3.685 \(\int \frac {\sqrt {-1+x^2} \sec ^{-1}(x)}{x^4} \, dx\)

Optimal. Leaf size=41 \[ \frac {1}{3 \sqrt {x^2}}-\frac {1}{9 \left (x^2\right )^{3/2}}+\frac {\left (x^2-1\right )^{3/2} \sec ^{-1}(x)}{3 x^3} \]

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Rubi [A]  time = 0.05, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {264, 5238, 12, 14} \[ \frac {1}{3 \sqrt {x^2}}-\frac {1}{9 \left (x^2\right )^{3/2}}+\frac {\left (x^2-1\right )^{3/2} \sec ^{-1}(x)}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Int[(Sqrt[-1 + x^2]*ArcSec[x])/x^4,x]

[Out]

-1/(9*(x^2)^(3/2)) + 1/(3*Sqrt[x^2]) + ((-1 + x^2)^(3/2)*ArcSec[x])/(3*x^3)

Rule 12

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

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

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 5238

Int[((a_.) + ArcSec[(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*ArcSec[c*x], u, x] - Dist[(b*c*x)/Sqrt[c^2*x^2], Int[SimplifyI
ntegrand[u/(x*Sqrt[c^2*x^2 - 1]), 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])) || (I
LtQ[(m + 2*p + 1)/2, 0] &&  !ILtQ[(m - 1)/2, 0]))

Rubi steps

\begin {align*} \int \frac {\sqrt {-1+x^2} \sec ^{-1}(x)}{x^4} \, dx &=\frac {\left (-1+x^2\right )^{3/2} \sec ^{-1}(x)}{3 x^3}-\frac {x \int \frac {-1+x^2}{3 x^4} \, dx}{\sqrt {x^2}}\\ &=\frac {\left (-1+x^2\right )^{3/2} \sec ^{-1}(x)}{3 x^3}-\frac {x \int \frac {-1+x^2}{x^4} \, dx}{3 \sqrt {x^2}}\\ &=\frac {\left (-1+x^2\right )^{3/2} \sec ^{-1}(x)}{3 x^3}-\frac {x \int \left (-\frac {1}{x^4}+\frac {1}{x^2}\right ) \, dx}{3 \sqrt {x^2}}\\ &=-\frac {1}{9 \left (x^2\right )^{3/2}}+\frac {1}{3 \sqrt {x^2}}+\frac {\left (-1+x^2\right )^{3/2} \sec ^{-1}(x)}{3 x^3}\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 48, normalized size = 1.17 \[ \frac {\sqrt {1-\frac {1}{x^2}} x \left (3 x^2-1\right )+3 \left (x^2-1\right )^2 \sec ^{-1}(x)}{9 x^3 \sqrt {x^2-1}} \]

Antiderivative was successfully verified.

[In]

Integrate[(Sqrt[-1 + x^2]*ArcSec[x])/x^4,x]

[Out]

(Sqrt[1 - x^(-2)]*x*(-1 + 3*x^2) + 3*(-1 + x^2)^2*ArcSec[x])/(9*x^3*Sqrt[-1 + x^2])

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {-1+x^2} \sec ^{-1}(x)}{x^4} \, dx \]

Verification is Not applicable to the result.

[In]

IntegrateAlgebraic[(Sqrt[-1 + x^2]*ArcSec[x])/x^4,x]

[Out]

Could not integrate

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fricas [A]  time = 1.14, size = 23, normalized size = 0.56 \[ \frac {3 \, {\left (x^{2} - 1\right )}^{\frac {3}{2}} \operatorname {arcsec}\relax (x) + 3 \, x^{2} - 1}{9 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsec(x)*(x^2-1)^(1/2)/x^4,x, algorithm="fricas")

[Out]

1/9*(3*(x^2 - 1)^(3/2)*arcsec(x) + 3*x^2 - 1)/x^3

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giac [B]  time = 1.16, size = 75, normalized size = 1.83 \[ -\frac {2 \, \arctan \left (-x + \sqrt {x^{2} - 1}\right )}{3 \, \mathrm {sgn}\relax (x)} + \frac {2 \, {\left (3 \, {\left (x - \sqrt {x^{2} - 1}\right )}^{4} + 1\right )} \arccos \left (\frac {1}{x}\right )}{3 \, {\left ({\left (x - \sqrt {x^{2} - 1}\right )}^{2} + 1\right )}^{3}} + \frac {3 \, x^{2} - 1}{9 \, x^{3} \mathrm {sgn}\relax (x)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsec(x)*(x^2-1)^(1/2)/x^4,x, algorithm="giac")

[Out]

-2/3*arctan(-x + sqrt(x^2 - 1))/sgn(x) + 2/3*(3*(x - sqrt(x^2 - 1))^4 + 1)*arccos(1/x)/((x - sqrt(x^2 - 1))^2
+ 1)^3 + 1/9*(3*x^2 - 1)/(x^3*sgn(x))

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maple [C]  time = 0.78, size = 329, normalized size = 8.02

method	result	size
default	\(-\frac {\sqrt {x^{2}-1}\, \left (\sqrt {\frac {x^{2}-1}{x^{2}}}\, x^{5}-5 i x^{4}-12 \sqrt {\frac {x^{2}-1}{x^{2}}}\, x^{3}+20 i x^{2}+16 \sqrt {\frac {x^{2}-1}{x^{2}}}\, x -16 i\right )}{144 \left (-i \sqrt {\frac {x^{2}-1}{x^{2}}}\, x +x^{2}-1\right ) x^{3}}+\frac {\left (i \sqrt {\frac {x^{2}-1}{x^{2}}}\, x^{5}-8 i \sqrt {\frac {x^{2}-1}{x^{2}}}\, x^{3}+4 x^{4}+8 i \sqrt {\frac {x^{2}-1}{x^{2}}}\, x -12 x^{2}+8\right ) \mathrm {arcsec}\relax (x )}{48 \sqrt {x^{2}-1}\, x^{3}}+\frac {\left (i \sqrt {\frac {x^{2}-1}{x^{2}}}\, x +x^{2}-1\right ) \left (-13 i+3 \,\mathrm {arcsec}\relax (x )\right )}{72 \sqrt {x^{2}-1}\, x}-\frac {\left (5 i+3 \,\mathrm {arcsec}\relax (x )\right ) \left (i \sqrt {\frac {x^{2}-1}{x^{2}}}\, x -1\right ) \sqrt {x^{2}-1}}{72 x}-\frac {\left (i \sqrt {\frac {x^{2}-1}{x^{2}}}\, x +x^{2}-1\right ) \left (7 i+9 \,\mathrm {arcsec}\relax (x )\right ) \cos \left (3 \,\mathrm {arcsec}\relax (x )\right )}{144 \sqrt {x^{2}-1}}-\frac {\left (i x^{2}-\sqrt {\frac {x^{2}-1}{x^{2}}}\, x -i\right ) \left (3 i+\mathrm {arcsec}\relax (x )\right ) \sin \left (3 \,\mathrm {arcsec}\relax (x )\right )}{48 \sqrt {x^{2}-1}}\)	\(329\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arcsec(x)*(x^2-1)^(1/2)/x^4,x,method=_RETURNVERBOSE)

[Out]

-1/144*(x^2-1)^(1/2)*(((x^2-1)/x^2)^(1/2)*x^5-5*I*x^4-12*((x^2-1)/x^2)^(1/2)*x^3+20*I*x^2+16*((x^2-1)/x^2)^(1/
2)*x-16*I)/(-I*((x^2-1)/x^2)^(1/2)*x+x^2-1)/x^3+1/48/(x^2-1)^(1/2)/x^3*(I*((x^2-1)/x^2)^(1/2)*x^5-8*I*((x^2-1)
/x^2)^(1/2)*x^3+4*x^4+8*I*((x^2-1)/x^2)^(1/2)*x-12*x^2+8)*arcsec(x)+1/72/(x^2-1)^(1/2)*(I*((x^2-1)/x^2)^(1/2)*
x+x^2-1)*(-13*I+3*arcsec(x))/x-1/72*(5*I+3*arcsec(x))*(I*((x^2-1)/x^2)^(1/2)*x-1)*(x^2-1)^(1/2)/x-1/144/(x^2-1
)^(1/2)*(I*((x^2-1)/x^2)^(1/2)*x+x^2-1)*(7*I+9*arcsec(x))*cos(3*arcsec(x))-1/48/(x^2-1)^(1/2)*(I*x^2-((x^2-1)/
x^2)^(1/2)*x-I)*(3*I+arcsec(x))*sin(3*arcsec(x))

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maxima [A]  time = 0.99, size = 27, normalized size = 0.66 \[ \frac {{\left (x^{2} - 1\right )}^{\frac {3}{2}} \operatorname {arcsec}\relax (x)}{3 \, x^{3}} + \frac {3 \, x^{2} - 1}{9 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsec(x)*(x^2-1)^(1/2)/x^4,x, algorithm="maxima")

[Out]

1/3*(x^2 - 1)^(3/2)*arcsec(x)/x^3 + 1/9*(3*x^2 - 1)/x^3

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\mathrm {acos}\left (\frac {1}{x}\right )\,\sqrt {x^2-1}}{x^4} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((acos(1/x)*(x^2 - 1)^(1/2))/x^4,x)

[Out]

int((acos(1/x)*(x^2 - 1)^(1/2))/x^4, x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(asec(x)*(x**2-1)**(1/2)/x**4,x)

[Out]

Timed out

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