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 = 0.05, antiderivative size = 45, 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 = {3308, 272, 52,
65, 214} \begin {gather*} \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 {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 214
Rule 272
Rule 3308
Rubi steps
\begin {align*} \int \sqrt {a+b \cos ^4(x)} \tan (x) \, dx &=-\left (\frac {1}{2} \text {Subst}\left (\int \frac {\sqrt {a+b x^2}}{x} \, dx,x,\cos ^2(x)\right )\right )\\ &=-\left (\frac {1}{4} \text {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 \text {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 \text {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*}
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Mathematica [A]
time = 0.04, size = 45, normalized size = 1.00 \begin {gather*} \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 {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 44, normalized size = 0.98
method | result | size |
derivativedivides | \(-\frac {\sqrt {a +b \left (\cos ^{4}\left (x \right )\right )}}{2}+\frac {\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {a +b \left (\cos ^{4}\left (x \right )\right )}}{\cos \left (x \right )^{2}}\right )}{2}\) | \(44\) |
default | \(-\frac {\sqrt {a +b \left (\cos ^{4}\left (x \right )\right )}}{2}+\frac {\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {a +b \left (\cos ^{4}\left (x \right )\right )}}{\cos \left (x \right )^{2}}\right )}{2}\) | \(44\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 43, normalized size = 0.96 \begin {gather*} \frac {1}{2} \, \sqrt {a} \operatorname {arsinh}\left (-\frac {a}{\sqrt {a b} {\left (\sin \left (x\right )^{2} - 1\right )}}\right ) - \frac {1}{2} \, \sqrt {b \sin \left (x\right )^{4} - 2 \, b \sin \left (x\right )^{2} + a + b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.48, size = 90, normalized size = 2.00 \begin {gather*} \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 {gather*}
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 \begin {gather*} \int \sqrt {a + b \cos ^{4}{\left (x \right )}} \tan {\left (x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.48, size = 38, normalized size = 0.84 \begin {gather*} -\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 {gather*}
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 \begin {gather*} \int \mathrm {tan}\left (x\right )\,\sqrt {b\,{\cos \left (x\right )}^4+a} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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