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 = 0.06, antiderivative size = 40, normalized size of antiderivative = 1.00, 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 50
Rule 63
Rule 208
Rule 3194
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*}
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Mathematica [A] time = 0.03, size = 40, normalized size = 1.00 \[ \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b \cos ^2(x)}}{\sqrt {a}}\right )-\sqrt {a+b \cos ^2(x)} \]
Antiderivative was successfully verified.
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fricas [A] time = 2.08, size = 90, normalized size = 2.25 \[ \left [\frac {1}{2} \, \sqrt {a} \log \left (\frac {b \cos \relax (x)^{2} + 2 \, \sqrt {b \cos \relax (x)^{2} + a} \sqrt {a} + 2 \, a}{\cos \relax (x)^{2}}\right ) - \sqrt {b \cos \relax (x)^{2} + a}, -\sqrt {-a} \arctan \left (\frac {\sqrt {b \cos \relax (x)^{2} + a} \sqrt {-a}}{a}\right ) - \sqrt {b \cos \relax (x)^{2} + a}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.40, size = 38, normalized size = 0.95 \[ -\frac {a \arctan \left (\frac {\sqrt {b \cos \relax (x)^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} - \sqrt {b \cos \relax (x)^{2} + a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 43, normalized size = 1.08 \[ -\sqrt {a +b \left (\cos ^{2}\relax (x )\right )}+\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {a +b \left (\cos ^{2}\relax (x )\right )}}{\cos \relax (x )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.14, size = 95, normalized size = 2.38 \[ \frac {1}{2} \, \sqrt {a} \log \left (b - \frac {\sqrt {-b \sin \relax (x)^{2} + a + b} \sqrt {a}}{\sin \relax (x) - 1} - \frac {a}{\sin \relax (x) - 1}\right ) + \frac {1}{2} \, \sqrt {a} \log \left (-b + \frac {\sqrt {-b \sin \relax (x)^{2} + a + b} \sqrt {a}}{\sin \relax (x) + 1} + \frac {a}{\sin \relax (x) + 1}\right ) - \sqrt {-b \sin \relax (x)^{2} + a + b} \]
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 \[ \int \mathrm {tan}\relax (x)\,\sqrt {b\,{\cos \relax (x)}^2+a} \,d x \]
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 \[ \int \sqrt {a + b \cos ^{2}{\relax (x )}} \tan {\relax (x )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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