Optimal. Leaf size=54 \[ \frac {\sqrt {a+b \sin ^2(e+f x)}}{f}-\frac {\sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b \sin ^2(e+f x)}}{\sqrt {a}}\right )}{f} \]
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Rubi [A] time = 0.06, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3194, 50, 63, 208} \[ \frac {\sqrt {a+b \sin ^2(e+f x)}}{f}-\frac {\sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b \sin ^2(e+f x)}}{\sqrt {a}}\right )}{f} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 208
Rule 3194
Rubi steps
\begin {align*} \int \cot (e+f x) \sqrt {a+b \sin ^2(e+f x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\sqrt {a+b x}}{x} \, dx,x,\sin ^2(e+f x)\right )}{2 f}\\ &=\frac {\sqrt {a+b \sin ^2(e+f x)}}{f}+\frac {a \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\sin ^2(e+f x)\right )}{2 f}\\ &=\frac {\sqrt {a+b \sin ^2(e+f x)}}{f}+\frac {a \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sin ^2(e+f x)}\right )}{b f}\\ &=-\frac {\sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b \sin ^2(e+f x)}}{\sqrt {a}}\right )}{f}+\frac {\sqrt {a+b \sin ^2(e+f x)}}{f}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 53, normalized size = 0.98 \[ -\frac {\sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b \sin ^2(e+f x)}}{\sqrt {a}}\right )-\sqrt {a+b \sin ^2(e+f x)}}{f} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.13, size = 135, normalized size = 2.50 \[ \left [\frac {\sqrt {a} \log \left (\frac {2 \, {\left (b \cos \left (f x + e\right )^{2} + 2 \, \sqrt {-b \cos \left (f x + e\right )^{2} + a + b} \sqrt {a} - 2 \, a - b\right )}}{\cos \left (f x + e\right )^{2} - 1}\right ) + 2 \, \sqrt {-b \cos \left (f x + e\right )^{2} + a + b}}{2 \, f}, \frac {\sqrt {-a} \arctan \left (\frac {\sqrt {-b \cos \left (f x + e\right )^{2} + a + b} \sqrt {-a}}{a}\right ) + \sqrt {-b \cos \left (f x + e\right )^{2} + a + b}}{f}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 49, normalized size = 0.91 \[ \frac {\frac {a \arctan \left (\frac {\sqrt {b \sin \left (f x + e\right )^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} + \sqrt {b \sin \left (f x + e\right )^{2} + a}}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.46, size = 61, normalized size = 1.13 \[ -\frac {\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}}{\sin \left (f x +e \right )}\right )}{f}+\frac {\sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 43, normalized size = 0.80 \[ -\frac {\sqrt {a} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | \sin \left (f x + e\right ) \right |}}\right ) - \sqrt {b \sin \left (f x + e\right )^{2} + a}}{f} \]
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 \[ \int \mathrm {cot}\left (e+f\,x\right )\,\sqrt {b\,{\sin \left (e+f\,x\right )}^2+a} \,d x \]
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 \[ \int \sqrt {a + b \sin ^{2}{\left (e + f x \right )}} \cot {\left (e + f x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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