Optimal. Leaf size=45 \[ \frac{1}{2} \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b \cos ^4(x)}}{\sqrt{a}}\right )-\frac{1}{2} \sqrt{a+b \cos ^4(x)} \]
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Rubi [A] time = 0.0765254, antiderivative size = 45, 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 = {3229, 266, 50, 63, 208} \[ \frac{1}{2} \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b \cos ^4(x)}}{\sqrt{a}}\right )-\frac{1}{2} \sqrt{a+b \cos ^4(x)} \]
Antiderivative was successfully verified.
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Rule 3229
Rule 266
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \sqrt{a+b \cos ^4(x)} \tan (x) \, dx &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \frac{\sqrt{a+b x^2}}{x} \, dx,x,\cos ^2(x)\right )\right )\\ &=-\left (\frac{1}{4} \operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{x} \, dx,x,\cos ^4(x)\right )\right )\\ &=-\frac{1}{2} \sqrt{a+b \cos ^4(x)}-\frac{1}{4} a \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\cos ^4(x)\right )\\ &=-\frac{1}{2} \sqrt{a+b \cos ^4(x)}-\frac{a \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \cos ^4(x)}\right )}{2 b}\\ &=\frac{1}{2} \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b \cos ^4(x)}}{\sqrt{a}}\right )-\frac{1}{2} \sqrt{a+b \cos ^4(x)}\\ \end{align*}
Mathematica [A] time = 0.0314078, size = 45, normalized size = 1. \[ \frac{1}{2} \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b \cos ^4(x)}}{\sqrt{a}}\right )-\frac{1}{2} \sqrt{a+b \cos ^4(x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.032, size = 44, normalized size = 1. \begin{align*} -{\frac{1}{2}\sqrt{a+b \left ( \cos \left ( x \right ) \right ) ^{4}}}+{\frac{1}{2}\sqrt{a}\ln \left ({\frac{1}{ \left ( \cos \left ( x \right ) \right ) ^{2}} \left ( 2\,a+2\,\sqrt{a}\sqrt{a+b \left ( \cos \left ( x \right ) \right ) ^{4}} \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 = 3.21236, size = 259, normalized size = 5.76 \begin{align*} \left [\frac{1}{4} \, \sqrt{a} \log \left (\frac{b \cos \left (x\right )^{4} + 2 \, \sqrt{b \cos \left (x\right )^{4} + a} \sqrt{a} + 2 \, a}{\cos \left (x\right )^{4}}\right ) - \frac{1}{2} \, \sqrt{b \cos \left (x\right )^{4} + a}, -\frac{1}{2} \, \sqrt{-a} \arctan \left (\frac{\sqrt{b \cos \left (x\right )^{4} + a} \sqrt{-a}}{a}\right ) - \frac{1}{2} \, \sqrt{b \cos \left (x\right )^{4} + 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 ^{4}{\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 = 1.15677, size = 51, normalized size = 1.13 \begin{align*} -\frac{a \arctan \left (\frac{\sqrt{b \cos \left (x\right )^{4} + a}}{\sqrt{-a}}\right )}{2 \, \sqrt{-a}} - \frac{1}{2} \, \sqrt{b \cos \left (x\right )^{4} + a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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