Optimal. Leaf size=58 \[ -\frac {\text {Ci}\left (\frac {2 \sqrt {1-a x}}{\sqrt {a x+1}}\right )}{2 a}-\frac {\log \left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )}{2 a} \]
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Rubi [A] time = 0.08, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {6681, 3312, 3302} \[ -\frac {\text {CosIntegral}\left (\frac {2 \sqrt {1-a x}}{\sqrt {a x+1}}\right )}{2 a}-\frac {\log \left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )}{2 a} \]
Antiderivative was successfully verified.
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Rule 3302
Rule 3312
Rule 6681
Rubi steps
\begin {align*} \int \frac {\cos ^2\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{1-a^2 x^2} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {\cos ^2(x)}{x} \, dx,x,\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{a}\\ &=-\frac {\operatorname {Subst}\left (\int \left (\frac {1}{2 x}+\frac {\cos (2 x)}{2 x}\right ) \, dx,x,\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{a}\\ &=-\frac {\log \left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{2 a}-\frac {\operatorname {Subst}\left (\int \frac {\cos (2 x)}{x} \, dx,x,\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{2 a}\\ &=-\frac {\text {Ci}\left (\frac {2 \sqrt {1-a x}}{\sqrt {1+a x}}\right )}{2 a}-\frac {\log \left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{2 a}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 51, normalized size = 0.88 \[ -\frac {\text {Ci}\left (\frac {2 \sqrt {1-a x}}{\sqrt {a x+1}}\right )+\log \left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )}{2 a} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\cos \left (\frac {\sqrt {-a x + 1}}{\sqrt {a x + 1}}\right )^{2}}{a^{2} x^{2} - 1}, x\right ) \]
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 \[ \int -\frac {\cos \left (\frac {\sqrt {-a x + 1}}{\sqrt {a x + 1}}\right )^{2}}{a^{2} x^{2} - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.36, size = 0, normalized size = 0.00 \[ \int \frac {\cos ^{2}\left (\frac {\sqrt {-a x +1}}{\sqrt {a x +1}}\right )}{-a^{2} x^{2}+1}\, dx \]
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 \[ -\frac {a \int \frac {\cos \left (\frac {2 \, \sqrt {-a x + 1}}{\sqrt {a x + 1}}\right )}{a^{2} x^{2} - 1}\,{d x} + a \int \frac {\cos \left (\frac {2 \, \sqrt {-a x + 1}}{\sqrt {a x + 1}}\right )}{{\left (a^{2} x^{2} - 1\right )} \cos \left (\frac {2 \, \sqrt {-a x + 1}}{\sqrt {a x + 1}}\right )^{2} + {\left (a^{2} x^{2} - 1\right )} \sin \left (\frac {2 \, \sqrt {-a x + 1}}{\sqrt {a x + 1}}\right )^{2}}\,{d x} - \log \left (a x + 1\right ) + \log \left (a x - 1\right )}{4 \, a} \]
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 \[ -\int \frac {{\cos \left (\frac {\sqrt {1-a\,x}}{\sqrt {a\,x+1}}\right )}^2}{a^2\,x^2-1} \,d x \]
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 \[ - \int \frac {\cos ^{2}{\left (\frac {\sqrt {- a x + 1}}{\sqrt {a x + 1}} \right )}}{a^{2} x^{2} - 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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