Optimal. Leaf size=50 \[ \frac{\tanh ^{-1}\left (\frac{x \sqrt{b-c}}{\sqrt{c} \sqrt{a-d}}\right )}{\sqrt{c} \sqrt{a-d} \sqrt{b-c}} \]
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Rubi [A] time = 0.0533083, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {208} \[ \frac{\tanh ^{-1}\left (\frac{x \sqrt{b-c}}{\sqrt{c} \sqrt{a-d}}\right )}{\sqrt{c} \sqrt{a-d} \sqrt{b-c}} \]
Antiderivative was successfully verified.
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Rule 208
Rubi steps
\begin{align*} \int \frac{1}{c (a-d)-(b-c) x^2} \, dx &=\frac{\tanh ^{-1}\left (\frac{\sqrt{b-c} x}{\sqrt{c} \sqrt{a-d}}\right )}{\sqrt{b-c} \sqrt{c} \sqrt{a-d}}\\ \end{align*}
Mathematica [A] time = 0.0202594, size = 50, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{x \sqrt{c-b}}{\sqrt{c} \sqrt{a-d}}\right )}{\sqrt{c} \sqrt{a-d} \sqrt{c-b}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 38, normalized size = 0.8 \begin{align*}{{\it Artanh} \left ({ \left ( b-c \right ) x{\frac{1}{\sqrt{c \left ( a-d \right ) \left ( b-c \right ) }}}} \right ){\frac{1}{\sqrt{c \left ( a-d \right ) \left ( b-c \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.52494, size = 370, normalized size = 7.4 \begin{align*} \left [\frac{\log \left (\frac{{\left (b - c\right )} x^{2} + a c - c d + 2 \, \sqrt{a b c - a c^{2} -{\left (b c - c^{2}\right )} d} x}{{\left (b - c\right )} x^{2} - a c + c d}\right )}{2 \, \sqrt{a b c - a c^{2} -{\left (b c - c^{2}\right )} d}}, \frac{\sqrt{-a b c + a c^{2} +{\left (b c - c^{2}\right )} d} \arctan \left (-\frac{\sqrt{-a b c + a c^{2} +{\left (b c - c^{2}\right )} d} x}{a c - c d}\right )}{a b c - a c^{2} -{\left (b c - c^{2}\right )} d}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.352276, size = 104, normalized size = 2.08 \begin{align*} - \frac{\sqrt{\frac{1}{c \left (a - d\right ) \left (b - c\right )}} \log{\left (- a c \sqrt{\frac{1}{c \left (a - d\right ) \left (b - c\right )}} + c d \sqrt{\frac{1}{c \left (a - d\right ) \left (b - c\right )}} + x \right )}}{2} + \frac{\sqrt{\frac{1}{c \left (a - d\right ) \left (b - c\right )}} \log{\left (a c \sqrt{\frac{1}{c \left (a - d\right ) \left (b - c\right )}} - c d \sqrt{\frac{1}{c \left (a - d\right ) \left (b - c\right )}} + x \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.47143, size = 78, normalized size = 1.56 \begin{align*} -\frac{\arctan \left (\frac{b x - c x}{\sqrt{-a b c + a c^{2} + b c d - c^{2} d}}\right )}{\sqrt{-a b c + a c^{2} + b c d - c^{2} d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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