Optimal. Leaf size=34 \[ \frac {1}{b c \left (a+b \log \left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )} \]
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Rubi [A] time = 0.07, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.075, Rules used = {2512, 2302, 30} \[ \frac {1}{b c \left (a+b \log \left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2302
Rule 2512
Rubi steps
\begin {align*} \int \frac {1}{\left (1-c^2 x^2\right ) \left (a+b \log \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {1}{x (a+b \log (x))^2} \, dx,x,\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}{c}\\ &=-\frac {\operatorname {Subst}\left (\int \frac {1}{x^2} \, dx,x,a+b \log \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )}{b c}\\ &=\frac {1}{b c \left (a+b \log \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 34, normalized size = 1.00 \[ \frac {1}{b c \left (a+b \log \left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 29, normalized size = 0.85 \[ \frac {1}{b^{2} c \log \left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right ) + a b c} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 34, normalized size = 1.00 \[ -\frac {2}{b^{2} c \log \left (c x + 1\right ) - b^{2} c \log \left (-c x + 1\right ) - 2 \, a b c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.31, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (-c^{2} x^{2}+1\right ) \left (b \ln \left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )+a \right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.84, size = 34, normalized size = 1.00 \[ -\frac {2}{b^{2} c \log \left (c x + 1\right ) - b^{2} c \log \left (-c x + 1\right ) - 2 \, a b c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ -\int \frac {1}{{\left (a+b\,\ln \left (\frac {\sqrt {1-c\,x}}{\sqrt {c\,x+1}}\right )\right )}^2\,\left (c^2\,x^2-1\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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