Integrand size = 36, antiderivative size = 58 \[ \int \frac {\cosh ^3\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{1-a^2 x^2} \, dx=-\frac {3 \text {Chi}\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{4 a}-\frac {\text {Chi}\left (\frac {3 \sqrt {1-a x}}{\sqrt {1+a x}}\right )}{4 a} \]
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Time = 0.08 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {6813, 3393, 3382} \[ \int \frac {\cosh ^3\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{1-a^2 x^2} \, dx=-\frac {3 \text {Chi}\left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )}{4 a}-\frac {\text {Chi}\left (\frac {3 \sqrt {1-a x}}{\sqrt {a x+1}}\right )}{4 a} \]
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Rule 3382
Rule 3393
Rule 6813
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {\cosh ^3(x)}{x} \, dx,x,\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{a} \\ & = -\frac {\text {Subst}\left (\int \left (\frac {3 \cosh (x)}{4 x}+\frac {\cosh (3 x)}{4 x}\right ) \, dx,x,\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{a} \\ & = -\frac {\text {Subst}\left (\int \frac {\cosh (3 x)}{x} \, dx,x,\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{4 a}-\frac {3 \text {Subst}\left (\int \frac {\cosh (x)}{x} \, dx,x,\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{4 a} \\ & = -\frac {3 \text {Chi}\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{4 a}-\frac {\text {Chi}\left (\frac {3 \sqrt {1-a x}}{\sqrt {1+a x}}\right )}{4 a} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.95 \[ \int \frac {\cosh ^3\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{1-a^2 x^2} \, dx=\frac {-3 \text {Chi}\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )-\text {Chi}\left (\frac {3 \sqrt {1-a x}}{\sqrt {1+a x}}\right )}{4 a} \]
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\[\int \frac {\cosh \left (\frac {\sqrt {-a x +1}}{\sqrt {a x +1}}\right )^{3}}{-a^{2} x^{2}+1}d x\]
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\[ \int \frac {\cosh ^3\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{1-a^2 x^2} \, dx=\int { -\frac {\cosh \left (\frac {\sqrt {-a x + 1}}{\sqrt {a x + 1}}\right )^{3}}{a^{2} x^{2} - 1} \,d x } \]
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\[ \int \frac {\cosh ^3\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{1-a^2 x^2} \, dx=- \int \frac {\cosh ^{3}{\left (\frac {\sqrt {- a x + 1}}{\sqrt {a x + 1}} \right )}}{a^{2} x^{2} - 1}\, dx \]
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\[ \int \frac {\cosh ^3\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{1-a^2 x^2} \, dx=\int { -\frac {\cosh \left (\frac {\sqrt {-a x + 1}}{\sqrt {a x + 1}}\right )^{3}}{a^{2} x^{2} - 1} \,d x } \]
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\[ \int \frac {\cosh ^3\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{1-a^2 x^2} \, dx=\int { -\frac {\cosh \left (\frac {\sqrt {-a x + 1}}{\sqrt {a x + 1}}\right )^{3}}{a^{2} x^{2} - 1} \,d x } \]
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Timed out. \[ \int \frac {\cosh ^3\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{1-a^2 x^2} \, dx=-\int \frac {{\mathrm {cosh}\left (\frac {\sqrt {1-a\,x}}{\sqrt {a\,x+1}}\right )}^3}{a^2\,x^2-1} \,d x \]
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