Optimal. Leaf size=45 \[ \frac {2}{3} \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b \cos ^3(x)}}{\sqrt {a}}\right )-\frac {2}{3} \sqrt {a+b \cos ^3(x)} \]
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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 = {3309, 272, 52,
65, 214} \begin {gather*} \frac {2}{3} \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b \cos ^3(x)}}{\sqrt {a}}\right )-\frac {2}{3} \sqrt {a+b \cos ^3(x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 214
Rule 272
Rule 3309
Rubi steps
\begin {align*} \int \sqrt {a+b \cos ^3(x)} \tan (x) \, dx &=-\text {Subst}\left (\int \frac {\sqrt {a+b x^3}}{x} \, dx,x,\cos (x)\right )\\ &=-\left (\frac {1}{3} \text {Subst}\left (\int \frac {\sqrt {a+b x}}{x} \, dx,x,\cos ^3(x)\right )\right )\\ &=-\frac {2}{3} \sqrt {a+b \cos ^3(x)}-\frac {1}{3} a \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\cos ^3(x)\right )\\ &=-\frac {2}{3} \sqrt {a+b \cos ^3(x)}-\frac {(2 a) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \cos ^3(x)}\right )}{3 b}\\ &=\frac {2}{3} \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b \cos ^3(x)}}{\sqrt {a}}\right )-\frac {2}{3} \sqrt {a+b \cos ^3(x)}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 45, normalized size = 1.00 \begin {gather*} \frac {2}{3} \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b \cos ^3(x)}}{\sqrt {a}}\right )-\frac {2}{3} \sqrt {a+b \cos ^3(x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.82, size = 34, normalized size = 0.76
method | result | size |
derivativedivides | \(\frac {2 \arctanh \left (\frac {\sqrt {a +b \left (\cos ^{3}\left (x \right )\right )}}{\sqrt {a}}\right ) \sqrt {a}}{3}-\frac {2 \sqrt {a +b \left (\cos ^{3}\left (x \right )\right )}}{3}\) | \(34\) |
default | \(\frac {2 \arctanh \left (\frac {\sqrt {a +b \left (\cos ^{3}\left (x \right )\right )}}{\sqrt {a}}\right ) \sqrt {a}}{3}-\frac {2 \sqrt {a +b \left (\cos ^{3}\left (x \right )\right )}}{3}\) | \(34\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 52, normalized size = 1.16 \begin {gather*} -\frac {1}{3} \, \sqrt {a} \log \left (\frac {\sqrt {b \cos \left (x\right )^{3} + a} - \sqrt {a}}{\sqrt {b \cos \left (x\right )^{3} + a} + \sqrt {a}}\right ) - \frac {2}{3} \, \sqrt {b \cos \left (x\right )^{3} + a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.76, size = 123, normalized size = 2.73 \begin {gather*} \left [\frac {1}{6} \, \sqrt {a} \log \left (-\frac {b^{2} \cos \left (x\right )^{6} + 8 \, a b \cos \left (x\right )^{3} + 4 \, {\left (b \cos \left (x\right )^{3} + 2 \, a\right )} \sqrt {b \cos \left (x\right )^{3} + a} \sqrt {a} + 8 \, a^{2}}{\cos \left (x\right )^{6}}\right ) - \frac {2}{3} \, \sqrt {b \cos \left (x\right )^{3} + a}, -\frac {1}{3} \, \sqrt {-a} \arctan \left (\frac {2 \, \sqrt {b \cos \left (x\right )^{3} + a} \sqrt {-a}}{b \cos \left (x\right )^{3} + 2 \, a}\right ) - \frac {2}{3} \, \sqrt {b \cos \left (x\right )^{3} + 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 ^{3}{\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 = 0.44, size = 38, normalized size = 0.84 \begin {gather*} -\frac {2 \, a \arctan \left (\frac {\sqrt {b \cos \left (x\right )^{3} + a}}{\sqrt {-a}}\right )}{3 \, \sqrt {-a}} - \frac {2}{3} \, \sqrt {b \cos \left (x\right )^{3} + 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 )}^3+a} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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