3.2.67 \(\int \frac {a+b \text {sech}^{-1}(c x)}{x^2 (d+e x^2)^{3/2}} \, dx\) [167]

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

[Out]

(-a-b*arcsech(c*x))/d/x/(e*x^2+d)^(1/2)-2*e*x*(a+b*arcsech(c*x))/d^2/(e*x^2+d)^(1/2)+b*(1/(c*x+1))^(1/2)*(c*x+
1)^(1/2)*(-c^2*x^2+1)^(1/2)*(e*x^2+d)^(1/2)/d^2/x+b*c*EllipticE(c*x,(-e/c^2/d)^(1/2))*(1/(c*x+1))^(1/2)*(c*x+1
)^(1/2)*(e*x^2+d)^(1/2)/d^2/(1+e*x^2/d)^(1/2)-b*(c^2*d+2*e)*EllipticF(c*x,(-e/c^2/d)^(1/2))*(1/(c*x+1))^(1/2)*
(c*x+1)^(1/2)*(1+e*x^2/d)^(1/2)/c/d^2/(e*x^2+d)^(1/2)

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Rubi [A]
time = 0.19, antiderivative size = 249, normalized size of antiderivative = 1.00, 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 = {277, 197, 6436, 12, 597, 538, 437, 435, 432, 430} \begin {gather*} -\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} \left (c^2 d+2 e\right ) \sqrt {\frac {e x^2}{d}+1} F\left (\text {ArcSin}(c x)\left |-\frac {e}{c^2 d}\right .\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 (\text {ArcSin}(c x)\left |-\frac {e}{c^2 d}\right .\right )}{d^2 \sqrt {\frac {e x^2}{d}+1}}+\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} \end {gather*}

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 12

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

Rule 197

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 277

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 430

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

Rule 432

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

Rule 435

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

Rule 437

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

Rule 538

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 597

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 6436

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]))

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*}

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Mathematica [C] Result contains complex when optimal does not.
time = 22.40, size = 501, normalized size = 2.01 \begin {gather*} \frac {\frac {b \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (d+e x^2\right )}{x}-\frac {a \left (d+2 e x^2\right )}{x}-\frac {b \left (d+2 e x^2\right ) \text {sech}^{-1}(c x)}{x}+\frac {b \sqrt {\frac {1-c x}{1+c x}} \left (-c^2 \left (d+e x^2\right )+\frac {(1+c x) \sqrt {\frac {c \left (\sqrt {d}-i \sqrt {e} x\right )}{\left (c \sqrt {d}-i \sqrt {e}\right ) (1+c x)}} \sqrt {\frac {c \left (\sqrt {d}+i \sqrt {e} x\right )}{\left (c \sqrt {d}+i \sqrt {e}\right ) (1+c x)}} \left (-i \left (c \sqrt {d}-i \sqrt {e}\right )^2 E\left (i \sinh ^{-1}\left (\sqrt {\frac {\left (c^2 d+e\right ) (1-c x)}{\left (c \sqrt {d}+i \sqrt {e}\right )^2 (1+c x)}}\right )|\frac {\left (c \sqrt {d}+i \sqrt {e}\right )^2}{\left (c \sqrt {d}-i \sqrt {e}\right )^2}\right )+2 \left (c \sqrt {d}-2 i \sqrt {e}\right ) \sqrt {e} F\left (i \sinh ^{-1}\left (\sqrt {\frac {\left (c^2 d+e\right ) (1-c x)}{\left (c \sqrt {d}+i \sqrt {e}\right )^2 (1+c x)}}\right )|\frac {\left (c \sqrt {d}+i \sqrt {e}\right )^2}{\left (c \sqrt {d}-i \sqrt {e}\right )^2}\right )\right )}{\sqrt {-\frac {\left (c \sqrt {d}-i \sqrt {e}\right ) (-1+c x)}{\left (c \sqrt {d}+i \sqrt {e}\right ) (1+c x)}}}\right )}{c}}{d^2 \sqrt {d+e x^2}} \end {gather*}

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 = 0.40, size = 0, normalized size = 0.00 \[\int \frac {a +b \,\mathrm {arcsech}\left (c x \right )}{x^{2} \left (e \,x^{2}+d \right )^{\frac {3}{2}}}\, dx\]

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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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]

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

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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

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]

Exception raised: TypeError >> Symbolic function elliptic_ec takes exactly 1 arguments (2 given)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a + b \operatorname {asech}{\left (c x \right )}}{x^{2} \left (d + e x^{2}\right )^{\frac {3}{2}}}\, dx \end {gather*}

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]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )}{x^2\,{\left (e\,x^2+d\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*acosh(1/(c*x)))/(x^2*(d + e*x^2)^(3/2)),x)

[Out]

int((a + b*acosh(1/(c*x)))/(x^2*(d + e*x^2)^(3/2)), x)

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