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.
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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.
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IntegrateAlgebraic [A] time = 1.96, size = 511, normalized size = 1.10 \begin {gather*} \frac {2 (a-1) \tan ^{-1}\left (\frac {(a-1) \left (x^2+1\right )}{\sqrt {1-a} \sqrt {a+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 {1-a^2}}+\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 {\log \left (\left (x^2-1\right ) \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}-\frac {\log \left (a x^2+\left (a x^2-a\right ) \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}}+a\right )}{a}+\frac {2 \sqrt {-a-1} \tan ^{-1}\left (\frac {\frac {a x^2}{\sqrt {-a-1} \sqrt {a-1}}+\frac {x^2}{\sqrt {-a-1} \sqrt {a-1}}+\frac {a}{\sqrt {-a-1} \sqrt {a-1}}+\frac {1}{\sqrt {-a-1} \sqrt {a-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-1}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.65, 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.
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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.
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maple [A] time = 0.07, size = 475, normalized size = 1.02 \begin {gather*} -\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}-1}\, \sqrt {a^{2}}\, 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}-1}\, \sqrt {a^{2}}\, a^{2}+\sqrt {\left (a x +1\right ) \left (a x -1\right )}\, \sqrt {a^{2}}\, a^{2}+2 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{3}-\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}-1}\, \sqrt {a^{2}}\, 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}-1}\, \sqrt {a^{2}}\, a -\ln \left (\frac {a^{2} x +\sqrt {\left (a x +1\right ) \left (a x -1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{3}-2 \sqrt {a^{2}}\, \sqrt {a^{2} x^{2}-1}\, a^{2}+\sqrt {\left (a x +1\right ) \left (a x -1\right )}\, \sqrt {a^{2}}-a \ln \left (\frac {a^{2} x +\sqrt {\left (a x +1\right ) \left (a x -1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\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}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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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.
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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.
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