Optimal. Leaf size=72 \[ \frac{2 a \sqrt{d \tan (e+f x)}}{f}-\frac{\sqrt{2} a \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{d} \tan (e+f x)+\sqrt{d}}{\sqrt{2} \sqrt{d \tan (e+f x)}}\right )}{f} \]
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Rubi [A] time = 0.0730247, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {3528, 3532, 208} \[ \frac{2 a \sqrt{d \tan (e+f x)}}{f}-\frac{\sqrt{2} a \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{d} \tan (e+f x)+\sqrt{d}}{\sqrt{2} \sqrt{d \tan (e+f x)}}\right )}{f} \]
Antiderivative was successfully verified.
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Rule 3528
Rule 3532
Rule 208
Rubi steps
\begin{align*} \int \sqrt{d \tan (e+f x)} (a+a \tan (e+f x)) \, dx &=\frac{2 a \sqrt{d \tan (e+f x)}}{f}+\int \frac{-a d+a d \tan (e+f x)}{\sqrt{d \tan (e+f x)}} \, dx\\ &=\frac{2 a \sqrt{d \tan (e+f x)}}{f}-\frac{\left (2 a^2 d^2\right ) \operatorname{Subst}\left (\int \frac{1}{-2 a^2 d^2+d x^2} \, dx,x,\frac{-a d-a d \tan (e+f x)}{\sqrt{d \tan (e+f x)}}\right )}{f}\\ &=-\frac{\sqrt{2} a \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{d}+\sqrt{d} \tan (e+f x)}{\sqrt{2} \sqrt{d \tan (e+f x)}}\right )}{f}+\frac{2 a \sqrt{d \tan (e+f x)}}{f}\\ \end{align*}
Mathematica [C] time = 0.10091, size = 92, normalized size = 1.28 \[ \frac{(1+i) a \sqrt{d \tan (e+f x)} \left (\sqrt [4]{-1} \tan ^{-1}\left ((-1)^{3/4} \sqrt{\tan (e+f x)}\right )+(1-i) \sqrt{\tan (e+f x)}-(-1)^{3/4} \tanh ^{-1}\left ((-1)^{3/4} \sqrt{\tan (e+f x)}\right )\right )}{f \sqrt{\tan (e+f x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.018, size = 337, normalized size = 4.7 \begin{align*} 2\,{\frac{a\sqrt{d\tan \left ( fx+e \right ) }}{f}}-{\frac{a\sqrt{2}}{4\,f}\sqrt [4]{{d}^{2}}\ln \left ({ \left ( d\tan \left ( fx+e \right ) +\sqrt [4]{{d}^{2}}\sqrt{d\tan \left ( fx+e \right ) }\sqrt{2}+\sqrt{{d}^{2}} \right ) \left ( d\tan \left ( fx+e \right ) -\sqrt [4]{{d}^{2}}\sqrt{d\tan \left ( fx+e \right ) }\sqrt{2}+\sqrt{{d}^{2}} \right ) ^{-1}} \right ) }-{\frac{a\sqrt{2}}{2\,f}\sqrt [4]{{d}^{2}}\arctan \left ({\sqrt{2}\sqrt{d\tan \left ( fx+e \right ) }{\frac{1}{\sqrt [4]{{d}^{2}}}}}+1 \right ) }+{\frac{a\sqrt{2}}{2\,f}\sqrt [4]{{d}^{2}}\arctan \left ( -{\sqrt{2}\sqrt{d\tan \left ( fx+e \right ) }{\frac{1}{\sqrt [4]{{d}^{2}}}}}+1 \right ) }+{\frac{ad\sqrt{2}}{4\,f}\ln \left ({ \left ( d\tan \left ( fx+e \right ) -\sqrt [4]{{d}^{2}}\sqrt{d\tan \left ( fx+e \right ) }\sqrt{2}+\sqrt{{d}^{2}} \right ) \left ( d\tan \left ( fx+e \right ) +\sqrt [4]{{d}^{2}}\sqrt{d\tan \left ( fx+e \right ) }\sqrt{2}+\sqrt{{d}^{2}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{d}^{2}}}}}+{\frac{ad\sqrt{2}}{2\,f}\arctan \left ({\sqrt{2}\sqrt{d\tan \left ( fx+e \right ) }{\frac{1}{\sqrt [4]{{d}^{2}}}}}+1 \right ){\frac{1}{\sqrt [4]{{d}^{2}}}}}-{\frac{ad\sqrt{2}}{2\,f}\arctan \left ( -{\sqrt{2}\sqrt{d\tan \left ( fx+e \right ) }{\frac{1}{\sqrt [4]{{d}^{2}}}}}+1 \right ){\frac{1}{\sqrt [4]{{d}^{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 = 1.69901, size = 436, normalized size = 6.06 \begin{align*} \left [\frac{\sqrt{2} a \sqrt{d} \log \left (\frac{d \tan \left (f x + e\right )^{2} - 2 \, \sqrt{2} \sqrt{d \tan \left (f x + e\right )} \sqrt{d}{\left (\tan \left (f x + e\right ) + 1\right )} + 4 \, d \tan \left (f x + e\right ) + d}{\tan \left (f x + e\right )^{2} + 1}\right ) + 4 \, \sqrt{d \tan \left (f x + e\right )} a}{2 \, f}, \frac{\sqrt{2} a \sqrt{-d} \arctan \left (\frac{\sqrt{2} \sqrt{d \tan \left (f x + e\right )} \sqrt{-d}{\left (\tan \left (f x + e\right ) + 1\right )}}{2 \, d \tan \left (f x + e\right )}\right ) + 2 \, \sqrt{d \tan \left (f x + e\right )} a}{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*} a \left (\int \sqrt{d \tan{\left (e + f x \right )}}\, dx + \int \sqrt{d \tan{\left (e + f x \right )}} \tan{\left (e + f x \right )}\, dx\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.30784, size = 336, normalized size = 4.67 \begin{align*} \frac{2 \, \sqrt{d \tan \left (f x + e\right )} a}{f} - \frac{\sqrt{2}{\left (a d \sqrt{{\left | d \right |}} - a{\left | d \right |}^{\frac{3}{2}}\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \sqrt{{\left | d \right |}} + 2 \, \sqrt{d \tan \left (f x + e\right )}\right )}}{2 \, \sqrt{{\left | d \right |}}}\right )}{2 \, d f} - \frac{\sqrt{2}{\left (a d \sqrt{{\left | d \right |}} - a{\left | d \right |}^{\frac{3}{2}}\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \sqrt{{\left | d \right |}} - 2 \, \sqrt{d \tan \left (f x + e\right )}\right )}}{2 \, \sqrt{{\left | d \right |}}}\right )}{2 \, d f} - \frac{\sqrt{2}{\left (a d \sqrt{{\left | d \right |}} + a{\left | d \right |}^{\frac{3}{2}}\right )} \log \left (d \tan \left (f x + e\right ) + \sqrt{2} \sqrt{d \tan \left (f x + e\right )} \sqrt{{\left | d \right |}} +{\left | d \right |}\right )}{4 \, d f} + \frac{\sqrt{2}{\left (a d \sqrt{{\left | d \right |}} + a{\left | d \right |}^{\frac{3}{2}}\right )} \log \left (d \tan \left (f x + e\right ) - \sqrt{2} \sqrt{d \tan \left (f x + e\right )} \sqrt{{\left | d \right |}} +{\left | d \right |}\right )}{4 \, d f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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