Optimal. Leaf size=47 \[ \frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b \cos ^n(x)}}{\sqrt{a}}\right )}{n}-\frac{2 \sqrt{a+b \cos ^n(x)}}{n} \]
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Rubi [A] time = 0.0788629, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {3230, 266, 50, 63, 208} \[ \frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b \cos ^n(x)}}{\sqrt{a}}\right )}{n}-\frac{2 \sqrt{a+b \cos ^n(x)}}{n} \]
Antiderivative was successfully verified.
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Rule 3230
Rule 266
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \sqrt{a+b \cos ^n(x)} \tan (x) \, dx &=-\operatorname{Subst}\left (\int \frac{\sqrt{a+b x^n}}{x} \, dx,x,\cos (x)\right )\\ &=-\frac{\operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{x} \, dx,x,\cos ^n(x)\right )}{n}\\ &=-\frac{2 \sqrt{a+b \cos ^n(x)}}{n}-\frac{a \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\cos ^n(x)\right )}{n}\\ &=-\frac{2 \sqrt{a+b \cos ^n(x)}}{n}-\frac{(2 a) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \cos ^n(x)}\right )}{b n}\\ &=\frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b \cos ^n(x)}}{\sqrt{a}}\right )}{n}-\frac{2 \sqrt{a+b \cos ^n(x)}}{n}\\ \end{align*}
Mathematica [A] time = 0.0303867, size = 46, normalized size = 0.98 \[ -\frac{2 \sqrt{a+b \cos ^n(x)}-2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b \cos ^n(x)}}{\sqrt{a}}\right )}{n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.02, size = 39, normalized size = 0.8 \begin{align*} -{\frac{1}{n} \left ( 2\,\sqrt{a+b \left ( \cos \left ( x \right ) \right ) ^{n}}-2\,\sqrt{a}{\it Artanh} \left ({\frac{\sqrt{a+b \left ( \cos \left ( x \right ) \right ) ^{n}}}{\sqrt{a}}} \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 = 2.10222, size = 254, normalized size = 5.4 \begin{align*} \left [\frac{\sqrt{a} \log \left (\frac{b \cos \left (x\right )^{n} + 2 \, \sqrt{b \cos \left (x\right )^{n} + a} \sqrt{a} + 2 \, a}{\cos \left (x\right )^{n}}\right ) - 2 \, \sqrt{b \cos \left (x\right )^{n} + a}}{n}, -\frac{2 \,{\left (\sqrt{-a} \arctan \left (\frac{\sqrt{b \cos \left (x\right )^{n} + a} \sqrt{-a}}{a}\right ) + \sqrt{b \cos \left (x\right )^{n} + a}\right )}}{n}\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 ^{n}{\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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{b \cos \left (x\right )^{n} + a} \tan \left (x\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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