Optimal. Leaf size=138 \[ \frac{3 i \sqrt{c-i c \tan (e+f x)}}{16 a^2 f (1+i \tan (e+f x))}+\frac{i \sqrt{c-i c \tan (e+f x)}}{4 a^2 f (1+i \tan (e+f x))^2}+\frac{3 i \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{2} \sqrt{c}}\right )}{16 \sqrt{2} a^2 f} \]
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Rubi [A] time = 0.193482, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.152, Rules used = {3522, 3487, 51, 63, 206} \[ \frac{3 i \sqrt{c-i c \tan (e+f x)}}{16 a^2 f (1+i \tan (e+f x))}+\frac{i \sqrt{c-i c \tan (e+f x)}}{4 a^2 f (1+i \tan (e+f x))^2}+\frac{3 i \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{2} \sqrt{c}}\right )}{16 \sqrt{2} a^2 f} \]
Antiderivative was successfully verified.
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Rule 3522
Rule 3487
Rule 51
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{\sqrt{c-i c \tan (e+f x)}}{(a+i a \tan (e+f x))^2} \, dx &=\frac{\int \cos ^4(e+f x) (c-i c \tan (e+f x))^{5/2} \, dx}{a^2 c^2}\\ &=\frac{\left (i c^3\right ) \operatorname{Subst}\left (\int \frac{1}{(c-x)^3 \sqrt{c+x}} \, dx,x,-i c \tan (e+f x)\right )}{a^2 f}\\ &=\frac{i \sqrt{c-i c \tan (e+f x)}}{4 a^2 f (1+i \tan (e+f x))^2}+\frac{\left (3 i c^2\right ) \operatorname{Subst}\left (\int \frac{1}{(c-x)^2 \sqrt{c+x}} \, dx,x,-i c \tan (e+f x)\right )}{8 a^2 f}\\ &=\frac{i \sqrt{c-i c \tan (e+f x)}}{4 a^2 f (1+i \tan (e+f x))^2}+\frac{3 i \sqrt{c-i c \tan (e+f x)}}{16 a^2 f (1+i \tan (e+f x))}+\frac{(3 i c) \operatorname{Subst}\left (\int \frac{1}{(c-x) \sqrt{c+x}} \, dx,x,-i c \tan (e+f x)\right )}{32 a^2 f}\\ &=\frac{i \sqrt{c-i c \tan (e+f x)}}{4 a^2 f (1+i \tan (e+f x))^2}+\frac{3 i \sqrt{c-i c \tan (e+f x)}}{16 a^2 f (1+i \tan (e+f x))}+\frac{(3 i c) \operatorname{Subst}\left (\int \frac{1}{2 c-x^2} \, dx,x,\sqrt{c-i c \tan (e+f x)}\right )}{16 a^2 f}\\ &=\frac{3 i \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{2} \sqrt{c}}\right )}{16 \sqrt{2} a^2 f}+\frac{i \sqrt{c-i c \tan (e+f x)}}{4 a^2 f (1+i \tan (e+f x))^2}+\frac{3 i \sqrt{c-i c \tan (e+f x)}}{16 a^2 f (1+i \tan (e+f x))}\\ \end{align*}
Mathematica [A] time = 1.55741, size = 136, normalized size = 0.99 \[ \frac{(\sin (2 (e+f x))+i \cos (2 (e+f x))) \left (\sqrt{c-i c \tan (e+f x)} (3 i \sin (2 (e+f x))+7 \cos (2 (e+f x))+7)+3 \sqrt{2} \sqrt{c} (\cos (2 (e+f x))+i \sin (2 (e+f x))) \tanh ^{-1}\left (\frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{2} \sqrt{c}}\right )\right )}{32 a^2 f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.074, size = 121, normalized size = 0.9 \begin{align*}{\frac{-2\,i{c}^{3}}{f{a}^{2}} \left ( -{\frac{1}{8\,c \left ( -c-ic\tan \left ( fx+e \right ) \right ) ^{2}}\sqrt{c-ic\tan \left ( fx+e \right ) }}-{\frac{3}{8\,c} \left ( -{\frac{1}{4\,c \left ( -c-ic\tan \left ( fx+e \right ) \right ) }\sqrt{c-ic\tan \left ( fx+e \right ) }}+{\frac{\sqrt{2}}{8}{\it Artanh} \left ({\frac{\sqrt{2}}{2}\sqrt{c-ic\tan \left ( fx+e \right ) }{\frac{1}{\sqrt{c}}}} \right ){c}^{-{\frac{3}{2}}}} \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 = 1.40511, size = 748, normalized size = 5.42 \begin{align*} \frac{{\left (3 \, \sqrt{\frac{1}{2}} a^{2} f \sqrt{-\frac{c}{a^{4} f^{2}}} e^{\left (4 i \, f x + 4 i \, e\right )} \log \left (\frac{3 \,{\left (\sqrt{2} \sqrt{\frac{1}{2}}{\left (a^{2} f e^{\left (2 i \, f x + 2 i \, e\right )} + a^{2} f\right )} \sqrt{\frac{c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt{-\frac{c}{a^{4} f^{2}}} + i \, c\right )} e^{\left (-i \, f x - i \, e\right )}}{8 \, a^{2} f}\right ) - 3 \, \sqrt{\frac{1}{2}} a^{2} f \sqrt{-\frac{c}{a^{4} f^{2}}} e^{\left (4 i \, f x + 4 i \, e\right )} \log \left (-\frac{3 \,{\left (\sqrt{2} \sqrt{\frac{1}{2}}{\left (a^{2} f e^{\left (2 i \, f x + 2 i \, e\right )} + a^{2} f\right )} \sqrt{\frac{c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt{-\frac{c}{a^{4} f^{2}}} - i \, c\right )} e^{\left (-i \, f x - i \, e\right )}}{8 \, a^{2} f}\right ) + \sqrt{2} \sqrt{\frac{c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}{\left (5 i \, e^{\left (4 i \, f x + 4 i \, e\right )} + 7 i \, e^{\left (2 i \, f x + 2 i \, e\right )} + 2 i\right )}\right )} e^{\left (-4 i \, f x - 4 i \, e\right )}}{32 \, a^{2} 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{\sqrt{-i \, c \tan \left (f x + e\right ) + c}}{{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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