3.292 \(\int \frac{1}{\sqrt{-1+x} x^3} \, dx\)

Optimal. Leaf size=41 \[ \frac{\sqrt{x-1}}{2 x^2}+\frac{3 \sqrt{x-1}}{4 x}+\frac{3}{4} \tan ^{-1}\left (\sqrt{x-1}\right ) \]

[Out]

Sqrt[-1 + x]/(2*x^2) + (3*Sqrt[-1 + x])/(4*x) + (3*ArcTan[Sqrt[-1 + x]])/4

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Rubi [A]  time = 0.0085171, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {51, 63, 203} \[ \frac{\sqrt{x-1}}{2 x^2}+\frac{3 \sqrt{x-1}}{4 x}+\frac{3}{4} \tan ^{-1}\left (\sqrt{x-1}\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[-1 + x]*x^3),x]

[Out]

Sqrt[-1 + x]/(2*x^2) + (3*Sqrt[-1 + x])/(4*x) + (3*ArcTan[Sqrt[-1 + x]])/4

Rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, 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 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{-1+x} x^3} \, dx &=\frac{\sqrt{-1+x}}{2 x^2}+\frac{3}{4} \int \frac{1}{\sqrt{-1+x} x^2} \, dx\\ &=\frac{\sqrt{-1+x}}{2 x^2}+\frac{3 \sqrt{-1+x}}{4 x}+\frac{3}{8} \int \frac{1}{\sqrt{-1+x} x} \, dx\\ &=\frac{\sqrt{-1+x}}{2 x^2}+\frac{3 \sqrt{-1+x}}{4 x}+\frac{3}{4} \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sqrt{-1+x}\right )\\ &=\frac{\sqrt{-1+x}}{2 x^2}+\frac{3 \sqrt{-1+x}}{4 x}+\frac{3}{4} \tan ^{-1}\left (\sqrt{-1+x}\right )\\ \end{align*}

Mathematica [C]  time = 0.0039461, size = 22, normalized size = 0.54 \[ 2 \sqrt{x-1} \, _2F_1\left (\frac{1}{2},3;\frac{3}{2};1-x\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[-1 + x]*x^3),x]

[Out]

2*Sqrt[-1 + x]*Hypergeometric2F1[1/2, 3, 3/2, 1 - x]

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Maple [A]  time = 0.007, size = 30, normalized size = 0.7 \begin{align*}{\frac{3}{4}\arctan \left ( \sqrt{-1+x} \right ) }+{\frac{1}{2\,{x}^{2}}\sqrt{-1+x}}+{\frac{3}{4\,x}\sqrt{-1+x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^3/(-1+x)^(1/2),x)

[Out]

3/4*arctan((-1+x)^(1/2))+1/2*(-1+x)^(1/2)/x^2+3/4*(-1+x)^(1/2)/x

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Maxima [A]  time = 1.44212, size = 51, normalized size = 1.24 \begin{align*} \frac{3 \,{\left (x - 1\right )}^{\frac{3}{2}} + 5 \, \sqrt{x - 1}}{4 \,{\left ({\left (x - 1\right )}^{2} + 2 \, x - 1\right )}} + \frac{3}{4} \, \arctan \left (\sqrt{x - 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(-1+x)^(1/2),x, algorithm="maxima")

[Out]

1/4*(3*(x - 1)^(3/2) + 5*sqrt(x - 1))/((x - 1)^2 + 2*x - 1) + 3/4*arctan(sqrt(x - 1))

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Fricas [A]  time = 1.89877, size = 82, normalized size = 2. \begin{align*} \frac{3 \, x^{2} \arctan \left (\sqrt{x - 1}\right ) +{\left (3 \, x + 2\right )} \sqrt{x - 1}}{4 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(-1+x)^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 2.67197, size = 131, normalized size = 3.2 \begin{align*} \begin{cases} \frac{3 i \operatorname{acosh}{\left (\frac{1}{\sqrt{x}} \right )}}{4} - \frac{3 i}{4 \sqrt{x} \sqrt{-1 + \frac{1}{x}}} + \frac{i}{4 x^{\frac{3}{2}} \sqrt{-1 + \frac{1}{x}}} + \frac{i}{2 x^{\frac{5}{2}} \sqrt{-1 + \frac{1}{x}}} & \text{for}\: \frac{1}{\left |{x}\right |} > 1 \\- \frac{3 \operatorname{asin}{\left (\frac{1}{\sqrt{x}} \right )}}{4} + \frac{3}{4 \sqrt{x} \sqrt{1 - \frac{1}{x}}} - \frac{1}{4 x^{\frac{3}{2}} \sqrt{1 - \frac{1}{x}}} - \frac{1}{2 x^{\frac{5}{2}} \sqrt{1 - \frac{1}{x}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**3/(-1+x)**(1/2),x)

[Out]

Piecewise((3*I*acosh(1/sqrt(x))/4 - 3*I/(4*sqrt(x)*sqrt(-1 + 1/x)) + I/(4*x**(3/2)*sqrt(-1 + 1/x)) + I/(2*x**(
5/2)*sqrt(-1 + 1/x)), 1/Abs(x) > 1), (-3*asin(1/sqrt(x))/4 + 3/(4*sqrt(x)*sqrt(1 - 1/x)) - 1/(4*x**(3/2)*sqrt(
1 - 1/x)) - 1/(2*x**(5/2)*sqrt(1 - 1/x)), True))

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Giac [A]  time = 1.07476, size = 39, normalized size = 0.95 \begin{align*} \frac{3 \,{\left (x - 1\right )}^{\frac{3}{2}} + 5 \, \sqrt{x - 1}}{4 \, x^{2}} + \frac{3}{4} \, \arctan \left (\sqrt{x - 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(-1+x)^(1/2),x, algorithm="giac")

[Out]

1/4*(3*(x - 1)^(3/2) + 5*sqrt(x - 1))/x^2 + 3/4*arctan(sqrt(x - 1))