Optimal. Leaf size=62 \[ \frac {4 i a^2 (c-i c \tan (e+f x))^{3/2}}{3 f}-\frac {2 i a^2 (c-i c \tan (e+f x))^{5/2}}{5 c f} \]
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Rubi [A] time = 0.15, antiderivative size = 62, normalized size of antiderivative = 1.00, 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} \[ \frac {4 i a^2 (c-i c \tan (e+f x))^{3/2}}{3 f}-\frac {2 i a^2 (c-i c \tan (e+f x))^{5/2}}{5 c f} \]
Antiderivative was successfully verified.
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Rule 43
Rule 3487
Rule 3522
Rubi steps
\begin {align*} \int (a+i a \tan (e+f x))^2 (c-i c \tan (e+f x))^{3/2} \, dx &=\left (a^2 c^2\right ) \int \frac {\sec ^4(e+f x)}{\sqrt {c-i c \tan (e+f x)}} \, dx\\ &=\frac {\left (i a^2\right ) \operatorname {Subst}\left (\int (c-x) \sqrt {c+x} \, dx,x,-i c \tan (e+f x)\right )}{c f}\\ &=\frac {\left (i a^2\right ) \operatorname {Subst}\left (\int \left (2 c \sqrt {c+x}-(c+x)^{3/2}\right ) \, dx,x,-i c \tan (e+f x)\right )}{c f}\\ &=\frac {4 i a^2 (c-i c \tan (e+f x))^{3/2}}{3 f}-\frac {2 i a^2 (c-i c \tan (e+f x))^{5/2}}{5 c f}\\ \end {align*}
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Mathematica [A] time = 2.45, size = 80, normalized size = 1.29 \[ -\frac {2 a^2 c (3 \tan (e+f x)-7 i) \sec (e+f x) \sqrt {c-i c \tan (e+f x)} (\cos (e-f x)-i \sin (e-f x))}{15 f (\cos (f x)+i \sin (f x))^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 71, normalized size = 1.15 \[ \frac {\sqrt {2} {\left (40 i \, a^{2} c e^{\left (2 i \, f x + 2 i \, e\right )} + 16 i \, a^{2} c\right )} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{15 \, {\left (f e^{\left (4 i \, f x + 4 i \, e\right )} + 2 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )}} \]
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 \[ \int {\left (i \, a \tan \left (f x + e\right ) + a\right )}^{2} {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.20, size = 47, normalized size = 0.76 \[ -\frac {2 i a^{2} \left (\frac {\left (c -i c \tan \left (f x +e \right )\right )^{\frac {5}{2}}}{5}-\frac {2 \left (c -i c \tan \left (f x +e \right )\right )^{\frac {3}{2}} c}{3}\right )}{f c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.58, size = 46, normalized size = 0.74 \[ -\frac {2 i \, {\left (3 \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}} a^{2} - 10 \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}} a^{2} c\right )}}{15 \, c f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.79, size = 168, normalized size = 2.71 \[ \frac {8\,a^2\,c\,\sqrt {\frac {2\,c}{{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1}}\,\left ({\mathrm {e}}^{-e\,2{}\mathrm {i}-f\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}+\frac {{\mathrm {e}}^{-e\,4{}\mathrm {i}-f\,x\,4{}\mathrm {i}}\,1{}\mathrm {i}}{2}+\frac {1}{2}{}\mathrm {i}\right )\,\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,7{}\mathrm {i}+{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}\,5{}\mathrm {i}+2{}\mathrm {i}\right )}{15\,f\,\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}+1{}\mathrm {i}\right )\,\left (2\,{\mathrm {e}}^{-e\,2{}\mathrm {i}-f\,x\,2{}\mathrm {i}}+2\,{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+\frac {{\mathrm {e}}^{-e\,4{}\mathrm {i}-f\,x\,4{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}}{2}+3\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - a^{2} \left (\int \left (- c \sqrt {- i c \tan {\left (e + f x \right )} + c}\right )\, dx + \int \left (- c \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{2}{\left (e + f x \right )}\right )\, dx + \int \left (- i c \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan {\left (e + f x \right )}\right )\, dx + \int \left (- i c \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{3}{\left (e + f x \right )}\right )\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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