Optimal. Leaf size=58 \[ \frac{\text{CosIntegral}\left (\frac{2 \sqrt{1-a x}}{\sqrt{a x+1}}\right )}{2 a}-\frac{\log \left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right )}{2 a} \]
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Rubi [A] time = 0.0817573, antiderivative size = 58, normalized size of antiderivative = 1., 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 6681
Rule 3312
Rule 3302
Rubi steps
\begin{align*} \int \frac{\sin ^2\left (\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right )}{1-a^2 x^2} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{\sin ^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*}
Mathematica [A] time = 0.0749969, size = 57, normalized size = 0.98 \[ \frac{\text{CosIntegral}\left (\frac{2 \sqrt{1-a x}}{\sqrt{a x+1}}\right )}{2 a}-\frac{\log (1-a x)}{4 a}+\frac{\log (a x+1)}{4 a} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.07, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{-{a}^{2}{x}^{2}+1} \left ( \sin \left ({\sqrt{-ax+1}{\frac{1}{\sqrt{ax+1}}}} \right ) \right ) ^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\cos \left (\frac{\sqrt{-a x + 1}}{\sqrt{a x + 1}}\right )^{2} - 1}{a^{2} x^{2} - 1}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{\sin ^{2}{\left (\frac{\sqrt{- a x + 1}}{\sqrt{a x + 1}} \right )}}{a^{2} x^{2} - 1}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{\sin \left (\frac{\sqrt{-a x + 1}}{\sqrt{a x + 1}}\right )^{2}}{a^{2} x^{2} - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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