Optimal. Leaf size=193 \[ -\frac{i c^4 \sqrt{c-i c \tan (e+f x)}}{4 a^3 f (c+i c \tan (e+f x))^2}+\frac{i c^4 (c-i c \tan (e+f x))^{3/2}}{3 a^3 f (c+i c \tan (e+f x))^3}+\frac{i c^3 \sqrt{c-i c \tan (e+f x)}}{16 a^3 f (c+i c \tan (e+f x))}+\frac{i c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{2} \sqrt{c}}\right )}{16 \sqrt{2} a^3 f} \]
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Rubi [A] time = 0.204621, antiderivative size = 193, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {3522, 3487, 47, 51, 63, 206} \[ -\frac{i c^4 \sqrt{c-i c \tan (e+f x)}}{4 a^3 f (c+i c \tan (e+f x))^2}+\frac{i c^4 (c-i c \tan (e+f x))^{3/2}}{3 a^3 f (c+i c \tan (e+f x))^3}+\frac{i c^3 \sqrt{c-i c \tan (e+f x)}}{16 a^3 f (c+i c \tan (e+f x))}+\frac{i c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{2} \sqrt{c}}\right )}{16 \sqrt{2} a^3 f} \]
Antiderivative was successfully verified.
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Rule 3522
Rule 3487
Rule 47
Rule 51
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{(c-i c \tan (e+f x))^{5/2}}{(a+i a \tan (e+f x))^3} \, dx &=\frac{\int \cos ^6(e+f x) (c-i c \tan (e+f x))^{11/2} \, dx}{a^3 c^3}\\ &=\frac{\left (i c^4\right ) \operatorname{Subst}\left (\int \frac{(c+x)^{3/2}}{(c-x)^4} \, dx,x,-i c \tan (e+f x)\right )}{a^3 f}\\ &=\frac{i c^4 (c-i c \tan (e+f x))^{3/2}}{3 a^3 f (c+i c \tan (e+f x))^3}-\frac{\left (i c^4\right ) \operatorname{Subst}\left (\int \frac{\sqrt{c+x}}{(c-x)^3} \, dx,x,-i c \tan (e+f x)\right )}{2 a^3 f}\\ &=\frac{i c^4 (c-i c \tan (e+f x))^{3/2}}{3 a^3 f (c+i c \tan (e+f x))^3}-\frac{i c^4 \sqrt{c-i c \tan (e+f x)}}{4 a^3 f (c+i c \tan (e+f x))^2}+\frac{\left (i c^4\right ) \operatorname{Subst}\left (\int \frac{1}{(c-x)^2 \sqrt{c+x}} \, dx,x,-i c \tan (e+f x)\right )}{8 a^3 f}\\ &=\frac{i c^4 (c-i c \tan (e+f x))^{3/2}}{3 a^3 f (c+i c \tan (e+f x))^3}-\frac{i c^4 \sqrt{c-i c \tan (e+f x)}}{4 a^3 f (c+i c \tan (e+f x))^2}+\frac{i c^3 \sqrt{c-i c \tan (e+f x)}}{16 a^3 f (c+i c \tan (e+f x))}+\frac{\left (i c^3\right ) \operatorname{Subst}\left (\int \frac{1}{(c-x) \sqrt{c+x}} \, dx,x,-i c \tan (e+f x)\right )}{32 a^3 f}\\ &=\frac{i c^4 (c-i c \tan (e+f x))^{3/2}}{3 a^3 f (c+i c \tan (e+f x))^3}-\frac{i c^4 \sqrt{c-i c \tan (e+f x)}}{4 a^3 f (c+i c \tan (e+f x))^2}+\frac{i c^3 \sqrt{c-i c \tan (e+f x)}}{16 a^3 f (c+i c \tan (e+f x))}+\frac{\left (i c^3\right ) \operatorname{Subst}\left (\int \frac{1}{2 c-x^2} \, dx,x,\sqrt{c-i c \tan (e+f x)}\right )}{16 a^3 f}\\ &=\frac{i c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{2} \sqrt{c}}\right )}{16 \sqrt{2} a^3 f}+\frac{i c^4 (c-i c \tan (e+f x))^{3/2}}{3 a^3 f (c+i c \tan (e+f x))^3}-\frac{i c^4 \sqrt{c-i c \tan (e+f x)}}{4 a^3 f (c+i c \tan (e+f x))^2}+\frac{i c^3 \sqrt{c-i c \tan (e+f x)}}{16 a^3 f (c+i c \tan (e+f x))}\\ \end{align*}
Mathematica [A] time = 3.65219, size = 152, normalized size = 0.79 \[ \frac{c^2 (\sin (3 (e+f x))+i \cos (3 (e+f x))) \left (\sqrt{c-i c \tan (e+f x)} \left (9 \cos (e+f x)+5 \cos (3 (e+f x))-44 i \sin (e+f x) \cos ^2(e+f x)\right )+3 \sqrt{2} \sqrt{c} (\cos (3 (e+f x))+i \sin (3 (e+f x))) \tanh ^{-1}\left (\frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{2} \sqrt{c}}\right )\right )}{96 a^3 f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.04, size = 115, normalized size = 0.6 \begin{align*}{\frac{2\,i{c}^{4}}{f{a}^{3}} \left ({\frac{1}{ \left ( -c-ic\tan \left ( fx+e \right ) \right ) ^{3}} \left ( -{\frac{1}{32\,c} \left ( c-ic\tan \left ( fx+e \right ) \right ) ^{{\frac{5}{2}}}}-{\frac{1}{6} \left ( c-ic\tan \left ( fx+e \right ) \right ) ^{{\frac{3}{2}}}}+{\frac{c}{8}\sqrt{c-ic\tan \left ( fx+e \right ) }} \right ) }+{\frac{\sqrt{2}}{64}{\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 ) } \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 = 1.34733, size = 819, normalized size = 4.24 \begin{align*} \frac{{\left (3 \, \sqrt{\frac{1}{2}} a^{3} f \sqrt{-\frac{c^{5}}{a^{6} f^{2}}} e^{\left (6 i \, f x + 6 i \, e\right )} \log \left (\frac{{\left (i \, c^{3} + \sqrt{2} \sqrt{\frac{1}{2}}{\left (a^{3} f e^{\left (2 i \, f x + 2 i \, e\right )} + a^{3} f\right )} \sqrt{\frac{c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt{-\frac{c^{5}}{a^{6} f^{2}}}\right )} e^{\left (-i \, f x - i \, e\right )}}{8 \, a^{3} f}\right ) - 3 \, \sqrt{\frac{1}{2}} a^{3} f \sqrt{-\frac{c^{5}}{a^{6} f^{2}}} e^{\left (6 i \, f x + 6 i \, e\right )} \log \left (\frac{{\left (i \, c^{3} - \sqrt{2} \sqrt{\frac{1}{2}}{\left (a^{3} f e^{\left (2 i \, f x + 2 i \, e\right )} + a^{3} f\right )} \sqrt{\frac{c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt{-\frac{c^{5}}{a^{6} f^{2}}}\right )} e^{\left (-i \, f x - i \, e\right )}}{8 \, a^{3} f}\right ) + \sqrt{2}{\left (-3 i \, c^{2} e^{\left (6 i \, f x + 6 i \, e\right )} - i \, c^{2} e^{\left (4 i \, f x + 4 i \, e\right )} + 10 i \, c^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + 8 i \, c^{2}\right )} \sqrt{\frac{c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}\right )} e^{\left (-6 i \, f x - 6 i \, e\right )}}{96 \, a^{3} 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{{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac{5}{2}}}{{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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