Optimal. Leaf size=27 \[ \frac {2 i a (c-i c \tan (e+f x))^{5/2}}{5 f} \]
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Rubi [A] time = 0.10, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {3522, 3487, 32} \[ \frac {2 i a (c-i c \tan (e+f x))^{5/2}}{5 f} \]
Antiderivative was successfully verified.
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Rule 32
Rule 3487
Rule 3522
Rubi steps
\begin {align*} \int (a+i a \tan (e+f x)) (c-i c \tan (e+f x))^{5/2} \, dx &=(a c) \int \sec ^2(e+f x) (c-i c \tan (e+f x))^{3/2} \, dx\\ &=\frac {(i a) \operatorname {Subst}\left (\int (c+x)^{3/2} \, dx,x,-i c \tan (e+f x)\right )}{f}\\ &=\frac {2 i a (c-i c \tan (e+f x))^{5/2}}{5 f}\\ \end {align*}
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Mathematica [B] time = 1.57, size = 70, normalized size = 2.59 \[ \frac {2 a c^2 \sec ^2(e+f x) (\cos (f x)-i \sin (f x)) \sqrt {c-i c \tan (e+f x)} (\sin (2 e+f x)+i \cos (2 e+f x))}{5 f} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.52, size = 53, normalized size = 1.96 \[ \frac {8 i \, \sqrt {2} a c^{2} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{5 \, {\left (f e^{\left (4 i \, f x + 4 i \, e\right )} + 2 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )}} \]
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 \[ \int {\left (i \, a \tan \left (f x + e\right ) + a\right )} {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.18, size = 22, normalized size = 0.81 \[ \frac {2 i a \left (c -i c \tan \left (f x +e \right )\right )^{\frac {5}{2}}}{5 f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 19, normalized size = 0.70 \[ \frac {2 i \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}} a}{5 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.31, size = 120, normalized size = 4.44 \[ \frac {a\,c^2\,\sqrt {\frac {c\,\left (\cos \left (2\,e+2\,f\,x\right )+1-\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,e+2\,f\,x\right )+1}}\,\left (2\,\cos \left (2\,e+2\,f\,x\right )+\cos \left (4\,e+4\,f\,x\right )+1-\sin \left (2\,e+2\,f\,x\right )\,2{}\mathrm {i}-\sin \left (4\,e+4\,f\,x\right )\,1{}\mathrm {i}\right )\,4{}\mathrm {i}}{5\,f\,\left (4\,\cos \left (2\,e+2\,f\,x\right )+\cos \left (4\,e+4\,f\,x\right )+3\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 8.23, size = 44, normalized size = 1.63 \[ \begin {cases} \frac {2 i a \left (- i c \tan {\left (e + f x \right )} + c\right )^{\frac {5}{2}}}{5 f} & \text {for}\: f \neq 0 \\x \left (i a \tan {\relax (e )} + a\right ) \left (- i c \tan {\relax (e )} + c\right )^{\frac {5}{2}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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