3.21.93 \(\int \frac {1}{x \sqrt {a^2-2 a b x+b^2 x^2}} \, dx\)

Optimal. Leaf size=71 \[ \frac {\log (x) (a-b x)}{a \sqrt {a^2-2 a b x+b^2 x^2}}-\frac {(a-b x) \log (a-b x)}{a \sqrt {a^2-2 a b x+b^2 x^2}} \]

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Rubi [A]  time = 0.02, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {646, 36, 29, 31} \begin {gather*} \frac {\log (x) (a-b x)}{a \sqrt {a^2-2 a b x+b^2 x^2}}-\frac {(a-b x) \log (a-b x)}{a \sqrt {a^2-2 a b x+b^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x*Sqrt[a^2 - 2*a*b*x + b^2*x^2]),x]

[Out]

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

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 36

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

Rule 646

Int[((d_.) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[(a + b*x + c*x^2)^Fra
cPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b,
c, d, e, m, p}, x] && EqQ[b^2 - 4*a*c, 0] &&  !IntegerQ[p] && NeQ[2*c*d - b*e, 0]

Rubi steps

\begin {align*} \int \frac {1}{x \sqrt {a^2-2 a b x+b^2 x^2}} \, dx &=\frac {\left (-a b+b^2 x\right ) \int \frac {1}{x \left (-a b+b^2 x\right )} \, dx}{\sqrt {a^2-2 a b x+b^2 x^2}}\\ &=-\frac {\left (-a b+b^2 x\right ) \int \frac {1}{x} \, dx}{a b \sqrt {a^2-2 a b x+b^2 x^2}}+\frac {\left (b \left (-a b+b^2 x\right )\right ) \int \frac {1}{-a b+b^2 x} \, dx}{a \sqrt {a^2-2 a b x+b^2 x^2}}\\ &=\frac {(a-b x) \log (x)}{a \sqrt {a^2-2 a b x+b^2 x^2}}-\frac {(a-b x) \log (a-b x)}{a \sqrt {a^2-2 a b x+b^2 x^2}}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 34, normalized size = 0.48 \begin {gather*} \frac {(a-b x) (\log (x)-\log (a-b x))}{a \sqrt {(a-b x)^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x*Sqrt[a^2 - 2*a*b*x + b^2*x^2]),x]

[Out]

((a - b*x)*(Log[x] - Log[a - b*x]))/(a*Sqrt[(a - b*x)^2])

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IntegrateAlgebraic [A]  time = 0.15, size = 44, normalized size = 0.62 \begin {gather*} \frac {2 \tanh ^{-1}\left (\frac {\sqrt {b^2} x}{a}-\frac {\sqrt {a^2-2 a b x+b^2 x^2}}{a}\right )}{a} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/(x*Sqrt[a^2 - 2*a*b*x + b^2*x^2]),x]

[Out]

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

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fricas [A]  time = 0.41, size = 17, normalized size = 0.24 \begin {gather*} \frac {\log \left (b x - a\right ) - \log \relax (x)}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(log(b*x - a) - log(x))/a

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giac [A]  time = 0.15, size = 31, normalized size = 0.44 \begin {gather*} {\left (\frac {\log \left ({\left | b x - a \right |}\right )}{a} - \frac {\log \left ({\left | x \right |}\right )}{a}\right )} \mathrm {sgn}\left (b x - a\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(log(abs(b*x - a))/a - log(abs(x))/a)*sgn(b*x - a)

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maple [A]  time = 0.16, size = 36, normalized size = 0.51 \begin {gather*} \frac {\left (b x -a \right ) \left (-\ln \relax (x )+\ln \left (b x -a \right )\right )}{\sqrt {\left (b x -a \right )^{2}}\, a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.81, size = 38, normalized size = 0.54 \begin {gather*} -\frac {\left (-1\right )^{-2 \, a b x + 2 \, a^{2}} \log \left (-\frac {2 \, a b x}{{\left | x \right |}} + \frac {2 \, a^{2}}{{\left | x \right |}}\right )}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(-1)^(-2*a*b*x + 2*a^2)*log(-2*a*b*x/abs(x) + 2*a^2/abs(x))/a

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mupad [B]  time = 1.17, size = 47, normalized size = 0.66 \begin {gather*} -\frac {\ln \left (\frac {a^2}{x}-a\,b+\frac {\sqrt {a^2}\,\sqrt {a^2-2\,a\,b\,x+b^2\,x^2}}{x}\right )}{\sqrt {a^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*((a - b*x)^2)^(1/2)),x)

[Out]

-log(a^2/x - a*b + ((a^2)^(1/2)*(a^2 + b^2*x^2 - 2*a*b*x)^(1/2))/x)/(a^2)^(1/2)

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sympy [A]  time = 0.17, size = 10, normalized size = 0.14 \begin {gather*} \frac {- \log {\relax (x )} + \log {\left (- \frac {a}{b} + x \right )}}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-log(x) + log(-a/b + x))/a

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