Optimal. Leaf size=156 \[ \frac{5 i \tanh ^{-1}\left (\frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{2} \sqrt{c}}\right )}{8 \sqrt{2} a c^{3/2} f}-\frac{5 i}{8 a c f \sqrt{c-i c \tan (e+f x)}}-\frac{5 i}{12 a f (c-i c \tan (e+f x))^{3/2}}+\frac{i}{2 a f (1+i \tan (e+f x)) (c-i c \tan (e+f x))^{3/2}} \]
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Rubi [A] time = 0.199666, antiderivative size = 156, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ \frac{5 i \tanh ^{-1}\left (\frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{2} \sqrt{c}}\right )}{8 \sqrt{2} a c^{3/2} f}-\frac{5 i}{8 a c f \sqrt{c-i c \tan (e+f x)}}-\frac{5 i}{12 a f (c-i c \tan (e+f x))^{3/2}}+\frac{i}{2 a f (1+i \tan (e+f x)) (c-i c \tan (e+f x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 3522
Rule 3487
Rule 51
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{(a+i a \tan (e+f x)) (c-i c \tan (e+f x))^{3/2}} \, dx &=\frac{\int \frac{\cos ^2(e+f x)}{\sqrt{c-i c \tan (e+f x)}} \, dx}{a c}\\ &=\frac{\left (i c^2\right ) \operatorname{Subst}\left (\int \frac{1}{(c-x)^2 (c+x)^{5/2}} \, dx,x,-i c \tan (e+f x)\right )}{a f}\\ &=\frac{i}{2 a f (1+i \tan (e+f x)) (c-i c \tan (e+f x))^{3/2}}+\frac{(5 i c) \operatorname{Subst}\left (\int \frac{1}{(c-x) (c+x)^{5/2}} \, dx,x,-i c \tan (e+f x)\right )}{4 a f}\\ &=-\frac{5 i}{12 a f (c-i c \tan (e+f x))^{3/2}}+\frac{i}{2 a f (1+i \tan (e+f x)) (c-i c \tan (e+f x))^{3/2}}+\frac{(5 i) \operatorname{Subst}\left (\int \frac{1}{(c-x) (c+x)^{3/2}} \, dx,x,-i c \tan (e+f x)\right )}{8 a f}\\ &=-\frac{5 i}{12 a f (c-i c \tan (e+f x))^{3/2}}+\frac{i}{2 a f (1+i \tan (e+f x)) (c-i c \tan (e+f x))^{3/2}}-\frac{5 i}{8 a c f \sqrt{c-i c \tan (e+f x)}}+\frac{(5 i) \operatorname{Subst}\left (\int \frac{1}{(c-x) \sqrt{c+x}} \, dx,x,-i c \tan (e+f x)\right )}{16 a c f}\\ &=-\frac{5 i}{12 a f (c-i c \tan (e+f x))^{3/2}}+\frac{i}{2 a f (1+i \tan (e+f x)) (c-i c \tan (e+f x))^{3/2}}-\frac{5 i}{8 a c f \sqrt{c-i c \tan (e+f x)}}+\frac{(5 i) \operatorname{Subst}\left (\int \frac{1}{2 c-x^2} \, dx,x,\sqrt{c-i c \tan (e+f x)}\right )}{8 a c f}\\ &=\frac{5 i \tanh ^{-1}\left (\frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{2} \sqrt{c}}\right )}{8 \sqrt{2} a c^{3/2} f}-\frac{5 i}{12 a f (c-i c \tan (e+f x))^{3/2}}+\frac{i}{2 a f (1+i \tan (e+f x)) (c-i c \tan (e+f x))^{3/2}}-\frac{5 i}{8 a c f \sqrt{c-i c \tan (e+f x)}}\\ \end{align*}
Mathematica [A] time = 2.59563, size = 138, normalized size = 0.88 \[ \frac{\sqrt{c-i c \tan (e+f x)} (\cos (e+f x)+i \sin (e+f x)) \left (5 \sin (e+f x)+5 \sin (3 (e+f x))-27 i \cos (e+f x)+i \cos (3 (e+f x))+15 i e^{-i (e+f x)} \sqrt{1+e^{2 i (e+f x)}} \tanh ^{-1}\left (\sqrt{1+e^{2 i (e+f x)}}\right )\right )}{48 a c^2 f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.037, size = 121, normalized size = 0.8 \begin{align*}{\frac{2\,i{c}^{2}}{fa} \left ( -{\frac{1}{4\,{c}^{3}} \left ({\frac{1}{-4\,c-4\,ic\tan \left ( fx+e \right ) }\sqrt{c-ic\tan \left ( fx+e \right ) }}-{\frac{5\,\sqrt{2}}{8}{\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{1}{4\,{c}^{3}}{\frac{1}{\sqrt{c-ic\tan \left ( fx+e \right ) }}}}-{\frac{1}{12\,{c}^{2}} \left ( c-ic\tan \left ( fx+e \right ) \right ) ^{-{\frac{3}{2}}}} \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 = 1.4397, size = 852, normalized size = 5.46 \begin{align*} \frac{{\left (15 i \, \sqrt{\frac{1}{2}} a c^{2} f \sqrt{\frac{1}{a^{2} c^{3} f^{2}}} e^{\left (2 i \, f x + 2 i \, e\right )} \log \left (\frac{{\left (\sqrt{2} \sqrt{\frac{1}{2}}{\left (20 i \, a c f e^{\left (2 i \, f x + 2 i \, e\right )} + 20 i \, a c f\right )} \sqrt{\frac{c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt{\frac{1}{a^{2} c^{3} f^{2}}} + 20 i\right )} e^{\left (-i \, f x - i \, e\right )}}{16 \, a c f}\right ) - 15 i \, \sqrt{\frac{1}{2}} a c^{2} f \sqrt{\frac{1}{a^{2} c^{3} f^{2}}} e^{\left (2 i \, f x + 2 i \, e\right )} \log \left (\frac{{\left (\sqrt{2} \sqrt{\frac{1}{2}}{\left (-20 i \, a c f e^{\left (2 i \, f x + 2 i \, e\right )} - 20 i \, a c f\right )} \sqrt{\frac{c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt{\frac{1}{a^{2} c^{3} f^{2}}} + 20 i\right )} e^{\left (-i \, f x - i \, e\right )}}{16 \, a c f}\right ) + \sqrt{2} \sqrt{\frac{c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}{\left (-2 i \, e^{\left (6 i \, f x + 6 i \, e\right )} - 16 i \, e^{\left (4 i \, f x + 4 i \, e\right )} - 11 i \, e^{\left (2 i \, f x + 2 i \, e\right )} + 3 i\right )}\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}}{48 \, a c^{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{1}{{\left (i \, a \tan \left (f x + e\right ) + a\right )}{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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