Optimal. Leaf size=29 \[ -\frac {\text {Ei}\left (-\frac {2 \sqrt {1-a x} \log (F)}{\sqrt {a x+1}}\right )}{a} \]
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Rubi [A] time = 0.10, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {2291, 2178} \[ -\frac {\text {Ei}\left (-\frac {2 \sqrt {1-a x} \log (F)}{\sqrt {a x+1}}\right )}{a} \]
Antiderivative was successfully verified.
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Rule 2178
Rule 2291
Rubi steps
\begin {align*} \int \frac {F^{-\frac {2 \sqrt {1-a x}}{\sqrt {1+a x}}}}{1-a^2 x^2} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {F^{-2 x}}{x} \, dx,x,\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{a}\\ &=-\frac {\text {Ei}\left (-\frac {2 \sqrt {1-a x} \log (F)}{\sqrt {1+a x}}\right )}{a}\\ \end {align*}
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Mathematica [A] time = 0.30, size = 29, normalized size = 1.00 \[ -\frac {\text {Ei}\left (-\frac {2 \sqrt {1-a x} \log (F)}{\sqrt {a x+1}}\right )}{a} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.13, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {1}{{\left (a^{2} x^{2} - 1\right )} F^{\frac {2 \, \sqrt {-a x + 1}}{\sqrt {a x + 1}}}}, x\right ) \]
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 \[ \int -\frac {1}{{\left (a^{2} x^{2} - 1\right )} F^{\frac {2 \, \sqrt {-a x + 1}}{\sqrt {a x + 1}}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.10, size = 0, normalized size = 0.00 \[ \int \frac {F^{-\frac {2 \sqrt {-a x +1}}{\sqrt {a x +1}}}}{-a^{2} x^{2}+1}\, dx \]
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 \[ -\int \frac {1}{{\left (a^{2} x^{2} - 1\right )} F^{\frac {2 \, \sqrt {-a x + 1}}{\sqrt {a x + 1}}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int -\frac {1}{F^{\frac {2\,\sqrt {1-a\,x}}{\sqrt {a\,x+1}}}\,\left (a^2\,x^2-1\right )} \,d x \]
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 \[ - \int \frac {1}{F^{\frac {2 \sqrt {- a x + 1}}{\sqrt {a x + 1}}} a^{2} x^{2} - F^{\frac {2 \sqrt {- a x + 1}}{\sqrt {a x + 1}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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