Optimal. Leaf size=82 \[ \frac{2 \sqrt [4]{-1} a d^{3/2} \tan ^{-1}\left (\frac{(-1)^{3/4} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right )}{f}+\frac{2 a d \sqrt{d \tan (e+f x)}}{f}+\frac{2 i a (d \tan (e+f x))^{3/2}}{3 f} \]
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Rubi [A] time = 0.116535, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {3528, 3533, 205} \[ \frac{2 \sqrt [4]{-1} a d^{3/2} \tan ^{-1}\left (\frac{(-1)^{3/4} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right )}{f}+\frac{2 a d \sqrt{d \tan (e+f x)}}{f}+\frac{2 i a (d \tan (e+f x))^{3/2}}{3 f} \]
Antiderivative was successfully verified.
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Rule 3528
Rule 3533
Rule 205
Rubi steps
\begin{align*} \int (d \tan (e+f x))^{3/2} (a+i a \tan (e+f x)) \, dx &=\frac{2 i a (d \tan (e+f x))^{3/2}}{3 f}+\int \sqrt{d \tan (e+f x)} (-i a d+a d \tan (e+f x)) \, dx\\ &=\frac{2 a d \sqrt{d \tan (e+f x)}}{f}+\frac{2 i a (d \tan (e+f x))^{3/2}}{3 f}+\int \frac{-a d^2-i 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 i a (d \tan (e+f x))^{3/2}}{3 f}+\frac{\left (2 a^2 d^4\right ) \operatorname{Subst}\left (\int \frac{1}{-a d^3+i a d^2 x^2} \, dx,x,\sqrt{d \tan (e+f x)}\right )}{f}\\ &=\frac{2 \sqrt [4]{-1} a d^{3/2} \tan ^{-1}\left (\frac{(-1)^{3/4} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right )}{f}+\frac{2 a d \sqrt{d \tan (e+f x)}}{f}+\frac{2 i a (d \tan (e+f x))^{3/2}}{3 f}\\ \end{align*}
Mathematica [A] time = 0.988392, size = 148, normalized size = 1.8 \[ -\frac{2 a d \left (2 e^{2 i (e+f x)}-4 e^{4 i (e+f x)}+3 \sqrt{\frac{-1+e^{2 i (e+f x)}}{1+e^{2 i (e+f x)}}} \left (1+e^{2 i (e+f x)}\right )^2 \tanh ^{-1}\left (\sqrt{\frac{-1+e^{2 i (e+f x)}}{1+e^{2 i (e+f x)}}}\right )+2\right ) \sqrt{d \tan (e+f x)}}{3 f \left (-1+e^{4 i (e+f x)}\right )} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.018, size = 367, normalized size = 4.5 \begin{align*}{\frac{{\frac{2\,i}{3}}a}{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{{\frac{i}{4}}a{d}^{2}\sqrt{2}}{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{{\frac{i}{2}}a{d}^{2}\sqrt{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{{\frac{i}{2}}a{d}^{2}\sqrt{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 [B] time = 2.1126, size = 830, normalized size = 10.12 \begin{align*} -\frac{3 \, \sqrt{-\frac{4 i \, a^{2} d^{3}}{f^{2}}}{\left (f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )} \log \left (\frac{{\left (-2 i \, a d^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + \sqrt{-\frac{4 i \, a^{2} d^{3}}{f^{2}}}{\left (f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )} \sqrt{\frac{-i \, d e^{\left (2 i \, f x + 2 i \, e\right )} + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}}{a d}\right ) - 3 \, \sqrt{-\frac{4 i \, a^{2} d^{3}}{f^{2}}}{\left (f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )} \log \left (\frac{{\left (-2 i \, a d^{2} e^{\left (2 i \, f x + 2 i \, e\right )} - \sqrt{-\frac{4 i \, a^{2} d^{3}}{f^{2}}}{\left (f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )} \sqrt{\frac{-i \, d e^{\left (2 i \, f x + 2 i \, e\right )} + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}}{a d}\right ) - 16 \,{\left (2 \, a d e^{\left (2 i \, f x + 2 i \, e\right )} + a d\right )} \sqrt{\frac{-i \, d e^{\left (2 i \, f x + 2 i \, e\right )} + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{12 \,{\left (f e^{\left (2 i \, f x + 2 i \, e\right )} + 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 i \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 [A] time = 1.13873, size = 157, normalized size = 1.91 \begin{align*} -\frac{2}{3} \, a{\left (\frac{3 i \, \sqrt{2} d^{\frac{3}{2}} \arctan \left (\frac{16 \, \sqrt{d^{2}} \sqrt{d \tan \left (f x + e\right )}}{8 i \, \sqrt{2} d^{\frac{3}{2}} + 8 \, \sqrt{2} \sqrt{d^{2}} \sqrt{d}}\right )}{f{\left (\frac{i \, d}{\sqrt{d^{2}}} + 1\right )}} + \frac{-i \, \sqrt{d \tan \left (f x + e\right )} d f^{2} \tan \left (f x + e\right ) - 3 \, \sqrt{d \tan \left (f x + e\right )} d f^{2}}{f^{3}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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