3.1015 \(\int \frac{-\sqrt{-4+x}-4 \sqrt{-1+x}+\sqrt{-4+x} x+\sqrt{-1+x} x}{(1+\sqrt{-4+x}+\sqrt{-1+x}) (4-5 x+x^2)} \, dx\)

Optimal. Leaf size=19 \[ 2 \log \left (\sqrt{x-4}+\sqrt{x-1}+1\right ) \]

[Out]

2*Log[1 + Sqrt[-4 + x] + Sqrt[-1 + x]]

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Rubi [A]  time = 0.556945, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 66, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {6688, 1586, 6684} \[ 2 \log \left (\sqrt{x-4}+\sqrt{x-1}+1\right ) \]

Antiderivative was successfully verified.

[In]

Int[(-Sqrt[-4 + x] - 4*Sqrt[-1 + x] + Sqrt[-4 + x]*x + Sqrt[-1 + x]*x)/((1 + Sqrt[-4 + x] + Sqrt[-1 + x])*(4 -
5*x + x^2)),x]

[Out]

2*Log[1 + Sqrt[-4 + x] + Sqrt[-1 + x]]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rule 1586

Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px, Qx, x]^p*Qx^(p + q), x] /; FreeQ[
q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]

Rule 6684

Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*Log[RemoveContent[y, x]], x] /;  !Fa
lseQ[q]]

Rubi steps

\begin{align*} \int \frac{-\sqrt{-4+x}-4 \sqrt{-1+x}+\sqrt{-4+x} x+\sqrt{-1+x} x}{\left (1+\sqrt{-4+x}+\sqrt{-1+x}\right ) \left (4-5 x+x^2\right )} \, dx &=\int \frac{\sqrt{-1+x} \left (-4+\sqrt{-4+x} \sqrt{-1+x}+x\right )}{\left (1+\sqrt{-4+x}+\sqrt{-1+x}\right ) \left (4-5 x+x^2\right )} \, dx\\ &=\int \frac{-4+\sqrt{-4+x} \sqrt{-1+x}+x}{\left (1+\sqrt{-4+x}+\sqrt{-1+x}\right ) (-4+x) \sqrt{-1+x}} \, dx\\ &=2 \log \left (1+\sqrt{-4+x}+\sqrt{-1+x}\right )\\ \end{align*}

Mathematica [B]  time = 1.34478, size = 75, normalized size = 3.95 \[ \frac{1}{2} \log \left (-5 x-4 \sqrt{x-4} \sqrt{x-1}+17\right )+\frac{1}{2} \log \left (-2 x-2 \sqrt{x-4} \sqrt{x-1}+5\right )-\tanh ^{-1}\left (\sqrt{x-4}\right )+\tanh ^{-1}\left (\frac{\sqrt{x-1}}{2}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(-Sqrt[-4 + x] - 4*Sqrt[-1 + x] + Sqrt[-4 + x]*x + Sqrt[-1 + x]*x)/((1 + Sqrt[-4 + x] + Sqrt[-1 + x]
)*(4 - 5*x + x^2)),x]

[Out]

-ArcTanh[Sqrt[-4 + x]] + ArcTanh[Sqrt[-1 + x]/2] + Log[17 - 4*Sqrt[-4 + x]*Sqrt[-1 + x] - 5*x]/2 + Log[5 - 2*S
qrt[-4 + x]*Sqrt[-1 + x] - 2*x]/2

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Maple [B]  time = 0.049, size = 147, normalized size = 7.7 \begin{align*}{\frac{\ln \left ( x-5 \right ) }{2}}+{\frac{1}{2}\ln \left ( -1+\sqrt{x-4} \right ) }-{\frac{1}{2}\ln \left ( 1+\sqrt{x-4} \right ) }+{\frac{1}{2}\ln \left ( \sqrt{x-1}+2 \right ) }-{\frac{1}{2}\ln \left ( -2+\sqrt{x-1} \right ) }+{\frac{7}{4}\sqrt{x-4}\sqrt{x-1}{\it Artanh} \left ({\frac{-17+5\,x}{4}{\frac{1}{\sqrt{{x}^{2}-5\,x+4}}}} \right ){\frac{1}{\sqrt{{x}^{2}-5\,x+4}}}}+{\frac{1}{4}\sqrt{x-4}\sqrt{x-1} \left ( 2\,\ln \left ( -5/2+x+\sqrt{{x}^{2}-5\,x+4} \right ) -5\,{\it Artanh} \left ( 1/4\,{\frac{-17+5\,x}{\sqrt{{x}^{2}-5\,x+4}}} \right ) \right ){\frac{1}{\sqrt{{x}^{2}-5\,x+4}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-(x-4)^(1/2)+x*(x-4)^(1/2)-4*(x-1)^(1/2)+x*(x-1)^(1/2))/(x^2-5*x+4)/(1+(x-4)^(1/2)+(x-1)^(1/2)),x)

[Out]

1/2*ln(x-5)+1/2*ln(-1+(x-4)^(1/2))-1/2*ln(1+(x-4)^(1/2))+1/2*ln((x-1)^(1/2)+2)-1/2*ln(-2+(x-1)^(1/2))+7/4*(x-4
)^(1/2)*(x-1)^(1/2)/(x^2-5*x+4)^(1/2)*arctanh(1/4*(-17+5*x)/(x^2-5*x+4)^(1/2))+1/4*(x-4)^(1/2)*(x-1)^(1/2)*(2*
ln(-5/2+x+(x^2-5*x+4)^(1/2))-5*arctanh(1/4*(-17+5*x)/(x^2-5*x+4)^(1/2)))/(x^2-5*x+4)^(1/2)

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Maxima [B]  time = 1.23879, size = 127, normalized size = 6.68 \begin{align*} \frac{1}{2} \, \log \left (x - 1\right ) + \frac{1}{2} \, \log \left (\frac{2 \, x^{2} + 2 \,{\left ({\left (x - 1\right )} \sqrt{x - 4} + 2 \, x - 6\right )} \sqrt{x - 1} + 2 \,{\left (2 \, x - 3\right )} \sqrt{x - 4} - 7 \, x + 3}{2 \,{\left ({\left (x - 1\right )} \sqrt{x - 4} + 2 \, x - 6\right )}}\right ) + \frac{1}{2} \, \log \left (\frac{{\left (x - 1\right )} \sqrt{x - 4} + 2 \, x - 6}{x - 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-(-4+x)^(1/2)+x*(-4+x)^(1/2)-4*(-1+x)^(1/2)+x*(-1+x)^(1/2))/(x^2-5*x+4)/(1+(-4+x)^(1/2)+(-1+x)^(1/2
)),x, algorithm="maxima")

[Out]

1/2*log(x - 1) + 1/2*log(1/2*(2*x^2 + 2*((x - 1)*sqrt(x - 4) + 2*x - 6)*sqrt(x - 1) + 2*(2*x - 3)*sqrt(x - 4)
- 7*x + 3)/((x - 1)*sqrt(x - 4) + 2*x - 6)) + 1/2*log(((x - 1)*sqrt(x - 4) + 2*x - 6)/(x - 1))

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Fricas [B]  time = 1.87414, size = 317, normalized size = 16.68 \begin{align*} -\frac{1}{2} \, \log \left (-{\left (4 \, x - 11\right )} \sqrt{x - 1} \sqrt{x - 4} + 4 \, x^{2} - 21 \, x + 23\right ) + \frac{1}{2} \, \log \left (\sqrt{x - 1} \sqrt{x - 4} - x + 7\right ) + \frac{1}{2} \, \log \left (x - 5\right ) + \frac{1}{2} \, \log \left (\sqrt{x - 1} + 2\right ) - \frac{1}{2} \, \log \left (\sqrt{x - 1} - 2\right ) - \frac{1}{2} \, \log \left (\sqrt{x - 4} + 1\right ) + \frac{1}{2} \, \log \left (\sqrt{x - 4} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-(-4+x)^(1/2)+x*(-4+x)^(1/2)-4*(-1+x)^(1/2)+x*(-1+x)^(1/2))/(x^2-5*x+4)/(1+(-4+x)^(1/2)+(-1+x)^(1/2
)),x, algorithm="fricas")

[Out]

-1/2*log(-(4*x - 11)*sqrt(x - 1)*sqrt(x - 4) + 4*x^2 - 21*x + 23) + 1/2*log(sqrt(x - 1)*sqrt(x - 4) - x + 7) +
1/2*log(x - 5) + 1/2*log(sqrt(x - 1) + 2) - 1/2*log(sqrt(x - 1) - 2) - 1/2*log(sqrt(x - 4) + 1) + 1/2*log(sqr
t(x - 4) - 1)

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Sympy [A]  time = 171.598, size = 17, normalized size = 0.89 \begin{align*} 2 \log{\left (\sqrt{x - 4} + \sqrt{x - 1} + 1 \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-(-4+x)**(1/2)+x*(-4+x)**(1/2)-4*(-1+x)**(1/2)+x*(-1+x)**(1/2))/(x**2-5*x+4)/(1+(-4+x)**(1/2)+(-1+x
)**(1/2)),x)

[Out]

2*log(sqrt(x - 4) + sqrt(x - 1) + 1)

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Giac [B]  time = 1.40703, size = 81, normalized size = 4.26 \begin{align*} \log \left (\sqrt{x - 1} + 2\right ) - \log \left ({\left | -\sqrt{x - 1} + \sqrt{x - 4} \right |}\right ) - \log \left ({\left | -\sqrt{x - 1} + \sqrt{x - 4} - 1 \right |}\right ) + \log \left ({\left | -\sqrt{x - 1} + \sqrt{x - 4} - 3 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-(-4+x)^(1/2)+x*(-4+x)^(1/2)-4*(-1+x)^(1/2)+x*(-1+x)^(1/2))/(x^2-5*x+4)/(1+(-4+x)^(1/2)+(-1+x)^(1/2
)),x, algorithm="giac")

[Out]

log(sqrt(x - 1) + 2) - log(abs(-sqrt(x - 1) + sqrt(x - 4))) - log(abs(-sqrt(x - 1) + sqrt(x - 4) - 1)) + log(a
bs(-sqrt(x - 1) + sqrt(x - 4) - 3))