3.1.97 \(\int \frac {x^3 (a+b \sec ^{-1}(c x))}{(d+e x^2)^2} \, dx\) [97]

Optimal. Leaf size=570 \[ -\frac {a+b \sec ^{-1}(c x)}{2 e \left (e+\frac {d}{x^2}\right )}-\frac {b \text {ArcTan}\left (\frac {\sqrt {c^2 d+e}}{c \sqrt {e} \sqrt {1-\frac {1}{c^2 x^2}} x}\right )}{2 e^{3/2} \sqrt {c^2 d+e}}+\frac {\left (a+b \sec ^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{i \sec ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 e^2}+\frac {\left (a+b \sec ^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{i \sec ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 e^2}+\frac {\left (a+b \sec ^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{i \sec ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 e^2}+\frac {\left (a+b \sec ^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{i \sec ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 e^2}-\frac {\left (a+b \sec ^{-1}(c x)\right ) \log \left (1+e^{2 i \sec ^{-1}(c x)}\right )}{e^2}-\frac {i b \text {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{i \sec ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 e^2}-\frac {i b \text {PolyLog}\left (2,\frac {c \sqrt {-d} e^{i \sec ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 e^2}-\frac {i b \text {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{i \sec ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 e^2}-\frac {i b \text {PolyLog}\left (2,\frac {c \sqrt {-d} e^{i \sec ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 e^2}+\frac {i b \text {PolyLog}\left (2,-e^{2 i \sec ^{-1}(c x)}\right )}{2 e^2} \]

[Out]

1/2*(-a-b*arcsec(c*x))/e/(e+d/x^2)-(a+b*arcsec(c*x))*ln(1+(1/c/x+I*(1-1/c^2/x^2)^(1/2))^2)/e^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^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^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^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^2+1/2*I*b*polylog(2,-(1/c/x+I*(1-1/c^2/x^2)^(1/2))^2)
/e^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^2-1/2*I*b*poly
log(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^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^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^2-1/2*b*arctan((c^2*d+e)^(1/2)/c/x/e^(1/2)/(1-1/c^2/x^2)^(1/2))/e^(
3/2)/(c^2*d+e)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 1.12, antiderivative size = 570, normalized size of antiderivative = 1.00, number of steps used = 29, number of rules used = 12, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {5348, 4818, 4722, 3800, 2221, 2317, 2438, 4814, 385, 211, 4826, 4616} \begin {gather*} \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 )}{2 e^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 )}{2 e^2}+\frac {\left (a+b \sec ^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{i \sec ^{-1}(c x)}}{\sqrt {c^2 d+e}+\sqrt {e}}\right )}{2 e^2}+\frac {\left (a+b \sec ^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{i \sec ^{-1}(c x)}}{\sqrt {c^2 d+e}+\sqrt {e}}\right )}{2 e^2}-\frac {a+b \sec ^{-1}(c x)}{2 e \left (\frac {d}{x^2}+e\right )}-\frac {\log \left (1+e^{2 i \sec ^{-1}(c x)}\right ) \left (a+b \sec ^{-1}(c x)\right )}{e^2}-\frac {b \text {ArcTan}\left (\frac {\sqrt {c^2 d+e}}{c \sqrt {e} x \sqrt {1-\frac {1}{c^2 x^2}}}\right )}{2 e^{3/2} \sqrt {c^2 d+e}}-\frac {i b \text {Li}_2\left (-\frac {c \sqrt {-d} e^{i \sec ^{-1}(c x)}}{\sqrt {e}-\sqrt {d c^2+e}}\right )}{2 e^2}-\frac {i b \text {Li}_2\left (\frac {c \sqrt {-d} e^{i \sec ^{-1}(c x)}}{\sqrt {e}-\sqrt {d c^2+e}}\right )}{2 e^2}-\frac {i b \text {Li}_2\left (-\frac {c \sqrt {-d} e^{i \sec ^{-1}(c x)}}{\sqrt {e}+\sqrt {d c^2+e}}\right )}{2 e^2}-\frac {i b \text {Li}_2\left (\frac {c \sqrt {-d} e^{i \sec ^{-1}(c x)}}{\sqrt {e}+\sqrt {d c^2+e}}\right )}{2 e^2}+\frac {i b \text {Li}_2\left (-e^{2 i \sec ^{-1}(c x)}\right )}{2 e^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^3*(a + b*ArcSec[c*x]))/(d + e*x^2)^2,x]

[Out]

-1/2*(a + b*ArcSec[c*x])/(e*(e + d/x^2)) - (b*ArcTan[Sqrt[c^2*d + e]/(c*Sqrt[e]*Sqrt[1 - 1/(c^2*x^2)]*x)])/(2*
e^(3/2)*Sqrt[c^2*d + e]) + ((a + b*ArcSec[c*x])*Log[1 - (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] - Sqrt[c^2*d +
e])])/(2*e^2) + ((a + b*ArcSec[c*x])*Log[1 + (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*
e^2) + ((a + b*ArcSec[c*x])*Log[1 - (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*e^2) + ((a
+ b*ArcSec[c*x])*Log[1 + (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*e^2) - ((a + b*ArcSe
c[c*x])*Log[1 + E^((2*I)*ArcSec[c*x])])/e^2 - ((I/2)*b*PolyLog[2, -((c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] -
Sqrt[c^2*d + e]))])/e^2 - ((I/2)*b*PolyLog[2, (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/e^2
- ((I/2)*b*PolyLog[2, -((c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e]))])/e^2 - ((I/2)*b*PolyLog[
2, (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/e^2 + ((I/2)*b*PolyLog[2, -E^((2*I)*ArcSec[c*x
])])/e^2

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 385

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rule 2221

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x]
- Dist[d*(m/(b*f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 3800

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[I*((c + d*x)^(m + 1)/(d*(m + 1))), x
] - Dist[2*I, Int[(c + d*x)^m*(E^(2*I*(e + f*x))/(1 + E^(2*I*(e + f*x)))), x], x] /; FreeQ[{c, d, e, f}, x] &&
IGtQ[m, 0]

Rule 4616

Int[(((e_.) + (f_.)*(x_))^(m_.)*Sin[(c_.) + (d_.)*(x_)])/(Cos[(c_.) + (d_.)*(x_)]*(b_.) + (a_)), x_Symbol] :>
Simp[I*((e + f*x)^(m + 1)/(b*f*(m + 1))), x] + (-Dist[I, Int[(e + f*x)^m*(E^(I*(c + d*x))/(a - Rt[a^2 - b^2, 2
] + b*E^(I*(c + d*x)))), x], x] - Dist[I, Int[(e + f*x)^m*(E^(I*(c + d*x))/(a + Rt[a^2 - b^2, 2] + b*E^(I*(c +
d*x)))), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && PosQ[a^2 - b^2]

Rule 4722

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> -Subst[Int[(a + b*x)^n*Tan[x], x], x, ArcCos[c
*x]] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0]

Rule 4814

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))*(x_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)
*((a + b*ArcCos[c*x])/(2*e*(p + 1))), x] + Dist[b*(c/(2*e*(p + 1))), Int[(d + e*x^2)^(p + 1)/Sqrt[1 - c^2*x^2]
, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[c^2*d + e, 0] && NeQ[p, -1]

Rule 4818

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]

Rule 4826

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)), x_Symbol] :> -Subst[Int[(a + b*x)^n*(Sin[x]/
(c*d + e*Cos[x])), x], x, ArcCos[c*x]] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[n, 0]

Rule 5348

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]

Rubi steps

\begin {align*} \int \frac {x^3 \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx &=-\text {Subst}\left (\int \frac {a+b \cos ^{-1}\left (\frac {x}{c}\right )}{x \left (e+d x^2\right )^2} \, dx,x,\frac {1}{x}\right )\\ &=-\text {Subst}\left (\int \left (\frac {a+b \cos ^{-1}\left (\frac {x}{c}\right )}{e^2 x}-\frac {d x \left (a+b \cos ^{-1}\left (\frac {x}{c}\right )\right )}{e \left (e+d x^2\right )^2}-\frac {d x \left (a+b \cos ^{-1}\left (\frac {x}{c}\right )\right )}{e^2 \left (e+d x^2\right )}\right ) \, dx,x,\frac {1}{x}\right )\\ &=-\frac {\text {Subst}\left (\int \frac {a+b \cos ^{-1}\left (\frac {x}{c}\right )}{x} \, dx,x,\frac {1}{x}\right )}{e^2}+\frac {d \text {Subst}\left (\int \frac {x \left (a+b \cos ^{-1}\left (\frac {x}{c}\right )\right )}{e+d x^2} \, dx,x,\frac {1}{x}\right )}{e^2}+\frac {d \text {Subst}\left (\int \frac {x \left (a+b \cos ^{-1}\left (\frac {x}{c}\right )\right )}{\left (e+d x^2\right )^2} \, dx,x,\frac {1}{x}\right )}{e}\\ &=-\frac {a+b \sec ^{-1}(c x)}{2 e \left (e+\frac {d}{x^2}\right )}+\frac {\text {Subst}\left (\int (a+b x) \tan (x) \, dx,x,\sec ^{-1}(c x)\right )}{e^2}+\frac {d \text {Subst}\left (\int \left (-\frac {\sqrt {-d} \left (a+b \cos ^{-1}\left (\frac {x}{c}\right )\right )}{2 d \left (\sqrt {e}-\sqrt {-d} x\right )}+\frac {\sqrt {-d} \left (a+b \cos ^{-1}\left (\frac {x}{c}\right )\right )}{2 d \left (\sqrt {e}+\sqrt {-d} x\right )}\right ) \, dx,x,\frac {1}{x}\right )}{e^2}-\frac {b \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{c^2}} \left (e+d x^2\right )} \, dx,x,\frac {1}{x}\right )}{2 c e}\\ &=-\frac {a+b \sec ^{-1}(c x)}{2 e \left (e+\frac {d}{x^2}\right )}+\frac {i \left (a+b \sec ^{-1}(c x)\right )^2}{2 b e^2}-\frac {(2 i) \text {Subst}\left (\int \frac {e^{2 i x} (a+b x)}{1+e^{2 i x}} \, dx,x,\sec ^{-1}(c x)\right )}{e^2}-\frac {\sqrt {-d} \text {Subst}\left (\int \frac {a+b \cos ^{-1}\left (\frac {x}{c}\right )}{\sqrt {e}-\sqrt {-d} x} \, dx,x,\frac {1}{x}\right )}{2 e^2}+\frac {\sqrt {-d} \text {Subst}\left (\int \frac {a+b \cos ^{-1}\left (\frac {x}{c}\right )}{\sqrt {e}+\sqrt {-d} x} \, dx,x,\frac {1}{x}\right )}{2 e^2}-\frac {b \text {Subst}\left (\int \frac {1}{e-\left (-d-\frac {e}{c^2}\right ) x^2} \, dx,x,\frac {1}{\sqrt {1-\frac {1}{c^2 x^2}} x}\right )}{2 c e}\\ &=-\frac {a+b \sec ^{-1}(c x)}{2 e \left (e+\frac {d}{x^2}\right )}+\frac {i \left (a+b \sec ^{-1}(c x)\right )^2}{2 b e^2}-\frac {b \tan ^{-1}\left (\frac {\sqrt {c^2 d+e}}{c \sqrt {e} \sqrt {1-\frac {1}{c^2 x^2}} x}\right )}{2 e^{3/2} \sqrt {c^2 d+e}}-\frac {\left (a+b \sec ^{-1}(c x)\right ) \log \left (1+e^{2 i \sec ^{-1}(c x)}\right )}{e^2}+\frac {b \text {Subst}\left (\int \log \left (1+e^{2 i x}\right ) \, dx,x,\sec ^{-1}(c x)\right )}{e^2}+\frac {\sqrt {-d} \text {Subst}\left (\int \frac {(a+b x) \sin (x)}{\frac {\sqrt {e}}{c}-\sqrt {-d} \cos (x)} \, dx,x,\sec ^{-1}(c x)\right )}{2 e^2}-\frac {\sqrt {-d} \text {Subst}\left (\int \frac {(a+b x) \sin (x)}{\frac {\sqrt {e}}{c}+\sqrt {-d} \cos (x)} \, dx,x,\sec ^{-1}(c x)\right )}{2 e^2}\\ &=-\frac {a+b \sec ^{-1}(c x)}{2 e \left (e+\frac {d}{x^2}\right )}-\frac {b \tan ^{-1}\left (\frac {\sqrt {c^2 d+e}}{c \sqrt {e} \sqrt {1-\frac {1}{c^2 x^2}} x}\right )}{2 e^{3/2} \sqrt {c^2 d+e}}-\frac {\left (a+b \sec ^{-1}(c x)\right ) \log \left (1+e^{2 i \sec ^{-1}(c x)}\right )}{e^2}-\frac {(i b) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 i \sec ^{-1}(c x)}\right )}{2 e^2}-\frac {\left (i \sqrt {-d}\right ) \text {Subst}\left (\int \frac {e^{i x} (a+b x)}{\frac {\sqrt {e}}{c}-\frac {\sqrt {c^2 d+e}}{c}-\sqrt {-d} e^{i x}} \, dx,x,\sec ^{-1}(c x)\right )}{2 e^2}-\frac {\left (i \sqrt {-d}\right ) \text {Subst}\left (\int \frac {e^{i x} (a+b x)}{\frac {\sqrt {e}}{c}+\frac {\sqrt {c^2 d+e}}{c}-\sqrt {-d} e^{i x}} \, dx,x,\sec ^{-1}(c x)\right )}{2 e^2}+\frac {\left (i \sqrt {-d}\right ) \text {Subst}\left (\int \frac {e^{i x} (a+b x)}{\frac {\sqrt {e}}{c}-\frac {\sqrt {c^2 d+e}}{c}+\sqrt {-d} e^{i x}} \, dx,x,\sec ^{-1}(c x)\right )}{2 e^2}+\frac {\left (i \sqrt {-d}\right ) \text {Subst}\left (\int \frac {e^{i x} (a+b x)}{\frac {\sqrt {e}}{c}+\frac {\sqrt {c^2 d+e}}{c}+\sqrt {-d} e^{i x}} \, dx,x,\sec ^{-1}(c x)\right )}{2 e^2}\\ &=-\frac {a+b \sec ^{-1}(c x)}{2 e \left (e+\frac {d}{x^2}\right )}-\frac {b \tan ^{-1}\left (\frac {\sqrt {c^2 d+e}}{c \sqrt {e} \sqrt {1-\frac {1}{c^2 x^2}} x}\right )}{2 e^{3/2} \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 )}{2 e^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 )}{2 e^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 )}{2 e^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 )}{2 e^2}-\frac {\left (a+b \sec ^{-1}(c x)\right ) \log \left (1+e^{2 i \sec ^{-1}(c x)}\right )}{e^2}+\frac {i b \text {Li}_2\left (-e^{2 i \sec ^{-1}(c x)}\right )}{2 e^2}-\frac {b \text {Subst}\left (\int \log \left (1-\frac {\sqrt {-d} e^{i x}}{\frac {\sqrt {e}}{c}-\frac {\sqrt {c^2 d+e}}{c}}\right ) \, dx,x,\sec ^{-1}(c x)\right )}{2 e^2}-\frac {b \text {Subst}\left (\int \log \left (1+\frac {\sqrt {-d} e^{i x}}{\frac {\sqrt {e}}{c}-\frac {\sqrt {c^2 d+e}}{c}}\right ) \, dx,x,\sec ^{-1}(c x)\right )}{2 e^2}-\frac {b \text {Subst}\left (\int \log \left (1-\frac {\sqrt {-d} e^{i x}}{\frac {\sqrt {e}}{c}+\frac {\sqrt {c^2 d+e}}{c}}\right ) \, dx,x,\sec ^{-1}(c x)\right )}{2 e^2}-\frac {b \text {Subst}\left (\int \log \left (1+\frac {\sqrt {-d} e^{i x}}{\frac {\sqrt {e}}{c}+\frac {\sqrt {c^2 d+e}}{c}}\right ) \, dx,x,\sec ^{-1}(c x)\right )}{2 e^2}\\ &=-\frac {a+b \sec ^{-1}(c x)}{2 e \left (e+\frac {d}{x^2}\right )}-\frac {b \tan ^{-1}\left (\frac {\sqrt {c^2 d+e}}{c \sqrt {e} \sqrt {1-\frac {1}{c^2 x^2}} x}\right )}{2 e^{3/2} \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 )}{2 e^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 )}{2 e^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 )}{2 e^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 )}{2 e^2}-\frac {\left (a+b \sec ^{-1}(c x)\right ) \log \left (1+e^{2 i \sec ^{-1}(c x)}\right )}{e^2}+\frac {i b \text {Li}_2\left (-e^{2 i \sec ^{-1}(c x)}\right )}{2 e^2}+\frac {(i b) \text {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {-d} x}{\frac {\sqrt {e}}{c}-\frac {\sqrt {c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{i \sec ^{-1}(c x)}\right )}{2 e^2}+\frac {(i b) \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {-d} x}{\frac {\sqrt {e}}{c}-\frac {\sqrt {c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{i \sec ^{-1}(c x)}\right )}{2 e^2}+\frac {(i b) \text {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {-d} x}{\frac {\sqrt {e}}{c}+\frac {\sqrt {c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{i \sec ^{-1}(c x)}\right )}{2 e^2}+\frac {(i b) \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {-d} x}{\frac {\sqrt {e}}{c}+\frac {\sqrt {c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{i \sec ^{-1}(c x)}\right )}{2 e^2}\\ &=-\frac {a+b \sec ^{-1}(c x)}{2 e \left (e+\frac {d}{x^2}\right )}-\frac {b \tan ^{-1}\left (\frac {\sqrt {c^2 d+e}}{c \sqrt {e} \sqrt {1-\frac {1}{c^2 x^2}} x}\right )}{2 e^{3/2} \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 )}{2 e^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 )}{2 e^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 )}{2 e^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 )}{2 e^2}-\frac {\left (a+b \sec ^{-1}(c x)\right ) \log \left (1+e^{2 i \sec ^{-1}(c x)}\right )}{e^2}-\frac {i b \text {Li}_2\left (-\frac {c \sqrt {-d} e^{i \sec ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 e^2}-\frac {i b \text {Li}_2\left (\frac {c \sqrt {-d} e^{i \sec ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 e^2}-\frac {i b \text {Li}_2\left (-\frac {c \sqrt {-d} e^{i \sec ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 e^2}-\frac {i b \text {Li}_2\left (\frac {c \sqrt {-d} e^{i \sec ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 e^2}+\frac {i b \text {Li}_2\left (-e^{2 i \sec ^{-1}(c x)}\right )}{2 e^2}\\ \end {align*}

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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1213\) vs. \(2(570)=1140\).
time = 0.90, size = 1213, normalized size = 2.13 \begin {gather*} \frac {\frac {2 a d}{d+e x^2}+\frac {b \sqrt {d} \sec ^{-1}(c x)}{\sqrt {d}-i \sqrt {e} x}+\frac {b \sqrt {d} \sec ^{-1}(c x)}{\sqrt {d}+i \sqrt {e} x}+2 b \text {ArcSin}\left (\frac {1}{c x}\right )+8 i b \text {ArcSin}\left (\frac {\sqrt {1-\frac {i \sqrt {e}}{c \sqrt {d}}}}{\sqrt {2}}\right ) \text {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 )+8 i b \text {ArcSin}\left (\frac {\sqrt {1+\frac {i \sqrt {e}}{c \sqrt {d}}}}{\sqrt {2}}\right ) \text {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 b \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 b \text {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 b \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 b \text {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 b \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 b \text {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 b \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 b \text {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 )-4 b \sec ^{-1}(c x) \log \left (1+e^{2 i \sec ^{-1}(c x)}\right )-\frac {b \sqrt {e} \log \left (\frac {2 \sqrt {d} \sqrt {e} \left (\sqrt {e}+c \left (i c \sqrt {d}-\sqrt {-c^2 d-e} \sqrt {1-\frac {1}{c^2 x^2}}\right ) x\right )}{\sqrt {-c^2 d-e} \left (\sqrt {d}-i \sqrt {e} x\right )}\right )}{\sqrt {-c^2 d-e}}-\frac {b \sqrt {e} \log \left (\frac {2 \sqrt {d} \sqrt {e} \left (-\sqrt {e}+c \left (i c \sqrt {d}+\sqrt {-c^2 d-e} \sqrt {1-\frac {1}{c^2 x^2}}\right ) x\right )}{\sqrt {-c^2 d-e} \left (\sqrt {d}+i \sqrt {e} x\right )}\right )}{\sqrt {-c^2 d-e}}+2 a \log \left (d+e x^2\right )-2 i b \text {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 i b \text {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 i b \text {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 i b \text {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 i b \text {PolyLog}\left (2,-e^{2 i \sec ^{-1}(c x)}\right )}{4 e^2} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(x^3*(a + b*ArcSec[c*x]))/(d + e*x^2)^2,x]

[Out]

((2*a*d)/(d + e*x^2) + (b*Sqrt[d]*ArcSec[c*x])/(Sqrt[d] - I*Sqrt[e]*x) + (b*Sqrt[d]*ArcSec[c*x])/(Sqrt[d] + I*
Sqrt[e]*x) + 2*b*ArcSin[1/(c*x)] + (8*I)*b*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[(((-I)*c*S
qrt[d] + Sqrt[e])*Tan[ArcSec[c*x]/2])/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])*Tan[ArcSec[c*x]/2])/Sqrt[c^2*d + e]] + 2*b*ArcSec[c*x]*Log[1 + (I*(Sqrt[e
] - Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] + 4*b*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*L
og[1 + (I*(Sqrt[e] - Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] + 2*b*ArcSec[c*x]*Log[1 + (I*(-Sqrt[e] +
Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] + 4*b*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*b*ArcSec[c*x]*Log[1 - (I*(Sqrt[e] + Sq
rt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] - 4*b*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*b*ArcSec[c*x]*Log[1 + (I*(Sqrt[e] + Sqrt[c
^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] - 4*b*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])] - 4*b*ArcSec[c*x]*Log[1 + E^((2*I)*ArcSec[c*x])] -
(b*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] - (b*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] + 2*a*Log[d + e*x^2] - (2*I)*b*PolyLog[2, ((-I)*(-Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec
[c*x]))/(c*Sqrt[d])] - (2*I)*b*PolyLog[2, (I*(-Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] - (2
*I)*b*PolyLog[2, ((-I)*(Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] - (2*I)*b*PolyLog[2, (I*(Sq
rt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] + (2*I)*b*PolyLog[2, -E^((2*I)*ArcSec[c*x])])/(4*e^2)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.99, size = 619, normalized size = 1.09

method	result	size
derivativedivides	\(\frac {\frac {a \,c^{6} d}{2 e^{2} \left (c^{2} e \,x^{2}+c^{2} d \right )}+\frac {a \,c^{4} \ln \left (c^{2} e \,x^{2}+c^{2} d \right )}{2 e^{2}}-\frac {b \,c^{6} x^{2} \mathrm {arcsec}\left (c x \right )}{2 \left (c^{2} e \,x^{2}+c^{2} d \right ) e}+\frac {i b \,c^{4} \sqrt {e \left (c^{2} d +e \right )}\, \arctanh \left (\frac {2 c^{2} d \left (\frac {1}{c x}+i \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 )}{2 e^{2} \left (c^{2} d +e \right )}-\frac {i b \,c^{4} \left (\munderset {\textit {\_R1} =\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 \mathrm {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 )+\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 )}{4 e^{2}}-\frac {b \,c^{4} \mathrm {arcsec}\left (c x \right ) \ln \left (1+i \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )}{e^{2}}-\frac {b \,c^{4} \mathrm {arcsec}\left (c x \right ) \ln \left (1-i \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )}{e^{2}}+\frac {i b \,c^{4} \dilog \left (1+i \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )}{e^{2}}+\frac {i b \,c^{4} \dilog \left (1-i \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )}{e^{2}}-\frac {i b \,c^{6} \left (\munderset {\textit {\_R1} =\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 \mathrm {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 )+\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 ) d}{4 e^{2}}}{c^{4}}\)	\(619\)
default	\(\frac {\frac {a \,c^{6} d}{2 e^{2} \left (c^{2} e \,x^{2}+c^{2} d \right )}+\frac {a \,c^{4} \ln \left (c^{2} e \,x^{2}+c^{2} d \right )}{2 e^{2}}-\frac {b \,c^{6} x^{2} \mathrm {arcsec}\left (c x \right )}{2 \left (c^{2} e \,x^{2}+c^{2} d \right ) e}+\frac {i b \,c^{4} \sqrt {e \left (c^{2} d +e \right )}\, \arctanh \left (\frac {2 c^{2} d \left (\frac {1}{c x}+i \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 )}{2 e^{2} \left (c^{2} d +e \right )}-\frac {i b \,c^{4} \left (\munderset {\textit {\_R1} =\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 \mathrm {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 )+\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 )}{4 e^{2}}-\frac {b \,c^{4} \mathrm {arcsec}\left (c x \right ) \ln \left (1+i \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )}{e^{2}}-\frac {b \,c^{4} \mathrm {arcsec}\left (c x \right ) \ln \left (1-i \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )}{e^{2}}+\frac {i b \,c^{4} \dilog \left (1+i \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )}{e^{2}}+\frac {i b \,c^{4} \dilog \left (1-i \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )}{e^{2}}-\frac {i b \,c^{6} \left (\munderset {\textit {\_R1} =\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 \mathrm {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 )+\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 ) d}{4 e^{2}}}{c^{4}}\)	\(619\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a+b*arcsec(c*x))/(e*x^2+d)^2,x,method=_RETURNVERBOSE)

[Out]

1/c^4*(1/2*a*c^6*d/e^2/(c^2*e*x^2+c^2*d)+1/2*a*c^4/e^2*ln(c^2*e*x^2+c^2*d)-1/2*b*c^6*x^2*arcsec(c*x)/(c^2*e*x^
2+c^2*d)/e+1/2*I*b*c^4*(e*(c^2*d+e))^(1/2)/e^2/(c^2*d+e)*arctanh(1/4*(2*c^2*d*(1/c/x+I*(1-1/c^2/x^2)^(1/2))^2+
2*c^2*d+4*e)/(c^2*d*e+e^2)^(1/2))-1/4*I*b*c^4/e^2*sum((_R1^2*c^2*d+c^2*d+4*e)/(_R1^2*c^2*d+c^2*d+2*e)*(I*arcse
c(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))-b*c^4/e^2*arcsec(c*x)*ln(1+I*(1/c/x+I*(1-1/c^2/x^2)^(1/2)))-b*c^4/e^2*arcsec
(c*x)*ln(1-I*(1/c/x+I*(1-1/c^2/x^2)^(1/2)))+I*b*c^4/e^2*dilog(1+I*(1/c/x+I*(1-1/c^2/x^2)^(1/2)))+I*b*c^4/e^2*d
ilog(1-I*(1/c/x+I*(1-1/c^2/x^2)^(1/2)))-1/4*I*b*c^6/e^2*sum((_R1^2+1)/(_R1^2*c^2*d+c^2*d+2*e)*(I*arcsec(c*x)*l
n((_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))*d)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*arcsec(c*x))/(e*x^2+d)^2,x, algorithm="maxima")

[Out]

1/2*(e^(-2)*log(x^2*e + d) + d/(x^2*e^3 + d*e^2))*a + b*integrate(x^3*arctan(sqrt(c*x + 1)*sqrt(c*x - 1))/(x^4
*e^2 + 2*d*x^2*e + d^2), x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*arcsec(c*x))/(e*x^2+d)^2,x, algorithm="fricas")

[Out]

integral((b*x^3*arcsec(c*x) + a*x^3)/(x^4*e^2 + 2*d*x^2*e + d^2), x)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(a+b*asec(c*x))/(e*x**2+d)**2,x)

[Out]

Timed out

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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*arcsec(c*x))/(e*x^2+d)^2,x, algorithm="giac")

[Out]

Timed out

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^3\,\left (a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )\right )}{{\left (e\,x^2+d\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3*(a + b*acos(1/(c*x))))/(d + e*x^2)^2,x)

[Out]

int((x^3*(a + b*acos(1/(c*x))))/(d + e*x^2)^2, x)

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