Optimal. Leaf size=93 \[ \frac{\sqrt{2} a d^{3/2} \tan ^{-1}\left (\frac{\sqrt{d}-\sqrt{d} \tan (e+f x)}{\sqrt{2} \sqrt{d \tan (e+f x)}}\right )}{f}+\frac{2 a d \sqrt{d \tan (e+f x)}}{f}+\frac{2 a (d \tan (e+f x))^{3/2}}{3 f} \]
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Rubi [A] time = 0.108572, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {3528, 3532, 205} \[ \frac{\sqrt{2} a d^{3/2} \tan ^{-1}\left (\frac{\sqrt{d}-\sqrt{d} \tan (e+f x)}{\sqrt{2} \sqrt{d \tan (e+f x)}}\right )}{f}+\frac{2 a d \sqrt{d \tan (e+f x)}}{f}+\frac{2 a (d \tan (e+f x))^{3/2}}{3 f} \]
Antiderivative was successfully verified.
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Rule 3528
Rule 3532
Rule 205
Rubi steps
\begin{align*} \int (d \tan (e+f x))^{3/2} (a+a \tan (e+f x)) \, dx &=\frac{2 a (d \tan (e+f x))^{3/2}}{3 f}+\int \sqrt{d \tan (e+f x)} (-a d+a d \tan (e+f x)) \, dx\\ &=\frac{2 a d \sqrt{d \tan (e+f x)}}{f}+\frac{2 a (d \tan (e+f x))^{3/2}}{3 f}+\int \frac{-a d^2-a d^2 \tan (e+f x)}{\sqrt{d \tan (e+f x)}} \, dx\\ &=\frac{2 a d \sqrt{d \tan (e+f x)}}{f}+\frac{2 a (d \tan (e+f x))^{3/2}}{3 f}-\frac{\left (2 a^2 d^4\right ) \operatorname{Subst}\left (\int \frac{1}{2 a^2 d^4+d x^2} \, dx,x,\frac{-a d^2+a d^2 \tan (e+f x)}{\sqrt{d \tan (e+f x)}}\right )}{f}\\ &=\frac{\sqrt{2} a d^{3/2} \tan ^{-1}\left (\frac{\sqrt{d}-\sqrt{d} \tan (e+f x)}{\sqrt{2} \sqrt{d \tan (e+f x)}}\right )}{f}+\frac{2 a d \sqrt{d \tan (e+f x)}}{f}+\frac{2 a (d \tan (e+f x))^{3/2}}{3 f}\\ \end{align*}
Mathematica [C] time = 0.422685, size = 105, normalized size = 1.13 \[ \frac{\left (\frac{1}{3}+\frac{i}{3}\right ) a (d \tan (e+f x))^{3/2} \left (-3 (-1)^{3/4} \tan ^{-1}\left ((-1)^{3/4} \sqrt{\tan (e+f x)}\right )+(1-i) \sqrt{\tan (e+f x)} (\tan (e+f x)+3)+3 \sqrt [4]{-1} \tanh ^{-1}\left ((-1)^{3/4} \sqrt{\tan (e+f x)}\right )\right )}{f \tan ^{\frac{3}{2}}(e+f x)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.02, size = 363, normalized size = 3.9 \begin{align*}{\frac{2\,a}{3\,f} \left ( d\tan \left ( fx+e \right ) \right ) ^{{\frac{3}{2}}}}+2\,{\frac{ad\sqrt{d\tan \left ( fx+e \right ) }}{f}}-{\frac{ad\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{ad\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}}{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{d}^{2}\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{a{d}^{2}\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{a{d}^{2}\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 = 1.67184, size = 516, normalized size = 5.55 \begin{align*} \left [\frac{3 \, \sqrt{2} a \sqrt{-d} 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 \,{\left (a d \tan \left (f x + e\right ) + 3 \, a d\right )} \sqrt{d \tan \left (f x + e\right )}}{6 \, f}, -\frac{3 \, \sqrt{2} a d^{\frac{3}{2}} \arctan \left (\frac{\sqrt{2} \sqrt{d \tan \left (f x + e\right )}{\left (\tan \left (f x + e\right ) - 1\right )}}{2 \, \sqrt{d} \tan \left (f x + e\right )}\right ) - 2 \,{\left (a d \tan \left (f x + e\right ) + 3 \, a d\right )} \sqrt{d \tan \left (f x + e\right )}}{3 \, 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 \left (d \tan{\left (e + f x \right )}\right )^{\frac{3}{2}}\, dx + \int \left (d \tan{\left (e + f x \right )}\right )^{\frac{3}{2}} \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 = 1.25996, size = 392, normalized size = 4.22 \begin{align*} -\frac{1}{12} \, d{\left (\frac{6 \, \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 f} + \frac{6 \, \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 f} + \frac{3 \, \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 f} - \frac{3 \, \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 f} - \frac{8 \,{\left (\sqrt{d \tan \left (f x + e\right )} a d^{3} f^{2} \tan \left (f x + e\right ) + 3 \, \sqrt{d \tan \left (f x + e\right )} a d^{3} f^{2}\right )}}{d^{3} f^{3}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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