Optimal. Leaf size=26 \[ -\frac {\text {Shi}\left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )}{a} \]
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Rubi [A] time = 0.04, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {6681, 3298} \[ -\frac {\text {Shi}\left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )}{a} \]
Antiderivative was successfully verified.
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Rule 3298
Rule 6681
Rubi steps
\begin {align*} \int \frac {\sinh \left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{1-a^2 x^2} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {\sinh (x)}{x} \, dx,x,\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{a}\\ &=-\frac {\text {Shi}\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{a}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 26, normalized size = 1.00 \[ -\frac {\text {Shi}\left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )}{a} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.08, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sinh \left (\frac {\sqrt {-a x + 1}}{\sqrt {a x + 1}}\right )}{a^{2} x^{2} - 1}, x\right ) \]
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 \[ \int -\frac {\sinh \left (\frac {\sqrt {-a x + 1}}{\sqrt {a x + 1}}\right )}{a^{2} x^{2} - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.10, size = 0, normalized size = 0.00 \[ \int \frac {\sinh \left (\frac {\sqrt {-a x +1}}{\sqrt {a x +1}}\right )}{-a^{2} x^{2}+1}\, dx \]
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 \[ -\int \frac {\sinh \left (\frac {\sqrt {-a x + 1}}{\sqrt {a x + 1}}\right )}{a^{2} x^{2} - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \[ -\int \frac {\mathrm {sinh}\left (\frac {\sqrt {1-a\,x}}{\sqrt {a\,x+1}}\right )}{a^2\,x^2-1} \,d x \]
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 \[ - \int \frac {\sinh {\left (\frac {\sqrt {- a x + 1}}{\sqrt {a x + 1}} \right )}}{a^{2} x^{2} - 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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