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

Optimal. Leaf size=519 \[ -\frac{b \sqrt{-d} \text{PolyLog}\left (2,-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{2 e^{3/2}}+\frac{b \sqrt{-d} \text{PolyLog}\left (2,\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{2 e^{3/2}}-\frac{b \sqrt{-d} \text{PolyLog}\left (2,-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right )}{2 e^{3/2}}+\frac{b \sqrt{-d} \text{PolyLog}\left (2,\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right )}{2 e^{3/2}}+\frac{\sqrt{-d} \left (a+b \text{sech}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{2 e^{3/2}}-\frac{\sqrt{-d} \left (a+b \text{sech}^{-1}(c x)\right ) \log \left (\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}+1\right )}{2 e^{3/2}}+\frac{\sqrt{-d} \left (a+b \text{sech}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right )}{2 e^{3/2}}-\frac{\sqrt{-d} \left (a+b \text{sech}^{-1}(c x)\right ) \log \left (\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}+1\right )}{2 e^{3/2}}+\frac{x \left (a+b \text{sech}^{-1}(c x)\right )}{e}-\frac{b \tan ^{-1}\left (\sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}\right )}{c e} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 1.2733, antiderivative size = 519, normalized size of antiderivative = 1., number of steps used = 24, number of rules used = 11, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.524, Rules used = {6303, 5792, 5662, 92, 205, 5707, 5800, 5562, 2190, 2279, 2391} \[ -\frac{b \sqrt{-d} \text{PolyLog}\left (2,-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{2 e^{3/2}}+\frac{b \sqrt{-d} \text{PolyLog}\left (2,\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{2 e^{3/2}}-\frac{b \sqrt{-d} \text{PolyLog}\left (2,-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right )}{2 e^{3/2}}+\frac{b \sqrt{-d} \text{PolyLog}\left (2,\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right )}{2 e^{3/2}}+\frac{\sqrt{-d} \left (a+b \text{sech}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{2 e^{3/2}}-\frac{\sqrt{-d} \left (a+b \text{sech}^{-1}(c x)\right ) \log \left (\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}+1\right )}{2 e^{3/2}}+\frac{\sqrt{-d} \left (a+b \text{sech}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right )}{2 e^{3/2}}-\frac{\sqrt{-d} \left (a+b \text{sech}^{-1}(c x)\right ) \log \left (\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}+1\right )}{2 e^{3/2}}+\frac{x \left (a+b \text{sech}^{-1}(c x)\right )}{e}-\frac{b \tan ^{-1}\left (\sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}\right )}{c e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 5662

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

Rule 92

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))), x_Symbol] :> Dist[b*f, Subst[I
nt[1/(d*(b*e - a*f)^2 + b*f^2*x^2), x], x, Sqrt[a + b*x]*Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] &&
EqQ[2*b*d*e - f*(b*c + a*d), 0]

Rule 205

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

Rule 5707

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a
+ b*ArcCosh[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
] && (p > 0 || IGtQ[n, 0])

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]

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

Mathematica [C]  time = 1.4199, size = 921, normalized size = 1.77 \[ \frac{4 a c \sqrt{e} x-4 a c \sqrt{d} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )+b \left (4 \sqrt{e} \left (c x \text{sech}^{-1}(c x)-2 \tan ^{-1}\left (\tanh \left (\frac{1}{2} \text{sech}^{-1}(c x)\right )\right )\right )-2 i c \sqrt{d} \left (-4 i \sin ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right ) \tanh ^{-1}\left (\frac{\left (i \sqrt{d} c+\sqrt{e}\right ) \tanh \left (\frac{1}{2} \text{sech}^{-1}(c x)\right )}{\sqrt{d c^2+e}}\right )+\text{sech}^{-1}(c x) \log \left (1+e^{-2 \text{sech}^{-1}(c x)}\right )-\text{sech}^{-1}(c x) \log \left (\frac{i e^{-\text{sech}^{-1}(c x)} \left (\sqrt{e}-\sqrt{d c^2+e}\right )}{c \sqrt{d}}+1\right )+2 i \sin ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right ) \log \left (\frac{i e^{-\text{sech}^{-1}(c x)} \left (\sqrt{e}-\sqrt{d c^2+e}\right )}{c \sqrt{d}}+1\right )-\text{sech}^{-1}(c x) \log \left (\frac{i e^{-\text{sech}^{-1}(c x)} \left (\sqrt{e}+\sqrt{d c^2+e}\right )}{c \sqrt{d}}+1\right )-2 i \sin ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right ) \log \left (\frac{i e^{-\text{sech}^{-1}(c x)} \left (\sqrt{e}+\sqrt{d c^2+e}\right )}{c \sqrt{d}}+1\right )+\text{PolyLog}\left (2,\frac{i \left (\sqrt{d c^2+e}-\sqrt{e}\right ) e^{-\text{sech}^{-1}(c x)}}{c \sqrt{d}}\right )+\text{PolyLog}\left (2,-\frac{i \left (\sqrt{e}+\sqrt{d c^2+e}\right ) e^{-\text{sech}^{-1}(c x)}}{c \sqrt{d}}\right )\right )+2 i c \sqrt{d} \left (-4 i \sin ^{-1}\left (\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right ) \tanh ^{-1}\left (\frac{\left (\sqrt{e}-i c \sqrt{d}\right ) \tanh \left (\frac{1}{2} \text{sech}^{-1}(c x)\right )}{\sqrt{d c^2+e}}\right )+\text{sech}^{-1}(c x) \log \left (1+e^{-2 \text{sech}^{-1}(c x)}\right )-\text{sech}^{-1}(c x) \log \left (\frac{i e^{-\text{sech}^{-1}(c x)} \left (\sqrt{d c^2+e}-\sqrt{e}\right )}{c \sqrt{d}}+1\right )+2 i \sin ^{-1}\left (\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right ) \log \left (\frac{i e^{-\text{sech}^{-1}(c x)} \left (\sqrt{d c^2+e}-\sqrt{e}\right )}{c \sqrt{d}}+1\right )-\text{sech}^{-1}(c x) \log \left (1-\frac{i \left (\sqrt{e}+\sqrt{d c^2+e}\right ) e^{-\text{sech}^{-1}(c x)}}{c \sqrt{d}}\right )-2 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^{-\text{sech}^{-1}(c x)}}{c \sqrt{d}}\right )+\text{PolyLog}\left (2,\frac{i \left (\sqrt{e}-\sqrt{d c^2+e}\right ) e^{-\text{sech}^{-1}(c x)}}{c \sqrt{d}}\right )+\text{PolyLog}\left (2,\frac{i \left (\sqrt{e}+\sqrt{d c^2+e}\right ) e^{-\text{sech}^{-1}(c x)}}{c \sqrt{d}}\right )\right )\right )}{4 c e^{3/2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(4*a*c*Sqrt[e]*x - 4*a*c*Sqrt[d]*ArcTan[(Sqrt[e]*x)/Sqrt[d]] + b*(4*Sqrt[e]*(c*x*ArcSech[c*x] - 2*ArcTan[Tanh[
ArcSech[c*x]/2]]) - (2*I)*c*Sqrt[d]*((-4*I)*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTanh[((I*c*Sq
rt[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] - 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)*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])] + PolyLog[2, ((-I)*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech
[c*x])]) + (2*I)*c*Sqrt[d]*((-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]*Log[1 + E^(-2*ArcSech[c*x])] - ArcSech[c*x]*L
og[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*Sqr
t[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)*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqr
t[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*c*e^(3/2))

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Maple [C]  time = 2.277, size = 411, normalized size = 0.8 \begin{align*}{\frac{ax}{e}}-{\frac{ad}{e}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}+{\frac{bx{\rm arcsech} \left (cx\right )}{e}}+{\frac{bcd}{8\,{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 ({\rm arcsech} \left (cx\right )\ln \left ({\frac{1}{{\it \_R1}} \left ({\it \_R1}-{\frac{1}{cx}}-\sqrt{-1+{\frac{1}{cx}}}\sqrt{1+{\frac{1}{cx}}} \right ) } \right ) +{\it dilog} \left ({\frac{1}{{\it \_R1}} \left ({\it \_R1}-{\frac{1}{cx}}-\sqrt{-1+{\frac{1}{cx}}}\sqrt{1+{\frac{1}{cx}}} \right ) } \right ) \right ) }}-2\,{\frac{b}{ce}\arctan \left ({\frac{1}{cx}}+\sqrt{-1+{\frac{1}{cx}}}\sqrt{1+{\frac{1}{cx}}} \right ) }-{\frac{bcd}{8\,{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 ({\rm arcsech} \left (cx\right )\ln \left ({\frac{1}{{\it \_R1}} \left ({\it \_R1}-{\frac{1}{cx}}-\sqrt{-1+{\frac{1}{cx}}}\sqrt{1+{\frac{1}{cx}}} \right ) } \right ) +{\it dilog} \left ({\frac{1}{{\it \_R1}} \left ({\it \_R1}-{\frac{1}{cx}}-\sqrt{-1+{\frac{1}{cx}}}\sqrt{1+{\frac{1}{cx}}} \right ) } \right ) \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a/e*x-a*d/e/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))+b*arcsech(c*x)/e*x+1/8*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)*(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))-2/c*b/e*arctan
(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))-1/8*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)
*(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))

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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*arcsech(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{arsech}\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*arcsech(c*x))/(e*x^2+d),x, algorithm="fricas")

[Out]

integral((b*x^2*arcsech(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{asech}{\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*asech(c*x))/(e*x**2+d),x)

[Out]

Integral(x**2*(a + b*asech(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{arsech}\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*arcsech(c*x))/(e*x^2+d),x, algorithm="giac")

[Out]

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