Optimal. Leaf size=38 \[ \frac {a}{f \sqrt {a \cos ^2(e+f x)}}+\frac {\sqrt {a \cos ^2(e+f x)}}{f} \]
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Rubi [A]
time = 0.07, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3255, 3284, 16,
45} \begin {gather*} \frac {a}{f \sqrt {a \cos ^2(e+f x)}}+\frac {\sqrt {a \cos ^2(e+f x)}}{f} \end {gather*}
Antiderivative was successfully verified.
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Rule 16
Rule 45
Rule 3255
Rule 3284
Rubi steps
\begin {align*} \int \sqrt {a-a \sin ^2(e+f x)} \tan ^3(e+f x) \, dx &=\int \sqrt {a \cos ^2(e+f x)} \tan ^3(e+f x) \, dx\\ &=-\frac {\text {Subst}\left (\int \frac {(1-x) \sqrt {a x}}{x^2} \, dx,x,\cos ^2(e+f x)\right )}{2 f}\\ &=-\frac {a^2 \text {Subst}\left (\int \frac {1-x}{(a x)^{3/2}} \, dx,x,\cos ^2(e+f x)\right )}{2 f}\\ &=-\frac {a^2 \text {Subst}\left (\int \left (\frac {1}{(a x)^{3/2}}-\frac {1}{a \sqrt {a x}}\right ) \, dx,x,\cos ^2(e+f x)\right )}{2 f}\\ &=\frac {a}{f \sqrt {a \cos ^2(e+f x)}}+\frac {\sqrt {a \cos ^2(e+f x)}}{f}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 29, normalized size = 0.76 \begin {gather*} \frac {a \left (1+\cos ^2(e+f x)\right )}{f \sqrt {a \cos ^2(e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 13.21, size = 35, normalized size = 0.92
method | result | size |
default | \(\frac {\sqrt {a \left (\cos ^{2}\left (f x +e \right )\right )}\, \left (\cos ^{2}\left (f x +e \right )+1\right )}{\cos \left (f x +e \right )^{2} f}\) | \(35\) |
risch | \(\frac {\sqrt {\left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{2} a \,{\mathrm e}^{-2 i \left (f x +e \right )}}\, {\mathrm e}^{2 i \left (f x +e \right )}}{2 f \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )}+\frac {\sqrt {\left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{2} a \,{\mathrm e}^{-2 i \left (f x +e \right )}}}{2 f \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )}+\frac {2 \sqrt {\left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{2} a \,{\mathrm e}^{-2 i \left (f x +e \right )}}\, {\mathrm e}^{2 i \left (f x +e \right )}}{f \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{2}}\) | \(152\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 48, normalized size = 1.26 \begin {gather*} \frac {\sqrt {-a \sin \left (f x + e\right )^{2} + a} a^{2} + \frac {a^{3}}{\sqrt {-a \sin \left (f x + e\right )^{2} + a}}}{a^{2} f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 34, normalized size = 0.89 \begin {gather*} \frac {\sqrt {a \cos \left (f x + e\right )^{2}} {\left (\cos \left (f x + e\right )^{2} + 1\right )}}{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 \sqrt {- a \left (\sin {\left (e + f x \right )} - 1\right ) \left (\sin {\left (e + f x \right )} + 1\right )} \tan ^{3}{\left (e + f x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.71, size = 39, normalized size = 1.03 \begin {gather*} \frac {4 \, \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 )^{4} - 1\right )} f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.72, size = 69, normalized size = 1.82 \begin {gather*} \frac {\sqrt {2}\,\sqrt {a\,\left (\cos \left (2\,e+2\,f\,x\right )+1\right )}\,\left (8\,\cos \left (2\,e+2\,f\,x\right )+\cos \left (4\,e+4\,f\,x\right )+7\right )}{2\,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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