Integrand size = 21, antiderivative size = 98 \[ \int \frac {\sin ^2(e+f x)}{(b \sec (e+f x))^{3/2}} \, dx=\frac {4 \sqrt {\cos (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),2\right ) \sqrt {b \sec (e+f x)}}{21 b^2 f}-\frac {2 b \sin (e+f x)}{7 f (b \sec (e+f x))^{5/2}}+\frac {4 \sin (e+f x)}{21 b f \sqrt {b \sec (e+f x)}} \]
[Out]
Time = 0.06 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2707, 3854, 3856, 2720} \[ \int \frac {\sin ^2(e+f x)}{(b \sec (e+f x))^{3/2}} \, dx=\frac {4 \sqrt {\cos (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),2\right ) \sqrt {b \sec (e+f x)}}{21 b^2 f}+\frac {4 \sin (e+f x)}{21 b f \sqrt {b \sec (e+f x)}}-\frac {2 b \sin (e+f x)}{7 f (b \sec (e+f x))^{5/2}} \]
[In]
[Out]
Rule 2707
Rule 2720
Rule 3854
Rule 3856
Rubi steps \begin{align*} \text {integral}& = -\frac {2 b \sin (e+f x)}{7 f (b \sec (e+f x))^{5/2}}+\frac {2}{7} \int \frac {1}{(b \sec (e+f x))^{3/2}} \, dx \\ & = -\frac {2 b \sin (e+f x)}{7 f (b \sec (e+f x))^{5/2}}+\frac {4 \sin (e+f x)}{21 b f \sqrt {b \sec (e+f x)}}+\frac {2 \int \sqrt {b \sec (e+f x)} \, dx}{21 b^2} \\ & = -\frac {2 b \sin (e+f x)}{7 f (b \sec (e+f x))^{5/2}}+\frac {4 \sin (e+f x)}{21 b f \sqrt {b \sec (e+f x)}}+\frac {\left (2 \sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)}\right ) \int \frac {1}{\sqrt {\cos (e+f x)}} \, dx}{21 b^2} \\ & = \frac {4 \sqrt {\cos (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),2\right ) \sqrt {b \sec (e+f x)}}{21 b^2 f}-\frac {2 b \sin (e+f x)}{7 f (b \sec (e+f x))^{5/2}}+\frac {4 \sin (e+f x)}{21 b f \sqrt {b \sec (e+f x)}} \\ \end{align*}
Time = 0.22 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.72 \[ \int \frac {\sin ^2(e+f x)}{(b \sec (e+f x))^{3/2}} \, dx=\frac {\sec ^2(e+f x) \left (16 \sqrt {\cos (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),2\right )+2 \sin (2 (e+f x))-3 \sin (4 (e+f x))\right )}{84 f (b \sec (e+f x))^{3/2}} \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 0.65 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.63
method | result | size |
default | \(-\frac {2 \left (2 i \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, F\left (i \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right ), i\right )+2 i \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, F\left (i \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right ), i\right ) \sec \left (f x +e \right )+3 \sin \left (f x +e \right ) \left (\cos ^{2}\left (f x +e \right )\right )-2 \sin \left (f x +e \right )\right )}{21 f \sqrt {b \sec \left (f x +e \right )}\, b}\) | \(160\) |
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.01 \[ \int \frac {\sin ^2(e+f x)}{(b \sec (e+f x))^{3/2}} \, dx=-\frac {2 \, {\left ({\left (3 \, \cos \left (f x + e\right )^{3} - 2 \, \cos \left (f x + e\right )\right )} \sqrt {\frac {b}{\cos \left (f x + e\right )}} \sin \left (f x + e\right ) + i \, \sqrt {2} \sqrt {b} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right ) - i \, \sqrt {2} \sqrt {b} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right )}}{21 \, b^{2} f} \]
[In]
[Out]
\[ \int \frac {\sin ^2(e+f x)}{(b \sec (e+f x))^{3/2}} \, dx=\int \frac {\sin ^{2}{\left (e + f x \right )}}{\left (b \sec {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \]
[In]
[Out]
\[ \int \frac {\sin ^2(e+f x)}{(b \sec (e+f x))^{3/2}} \, dx=\int { \frac {\sin \left (f x + e\right )^{2}}{\left (b \sec \left (f x + e\right )\right )^{\frac {3}{2}}} \,d x } \]
[In]
[Out]
\[ \int \frac {\sin ^2(e+f x)}{(b \sec (e+f x))^{3/2}} \, dx=\int { \frac {\sin \left (f x + e\right )^{2}}{\left (b \sec \left (f x + e\right )\right )^{\frac {3}{2}}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {\sin ^2(e+f x)}{(b \sec (e+f x))^{3/2}} \, dx=\int \frac {{\sin \left (e+f\,x\right )}^2}{{\left (\frac {b}{\cos \left (e+f\,x\right )}\right )}^{3/2}} \,d x \]
[In]
[Out]