Integrand size = 24, antiderivative size = 104 \[ \int \sec (c+d x) (a+i a \tan (c+d x))^{5/2} \, dx=\frac {64 i a^3 \sec (c+d x)}{15 d \sqrt {a+i a \tan (c+d x)}}+\frac {16 i a^2 \sec (c+d x) \sqrt {a+i a \tan (c+d x)}}{15 d}+\frac {2 i a \sec (c+d x) (a+i a \tan (c+d x))^{3/2}}{5 d} \]
[Out]
Time = 0.13 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {3575, 3574} \[ \int \sec (c+d x) (a+i a \tan (c+d x))^{5/2} \, dx=\frac {64 i a^3 \sec (c+d x)}{15 d \sqrt {a+i a \tan (c+d x)}}+\frac {16 i a^2 \sec (c+d x) \sqrt {a+i a \tan (c+d x)}}{15 d}+\frac {2 i a \sec (c+d x) (a+i a \tan (c+d x))^{3/2}}{5 d} \]
[In]
[Out]
Rule 3574
Rule 3575
Rubi steps \begin{align*} \text {integral}& = \frac {2 i a \sec (c+d x) (a+i a \tan (c+d x))^{3/2}}{5 d}+\frac {1}{5} (8 a) \int \sec (c+d x) (a+i a \tan (c+d x))^{3/2} \, dx \\ & = \frac {16 i a^2 \sec (c+d x) \sqrt {a+i a \tan (c+d x)}}{15 d}+\frac {2 i a \sec (c+d x) (a+i a \tan (c+d x))^{3/2}}{5 d}+\frac {1}{15} \left (32 a^2\right ) \int \sec (c+d x) \sqrt {a+i a \tan (c+d x)} \, dx \\ & = \frac {64 i a^3 \sec (c+d x)}{15 d \sqrt {a+i a \tan (c+d x)}}+\frac {16 i a^2 \sec (c+d x) \sqrt {a+i a \tan (c+d x)}}{15 d}+\frac {2 i a \sec (c+d x) (a+i a \tan (c+d x))^{3/2}}{5 d} \\ \end{align*}
Time = 0.59 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.89 \[ \int \sec (c+d x) (a+i a \tan (c+d x))^{5/2} \, dx=\frac {2 a^2 \sec ^2(c+d x) (i \cos (c-d x)+\sin (c-d x)) (20+23 \cos (2 (c+d x))+7 i \sin (2 (c+d x))) \sqrt {a+i a \tan (c+d x)}}{15 d (\cos (d x)+i \sin (d x))^2} \]
[In]
[Out]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 181 vs. \(2 (86 ) = 172\).
Time = 7.42 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.75
method | result | size |
default | \(\frac {2 i \left (-\tan \left (d x +c \right )+i\right )^{2} a^{2} \sqrt {a \left (1+i \tan \left (d x +c \right )\right )}\, \left (32 i \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )-14 i \cos \left (d x +c \right ) \sin \left (d x +c \right )-32 \left (\cos ^{3}\left (d x +c \right )\right )-3 i \sin \left (d x +c \right )-46 \left (\cos ^{2}\left (d x +c \right )\right )-11 \cos \left (d x +c \right )+3\right )}{15 d \left (4 \left (\cos ^{3}\left (d x +c \right )\right )+2 \left (\cos ^{2}\left (d x +c \right )\right )+4 i \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )-3 \cos \left (d x +c \right )+2 i \cos \left (d x +c \right ) \sin \left (d x +c \right )-1-i \sin \left (d x +c \right )\right )}\) | \(182\) |
[In]
[Out]
none
Time = 0.25 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.80 \[ \int \sec (c+d x) (a+i a \tan (c+d x))^{5/2} \, dx=-\frac {8 \, \sqrt {2} {\left (-15 i \, a^{2} e^{\left (4 i \, d x + 4 i \, c\right )} - 20 i \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} - 8 i \, a^{2}\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}}{15 \, {\left (d e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]
[In]
[Out]
\[ \int \sec (c+d x) (a+i a \tan (c+d x))^{5/2} \, dx=\int \left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{\frac {5}{2}} \sec {\left (c + d x \right )}\, dx \]
[In]
[Out]
\[ \int \sec (c+d x) (a+i a \tan (c+d x))^{5/2} \, dx=\int { {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sec \left (d x + c\right ) \,d x } \]
[In]
[Out]
\[ \int \sec (c+d x) (a+i a \tan (c+d x))^{5/2} \, dx=\int { {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sec \left (d x + c\right ) \,d x } \]
[In]
[Out]
Time = 7.17 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.01 \[ \int \sec (c+d x) (a+i a \tan (c+d x))^{5/2} \, dx=\frac {8\,a^2\,{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}\,\sqrt {a-\frac {a\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )\,1{}\mathrm {i}}{{\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1}}\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}\,20{}\mathrm {i}+{\mathrm {e}}^{c\,4{}\mathrm {i}+d\,x\,4{}\mathrm {i}}\,15{}\mathrm {i}+8{}\mathrm {i}\right )}{15\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^2} \]
[In]
[Out]