Optimal. Leaf size=74 \[ \frac{\sqrt{2} a \tanh ^{-1}\left (\frac{\sqrt{d} \tan (e+f x)+\sqrt{d}}{\sqrt{2} \sqrt{d \tan (e+f x)}}\right )}{d^{3/2} f}-\frac{2 a}{d f \sqrt{d \tan (e+f x)}} \]
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Rubi [A] time = 0.0792321, antiderivative size = 74, 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 = {3529, 3532, 208} \[ \frac{\sqrt{2} a \tanh ^{-1}\left (\frac{\sqrt{d} \tan (e+f x)+\sqrt{d}}{\sqrt{2} \sqrt{d \tan (e+f x)}}\right )}{d^{3/2} f}-\frac{2 a}{d f \sqrt{d \tan (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 3529
Rule 3532
Rule 208
Rubi steps
\begin{align*} \int \frac{a+a \tan (e+f x)}{(d \tan (e+f x))^{3/2}} \, dx &=-\frac{2 a}{d f \sqrt{d \tan (e+f x)}}+\frac{\int \frac{a d-a d \tan (e+f x)}{\sqrt{d \tan (e+f x)}} \, dx}{d^2}\\ &=-\frac{2 a}{d f \sqrt{d \tan (e+f x)}}-\frac{\left (2 a^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 \tanh ^{-1}\left (\frac{\sqrt{d}+\sqrt{d} \tan (e+f x)}{\sqrt{2} \sqrt{d \tan (e+f x)}}\right )}{d^{3/2} f}-\frac{2 a}{d f \sqrt{d \tan (e+f x)}}\\ \end{align*}
Mathematica [C] time = 0.124292, size = 64, normalized size = 0.86 \[ -\frac{(1+i) a \left (\, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};-i \tan (e+f x)\right )-i \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};i \tan (e+f x)\right )\right )}{d f \sqrt{d \tan (e+f x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.02, size = 355, normalized size = 4.8 \begin{align*} -2\,{\frac{a}{df\sqrt{d\tan \left ( fx+e \right ) }}}+{\frac{a\sqrt{2}}{4\,f{d}^{2}}\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{d}^{2}}\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{d}^{2}}\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}}{4\,df}\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{a\sqrt{2}}{2\,df}\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{a\sqrt{2}}{2\,df}\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 = 1.59965, size = 521, normalized size = 7.04 \begin{align*} \left [\frac{\sqrt{2} a \sqrt{d} \log \left (\frac{\tan \left (f x + e\right )^{2} + \frac{2 \, \sqrt{2} \sqrt{d \tan \left (f x + e\right )}{\left (\tan \left (f x + e\right ) + 1\right )}}{\sqrt{d}} + 4 \, \tan \left (f x + e\right ) + 1}{\tan \left (f x + e\right )^{2} + 1}\right ) \tan \left (f x + e\right ) - 4 \, \sqrt{d \tan \left (f x + e\right )} a}{2 \, d^{2} f \tan \left (f x + e\right )}, -\frac{\sqrt{2} a d \sqrt{-\frac{1}{d}} \arctan \left (\frac{\sqrt{2} \sqrt{d \tan \left (f x + e\right )} \sqrt{-\frac{1}{d}}{\left (\tan \left (f x + e\right ) + 1\right )}}{2 \, \tan \left (f x + e\right )}\right ) \tan \left (f x + e\right ) + 2 \, \sqrt{d \tan \left (f x + e\right )} a}{d^{2} f \tan \left (f x + e\right )}\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 \frac{1}{\left (d \tan{\left (e + f x \right )}\right )^{\frac{3}{2}}}\, dx + \int \frac{\tan{\left (e + f x \right )}}{\left (d \tan{\left (e + f x \right )}\right )^{\frac{3}{2}}}\, dx\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.28527, size = 342, normalized size = 4.62 \begin{align*} -\frac{\frac{8 \, a}{\sqrt{d \tan \left (f x + e\right )} f} - \frac{2 \, \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 )}{d^{2} f} - \frac{2 \, \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 )}{d^{2} 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 )}{d^{2} 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 )}{d^{2} f}}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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