Optimal. Leaf size=67 \[ -\frac{i (a+i a \tan (e+f x))^m \, _2F_1\left (1,m-\frac{3}{2};-\frac{1}{2};\frac{1}{2} (1-i \tan (e+f x))\right )}{3 f (c-i c \tan (e+f x))^{3/2}} \]
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Rubi [A] time = 0.112425, antiderivative size = 88, normalized size of antiderivative = 1.31, number of steps used = 3, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {3523, 70, 69} \[ -\frac{i 2^m (1+i \tan (e+f x))^{-m} (a+i a \tan (e+f x))^m \, _2F_1\left (-\frac{3}{2},1-m;-\frac{1}{2};\frac{1}{2} (1-i \tan (e+f x))\right )}{3 f (c-i c \tan (e+f x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 3523
Rule 70
Rule 69
Rubi steps
\begin{align*} \int \frac{(a+i a \tan (e+f x))^m}{(c-i c \tan (e+f x))^{3/2}} \, dx &=\frac{(a c) \operatorname{Subst}\left (\int \frac{(a+i a x)^{-1+m}}{(c-i c x)^{5/2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{\left (2^{-1+m} c (a+i a \tan (e+f x))^m \left (\frac{a+i a \tan (e+f x)}{a}\right )^{-m}\right ) \operatorname{Subst}\left (\int \frac{\left (\frac{1}{2}+\frac{i x}{2}\right )^{-1+m}}{(c-i c x)^{5/2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac{i 2^m \, _2F_1\left (-\frac{3}{2},1-m;-\frac{1}{2};\frac{1}{2} (1-i \tan (e+f x))\right ) (1+i \tan (e+f x))^{-m} (a+i a \tan (e+f x))^m}{3 f (c-i c \tan (e+f x))^{3/2}}\\ \end{align*}
Mathematica [B] time = 107.813, size = 141, normalized size = 2.1 \[ -\frac{i c 2^{m-\frac{5}{2}} \left (e^{i f x}\right )^m \left (\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}\right )^m \, _2F_1\left (1,\frac{5}{2};m+1;-e^{2 i (e+f x)}\right ) \sec ^{-m}(e+f x) (\cos (f x)+i \sin (f x))^{-m} (a+i a \tan (e+f x))^m}{f m \left (\frac{c}{1+e^{2 i (e+f x)}}\right )^{5/2}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.409, size = 0, normalized size = 0. \begin{align*} \int{ \left ( a+ia\tan \left ( fx+e \right ) \right ) ^{m} \left ( c-ic\tan \left ( fx+e \right ) \right ) ^{-{\frac{3}{2}}}}\, dx \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: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{2} \left (\frac{2 \, a e^{\left (2 i \, f x + 2 i \, e\right )}}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}\right )^{m} \sqrt{\frac{c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}{\left (e^{\left (4 i \, f x + 4 i \, e\right )} + 2 \, e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right )}}{4 \, c^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 \, a \tan \left (f x + e\right ) + a\right )}^{m}}{{\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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