3.3.92 \(\int \frac {1}{\sqrt {-1+x} x^3} \, dx\) [292]

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

[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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Rubi [A]
time = 0.01, antiderivative size = 41, normalized size of antiderivative = 1.00, 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 = {44, 65, 209} \begin {gather*} \frac {3}{4} \text {ArcTan}\left (\sqrt {x-1}\right )+\frac {\sqrt {x-1}}{2 x^2}+\frac {3 \sqrt {x-1}}{4 x} \end {gather*}

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 44

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] && ILtQ[m, -1] &&  !IntegerQ[n] && LtQ[n, 0]

Rule 65

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 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 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} \text {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*}

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Mathematica [A]
time = 0.03, size = 31, normalized size = 0.76 \begin {gather*} \frac {1}{4} \left (\frac {\sqrt {-1+x} (2+3 x)}{x^2}+3 \tan ^{-1}\left (\sqrt {-1+x}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[-1 + x]*x^3),x]

[Out]

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

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Maple [A]
time = 0.11, size = 30, normalized size = 0.73

method	result	size
derivativedivides	\(\frac {3 \arctan \left (\sqrt {-1+x}\right )}{4}+\frac {\sqrt {-1+x}}{2 x^{2}}+\frac {3 \sqrt {-1+x}}{4 x}\)	\(30\)
default	\(\frac {3 \arctan \left (\sqrt {-1+x}\right )}{4}+\frac {\sqrt {-1+x}}{2 x^{2}}+\frac {3 \sqrt {-1+x}}{4 x}\)	\(30\)
risch	\(\frac {3 x^{2}-x -2}{4 x^{2} \sqrt {-1+x}}+\frac {3 \arctan \left (\sqrt {-1+x}\right )}{4}\)	\(30\)
trager	\(\frac {\left (2+3 x \right ) \sqrt {-1+x}}{4 x^{2}}+\frac {3 \RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {-1+x}+x -2}{x}\right )}{8}\)	\(46\)
meijerg	\(\frac {\sqrt {-\mathrm {signum}\left (-1+x \right )}\, \left (-\frac {\sqrt {\pi }}{2 x^{2}}-\frac {\sqrt {\pi }}{2 x}+\frac {3 \left (\frac {7}{6}-2 \ln \left (2\right )+\ln \left (x \right )+i \pi \right ) \sqrt {\pi }}{8}+\frac {\sqrt {\pi }\, \left (-7 x^{2}+8 x +8\right )}{16 x^{2}}-\frac {\sqrt {\pi }\, \left (12 x +8\right ) \sqrt {1-x}}{16 x^{2}}-\frac {3 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {1-x}}{2}\right )}{4}\right )}{\sqrt {\pi }\, \sqrt {\mathrm {signum}\left (-1+x \right )}}\)	\(108\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 2.31, size = 38, normalized size = 0.93 \begin {gather*} \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 {gather*}

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 = 1.70, size = 28, normalized size = 0.68 \begin {gather*} \frac {3 \, x^{2} \arctan \left (\sqrt {x - 1}\right ) + {\left (3 \, x + 2\right )} \sqrt {x - 1}}{4 \, x^{2}} \end {gather*}

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 [C] Result contains complex when optimal does not.
time = 1.95, size = 131, normalized size = 3.20 \begin {gather*} \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 {gather*}

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 = 0.92, size = 29, normalized size = 0.71 \begin {gather*} \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 {gather*}

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))

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Mupad [B]
time = 0.04, size = 29, normalized size = 0.71 \begin {gather*} \frac {3\,\mathrm {atan}\left (\sqrt {x-1}\right )}{4}+\frac {3\,\sqrt {x-1}}{4\,x}+\frac {\sqrt {x-1}}{2\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^3*(x - 1)^(1/2)),x)

[Out]

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

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