Optimal. Leaf size=19 \[ \frac {2}{3} \cos (x) \sin (x) \sqrt {\sec ^4(x) \tan (x)} \]
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Rubi [A]
time = 0.08, antiderivative size = 29, normalized size of antiderivative = 1.53, number of steps
used = 5, number of rules used = 4, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {2024, 1968,
1264, 30} \begin {gather*} \frac {2 \tan ^2(x) \sec ^2(x)}{3 \sqrt {\tan ^5(x)+2 \tan ^3(x)+\tan (x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 1264
Rule 1968
Rule 2024
Rubi steps
\begin {align*} \int \sqrt {\sec ^4(x) \tan (x)} \, dx &=\text {Subst}\left (\int \frac {x \left (1+x^2\right )}{\sqrt {x \left (1+x^2\right )^2}} \, dx,x,\tan (x)\right )\\ &=\text {Subst}\left (\int \frac {x \left (1+x^2\right )}{\sqrt {x+2 x^3+x^5}} \, dx,x,\tan (x)\right )\\ &=\frac {\left (\sqrt {\tan (x)} \sqrt {1+2 \tan ^2(x)+\tan ^4(x)}\right ) \text {Subst}\left (\int \frac {\sqrt {x} \left (1+x^2\right )}{\sqrt {1+2 x^2+x^4}} \, dx,x,\tan (x)\right )}{\sqrt {\tan (x)+2 \tan ^3(x)+\tan ^5(x)}}\\ &=\frac {\left (\sec ^2(x) \sqrt {\tan (x)}\right ) \text {Subst}\left (\int \sqrt {x} \, dx,x,\tan (x)\right )}{\sqrt {\tan (x)+2 \tan ^3(x)+\tan ^5(x)}}\\ &=\frac {2 \sec ^2(x) \tan ^2(x)}{3 \sqrt {\tan (x)+2 \tan ^3(x)+\tan ^5(x)}}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 19, normalized size = 1.00 \begin {gather*} \frac {2}{3} \cos (x) \sin (x) \sqrt {\sec ^4(x) \tan (x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.20, size = 16, normalized size = 0.84
method | result | size |
default | \(\frac {2 \cos \left (x \right ) \sin \left (x \right ) \sqrt {\frac {\sin \left (x \right )}{\cos \left (x \right )^{5}}}}{3}\) | \(16\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 1.05, size = 6, normalized size = 0.32 \begin {gather*} \frac {2}{3} \, \tan \left (x\right )^{\frac {3}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.98, size = 15, normalized size = 0.79 \begin {gather*} \frac {2}{3} \, \sqrt {\frac {\sin \left (x\right )}{\cos \left (x\right )^{5}}} \cos \left (x\right ) \sin \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.51, size = 15, normalized size = 0.79 \begin {gather*} \frac {\sin \left (2\,x\right )\,\sqrt {\frac {\sin \left (x\right )}{{\cos \left (x\right )}^5}}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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