Optimal. Leaf size=67 \[ \frac {i (c-i c \tan (e+f x))^{5/2} (a+i a \tan (e+f x))^m \, _2F_1\left (1,m+\frac {5}{2};\frac {7}{2};\frac {1}{2} (1-i \tan (e+f x))\right )}{5 f} \]
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Rubi [A] time = 0.11, 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} \[ \frac {i 2^m (c-i c \tan (e+f x))^{5/2} (1+i \tan (e+f x))^{-m} (a+i a \tan (e+f x))^m \, _2F_1\left (\frac {5}{2},1-m;\frac {7}{2};\frac {1}{2} (1-i \tan (e+f x))\right )}{5 f} \]
Antiderivative was successfully verified.
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Rule 69
Rule 70
Rule 3523
Rubi steps
\begin {align*} \int (a+i a \tan (e+f x))^m (c-i c \tan (e+f x))^{5/2} \, dx &=\frac {(a c) \operatorname {Subst}\left (\int (a+i a x)^{-1+m} (c-i c x)^{3/2} \, 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 \left (\frac {1}{2}+\frac {i x}{2}\right )^{-1+m} (c-i c x)^{3/2} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {i 2^m \, _2F_1\left (\frac {5}{2},1-m;\frac {7}{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 (c-i c \tan (e+f x))^{5/2}}{5 f}\\ \end {align*}
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Mathematica [B] time = 69.86, size = 141, normalized size = 2.10 \[ -\frac {i c 2^{m+\frac {3}{2}} \left (e^{i f x}\right )^m \left (\frac {c}{1+e^{2 i (e+f x)}}\right )^{3/2} \left (\frac {e^{i (e+f x)}}{1+e^{2 i (e+f x)}}\right )^m \, _2F_1\left (-\frac {3}{2},1;m+1;-e^{2 i (e+f x)}\right ) \sec ^{-m}(e+f x) (\cos (f x)+i \sin (f x))^{-m} (a+i a \tan (e+f x))^m}{f m} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {4 \, \sqrt {2} c^{2} \left (\frac {2 \, a e^{\left (2 i \, f x + 2 i \, e\right )}}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}\right )^{m} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{e^{\left (4 i \, f x + 4 i \, e\right )} + 2 \, e^{\left (2 i \, f x + 2 i \, e\right )} + 1}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}} {\left (i \, a \tan \left (f x + e\right ) + a\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 4.48, size = 0, normalized size = 0.00 \[ \int \left (a +i a \tan \left (f x +e \right )\right )^{m} \left (c -i c \tan \left (f x +e \right )\right )^{\frac {5}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}} {\left (i \, a \tan \left (f x + e\right ) + a\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^m\,{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{5/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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