Integrand size = 31, antiderivative size = 27 \[ \int \frac {a+i a \tan (e+f x)}{(c-i c \tan (e+f x))^{5/2}} \, dx=-\frac {2 i a}{5 f (c-i c \tan (e+f x))^{5/2}} \]
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Time = 0.17 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {3603, 3568, 32} \[ \int \frac {a+i a \tan (e+f x)}{(c-i c \tan (e+f x))^{5/2}} \, dx=-\frac {2 i a}{5 f (c-i c \tan (e+f x))^{5/2}} \]
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Rule 32
Rule 3568
Rule 3603
Rubi steps \begin{align*} \text {integral}& = (a c) \int \frac {\sec ^2(e+f x)}{(c-i c \tan (e+f x))^{7/2}} \, dx \\ & = \frac {(i a) \text {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*}
Time = 1.40 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {a+i a \tan (e+f x)}{(c-i c \tan (e+f x))^{5/2}} \, dx=-\frac {2 i a}{5 f (c-i c \tan (e+f x))^{5/2}} \]
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Time = 0.42 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81
method | result | size |
derivativedivides | \(-\frac {2 i a}{5 f \left (c -i c \tan \left (f x +e \right )\right )^{\frac {5}{2}}}\) | \(22\) |
default | \(-\frac {2 i a}{5 f \left (c -i c \tan \left (f x +e \right )\right )^{\frac {5}{2}}}\) | \(22\) |
risch | \(-\frac {i a \left ({\mathrm e}^{4 i \left (f x +e \right )}+2 \,{\mathrm e}^{2 i \left (f x +e \right )}+1\right ) \sqrt {2}}{20 c^{2} \sqrt {\frac {c}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, f}\) | \(53\) |
parts | \(\frac {2 i a c \left (\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c -i c \tan \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right )}{16 c^{\frac {7}{2}}}-\frac {1}{8 c^{3} \sqrt {c -i c \tan \left (f x +e \right )}}-\frac {1}{12 c^{2} \left (c -i c \tan \left (f x +e \right )\right )^{\frac {3}{2}}}-\frac {1}{10 c \left (c -i c \tan \left (f x +e \right )\right )^{\frac {5}{2}}}\right )}{f}+\frac {i a \left (-\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c -i c \tan \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right )}{8 c^{\frac {5}{2}}}-\frac {1}{5 \left (c -i c \tan \left (f x +e \right )\right )^{\frac {5}{2}}}+\frac {1}{4 c^{2} \sqrt {c -i c \tan \left (f x +e \right )}}+\frac {1}{6 c \left (c -i c \tan \left (f x +e \right )\right )^{\frac {3}{2}}}\right )}{f}\) | \(192\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (19) = 38\).
Time = 0.23 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.52 \[ \int \frac {a+i a \tan (e+f x)}{(c-i c \tan (e+f x))^{5/2}} \, dx=\frac {\sqrt {2} {\left (-i \, a e^{\left (6 i \, f x + 6 i \, e\right )} - 3 i \, a e^{\left (4 i \, f x + 4 i \, e\right )} - 3 i \, a e^{\left (2 i \, f x + 2 i \, e\right )} - i \, a\right )} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{20 \, c^{3} f} \]
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Time = 8.45 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.70 \[ \int \frac {a+i a \tan (e+f x)}{(c-i c \tan (e+f x))^{5/2}} \, dx=\begin {cases} - \frac {2 i a}{5 f \left (- i c \tan {\left (e + f x \right )} + c\right )^{\frac {5}{2}}} & \text {for}\: f \neq 0 \\\frac {x \left (i a \tan {\left (e \right )} + a\right )}{\left (- i c \tan {\left (e \right )} + c\right )^{\frac {5}{2}}} & \text {otherwise} \end {cases} \]
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none
Time = 0.21 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.70 \[ \int \frac {a+i a \tan (e+f x)}{(c-i c \tan (e+f x))^{5/2}} \, dx=-\frac {2 i \, a}{5 \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}} f} \]
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\[ \int \frac {a+i a \tan (e+f x)}{(c-i c \tan (e+f x))^{5/2}} \, dx=\int { \frac {i \, a \tan \left (f x + e\right ) + a}{{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}}} \,d x } \]
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Time = 6.95 (sec) , antiderivative size = 118, normalized size of antiderivative = 4.37 \[ \int \frac {a+i a \tan (e+f x)}{(c-i c \tan (e+f x))^{5/2}} \, dx=\frac {a\,\sqrt {-\frac {c\,\left (-2\,{\cos \left (e+f\,x\right )}^2+\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{2\,{\cos \left (e+f\,x\right )}^2}}\,\left (-{\cos \left (e+f\,x\right )}^2\,6{}\mathrm {i}-{\cos \left (2\,e+2\,f\,x\right )}^2\,6{}\mathrm {i}-{\cos \left (3\,e+3\,f\,x\right )}^2\,2{}\mathrm {i}+3\,\sin \left (2\,e+2\,f\,x\right )+3\,\sin \left (4\,e+4\,f\,x\right )+\sin \left (6\,e+6\,f\,x\right )+6{}\mathrm {i}\right )}{20\,c^3\,f} \]
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