Optimal. Leaf size=69 \[ \frac{2 i a \sqrt{c+d \tan (e+f x)}}{f}-\frac{2 i a \sqrt{c-i d} \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right )}{f} \]
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Rubi [A] time = 0.143694, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3528, 3537, 63, 208} \[ \frac{2 i a \sqrt{c+d \tan (e+f x)}}{f}-\frac{2 i a \sqrt{c-i d} \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right )}{f} \]
Antiderivative was successfully verified.
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Rule 3528
Rule 3537
Rule 63
Rule 208
Rubi steps
\begin{align*} \int (a+i a \tan (e+f x)) \sqrt{c+d \tan (e+f x)} \, dx &=\frac{2 i a \sqrt{c+d \tan (e+f x)}}{f}+\int \frac{a (c-i d)+a (i c+d) \tan (e+f x)}{\sqrt{c+d \tan (e+f x)}} \, dx\\ &=\frac{2 i a \sqrt{c+d \tan (e+f x)}}{f}+\frac{\left (i a^2 (c-i d)^2\right ) \operatorname{Subst}\left (\int \frac{1}{\left (a^2 (i c+d)^2+a (c-i d) x\right ) \sqrt{c+\frac{d x}{a (i c+d)}}} \, dx,x,a (i c+d) \tan (e+f x)\right )}{f}\\ &=\frac{2 i a \sqrt{c+d \tan (e+f x)}}{f}-\frac{\left (2 a^3 (c-i d)^3\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{a^2 c (c-i d) (i c+d)}{d}+a^2 (i c+d)^2+\frac{a^2 (c-i d) (i c+d) x^2}{d}} \, dx,x,\sqrt{c+d \tan (e+f x)}\right )}{d f}\\ &=-\frac{2 i a \sqrt{c-i d} \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right )}{f}+\frac{2 i a \sqrt{c+d \tan (e+f x)}}{f}\\ \end{align*}
Mathematica [A] time = 1.27815, size = 88, normalized size = 1.28 \[ \frac{2 i a \left (\sqrt{c+d \tan (e+f x)}-\sqrt{c-i d} \tanh ^{-1}\left (\frac{\sqrt{c-\frac{i d \left (-1+e^{2 i (e+f x)}\right )}{1+e^{2 i (e+f x)}}}}{\sqrt{c-i d}}\right )\right )}{f} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.025, size = 1173, normalized size = 17. \begin{align*} \text{result too large to display} \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 = 1.89148, size = 807, normalized size = 11.7 \begin{align*} \frac{f \sqrt{-\frac{4 \, a^{2} c - 4 i \, a^{2} d}{f^{2}}} \log \left (\frac{{\left (2 \, a c +{\left (i \, f e^{\left (2 i \, f x + 2 i \, e\right )} + i \, f\right )} \sqrt{\frac{{\left (c - i \, d\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + c + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt{-\frac{4 \, a^{2} c - 4 i \, a^{2} d}{f^{2}}} +{\left (2 \, a c - 2 i \, a d\right )} e^{\left (2 i \, f x + 2 i \, e\right )}\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}}{a}\right ) - f \sqrt{-\frac{4 \, a^{2} c - 4 i \, a^{2} d}{f^{2}}} \log \left (\frac{{\left (2 \, a c +{\left (-i \, f e^{\left (2 i \, f x + 2 i \, e\right )} - i \, f\right )} \sqrt{\frac{{\left (c - i \, d\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + c + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt{-\frac{4 \, a^{2} c - 4 i \, a^{2} d}{f^{2}}} +{\left (2 \, a c - 2 i \, a d\right )} e^{\left (2 i \, f x + 2 i \, e\right )}\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}}{a}\right ) + 8 i \, a \sqrt{\frac{{\left (c - i \, d\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + c + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{4 \, f} \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 i \sqrt{c + d \tan{\left (e + f x \right )}} \tan{\left (e + f x \right )}\, dx + \int \sqrt{c + 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 [B] time = 1.3795, size = 251, normalized size = 3.64 \begin{align*} 2 \, a{\left (\frac{i \, \sqrt{d \tan \left (f x + e\right ) + c}}{f} - \frac{2 \,{\left (-2 i \, c - 2 \, d\right )} \arctan \left (\frac{4 \,{\left (\sqrt{d \tan \left (f x + e\right ) + c} c - \sqrt{c^{2} + d^{2}} \sqrt{d \tan \left (f x + e\right ) + c}\right )}}{c \sqrt{-8 \, c + 8 \, \sqrt{c^{2} + d^{2}}} - i \, \sqrt{-8 \, c + 8 \, \sqrt{c^{2} + d^{2}}} d - \sqrt{c^{2} + d^{2}} \sqrt{-8 \, c + 8 \, \sqrt{c^{2} + d^{2}}}}\right )}{\sqrt{-8 \, c + 8 \, \sqrt{c^{2} + d^{2}}} f{\left (-\frac{i \, d}{c - \sqrt{c^{2} + d^{2}}} + 1\right )}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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