Integrand size = 35, antiderivative size = 43 \[ \int \frac {(c-i c \tan (e+f x))^{3/2}}{(a+i a \tan (e+f x))^{3/2}} \, dx=\frac {i (c-i c \tan (e+f x))^{3/2}}{3 f (a+i a \tan (e+f x))^{3/2}} \]
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Time = 0.13 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {3604, 37} \[ \int \frac {(c-i c \tan (e+f x))^{3/2}}{(a+i a \tan (e+f x))^{3/2}} \, dx=\frac {i (c-i c \tan (e+f x))^{3/2}}{3 f (a+i a \tan (e+f x))^{3/2}} \]
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Rule 37
Rule 3604
Rubi steps \begin{align*} \text {integral}& = \frac {(a c) \text {Subst}\left (\int \frac {\sqrt {c-i c x}}{(a+i a x)^{5/2}} \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {i (c-i c \tan (e+f x))^{3/2}}{3 f (a+i a \tan (e+f x))^{3/2}} \\ \end{align*}
Time = 1.12 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00 \[ \int \frac {(c-i c \tan (e+f x))^{3/2}}{(a+i a \tan (e+f x))^{3/2}} \, dx=\frac {i (c-i c \tan (e+f x))^{3/2}}{3 f (a+i a \tan (e+f x))^{3/2}} \]
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Time = 0.97 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.49
method | result | size |
derivativedivides | \(\frac {\sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, c \left (1+\tan ^{2}\left (f x +e \right )\right )}{3 f \,a^{2} \left (-\tan \left (f x +e \right )+i\right )^{3}}\) | \(64\) |
default | \(\frac {\sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, c \left (1+\tan ^{2}\left (f x +e \right )\right )}{3 f \,a^{2} \left (-\tan \left (f x +e \right )+i\right )^{3}}\) | \(64\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 67 vs. \(2 (31) = 62\).
Time = 0.23 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.56 \[ \int \frac {(c-i c \tan (e+f x))^{3/2}}{(a+i a \tan (e+f x))^{3/2}} \, dx=\frac {{\left (i \, c e^{\left (2 i \, f x + 2 i \, e\right )} + i \, c\right )} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} e^{\left (-3 i \, f x - 3 i \, e\right )}}{3 \, a^{2} f} \]
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\[ \int \frac {(c-i c \tan (e+f x))^{3/2}}{(a+i a \tan (e+f x))^{3/2}} \, dx=\int \frac {\left (- i c \left (\tan {\left (e + f x \right )} + i\right )\right )^{\frac {3}{2}}}{\left (i a \left (\tan {\left (e + f x \right )} - i\right )\right )^{\frac {3}{2}}}\, dx \]
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none
Time = 0.39 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.81 \[ \int \frac {(c-i c \tan (e+f x))^{3/2}}{(a+i a \tan (e+f x))^{3/2}} \, dx=\frac {{\left (i \, c \cos \left (3 \, f x + 3 \, e\right ) + c \sin \left (3 \, f x + 3 \, e\right )\right )} \sqrt {c}}{3 \, a^{\frac {3}{2}} f} \]
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\[ \int \frac {(c-i c \tan (e+f x))^{3/2}}{(a+i a \tan (e+f x))^{3/2}} \, dx=\int { \frac {{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}}}{{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
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Time = 7.06 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.44 \[ \int \frac {(c-i c \tan (e+f x))^{3/2}}{(a+i a \tan (e+f x))^{3/2}} \, dx=-\frac {c\,\left (-1+\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )\,\sqrt {-c\,\left (-1+\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}}{3\,a\,f\,\left (\mathrm {tan}\left (e+f\,x\right )-\mathrm {i}\right )\,\sqrt {a\,\left (1+\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}} \]
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