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

Optimal. Leaf size=487 \[ -\frac{i b \text{PolyLog}\left (2,-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{2 e}-\frac{i b \text{PolyLog}\left (2,\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{2 e}-\frac{i b \text{PolyLog}\left (2,-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right )}{2 e}-\frac{i b \text{PolyLog}\left (2,\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right )}{2 e}+\frac{i b \text{PolyLog}\left (2,-e^{2 i \sec ^{-1}(c x)}\right )}{2 e}+\frac{\left (a+b \sec ^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{2 e}+\frac{\left (a+b \sec ^{-1}(c x)\right ) \log \left (1+\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{2 e}+\frac{\left (a+b \sec ^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right )}{2 e}+\frac{\left (a+b \sec ^{-1}(c x)\right ) \log \left (1+\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right )}{2 e}-\frac{\log \left (1+e^{2 i \sec ^{-1}(c x)}\right ) \left (a+b \sec ^{-1}(c x)\right )}{e} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 1.15161, antiderivative size = 487, normalized size of antiderivative = 1., number of steps used = 26, number of rules used = 9, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.474, Rules used = {5240, 4734, 4626, 3719, 2190, 2279, 2391, 4742, 4520} \[ -\frac{i b \text{PolyLog}\left (2,-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{2 e}-\frac{i b \text{PolyLog}\left (2,\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{2 e}-\frac{i b \text{PolyLog}\left (2,-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right )}{2 e}-\frac{i b \text{PolyLog}\left (2,\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right )}{2 e}+\frac{i b \text{PolyLog}\left (2,-e^{2 i \sec ^{-1}(c x)}\right )}{2 e}+\frac{\left (a+b \sec ^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{2 e}+\frac{\left (a+b \sec ^{-1}(c x)\right ) \log \left (1+\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{2 e}+\frac{\left (a+b \sec ^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right )}{2 e}+\frac{\left (a+b \sec ^{-1}(c x)\right ) \log \left (1+\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right )}{2 e}-\frac{\log \left (1+e^{2 i \sec ^{-1}(c x)}\right ) \left (a+b \sec ^{-1}(c x)\right )}{e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 4626

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

Rule 3719

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 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]

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]

Rubi steps

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

Mathematica [A]  time = 0.376304, size = 891, normalized size = 1.83 \[ \frac{4 i b \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 )+4 i b \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 )+b \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 )+2 b \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 )+b \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 )+2 b \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 )+b \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 )-2 b \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 )+b \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 )-2 b \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 b \sec ^{-1}(c x) \log \left (1+e^{2 i \sec ^{-1}(c x)}\right )+a \log \left (e x^2+d\right )-i b \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 )-i b \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 )-i b \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 )-i b \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 )+i b \text{PolyLog}\left (2,-e^{2 i \sec ^{-1}(c x)}\right )}{2 e} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((4*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]] + (4*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]] + b*ArcSec[c*x]*Log[1 + (I*(Sqrt[e] - Sqrt[c^2*d + e])*E^(I*ArcSec[c*x
]))/(c*Sqrt[d])] + 2*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])] + b*ArcSec[c*x]*Log[1 + (I*(-Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(
c*Sqrt[d])] + 2*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])] + b*ArcSec[c*x]*Log[1 - (I*(Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqr
t[d])] - 2*b*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 - (I*(Sqrt[e] + Sqrt[c^2*d + e])*E^(I*Arc
Sec[c*x]))/(c*Sqrt[d])] + b*ArcSec[c*x]*Log[1 + (I*(Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])]
- 2*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 + E^((2*I)*ArcSec[c*x])] + a*Log[d + e*x^2] - I*b*PolyLog[2, (I*(Sqr
t[e] - Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] - I*b*PolyLog[2, (I*(-Sqrt[e] + Sqrt[c^2*d + e])*E^(I*
ArcSec[c*x]))/(c*Sqrt[d])] - I*b*PolyLog[2, ((-I)*(Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])]
- I*b*PolyLog[2, (I*(Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] + I*b*PolyLog[2, -E^((2*I)*Arc
Sec[c*x])])/(2*e)

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Maple [C]  time = 0.5, size = 453, normalized size = 0.9 \begin{align*}{\frac{a\ln \left ({c}^{2}e{x}^{2}+{c}^{2}d \right ) }{2\,e}}-{\frac{{\frac{i}{4}}{c}^{2}bd}{e}\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}+1}{{{\it \_R1}}^{2}{c}^{2}d+{c}^{2}d+2\,e} \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{b{\rm arcsec} \left (cx\right )}{e}\ln \left ( 1+i \left ({\frac{1}{cx}}+i\sqrt{1-{\frac{1}{{c}^{2}{x}^{2}}}} \right ) \right ) }-{\frac{b{\rm arcsec} \left (cx\right )}{e}\ln \left ( 1-i \left ({\frac{1}{cx}}+i\sqrt{1-{\frac{1}{{c}^{2}{x}^{2}}}} \right ) \right ) }+{\frac{ib}{e}{\it dilog} \left ( 1+i \left ({\frac{1}{cx}}+i\sqrt{1-{\frac{1}{{c}^{2}{x}^{2}}}} \right ) \right ) }+{\frac{ib}{e}{\it dilog} \left ( 1-i \left ({\frac{1}{cx}}+i\sqrt{1-{\frac{1}{{c}^{2}{x}^{2}}}} \right ) \right ) }-{\frac{{\frac{i}{4}}b}{e}\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}}^{2}{c}^{2}d+{c}^{2}d+2\,e} \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*(a+b*arcsec(c*x))/(e*x^2+d),x)

[Out]

1/2*a/e*ln(c^2*e*x^2+c^2*d)-1/4*I*c^2*b*d/e*sum((_R1^2+1)/(_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))-b/e*arcsec(c*x)*ln(1+I*(1/c/x+I*(1-1/c^2/x^2)^(1/2)))-b/e*arcsec(c*x)*ln(1-I*(1/c/x+I*(1-1/c^2/x
^2)^(1/2)))+I*b/e*dilog(1+I*(1/c/x+I*(1-1/c^2/x^2)^(1/2)))+I*b/e*dilog(1-I*(1/c/x+I*(1-1/c^2/x^2)^(1/2)))-1/4*
I*b/e*sum((_R1^2*c^2*d+c^2*d+4*e)/(_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]  time = 0., size = 0, normalized size = 0. \begin{align*} b \int \frac{x \arctan \left (\sqrt{c x + 1} \sqrt{c x - 1}\right )}{e x^{2} + d}\,{d x} + \frac{a \log \left (e x^{2} + d\right )}{2 \, e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Integral(x*(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}{e x^{2} + d}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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