Optimal. Leaf size=19 \[ -\frac {\sqrt {a \cos ^2(e+f x)}}{f} \]
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Rubi [A]
time = 0.04, antiderivative size = 19, 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 {\sqrt {a \cos ^2(e+f x)}}{f} \end {gather*}
Antiderivative was successfully verified.
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Rule 16
Rule 32
Rule 3255
Rule 3284
Rubi steps
\begin {align*} \int \sqrt {a-a \sin ^2(e+f x)} \tan (e+f x) \, dx &=\int \sqrt {a \cos ^2(e+f x)} \tan (e+f x) \, dx\\ &=-\frac {\text {Subst}\left (\int \frac {\sqrt {a x}}{x} \, dx,x,\cos ^2(e+f x)\right )}{2 f}\\ &=-\frac {a \text {Subst}\left (\int \frac {1}{\sqrt {a x}} \, dx,x,\cos ^2(e+f x)\right )}{2 f}\\ &=-\frac {\sqrt {a \cos ^2(e+f x)}}{f}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 19, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {a \cos ^2(e+f x)}}{f} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 21, normalized size = 1.11
method | result | size |
derivativedivides | \(-\frac {\sqrt {a -a \left (\sin ^{2}\left (f x +e \right )\right )}}{f}\) | \(21\) |
default | \(-\frac {\sqrt {a -a \left (\sin ^{2}\left (f x +e \right )\right )}}{f}\) | \(21\) |
risch | \(-\frac {\sqrt {\left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{2} a \,{\mathrm e}^{-2 i \left (f x +e \right )}}\, {\mathrm e}^{2 i \left (f x +e \right )}}{2 f \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )}-\frac {\sqrt {\left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{2} a \,{\mathrm e}^{-2 i \left (f x +e \right )}}}{2 f \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )}\) | \(99\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.54, size = 21, normalized size = 1.11 \begin {gather*} -\frac {\sqrt {-a \sin \left (f x + e\right )^{2} + a}}{f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 17, normalized size = 0.89 \begin {gather*} -\frac {\sqrt {a \cos \left (f x + e\right )^{2}}}{f} \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 \left (\sin {\left (e + f x \right )} - 1\right ) \left (\sin {\left (e + f x \right )} + 1\right )} \tan {\left (e + f x \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. 39 vs.
\(2 (18) = 36\).
time = 0.49, size = 39, normalized size = 2.05 \begin {gather*} \frac {2 \, \sqrt {a} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 1\right )}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 1\right )} f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 15.25, size = 20, normalized size = 1.05 \begin {gather*} -\frac {\sqrt {a-a\,{\sin \left (e+f\,x\right )}^2}}{f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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