Optimal. Leaf size=37 \[ \frac{1}{2 b c \left (a+b \log \left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )^2} \]
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Rubi [A] time = 0.0651834, antiderivative size = 37, normalized size of antiderivative = 1., 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} \[ \frac{1}{2 b c \left (a+b \log \left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )^2} \]
Antiderivative was successfully verified.
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Rule 2512
Rule 2302
Rule 30
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 )^3} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{1}{x (a+b \log (x))^3} \, dx,x,\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )}{c}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{1}{x^3} \, dx,x,a+b \log \left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )}{b c}\\ &=\frac{1}{2 b c \left (a+b \log \left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^2}\\ \end{align*}
Mathematica [A] time = 0.0105605, size = 37, normalized size = 1. \[ \frac{1}{2 b c \left (a+b \log \left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.349, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{-{c}^{2}{x}^{2}+1} \left ( a+b\ln \left ({\sqrt{-cx+1}{\frac{1}{\sqrt{cx+1}}}} \right ) \right ) ^{-3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.60891, size = 108, normalized size = 2.92 \begin{align*} \frac{2}{b^{3} c \log \left (c x + 1\right )^{2} + b^{3} c \log \left (-c x + 1\right )^{2} - 4 \, a b^{2} c \log \left (c x + 1\right ) + 4 \, a^{2} b c - 2 \,{\left (b^{3} c \log \left (c x + 1\right ) - 2 \, a b^{2} c\right )} \log \left (-c x + 1\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.84696, size = 142, normalized size = 3.84 \begin{align*} \frac{1}{2 \,{\left (b^{3} c \log \left (\frac{\sqrt{-c x + 1}}{\sqrt{c x + 1}}\right )^{2} + 2 \, a b^{2} c \log \left (\frac{\sqrt{-c x + 1}}{\sqrt{c x + 1}}\right ) + a^{2} b c\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.30372, size = 115, normalized size = 3.11 \begin{align*} \frac{2}{b^{3} c \log \left (c x + 1\right )^{2} - 2 \, b^{3} c \log \left (c x + 1\right ) \log \left (-c x + 1\right ) + b^{3} c \log \left (-c x + 1\right )^{2} - 4 \, a b^{2} c \log \left (c x + 1\right ) + 4 \, a b^{2} c \log \left (-c x + 1\right ) + 4 \, a^{2} b c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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