3.91 \(\int \frac{x^2 (a+b \sec ^{-1}(c x))}{d+e x^2} \, dx\)

Optimal. Leaf size=546 \[ \frac{i b \sqrt{-d} \text{PolyLog}\left (2,-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{2 e^{3/2}}-\frac{i b \sqrt{-d} \text{PolyLog}\left (2,\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{2 e^{3/2}}+\frac{i b \sqrt{-d} \text{PolyLog}\left (2,-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right )}{2 e^{3/2}}-\frac{i b \sqrt{-d} \text{PolyLog}\left (2,\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right )}{2 e^{3/2}}+\frac{\sqrt{-d} \left (a+b \sec ^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{2 e^{3/2}}-\frac{\sqrt{-d} \left (a+b \sec ^{-1}(c x)\right ) \log \left (1+\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{2 e^{3/2}}+\frac{\sqrt{-d} \left (a+b \sec ^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right )}{2 e^{3/2}}-\frac{\sqrt{-d} \left (a+b \sec ^{-1}(c x)\right ) \log \left (1+\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right )}{2 e^{3/2}}+\frac{x \left (a+b \sec ^{-1}(c x)\right )}{e}-\frac{b \tanh ^{-1}\left (\sqrt{1-\frac{1}{c^2 x^2}}\right )}{c e} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 1.2943, antiderivative size = 546, normalized size of antiderivative = 1., number of steps used = 25, number of rules used = 12, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.571, Rules used = {5240, 4734, 4628, 266, 63, 208, 4668, 4742, 4520, 2190, 2279, 2391} \[ \frac{i b \sqrt{-d} \text{PolyLog}\left (2,-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{2 e^{3/2}}-\frac{i b \sqrt{-d} \text{PolyLog}\left (2,\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{2 e^{3/2}}+\frac{i b \sqrt{-d} \text{PolyLog}\left (2,-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right )}{2 e^{3/2}}-\frac{i b \sqrt{-d} \text{PolyLog}\left (2,\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right )}{2 e^{3/2}}+\frac{\sqrt{-d} \left (a+b \sec ^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{2 e^{3/2}}-\frac{\sqrt{-d} \left (a+b \sec ^{-1}(c x)\right ) \log \left (1+\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{2 e^{3/2}}+\frac{\sqrt{-d} \left (a+b \sec ^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right )}{2 e^{3/2}}-\frac{\sqrt{-d} \left (a+b \sec ^{-1}(c x)\right ) \log \left (1+\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right )}{2 e^{3/2}}+\frac{x \left (a+b \sec ^{-1}(c x)\right )}{e}-\frac{b \tanh ^{-1}\left (\sqrt{1-\frac{1}{c^2 x^2}}\right )}{c e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 5240

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]

Rule 4734

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 4628

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcCo
s[c*x])^n)/(d*(m + 1)), x] + Dist[(b*c*n)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcCos[c*x])^(n - 1))/Sqrt[1
- c^2*x^2], x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

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

Rule 4668

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a
+ b*ArcCos[c*x])^n, (d + e*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, n}, x] && NeQ[c^2*d + e, 0] && IntegerQ[p]
&& (GtQ[p, 0] || IGtQ[n, 0])

Rule 4742

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 4520

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 2190

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*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), 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 2279

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 2391

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]

Rubi steps

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

Mathematica [A]  time = 1.36247, size = 1023, normalized size = 1.87 \[ \frac{a x}{e}-\frac{a \sqrt{d} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{e^{3/2}}+b \left (\frac{c x \sec ^{-1}(c x)+\log \left (\cos \left (\frac{1}{2} \sec ^{-1}(c x)\right )-\sin \left (\frac{1}{2} \sec ^{-1}(c x)\right )\right )-\log \left (\cos \left (\frac{1}{2} \sec ^{-1}(c x)\right )+\sin \left (\frac{1}{2} \sec ^{-1}(c x)\right )\right )}{c e}-\frac{\sqrt{d} \left (8 \sin ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right ) \tan ^{-1}\left (\frac{\left (i \sqrt{d} c+\sqrt{e}\right ) \tan \left (\frac{1}{2} \sec ^{-1}(c x)\right )}{\sqrt{d c^2+e}}\right )-2 i \sec ^{-1}(c x) \log \left (\frac{i e^{i \sec ^{-1}(c x)} \left (\sqrt{e}-\sqrt{d c^2+e}\right )}{c \sqrt{d}}+1\right )-4 i \sin ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right ) \log \left (\frac{i e^{i \sec ^{-1}(c x)} \left (\sqrt{e}-\sqrt{d c^2+e}\right )}{c \sqrt{d}}+1\right )-2 i \sec ^{-1}(c x) \log \left (\frac{i e^{i \sec ^{-1}(c x)} \left (\sqrt{e}+\sqrt{d c^2+e}\right )}{c \sqrt{d}}+1\right )+4 i \sin ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right ) \log \left (\frac{i e^{i \sec ^{-1}(c x)} \left (\sqrt{e}+\sqrt{d c^2+e}\right )}{c \sqrt{d}}+1\right )+2 i \sec ^{-1}(c x) \log \left (1+e^{2 i \sec ^{-1}(c x)}\right )-2 \text{PolyLog}\left (2,\frac{i \left (\sqrt{d c^2+e}-\sqrt{e}\right ) e^{i \sec ^{-1}(c x)}}{c \sqrt{d}}\right )-2 \text{PolyLog}\left (2,-\frac{i \left (\sqrt{e}+\sqrt{d c^2+e}\right ) e^{i \sec ^{-1}(c x)}}{c \sqrt{d}}\right )+\text{PolyLog}\left (2,-e^{2 i \sec ^{-1}(c x)}\right )\right )}{4 e^{3/2}}+\frac{\sqrt{d} \left (8 \sin ^{-1}\left (\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right ) \tan ^{-1}\left (\frac{\left (\sqrt{e}-i c \sqrt{d}\right ) \tan \left (\frac{1}{2} \sec ^{-1}(c x)\right )}{\sqrt{d c^2+e}}\right )-2 i \sec ^{-1}(c x) \log \left (\frac{i e^{i \sec ^{-1}(c x)} \left (\sqrt{d c^2+e}-\sqrt{e}\right )}{c \sqrt{d}}+1\right )-4 i \sin ^{-1}\left (\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right ) \log \left (\frac{i e^{i \sec ^{-1}(c x)} \left (\sqrt{d c^2+e}-\sqrt{e}\right )}{c \sqrt{d}}+1\right )-2 i \sec ^{-1}(c x) \log \left (1-\frac{i \left (\sqrt{e}+\sqrt{d c^2+e}\right ) e^{i \sec ^{-1}(c x)}}{c \sqrt{d}}\right )+4 i \sin ^{-1}\left (\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right ) \log \left (1-\frac{i \left (\sqrt{e}+\sqrt{d c^2+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 \text{PolyLog}\left (2,\frac{i \left (\sqrt{e}-\sqrt{d c^2+e}\right ) e^{i \sec ^{-1}(c x)}}{c \sqrt{d}}\right )-2 \text{PolyLog}\left (2,\frac{i \left (\sqrt{e}+\sqrt{d c^2+e}\right ) e^{i \sec ^{-1}(c x)}}{c \sqrt{d}}\right )+\text{PolyLog}\left (2,-e^{2 i \sec ^{-1}(c x)}\right )\right )}{4 e^{3/2}}\right ) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(a*x)/e - (a*Sqrt[d]*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/e^(3/2) + b*((c*x*ArcSec[c*x] + Log[Cos[ArcSec[c*x]/2] - Sin
[ArcSec[c*x]/2]] - Log[Cos[ArcSec[c*x]/2] + Sin[ArcSec[c*x]/2]])/(c*e) - (Sqrt[d]*(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*PolyLog[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)*Ar
cSec[c*x])]))/(4*e^(3/2)) + (Sqrt[d]*(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*P
olyLog[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*e^(3/2)))

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Maple [C]  time = 1.723, size = 374, normalized size = 0.7 \begin{align*}{\frac{ax}{e}}-{\frac{ad}{e}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}+{\frac{bx{\rm arcsec} \left (cx\right )}{e}}+{\frac{{\frac{i}{8}}cbd}{{e}^{2}}\sum _{{\it \_R1}={\it RootOf} \left ({c}^{2}d{{\it \_Z}}^{4}+ \left ( 2\,{c}^{2}d+4\,e \right ){{\it \_Z}}^{2}+{c}^{2}d \right ) }{\frac{{{\it \_R1}}^{2}{c}^{2}d+{c}^{2}d+4\,e}{{\it \_R1}\, \left ({{\it \_R1}}^{2}{c}^{2}d+{c}^{2}d+2\,e \right ) } \left ( i{\rm arcsec} \left (cx\right )\ln \left ({\frac{1}{{\it \_R1}} \left ({\it \_R1}-{\frac{1}{cx}}-i\sqrt{1-{\frac{1}{{c}^{2}{x}^{2}}}} \right ) } \right ) +{\it dilog} \left ({\frac{1}{{\it \_R1}} \left ({\it \_R1}-{\frac{1}{cx}}-i\sqrt{1-{\frac{1}{{c}^{2}{x}^{2}}}} \right ) } \right ) \right ) }}+{\frac{2\,ib}{ce}\arctan \left ({\frac{1}{cx}}+i\sqrt{1-{\frac{1}{{c}^{2}{x}^{2}}}} \right ) }-{\frac{{\frac{i}{8}}cbd}{{e}^{2}}\sum _{{\it \_R1}={\it RootOf} \left ({c}^{2}d{{\it \_Z}}^{4}+ \left ( 2\,{c}^{2}d+4\,e \right ){{\it \_Z}}^{2}+{c}^{2}d \right ) }{\frac{{{\it \_R1}}^{2}{c}^{2}d+4\,{{\it \_R1}}^{2}e+{c}^{2}d}{{\it \_R1}\, \left ({{\it \_R1}}^{2}{c}^{2}d+{c}^{2}d+2\,e \right ) } \left ( i{\rm arcsec} \left (cx\right )\ln \left ({\frac{1}{{\it \_R1}} \left ({\it \_R1}-{\frac{1}{cx}}-i\sqrt{1-{\frac{1}{{c}^{2}{x}^{2}}}} \right ) } \right ) +{\it dilog} \left ({\frac{1}{{\it \_R1}} \left ({\it \_R1}-{\frac{1}{cx}}-i\sqrt{1-{\frac{1}{{c}^{2}{x}^{2}}}} \right ) } \right ) \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x/e-a*d/e/(d*e)^(1/2)*arctan(e*x/(d*e)^(1/2))+b*arcsec(c*x)/e*x+1/8*I*c*b/e^2*d*sum((_R1^2*c^2*d+c^2*d+4*e)/
_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))+2*I/c*b/e*arctan(1/c/x+I*(1-1/c^2/x^2)^(
1/2))-1/8*I*c*b/e^2*d*sum((_R1^2*c^2*d+4*_R1^2*e+c^2*d)/_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))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b x^{2} \operatorname{arcsec}\left (c x\right ) + a x^{2}}{e x^{2} + d}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2} \left (a + b \operatorname{asec}{\left (c x \right )}\right )}{d + e x^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \operatorname{arcsec}\left (c x\right ) + a\right )} x^{2}}{e x^{2} + d}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*arcsec(c*x) + a)*x^2/(e*x^2 + d), x)