Optimal. Leaf size=91 \[ \frac{a^2 \sqrt{\sin (2 e+2 f x)} F\left (\left .e+f x-\frac{\pi }{4}\right |2\right ) \sqrt{b \sec (e+f x)}}{2 f \sqrt{a \sin (e+f x)}}-\frac{a b \sqrt{a \sin (e+f x)}}{f \sqrt{b \sec (e+f x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.150605, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {2583, 2585, 2573, 2641} \[ \frac{a^2 \sqrt{\sin (2 e+2 f x)} F\left (\left .e+f x-\frac{\pi }{4}\right |2\right ) \sqrt{b \sec (e+f x)}}{2 f \sqrt{a \sin (e+f x)}}-\frac{a b \sqrt{a \sin (e+f x)}}{f \sqrt{b \sec (e+f x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2583
Rule 2585
Rule 2573
Rule 2641
Rubi steps
\begin{align*} \int \sqrt{b \sec (e+f x)} (a \sin (e+f x))^{3/2} \, dx &=-\frac{a b \sqrt{a \sin (e+f x)}}{f \sqrt{b \sec (e+f x)}}+\frac{1}{2} a^2 \int \frac{\sqrt{b \sec (e+f x)}}{\sqrt{a \sin (e+f x)}} \, dx\\ &=-\frac{a b \sqrt{a \sin (e+f x)}}{f \sqrt{b \sec (e+f x)}}+\frac{1}{2} \left (a^2 \sqrt{b \cos (e+f x)} \sqrt{b \sec (e+f x)}\right ) \int \frac{1}{\sqrt{b \cos (e+f x)} \sqrt{a \sin (e+f x)}} \, dx\\ &=-\frac{a b \sqrt{a \sin (e+f x)}}{f \sqrt{b \sec (e+f x)}}+\frac{\left (a^2 \sqrt{b \sec (e+f x)} \sqrt{\sin (2 e+2 f x)}\right ) \int \frac{1}{\sqrt{\sin (2 e+2 f x)}} \, dx}{2 \sqrt{a \sin (e+f x)}}\\ &=-\frac{a b \sqrt{a \sin (e+f x)}}{f \sqrt{b \sec (e+f x)}}+\frac{a^2 F\left (\left .e-\frac{\pi }{4}+f x\right |2\right ) \sqrt{b \sec (e+f x)} \sqrt{\sin (2 e+2 f x)}}{2 f \sqrt{a \sin (e+f x)}}\\ \end{align*}
Mathematica [C] time = 1.2486, size = 66, normalized size = 0.73 \[ \frac{(a \sin (e+f x))^{5/2} (b \sec (e+f x))^{3/2} \, _2F_1\left (-\frac{1}{2},-\frac{1}{4};\frac{1}{2};\sec ^2(e+f x)\right )}{a b f \left (-\tan ^2(e+f x)\right )^{5/4}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.125, size = 184, normalized size = 2. \begin{align*} -{\frac{\sqrt{2}}{2\,f \left ( -1+\cos \left ( fx+e \right ) \right ) \sin \left ( fx+e \right ) } \left ( \sin \left ( fx+e \right ) \sqrt{{\frac{1-\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }}}\sqrt{{\frac{-1+\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }}}\sqrt{{\frac{-1+\cos \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }}}{\it EllipticF} \left ( \sqrt{{\frac{1-\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }}},{\frac{\sqrt{2}}{2}} \right ) +\sqrt{2} \left ( \cos \left ( fx+e \right ) \right ) ^{2}-\sqrt{2}\cos \left ( fx+e \right ) \right ) \left ( a\sin \left ( fx+e \right ) \right ) ^{{\frac{3}{2}}}\sqrt{{\frac{b}{\cos \left ( fx+e \right ) }}}} \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 \sqrt{b \sec \left (f x + e\right )} \left (a \sin \left (f x + e\right )\right )^{\frac{3}{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 (\sqrt{b \sec \left (f x + e\right )} \sqrt{a \sin \left (f x + e\right )} a \sin \left (f x + e\right ), 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 \sqrt{b \sec \left (f x + e\right )} \left (a \sin \left (f x + e\right )\right )^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]