Optimal. Leaf size=55 \[ \frac{\sqrt{a-c^2 x^2} \log \left (\tan ^{-1}\left (\frac{c x}{\sqrt{a-c^2 x^2}}\right )\right )}{c \sqrt{d-\frac{c^2 d x^2}{a}}} \]
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Rubi [A] time = 0.107999, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.051, Rules used = {5157, 5153} \[ \frac{\sqrt{a-c^2 x^2} \log \left (\tan ^{-1}\left (\frac{c x}{\sqrt{a-c^2 x^2}}\right )\right )}{c \sqrt{d-\frac{c^2 d x^2}{a}}} \]
Antiderivative was successfully verified.
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Rule 5157
Rule 5153
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{d-\frac{c^2 d x^2}{a}} \tan ^{-1}\left (\frac{c x}{\sqrt{a-c^2 x^2}}\right )} \, dx &=\frac{\sqrt{a-c^2 x^2} \int \frac{1}{\sqrt{a-c^2 x^2} \tan ^{-1}\left (\frac{c x}{\sqrt{a-c^2 x^2}}\right )} \, dx}{\sqrt{d-\frac{c^2 d x^2}{a}}}\\ &=\frac{\sqrt{a-c^2 x^2} \log \left (\tan ^{-1}\left (\frac{c x}{\sqrt{a-c^2 x^2}}\right )\right )}{c \sqrt{d-\frac{c^2 d x^2}{a}}}\\ \end{align*}
Mathematica [A] time = 0.0547002, size = 55, normalized size = 1. \[ \frac{\sqrt{a-c^2 x^2} \log \left (\tan ^{-1}\left (\frac{c x}{\sqrt{a-c^2 x^2}}\right )\right )}{c \sqrt{d-\frac{c^2 d x^2}{a}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 1.108, size = 71, normalized size = 1.3 \begin{align*} -{\frac{a}{d \left ({c}^{2}{x}^{2}-a \right ) c}\sqrt{-{\frac{d \left ({c}^{2}{x}^{2}-a \right ) }{a}}}\sqrt{-{c}^{2}{x}^{2}+a}\ln \left ( \arctan \left ({cx{\frac{1}{\sqrt{-{c}^{2}{x}^{2}+a}}}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-\frac{c^{2} d x^{2}}{a} + d} \arctan \left (\frac{c x}{\sqrt{-c^{2} x^{2} + a}}\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.80791, size = 165, normalized size = 3. \begin{align*} -\frac{\sqrt{-c^{2} x^{2} + a} a \sqrt{-\frac{c^{2} d x^{2} - a d}{a}} \log \left (2 \, \arctan \left (\frac{\sqrt{-c^{2} x^{2} + a} c x}{c^{2} x^{2} - a}\right )\right )}{c^{3} d x^{2} - a c d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{- d \left (-1 + \frac{c^{2} x^{2}}{a}\right )} \operatorname{atan}{\left (\frac{c x}{\sqrt{a - c^{2} x^{2}}} \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-\frac{c^{2} d x^{2}}{a} + d} \arctan \left (\frac{c x}{\sqrt{-c^{2} x^{2} + a}}\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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