Integrand size = 31, antiderivative size = 27 \[ \int (a+i a \tan (e+f x)) (c-i c \tan (e+f x))^{5/2} \, dx=\frac {2 i a (c-i c \tan (e+f x))^{5/2}}{5 f} \]
[Out]
Time = 0.12 (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} \[ \int (a+i a \tan (e+f x)) (c-i c \tan (e+f x))^{5/2} \, dx=\frac {2 i a (c-i c \tan (e+f x))^{5/2}}{5 f} \]
[In]
[Out]
Rule 32
Rule 3568
Rule 3603
Rubi steps \begin{align*} \text {integral}& = (a c) \int \sec ^2(e+f x) (c-i c \tan (e+f x))^{3/2} \, dx \\ & = \frac {(i a) \text {Subst}\left (\int (c+x)^{3/2} \, dx,x,-i c \tan (e+f x)\right )}{f} \\ & = \frac {2 i a (c-i c \tan (e+f x))^{5/2}}{5 f} \\ \end{align*}
Time = 0.90 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int (a+i a \tan (e+f x)) (c-i c \tan (e+f x))^{5/2} \, dx=\frac {2 i a (c-i c \tan (e+f x))^{5/2}}{5 f} \]
[In]
[Out]
Time = 0.42 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81
method | result | size |
derivativedivides | \(\frac {2 i a \left (c -i c \tan \left (f x +e \right )\right )^{\frac {5}{2}}}{5 f}\) | \(22\) |
default | \(\frac {2 i a \left (c -i c \tan \left (f x +e \right )\right )^{\frac {5}{2}}}{5 f}\) | \(22\) |
parts | \(\frac {2 i a c \left (-\frac {\left (c -i c \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3}-2 c \sqrt {c -i c \tan \left (f x +e \right )}+2 c^{\frac {3}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c -i c \tan \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right )\right )}{f}+\frac {i a \left (\frac {2 \left (c -i c \tan \left (f x +e \right )\right )^{\frac {5}{2}}}{5}+\frac {2 c \left (c -i c \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3}+4 c^{2} \sqrt {c -i c \tan \left (f x +e \right )}-4 c^{\frac {5}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c -i c \tan \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right )\right )}{f}\) | \(166\) |
[In]
[Out]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (19) = 38\).
Time = 0.24 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.96 \[ \int (a+i a \tan (e+f x)) (c-i c \tan (e+f x))^{5/2} \, dx=\frac {8 i \, \sqrt {2} a c^{2} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{5 \, {\left (f e^{\left (4 i \, f x + 4 i \, e\right )} + 2 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )}} \]
[In]
[Out]
Time = 3.38 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.63 \[ \int (a+i a \tan (e+f x)) (c-i c \tan (e+f x))^{5/2} \, dx=\begin {cases} \frac {2 i a \left (- i c \tan {\left (e + f x \right )} + c\right )^{\frac {5}{2}}}{5 f} & \text {for}\: f \neq 0 \\x \left (i a \tan {\left (e \right )} + a\right ) \left (- i c \tan {\left (e \right )} + c\right )^{\frac {5}{2}} & \text {otherwise} \end {cases} \]
[In]
[Out]
none
Time = 0.21 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.70 \[ \int (a+i a \tan (e+f x)) (c-i c \tan (e+f x))^{5/2} \, dx=\frac {2 i \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}} a}{5 \, f} \]
[In]
[Out]
\[ \int (a+i a \tan (e+f x)) (c-i c \tan (e+f x))^{5/2} \, dx=\int { {\left (i \, a \tan \left (f x + e\right ) + a\right )} {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}} \,d x } \]
[In]
[Out]
Time = 0.33 (sec) , antiderivative size = 120, normalized size of antiderivative = 4.44 \[ \int (a+i a \tan (e+f x)) (c-i c \tan (e+f x))^{5/2} \, dx=\frac {a\,c^2\,\sqrt {\frac {c\,\left (\cos \left (2\,e+2\,f\,x\right )+1-\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,e+2\,f\,x\right )+1}}\,\left (2\,\cos \left (2\,e+2\,f\,x\right )+\cos \left (4\,e+4\,f\,x\right )+1-\sin \left (2\,e+2\,f\,x\right )\,2{}\mathrm {i}-\sin \left (4\,e+4\,f\,x\right )\,1{}\mathrm {i}\right )\,4{}\mathrm {i}}{5\,f\,\left (4\,\cos \left (2\,e+2\,f\,x\right )+\cos \left (4\,e+4\,f\,x\right )+3\right )} \]
[In]
[Out]