Optimal. Leaf size=92 \[ \frac{2 i a^3 (c-i c \tan (e+f x))^{5/2}}{5 c^2 f}-\frac{8 i a^3 (c-i c \tan (e+f x))^{3/2}}{3 c f}+\frac{8 i a^3 \sqrt{c-i c \tan (e+f x)}}{f} \]
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Rubi [A] time = 0.162748, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {3522, 3487, 43} \[ \frac{2 i a^3 (c-i c \tan (e+f x))^{5/2}}{5 c^2 f}-\frac{8 i a^3 (c-i c \tan (e+f x))^{3/2}}{3 c f}+\frac{8 i a^3 \sqrt{c-i c \tan (e+f x)}}{f} \]
Antiderivative was successfully verified.
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Rule 3522
Rule 3487
Rule 43
Rubi steps
\begin{align*} \int (a+i a \tan (e+f x))^3 \sqrt{c-i c \tan (e+f x)} \, dx &=\left (a^3 c^3\right ) \int \frac{\sec ^6(e+f x)}{(c-i c \tan (e+f x))^{5/2}} \, dx\\ &=\frac{\left (i a^3\right ) \operatorname{Subst}\left (\int \frac{(c-x)^2}{\sqrt{c+x}} \, dx,x,-i c \tan (e+f x)\right )}{c^2 f}\\ &=\frac{\left (i a^3\right ) \operatorname{Subst}\left (\int \left (\frac{4 c^2}{\sqrt{c+x}}-4 c \sqrt{c+x}+(c+x)^{3/2}\right ) \, dx,x,-i c \tan (e+f x)\right )}{c^2 f}\\ &=\frac{8 i a^3 \sqrt{c-i c \tan (e+f x)}}{f}-\frac{8 i a^3 (c-i c \tan (e+f x))^{3/2}}{3 c f}+\frac{2 i a^3 (c-i c \tan (e+f x))^{5/2}}{5 c^2 f}\\ \end{align*}
Mathematica [A] time = 2.90233, size = 61, normalized size = 0.66 \[ \frac{2 i a^3 \sec ^2(e+f x) \sqrt{c-i c \tan (e+f x)} (7 i \sin (2 (e+f x))+23 \cos (2 (e+f x))+20)}{15 f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.101, size = 66, normalized size = 0.7 \begin{align*}{\frac{2\,i{a}^{3}}{f{c}^{2}} \left ({\frac{1}{5} \left ( c-ic\tan \left ( fx+e \right ) \right ) ^{{\frac{5}{2}}}}-{\frac{4\,c}{3} \left ( c-ic\tan \left ( fx+e \right ) \right ) ^{{\frac{3}{2}}}}+4\,{c}^{2}\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 [A] time = 1.15751, size = 90, normalized size = 0.98 \begin{align*} \frac{2 i \,{\left (3 \,{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac{5}{2}} a^{3} - 20 \,{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac{3}{2}} a^{3} c + 60 \, \sqrt{-i \, c \tan \left (f x + e\right ) + c} a^{3} c^{2}\right )}}{15 \, c^{2} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.3721, size = 239, normalized size = 2.6 \begin{align*} \frac{\sqrt{2}{\left (120 i \, a^{3} e^{\left (4 i \, f x + 4 i \, e\right )} + 160 i \, a^{3} e^{\left (2 i \, f x + 2 i \, e\right )} + 64 i \, a^{3}\right )} \sqrt{\frac{c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{15 \,{\left (f e^{\left (4 i \, f x + 4 i \, e\right )} + 2 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )}} \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*} a^{3} \left (\int - 3 \sqrt{- i c \tan{\left (e + f x \right )} + c} \tan ^{2}{\left (e + f x \right )}\, dx + \int 3 i \sqrt{- i c \tan{\left (e + f x \right )} + c} \tan{\left (e + f x \right )}\, dx + \int - i \sqrt{- i c \tan{\left (e + f x \right )} + c} \tan ^{3}{\left (e + f x \right )}\, dx + \int \sqrt{- i c \tan{\left (e + f x \right )} + c}\, dx\right ) \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{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{3} \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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