Optimal. Leaf size=70 \[ \frac{2 b \sin (e+f x) (b \sec (e+f x))^{3/2}}{3 f}-\frac{4 b^2 \sqrt{\cos (e+f x)} F\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{b \sec (e+f x)}}{3 f} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0646299, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2624, 3771, 2641} \[ \frac{2 b \sin (e+f x) (b \sec (e+f x))^{3/2}}{3 f}-\frac{4 b^2 \sqrt{\cos (e+f x)} F\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{b \sec (e+f x)}}{3 f} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2624
Rule 3771
Rule 2641
Rubi steps
\begin{align*} \int (b \sec (e+f x))^{5/2} \sin ^2(e+f x) \, dx &=\frac{2 b (b \sec (e+f x))^{3/2} \sin (e+f x)}{3 f}-\frac{1}{3} \left (2 b^2\right ) \int \sqrt{b \sec (e+f x)} \, dx\\ &=\frac{2 b (b \sec (e+f x))^{3/2} \sin (e+f x)}{3 f}-\frac{1}{3} \left (2 b^2 \sqrt{\cos (e+f x)} \sqrt{b \sec (e+f x)}\right ) \int \frac{1}{\sqrt{\cos (e+f x)}} \, dx\\ &=-\frac{4 b^2 \sqrt{\cos (e+f x)} F\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{b \sec (e+f x)}}{3 f}+\frac{2 b (b \sec (e+f x))^{3/2} \sin (e+f x)}{3 f}\\ \end{align*}
Mathematica [A] time = 0.124455, size = 52, normalized size = 0.74 \[ \frac{2 b^2 \sqrt{b \sec (e+f x)} \left (\tan (e+f x)-2 \sqrt{\cos (e+f x)} F\left (\left .\frac{1}{2} (e+f x)\right |2\right )\right )}{3 f} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.129, size = 126, normalized size = 1.8 \begin{align*}{\frac{ \left ( -2+2\,\cos \left ( fx+e \right ) \right ) \cos \left ( fx+e \right ) \left ( \cos \left ( fx+e \right ) +1 \right ) ^{2}}{3\,f \left ( \sin \left ( fx+e \right ) \right ) ^{3}} \left ( 2\,i\cos \left ( fx+e \right ){\it EllipticF} \left ({\frac{i \left ( -1+\cos \left ( fx+e \right ) \right ) }{\sin \left ( fx+e \right ) }},i \right ) \sqrt{ \left ( \cos \left ( fx+e \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( fx+e \right ) }{\cos \left ( fx+e \right ) +1}}}\sin \left ( fx+e \right ) +\cos \left ( fx+e \right ) -1 \right ) \left ({\frac{b}{\cos \left ( fx+e \right ) }} \right ) ^{{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \sec \left (f x + e\right )\right )^{\frac{5}{2}} \sin \left (f x + e\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-{\left (b^{2} \cos \left (f x + e\right )^{2} - b^{2}\right )} \sqrt{b \sec \left (f x + e\right )} \sec \left (f x + e\right )^{2}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \sec \left (f x + e\right )\right )^{\frac{5}{2}} \sin \left (f x + e\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]