Optimal. Leaf size=25 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{a+b \cos ^2(x)}}{\sqrt{a}}\right )}{\sqrt{a}} \]
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Rubi [A] time = 0.0599249, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3194, 63, 208} \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{a+b \cos ^2(x)}}{\sqrt{a}}\right )}{\sqrt{a}} \]
Antiderivative was successfully verified.
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Rule 3194
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\tan (x)}{\sqrt{a+b \cos ^2(x)}} \, dx &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\cos ^2(x)\right )\right )\\ &=-\frac{\operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \cos ^2(x)}\right )}{b}\\ &=\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b \cos ^2(x)}}{\sqrt{a}}\right )}{\sqrt{a}}\\ \end{align*}
Mathematica [A] time = 0.0123528, size = 25, normalized size = 1. \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{a+b \cos ^2(x)}}{\sqrt{a}}\right )}{\sqrt{a}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.02, size = 30, normalized size = 1.2 \begin{align*}{\ln \left ({\frac{1}{\cos \left ( x \right ) } \left ( 2\,a+2\,\sqrt{a}\sqrt{a+b \left ( \cos \left ( x \right ) \right ) ^{2}} \right ) } \right ){\frac{1}{\sqrt{a}}}} \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 [A] time = 2.12165, size = 184, normalized size = 7.36 \begin{align*} \left [\frac{\log \left (\frac{b \cos \left (x\right )^{2} + 2 \, \sqrt{b \cos \left (x\right )^{2} + a} \sqrt{a} + 2 \, a}{\cos \left (x\right )^{2}}\right )}{2 \, \sqrt{a}}, -\frac{\sqrt{-a} \arctan \left (\frac{\sqrt{b \cos \left (x\right )^{2} + a} \sqrt{-a}}{a}\right )}{a}\right ] \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{\tan{\left (x \right )}}{\sqrt{a + b \cos ^{2}{\left (x \right )}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15484, size = 32, normalized size = 1.28 \begin{align*} -\frac{\arctan \left (\frac{\sqrt{b \cos \left (x\right )^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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