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

Optimal. Leaf size=217 \[ -\frac{a+b \text{sech}^{-1}(c x)}{4 e \left (d+e x^2\right )^2}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )}{4 d^2 e}-\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left (3 c^2 d+2 e\right ) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{1-c^2 x^2}}{\sqrt{c^2 d+e}}\right )}{8 d^2 \sqrt{e} \left (c^2 d+e\right )^{3/2}}-\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2}}{8 d \left (c^2 d+e\right ) \left (d+e x^2\right )} \]

[Out]

-(b*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(8*d*(c^2*d + e)*(d + e*x^2)) - (a + b*ArcSech[c*x])
/(4*e*(d + e*x^2)^2) + (b*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*ArcTanh[Sqrt[1 - c^2*x^2]])/(4*d^2*e) - (b*(3*c^2
*d + 2*e)*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*ArcTanh[(Sqrt[e]*Sqrt[1 - c^2*x^2])/Sqrt[c^2*d + e]])/(8*d^2*Sqrt
[e]*(c^2*d + e)^(3/2))

________________________________________________________________________________________

Rubi [A]  time = 0.292666, antiderivative size = 217, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.368, Rules used = {6299, 517, 446, 103, 156, 63, 208} \[ -\frac{a+b \text{sech}^{-1}(c x)}{4 e \left (d+e x^2\right )^2}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )}{4 d^2 e}-\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left (3 c^2 d+2 e\right ) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{1-c^2 x^2}}{\sqrt{c^2 d+e}}\right )}{8 d^2 \sqrt{e} \left (c^2 d+e\right )^{3/2}}-\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2}}{8 d \left (c^2 d+e\right ) \left (d+e x^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(8*d*(c^2*d + e)*(d + e*x^2)) - (a + b*ArcSech[c*x])
/(4*e*(d + e*x^2)^2) + (b*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*ArcTanh[Sqrt[1 - c^2*x^2]])/(4*d^2*e) - (b*(3*c^2
*d + 2*e)*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*ArcTanh[(Sqrt[e]*Sqrt[1 - c^2*x^2])/Sqrt[c^2*d + e]])/(8*d^2*Sqrt
[e]*(c^2*d + e)^(3/2))

Rule 6299

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

Rule 517

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

Rule 446

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

Rule 103

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] + Dist[1/((m + 1)*(b*
c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] &&
IntegerQ[m] && (IntegerQ[n] || IntegersQ[2*n, 2*p])

Rule 156

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :>
Dist[(b*g - a*h)/(b*c - a*d), Int[(e + f*x)^p/(a + b*x), x], x] - Dist[(d*g - c*h)/(b*c - a*d), Int[(e + f*x)
^p/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]

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]

Rubi steps

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

Mathematica [C]  time = 1.02215, size = 486, normalized size = 2.24 \[ \frac{1}{16} \left (-\frac{4 a}{e \left (d+e x^2\right )^2}-\frac{b \left (3 c^2 d+2 e\right ) \log \left (\frac{16 d^2 \sqrt{e} \sqrt{c^2 d+e} \left (c x \sqrt{\frac{1-c x}{c x+1}} \sqrt{c^2 d+e}+\sqrt{\frac{1-c x}{c x+1}} \sqrt{c^2 d+e}-i c^2 \sqrt{d} x+\sqrt{e}\right )}{b \left (3 c^2 d+2 e\right ) \left (\sqrt{e} x-i \sqrt{d}\right )}\right )}{d^2 \sqrt{e} \left (c^2 d+e\right )^{3/2}}-\frac{b \left (3 c^2 d+2 e\right ) \log \left (\frac{16 d^2 \sqrt{e} \sqrt{c^2 d+e} \left (c x \sqrt{\frac{1-c x}{c x+1}} \sqrt{c^2 d+e}+\sqrt{\frac{1-c x}{c x+1}} \sqrt{c^2 d+e}+i c^2 \sqrt{d} x+\sqrt{e}\right )}{b \left (3 c^2 d+2 e\right ) \left (\sqrt{e} x+i \sqrt{d}\right )}\right )}{d^2 \sqrt{e} \left (c^2 d+e\right )^{3/2}}-\frac{2 \sqrt{\frac{1-c x}{c x+1}} (b c x+b)}{d \left (c^2 d+e\right ) \left (d+e x^2\right )}+\frac{4 b \log \left (c x \sqrt{\frac{1-c x}{c x+1}}+\sqrt{\frac{1-c x}{c x+1}}+1\right )}{d^2 e}-\frac{4 b \text{sech}^{-1}(c x)}{e \left (d+e x^2\right )^2}-\frac{4 b \log (x)}{d^2 e}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-4*a)/(e*(d + e*x^2)^2) - (2*Sqrt[(1 - c*x)/(1 + c*x)]*(b + b*c*x))/(d*(c^2*d + e)*(d + e*x^2)) - (4*b*ArcSe
ch[c*x])/(e*(d + e*x^2)^2) - (4*b*Log[x])/(d^2*e) + (4*b*Log[1 + Sqrt[(1 - c*x)/(1 + c*x)] + c*x*Sqrt[(1 - c*x
)/(1 + c*x)]])/(d^2*e) - (b*(3*c^2*d + 2*e)*Log[(16*d^2*Sqrt[e]*Sqrt[c^2*d + e]*(Sqrt[e] - I*c^2*Sqrt[d]*x + S
qrt[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)]))/(b*(3*c^2*d + 2*e)*
((-I)*Sqrt[d] + Sqrt[e]*x))])/(d^2*Sqrt[e]*(c^2*d + e)^(3/2)) - (b*(3*c^2*d + 2*e)*Log[(16*d^2*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)]))/(b*(3*c^2*d + 2*e)*(I*Sqrt[d] + Sqrt[e]*x))])/(d^2*Sqrt[e]*(c^2*d + e)^(3/2)))/16

________________________________________________________________________________________

Maple [B]  time = 0.354, size = 3301, normalized size = 15.2 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*c^4*a/e/(c^2*e*x^2+c^2*d)^2-1/4*c^4*b/e/(c^2*e*x^2+c^2*d)^2*arcsech(c*x)-1/4*c^7*b*(-(c*x-1)/c/x)^(1/2)*x
^3*((c*x+1)/c/x)^(1/2)*e^3/(-c*x*e+(-c^2*d*e)^(1/2))/(c*x*e+(-c^2*d*e)^(1/2))/((-c^2*d*e)^(1/2)-e)^2/((-c^2*d*
e)^(1/2)+e)^2/(-c^2*x^2+1)^(1/2)*arctanh(1/(-c^2*x^2+1)^(1/2))-1/4*c^7*b*(-(c*x-1)/c/x)^(1/2)*x*((c*x+1)/c/x)^
(1/2)*e^2/(-c*x*e+(-c^2*d*e)^(1/2))/(c*x*e+(-c^2*d*e)^(1/2))/((-c^2*d*e)^(1/2)-e)^2/((-c^2*d*e)^(1/2)+e)^2*d/(
-c^2*x^2+1)^(1/2)*arctanh(1/(-c^2*x^2+1)^(1/2))+3/16*c^7*b*(-(c*x-1)/c/x)^(1/2)*x^3*((c*x+1)/c/x)^(1/2)*e^3/(-
c*x*e+(-c^2*d*e)^(1/2))/(c*x*e+(-c^2*d*e)^(1/2))/((c^2*d+e)/e)^(1/2)/((-c^2*d*e)^(1/2)-e)^2/((-c^2*d*e)^(1/2)+
e)^2/(-c^2*x^2+1)^(1/2)*ln(2*(-(-c^2*x^2+1)^(1/2)*((c^2*d+e)/e)^(1/2)*e+(-c^2*d*e)^(1/2)*c*x-e)/(-c*x*e+(-c^2*
d*e)^(1/2)))+3/16*c^7*b*(-(c*x-1)/c/x)^(1/2)*x*((c*x+1)/c/x)^(1/2)*e^2/(-c*x*e+(-c^2*d*e)^(1/2))/(c*x*e+(-c^2*
d*e)^(1/2))/((c^2*d+e)/e)^(1/2)/((-c^2*d*e)^(1/2)-e)^2/((-c^2*d*e)^(1/2)+e)^2*d/(-c^2*x^2+1)^(1/2)*ln(2*(-(-c^
2*x^2+1)^(1/2)*((c^2*d+e)/e)^(1/2)*e+(-c^2*d*e)^(1/2)*c*x-e)/(-c*x*e+(-c^2*d*e)^(1/2)))+3/16*c^7*b*(-(c*x-1)/c
/x)^(1/2)*x^3*((c*x+1)/c/x)^(1/2)*e^3/(-c*x*e+(-c^2*d*e)^(1/2))/(c*x*e+(-c^2*d*e)^(1/2))/((c^2*d+e)/e)^(1/2)/(
(-c^2*d*e)^(1/2)-e)^2/((-c^2*d*e)^(1/2)+e)^2/(-c^2*x^2+1)^(1/2)*ln(2*((-c^2*x^2+1)^(1/2)*((c^2*d+e)/e)^(1/2)*e
+(-c^2*d*e)^(1/2)*c*x+e)/(c*x*e+(-c^2*d*e)^(1/2)))+3/16*c^7*b*(-(c*x-1)/c/x)^(1/2)*x*((c*x+1)/c/x)^(1/2)*e^2/(
-c*x*e+(-c^2*d*e)^(1/2))/(c*x*e+(-c^2*d*e)^(1/2))/((c^2*d+e)/e)^(1/2)/((-c^2*d*e)^(1/2)-e)^2/((-c^2*d*e)^(1/2)
+e)^2*d/(-c^2*x^2+1)^(1/2)*ln(2*((-c^2*x^2+1)^(1/2)*((c^2*d+e)/e)^(1/2)*e+(-c^2*d*e)^(1/2)*c*x+e)/(c*x*e+(-c^2
*d*e)^(1/2)))+1/8*c^5*b*(-(c*x-1)/c/x)^(1/2)*x*((c*x+1)/c/x)^(1/2)*e^3/(-c*x*e+(-c^2*d*e)^(1/2))/(c*x*e+(-c^2*
d*e)^(1/2))/((-c^2*d*e)^(1/2)-e)^2/((-c^2*d*e)^(1/2)+e)^2-1/2*c^5*b*(-(c*x-1)/c/x)^(1/2)*x^3*((c*x+1)/c/x)^(1/
2)*e^4/(-c*x*e+(-c^2*d*e)^(1/2))/(c*x*e+(-c^2*d*e)^(1/2))/((-c^2*d*e)^(1/2)-e)^2/((-c^2*d*e)^(1/2)+e)^2/d/(-c^
2*x^2+1)^(1/2)*arctanh(1/(-c^2*x^2+1)^(1/2))-1/2*c^5*b*(-(c*x-1)/c/x)^(1/2)*x*((c*x+1)/c/x)^(1/2)*e^3/(-c*x*e+
(-c^2*d*e)^(1/2))/(c*x*e+(-c^2*d*e)^(1/2))/((-c^2*d*e)^(1/2)-e)^2/((-c^2*d*e)^(1/2)+e)^2/(-c^2*x^2+1)^(1/2)*ar
ctanh(1/(-c^2*x^2+1)^(1/2))+5/16*c^5*b*(-(c*x-1)/c/x)^(1/2)*x^3*((c*x+1)/c/x)^(1/2)*e^4/(-c*x*e+(-c^2*d*e)^(1/
2))/(c*x*e+(-c^2*d*e)^(1/2))/((c^2*d+e)/e)^(1/2)/((-c^2*d*e)^(1/2)-e)^2/((-c^2*d*e)^(1/2)+e)^2/d/(-c^2*x^2+1)^
(1/2)*ln(2*(-(-c^2*x^2+1)^(1/2)*((c^2*d+e)/e)^(1/2)*e+(-c^2*d*e)^(1/2)*c*x-e)/(-c*x*e+(-c^2*d*e)^(1/2)))+5/16*
c^5*b*(-(c*x-1)/c/x)^(1/2)*x*((c*x+1)/c/x)^(1/2)*e^3/(-c*x*e+(-c^2*d*e)^(1/2))/(c*x*e+(-c^2*d*e)^(1/2))/((c^2*
d+e)/e)^(1/2)/((-c^2*d*e)^(1/2)-e)^2/((-c^2*d*e)^(1/2)+e)^2/(-c^2*x^2+1)^(1/2)*ln(2*(-(-c^2*x^2+1)^(1/2)*((c^2
*d+e)/e)^(1/2)*e+(-c^2*d*e)^(1/2)*c*x-e)/(-c*x*e+(-c^2*d*e)^(1/2)))+5/16*c^5*b*(-(c*x-1)/c/x)^(1/2)*x^3*((c*x+
1)/c/x)^(1/2)*e^4/(-c*x*e+(-c^2*d*e)^(1/2))/(c*x*e+(-c^2*d*e)^(1/2))/((c^2*d+e)/e)^(1/2)/((-c^2*d*e)^(1/2)-e)^
2/((-c^2*d*e)^(1/2)+e)^2/d/(-c^2*x^2+1)^(1/2)*ln(2*((-c^2*x^2+1)^(1/2)*((c^2*d+e)/e)^(1/2)*e+(-c^2*d*e)^(1/2)*
c*x+e)/(c*x*e+(-c^2*d*e)^(1/2)))+5/16*c^5*b*(-(c*x-1)/c/x)^(1/2)*x*((c*x+1)/c/x)^(1/2)*e^3/(-c*x*e+(-c^2*d*e)^
(1/2))/(c*x*e+(-c^2*d*e)^(1/2))/((c^2*d+e)/e)^(1/2)/((-c^2*d*e)^(1/2)-e)^2/((-c^2*d*e)^(1/2)+e)^2/(-c^2*x^2+1)
^(1/2)*ln(2*((-c^2*x^2+1)^(1/2)*((c^2*d+e)/e)^(1/2)*e+(-c^2*d*e)^(1/2)*c*x+e)/(c*x*e+(-c^2*d*e)^(1/2)))+1/8*c^
3*b*e^4*(-(c*x-1)/c/x)^(1/2)*x*((c*x+1)/c/x)^(1/2)/(c*x*e+(-c^2*d*e)^(1/2))/(-c*x*e+(-c^2*d*e)^(1/2))/((-c^2*d
*e)^(1/2)-e)^2/((-c^2*d*e)^(1/2)+e)^2/d-1/4*c^3*b*e^5*(-(c*x-1)/c/x)^(1/2)*x^3*((c*x+1)/c/x)^(1/2)/(c*x*e+(-c^
2*d*e)^(1/2))/(-c*x*e+(-c^2*d*e)^(1/2))/((-c^2*d*e)^(1/2)-e)^2/((-c^2*d*e)^(1/2)+e)^2/d^2/(-c^2*x^2+1)^(1/2)*a
rctanh(1/(-c^2*x^2+1)^(1/2))-1/4*c^3*b*e^4*(-(c*x-1)/c/x)^(1/2)*x*((c*x+1)/c/x)^(1/2)/(c*x*e+(-c^2*d*e)^(1/2))
/(-c*x*e+(-c^2*d*e)^(1/2))/((-c^2*d*e)^(1/2)-e)^2/((-c^2*d*e)^(1/2)+e)^2/d/(-c^2*x^2+1)^(1/2)*arctanh(1/(-c^2*
x^2+1)^(1/2))+1/8*c^3*b*e^5*(-(c*x-1)/c/x)^(1/2)*x^3*((c*x+1)/c/x)^(1/2)/(c*x*e+(-c^2*d*e)^(1/2))/((c^2*d+e)/e
)^(1/2)/(-c*x*e+(-c^2*d*e)^(1/2))/((-c^2*d*e)^(1/2)-e)^2/((-c^2*d*e)^(1/2)+e)^2/d^2/(-c^2*x^2+1)^(1/2)*ln(2*(-
(-c^2*x^2+1)^(1/2)*((c^2*d+e)/e)^(1/2)*e+(-c^2*d*e)^(1/2)*c*x-e)/(-c*x*e+(-c^2*d*e)^(1/2)))+1/8*c^3*b*e^4*(-(c
*x-1)/c/x)^(1/2)*x*((c*x+1)/c/x)^(1/2)/(c*x*e+(-c^2*d*e)^(1/2))/((c^2*d+e)/e)^(1/2)/(-c*x*e+(-c^2*d*e)^(1/2))/
((-c^2*d*e)^(1/2)-e)^2/((-c^2*d*e)^(1/2)+e)^2/d/(-c^2*x^2+1)^(1/2)*ln(2*(-(-c^2*x^2+1)^(1/2)*((c^2*d+e)/e)^(1/
2)*e+(-c^2*d*e)^(1/2)*c*x-e)/(-c*x*e+(-c^2*d*e)^(1/2)))+1/8*c^3*b*e^5*(-(c*x-1)/c/x)^(1/2)*x^3*((c*x+1)/c/x)^(
1/2)/(c*x*e+(-c^2*d*e)^(1/2))/((c^2*d+e)/e)^(1/2)/(-c*x*e+(-c^2*d*e)^(1/2))/((-c^2*d*e)^(1/2)-e)^2/((-c^2*d*e)
^(1/2)+e)^2/d^2/(-c^2*x^2+1)^(1/2)*ln(2*((-c^2*x^2+1)^(1/2)*((c^2*d+e)/e)^(1/2)*e+(-c^2*d*e)^(1/2)*c*x+e)/(c*x
*e+(-c^2*d*e)^(1/2)))+1/8*c^3*b*e^4*(-(c*x-1)/c/x)^(1/2)*x*((c*x+1)/c/x)^(1/2)/(c*x*e+(-c^2*d*e)^(1/2))/((c^2*
d+e)/e)^(1/2)/(-c*x*e+(-c^2*d*e)^(1/2))/((-c^2*d*e)^(1/2)-e)^2/((-c^2*d*e)^(1/2)+e)^2/d/(-c^2*x^2+1)^(1/2)*ln(
2*((-c^2*x^2+1)^(1/2)*((c^2*d+e)/e)^(1/2)*e+(-c^2*d*e)^(1/2)*c*x+e)/(c*x*e+(-c^2*d*e)^(1/2)))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError

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Fricas [B]  time = 2.89457, size = 2538, normalized size = 11.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/16*(4*a*c^4*d^4 + 2*(4*a + b)*c^2*d^3*e + 2*(2*a + b)*d^2*e^2 + 2*(b*c^2*d*e^3 + b*e^4)*x^4 + 4*(b*c^2*d^2
*e^2 + b*d*e^3)*x^2 - (3*b*c^2*d^3 + (3*b*c^2*d*e^2 + 2*b*e^3)*x^4 + 2*b*d^2*e + 2*(3*b*c^2*d^2*e + 2*b*d*e^2)
*x^2)*sqrt(c^2*d*e + e^2)*log((c^4*d^2 + 4*c^2*d*e - (c^4*d*e + 2*c^2*e^2)*x^2 + 4*(c^3*d*e + c*e^2)*x*sqrt(-(
c^2*x^2 - 1)/(c^2*x^2)) + 4*e^2 + 2*(c^2*e*x^2 - c^2*d - (c^3*d + 2*c*e)*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) - 2*
e)*sqrt(c^2*d*e + e^2))/(e*x^2 + d)) + 4*(b*c^4*d^4 + 2*b*c^2*d^3*e + b*d^2*e^2 + (b*c^4*d^2*e^2 + 2*b*c^2*d*e
^3 + b*e^4)*x^4 + 2*(b*c^4*d^3*e + 2*b*c^2*d^2*e^2 + b*d*e^3)*x^2)*log((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) - 1
)/x) + 4*(b*c^4*d^4 + 2*b*c^2*d^3*e + b*d^2*e^2)*log((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) + 1)/(c*x)) + 2*((b*c
^3*d^2*e^2 + b*c*d*e^3)*x^3 + (b*c^3*d^3*e + b*c*d^2*e^2)*x)*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)))/(c^4*d^6*e + 2*c^
2*d^5*e^2 + d^4*e^3 + (c^4*d^4*e^3 + 2*c^2*d^3*e^4 + d^2*e^5)*x^4 + 2*(c^4*d^5*e^2 + 2*c^2*d^4*e^3 + d^3*e^4)*
x^2), -1/8*(2*a*c^4*d^4 + (4*a + b)*c^2*d^3*e + (2*a + b)*d^2*e^2 + (b*c^2*d*e^3 + b*e^4)*x^4 + 2*(b*c^2*d^2*e
^2 + b*d*e^3)*x^2 + (3*b*c^2*d^3 + (3*b*c^2*d*e^2 + 2*b*e^3)*x^4 + 2*b*d^2*e + 2*(3*b*c^2*d^2*e + 2*b*d*e^2)*x
^2)*sqrt(-c^2*d*e - e^2)*arctan((sqrt(-c^2*d*e - e^2)*c*d*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) - sqrt(-c^2*d*e - e
^2)*(e*x^2 + d))/((c^2*d*e + e^2)*x^2)) + 2*(b*c^4*d^4 + 2*b*c^2*d^3*e + b*d^2*e^2 + (b*c^4*d^2*e^2 + 2*b*c^2*
d*e^3 + b*e^4)*x^4 + 2*(b*c^4*d^3*e + 2*b*c^2*d^2*e^2 + b*d*e^3)*x^2)*log((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2))
- 1)/x) + 2*(b*c^4*d^4 + 2*b*c^2*d^3*e + b*d^2*e^2)*log((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) + 1)/(c*x)) + ((b*
c^3*d^2*e^2 + b*c*d*e^3)*x^3 + (b*c^3*d^3*e + b*c*d^2*e^2)*x)*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)))/(c^4*d^6*e + 2*c
^2*d^5*e^2 + d^4*e^3 + (c^4*d^4*e^3 + 2*c^2*d^3*e^4 + d^2*e^5)*x^4 + 2*(c^4*d^5*e^2 + 2*c^2*d^4*e^3 + d^3*e^4)
*x^2)]

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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*(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}{{\left (e x^{2} + d\right )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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