3.123 \(\int \frac{x^5 (a+b \text{sech}^{-1}(c x))}{(d+e x^2)^3} \, dx\)

Optimal. Leaf size=778 \[ \frac{b \text{PolyLog}\left (2,-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{2 e^3}+\frac{b \text{PolyLog}\left (2,\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{2 e^3}+\frac{b \text{PolyLog}\left (2,-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right )}{2 e^3}+\frac{b \text{PolyLog}\left (2,\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right )}{2 e^3}+\frac{b \text{PolyLog}\left (2,-e^{-2 \text{sech}^{-1}(c x)}\right )}{2 e^3}+\frac{\left (a+b \text{sech}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{2 e^3}+\frac{\left (a+b \text{sech}^{-1}(c x)\right ) \log \left (\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}+1\right )}{2 e^3}+\frac{\left (a+b \text{sech}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right )}{2 e^3}+\frac{\left (a+b \text{sech}^{-1}(c x)\right ) \log \left (\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}+1\right )}{2 e^3}-\frac{a+b \text{sech}^{-1}(c x)}{2 e^2 \left (\frac{d}{x^2}+e\right )}-\frac{a+b \text{sech}^{-1}(c x)}{4 e \left (\frac{d}{x^2}+e\right )^2}-\frac{\left (a+b \text{sech}^{-1}(c x)\right )^2}{b e^3}-\frac{\log \left (e^{-2 \text{sech}^{-1}(c x)}+1\right ) \left (a+b \text{sech}^{-1}(c x)\right )}{e^3}+\frac{b d \left (c^2-\frac{1}{x^2}\right )}{8 c e^2 x \sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1} \left (c^2 d+e\right ) \left (\frac{d}{x^2}+e\right )}+\frac{b \sqrt{\frac{1}{c^2 x^2}-1} \left (c^2 d+2 e\right ) \tanh ^{-1}\left (\frac{\sqrt{c^2 d+e}}{c \sqrt{e} x \sqrt{\frac{1}{c^2 x^2}-1}}\right )}{8 e^{5/2} \sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1} \left (c^2 d+e\right )^{3/2}}+\frac{b \sqrt{\frac{1}{c^2 x^2}-1} \tanh ^{-1}\left (\frac{\sqrt{c^2 d+e}}{c \sqrt{e} x \sqrt{\frac{1}{c^2 x^2}-1}}\right )}{2 e^{5/2} \sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1} \sqrt{c^2 d+e}} \]

[Out]

(b*d*(c^2 - x^(-2)))/(8*c*e^2*(c^2*d + e)*(e + d/x^2)*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]*x) - (a + b*ArcSech
[c*x])/(4*e*(e + d/x^2)^2) - (a + b*ArcSech[c*x])/(2*e^2*(e + d/x^2)) - (a + b*ArcSech[c*x])^2/(b*e^3) + (b*Sq
rt[-1 + 1/(c^2*x^2)]*ArcTanh[Sqrt[c^2*d + e]/(c*Sqrt[e]*Sqrt[-1 + 1/(c^2*x^2)]*x)])/(2*e^(5/2)*Sqrt[c^2*d + e]
*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]) + (b*(c^2*d + 2*e)*Sqrt[-1 + 1/(c^2*x^2)]*ArcTanh[Sqrt[c^2*d + e]/(c*Sq
rt[e]*Sqrt[-1 + 1/(c^2*x^2)]*x)])/(8*e^(5/2)*(c^2*d + e)^(3/2)*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]) - ((a + b
*ArcSech[c*x])*Log[1 + E^(-2*ArcSech[c*x])])/e^3 + ((a + b*ArcSech[c*x])*Log[1 - (c*Sqrt[-d]*E^ArcSech[c*x])/(
Sqrt[e] - Sqrt[c^2*d + e])])/(2*e^3) + ((a + b*ArcSech[c*x])*Log[1 + (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] - Sq
rt[c^2*d + e])])/(2*e^3) + ((a + b*ArcSech[c*x])*Log[1 - (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e
])])/(2*e^3) + ((a + b*ArcSech[c*x])*Log[1 + (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*e^3)
+ (b*PolyLog[2, -E^(-2*ArcSech[c*x])])/(2*e^3) + (b*PolyLog[2, -((c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] - Sqrt[
c^2*d + e]))])/(2*e^3) + (b*PolyLog[2, (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*e^3) + (b*
PolyLog[2, -((c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e]))])/(2*e^3) + (b*PolyLog[2, (c*Sqrt[-d]*E^
ArcSech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*e^3)

________________________________________________________________________________________

Rubi [A]  time = 1.70935, antiderivative size = 760, normalized size of antiderivative = 0.98, number of steps used = 35, number of rules used = 14, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667, Rules used = {6303, 5792, 5660, 3718, 2190, 2279, 2391, 5788, 519, 382, 377, 208, 5800, 5562} \[ \frac{b \text{PolyLog}\left (2,-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{2 e^3}+\frac{b \text{PolyLog}\left (2,\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{2 e^3}+\frac{b \text{PolyLog}\left (2,-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right )}{2 e^3}+\frac{b \text{PolyLog}\left (2,\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right )}{2 e^3}-\frac{b \text{PolyLog}\left (2,-e^{2 \text{sech}^{-1}(c x)}\right )}{2 e^3}+\frac{\left (a+b \text{sech}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{2 e^3}+\frac{\left (a+b \text{sech}^{-1}(c x)\right ) \log \left (\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}+1\right )}{2 e^3}+\frac{\left (a+b \text{sech}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right )}{2 e^3}+\frac{\left (a+b \text{sech}^{-1}(c x)\right ) \log \left (\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}+1\right )}{2 e^3}-\frac{a+b \text{sech}^{-1}(c x)}{2 e^2 \left (\frac{d}{x^2}+e\right )}-\frac{a+b \text{sech}^{-1}(c x)}{4 e \left (\frac{d}{x^2}+e\right )^2}-\frac{\log \left (e^{2 \text{sech}^{-1}(c x)}+1\right ) \left (a+b \text{sech}^{-1}(c x)\right )}{e^3}+\frac{b d \left (c^2-\frac{1}{x^2}\right )}{8 c e^2 x \sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1} \left (c^2 d+e\right ) \left (\frac{d}{x^2}+e\right )}+\frac{b \sqrt{\frac{1}{c^2 x^2}-1} \left (c^2 d+2 e\right ) \tanh ^{-1}\left (\frac{\sqrt{c^2 d+e}}{c \sqrt{e} x \sqrt{\frac{1}{c^2 x^2}-1}}\right )}{8 e^{5/2} \sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1} \left (c^2 d+e\right )^{3/2}}+\frac{b \sqrt{\frac{1}{c^2 x^2}-1} \tanh ^{-1}\left (\frac{\sqrt{c^2 d+e}}{c \sqrt{e} x \sqrt{\frac{1}{c^2 x^2}-1}}\right )}{2 e^{5/2} \sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1} \sqrt{c^2 d+e}} \]

Warning: Unable to verify antiderivative.

[In]

Int[(x^5*(a + b*ArcSech[c*x]))/(d + e*x^2)^3,x]

[Out]

(b*d*(c^2 - x^(-2)))/(8*c*e^2*(c^2*d + e)*(e + d/x^2)*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]*x) - (a + b*ArcSech
[c*x])/(4*e*(e + d/x^2)^2) - (a + b*ArcSech[c*x])/(2*e^2*(e + d/x^2)) + (b*Sqrt[-1 + 1/(c^2*x^2)]*ArcTanh[Sqrt
[c^2*d + e]/(c*Sqrt[e]*Sqrt[-1 + 1/(c^2*x^2)]*x)])/(2*e^(5/2)*Sqrt[c^2*d + e]*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c
*x)]) + (b*(c^2*d + 2*e)*Sqrt[-1 + 1/(c^2*x^2)]*ArcTanh[Sqrt[c^2*d + e]/(c*Sqrt[e]*Sqrt[-1 + 1/(c^2*x^2)]*x)])
/(8*e^(5/2)*(c^2*d + e)^(3/2)*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]) + ((a + b*ArcSech[c*x])*Log[1 - (c*Sqrt[-d
]*E^ArcSech[c*x])/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*e^3) + ((a + b*ArcSech[c*x])*Log[1 + (c*Sqrt[-d]*E^ArcSech[
c*x])/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*e^3) + ((a + b*ArcSech[c*x])*Log[1 - (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[
e] + Sqrt[c^2*d + e])])/(2*e^3) + ((a + b*ArcSech[c*x])*Log[1 + (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + Sqrt[c^
2*d + e])])/(2*e^3) - ((a + b*ArcSech[c*x])*Log[1 + E^(2*ArcSech[c*x])])/e^3 + (b*PolyLog[2, -((c*Sqrt[-d]*E^A
rcSech[c*x])/(Sqrt[e] - Sqrt[c^2*d + e]))])/(2*e^3) + (b*PolyLog[2, (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] - Sqr
t[c^2*d + e])])/(2*e^3) + (b*PolyLog[2, -((c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e]))])/(2*e^3) +
(b*PolyLog[2, (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*e^3) - (b*PolyLog[2, -E^(2*ArcSech
[c*x])])/(2*e^3)

Rule 6303

Int[((a_.) + ArcSech[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_.) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> -Subst[Int
[((e + d*x^2)^p*(a + b*ArcCosh[x/c])^n)/x^(m + 2*(p + 1)), x], x, 1/x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ
[n, 0] && IntegersQ[m, p]

Rule 5792

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Int
[ExpandIntegrand[(a + b*ArcCosh[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 5660

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

Rule 3718

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (Complex[0, fz_])*(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*fz*x)))/(1 + E^(2*(-(I*e) + f*fz*x))), x],
x] /; FreeQ[{c, d, e, f, fz}, 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 5788

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

Rule 519

Int[(u_.)*((c_) + (d_.)*(x_)^(n_.))^(q_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p_)*((a2_) + (b2_.)*(x_)^(non2_.))^(
p_), x_Symbol] :> Dist[((a1 + b1*x^(n/2))^FracPart[p]*(a2 + b2*x^(n/2))^FracPart[p])/(a1*a2 + b1*b2*x^n)^FracP
art[p], Int[u*(a1*a2 + b1*b2*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a1, b1, a2, b2, c, d, n, p, q}, x] && EqQ[
non2, n/2] && EqQ[a2*b1 + a1*b2, 0] &&  !(EqQ[n, 2] && IGtQ[q, 0])

Rule 382

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[(b*x*(a + b*x^n)^(p + 1)*(
c + d*x^n)^(q + 1))/(a*n*(p + 1)*(b*c - a*d)), x] + Dist[(b*c + n*(p + 1)*(b*c - a*d))/(a*n*(p + 1)*(b*c - a*d
)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n, q}, x] && NeQ[b*c - a*d, 0] && EqQ[
n*(p + q + 2) + 1, 0] && (LtQ[p, -1] ||  !LtQ[q, -1]) && NeQ[p, -1]

Rule 377

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 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 5800

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

Rule 5562

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

Rubi steps

\begin{align*} \int \frac{x^5 \left (a+b \text{sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx &=-\operatorname{Subst}\left (\int \frac{a+b \cosh ^{-1}\left (\frac{x}{c}\right )}{x \left (e+d x^2\right )^3} \, dx,x,\frac{1}{x}\right )\\ &=-\operatorname{Subst}\left (\int \left (\frac{a+b \cosh ^{-1}\left (\frac{x}{c}\right )}{e^3 x}-\frac{d x \left (a+b \cosh ^{-1}\left (\frac{x}{c}\right )\right )}{e \left (e+d x^2\right )^3}-\frac{d x \left (a+b \cosh ^{-1}\left (\frac{x}{c}\right )\right )}{e^2 \left (e+d x^2\right )^2}-\frac{d x \left (a+b \cosh ^{-1}\left (\frac{x}{c}\right )\right )}{e^3 \left (e+d x^2\right )}\right ) \, dx,x,\frac{1}{x}\right )\\ &=-\frac{\operatorname{Subst}\left (\int \frac{a+b \cosh ^{-1}\left (\frac{x}{c}\right )}{x} \, dx,x,\frac{1}{x}\right )}{e^3}+\frac{d \operatorname{Subst}\left (\int \frac{x \left (a+b \cosh ^{-1}\left (\frac{x}{c}\right )\right )}{e+d x^2} \, dx,x,\frac{1}{x}\right )}{e^3}+\frac{d \operatorname{Subst}\left (\int \frac{x \left (a+b \cosh ^{-1}\left (\frac{x}{c}\right )\right )}{\left (e+d x^2\right )^2} \, dx,x,\frac{1}{x}\right )}{e^2}+\frac{d \operatorname{Subst}\left (\int \frac{x \left (a+b \cosh ^{-1}\left (\frac{x}{c}\right )\right )}{\left (e+d x^2\right )^3} \, dx,x,\frac{1}{x}\right )}{e}\\ &=-\frac{a+b \text{sech}^{-1}(c x)}{4 e \left (e+\frac{d}{x^2}\right )^2}-\frac{a+b \text{sech}^{-1}(c x)}{2 e^2 \left (e+\frac{d}{x^2}\right )}-\frac{\operatorname{Subst}\left (\int (a+b x) \tanh (x) \, dx,x,\text{sech}^{-1}(c x)\right )}{e^3}+\frac{d \operatorname{Subst}\left (\int \left (-\frac{\sqrt{-d} \left (a+b \cosh ^{-1}\left (\frac{x}{c}\right )\right )}{2 d \left (\sqrt{e}-\sqrt{-d} x\right )}+\frac{\sqrt{-d} \left (a+b \cosh ^{-1}\left (\frac{x}{c}\right )\right )}{2 d \left (\sqrt{e}+\sqrt{-d} x\right )}\right ) \, dx,x,\frac{1}{x}\right )}{e^3}+\frac{b \operatorname{Subst}\left (\int \frac{1}{\sqrt{-1+\frac{x}{c}} \sqrt{1+\frac{x}{c}} \left (e+d x^2\right )} \, dx,x,\frac{1}{x}\right )}{2 c e^2}+\frac{b \operatorname{Subst}\left (\int \frac{1}{\sqrt{-1+\frac{x}{c}} \sqrt{1+\frac{x}{c}} \left (e+d x^2\right )^2} \, dx,x,\frac{1}{x}\right )}{4 c e}\\ &=-\frac{a+b \text{sech}^{-1}(c x)}{4 e \left (e+\frac{d}{x^2}\right )^2}-\frac{a+b \text{sech}^{-1}(c x)}{2 e^2 \left (e+\frac{d}{x^2}\right )}+\frac{\left (a+b \text{sech}^{-1}(c x)\right )^2}{2 b e^3}-\frac{2 \operatorname{Subst}\left (\int \frac{e^{2 x} (a+b x)}{1+e^{2 x}} \, dx,x,\text{sech}^{-1}(c x)\right )}{e^3}-\frac{\sqrt{-d} \operatorname{Subst}\left (\int \frac{a+b \cosh ^{-1}\left (\frac{x}{c}\right )}{\sqrt{e}-\sqrt{-d} x} \, dx,x,\frac{1}{x}\right )}{2 e^3}+\frac{\sqrt{-d} \operatorname{Subst}\left (\int \frac{a+b \cosh ^{-1}\left (\frac{x}{c}\right )}{\sqrt{e}+\sqrt{-d} x} \, dx,x,\frac{1}{x}\right )}{2 e^3}+\frac{\left (b \sqrt{-1+\frac{1}{c^2 x^2}}\right ) \operatorname{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^2 \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}+\frac{\left (b \sqrt{-1+\frac{1}{c^2 x^2}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-1+\frac{x^2}{c^2}} \left (e+d x^2\right )^2} \, dx,x,\frac{1}{x}\right )}{4 c e \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}\\ &=\frac{b d \left (c^2-\frac{1}{x^2}\right )}{8 c e^2 \left (c^2 d+e\right ) \left (e+\frac{d}{x^2}\right ) \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}} x}-\frac{a+b \text{sech}^{-1}(c x)}{4 e \left (e+\frac{d}{x^2}\right )^2}-\frac{a+b \text{sech}^{-1}(c x)}{2 e^2 \left (e+\frac{d}{x^2}\right )}+\frac{\left (a+b \text{sech}^{-1}(c x)\right )^2}{2 b e^3}-\frac{\left (a+b \text{sech}^{-1}(c x)\right ) \log \left (1+e^{2 \text{sech}^{-1}(c x)}\right )}{e^3}+\frac{b \operatorname{Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\text{sech}^{-1}(c x)\right )}{e^3}-\frac{\sqrt{-d} \operatorname{Subst}\left (\int \frac{(a+b x) \sinh (x)}{\frac{\sqrt{e}}{c}-\sqrt{-d} \cosh (x)} \, dx,x,\text{sech}^{-1}(c x)\right )}{2 e^3}+\frac{\sqrt{-d} \operatorname{Subst}\left (\int \frac{(a+b x) \sinh (x)}{\frac{\sqrt{e}}{c}+\sqrt{-d} \cosh (x)} \, dx,x,\text{sech}^{-1}(c x)\right )}{2 e^3}+\frac{\left (b \sqrt{-1+\frac{1}{c^2 x^2}}\right ) \operatorname{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^2 \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}+\frac{\left (b \left (c^2 d+2 e\right ) \sqrt{-1+\frac{1}{c^2 x^2}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-1+\frac{x^2}{c^2}} \left (e+d x^2\right )} \, dx,x,\frac{1}{x}\right )}{8 c e^2 \left (c^2 d+e\right ) \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}\\ &=\frac{b d \left (c^2-\frac{1}{x^2}\right )}{8 c e^2 \left (c^2 d+e\right ) \left (e+\frac{d}{x^2}\right ) \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}} x}-\frac{a+b \text{sech}^{-1}(c x)}{4 e \left (e+\frac{d}{x^2}\right )^2}-\frac{a+b \text{sech}^{-1}(c x)}{2 e^2 \left (e+\frac{d}{x^2}\right )}+\frac{b \sqrt{-1+\frac{1}{c^2 x^2}} \tanh ^{-1}\left (\frac{\sqrt{c^2 d+e}}{c \sqrt{e} \sqrt{-1+\frac{1}{c^2 x^2}} x}\right )}{2 e^{5/2} \sqrt{c^2 d+e} \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}-\frac{\left (a+b \text{sech}^{-1}(c x)\right ) \log \left (1+e^{2 \text{sech}^{-1}(c x)}\right )}{e^3}+\frac{b \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 \text{sech}^{-1}(c x)}\right )}{2 e^3}-\frac{\sqrt{-d} \operatorname{Subst}\left (\int \frac{e^x (a+b x)}{\frac{\sqrt{e}}{c}-\frac{\sqrt{c^2 d+e}}{c}-\sqrt{-d} e^x} \, dx,x,\text{sech}^{-1}(c x)\right )}{2 e^3}-\frac{\sqrt{-d} \operatorname{Subst}\left (\int \frac{e^x (a+b x)}{\frac{\sqrt{e}}{c}+\frac{\sqrt{c^2 d+e}}{c}-\sqrt{-d} e^x} \, dx,x,\text{sech}^{-1}(c x)\right )}{2 e^3}+\frac{\sqrt{-d} \operatorname{Subst}\left (\int \frac{e^x (a+b x)}{\frac{\sqrt{e}}{c}-\frac{\sqrt{c^2 d+e}}{c}+\sqrt{-d} e^x} \, dx,x,\text{sech}^{-1}(c x)\right )}{2 e^3}+\frac{\sqrt{-d} \operatorname{Subst}\left (\int \frac{e^x (a+b x)}{\frac{\sqrt{e}}{c}+\frac{\sqrt{c^2 d+e}}{c}+\sqrt{-d} e^x} \, dx,x,\text{sech}^{-1}(c x)\right )}{2 e^3}+\frac{\left (b \left (c^2 d+2 e\right ) \sqrt{-1+\frac{1}{c^2 x^2}}\right ) \operatorname{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 )}{8 c e^2 \left (c^2 d+e\right ) \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}\\ &=\frac{b d \left (c^2-\frac{1}{x^2}\right )}{8 c e^2 \left (c^2 d+e\right ) \left (e+\frac{d}{x^2}\right ) \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}} x}-\frac{a+b \text{sech}^{-1}(c x)}{4 e \left (e+\frac{d}{x^2}\right )^2}-\frac{a+b \text{sech}^{-1}(c x)}{2 e^2 \left (e+\frac{d}{x^2}\right )}+\frac{b \sqrt{-1+\frac{1}{c^2 x^2}} \tanh ^{-1}\left (\frac{\sqrt{c^2 d+e}}{c \sqrt{e} \sqrt{-1+\frac{1}{c^2 x^2}} x}\right )}{2 e^{5/2} \sqrt{c^2 d+e} \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}+\frac{b \left (c^2 d+2 e\right ) \sqrt{-1+\frac{1}{c^2 x^2}} \tanh ^{-1}\left (\frac{\sqrt{c^2 d+e}}{c \sqrt{e} \sqrt{-1+\frac{1}{c^2 x^2}} x}\right )}{8 e^{5/2} \left (c^2 d+e\right )^{3/2} \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}+\frac{\left (a+b \text{sech}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{2 e^3}+\frac{\left (a+b \text{sech}^{-1}(c x)\right ) \log \left (1+\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{2 e^3}+\frac{\left (a+b \text{sech}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}+\sqrt{c^2 d+e}}\right )}{2 e^3}+\frac{\left (a+b \text{sech}^{-1}(c x)\right ) \log \left (1+\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}+\sqrt{c^2 d+e}}\right )}{2 e^3}-\frac{\left (a+b \text{sech}^{-1}(c x)\right ) \log \left (1+e^{2 \text{sech}^{-1}(c x)}\right )}{e^3}-\frac{b \text{Li}_2\left (-e^{2 \text{sech}^{-1}(c x)}\right )}{2 e^3}-\frac{b \operatorname{Subst}\left (\int \log \left (1-\frac{\sqrt{-d} e^x}{\frac{\sqrt{e}}{c}-\frac{\sqrt{c^2 d+e}}{c}}\right ) \, dx,x,\text{sech}^{-1}(c x)\right )}{2 e^3}-\frac{b \operatorname{Subst}\left (\int \log \left (1+\frac{\sqrt{-d} e^x}{\frac{\sqrt{e}}{c}-\frac{\sqrt{c^2 d+e}}{c}}\right ) \, dx,x,\text{sech}^{-1}(c x)\right )}{2 e^3}-\frac{b \operatorname{Subst}\left (\int \log \left (1-\frac{\sqrt{-d} e^x}{\frac{\sqrt{e}}{c}+\frac{\sqrt{c^2 d+e}}{c}}\right ) \, dx,x,\text{sech}^{-1}(c x)\right )}{2 e^3}-\frac{b \operatorname{Subst}\left (\int \log \left (1+\frac{\sqrt{-d} e^x}{\frac{\sqrt{e}}{c}+\frac{\sqrt{c^2 d+e}}{c}}\right ) \, dx,x,\text{sech}^{-1}(c x)\right )}{2 e^3}\\ &=\frac{b d \left (c^2-\frac{1}{x^2}\right )}{8 c e^2 \left (c^2 d+e\right ) \left (e+\frac{d}{x^2}\right ) \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}} x}-\frac{a+b \text{sech}^{-1}(c x)}{4 e \left (e+\frac{d}{x^2}\right )^2}-\frac{a+b \text{sech}^{-1}(c x)}{2 e^2 \left (e+\frac{d}{x^2}\right )}+\frac{b \sqrt{-1+\frac{1}{c^2 x^2}} \tanh ^{-1}\left (\frac{\sqrt{c^2 d+e}}{c \sqrt{e} \sqrt{-1+\frac{1}{c^2 x^2}} x}\right )}{2 e^{5/2} \sqrt{c^2 d+e} \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}+\frac{b \left (c^2 d+2 e\right ) \sqrt{-1+\frac{1}{c^2 x^2}} \tanh ^{-1}\left (\frac{\sqrt{c^2 d+e}}{c \sqrt{e} \sqrt{-1+\frac{1}{c^2 x^2}} x}\right )}{8 e^{5/2} \left (c^2 d+e\right )^{3/2} \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}+\frac{\left (a+b \text{sech}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{2 e^3}+\frac{\left (a+b \text{sech}^{-1}(c x)\right ) \log \left (1+\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{2 e^3}+\frac{\left (a+b \text{sech}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}+\sqrt{c^2 d+e}}\right )}{2 e^3}+\frac{\left (a+b \text{sech}^{-1}(c x)\right ) \log \left (1+\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}+\sqrt{c^2 d+e}}\right )}{2 e^3}-\frac{\left (a+b \text{sech}^{-1}(c x)\right ) \log \left (1+e^{2 \text{sech}^{-1}(c x)}\right )}{e^3}-\frac{b \text{Li}_2\left (-e^{2 \text{sech}^{-1}(c x)}\right )}{2 e^3}-\frac{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^{\text{sech}^{-1}(c x)}\right )}{2 e^3}-\frac{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^{\text{sech}^{-1}(c x)}\right )}{2 e^3}-\frac{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^{\text{sech}^{-1}(c x)}\right )}{2 e^3}-\frac{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^{\text{sech}^{-1}(c x)}\right )}{2 e^3}\\ &=\frac{b d \left (c^2-\frac{1}{x^2}\right )}{8 c e^2 \left (c^2 d+e\right ) \left (e+\frac{d}{x^2}\right ) \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}} x}-\frac{a+b \text{sech}^{-1}(c x)}{4 e \left (e+\frac{d}{x^2}\right )^2}-\frac{a+b \text{sech}^{-1}(c x)}{2 e^2 \left (e+\frac{d}{x^2}\right )}+\frac{b \sqrt{-1+\frac{1}{c^2 x^2}} \tanh ^{-1}\left (\frac{\sqrt{c^2 d+e}}{c \sqrt{e} \sqrt{-1+\frac{1}{c^2 x^2}} x}\right )}{2 e^{5/2} \sqrt{c^2 d+e} \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}+\frac{b \left (c^2 d+2 e\right ) \sqrt{-1+\frac{1}{c^2 x^2}} \tanh ^{-1}\left (\frac{\sqrt{c^2 d+e}}{c \sqrt{e} \sqrt{-1+\frac{1}{c^2 x^2}} x}\right )}{8 e^{5/2} \left (c^2 d+e\right )^{3/2} \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}+\frac{\left (a+b \text{sech}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{2 e^3}+\frac{\left (a+b \text{sech}^{-1}(c x)\right ) \log \left (1+\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{2 e^3}+\frac{\left (a+b \text{sech}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}+\sqrt{c^2 d+e}}\right )}{2 e^3}+\frac{\left (a+b \text{sech}^{-1}(c x)\right ) \log \left (1+\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}+\sqrt{c^2 d+e}}\right )}{2 e^3}-\frac{\left (a+b \text{sech}^{-1}(c x)\right ) \log \left (1+e^{2 \text{sech}^{-1}(c x)}\right )}{e^3}+\frac{b \text{Li}_2\left (-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{2 e^3}+\frac{b \text{Li}_2\left (\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{2 e^3}+\frac{b \text{Li}_2\left (-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}+\sqrt{c^2 d+e}}\right )}{2 e^3}+\frac{b \text{Li}_2\left (\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}+\sqrt{c^2 d+e}}\right )}{2 e^3}-\frac{b \text{Li}_2\left (-e^{2 \text{sech}^{-1}(c x)}\right )}{2 e^3}\\ \end{align*}

Mathematica [C]  time = 7.48651, size = 2000, normalized size = 2.57 \[ \text{result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(x^5*(a + b*ArcSech[c*x]))/(d + e*x^2)^3,x]

[Out]

-(a*d^2)/(4*e^3*(d + e*x^2)^2) + (a*d)/(e^3*(d + e*x^2)) + (a*Log[d + e*x^2])/(2*e^3) + b*(-(d*(((-I)*Sqrt[e]*
Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x))/(Sqrt[d]*(c^2*d + e)*((-I)*Sqrt[d] + Sqrt[e]*x)) - ArcSech[c*x]/(Sqrt[e]*
((-I)*Sqrt[d] + Sqrt[e]*x)^2) + Log[x]/(d*Sqrt[e]) - Log[1 + Sqrt[(1 - c*x)/(1 + c*x)] + c*x*Sqrt[(1 - c*x)/(1
+ c*x)]]/(d*Sqrt[e]) + ((2*c^2*d + e)*Log[(-4*d*Sqrt[e]*Sqrt[c^2*d + e]*(Sqrt[e] - I*c^2*Sqrt[d]*x + Sqrt[c^2
*d + e]*Sqrt[(1 - c*x)/(1 + c*x)] + c*Sqrt[c^2*d + e]*x*Sqrt[(1 - c*x)/(1 + c*x)]))/((2*c^2*d + e)*((-I)*Sqrt[
d] + Sqrt[e]*x))])/(d*(c^2*d + e)^(3/2))))/(16*e^(5/2)) - (d*((I*Sqrt[e]*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x))/
(Sqrt[d]*(c^2*d + e)*(I*Sqrt[d] + Sqrt[e]*x)) - ArcSech[c*x]/(Sqrt[e]*(I*Sqrt[d] + Sqrt[e]*x)^2) + Log[x]/(d*S
qrt[e]) - Log[1 + Sqrt[(1 - c*x)/(1 + c*x)] + c*x*Sqrt[(1 - c*x)/(1 + c*x)]]/(d*Sqrt[e]) + ((2*c^2*d + e)*Log[
(-4*d*Sqrt[e]*Sqrt[c^2*d + e]*(Sqrt[e] + I*c^2*Sqrt[d]*x + Sqrt[c^2*d + e]*Sqrt[(1 - c*x)/(1 + c*x)] + c*Sqrt[
c^2*d + e]*x*Sqrt[(1 - c*x)/(1 + c*x)]))/((2*c^2*d + e)*(I*Sqrt[d] + Sqrt[e]*x))])/(d*(c^2*d + e)^(3/2))))/(16
*e^(5/2)) - (((7*I)/16)*Sqrt[d]*(-(ArcSech[c*x]/(I*Sqrt[d]*Sqrt[e] + e*x)) + (I*(Log[x]/Sqrt[e] - Log[1 + Sqrt
[(1 - c*x)/(1 + c*x)] + c*x*Sqrt[(1 - c*x)/(1 + c*x)]]/Sqrt[e] + Log[((2*I)*Sqrt[e]*(Sqrt[d]*Sqrt[(1 - c*x)/(1
+ c*x)]*(1 + c*x) + (Sqrt[d]*Sqrt[e] + I*c^2*d*x)/Sqrt[c^2*d + e]))/(I*Sqrt[d] + Sqrt[e]*x)]/Sqrt[c^2*d + e])
)/Sqrt[d]))/e^(5/2) + (((7*I)/16)*Sqrt[d]*(-(ArcSech[c*x]/((-I)*Sqrt[d]*Sqrt[e] + e*x)) - (I*(Log[x]/Sqrt[e] -
Log[1 + Sqrt[(1 - c*x)/(1 + c*x)] + c*x*Sqrt[(1 - c*x)/(1 + c*x)]]/Sqrt[e] + Log[(2*Sqrt[e]*(I*Sqrt[d]*Sqrt[(
1 - c*x)/(1 + c*x)]*(1 + c*x) + (I*Sqrt[d]*Sqrt[e] + c^2*d*x)/Sqrt[c^2*d + e]))/((-I)*Sqrt[d] + Sqrt[e]*x)]/Sq
rt[c^2*d + e]))/Sqrt[d]))/e^(5/2) + (PolyLog[2, -E^(-2*ArcSech[c*x])] - 2*((-4*I)*ArcSin[Sqrt[1 + (I*Sqrt[e])/
(c*Sqrt[d])]/Sqrt[2]]*ArcTanh[((I*c*Sqrt[d] + Sqrt[e])*Tanh[ArcSech[c*x]/2])/Sqrt[c^2*d + e]] + ArcSech[c*x]*L
og[1 + E^(-2*ArcSech[c*x])] - ArcSech[c*x]*Log[1 + (I*(Sqrt[e] - Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])]
+ (2*I)*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(Sqrt[e] - Sqrt[c^2*d + e]))/(c*Sqrt[d]*
E^ArcSech[c*x])] - ArcSech[c*x]*Log[1 + (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] - (2*I)*Ar
cSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c
*x])] + PolyLog[2, (I*(-Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + PolyLog[2, ((-I)*(Sqrt[e] +
Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])]))/(4*e^3) - (-PolyLog[2, -E^(-2*ArcSech[c*x])] + 2*((-4*I)*ArcSi
n[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTanh[(((-I)*c*Sqrt[d] + Sqrt[e])*Tanh[ArcSech[c*x]/2])/Sqrt[c^
2*d + e]] + ArcSech[c*x]*Log[1 + E^(-2*ArcSech[c*x])] - ArcSech[c*x]*Log[1 + (I*(-Sqrt[e] + Sqrt[c^2*d + e]))/
(c*Sqrt[d]*E^ArcSech[c*x])] + (2*I)*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(-Sqrt[e] + S
qrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] - ArcSech[c*x]*Log[1 - (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]
*E^ArcSech[c*x])] - (2*I)*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 - (I*(Sqrt[e] + Sqrt[c^2*d +
e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + PolyLog[2, (I*(Sqrt[e] - Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + P
olyLog[2, (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])]))/(4*e^3))

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Maple [C]  time = 0.846, size = 1779, normalized size = 2.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(a+b*arcsech(c*x))/(e*x^2+d)^3,x)

[Out]

-c^2*b/e^3/(c^2*d+e)*d*arcsech(c*x)*ln(1-I*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2)))+1/4*c^4*b/e/(c^2*d+e)/(c^
2*e*x^2+c^2*d)^2*x^2*d-5/8*c^2*b*(e*(c^2*d+e))^(1/2)/e^3/(c^2*d+e)^2*d*arctanh(1/4*(2*c^2*d*(1/c/x+(-1+1/c/x)^
(1/2)*(1+1/c/x)^(1/2))^2+2*c^2*d+4*e)/(c^2*d*e+e^2)^(1/2))-c^2*b/e^3/(c^2*d+e)*d*arcsech(c*x)*ln(1+I*(1/c/x+(-
1+1/c/x)^(1/2)*(1+1/c/x)^(1/2)))-1/2*c^4*b/e/(c^2*d+e)/(c^2*e*x^2+c^2*d)^2*arcsech(c*x)*d*x^2-3/4*c^6*b/e/(c^2
*d+e)/(c^2*e*x^2+c^2*d)^2*arcsech(c*x)*d*x^4-1/2*c^6*b/e^2/(c^2*d+e)/(c^2*e*x^2+c^2*d)^2*arcsech(c*x)*d^2*x^2-
1/8*c^5*b/e^2/(c^2*d+e)/(c^2*e*x^2+c^2*d)^2*(-(c*x-1)/c/x)^(1/2)*((c*x+1)/c/x)^(1/2)*x*d^2-b/e^2/(c^2*d+e)*dil
og(1+I*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2)))-b/e^2/(c^2*d+e)*dilog(1-I*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(
1/2)))+1/2*a/e^3*ln(c^2*e*x^2+c^2*d)+1/8*c^4*b/e^2/(c^2*d+e)/(c^2*e*x^2+c^2*d)^2*d^2-c^2*b/e^3/(c^2*d+e)*d*dil
og(1+I*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2)))-c^2*b/e^3/(c^2*d+e)*d*dilog(1-I*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/
c/x)^(1/2)))+1/4*c^4*b/e^3/(c^2*d+e)*d^2*sum((_R1^2+1)/(_R1^2*c^2*d+c^2*d+2*e)*(arcsech(c*x)*ln((_R1-1/c/x-(-1
+1/c/x)^(1/2)*(1+1/c/x)^(1/2))/_R1)+dilog((_R1-1/c/x-(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))/_R1)),_R1=RootOf(c^2*d*
_Z^4+(2*c^2*d+4*e)*_Z^2+c^2*d))+1/4*c^2*b/e^2/(c^2*d+e)*d*sum((_R1^2+1)/(_R1^2*c^2*d+c^2*d+2*e)*(arcsech(c*x)*
ln((_R1-1/c/x-(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))/_R1)+dilog((_R1-1/c/x-(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))/_R1)),
_R1=RootOf(c^2*d*_Z^4+(2*c^2*d+4*e)*_Z^2+c^2*d))+1/4*c^2*b/e^3/(c^2*d+e)*d*sum((_R1^2*c^2*d+c^2*d+4*e)/(_R1^2*
c^2*d+c^2*d+2*e)*(arcsech(c*x)*ln((_R1-1/c/x-(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))/_R1)+dilog((_R1-1/c/x-(-1+1/c/x
)^(1/2)*(1+1/c/x)^(1/2))/_R1)),_R1=RootOf(c^2*d*_Z^4+(2*c^2*d+4*e)*_Z^2+c^2*d))-3/4*c^4*b/(c^2*d+e)/(c^2*e*x^2
+c^2*d)^2*arcsech(c*x)*x^4+1/4*b/e^2/(c^2*d+e)*sum((_R1^2*c^2*d+c^2*d+4*e)/(_R1^2*c^2*d+c^2*d+2*e)*(arcsech(c*
x)*ln((_R1-1/c/x-(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))/_R1)+dilog((_R1-1/c/x-(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))/_R1
)),_R1=RootOf(c^2*d*_Z^4+(2*c^2*d+4*e)*_Z^2+c^2*d))-1/8*c^5*b/e/(c^2*d+e)/(c^2*e*x^2+c^2*d)^2*(-(c*x-1)/c/x)^(
1/2)*((c*x+1)/c/x)^(1/2)*x^3*d-1/4*c^4*a*d^2/e^3/(c^2*e*x^2+c^2*d)^2-3/4*b*(e*(c^2*d+e))^(1/2)/e^2/(c^2*d+e)^2
*arctanh(1/4*(2*c^2*d*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))^2+2*c^2*d+4*e)/(c^2*d*e+e^2)^(1/2))-b/e^2/(c^2*
d+e)*arcsech(c*x)*ln(1+I*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2)))-b/e^2/(c^2*d+e)*arcsech(c*x)*ln(1-I*(1/c/x+
(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2)))+c^2*a/e^3*d/(c^2*e*x^2+c^2*d)+1/8*c^4*b/(c^2*d+e)/(c^2*e*x^2+c^2*d)^2*x^4

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{4} \, a{\left (\frac{4 \, d e x^{2} + 3 \, d^{2}}{e^{5} x^{4} + 2 \, d e^{4} x^{2} + d^{2} e^{3}} + \frac{2 \, \log \left (e x^{2} + d\right )}{e^{3}}\right )} + b \int \frac{x^{5} \log \left (\sqrt{\frac{1}{c x} + 1} \sqrt{\frac{1}{c x} - 1} + \frac{1}{c x}\right )}{e^{3} x^{6} + 3 \, d e^{2} x^{4} + 3 \, d^{2} e x^{2} + d^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(a+b*arcsech(c*x))/(e*x^2+d)^3,x, algorithm="maxima")

[Out]

1/4*a*((4*d*e*x^2 + 3*d^2)/(e^5*x^4 + 2*d*e^4*x^2 + d^2*e^3) + 2*log(e*x^2 + d)/e^3) + b*integrate(x^5*log(sqr
t(1/(c*x) + 1)*sqrt(1/(c*x) - 1) + 1/(c*x))/(e^3*x^6 + 3*d*e^2*x^4 + 3*d^2*e*x^2 + d^3), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b x^{5} \operatorname{arsech}\left (c x\right ) + a x^{5}}{e^{3} x^{6} + 3 \, d e^{2} x^{4} + 3 \, d^{2} e x^{2} + d^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(a+b*arcsech(c*x))/(e*x^2+d)^3,x, algorithm="fricas")

[Out]

integral((b*x^5*arcsech(c*x) + a*x^5)/(e^3*x^6 + 3*d*e^2*x^4 + 3*d^2*e*x^2 + d^3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*(a+b*asech(c*x))/(e*x**2+d)**3,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \operatorname{arsech}\left (c x\right ) + a\right )} x^{5}}{{\left (e x^{2} + d\right )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(a+b*arcsech(c*x))/(e*x^2+d)^3,x, algorithm="giac")

[Out]

integrate((b*arcsech(c*x) + a)*x^5/(e*x^2 + d)^3, x)