Optimal. Leaf size=40 \[ -\frac{2 \sqrt [4]{-1} a \tanh ^{-1}\left (\frac{(-1)^{3/4} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right )}{\sqrt{d} f} \]
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Rubi [A] time = 0.0391566, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {3533, 208} \[ -\frac{2 \sqrt [4]{-1} a \tanh ^{-1}\left (\frac{(-1)^{3/4} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right )}{\sqrt{d} f} \]
Antiderivative was successfully verified.
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Rule 3533
Rule 208
Rubi steps
\begin{align*} \int \frac{a-i a \tan (e+f x)}{\sqrt{d \tan (e+f x)}} \, dx &=\frac{\left (2 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{a d+i a x^2} \, dx,x,\sqrt{d \tan (e+f x)}\right )}{f}\\ &=-\frac{2 \sqrt [4]{-1} a \tanh ^{-1}\left (\frac{(-1)^{3/4} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right )}{\sqrt{d} f}\\ \end{align*}
Mathematica [A] time = 0.044987, size = 50, normalized size = 1.25 \[ -\frac{2 \sqrt [4]{-1} a \sqrt{\tan (e+f x)} \tanh ^{-1}\left ((-1)^{3/4} \sqrt{\tan (e+f x)}\right )}{f \sqrt{d \tan (e+f x)}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.031, size = 330, normalized size = 8.3 \begin{align*}{\frac{a\sqrt{2}}{4\,fd}\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\,fd}\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\,fd}\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\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\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\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 [C] time = 1.90574, size = 416, normalized size = 10.4 \begin{align*} \frac{1}{2} \, \sqrt{\frac{i \, a^{2}}{d f^{2}}} \log \left (\frac{d f \sqrt{\frac{i \, a^{2}}{d f^{2}}} +{\left (a e^{\left (2 i \, f x + 2 i \, e\right )} + a\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}}}{a}\right ) - \frac{1}{2} \, \sqrt{\frac{i \, a^{2}}{d f^{2}}} \log \left (-\frac{d f \sqrt{\frac{i \, a^{2}}{d f^{2}}} -{\left (a e^{\left (2 i \, f x + 2 i \, e\right )} + a\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}}}{a}\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}{\sqrt{d \tan{\left (e + f x \right )}}}\, dx + \int \frac{i \tan{\left (e + f x \right )}}{\sqrt{d \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 [C] time = 1.34241, size = 90, normalized size = 2.25 \begin{align*} \frac{2 \, \sqrt{2} a \arctan \left (\frac{16 i \, \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 )}{\sqrt{d} f{\left (\frac{i \, d}{\sqrt{d^{2}}} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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