Optimal. Leaf size=40 \[ \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b \cos ^2(x)}}{\sqrt{a}}\right )-\sqrt{a+b \cos ^2(x)} \]
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Rubi [A] time = 0.0627249, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {3194, 50, 63, 208} \[ \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b \cos ^2(x)}}{\sqrt{a}}\right )-\sqrt{a+b \cos ^2(x)} \]
Antiderivative was successfully verified.
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Rule 3194
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \sqrt{a+b \cos ^2(x)} \tan (x) \, dx &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{x} \, dx,x,\cos ^2(x)\right )\right )\\ &=-\sqrt{a+b \cos ^2(x)}-\frac{1}{2} a \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\cos ^2(x)\right )\\ &=-\sqrt{a+b \cos ^2(x)}-\frac{a \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \cos ^2(x)}\right )}{b}\\ &=\sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b \cos ^2(x)}}{\sqrt{a}}\right )-\sqrt{a+b \cos ^2(x)}\\ \end{align*}
Mathematica [A] time = 0.026397, size = 40, normalized size = 1. \[ \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b \cos ^2(x)}}{\sqrt{a}}\right )-\sqrt{a+b \cos ^2(x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.018, size = 43, normalized size = 1.1 \begin{align*} -\sqrt{a+b \left ( \cos \left ( x \right ) \right ) ^{2}}+\sqrt{a}\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 ) \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 = 3.37192, size = 243, normalized size = 6.08 \begin{align*} \left [\frac{1}{2} \, \sqrt{a} \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 ) - \sqrt{b \cos \left (x\right )^{2} + a}, -\sqrt{-a} \arctan \left (\frac{\sqrt{b \cos \left (x\right )^{2} + a} \sqrt{-a}}{a}\right ) - \sqrt{b \cos \left (x\right )^{2} + 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 \sqrt{a + b \cos ^{2}{\left (x \right )}} \tan{\left (x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14544, size = 51, normalized size = 1.27 \begin{align*} -\frac{a \arctan \left (\frac{\sqrt{b \cos \left (x\right )^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} - \sqrt{b \cos \left (x\right )^{2} + a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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