \left (\frac {1}{c x}\right )\right ) \log \left (\frac {\sqrt {-d} e^{i \arccos \left (\frac {1}{c x}\right )} c}{\sqrt {e}-\sqrt {d c^2+e}}+1\right )}{16 (-d)^{3/2} e^{3/2}}-\frac {\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 {d c^2+e}}\right )}{16 (-d)^{3/2} e^{3/2}}+\frac {\left (a+b \arccos \left (\frac {1}{c x}\right )\right ) \log \left (\frac {\sqrt {-d} e^{i \arccos \left (\frac {1}{c x}\right )} c}{\sqrt {e}+\sqrt {d c^2+e}}+1\right )}{16 (-d)^{3/2} e^{3/2}}-\frac {i b \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 )}{16 (-d)^{3/2} e^{3/2}}+\frac {i b \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 )}{16 (-d)^{3/2} e^{3/2}}-\frac {i b \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 )}{16 (-d)^{3/2} e^{3/2}}+\frac {i b \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 )}{16 (-d)^{3/2} e^{3/2}}\) |
(b*c*Sqrt[1 - 1/(c^2*x^2)])/(16*Sqrt[-d]*Sqrt[e]*(c^2*d + e)*(Sqrt[-d]*Sqr t[e] - d/x)) + (b*c*Sqrt[1 - 1/(c^2*x^2)])/(16*Sqrt[-d]*Sqrt[e]*(c^2*d + e )*(Sqrt[-d]*Sqrt[e] + d/x)) + (a + b*ArcCos[1/(c*x)])/(16*Sqrt[-d]*Sqrt[e] *(Sqrt[-d]*Sqrt[e] - d/x)^2) + (a + b*ArcCos[1/(c*x)])/(16*d*e*(Sqrt[-d]*S qrt[e] - d/x)) - (a + b*ArcCos[1/(c*x)])/(16*Sqrt[-d]*Sqrt[e]*(Sqrt[-d]*Sq rt[e] + d/x)^2) - (a + b*ArcCos[1/(c*x)])/(16*d*e*(Sqrt[-d]*Sqrt[e] + d/x) ) - (b*ArcTanh[(c^2*d - (Sqrt[-d]*Sqrt[e])/x)/(c*Sqrt[d]*Sqrt[c^2*d + e]*S qrt[1 - 1/(c^2*x^2)])])/(16*d^(3/2)*(c^2*d + e)^(3/2)) + (b*ArcTanh[(c^2*d - (Sqrt[-d]*Sqrt[e])/x)/(c*Sqrt[d]*Sqrt[c^2*d + e]*Sqrt[1 - 1/(c^2*x^2)]) ])/(16*d^(3/2)*e*Sqrt[c^2*d + e]) - (b*ArcTanh[(c^2*d + (Sqrt[-d]*Sqrt[e]) /x)/(c*Sqrt[d]*Sqrt[c^2*d + e]*Sqrt[1 - 1/(c^2*x^2)])])/(16*d^(3/2)*(c^2*d + e)^(3/2)) + (b*ArcTanh[(c^2*d + (Sqrt[-d]*Sqrt[e])/x)/(c*Sqrt[d]*Sqrt[c ^2*d + e]*Sqrt[1 - 1/(c^2*x^2)])])/(16*d^(3/2)*e*Sqrt[c^2*d + e]) - ((a + b*ArcCos[1/(c*x)])*Log[1 - (c*Sqrt[-d]*E^(I*ArcCos[1/(c*x)]))/(Sqrt[e] - S qrt[c^2*d + e])])/(16*(-d)^(3/2)*e^(3/2)) + ((a + b*ArcCos[1/(c*x)])*Log[1 + (c*Sqrt[-d]*E^(I*ArcCos[1/(c*x)]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(16*(- d)^(3/2)*e^(3/2)) - ((a + b*ArcCos[1/(c*x)])*Log[1 - (c*Sqrt[-d]*E^(I*ArcC os[1/(c*x)]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(16*(-d)^(3/2)*e^(3/2)) + ((a + b*ArcCos[1/(c*x)])*Log[1 + (c*Sqrt[-d]*E^(I*ArcCos[1/(c*x)]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(16*(-d)^(3/2)*e^(3/2)) - ((I/16)*b*PolyLog[2, -((c...
3.2.9.3.1 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 = 33.26 (sec) , antiderivative size = 1288, normalized size of antiderivative = 1.15
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(1288\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(1307\) |
| default | \(\text {Expression too large to display}\) | \(1307\) |
a*((1/8/d*x^3-1/8/e*x)/(e*x^2+d)^2+1/8/e/d/(d*e)^(1/2)*arctan(e*x/(d*e)^(1 /2)))+b/c^3*(1/8*x*c^5*(c^4*d*e*arcsec(c*x)*x^2-d^2*c^4*arcsec(c*x)-((c^2* x^2-1)/c^2/x^2)^(1/2)*e^2*c^3*x^3-((c^2*x^2-1)/c^2/x^2)^(1/2)*c^3*d*e*x+e^ 2*arcsec(c*x)*c^2*x^2-c^2*d*e*arcsec(c*x))/d/e/(c^2*d+e)/(c^2*e*x^2+c^2*d) ^2-1/8*I*((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)*c*arctan(c*d*(1/c/x+I*(1-1/c^2/x^2)^(1/2))/((c^2*d+2*(e*(c^2* d+e))^(1/2)+2*e)*d)^(1/2))/(c^2*d+e)/e/d^3+1/8*I*((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)*c*arctan(c*d*(1/c/x+I*(1-1/c^2/x^2)^(1/2))/((c^2*d+2*(e*(c^2 *d+e))^(1/2)+2*e)*d)^(1/2))/(c^2*d+e)^2/e/d^3-1/8*I*(-(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)*c*arctanh(c*d*(1/ c/x+I*(1-1/c^2/x^2)^(1/2))/((-c^2*d+2*(e*(c^2*d+e))^(1/2)-2*e)*d)^(1/2))/( c^2*d+e)/e/d^3+1/8*I*(-(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)*c*arctanh(c*d* (1/c/x+I*(1-1/c^2/x^2)^(1/2))/((-c^2*d+2*(e*(c^2*d+e))^(1/2)-2*e)*d)^(1/2) )/(c^2*d+e)^2/e/d^3+1/16*I/(c^2*d+e)/d*c^4*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))+1/16*I/(c^2*d+e)/e*c^6*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/...
\[ \int \frac {x^2 \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\int { \frac {{\left (b \operatorname {arcsec}\left (c x\right ) + a\right )} x^{2}}{{\left (e x^{2} + d\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {x^2 \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {x^2 \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\text {Exception raised: ValueError} \]
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. \[ \int \frac {x^2 \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\text {Exception raised: RuntimeError} \]
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. \[ \int \frac {x^2 \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\int \frac {x^2\,\left (a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )\right )}{{\left (e\,x^2+d\right )}^3} \,d x \]