Integrand size = 36, antiderivative size = 58 \[ \int \frac {\sin ^2\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{1-a^2 x^2} \, dx=\frac {\operatorname {CosIntegral}\left (\frac {2 \sqrt {1-a x}}{\sqrt {1+a x}}\right )}{2 a}-\frac {\log \left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{2 a} \]
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Time = 0.09 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {6813, 3393, 3383} \[ \int \frac {\sin ^2\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{1-a^2 x^2} \, dx=\frac {\operatorname {CosIntegral}\left (\frac {2 \sqrt {1-a x}}{\sqrt {a x+1}}\right )}{2 a}-\frac {\log \left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )}{2 a} \]
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Rule 3383
Rule 3393
Rule 6813
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {\sin ^2(x)}{x} \, dx,x,\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{a} \\ & = -\frac {\text {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 {\text {Subst}\left (\int \frac {\cos (2 x)}{x} \, dx,x,\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{2 a} \\ & = \frac {\operatorname {CosIntegral}\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*}
Time = 0.22 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.98 \[ \int \frac {\sin ^2\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{1-a^2 x^2} \, dx=\frac {\operatorname {CosIntegral}\left (\frac {2 \sqrt {1-a x}}{\sqrt {1+a x}}\right )}{2 a}-\frac {\log (1-a x)}{4 a}+\frac {\log (1+a x)}{4 a} \]
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\[\int \frac {\sin \left (\frac {\sqrt {-a x +1}}{\sqrt {a x +1}}\right )^{2}}{-a^{2} x^{2}+1}d x\]
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\[ \int \frac {\sin ^2\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{1-a^2 x^2} \, dx=\int { -\frac {\sin \left (\frac {\sqrt {-a x + 1}}{\sqrt {a x + 1}}\right )^{2}}{a^{2} x^{2} - 1} \,d x } \]
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\[ \int \frac {\sin ^2\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{1-a^2 x^2} \, dx=- \int \frac {\sin ^{2}{\left (\frac {\sqrt {- a x + 1}}{\sqrt {a x + 1}} \right )}}{a^{2} x^{2} - 1}\, dx \]
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\[ \int \frac {\sin ^2\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{1-a^2 x^2} \, dx=\int { -\frac {\sin \left (\frac {\sqrt {-a x + 1}}{\sqrt {a x + 1}}\right )^{2}}{a^{2} x^{2} - 1} \,d x } \]
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\[ \int \frac {\sin ^2\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{1-a^2 x^2} \, dx=\int { -\frac {\sin \left (\frac {\sqrt {-a x + 1}}{\sqrt {a x + 1}}\right )^{2}}{a^{2} x^{2} - 1} \,d x } \]
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Timed out. \[ \int \frac {\sin ^2\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{1-a^2 x^2} \, dx=-\int \frac {{\sin \left (\frac {\sqrt {1-a\,x}}{\sqrt {a\,x+1}}\right )}^2}{a^2\,x^2-1} \,d x \]
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