Optimal. Leaf size=50 \[ -\frac {\sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a \cos ^2(e+f x)}}{\sqrt {a}}\right )}{f}+\frac {\sqrt {a \cos ^2(e+f x)}}{f} \]
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Rubi [A]
time = 0.05, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {3255, 3284, 52,
65, 212} \begin {gather*} \frac {\sqrt {a \cos ^2(e+f x)}}{f}-\frac {\sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a \cos ^2(e+f x)}}{\sqrt {a}}\right )}{f} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 212
Rule 3255
Rule 3284
Rubi steps
\begin {align*} \int \cot (e+f x) \sqrt {a-a \sin ^2(e+f x)} \, dx &=\int \sqrt {a \cos ^2(e+f x)} \cot (e+f x) \, dx\\ &=-\frac {\text {Subst}\left (\int \frac {\sqrt {a x}}{1-x} \, dx,x,\cos ^2(e+f x)\right )}{2 f}\\ &=\frac {\sqrt {a \cos ^2(e+f x)}}{f}-\frac {a \text {Subst}\left (\int \frac {1}{(1-x) \sqrt {a x}} \, dx,x,\cos ^2(e+f x)\right )}{2 f}\\ &=\frac {\sqrt {a \cos ^2(e+f x)}}{f}-\frac {\text {Subst}\left (\int \frac {1}{1-\frac {x^2}{a}} \, dx,x,\sqrt {a \cos ^2(e+f x)}\right )}{f}\\ &=-\frac {\sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a \cos ^2(e+f x)}}{\sqrt {a}}\right )}{f}+\frac {\sqrt {a \cos ^2(e+f x)}}{f}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 55, normalized size = 1.10 \begin {gather*} \frac {\sqrt {a \cos ^2(e+f x)} \left (\cos (e+f x)-\log \left (\cos \left (\frac {1}{2} (e+f x)\right )\right )+\log \left (\sin \left (\frac {1}{2} (e+f x)\right )\right )\right ) \sec (e+f x)}{f} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 11.22, size = 52, normalized size = 1.04
method | result | size |
default | \(\frac {\sqrt {a \left (\cos ^{2}\left (f x +e \right )\right )}-\sqrt {a}\, \ln \left (\frac {2 \sqrt {a}\, \sqrt {a \left (\cos ^{2}\left (f x +e \right )\right )}+2 a}{\sin \left (f x +e \right )}\right )}{f}\) | \(52\) |
risch | \(\frac {\sqrt {\left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{2} a \,{\mathrm e}^{-2 i \left (f x +e \right )}}\, {\mathrm e}^{2 i \left (f x +e \right )}}{2 f \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )}+\frac {\sqrt {\left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{2} a \,{\mathrm e}^{-2 i \left (f x +e \right )}}}{2 f \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )}-\frac {\ln \left ({\mathrm e}^{i f x}+{\mathrm e}^{-i e}\right ) \sqrt {\left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{2} a \,{\mathrm e}^{-2 i \left (f x +e \right )}}\, {\mathrm e}^{i \left (f x +e \right )}}{f \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )}+\frac {\ln \left ({\mathrm e}^{i f x}-{\mathrm e}^{-i e}\right ) \sqrt {\left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{2} a \,{\mathrm e}^{-2 i \left (f x +e \right )}}\, {\mathrm e}^{i \left (f x +e \right )}}{f \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )}\) | \(232\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 74, normalized size = 1.48 \begin {gather*} -\frac {\sqrt {a} \log \left (\frac {2 \, \sqrt {-a \sin \left (f x + e\right )^{2} + a} \sqrt {a}}{{\left | \sin \left (f x + e\right ) \right |}} + \frac {2 \, a}{{\left | \sin \left (f x + e\right ) \right |}}\right ) - \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 = 0.40, size = 57, normalized size = 1.14 \begin {gather*} \frac {\sqrt {a \cos \left (f x + e\right )^{2}} {\left (2 \, \cos \left (f x + e\right ) - \log \left (-\frac {\cos \left (f x + e\right ) + 1}{\cos \left (f x + e\right ) - 1}\right )\right )}}{2 \, f \cos \left (f x + e\right )} \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 )} \cot {\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 = 0.45, size = 55, normalized size = 1.10 \begin {gather*} \frac {a {\left (\frac {\arctan \left (\frac {\sqrt {-a \sin \left (f x + e\right )^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} + \frac {\sqrt {-a \sin \left (f x + e\right )^{2} + a}}{a}\right )}}{f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \mathrm {cot}\left (e+f\,x\right )\,\sqrt {a-a\,{\sin \left (e+f\,x\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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