Optimal. Leaf size=110 \[ \frac{(-B+i A) \sqrt{c-i c \tan (e+f x)}}{f \sqrt{a+i a \tan (e+f x)}}-\frac{2 B \sqrt{c} \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{a+i a \tan (e+f x)}}{\sqrt{a} \sqrt{c-i c \tan (e+f x)}}\right )}{\sqrt{a} f} \]
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Rubi [A] time = 0.219041, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 45, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {3588, 78, 63, 217, 203} \[ \frac{(-B+i A) \sqrt{c-i c \tan (e+f x)}}{f \sqrt{a+i a \tan (e+f x)}}-\frac{2 B \sqrt{c} \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{a+i a \tan (e+f x)}}{\sqrt{a} \sqrt{c-i c \tan (e+f x)}}\right )}{\sqrt{a} f} \]
Antiderivative was successfully verified.
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Rule 3588
Rule 78
Rule 63
Rule 217
Rule 203
Rubi steps
\begin{align*} \int \frac{(A+B \tan (e+f x)) \sqrt{c-i c \tan (e+f x)}}{\sqrt{a+i a \tan (e+f x)}} \, dx &=\frac{(a c) \operatorname{Subst}\left (\int \frac{A+B x}{(a+i a x)^{3/2} \sqrt{c-i c x}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{(i A-B) \sqrt{c-i c \tan (e+f x)}}{f \sqrt{a+i a \tan (e+f x)}}-\frac{(i B c) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+i a x} \sqrt{c-i c x}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{(i A-B) \sqrt{c-i c \tan (e+f x)}}{f \sqrt{a+i a \tan (e+f x)}}-\frac{(2 B c) \operatorname{Subst}\left (\int \frac{1}{\sqrt{2 c-\frac{c x^2}{a}}} \, dx,x,\sqrt{a+i a \tan (e+f x)}\right )}{a f}\\ &=\frac{(i A-B) \sqrt{c-i c \tan (e+f x)}}{f \sqrt{a+i a \tan (e+f x)}}-\frac{(2 B c) \operatorname{Subst}\left (\int \frac{1}{1+\frac{c x^2}{a}} \, dx,x,\frac{\sqrt{a+i a \tan (e+f x)}}{\sqrt{c-i c \tan (e+f x)}}\right )}{a f}\\ &=-\frac{2 B \sqrt{c} \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{a+i a \tan (e+f x)}}{\sqrt{a} \sqrt{c-i c \tan (e+f x)}}\right )}{\sqrt{a} f}+\frac{(i A-B) \sqrt{c-i c \tan (e+f x)}}{f \sqrt{a+i a \tan (e+f x)}}\\ \end{align*}
Mathematica [A] time = 3.98718, size = 152, normalized size = 1.38 \[ \frac{c \sec (e+f x) \left (\sin \left (\frac{1}{2} (e+f x)\right )+i \cos \left (\frac{1}{2} (e+f x)\right )\right ) \left ((A+i B) \left (\cos \left (\frac{1}{2} (e+f x)\right )-i \sin \left (\frac{1}{2} (e+f x)\right )\right )+2 i B \left (\cos \left (\frac{1}{2} (e+f x)\right )+i \sin \left (\frac{1}{2} (e+f x)\right )\right ) \tan ^{-1}(\cos (e+f x)+i \sin (e+f x))\right )}{f \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.18, size = 323, normalized size = 2.9 \begin{align*}{\frac{-i}{af \left ( -\tan \left ( fx+e \right ) +i \right ) ^{2}}\sqrt{-c \left ( -1+i\tan \left ( fx+e \right ) \right ) }\sqrt{a \left ( 1+i\tan \left ( fx+e \right ) \right ) } \left ( -2\,iB\ln \left ({ \left ( ac\tan \left ( fx+e \right ) +\sqrt{ac \left ( 1+ \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) }\sqrt{ac} \right ){\frac{1}{\sqrt{ac}}}} \right ) \tan \left ( fx+e \right ) ac+B\ln \left ({ \left ( ac\tan \left ( fx+e \right ) +\sqrt{ac \left ( 1+ \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) }\sqrt{ac} \right ){\frac{1}{\sqrt{ac}}}} \right ) \left ( \tan \left ( fx+e \right ) \right ) ^{2}ac+iA\sqrt{ac \left ( 1+ \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) }\sqrt{ac}\tan \left ( fx+e \right ) +iB\sqrt{ac \left ( 1+ \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) }\sqrt{ac}-B\ln \left ({ \left ( ac\tan \left ( fx+e \right ) +\sqrt{ac \left ( 1+ \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) }\sqrt{ac} \right ){\frac{1}{\sqrt{ac}}}} \right ) ac-B\sqrt{ac \left ( 1+ \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) }\sqrt{ac}\tan \left ( fx+e \right ) +A\sqrt{ac \left ( 1+ \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) }\sqrt{ac} \right ){\frac{1}{\sqrt{ac \left ( 1+ \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) }}}{\frac{1}{\sqrt{ac}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.37535, size = 189, normalized size = 1.72 \begin{align*} -\frac{{\left (2 \, B \arctan \left (\cos \left (f x + e\right ), \sin \left (f x + e\right ) + 1\right ) + 2 \, B \arctan \left (\cos \left (f x + e\right ), -\sin \left (f x + e\right ) + 1\right ) - 2 \,{\left (i \, A - B\right )} \cos \left (f x + e\right ) + i \, B \log \left (\cos \left (f x + e\right )^{2} + \sin \left (f x + e\right )^{2} + 2 \, \sin \left (f x + e\right ) + 1\right ) - i \, B \log \left (\cos \left (f x + e\right )^{2} + \sin \left (f x + e\right )^{2} - 2 \, \sin \left (f x + e\right ) + 1\right ) -{\left (2 \, A + 2 i \, B\right )} \sin \left (f x + e\right )\right )} \sqrt{c}}{2 \, \sqrt{a} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.59364, size = 1013, normalized size = 9.21 \begin{align*} \frac{{\left (a f \sqrt{-\frac{B^{2} c}{a f^{2}}} e^{\left (2 i \, f x + 2 i \, e\right )} \log \left (-\frac{2 \,{\left (B e^{\left (2 i \, f x + 2 i \, e\right )} + B\right )} \sqrt{\frac{a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt{\frac{c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} e^{\left (i \, f x + i \, e\right )} +{\left (a f e^{\left (2 i \, f x + 2 i \, e\right )} - a f\right )} \sqrt{-\frac{B^{2} c}{a f^{2}}}}{2 \,{\left (B e^{\left (2 i \, f x + 2 i \, e\right )} + B\right )}}\right ) - a f \sqrt{-\frac{B^{2} c}{a f^{2}}} e^{\left (2 i \, f x + 2 i \, e\right )} \log \left (-\frac{2 \,{\left (B e^{\left (2 i \, f x + 2 i \, e\right )} + B\right )} \sqrt{\frac{a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt{\frac{c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} e^{\left (i \, f x + i \, e\right )} -{\left (a f e^{\left (2 i \, f x + 2 i \, e\right )} - a f\right )} \sqrt{-\frac{B^{2} c}{a f^{2}}}}{2 \,{\left (B e^{\left (2 i \, f x + 2 i \, e\right )} + B\right )}}\right ) +{\left ({\left (-2 i \, A + 2 \, B\right )} e^{\left (3 i \, f x + 3 i \, e\right )} +{\left (2 i \, A - 2 \, B\right )} e^{\left (2 i \, f x + 2 i \, e\right )} +{\left (-2 i \, A + 2 \, B\right )} e^{\left (i \, f x + i \, e\right )} + 2 i \, A - 2 \, B\right )} \sqrt{\frac{a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt{\frac{c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} e^{\left (i \, f x + i \, e\right )}\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}}{2 \, a 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*} \int \frac{\sqrt{- c \left (i \tan{\left (e + f x \right )} - 1\right )} \left (A + B \tan{\left (e + f x \right )}\right )}{\sqrt{a \left (i \tan{\left (e + f x \right )} + 1\right )}}\, dx \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{{\left (B \tan \left (f x + e\right ) + A\right )} \sqrt{-i \, c \tan \left (f x + e\right ) + c}}{\sqrt{i \, a \tan \left (f x + e\right ) + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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