Optimal. Leaf size=94 \[ \frac {2 i a^3 (c-i c \tan (e+f x))^{9/2}}{9 c^2 f}-\frac {8 i a^3 (c-i c \tan (e+f x))^{7/2}}{7 c f}+\frac {8 i a^3 (c-i c \tan (e+f x))^{5/2}}{5 f} \]
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Rubi [A] time = 0.16, antiderivative size = 94, 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} \[ \frac {2 i a^3 (c-i c \tan (e+f x))^{9/2}}{9 c^2 f}-\frac {8 i a^3 (c-i c \tan (e+f x))^{7/2}}{7 c f}+\frac {8 i a^3 (c-i c \tan (e+f x))^{5/2}}{5 f} \]
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))^3 (c-i c \tan (e+f x))^{5/2} \, dx &=\left (a^3 c^3\right ) \int \frac {\sec ^6(e+f x)}{\sqrt {c-i c \tan (e+f x)}} \, dx\\ &=\frac {\left (i a^3\right ) \operatorname {Subst}\left (\int (c-x)^2 (c+x)^{3/2} \, dx,x,-i c \tan (e+f x)\right )}{c^2 f}\\ &=\frac {\left (i a^3\right ) \operatorname {Subst}\left (\int \left (4 c^2 (c+x)^{3/2}-4 c (c+x)^{5/2}+(c+x)^{7/2}\right ) \, dx,x,-i c \tan (e+f x)\right )}{c^2 f}\\ &=\frac {8 i a^3 (c-i c \tan (e+f x))^{5/2}}{5 f}-\frac {8 i a^3 (c-i c \tan (e+f x))^{7/2}}{7 c f}+\frac {2 i a^3 (c-i c \tan (e+f x))^{9/2}}{9 c^2 f}\\ \end {align*}
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Mathematica [A] time = 4.83, size = 100, normalized size = 1.06 \[ \frac {2 a^3 c^2 \sec ^4(e+f x) \sqrt {c-i c \tan (e+f x)} (\sin (2 e-f x)+i \cos (2 e-f x)) (55 i \sin (2 (e+f x))+71 \cos (2 (e+f x))+36)}{315 f (\cos (f x)+i \sin (f x))^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.57, size = 116, normalized size = 1.23 \[ \frac {\sqrt {2} {\left (2016 i \, a^{3} c^{2} e^{\left (4 i \, f x + 4 i \, e\right )} + 1152 i \, a^{3} c^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + 256 i \, a^{3} c^{2}\right )} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{315 \, {\left (f e^{\left (8 i \, f x + 8 i \, e\right )} + 4 \, f e^{\left (6 i \, f x + 6 i \, e\right )} + 6 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 4 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )}} \]
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 \[ \int {\left (i \, a \tan \left (f x + e\right ) + a\right )}^{3} {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.31, size = 66, normalized size = 0.70 \[ \frac {2 i a^{3} \left (\frac {\left (c -i c \tan \left (f x +e \right )\right )^{\frac {9}{2}}}{9}-\frac {4 c \left (c -i c \tan \left (f x +e \right )\right )^{\frac {7}{2}}}{7}+\frac {4 c^{2} \left (c -i c \tan \left (f x +e \right )\right )^{\frac {5}{2}}}{5}\right )}{f \,c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.59, size = 67, normalized size = 0.71 \[ \frac {2 i \, {\left (35 \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {9}{2}} a^{3} - 180 \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {7}{2}} a^{3} c + 252 \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}} a^{3} c^{2}\right )}}{315 \, c^{2} f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.23, size = 97, normalized size = 1.03 \[ \frac {32\,a^3\,c^2\,\sqrt {c+\frac {c\,\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )\,1{}\mathrm {i}}{{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1}}\,\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,36{}\mathrm {i}+{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}\,63{}\mathrm {i}+8{}\mathrm {i}\right )}{315\,f\,{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1\right )}^4} \]
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 \[ - i a^{3} \left (\int i c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c}\, dx + \int \left (- c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan {\left (e + f x \right )}\right )\, dx + \int \left (- 2 c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{3}{\left (e + f x \right )}\right )\, dx + \int \left (- c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{5}{\left (e + f x \right )}\right )\, dx + \int 2 i c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{2}{\left (e + f x \right )}\, dx + \int i c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{4}{\left (e + f x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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