3.31.55 \(\int \sqrt {\frac {-1+a x-2 x^2+2 a x^3-x^4+a x^5}{1+a x-2 x^2-2 a x^3+x^4+a x^5}} \, dx\)

Optimal. Leaf size=466 \[ \frac {2 \sqrt {-(a+1)^2} \tan ^{-1}\left (\frac {\frac {(a+1)^2 x^2}{\sqrt {-a^2-2 a-1} \sqrt {a^2-1}}+\frac {(a+1)^2}{\sqrt {-a^2-2 a-1} \sqrt {a^2-1}}}{(x-1) (x+1) \sqrt {\frac {a x^5+2 a x^3+a x-x^4-2 x^2-1}{a x^5-2 a x^3+a x+x^4-2 x^2+1}}}\right )}{\sqrt {a^2-1}}-\frac {2 \sqrt {-(a-1)^2} \tan ^{-1}\left (\frac {\frac {(a-1)^2 x^2}{\sqrt {-a^2+2 a-1} \sqrt {a^2-1}}+\frac {(a-1)^2}{\sqrt {-a^2+2 a-1} \sqrt {a^2-1}}}{(x-1) (x+1) \sqrt {\frac {a x^5+2 a x^3+a x-x^4-2 x^2-1}{a x^5-2 a x^3+a x+x^4-2 x^2+1}}}\right )}{\sqrt {a^2-1}}+\frac {\sqrt {\frac {a x^5+2 a x^3+a x-x^4-2 x^2-1}{a x^5-2 a x^3+a x+x^4-2 x^2+1}} \left (a x^3-a x+x^2-1\right )}{a \left (x^2+1\right )}-\frac {2 \tanh ^{-1}\left (\frac {(x-1) (x+1) \sqrt {\frac {a x^5+2 a x^3+a x-x^4-2 x^2-1}{a x^5-2 a x^3+a x+x^4-2 x^2+1}}}{x^2+1}\right )}{a} \]

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Rubi [F]  time = 0.06, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \sqrt {\frac {-1+a x-2 x^2+2 a x^3-x^4+a x^5}{1+a x-2 x^2-2 a x^3+x^4+a x^5}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[Sqrt[(-1 + a*x - 2*x^2 + 2*a*x^3 - x^4 + a*x^5)/(1 + a*x - 2*x^2 - 2*a*x^3 + x^4 + a*x^5)],x]

[Out]

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

Rubi steps

\begin {align*} \int \sqrt {\frac {-1+a x-2 x^2+2 a x^3-x^4+a x^5}{1+a x-2 x^2-2 a x^3+x^4+a x^5}} \, dx &=\int \sqrt {\frac {(-1+a x) \left (1+x^2\right )^2}{(1+a x) \left (-1+x^2\right )^2}} \, dx\\ \end {align*}

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Mathematica [A]  time = 0.65, size = 238, normalized size = 0.51 \begin {gather*} \frac {\left (x^2-1\right ) \sqrt {a x+1} \sqrt {\frac {\left (x^2+1\right )^2 (a x-1)}{\left (x^2-1\right )^2 (a x+1)}} \left ((a+1) \left (\sqrt {\frac {a-1}{a+1}} \left (\sqrt {a x+1} (a x-1)^{3/2}+2 \sqrt {-(a x-1)^2} \sin ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )+2 a (a x-1) \tanh ^{-1}\left (\sqrt {\frac {a-1}{a+1}} \sqrt {\frac {a x-1}{a x+1}}\right )\right )-2 (a-1) a (a x-1) \tanh ^{-1}\left (\frac {\sqrt {\frac {a x-1}{a x+1}}}{\sqrt {\frac {a-1}{a+1}}}\right )\right )}{a \sqrt {\frac {a-1}{a+1}} (a+1) \left (x^2+1\right ) (a x-1)^{3/2}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[Sqrt[(-1 + a*x - 2*x^2 + 2*a*x^3 - x^4 + a*x^5)/(1 + a*x - 2*x^2 - 2*a*x^3 + x^4 + a*x^5)],x]

[Out]

(Sqrt[1 + a*x]*(-1 + x^2)*Sqrt[((-1 + a*x)*(1 + x^2)^2)/((1 + a*x)*(-1 + x^2)^2)]*(-2*(-1 + a)*a*(-1 + a*x)*Ar
cTanh[Sqrt[(-1 + a*x)/(1 + a*x)]/Sqrt[(-1 + a)/(1 + a)]] + (1 + a)*(Sqrt[(-1 + a)/(1 + a)]*((-1 + a*x)^(3/2)*S
qrt[1 + a*x] + 2*Sqrt[-(-1 + a*x)^2]*ArcSin[Sqrt[1 - a*x]/Sqrt[2]]) + 2*a*(-1 + a*x)*ArcTanh[Sqrt[(-1 + a)/(1
+ a)]*Sqrt[(-1 + a*x)/(1 + a*x)]])))/(a*Sqrt[(-1 + a)/(1 + a)]*(1 + a)*(-1 + a*x)^(3/2)*(1 + x^2))

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IntegrateAlgebraic [A]  time = 1.99, size = 511, normalized size = 1.10 \begin {gather*} \frac {\left (-1-a x+x^2+a x^3\right ) \sqrt {\frac {-1+a x-2 x^2+2 a x^3-x^4+a x^5}{1+a x-2 x^2-2 a x^3+x^4+a x^5}}}{a \left (1+x^2\right )}+\frac {2 (-1+a) \tan ^{-1}\left (\frac {(-1+a) \left (1+x^2\right )}{\sqrt {1-a} \sqrt {1+a} (-1+x) (1+x) \sqrt {\frac {-1+a x-2 x^2+2 a x^3-x^4+a x^5}{1+a x-2 x^2-2 a x^3+x^4+a x^5}}}\right )}{\sqrt {1-a^2}}+\frac {2 \sqrt {-1-a} \tan ^{-1}\left (\frac {\frac {1}{\sqrt {-1-a} \sqrt {-1+a}}+\frac {a}{\sqrt {-1-a} \sqrt {-1+a}}+\frac {x^2}{\sqrt {-1-a} \sqrt {-1+a}}+\frac {a x^2}{\sqrt {-1-a} \sqrt {-1+a}}}{(-1+x) (1+x) \sqrt {\frac {-1+a x-2 x^2+2 a x^3-x^4+a x^5}{1+a x-2 x^2-2 a x^3+x^4+a x^5}}}\right )}{\sqrt {-1+a}}+\frac {\log \left (-1-x^2+\left (-1+x^2\right ) \sqrt {\frac {-1+a x-2 x^2+2 a x^3-x^4+a x^5}{1+a x-2 x^2-2 a x^3+x^4+a x^5}}\right )}{a}-\frac {\log \left (a+a x^2+\left (-a+a x^2\right ) \sqrt {\frac {-1+a x-2 x^2+2 a x^3-x^4+a x^5}{1+a x-2 x^2-2 a x^3+x^4+a x^5}}\right )}{a} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[Sqrt[(-1 + a*x - 2*x^2 + 2*a*x^3 - x^4 + a*x^5)/(1 + a*x - 2*x^2 - 2*a*x^3 + x^4 + a*x^5)],
x]

[Out]

((-1 - a*x + x^2 + a*x^3)*Sqrt[(-1 + a*x - 2*x^2 + 2*a*x^3 - x^4 + a*x^5)/(1 + a*x - 2*x^2 - 2*a*x^3 + x^4 + a
*x^5)])/(a*(1 + x^2)) + (2*(-1 + a)*ArcTan[((-1 + a)*(1 + x^2))/(Sqrt[1 - a]*Sqrt[1 + a]*(-1 + x)*(1 + x)*Sqrt
[(-1 + a*x - 2*x^2 + 2*a*x^3 - x^4 + a*x^5)/(1 + a*x - 2*x^2 - 2*a*x^3 + x^4 + a*x^5)])])/Sqrt[1 - a^2] + (2*S
qrt[-1 - a]*ArcTan[(1/(Sqrt[-1 - a]*Sqrt[-1 + a]) + a/(Sqrt[-1 - a]*Sqrt[-1 + a]) + x^2/(Sqrt[-1 - a]*Sqrt[-1
+ a]) + (a*x^2)/(Sqrt[-1 - a]*Sqrt[-1 + a]))/((-1 + x)*(1 + x)*Sqrt[(-1 + a*x - 2*x^2 + 2*a*x^3 - x^4 + a*x^5)
/(1 + a*x - 2*x^2 - 2*a*x^3 + x^4 + a*x^5)])])/Sqrt[-1 + a] + Log[-1 - x^2 + (-1 + x^2)*Sqrt[(-1 + a*x - 2*x^2
+ 2*a*x^3 - x^4 + a*x^5)/(1 + a*x - 2*x^2 - 2*a*x^3 + x^4 + a*x^5)]]/a - Log[a + a*x^2 + (-a + a*x^2)*Sqrt[(-
1 + a*x - 2*x^2 + 2*a*x^3 - x^4 + a*x^5)/(1 + a*x - 2*x^2 - 2*a*x^3 + x^4 + a*x^5)]]/a

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fricas [A]  time = 0.87, size = 1023, normalized size = 2.20 \begin {gather*} \left [\frac {{\left (a x^{2} + a\right )} \sqrt {\frac {a + 1}{a - 1}} \log \left (-\frac {a^{2} x^{3} + a^{2} x + x^{2} + {\left ({\left (a^{2} - a\right )} x^{3} + {\left (a - 1\right )} x^{2} - {\left (a^{2} - a\right )} x - a + 1\right )} \sqrt {\frac {a x^{5} + 2 \, a x^{3} - x^{4} + a x - 2 \, x^{2} - 1}{a x^{5} - 2 \, a x^{3} + x^{4} + a x - 2 \, x^{2} + 1}} \sqrt {\frac {a + 1}{a - 1}} + 1}{x^{3} + x^{2} + x + 1}\right ) + {\left (a x^{2} + a\right )} \sqrt {\frac {a - 1}{a + 1}} \log \left (\frac {a^{2} x^{3} + a^{2} x - x^{2} - {\left ({\left (a^{2} + a\right )} x^{3} + {\left (a + 1\right )} x^{2} - {\left (a^{2} + a\right )} x - a - 1\right )} \sqrt {\frac {a x^{5} + 2 \, a x^{3} - x^{4} + a x - 2 \, x^{2} - 1}{a x^{5} - 2 \, a x^{3} + x^{4} + a x - 2 \, x^{2} + 1}} \sqrt {\frac {a - 1}{a + 1}} - 1}{x^{3} - x^{2} + x - 1}\right ) - {\left (x^{2} + 1\right )} \log \left (\frac {x^{2} + {\left (x^{2} - 1\right )} \sqrt {\frac {a x^{5} + 2 \, a x^{3} - x^{4} + a x - 2 \, x^{2} - 1}{a x^{5} - 2 \, a x^{3} + x^{4} + a x - 2 \, x^{2} + 1}} + 1}{x^{2} + 1}\right ) + {\left (x^{2} + 1\right )} \log \left (-\frac {x^{2} - {\left (x^{2} - 1\right )} \sqrt {\frac {a x^{5} + 2 \, a x^{3} - x^{4} + a x - 2 \, x^{2} - 1}{a x^{5} - 2 \, a x^{3} + x^{4} + a x - 2 \, x^{2} + 1}} + 1}{x^{2} + 1}\right ) + {\left (a x^{3} - a x + x^{2} - 1\right )} \sqrt {\frac {a x^{5} + 2 \, a x^{3} - x^{4} + a x - 2 \, x^{2} - 1}{a x^{5} - 2 \, a x^{3} + x^{4} + a x - 2 \, x^{2} + 1}}}{a x^{2} + a}, -\frac {2 \, {\left (a x^{2} + a\right )} \sqrt {-\frac {a + 1}{a - 1}} \arctan \left (\frac {{\left ({\left (a - 1\right )} x^{2} - a + 1\right )} \sqrt {\frac {a x^{5} + 2 \, a x^{3} - x^{4} + a x - 2 \, x^{2} - 1}{a x^{5} - 2 \, a x^{3} + x^{4} + a x - 2 \, x^{2} + 1}} \sqrt {-\frac {a + 1}{a - 1}}}{{\left (a + 1\right )} x^{2} + a + 1}\right ) - 2 \, {\left (a x^{2} + a\right )} \sqrt {-\frac {a - 1}{a + 1}} \arctan \left (\frac {{\left ({\left (a + 1\right )} x^{2} - a - 1\right )} \sqrt {\frac {a x^{5} + 2 \, a x^{3} - x^{4} + a x - 2 \, x^{2} - 1}{a x^{5} - 2 \, a x^{3} + x^{4} + a x - 2 \, x^{2} + 1}} \sqrt {-\frac {a - 1}{a + 1}}}{{\left (a - 1\right )} x^{2} + a - 1}\right ) + {\left (x^{2} + 1\right )} \log \left (\frac {x^{2} + {\left (x^{2} - 1\right )} \sqrt {\frac {a x^{5} + 2 \, a x^{3} - x^{4} + a x - 2 \, x^{2} - 1}{a x^{5} - 2 \, a x^{3} + x^{4} + a x - 2 \, x^{2} + 1}} + 1}{x^{2} + 1}\right ) - {\left (x^{2} + 1\right )} \log \left (-\frac {x^{2} - {\left (x^{2} - 1\right )} \sqrt {\frac {a x^{5} + 2 \, a x^{3} - x^{4} + a x - 2 \, x^{2} - 1}{a x^{5} - 2 \, a x^{3} + x^{4} + a x - 2 \, x^{2} + 1}} + 1}{x^{2} + 1}\right ) - {\left (a x^{3} - a x + x^{2} - 1\right )} \sqrt {\frac {a x^{5} + 2 \, a x^{3} - x^{4} + a x - 2 \, x^{2} - 1}{a x^{5} - 2 \, a x^{3} + x^{4} + a x - 2 \, x^{2} + 1}}}{a x^{2} + a}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((a*x^5+2*a*x^3-x^4+a*x-2*x^2-1)/(a*x^5-2*a*x^3+x^4+a*x-2*x^2+1))^(1/2),x, algorithm="fricas")

[Out]

[((a*x^2 + a)*sqrt((a + 1)/(a - 1))*log(-(a^2*x^3 + a^2*x + x^2 + ((a^2 - a)*x^3 + (a - 1)*x^2 - (a^2 - a)*x -
a + 1)*sqrt((a*x^5 + 2*a*x^3 - x^4 + a*x - 2*x^2 - 1)/(a*x^5 - 2*a*x^3 + x^4 + a*x - 2*x^2 + 1))*sqrt((a + 1)
/(a - 1)) + 1)/(x^3 + x^2 + x + 1)) + (a*x^2 + a)*sqrt((a - 1)/(a + 1))*log((a^2*x^3 + a^2*x - x^2 - ((a^2 + a
)*x^3 + (a + 1)*x^2 - (a^2 + a)*x - a - 1)*sqrt((a*x^5 + 2*a*x^3 - x^4 + a*x - 2*x^2 - 1)/(a*x^5 - 2*a*x^3 + x
^4 + a*x - 2*x^2 + 1))*sqrt((a - 1)/(a + 1)) - 1)/(x^3 - x^2 + x - 1)) - (x^2 + 1)*log((x^2 + (x^2 - 1)*sqrt((
a*x^5 + 2*a*x^3 - x^4 + a*x - 2*x^2 - 1)/(a*x^5 - 2*a*x^3 + x^4 + a*x - 2*x^2 + 1)) + 1)/(x^2 + 1)) + (x^2 + 1
)*log(-(x^2 - (x^2 - 1)*sqrt((a*x^5 + 2*a*x^3 - x^4 + a*x - 2*x^2 - 1)/(a*x^5 - 2*a*x^3 + x^4 + a*x - 2*x^2 +
1)) + 1)/(x^2 + 1)) + (a*x^3 - a*x + x^2 - 1)*sqrt((a*x^5 + 2*a*x^3 - x^4 + a*x - 2*x^2 - 1)/(a*x^5 - 2*a*x^3
+ x^4 + a*x - 2*x^2 + 1)))/(a*x^2 + a), -(2*(a*x^2 + a)*sqrt(-(a + 1)/(a - 1))*arctan(((a - 1)*x^2 - a + 1)*sq
rt((a*x^5 + 2*a*x^3 - x^4 + a*x - 2*x^2 - 1)/(a*x^5 - 2*a*x^3 + x^4 + a*x - 2*x^2 + 1))*sqrt(-(a + 1)/(a - 1))
/((a + 1)*x^2 + a + 1)) - 2*(a*x^2 + a)*sqrt(-(a - 1)/(a + 1))*arctan(((a + 1)*x^2 - a - 1)*sqrt((a*x^5 + 2*a*
x^3 - x^4 + a*x - 2*x^2 - 1)/(a*x^5 - 2*a*x^3 + x^4 + a*x - 2*x^2 + 1))*sqrt(-(a - 1)/(a + 1))/((a - 1)*x^2 +
a - 1)) + (x^2 + 1)*log((x^2 + (x^2 - 1)*sqrt((a*x^5 + 2*a*x^3 - x^4 + a*x - 2*x^2 - 1)/(a*x^5 - 2*a*x^3 + x^4
+ a*x - 2*x^2 + 1)) + 1)/(x^2 + 1)) - (x^2 + 1)*log(-(x^2 - (x^2 - 1)*sqrt((a*x^5 + 2*a*x^3 - x^4 + a*x - 2*x
^2 - 1)/(a*x^5 - 2*a*x^3 + x^4 + a*x - 2*x^2 + 1)) + 1)/(x^2 + 1)) - (a*x^3 - a*x + x^2 - 1)*sqrt((a*x^5 + 2*a
*x^3 - x^4 + a*x - 2*x^2 - 1)/(a*x^5 - 2*a*x^3 + x^4 + a*x - 2*x^2 + 1)))/(a*x^2 + a)]

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((a*x^5+2*a*x^3-x^4+a*x-2*x^2-1)/(a*x^5-2*a*x^3+x^4+a*x-2*x^2+1))^(1/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Warn
ing, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Check [ab
s(x^4-1),abs(a*x^5-2*a*x^3+a*x+x^4-2*x^2+1)]sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l)
Error: Bad Argument Value

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maple [A]  time = 0.24, size = 402, normalized size = 0.86

method	result	size
risch	\(\frac {\left (a x +1\right ) \sqrt {\frac {\left (x^{2}+1\right )^{2} \left (a x -1\right )}{\left (x^{2}-1\right )^{2} \left (a x +1\right )}}\, \left (x^{2}-1\right )}{a \left (x^{2}+1\right )}+\frac {\left (-\frac {\ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{\sqrt {a^{2}}}+\frac {\ln \left (\frac {2 a^{2}-2+2 a^{2} \left (-1+x \right )+2 \sqrt {a^{2}-1}\, \sqrt {a^{2} \left (-1+x \right )^{2}+2 a^{2} \left (-1+x \right )+a^{2}-1}}{-1+x}\right )}{\sqrt {a^{2}-1}}-\frac {\ln \left (\frac {2 a^{2}-2+2 a^{2} \left (-1+x \right )+2 \sqrt {a^{2}-1}\, \sqrt {a^{2} \left (-1+x \right )^{2}+2 a^{2} \left (-1+x \right )+a^{2}-1}}{-1+x}\right ) a}{\sqrt {a^{2}-1}}-\frac {\ln \left (\frac {2 a^{2}-2-2 a^{2} \left (1+x \right )+2 \sqrt {a^{2}-1}\, \sqrt {a^{2} \left (1+x \right )^{2}-2 a^{2} \left (1+x \right )+a^{2}-1}}{1+x}\right ) a}{\sqrt {a^{2}-1}}-\frac {\ln \left (\frac {2 a^{2}-2-2 a^{2} \left (1+x \right )+2 \sqrt {a^{2}-1}\, \sqrt {a^{2} \left (1+x \right )^{2}-2 a^{2} \left (1+x \right )+a^{2}-1}}{1+x}\right )}{\sqrt {a^{2}-1}}\right ) \sqrt {\frac {\left (x^{2}+1\right )^{2} \left (a x -1\right )}{\left (x^{2}-1\right )^{2} \left (a x +1\right )}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, \left (x^{2}-1\right )}{\left (x^{2}+1\right ) \left (a x -1\right )}\)	\(402\)
default	\(-\frac {\sqrt {\frac {a \,x^{5}+2 a \,x^{3}-x^{4}+a x -2 x^{2}-1}{a \,x^{5}-2 a \,x^{3}+x^{4}+a x -2 x^{2}+1}}\, \left (x^{2}-1\right ) \left (a x +1\right ) \left (\ln \left (\frac {-2 a^{2} x +2 \sqrt {a^{2}-1}\, \sqrt {a^{2} x^{2}-1}-2}{1+x}\right ) \sqrt {a^{2}}\, \sqrt {a^{2}-1}\, a^{2}+\ln \left (\frac {2 a^{2} x +2 \sqrt {a^{2}-1}\, \sqrt {a^{2} x^{2}-1}-2}{-1+x}\right ) \sqrt {a^{2}}\, \sqrt {a^{2}-1}\, a^{2}+\ln \left (\frac {-2 a^{2} x +2 \sqrt {a^{2}-1}\, \sqrt {a^{2} x^{2}-1}-2}{1+x}\right ) \sqrt {a^{2}}\, \sqrt {a^{2}-1}\, a -\ln \left (\frac {2 a^{2} x +2 \sqrt {a^{2}-1}\, \sqrt {a^{2} x^{2}-1}-2}{-1+x}\right ) \sqrt {a^{2}}\, \sqrt {a^{2}-1}\, a -\ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{\sqrt {a^{2}}}\right ) a^{3}+2 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{3}-2 \sqrt {a^{2}}\, \sqrt {a^{2} x^{2}-1}\, a^{2}+\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, a^{2}-a \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{\sqrt {a^{2}}}\right )+\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}\right )}{\left (x^{2}+1\right ) \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, \left (-1+a \right ) \left (1+a \right ) a \sqrt {a^{2}}}\)	\(475\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a*x^5+2*a*x^3-x^4+a*x-2*x^2-1)/(a*x^5-2*a*x^3+x^4+a*x-2*x^2+1))^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {\frac {a x^{5} + 2 \, a x^{3} - x^{4} + a x - 2 \, x^{2} - 1}{a x^{5} - 2 \, a x^{3} + x^{4} + a x - 2 \, x^{2} + 1}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((a*x^5+2*a*x^3-x^4+a*x-2*x^2-1)/(a*x^5-2*a*x^3+x^4+a*x-2*x^2+1))^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt((a*x^5 + 2*a*x^3 - x^4 + a*x - 2*x^2 - 1)/(a*x^5 - 2*a*x^3 + x^4 + a*x - 2*x^2 + 1)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \sqrt {\frac {a\,x^5-x^4+2\,a\,x^3-2\,x^2+a\,x-1}{a\,x^5+x^4-2\,a\,x^3-2\,x^2+a\,x+1}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a*x + 2*a*x^3 + a*x^5 - 2*x^2 - x^4 - 1)/(a*x - 2*a*x^3 + a*x^5 - 2*x^2 + x^4 + 1))^(1/2),x)

[Out]

int(((a*x + 2*a*x^3 + a*x^5 - 2*x^2 - x^4 - 1)/(a*x - 2*a*x^3 + a*x^5 - 2*x^2 + x^4 + 1))^(1/2), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((a*x**5+2*a*x**3-x**4+a*x-2*x**2-1)/(a*x**5-2*a*x**3+x**4+a*x-2*x**2+1))**(1/2),x)

[Out]

Timed out

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