Optimal. Leaf size=18 \[ \frac {1}{f \sqrt {a \cos ^2(e+f x)}} \]
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Rubi [A]
time = 0.04, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3255, 3284, 16,
32} \begin {gather*} \frac {1}{f \sqrt {a \cos ^2(e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 16
Rule 32
Rule 3255
Rule 3284
Rubi steps
\begin {align*} \int \frac {\tan (e+f x)}{\sqrt {a-a \sin ^2(e+f x)}} \, dx &=\int \frac {\tan (e+f x)}{\sqrt {a \cos ^2(e+f x)}} \, dx\\ &=-\frac {\text {Subst}\left (\int \frac {1}{x \sqrt {a x}} \, dx,x,\cos ^2(e+f x)\right )}{2 f}\\ &=-\frac {a \text {Subst}\left (\int \frac {1}{(a x)^{3/2}} \, dx,x,\cos ^2(e+f x)\right )}{2 f}\\ &=\frac {1}{f \sqrt {a \cos ^2(e+f x)}}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 18, normalized size = 1.00 \begin {gather*} \frac {1}{f \sqrt {a \cos ^2(e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 20, normalized size = 1.11
method | result | size |
derivativedivides | \(\frac {1}{\sqrt {a -a \left (\sin ^{2}\left (f x +e \right )\right )}\, f}\) | \(20\) |
default | \(\frac {1}{\sqrt {a -a \left (\sin ^{2}\left (f x +e \right )\right )}\, f}\) | \(20\) |
risch | \(\frac {2}{\sqrt {\left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{2} a \,{\mathrm e}^{-2 i \left (f x +e \right )}}\, f}\) | \(32\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 69 vs.
\(2 (17) = 34\).
time = 0.52, size = 69, normalized size = 3.83 \begin {gather*} \frac {\frac {\sqrt {-a \sin \left (f x + e\right )^{2} + a}}{a \sin \left (f x + e\right ) + a} - \frac {\sqrt {-a \sin \left (f x + e\right )^{2} + a}}{a \sin \left (f x + e\right ) - a}}{2 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 27, normalized size = 1.50 \begin {gather*} \frac {\sqrt {a \cos \left (f x + e\right )^{2}}}{a f \cos \left (f x + e\right )^{2}} \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 \frac {\tan {\left (e + f x \right )}}{\sqrt {- a \left (\sin {\left (e + f x \right )} - 1\right ) \left (\sin {\left (e + f x \right )} + 1\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 41 vs.
\(2 (17) = 34\).
time = 0.58, size = 41, normalized size = 2.28 \begin {gather*} \frac {2}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )} \sqrt {a} f \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.39, size = 61, normalized size = 3.39 \begin {gather*} \frac {2\,\sqrt {2}\,\left (\cos \left (2\,e+2\,f\,x\right )+1\right )\,\sqrt {a\,\left (\cos \left (2\,e+2\,f\,x\right )+1\right )}}{a\,f\,\left (4\,\cos \left (2\,e+2\,f\,x\right )+\cos \left (4\,e+4\,f\,x\right )+3\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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