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

Optimal. Leaf size=21 \[ \log \left (\sqrt{x^2-4}+\sqrt{x^2-1}+1\right ) \]

[Out]

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

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Rubi [A]  time = 0.285303, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 85, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.012, Rules used = {6684} \[ \log \left (\sqrt{x^2-4}+\sqrt{x^2-1}+1\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [B]  time = 1.23474, size = 97, normalized size = 4.62 \[ \frac{1}{4} \log \left (-5 x^2-4 \sqrt{x^2-4} \sqrt{x^2-1}+17\right )+\frac{1}{4} \log \left (-2 x^2-2 \sqrt{x^2-4} \sqrt{x^2-1}+5\right )-\frac{1}{2} \tanh ^{-1}\left (\sqrt{x^2-4}\right )+\frac{1}{2} \tanh ^{-1}\left (\frac{\sqrt{x^2-1}}{2}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.136, size = 1088, normalized size = 51.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/(-2+5^(1/2))/(5^(1/2)-1)/(5^(1/2)+1)/(2+5^(1/2))*5^(1/2)*ln(x+((x-5^(1/2))^2+2*5^(1/2)*(x-5^(1/2))+1)^(1/2))
-1/(-2+5^(1/2))/(5^(1/2)-1)/(5^(1/2)+1)/(2+5^(1/2))*5^(1/2)*ln(x+((x+5^(1/2))^2-2*5^(1/2)*(x+5^(1/2))+1)^(1/2)
)-1/2/(5^(1/2)+1)/(5^(1/2)-1)*ln(x+((1+x)^2-2-2*x)^(1/2))+1/2/(5^(1/2)+1)/(5^(1/2)-1)*((-1+x)^2-2+2*x)^(1/2)+1
/2/(5^(1/2)+1)/(5^(1/2)-1)*ln(x+((-1+x)^2-2+2*x)^(1/2))+1/2/(5^(1/2)+1)/(5^(1/2)-1)*((1+x)^2-2-2*x)^(1/2)-1/4/
(2+5^(1/2))/(-2+5^(1/2))*((-2+x)^2+4*x-8)^(1/2)-1/2/(2+5^(1/2))/(-2+5^(1/2))*ln(x+((-2+x)^2+4*x-8)^(1/2))-1/4/
(2+5^(1/2))/(-2+5^(1/2))*((2+x)^2-4*x-8)^(1/2)+1/2/(2+5^(1/2))/(-2+5^(1/2))*ln(x+((2+x)^2-4*x-8)^(1/2))-1/2/(-
2+5^(1/2))/(5^(1/2)-1)/(5^(1/2)+1)/(2+5^(1/2))*5^(1/2)*ln(x+((x-5^(1/2))^2+2*5^(1/2)*(x-5^(1/2))+4)^(1/2))+1/2
/(-2+5^(1/2))/(5^(1/2)-1)/(5^(1/2)+1)/(2+5^(1/2))*5^(1/2)*ln(x+((x+5^(1/2))^2-2*5^(1/2)*(x+5^(1/2))+4)^(1/2))+
1/4*ln(x^2-5)-1/(-2+5^(1/2))/(5^(1/2)-1)/(5^(1/2)+1)/(2+5^(1/2))*arctanh(1/2*(2+2*5^(1/2)*(x-5^(1/2)))/((x-5^(
1/2))^2+2*5^(1/2)*(x-5^(1/2))+1)^(1/2))+1/(-2+5^(1/2))/(5^(1/2)-1)/(5^(1/2)+1)/(2+5^(1/2))*((x-5^(1/2))^2+2*5^
(1/2)*(x-5^(1/2))+1)^(1/2)-1/(-2+5^(1/2))/(5^(1/2)-1)/(5^(1/2)+1)/(2+5^(1/2))*arctanh(1/2*(2-2*5^(1/2)*(x+5^(1
/2)))/((x+5^(1/2))^2-2*5^(1/2)*(x+5^(1/2))+1)^(1/2))+1/(-2+5^(1/2))/(5^(1/2)-1)/(5^(1/2)+1)/(2+5^(1/2))*((x+5^
(1/2))^2-2*5^(1/2)*(x+5^(1/2))+1)^(1/2)+1/(-2+5^(1/2))/(5^(1/2)-1)/(5^(1/2)+1)/(2+5^(1/2))*arctanh(1/4*(8+2*5^
(1/2)*(x-5^(1/2)))/((x-5^(1/2))^2+2*5^(1/2)*(x-5^(1/2))+4)^(1/2))-1/2/(-2+5^(1/2))/(5^(1/2)-1)/(5^(1/2)+1)/(2+
5^(1/2))*((x-5^(1/2))^2+2*5^(1/2)*(x-5^(1/2))+4)^(1/2)+1/(-2+5^(1/2))/(5^(1/2)-1)/(5^(1/2)+1)/(2+5^(1/2))*arct
anh(1/4*(8-2*5^(1/2)*(x+5^(1/2)))/((x+5^(1/2))^2-2*5^(1/2)*(x+5^(1/2))+4)^(1/2))-1/2/(-2+5^(1/2))/(5^(1/2)-1)/
(5^(1/2)+1)/(2+5^(1/2))*((x+5^(1/2))^2-2*5^(1/2)*(x+5^(1/2))+4)^(1/2)+7/8*(x^2-4)^(1/2)*(x^2-1)^(1/2)/(x^4-5*x
^2+4)^(1/2)*arctanh(1/4*(5*x^2-17)/(x^4-5*x^2+4)^(1/2))+1/8*(x^2-4)^(1/2)*(x^2-1)^(1/2)*(2*ln(-5/2+x^2+(x^4-5*
x^2+4)^(1/2))-5*arctanh(1/4*(5*x^2-17)/(x^4-5*x^2+4)^(1/2)))/(x^4-5*x^2+4)^(1/2)

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Maxima [B]  time = 1.20735, size = 231, normalized size = 11. \begin{align*} \frac{1}{4} \, \log \left (x + 1\right ) + \frac{3}{8} \, \log \left (x - 1\right ) + \frac{1}{8} \, \log \left (x - 2\right ) + \frac{1}{4} \, \log \left (\frac{2 \, x^{4} + 4 \,{\left (x^{2} - 3\right )} \sqrt{x + 1} \sqrt{x - 1} - 7 \, x^{2} + 2 \,{\left ({\left (x^{2} - 1\right )} \sqrt{x + 1} \sqrt{x - 1} \sqrt{x - 2} +{\left (2 \, x^{2} - 3\right )} \sqrt{x - 2}\right )} \sqrt{x + 2} + 3}{2 \,{\left ({\left (x^{2} - 1\right )} \sqrt{x + 1} \sqrt{x - 1} \sqrt{x - 2} +{\left (2 \, x^{2} - 3\right )} \sqrt{x - 2}\right )}}\right ) + \frac{1}{4} \, \log \left (\frac{{\left (x^{2} - 1\right )} \sqrt{x + 1} \sqrt{x - 1} + 2 \, x^{2} - 3}{{\left (x^{2} - 1\right )} \sqrt{x - 1}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*log(x + 1) + 3/8*log(x - 1) + 1/8*log(x - 2) + 1/4*log(1/2*(2*x^4 + 4*(x^2 - 3)*sqrt(x + 1)*sqrt(x - 1) -
7*x^2 + 2*((x^2 - 1)*sqrt(x + 1)*sqrt(x - 1)*sqrt(x - 2) + (2*x^2 - 3)*sqrt(x - 2))*sqrt(x + 2) + 3)/((x^2 - 1
)*sqrt(x + 1)*sqrt(x - 1)*sqrt(x - 2) + (2*x^2 - 3)*sqrt(x - 2))) + 1/4*log(((x^2 - 1)*sqrt(x + 1)*sqrt(x - 1)
+ 2*x^2 - 3)/((x^2 - 1)*sqrt(x - 1)))

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Fricas [B]  time = 1.92392, size = 452, normalized size = 21.52 \begin{align*} -\frac{1}{4} \, \log \left (4 \, x^{4} -{\left (4 \, x^{2} - 11\right )} \sqrt{x^{2} - 1} \sqrt{x^{2} - 4} - 21 \, x^{2} + 23\right ) - \frac{1}{4} \, \log \left (x^{2} - \sqrt{x^{2} - 1}{\left (x + 2\right )} + 2 \, x - 1\right ) + \frac{1}{4} \, \log \left (x^{2} - \sqrt{x^{2} - 4}{\left (x + 1\right )} + x - 4\right ) - \frac{1}{4} \, \log \left (x^{2} - \sqrt{x^{2} - 4}{\left (x - 1\right )} - x - 4\right ) + \frac{1}{4} \, \log \left (x^{2} - \sqrt{x^{2} - 1}{\left (x - 2\right )} - 2 \, x - 1\right ) + \frac{1}{4} \, \log \left (x^{2} - 5\right ) + \frac{1}{4} \, \log \left (-x^{2} + \sqrt{x^{2} - 1} \sqrt{x^{2} - 4} + 7\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B]  time = 1.38784, size = 105, normalized size = 5. \begin{align*} \frac{1}{2} \, \log \left (\sqrt{x^{2} - 1} + 2\right ) - \frac{1}{2} \, \log \left ({\left | -\sqrt{x^{2} - 1} + \sqrt{x^{2} - 4} \right |}\right ) - \frac{1}{2} \, \log \left ({\left | -\sqrt{x^{2} - 1} + \sqrt{x^{2} - 4} - 1 \right |}\right ) + \frac{1}{2} \, \log \left ({\left | -\sqrt{x^{2} - 1} + \sqrt{x^{2} - 4} - 3 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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