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 = 0.06, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, 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 63
Rule 208
Rule 3194
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*}
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Mathematica [A] time = 0.02, size = 25, normalized size = 1.00 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \cos ^2(x)}}{\sqrt {a}}\right )}{\sqrt {a}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.80, size = 67, normalized size = 2.68 \[ \left [\frac {\log \left (\frac {b \cos \relax (x)^{2} + 2 \, \sqrt {b \cos \relax (x)^{2} + a} \sqrt {a} + 2 \, a}{\cos \relax (x)^{2}}\right )}{2 \, \sqrt {a}}, -\frac {\sqrt {-a} \arctan \left (\frac {\sqrt {b \cos \relax (x)^{2} + a} \sqrt {-a}}{a}\right )}{a}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.31, size = 24, normalized size = 0.96 \[ -\frac {\arctan \left (\frac {\sqrt {b \cos \relax (x)^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 30, normalized size = 1.20 \[ \frac {\ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {a +b \left (\cos ^{2}\relax (x )\right )}}{\cos \relax (x )}\right )}{\sqrt {a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.89, size = 81, normalized size = 3.24 \[ \frac {\log \left (b - \frac {\sqrt {-b \sin \relax (x)^{2} + a + b} \sqrt {a}}{\sin \relax (x) - 1} - \frac {a}{\sin \relax (x) - 1}\right )}{2 \, \sqrt {a}} + \frac {\log \left (-b + \frac {\sqrt {-b \sin \relax (x)^{2} + a + b} \sqrt {a}}{\sin \relax (x) + 1} + \frac {a}{\sin \relax (x) + 1}\right )}{2 \, \sqrt {a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {\mathrm {tan}\relax (x)}{\sqrt {b\,{\cos \relax (x)}^2+a}} \,d x \]
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 \[ \int \frac {\tan {\relax (x )}}{\sqrt {a + b \cos ^{2}{\relax (x )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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