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 = 0.08, antiderivative size = 47, normalized size of antiderivative = 1.00, 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 50
Rule 63
Rule 208
Rule 266
Rule 3230
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*}
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Mathematica [A] time = 0.03, 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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fricas [A] time = 0.53, size = 97, normalized size = 2.06 \[ \left [\frac {\sqrt {a} \log \left (\frac {b \cos \relax (x)^{n} + 2 \, \sqrt {b \cos \relax (x)^{n} + a} \sqrt {a} + 2 \, a}{\cos \relax (x)^{n}}\right ) - 2 \, \sqrt {b \cos \relax (x)^{n} + a}}{n}, -\frac {2 \, {\left (\sqrt {-a} \arctan \left (\frac {\sqrt {b \cos \relax (x)^{n} + a} \sqrt {-a}}{a}\right ) + \sqrt {b \cos \relax (x)^{n} + a}\right )}}{n}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.56, size = 46, normalized size = 0.98 \[ -\frac {2 \, {\left (\frac {a b \arctan \left (\frac {\sqrt {b \cos \relax (x)^{n} + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} + \sqrt {b \cos \relax (x)^{n} + a} b\right )}}{b n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 39, normalized size = 0.83 \[ -\frac {2 \sqrt {a +b \left (\cos ^{n}\relax (x )\right )}-2 \sqrt {a}\, \arctanh \left (\frac {\sqrt {a +b \left (\cos ^{n}\relax (x )\right )}}{\sqrt {a}}\right )}{n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.81, size = 58, normalized size = 1.23 \[ -\frac {\sqrt {a} \log \left (\frac {\sqrt {b \cos \relax (x)^{n} + a} - \sqrt {a}}{\sqrt {b \cos \relax (x)^{n} + a} + \sqrt {a}}\right )}{n} - \frac {2 \, \sqrt {b \cos \relax (x)^{n} + a}}{n} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \mathrm {tan}\relax (x)\,\sqrt {a+b\,{\cos \relax (x)}^n} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {a + b \cos ^{n}{\relax (x )}} \tan {\relax (x )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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