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 = 0.0649755, antiderivative size = 54, normalized size of antiderivative = 1., 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 3194
Rule 50
Rule 63
Rule 208
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*}
Mathematica [A] time = 0.0465664, 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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Maple [A] time = 1.001, size = 61, normalized size = 1.1 \begin{align*} -{\frac{1}{f}\sqrt{a}\ln \left ({\frac{1}{\sin \left ( fx+e \right ) } \left ( 2\,a+2\,\sqrt{a}\sqrt{a+b \left ( \sin \left ( fx+e \right ) \right ) ^{2}} \right ) } \right ) }+{\frac{1}{f}\sqrt{a+b \left ( \sin \left ( fx+e \right ) \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 6.16947, size = 347, normalized size = 6.43 \begin{align*} \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 ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a + b \sin ^{2}{\left (e + f x \right )}} \cot{\left (e + f x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09873, size = 66, normalized size = 1.22 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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