Optimal. Leaf size=87 \[ \frac{2 i a}{d^2 f \sqrt{d \tan (e+f x)}}+\frac{2 \sqrt [4]{-1} a \tanh ^{-1}\left (\frac{(-1)^{3/4} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right )}{d^{5/2} f}-\frac{2 a}{3 d f (d \tan (e+f x))^{3/2}} \]
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Rubi [A] time = 0.125183, antiderivative size = 87, 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 = {3529, 3533, 208} \[ \frac{2 i a}{d^2 f \sqrt{d \tan (e+f x)}}+\frac{2 \sqrt [4]{-1} a \tanh ^{-1}\left (\frac{(-1)^{3/4} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right )}{d^{5/2} f}-\frac{2 a}{3 d f (d \tan (e+f x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 3529
Rule 3533
Rule 208
Rubi steps
\begin{align*} \int \frac{a-i a \tan (e+f x)}{(d \tan (e+f x))^{5/2}} \, dx &=-\frac{2 a}{3 d f (d \tan (e+f x))^{3/2}}+\frac{\int \frac{-i a d-a d \tan (e+f x)}{(d \tan (e+f x))^{3/2}} \, dx}{d^2}\\ &=-\frac{2 a}{3 d f (d \tan (e+f x))^{3/2}}+\frac{2 i a}{d^2 f \sqrt{d \tan (e+f x)}}+\frac{\int \frac{-a d^2+i a d^2 \tan (e+f x)}{\sqrt{d \tan (e+f x)}} \, dx}{d^4}\\ &=-\frac{2 a}{3 d f (d \tan (e+f x))^{3/2}}+\frac{2 i a}{d^2 f \sqrt{d \tan (e+f x)}}+\frac{\left (2 a^2\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 \tanh ^{-1}\left (\frac{(-1)^{3/4} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right )}{d^{5/2} f}-\frac{2 a}{3 d f (d \tan (e+f x))^{3/2}}+\frac{2 i a}{d^2 f \sqrt{d \tan (e+f x)}}\\ \end{align*}
Mathematica [C] time = 0.114292, size = 41, normalized size = 0.47 \[ -\frac{2 a \, _2F_1\left (-\frac{3}{2},1;-\frac{1}{2};-i \tan (e+f x)\right )}{3 d f (d \tan (e+f x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.022, size = 378, normalized size = 4.3 \begin{align*}{\frac{2\,ia}{{d}^{2}f}{\frac{1}{\sqrt{d\tan \left ( fx+e \right ) }}}}-{\frac{2\,a}{3\,df} \left ( d\tan \left ( fx+e \right ) \right ) ^{-{\frac{3}{2}}}}-{\frac{a\sqrt{2}}{4\,f{d}^{3}}\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}^{3}}\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}^{3}}\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}}{{d}^{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}}{{d}^{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}}{{d}^{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 = 2.30662, size = 977, normalized size = 11.23 \begin{align*} -\frac{3 \,{\left (d^{3} f e^{\left (4 i \, f x + 4 i \, e\right )} - 2 \, d^{3} f e^{\left (2 i \, f x + 2 i \, e\right )} + d^{3} f\right )} \sqrt{\frac{4 i \, a^{2}}{d^{5} f^{2}}} \log \left (-\frac{{\left ({\left (d^{2} f e^{\left (2 i \, f x + 2 i \, e\right )} + d^{2} 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}} \sqrt{\frac{4 i \, a^{2}}{d^{5} f^{2}}} + 2 i \, a\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}}{d^{2} f}\right ) - 3 \,{\left (d^{3} f e^{\left (4 i \, f x + 4 i \, e\right )} - 2 \, d^{3} f e^{\left (2 i \, f x + 2 i \, e\right )} + d^{3} f\right )} \sqrt{\frac{4 i \, a^{2}}{d^{5} f^{2}}} \log \left (\frac{{\left ({\left (d^{2} f e^{\left (2 i \, f x + 2 i \, e\right )} + d^{2} 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}} \sqrt{\frac{4 i \, a^{2}}{d^{5} f^{2}}} - 2 i \, a\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}}{d^{2} f}\right ) + 16 \,{\left (a e^{\left (4 i \, f x + 4 i \, e\right )} - a e^{\left (2 i \, f x + 2 i \, e\right )} - 2 \, 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}}}{12 \,{\left (d^{3} f e^{\left (4 i \, f x + 4 i \, e\right )} - 2 \, d^{3} f e^{\left (2 i \, f x + 2 i \, e\right )} + d^{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 - \frac{1}{\left (d \tan{\left (e + f x \right )}\right )^{\frac{5}{2}}}\, dx + \int \frac{i \tan{\left (e + f x \right )}}{\left (d \tan{\left (e + f x \right )}\right )^{\frac{5}{2}}}\, dx\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19358, size = 147, normalized size = 1.69 \begin{align*} -\frac{2}{3} \, a{\left (-\frac{3 i \, \sqrt{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 )}{d^{\frac{5}{2}} f{\left (-\frac{i \, d}{\sqrt{d^{2}}} + 1\right )}} + \frac{-3 i \, d \tan \left (f x + e\right ) + d}{\sqrt{d \tan \left (f x + e\right )} d^{3} f \tan \left (f x + e\right )}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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