Optimal. Leaf size=27 \[ -\frac {2 i a}{5 f (c-i c \tan (e+f x))^{5/2}} \]
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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} \[ -\frac {2 i a}{5 f (c-i c \tan (e+f x))^{5/2}} \]
Antiderivative was successfully verified.
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Rule 32
Rule 3487
Rule 3522
Rubi steps
\begin {align*} \int \frac {a+i a \tan (e+f x)}{(c-i c \tan (e+f x))^{5/2}} \, dx &=(a c) \int \frac {\sec ^2(e+f x)}{(c-i c \tan (e+f x))^{7/2}} \, dx\\ &=\frac {(i a) \operatorname {Subst}\left (\int \frac {1}{(c+x)^{7/2}} \, dx,x,-i c \tan (e+f x)\right )}{f}\\ &=-\frac {2 i a}{5 f (c-i c \tan (e+f x))^{5/2}}\\ \end {align*}
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Mathematica [B] time = 2.57, size = 72, normalized size = 2.67 \[ \frac {2 a \cos ^3(e+f x) (\cos (f x)-i \sin (f x)) \sqrt {c-i c \tan (e+f x)} (\sin (3 e+4 f x)-i \cos (3 e+4 f x))}{5 c^3 f} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.47, size = 68, normalized size = 2.52 \[ \frac {\sqrt {2} {\left (-i \, a e^{\left (6 i \, f x + 6 i \, e\right )} - 3 i \, a e^{\left (4 i \, f x + 4 i \, e\right )} - 3 i \, a e^{\left (2 i \, f x + 2 i \, e\right )} - i \, a\right )} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{20 \, c^{3} f} \]
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 \[ \int \frac {i \, a \tan \left (f x + e\right ) + a}{{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.17, size = 22, normalized size = 0.81 \[ -\frac {2 i a}{5 f \left (c -i c \tan \left (f x +e \right )\right )^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.62, size = 19, normalized size = 0.70 \[ -\frac {2 i \, a}{5 \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}} f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.49, size = 118, normalized size = 4.37 \[ \frac {a\,\sqrt {-\frac {c\,\left (-2\,{\cos \left (e+f\,x\right )}^2+\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{2\,{\cos \left (e+f\,x\right )}^2}}\,\left (-{\cos \left (e+f\,x\right )}^2\,6{}\mathrm {i}-{\cos \left (2\,e+2\,f\,x\right )}^2\,6{}\mathrm {i}-{\cos \left (3\,e+3\,f\,x\right )}^2\,2{}\mathrm {i}+3\,\sin \left (2\,e+2\,f\,x\right )+3\,\sin \left (4\,e+4\,f\,x\right )+\sin \left (6\,e+6\,f\,x\right )+6{}\mathrm {i}\right )}{20\,c^3\,f} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 13.27, size = 46, normalized size = 1.70 \[ \begin {cases} - \frac {2 i a}{5 f \left (- i c \tan {\left (e + f x \right )} + c\right )^{\frac {5}{2}}} & \text {for}\: f \neq 0 \\\frac {x \left (i a \tan {\relax (e )} + a\right )}{\left (- i c \tan {\relax (e )} + c\right )^{\frac {5}{2}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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