Optimal. Leaf size=141 \[ \frac{-B+i A}{2 a f (1+i \tan (e+f x)) \sqrt{c-i c \tan (e+f x)}}-\frac{B+3 i A}{4 a f \sqrt{c-i c \tan (e+f x)}}+\frac{(B+3 i A) \tanh ^{-1}\left (\frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{2} \sqrt{c}}\right )}{4 \sqrt{2} a \sqrt{c} f} \]
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Rubi [A] time = 0.219454, antiderivative size = 141, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.116, Rules used = {3588, 78, 51, 63, 208} \[ \frac{-B+i A}{2 a f (1+i \tan (e+f x)) \sqrt{c-i c \tan (e+f x)}}-\frac{B+3 i A}{4 a f \sqrt{c-i c \tan (e+f x)}}+\frac{(B+3 i A) \tanh ^{-1}\left (\frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{2} \sqrt{c}}\right )}{4 \sqrt{2} a \sqrt{c} f} \]
Antiderivative was successfully verified.
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Rule 3588
Rule 78
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{A+B \tan (e+f x)}{(a+i a \tan (e+f x)) \sqrt{c-i c \tan (e+f x)}} \, dx &=\frac{(a c) \operatorname{Subst}\left (\int \frac{A+B x}{(a+i a x)^2 (c-i c x)^{3/2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{i A-B}{2 a f (1+i \tan (e+f x)) \sqrt{c-i c \tan (e+f x)}}+\frac{((3 A-i B) c) \operatorname{Subst}\left (\int \frac{1}{(a+i a x) (c-i c x)^{3/2}} \, dx,x,\tan (e+f x)\right )}{4 f}\\ &=-\frac{3 i A+B}{4 a f \sqrt{c-i c \tan (e+f x)}}+\frac{i A-B}{2 a f (1+i \tan (e+f x)) \sqrt{c-i c \tan (e+f x)}}+\frac{(3 A-i B) \operatorname{Subst}\left (\int \frac{1}{(a+i a x) \sqrt{c-i c x}} \, dx,x,\tan (e+f x)\right )}{8 f}\\ &=-\frac{3 i A+B}{4 a f \sqrt{c-i c \tan (e+f x)}}+\frac{i A-B}{2 a f (1+i \tan (e+f x)) \sqrt{c-i c \tan (e+f x)}}+\frac{(3 i A+B) \operatorname{Subst}\left (\int \frac{1}{2 a-\frac{a x^2}{c}} \, dx,x,\sqrt{c-i c \tan (e+f x)}\right )}{4 c f}\\ &=\frac{(3 i A+B) \tanh ^{-1}\left (\frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{2} \sqrt{c}}\right )}{4 \sqrt{2} a \sqrt{c} f}-\frac{3 i A+B}{4 a f \sqrt{c-i c \tan (e+f x)}}+\frac{i A-B}{2 a f (1+i \tan (e+f x)) \sqrt{c-i c \tan (e+f x)}}\\ \end{align*}
Mathematica [A] time = 3.62969, size = 160, normalized size = 1.13 \[ \frac{e^{-2 i (e+f x)} \sqrt{\frac{c}{1+e^{2 i (e+f x)}}} \left ((B+3 i A) e^{2 i (e+f x)} \sqrt{1+e^{2 i (e+f x)}} \tanh ^{-1}\left (\sqrt{1+e^{2 i (e+f x)}}\right )-i \left (1+e^{2 i (e+f x)}\right ) \left (A \left (-1+2 e^{2 i (e+f x)}\right )-i B \left (1+2 e^{2 i (e+f x)}\right )\right )\right )}{4 \sqrt{2} a c f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.16, size = 121, normalized size = 0.9 \begin{align*}{\frac{2\,ic}{af} \left ( -{\frac{1}{4\,c} \left ({\frac{1}{-c-ic\tan \left ( fx+e \right ) } \left ({\frac{i}{2}}B+{\frac{A}{2}} \right ) \sqrt{c-ic\tan \left ( fx+e \right ) }}-{\frac{ \left ( 3\,A-iB \right ) \sqrt{2}}{4}{\it Artanh} \left ({\frac{\sqrt{2}}{2}\sqrt{c-ic\tan \left ( fx+e \right ) }{\frac{1}{\sqrt{c}}}} \right ){\frac{1}{\sqrt{c}}}} \right ) }-{\frac{A-iB}{4\,c}{\frac{1}{\sqrt{c-ic\tan \left ( fx+e \right ) }}}} \right ) } \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 = 1.11735, size = 903, normalized size = 6.4 \begin{align*} \frac{{\left (\sqrt{\frac{1}{2}} a c f \sqrt{-\frac{9 \, A^{2} - 6 i \, A B - B^{2}}{a^{2} c f^{2}}} e^{\left (2 i \, f x + 2 i \, e\right )} \log \left (\frac{{\left (\sqrt{2} \sqrt{\frac{1}{2}}{\left (a f e^{\left (2 i \, f x + 2 i \, e\right )} + a f\right )} \sqrt{\frac{c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt{-\frac{9 \, A^{2} - 6 i \, A B - B^{2}}{a^{2} c f^{2}}} + 3 i \, A + B\right )} e^{\left (-i \, f x - i \, e\right )}}{2 \, a f}\right ) - \sqrt{\frac{1}{2}} a c f \sqrt{-\frac{9 \, A^{2} - 6 i \, A B - B^{2}}{a^{2} c f^{2}}} e^{\left (2 i \, f x + 2 i \, e\right )} \log \left (-\frac{{\left (\sqrt{2} \sqrt{\frac{1}{2}}{\left (a f e^{\left (2 i \, f x + 2 i \, e\right )} + a f\right )} \sqrt{\frac{c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt{-\frac{9 \, A^{2} - 6 i \, A B - B^{2}}{a^{2} c f^{2}}} - 3 i \, A - B\right )} e^{\left (-i \, f x - i \, e\right )}}{2 \, a f}\right ) + \sqrt{2}{\left ({\left (-2 i \, A - 2 \, B\right )} e^{\left (4 i \, f x + 4 i \, e\right )} +{\left (-i \, A - 3 \, B\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + i \, A - B\right )} \sqrt{\frac{c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}}{8 \, a c f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{B \tan \left (f x + e\right ) + A}{{\left (i \, a \tan \left (f x + e\right ) + a\right )} \sqrt{-i \, c \tan \left (f x + e\right ) + c}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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