Optimal. Leaf size=18 \[ \frac {1}{f \sqrt {a \cos ^2(e+f x)}} \]
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Rubi [A] time = 0.06, 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 = {3176, 3205, 16, 32} \[ \frac {1}{f \sqrt {a \cos ^2(e+f x)}} \]
Antiderivative was successfully verified.
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Rule 16
Rule 32
Rule 3176
Rule 3205
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 {\operatorname {Subst}\left (\int \frac {1}{x \sqrt {a x}} \, dx,x,\cos ^2(e+f x)\right )}{2 f}\\ &=-\frac {a \operatorname {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 = 0.03, size = 18, normalized size = 1.00 \[ \frac {1}{f \sqrt {a \cos ^2(e+f x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 27, normalized size = 1.50 \[ \frac {\sqrt {a \cos \left (f x + e\right )^{2}}}{a f \cos \left (f x + e\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 20, normalized size = 1.11 \[ \frac {1}{f \sqrt {a -a \left (\sin ^{2}\left (f x +e \right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.44, size = 65, normalized size = 3.61 \[ \frac {\frac {\sqrt {-a \sin \left (f x + e\right )^{2} + a}}{a \sin \left (f x + e\right ) + a} - \frac {\sqrt {-a \sin \left (f x + e\right )^{2} + a}}{a \sin \left (f x + e\right ) - a}}{2 \, f} \]
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 \[ \frac {2\,\sqrt {2}\,\left (\cos \left (2\,e+2\,f\,x\right )+1\right )\,\sqrt {a\,\left (\cos \left (2\,e+2\,f\,x\right )+1\right )}}{a\,f\,\left (4\,\cos \left (2\,e+2\,f\,x\right )+\cos \left (4\,e+4\,f\,x\right )+3\right )} \]
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 \[ \int \frac {\tan {\left (e + f x \right )}}{\sqrt {- a \left (\sin {\left (e + f x \right )} - 1\right ) \left (\sin {\left (e + f x \right )} + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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