Integrand size = 25, antiderivative size = 34 \[ \int \frac {(b \tan (e+f x))^{3/2}}{(d \sec (e+f x))^{5/2}} \, dx=\frac {2 (b \tan (e+f x))^{5/2}}{5 b f (d \sec (e+f x))^{5/2}} \]
[Out]
Time = 0.06 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {2685} \[ \int \frac {(b \tan (e+f x))^{3/2}}{(d \sec (e+f x))^{5/2}} \, dx=\frac {2 (b \tan (e+f x))^{5/2}}{5 b f (d \sec (e+f x))^{5/2}} \]
[In]
[Out]
Rule 2685
Rubi steps \begin{align*} \text {integral}& = \frac {2 (b \tan (e+f x))^{5/2}}{5 b f (d \sec (e+f x))^{5/2}} \\ \end{align*}
Time = 0.95 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.26 \[ \int \frac {(b \tan (e+f x))^{3/2}}{(d \sec (e+f x))^{5/2}} \, dx=\frac {2 b \sin ^2(e+f x) \sqrt {b \tan (e+f x)}}{5 d^2 f \sqrt {d \sec (e+f x)}} \]
[In]
[Out]
Time = 1.46 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.12
method | result | size |
default | \(\frac {2 \left (\sin ^{2}\left (f x +e \right )\right ) \sqrt {b \tan \left (f x +e \right )}\, b}{5 f \,d^{2} \sqrt {d \sec \left (f x +e \right )}}\) | \(38\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (28) = 56\).
Time = 0.25 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.71 \[ \int \frac {(b \tan (e+f x))^{3/2}}{(d \sec (e+f x))^{5/2}} \, dx=-\frac {2 \, {\left (b \cos \left (f x + e\right )^{3} - b \cos \left (f x + e\right )\right )} \sqrt {\frac {b \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \sqrt {\frac {d}{\cos \left (f x + e\right )}}}{5 \, d^{3} f} \]
[In]
[Out]
Time = 45.79 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.56 \[ \int \frac {(b \tan (e+f x))^{3/2}}{(d \sec (e+f x))^{5/2}} \, dx=\begin {cases} \frac {2 \left (b \tan {\left (e + f x \right )}\right )^{\frac {3}{2}} \tan {\left (e + f x \right )}}{5 f \left (d \sec {\left (e + f x \right )}\right )^{\frac {5}{2}}} & \text {for}\: f \neq 0 \\\frac {x \left (b \tan {\left (e \right )}\right )^{\frac {3}{2}}}{\left (d \sec {\left (e \right )}\right )^{\frac {5}{2}}} & \text {otherwise} \end {cases} \]
[In]
[Out]
\[ \int \frac {(b \tan (e+f x))^{3/2}}{(d \sec (e+f x))^{5/2}} \, dx=\int { \frac {\left (b \tan \left (f x + e\right )\right )^{\frac {3}{2}}}{\left (d \sec \left (f x + e\right )\right )^{\frac {5}{2}}} \,d x } \]
[In]
[Out]
\[ \int \frac {(b \tan (e+f x))^{3/2}}{(d \sec (e+f x))^{5/2}} \, dx=\int { \frac {\left (b \tan \left (f x + e\right )\right )^{\frac {3}{2}}}{\left (d \sec \left (f x + e\right )\right )^{\frac {5}{2}}} \,d x } \]
[In]
[Out]
Time = 3.71 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.91 \[ \int \frac {(b \tan (e+f x))^{3/2}}{(d \sec (e+f x))^{5/2}} \, dx=\frac {b\,\sqrt {\frac {d}{\cos \left (e+f\,x\right )}}\,\left (\cos \left (e+f\,x\right )-\cos \left (3\,e+3\,f\,x\right )\right )\,\sqrt {\frac {b\,\sin \left (2\,e+2\,f\,x\right )}{\cos \left (2\,e+2\,f\,x\right )+1}}}{10\,d^3\,f} \]
[In]
[Out]