Optimal. Leaf size=62 \[ -\frac {2 \tan ^{-1}\left (\frac {a x}{\sqrt {\left (15-a^2\right ) x^2+x^6+6 x^5+15 x^4+20 x^3+6 x+1}+x^3+3 x^2+3 x+1}\right )}{a} \]
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Rubi [F] time = 0.50, 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 \frac {-1+2 x}{(1+x) \sqrt {-a^2 x^2+(1+x)^6}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {-1+2 x}{(1+x) \sqrt {-a^2 x^2+(1+x)^6}} \, dx &=\int \left (\frac {2}{\sqrt {-a^2 x^2+(1+x)^6}}-\frac {3}{(1+x) \sqrt {-a^2 x^2+(1+x)^6}}\right ) \, dx\\ &=2 \int \frac {1}{\sqrt {-a^2 x^2+(1+x)^6}} \, dx-3 \int \frac {1}{(1+x) \sqrt {-a^2 x^2+(1+x)^6}} \, dx\\ \end {align*}
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Mathematica [F] time = 0.31, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-1+2 x}{(1+x) \sqrt {-a^2 x^2+(1+x)^6}} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.44, size = 62, normalized size = 1.00 \begin {gather*} -\frac {2 \tan ^{-1}\left (\frac {a x}{\sqrt {\left (15-a^2\right ) x^2+x^6+6 x^5+15 x^4+20 x^3+6 x+1}+x^3+3 x^2+3 x+1}\right )}{a} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 47, normalized size = 0.76 \begin {gather*} \frac {\arctan \left (\frac {\sqrt {x^{6} + 6 \, x^{5} + 15 \, x^{4} - {\left (a^{2} - 15\right )} x^{2} + 20 \, x^{3} + 6 \, x + 1}}{a x}\right )}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 \, x - 1}{\sqrt {{\left (x + 1\right )}^{6} - a^{2} x^{2}} {\left (x + 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.34, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-1+2 x}{\left (1+x \right ) \sqrt {-a^{2} x^{2}+\left (1+x \right )^{6}}}\, dx \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 \frac {2 \, x - 1}{\sqrt {{\left (x + 1\right )}^{6} - a^{2} x^{2}} {\left (x + 1\right )}}\,{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 = -0.02 \begin {gather*} \int \frac {2\,x-1}{\sqrt {{\left (x+1\right )}^6-a^2\,x^2}\,\left (x+1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 x - 1}{\sqrt {\left (- a x + x^{3} + 3 x^{2} + 3 x + 1\right ) \left (a x + x^{3} + 3 x^{2} + 3 x + 1\right )} \left (x + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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