Integrand size = 33, antiderivative size = 62 \[ \int (a+i a \tan (e+f x))^2 (c-i c \tan (e+f x))^{3/2} \, dx=\frac {4 i a^2 (c-i c \tan (e+f x))^{3/2}}{3 f}-\frac {2 i a^2 (c-i c \tan (e+f x))^{5/2}}{5 c f} \]
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Time = 0.18 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {3603, 3568, 45} \[ \int (a+i a \tan (e+f x))^2 (c-i c \tan (e+f x))^{3/2} \, dx=\frac {4 i a^2 (c-i c \tan (e+f x))^{3/2}}{3 f}-\frac {2 i a^2 (c-i c \tan (e+f x))^{5/2}}{5 c f} \]
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Rule 45
Rule 3568
Rule 3603
Rubi steps \begin{align*} \text {integral}& = \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 ) \text {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 ) \text {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*}
Time = 1.54 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.81 \[ \int (a+i a \tan (e+f x))^2 (c-i c \tan (e+f x))^{3/2} \, dx=\frac {2 a^2 c (7+3 i \tan (e+f x)) (i+\tan (e+f x)) \sqrt {c-i c \tan (e+f x)}}{15 f} \]
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Time = 0.58 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.76
method | result | size |
derivativedivides | \(\frac {2 i a^{2} \left (-\frac {\left (c -i c \tan \left (f x +e \right )\right )^{\frac {5}{2}}}{5}+\frac {2 c \left (c -i c \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3}\right )}{f c}\) | \(47\) |
default | \(\frac {2 i a^{2} \left (-\frac {\left (c -i c \tan \left (f x +e \right )\right )^{\frac {5}{2}}}{5}+\frac {2 c \left (c -i c \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3}\right )}{f c}\) | \(47\) |
parts | \(\frac {2 i a^{2} c \left (-\sqrt {c -i c \tan \left (f x +e \right )}+\sqrt {c}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c -i c \tan \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right )\right )}{f}+\frac {2 i a^{2} \left (\frac {2 \left (c -i c \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3}+2 c \sqrt {c -i c \tan \left (f x +e \right )}-2 c^{\frac {3}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c -i c \tan \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right )\right )}{f}-\frac {2 i a^{2} \left (\frac {\left (c -i c \tan \left (f x +e \right )\right )^{\frac {5}{2}}}{5}+c^{2} \sqrt {c -i c \tan \left (f x +e \right )}-c^{\frac {5}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c -i c \tan \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right )\right )}{f c}\) | \(211\) |
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Time = 0.24 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.15 \[ \int (a+i a \tan (e+f x))^2 (c-i c \tan (e+f x))^{3/2} \, dx=-\frac {8 \, \sqrt {2} {\left (-5 i \, a^{2} c e^{\left (2 i \, f x + 2 i \, e\right )} - 2 i \, a^{2} c\right )} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{15 \, {\left (f e^{\left (4 i \, f x + 4 i \, e\right )} + 2 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )}} \]
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\[ \int (a+i a \tan (e+f x))^2 (c-i c \tan (e+f x))^{3/2} \, dx=- a^{2} \left (\int \left (- c \sqrt {- i c \tan {\left (e + f x \right )} + c}\right )\, dx + \int \left (- c \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{2}{\left (e + f x \right )}\right )\, dx + \int \left (- i c \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan {\left (e + f x \right )}\right )\, dx + \int \left (- i c \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{3}{\left (e + f x \right )}\right )\, dx\right ) \]
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Time = 0.32 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.74 \[ \int (a+i a \tan (e+f x))^2 (c-i c \tan (e+f x))^{3/2} \, dx=-\frac {2 i \, {\left (3 \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}} a^{2} - 10 \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}} a^{2} c\right )}}{15 \, c f} \]
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\[ \int (a+i a \tan (e+f x))^2 (c-i c \tan (e+f x))^{3/2} \, dx=\int { {\left (i \, a \tan \left (f x + e\right ) + a\right )}^{2} {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}} \,d x } \]
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Time = 8.86 (sec) , antiderivative size = 168, normalized size of antiderivative = 2.71 \[ \int (a+i a \tan (e+f x))^2 (c-i c \tan (e+f x))^{3/2} \, dx=\frac {8\,a^2\,c\,\sqrt {\frac {2\,c}{{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1}}\,\left ({\mathrm {e}}^{-e\,2{}\mathrm {i}-f\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}+\frac {{\mathrm {e}}^{-e\,4{}\mathrm {i}-f\,x\,4{}\mathrm {i}}\,1{}\mathrm {i}}{2}+\frac {1}{2}{}\mathrm {i}\right )\,\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,7{}\mathrm {i}+{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}\,5{}\mathrm {i}+2{}\mathrm {i}\right )}{15\,f\,\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}+1{}\mathrm {i}\right )\,\left (2\,{\mathrm {e}}^{-e\,2{}\mathrm {i}-f\,x\,2{}\mathrm {i}}+2\,{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+\frac {{\mathrm {e}}^{-e\,4{}\mathrm {i}-f\,x\,4{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}}{2}+3\right )} \]
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