Optimal. Leaf size=58 \[ \frac{\sqrt{a+b} \tanh ^{-1}\left (\frac{\sqrt{a+b \sin ^2(e+f x)}}{\sqrt{a+b}}\right )}{f}-\frac{\sqrt{a+b \sin ^2(e+f x)}}{f} \]
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Rubi [A] time = 0.0559427, antiderivative size = 58, 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} \[ \frac{\sqrt{a+b} \tanh ^{-1}\left (\frac{\sqrt{a+b \sin ^2(e+f x)}}{\sqrt{a+b}}\right )}{f}-\frac{\sqrt{a+b \sin ^2(e+f x)}}{f} \]
Antiderivative was successfully verified.
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Rule 3194
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \sqrt{a+b \sin ^2(e+f x)} \tan (e+f x) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{1-x} \, dx,x,\sin ^2(e+f x)\right )}{2 f}\\ &=-\frac{\sqrt{a+b \sin ^2(e+f x)}}{f}+\frac{(a+b) \operatorname{Subst}\left (\int \frac{1}{(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+b) \operatorname{Subst}\left (\int \frac{1}{1+\frac{a}{b}-\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sin ^2(e+f x)}\right )}{b f}\\ &=\frac{\sqrt{a+b} \tanh ^{-1}\left (\frac{\sqrt{a+b \sin ^2(e+f x)}}{\sqrt{a+b}}\right )}{f}-\frac{\sqrt{a+b \sin ^2(e+f x)}}{f}\\ \end{align*}
Mathematica [A] time = 0.0579054, size = 60, normalized size = 1.03 \[ \frac{\sqrt{a+b} \tanh ^{-1}\left (\frac{\sqrt{a-b \cos ^2(e+f x)+b}}{\sqrt{a+b}}\right )-\sqrt{a-b \cos ^2(e+f x)+b}}{f} \]
Antiderivative was successfully verified.
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Maple [B] time = 3.81, size = 134, normalized size = 2.3 \begin{align*}{\frac{1}{2\,f}\sqrt{a+b}\ln \left ( 2\,{\frac{\sqrt{a+b}\sqrt{a+b-b \left ( \cos \left ( fx+e \right ) \right ) ^{2}}+b\sin \left ( fx+e \right ) +a}{-1+\sin \left ( fx+e \right ) }} \right ) }+{\frac{1}{2\,f}\sqrt{a+b}\ln \left ( 2\,{\frac{\sqrt{a+b}\sqrt{a+b-b \left ( \cos \left ( fx+e \right ) \right ) ^{2}}-b\sin \left ( fx+e \right ) +a}{1+\sin \left ( fx+e \right ) }} \right ) }-{\frac{1}{f}\sqrt{a+b-b \left ( \cos \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 = 3.36449, size = 370, normalized size = 6.38 \begin{align*} \left [\frac{\sqrt{a + b} \log \left (\frac{b \cos \left (f x + e\right )^{2} - 2 \, \sqrt{-b \cos \left (f x + e\right )^{2} + a + b} \sqrt{a + b} - 2 \, a - 2 \, b}{\cos \left (f x + e\right )^{2}}\right ) - 2 \, \sqrt{-b \cos \left (f x + e\right )^{2} + a + b}}{2 \, f}, -\frac{\sqrt{-a - b} \arctan \left (\frac{\sqrt{-b \cos \left (f x + e\right )^{2} + a + b} \sqrt{-a - b}}{a + b}\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 )}} \tan{\left (e + f x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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