Optimal. Leaf size=58 \[ -\frac {\text {Chi}\left (\frac {2 \sqrt {1-a x}}{\sqrt {1+a x}}\right )}{2 a}+\frac {\log \left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{2 a} \]
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Rubi [A]
time = 0.06, 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 = {6813, 3393,
3382} \begin {gather*} \frac {\log \left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )}{2 a}-\frac {\text {Chi}\left (\frac {2 \sqrt {1-a x}}{\sqrt {a x+1}}\right )}{2 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 3382
Rule 3393
Rule 6813
Rubi steps
\begin {align*} \int \frac {\sinh ^2\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{1-a^2 x^2} \, dx &=-\frac {\text {Subst}\left (\int \frac {\sinh ^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 {\cosh (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 {\cosh (2 x)}{x} \, dx,x,\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{2 a}\\ &=-\frac {\text {Chi}\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 = 0.03, size = 57, normalized size = 0.98 \begin {gather*} -\frac {\text {Chi}\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} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\sinh ^{2}\left (\frac {\sqrt {-a x +1}}{\sqrt {a x +1}}\right )}{-a^{2} x^{2}+1}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} - \int \frac {\sinh ^{2}{\left (\frac {\sqrt {- a x + 1}}{\sqrt {a x + 1}} \right )}}{a^{2} x^{2} - 1}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} -\int \frac {{\mathrm {sinh}\left (\frac {\sqrt {1-a\,x}}{\sqrt {a\,x+1}}\right )}^2}{a^2\,x^2-1} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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