\(\int \frac {1}{x \sqrt {-a+b x}} \, dx\) [359]

Optimal result

Integrand size = 15, antiderivative size = 25 \[ \int \frac {1}{x \sqrt {-a+b x}} \, dx=\frac {2 \arctan \left (\frac {\sqrt {-a+b x}}{\sqrt {a}}\right )}{\sqrt {a}} \]

[Out]

2*arctan((b*x-a)^(1/2)/a^(1/2))/a^(1/2)

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {65, 211} \[ \int \frac {1}{x \sqrt {-a+b x}} \, dx=\frac {2 \arctan \left (\frac {\sqrt {b x-a}}{\sqrt {a}}\right )}{\sqrt {a}} \]

[In]

Int[1/(x*Sqrt[-a + b*x]),x]

[Out]

(2*ArcTan[Sqrt[-a + b*x]/Sqrt[a]])/Sqrt[a]

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 211

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

Rubi steps \begin{align*} \text {integral}& = \frac {2 \text {Subst}\left (\int \frac {1}{\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {-a+b x}\right )}{b} \\ & = \frac {2 \tan ^{-1}\left (\frac {\sqrt {-a+b x}}{\sqrt {a}}\right )}{\sqrt {a}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \sqrt {-a+b x}} \, dx=\frac {2 \arctan \left (\frac {\sqrt {-a+b x}}{\sqrt {a}}\right )}{\sqrt {a}} \]

[In]

Integrate[1/(x*Sqrt[-a + b*x]),x]

[Out]

(2*ArcTan[Sqrt[-a + b*x]/Sqrt[a]])/Sqrt[a]

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80

method	result	size
derivativedivides	\(\frac {2 \arctan \left (\frac {\sqrt {b x -a}}{\sqrt {a}}\right )}{\sqrt {a}}\)	\(20\)
default	\(\frac {2 \arctan \left (\frac {\sqrt {b x -a}}{\sqrt {a}}\right )}{\sqrt {a}}\)	\(20\)
pseudoelliptic	\(\frac {2 \arctan \left (\frac {\sqrt {b x -a}}{\sqrt {a}}\right )}{\sqrt {a}}\)	\(20\)

[In]

int(1/x/(b*x-a)^(1/2),x,method=_RETURNVERBOSE)

[Out]

2*arctan((b*x-a)^(1/2)/a^(1/2))/a^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.32 \[ \int \frac {1}{x \sqrt {-a+b x}} \, dx=\left [-\frac {\sqrt {-a} \log \left (\frac {b x - 2 \, \sqrt {b x - a} \sqrt {-a} - 2 \, a}{x}\right )}{a}, \frac {2 \, \arctan \left (\frac {\sqrt {b x - a}}{\sqrt {a}}\right )}{\sqrt {a}}\right ] \]

[In]

integrate(1/x/(b*x-a)^(1/2),x, algorithm="fricas")

[Out]

[-sqrt(-a)*log((b*x - 2*sqrt(b*x - a)*sqrt(-a) - 2*a)/x)/a, 2*arctan(sqrt(b*x - a)/sqrt(a))/sqrt(a)]

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.93 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.16 \[ \int \frac {1}{x \sqrt {-a+b x}} \, dx=\begin {cases} \frac {2 i \operatorname {acosh}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )}}{\sqrt {a}} & \text {for}\: \left |{\frac {a}{b x}}\right | > 1 \\- \frac {2 \operatorname {asin}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )}}{\sqrt {a}} & \text {otherwise} \end {cases} \]

[In]

integrate(1/x/(b*x-a)**(1/2),x)

[Out]

Piecewise((2*I*acosh(sqrt(a)/(sqrt(b)*sqrt(x)))/sqrt(a), Abs(a/(b*x)) > 1), (-2*asin(sqrt(a)/(sqrt(b)*sqrt(x))
)/sqrt(a), True))

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76 \[ \int \frac {1}{x \sqrt {-a+b x}} \, dx=\frac {2 \, \arctan \left (\frac {\sqrt {b x - a}}{\sqrt {a}}\right )}{\sqrt {a}} \]

[In]

integrate(1/x/(b*x-a)^(1/2),x, algorithm="maxima")

[Out]

2*arctan(sqrt(b*x - a)/sqrt(a))/sqrt(a)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76 \[ \int \frac {1}{x \sqrt {-a+b x}} \, dx=\frac {2 \, \arctan \left (\frac {\sqrt {b x - a}}{\sqrt {a}}\right )}{\sqrt {a}} \]

[In]

integrate(1/x/(b*x-a)^(1/2),x, algorithm="giac")

[Out]

2*arctan(sqrt(b*x - a)/sqrt(a))/sqrt(a)

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76 \[ \int \frac {1}{x \sqrt {-a+b x}} \, dx=\frac {2\,\mathrm {atan}\left (\frac {\sqrt {b\,x-a}}{\sqrt {a}}\right )}{\sqrt {a}} \]

[In]

int(1/(x*(b*x - a)^(1/2)),x)

[Out]

(2*atan((b*x - a)^(1/2)/a^(1/2)))/a^(1/2)