Optimal. Leaf size=42 \[ \text {Int}\left (\frac {\text {csch}^2\left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )}{(1-a x) (a x+1)},x\right ) \]
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Rubi [A] time = 0.08, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\text {csch}^2\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{1-a^2 x^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\text {csch}^2\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{1-a^2 x^2} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {\text {csch}^2(x)}{x} \, dx,x,\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{a}\\ \end {align*}
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Mathematica [A] time = 38.06, size = 0, normalized size = 0.00 \[ \int \frac {\text {csch}^2\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{1-a^2 x^2} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.94, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {1}{{\left (a^{2} x^{2} - 1\right )} \sinh \left (\frac {\sqrt {-a x + 1}}{\sqrt {a x + 1}}\right )^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {1}{{\left (a^{2} x^{2} - 1\right )} \sinh \left (\frac {\sqrt {-a x + 1}}{\sqrt {a x + 1}}\right )^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (-a^{2} x^{2}+1\right ) \sinh \left (\frac {\sqrt {-a x +1}}{\sqrt {a x +1}}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {2 \, \sqrt {a x + 1}}{\sqrt {-a x + 1} a e^{\left (\frac {2 \, \sqrt {-a x + 1}}{\sqrt {a x + 1}}\right )} - \sqrt {-a x + 1} a} - \int \frac {\sqrt {a x + 1}}{{\left (a^{2} x^{2} - 1\right )} \sqrt {-a x + 1} e^{\left (\frac {\sqrt {-a x + 1}}{\sqrt {a x + 1}}\right )} + {\left (a^{2} x^{2} - 1\right )} \sqrt {-a x + 1}}\,{d x} + \int \frac {\sqrt {a x + 1}}{{\left (a^{2} x^{2} - 1\right )} \sqrt {-a x + 1} e^{\left (\frac {\sqrt {-a x + 1}}{\sqrt {a x + 1}}\right )} - {\left (a^{2} x^{2} - 1\right )} \sqrt {-a x + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.02 \[ -\int \frac {1}{{\mathrm {sinh}\left (\frac {\sqrt {1-a\,x}}{\sqrt {a\,x+1}}\right )}^2\,\left (a^2\,x^2-1\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ - \int \frac {1}{a^{2} x^{2} \sinh ^{2}{\left (\frac {\sqrt {- a x + 1}}{\sqrt {a x + 1}} \right )} - \sinh ^{2}{\left (\frac {\sqrt {- a x + 1}}{\sqrt {a x + 1}} \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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