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

Optimal. Leaf size=249 \[ -\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left (c^2 d+2 e\right ) \sqrt{\frac{e x^2}{d}+1} \text{EllipticF}\left (\sin ^{-1}(c x),-\frac{e}{c^2 d}\right )}{c d^2 \sqrt{d+e x^2}}-\frac{2 e x \left (a+b \text{sech}^{-1}(c x)\right )}{d^2 \sqrt{d+e x^2}}-\frac{a+b \text{sech}^{-1}(c x)}{d x \sqrt{d+e x^2}}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \sqrt{d+e x^2}}{d^2 x}+\frac{b c \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{d+e x^2} E\left (\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right )}{d^2 \sqrt{\frac{e x^2}{d}+1}} \]

[Out]

(b*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2])/(d^2*x) - (a + b*ArcSech[c*x])/(d*x*S
qrt[d + e*x^2]) - (2*e*x*(a + b*ArcSech[c*x]))/(d^2*Sqrt[d + e*x^2]) + (b*c*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]
*Sqrt[d + e*x^2]*EllipticE[ArcSin[c*x], -(e/(c^2*d))])/(d^2*Sqrt[1 + (e*x^2)/d]) - (b*(c^2*d + 2*e)*Sqrt[(1 +
c*x)^(-1)]*Sqrt[1 + c*x]*Sqrt[1 + (e*x^2)/d]*EllipticF[ArcSin[c*x], -(e/(c^2*d))])/(c*d^2*Sqrt[d + e*x^2])

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Rubi [A]  time = 0.286656, antiderivative size = 249, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 10, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.435, Rules used = {271, 191, 6301, 12, 583, 524, 426, 424, 421, 419} \[ -\frac{2 e x \left (a+b \text{sech}^{-1}(c x)\right )}{d^2 \sqrt{d+e x^2}}-\frac{a+b \text{sech}^{-1}(c x)}{d x \sqrt{d+e x^2}}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \sqrt{d+e x^2}}{d^2 x}-\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left (c^2 d+2 e\right ) \sqrt{\frac{e x^2}{d}+1} F\left (\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right )}{c d^2 \sqrt{d+e x^2}}+\frac{b c \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{d+e x^2} E\left (\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right )}{d^2 \sqrt{\frac{e x^2}{d}+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2])/(d^2*x) - (a + b*ArcSech[c*x])/(d*x*S
qrt[d + e*x^2]) - (2*e*x*(a + b*ArcSech[c*x]))/(d^2*Sqrt[d + e*x^2]) + (b*c*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]
*Sqrt[d + e*x^2]*EllipticE[ArcSin[c*x], -(e/(c^2*d))])/(d^2*Sqrt[1 + (e*x^2)/d]) - (b*(c^2*d + 2*e)*Sqrt[(1 +
c*x)^(-1)]*Sqrt[1 + c*x]*Sqrt[1 + (e*x^2)/d]*EllipticF[ArcSin[c*x], -(e/(c^2*d))])/(c*d^2*Sqrt[d + e*x^2])

Rule 271

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

Rule 191

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

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 583

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

Rule 524

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

Rule 426

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Dist[Sqrt[a + b*x^2]/Sqrt[1 + (b*x^2)/a]
, Int[Sqrt[1 + (b*x^2)/a]/Sqrt[c + d*x^2], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] &&  !GtQ
[a, 0]

Rule 424

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]*EllipticE[ArcSin[Rt[-(d/c)
, 2]*x], (b*c)/(a*d)])/(Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[
a, 0]

Rule 421

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Dist[Sqrt[1 + (d*x^2)/c]/Sqrt[c + d*
x^2], Int[1/(Sqrt[a + b*x^2]*Sqrt[1 + (d*x^2)/c]), x], x] /; FreeQ[{a, b, c, d}, x] &&  !GtQ[c, 0]

Rule 419

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(1*EllipticF[ArcSin[Rt[-(d/c),
2]*x], (b*c)/(a*d)])/(Sqrt[a]*Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] &
& GtQ[a, 0] &&  !(NegQ[b/a] && SimplerSqrtQ[-(b/a), -(d/c)])

Rubi steps

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

Mathematica [C]  time = 4.46121, size = 501, normalized size = 2.01 \[ \frac{\frac{b \sqrt{\frac{1-c x}{c x+1}} \left (-c^2 \left (d+e x^2\right )+\frac{(c x+1) \sqrt{\frac{c \left (\sqrt{d}-i \sqrt{e} x\right )}{(c x+1) \left (c \sqrt{d}-i \sqrt{e}\right )}} \sqrt{\frac{c \left (\sqrt{d}+i \sqrt{e} x\right )}{(c x+1) \left (c \sqrt{d}+i \sqrt{e}\right )}} \left (2 \sqrt{e} \left (c \sqrt{d}-2 i \sqrt{e}\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{\frac{(1-c x) \left (c^2 d+e\right )}{(c x+1) \left (c \sqrt{d}+i \sqrt{e}\right )^2}}\right ),\frac{\left (c \sqrt{d}+i \sqrt{e}\right )^2}{\left (c \sqrt{d}-i \sqrt{e}\right )^2}\right )-i \left (c \sqrt{d}-i \sqrt{e}\right )^2 E\left (i \sinh ^{-1}\left (\sqrt{\frac{\left (d c^2+e\right ) (1-c x)}{\left (\sqrt{d} c+i \sqrt{e}\right )^2 (c x+1)}}\right )|\frac{\left (\sqrt{d} c+i \sqrt{e}\right )^2}{\left (c \sqrt{d}-i \sqrt{e}\right )^2}\right )\right )}{\sqrt{-\frac{(c x-1) \left (c \sqrt{d}-i \sqrt{e}\right )}{(c x+1) \left (c \sqrt{d}+i \sqrt{e}\right )}}}\right )}{c}-\frac{a \left (d+2 e x^2\right )}{x}+\frac{b \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left (d+e x^2\right )}{x}-\frac{b \text{sech}^{-1}(c x) \left (d+2 e x^2\right )}{x}}{d^2 \sqrt{d+e x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((b*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*(d + e*x^2))/x - (a*(d + 2*e*x^2))/x - (b*(d + 2*e*x^2)*ArcSech[c*x])/
x + (b*Sqrt[(1 - c*x)/(1 + c*x)]*(-(c^2*(d + e*x^2)) + ((1 + c*x)*Sqrt[(c*(Sqrt[d] - I*Sqrt[e]*x))/((c*Sqrt[d]
- I*Sqrt[e])*(1 + c*x))]*Sqrt[(c*(Sqrt[d] + I*Sqrt[e]*x))/((c*Sqrt[d] + I*Sqrt[e])*(1 + c*x))]*((-I)*(c*Sqrt[
d] - I*Sqrt[e])^2*EllipticE[I*ArcSinh[Sqrt[((c^2*d + e)*(1 - c*x))/((c*Sqrt[d] + I*Sqrt[e])^2*(1 + c*x))]], (c
*Sqrt[d] + I*Sqrt[e])^2/(c*Sqrt[d] - I*Sqrt[e])^2] + 2*(c*Sqrt[d] - (2*I)*Sqrt[e])*Sqrt[e]*EllipticF[I*ArcSinh
[Sqrt[((c^2*d + e)*(1 - c*x))/((c*Sqrt[d] + I*Sqrt[e])^2*(1 + c*x))]], (c*Sqrt[d] + I*Sqrt[e])^2/(c*Sqrt[d] -
I*Sqrt[e])^2]))/Sqrt[-(((c*Sqrt[d] - I*Sqrt[e])*(-1 + c*x))/((c*Sqrt[d] + I*Sqrt[e])*(1 + c*x)))]))/c)/(d^2*Sq
rt[d + e*x^2])

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Maple [F]  time = 0.809, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b{\rm arcsech} \left (cx\right )}{{x}^{2}} \left ( e{x}^{2}+d \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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((a+b*arcsech(c*x))/x^2/(e*x^2+d)^(3/2),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{\sqrt{e x^{2} + d}{\left (b \operatorname{arsech}\left (c x\right ) + a\right )}}{e^{2} x^{6} + 2 \, d e x^{4} + d^{2} x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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