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

Optimal. Leaf size=173 \[ \frac{x^4 \left (a+b \text{sech}^{-1}(c x)\right )}{4 d \left (d+e x^2\right )^2}-\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left (c^2 d+2 e\right ) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{1-c^2 x^2}}{\sqrt{c^2 d+e}}\right )}{8 d e^{3/2} \left (c^2 d+e\right )^{3/2}}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2}}{8 e \left (c^2 d+e\right ) \left (d+e x^2\right )} \]

[Out]

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

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Rubi [A]  time = 0.189258, antiderivative size = 173, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {264, 6301, 12, 446, 78, 63, 208} \[ \frac{x^4 \left (a+b \text{sech}^{-1}(c x)\right )}{4 d \left (d+e x^2\right )^2}-\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left (c^2 d+2 e\right ) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{1-c^2 x^2}}{\sqrt{c^2 d+e}}\right )}{8 d e^{3/2} \left (c^2 d+e\right )^{3/2}}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2}}{8 e \left (c^2 d+e\right ) \left (d+e x^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[(x^3*(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*e*(c^2*d + e)*(d + e*x^2)) + (x^4*(a + b*ArcSech[c
*x]))/(4*d*(d + e*x^2)^2) - (b*(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*e^(3/2)*(c^2*d + e)^(3/2))

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rule 6301

Int[((a_.) + ArcSech[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_.)*((d_.) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u
= IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcSech[c*x], u, x] + Dist[b*Sqrt[1 + c*x]*Sqrt[1/(1 + c*x)],
Int[SimplifyIntegrand[u/(x*Sqrt[1 - c*x]*Sqrt[1 + c*x]), x], x], x]] /; FreeQ[{a, b, c, d, e, f, m, p}, x] &&
((IGtQ[p, 0] &&  !(ILtQ[(m - 1)/2, 0] && GtQ[m + 2*p + 3, 0])) || (IGtQ[(m + 1)/2, 0] &&  !(ILtQ[p, 0] && GtQ
[m + 2*p + 3, 0])) || (ILtQ[(m + 2*p + 1)/2, 0] &&  !ILtQ[(m - 1)/2, 0]))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 78

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

Rule 63

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

Rule 208

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

Rubi steps

\begin{align*} \int \frac{x^3 \left (a+b \text{sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx &=\frac{x^4 \left (a+b \text{sech}^{-1}(c x)\right )}{4 d \left (d+e x^2\right )^2}+\left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{x^3}{4 d \sqrt{1-c^2 x^2} \left (d+e x^2\right )^2} \, dx\\ &=\frac{x^4 \left (a+b \text{sech}^{-1}(c x)\right )}{4 d \left (d+e x^2\right )^2}+\frac{\left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{x^3}{\sqrt{1-c^2 x^2} \left (d+e x^2\right )^2} \, dx}{4 d}\\ &=\frac{x^4 \left (a+b \text{sech}^{-1}(c x)\right )}{4 d \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{x}{\sqrt{1-c^2 x} (d+e x)^2} \, dx,x,x^2\right )}{8 d}\\ &=\frac{b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{8 e \left (c^2 d+e\right ) \left (d+e x^2\right )}+\frac{x^4 \left (a+b \text{sech}^{-1}(c x)\right )}{4 d \left (d+e x^2\right )^2}+\frac{\left (b \left (c^2 d+2 e\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 )}{16 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 e \left (c^2 d+e\right ) \left (d+e x^2\right )}+\frac{x^4 \left (a+b \text{sech}^{-1}(c x)\right )}{4 d \left (d+e x^2\right )^2}-\frac{\left (b \left (c^2 d+2 e\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 )}{8 c^2 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 e \left (c^2 d+e\right ) \left (d+e x^2\right )}+\frac{x^4 \left (a+b \text{sech}^{-1}(c x)\right )}{4 d \left (d+e x^2\right )^2}-\frac{b \left (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 e^{3/2} \left (c^2 d+e\right )^{3/2}}\\ \end{align*}

Mathematica [C]  time = 1.48594, size = 486, normalized size = 2.81 \[ -\frac{\frac{8 a}{d+e x^2}-\frac{4 a d}{\left (d+e x^2\right )^2}+\frac{b \sqrt{e} \left (c^2 d+2 e\right ) \log \left (\frac{16 d e^{3/2} \sqrt{c^2 d+e} \left (c x \sqrt{\frac{1-c x}{c x+1}} \sqrt{c^2 d+e}+\sqrt{\frac{1-c x}{c x+1}} \sqrt{c^2 d+e}-i c^2 \sqrt{d} x+\sqrt{e}\right )}{b \left (c^2 d+2 e\right ) \left (\sqrt{e} x-i \sqrt{d}\right )}\right )}{d \left (c^2 d+e\right )^{3/2}}+\frac{b \sqrt{e} \left (c^2 d+2 e\right ) \log \left (\frac{16 d e^{3/2} \sqrt{c^2 d+e} \left (c x \sqrt{\frac{1-c x}{c x+1}} \sqrt{c^2 d+e}+\sqrt{\frac{1-c x}{c x+1}} \sqrt{c^2 d+e}+i c^2 \sqrt{d} x+\sqrt{e}\right )}{b \left (c^2 d+2 e\right ) \left (\sqrt{e} x+i \sqrt{d}\right )}\right )}{d \left (c^2 d+e\right )^{3/2}}-\frac{2 e \sqrt{\frac{1-c x}{c x+1}} (b c x+b)}{\left (c^2 d+e\right ) \left (d+e x^2\right )}+\frac{4 b \text{sech}^{-1}(c x) \left (d+2 e x^2\right )}{\left (d+e x^2\right )^2}-\frac{4 b \log \left (c x \sqrt{\frac{1-c x}{c x+1}}+\sqrt{\frac{1-c x}{c x+1}}+1\right )}{d}+\frac{4 b \log (x)}{d}}{16 e^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((-4*a*d)/(d + e*x^2)^2 + (8*a)/(d + e*x^2) - (2*e*Sqrt[(1 - c*x)/(1 + c*x)]*(b + b*c*x))/((c^2*d + e)*(d + e
*x^2)) + (4*b*(d + 2*e*x^2)*ArcSech[c*x])/(d + e*x^2)^2 + (4*b*Log[x])/d - (4*b*Log[1 + Sqrt[(1 - c*x)/(1 + c*
x)] + c*x*Sqrt[(1 - c*x)/(1 + c*x)]])/d + (b*Sqrt[e]*(c^2*d + 2*e)*Log[(16*d*e^(3/2)*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*(c^2*d + 2*e)*((-I)*Sqrt[d] + Sqrt[e]*x))])/(d*(c^2*d + e)^(3/2)) + (b*Sqrt[e]*(c^2*d + 2*e)*Log[(16*d*e^
(3/2)*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*(c^2*d + 2*e)*(I*Sqrt[d] + Sqrt[e]*x))])/(d*(c^2*d + e)^(3/2)))/(16*e^2)

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Maple [B]  time = 0.362, size = 3343, normalized size = 19.3 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*c^2*a/e^2/(c^2*e*x^2+c^2*d)+1/4*c^4*a/e^2*d/(c^2*e*x^2+c^2*d)^2-1/2*c^2*b*arcsech(c*x)/e^2/(c^2*e*x^2+c^2
*d)+1/4*c^4*b*arcsech(c*x)/e^2*d/(c^2*e*x^2+c^2*d)^2-1/4*c^7*b*(-(c*x-1)/c/x)^(1/2)*x^3*((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))-1/4*c^7*b*(-(c*x-1)/c/x)^(1/2)*x*((c*x+1)/c/x)^(1/2)*e/(-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)*arcta
nh(1/(-c^2*x^2+1)^(1/2))+1/16*c^7*b*(-(c*x-1)/c/x)^(1/2)*x^3*((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)))+1/16*c^7*b
*(-(c*x-1)/c/x)^(1/2)*x*((c*x+1)/c/x)^(1/2)*e/(-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^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)))+1/16*c^7*b*(-(c*x-1)/c/x)^(1/2)*x^3*((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)))+1/16*c^7*b*(-(c*x-1)/c/x)^(1/2)*x*((c*x+1)/c/x)^(1/2)*e/(-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^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)))-1/2*c
^5*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))-1/2*c^5*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))-1/8*c^5*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+3/16*c^5*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^5*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^5*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^5*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)))-1/4*c^3*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))-1/4*c^3*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)*arctanh(1/(-c^2*x^2+1)^(1/2))-1/8*c^3*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+1/8*c^3*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)))+1/8*c^3*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*(-(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)))+1/8*c^3*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)))

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

[Out]

Exception raised: RuntimeError

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Fricas [B]  time = 2.97766, size = 2743, normalized size = 15.86 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(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*(2*a*c^4*d
^3*e + (4*a - b)*c^2*d^2*e^2 + (2*a - b)*d*e^3)*x^2 - (b*c^2*d^3 + (b*c^2*d*e^2 + 2*b*e^3)*x^4 + 2*b*d^2*e + 2
*(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 + 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)/(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^5*e^2 + 2*c^2*d^4*e^3 + d^3*e^4 + (c^4*d^3*e
^4 + 2*c^2*d^2*e^5 + d*e^6)*x^4 + 2*(c^4*d^4*e^3 + 2*c^2*d^3*e^4 + d^2*e^5)*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*(2*a*c^4*d^3*e + (4*a - b)*c^2*d^2*e^2 + (2*a
- b)*d*e^3)*x^2 + (b*c^2*d^3 + (b*c^2*d*e^2 + 2*b*e^3)*x^4 + 2*b*d^2*e + 2*(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 + 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)/(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^5*e^2 + 2*c^2*d^4*e^3 + d^3*e^4 + (c^4*d^3*e^4 + 2*c^2*d^2*e^5 + d*e^6)*x^4 +
2*(c^4*d^4*e^3 + 2*c^2*d^3*e^4 + d^2*e^5)*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**3*(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^{3}}{{\left (e x^{2} + d\right )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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