Integrand size = 25, antiderivative size = 95 \[ \int \frac {1}{(d \sec (e+f x))^{3/2} \sqrt {b \tan (e+f x)}} \, dx=\frac {4 \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),2\right ) \sqrt {d \sec (e+f x)} \sqrt {\sin (e+f x)}}{3 d^2 f \sqrt {b \tan (e+f x)}}+\frac {2 \sqrt {b \tan (e+f x)}}{3 b f (d \sec (e+f x))^{3/2}} \]
[Out]
Time = 0.13 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {2692, 2696, 2721, 2720} \[ \int \frac {1}{(d \sec (e+f x))^{3/2} \sqrt {b \tan (e+f x)}} \, dx=\frac {4 \sqrt {\sin (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),2\right ) \sqrt {d \sec (e+f x)}}{3 d^2 f \sqrt {b \tan (e+f x)}}+\frac {2 \sqrt {b \tan (e+f x)}}{3 b f (d \sec (e+f x))^{3/2}} \]
[In]
[Out]
Rule 2692
Rule 2696
Rule 2720
Rule 2721
Rubi steps \begin{align*} \text {integral}& = \frac {2 \sqrt {b \tan (e+f x)}}{3 b f (d \sec (e+f x))^{3/2}}+\frac {2 \int \frac {\sqrt {d \sec (e+f x)}}{\sqrt {b \tan (e+f x)}} \, dx}{3 d^2} \\ & = \frac {2 \sqrt {b \tan (e+f x)}}{3 b f (d \sec (e+f x))^{3/2}}+\frac {\left (2 \sqrt {d \sec (e+f x)} \sqrt {b \sin (e+f x)}\right ) \int \frac {1}{\sqrt {b \sin (e+f x)}} \, dx}{3 d^2 \sqrt {b \tan (e+f x)}} \\ & = \frac {2 \sqrt {b \tan (e+f x)}}{3 b f (d \sec (e+f x))^{3/2}}+\frac {\left (2 \sqrt {d \sec (e+f x)} \sqrt {\sin (e+f x)}\right ) \int \frac {1}{\sqrt {\sin (e+f x)}} \, dx}{3 d^2 \sqrt {b \tan (e+f x)}} \\ & = \frac {4 \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),2\right ) \sqrt {d \sec (e+f x)} \sqrt {\sin (e+f x)}}{3 d^2 f \sqrt {b \tan (e+f x)}}+\frac {2 \sqrt {b \tan (e+f x)}}{3 b f (d \sec (e+f x))^{3/2}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.74 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.74 \[ \int \frac {1}{(d \sec (e+f x))^{3/2} \sqrt {b \tan (e+f x)}} \, dx=\frac {2 \left (1+2 \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {3}{4},\frac {5}{4},-\tan ^2(e+f x)\right ) \sec ^2(e+f x)^{3/4}\right ) \sqrt {b \tan (e+f x)}}{3 b f (d \sec (e+f x))^{3/2}} \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 1.93 (sec) , antiderivative size = 239, normalized size of antiderivative = 2.52
method | result | size |
default | \(\frac {\left (2 i \sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, \sqrt {i \left (-i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, \sqrt {i \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right )}\, F\left (\sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}, \frac {\sqrt {2}}{2}\right )+2 i \sec \left (f x +e \right ) \sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, \sqrt {i \left (-i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, \sqrt {i \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right )}\, F\left (\sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}, \frac {\sqrt {2}}{2}\right )+\sqrt {2}\, \sin \left (f x +e \right )\right ) \sqrt {2}}{3 f \sqrt {b \tan \left (f x +e \right )}\, \sqrt {d \sec \left (f x +e \right )}\, d}\) | \(239\) |
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.06 \[ \int \frac {1}{(d \sec (e+f x))^{3/2} \sqrt {b \tan (e+f x)}} \, dx=\frac {2 \, {\left (\sqrt {\frac {b \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \sqrt {\frac {d}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )^{2} + \sqrt {-2 i \, b d} {\rm weierstrassPInverse}\left (4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right ) + \sqrt {2 i \, b d} {\rm weierstrassPInverse}\left (4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right )}}{3 \, b d^{2} f} \]
[In]
[Out]
\[ \int \frac {1}{(d \sec (e+f x))^{3/2} \sqrt {b \tan (e+f x)}} \, dx=\int \frac {1}{\sqrt {b \tan {\left (e + f x \right )}} \left (d \sec {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \]
[In]
[Out]
\[ \int \frac {1}{(d \sec (e+f x))^{3/2} \sqrt {b \tan (e+f x)}} \, dx=\int { \frac {1}{\left (d \sec \left (f x + e\right )\right )^{\frac {3}{2}} \sqrt {b \tan \left (f x + e\right )}} \,d x } \]
[In]
[Out]
\[ \int \frac {1}{(d \sec (e+f x))^{3/2} \sqrt {b \tan (e+f x)}} \, dx=\int { \frac {1}{\left (d \sec \left (f x + e\right )\right )^{\frac {3}{2}} \sqrt {b \tan \left (f x + e\right )}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {1}{(d \sec (e+f x))^{3/2} \sqrt {b \tan (e+f x)}} \, dx=\int \frac {1}{\sqrt {b\,\mathrm {tan}\left (e+f\,x\right )}\,{\left (\frac {d}{\cos \left (e+f\,x\right )}\right )}^{3/2}} \,d x \]
[In]
[Out]