Optimal. Leaf size=172 \[ \frac{a^4 \sqrt{\sin (2 e+2 f x)} F\left (\left .e+f x-\frac{\pi }{4}\right |2\right ) \sqrt{b \sec (e+f x)}}{24 b^2 f \sqrt{a \sin (e+f x)}}-\frac{a^3 \sqrt{a \sin (e+f x)}}{12 b f \sqrt{b \sec (e+f x)}}+\frac{(a \sin (e+f x))^{9/2}}{5 a b f \sqrt{b \sec (e+f x)}}-\frac{a (a \sin (e+f x))^{5/2}}{30 b f \sqrt{b \sec (e+f x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.274728, antiderivative size = 172, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2582, 2583, 2585, 2573, 2641} \[ \frac{a^4 \sqrt{\sin (2 e+2 f x)} F\left (\left .e+f x-\frac{\pi }{4}\right |2\right ) \sqrt{b \sec (e+f x)}}{24 b^2 f \sqrt{a \sin (e+f x)}}-\frac{a^3 \sqrt{a \sin (e+f x)}}{12 b f \sqrt{b \sec (e+f x)}}+\frac{(a \sin (e+f x))^{9/2}}{5 a b f \sqrt{b \sec (e+f x)}}-\frac{a (a \sin (e+f x))^{5/2}}{30 b f \sqrt{b \sec (e+f x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2582
Rule 2583
Rule 2585
Rule 2573
Rule 2641
Rubi steps
\begin{align*} \int \frac{(a \sin (e+f x))^{7/2}}{(b \sec (e+f x))^{3/2}} \, dx &=\frac{(a \sin (e+f x))^{9/2}}{5 a b f \sqrt{b \sec (e+f x)}}+\frac{\int \sqrt{b \sec (e+f x)} (a \sin (e+f x))^{7/2} \, dx}{10 b^2}\\ &=-\frac{a (a \sin (e+f x))^{5/2}}{30 b f \sqrt{b \sec (e+f x)}}+\frac{(a \sin (e+f x))^{9/2}}{5 a b f \sqrt{b \sec (e+f x)}}+\frac{a^2 \int \sqrt{b \sec (e+f x)} (a \sin (e+f x))^{3/2} \, dx}{12 b^2}\\ &=-\frac{a^3 \sqrt{a \sin (e+f x)}}{12 b f \sqrt{b \sec (e+f x)}}-\frac{a (a \sin (e+f x))^{5/2}}{30 b f \sqrt{b \sec (e+f x)}}+\frac{(a \sin (e+f x))^{9/2}}{5 a b f \sqrt{b \sec (e+f x)}}+\frac{a^4 \int \frac{\sqrt{b \sec (e+f x)}}{\sqrt{a \sin (e+f x)}} \, dx}{24 b^2}\\ &=-\frac{a^3 \sqrt{a \sin (e+f x)}}{12 b f \sqrt{b \sec (e+f x)}}-\frac{a (a \sin (e+f x))^{5/2}}{30 b f \sqrt{b \sec (e+f x)}}+\frac{(a \sin (e+f x))^{9/2}}{5 a b f \sqrt{b \sec (e+f x)}}+\frac{\left (a^4 \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}{24 b^2}\\ &=-\frac{a^3 \sqrt{a \sin (e+f x)}}{12 b f \sqrt{b \sec (e+f x)}}-\frac{a (a \sin (e+f x))^{5/2}}{30 b f \sqrt{b \sec (e+f x)}}+\frac{(a \sin (e+f x))^{9/2}}{5 a b f \sqrt{b \sec (e+f x)}}+\frac{\left (a^4 \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}{24 b^2 \sqrt{a \sin (e+f x)}}\\ &=-\frac{a^3 \sqrt{a \sin (e+f x)}}{12 b f \sqrt{b \sec (e+f x)}}-\frac{a (a \sin (e+f x))^{5/2}}{30 b f \sqrt{b \sec (e+f x)}}+\frac{(a \sin (e+f x))^{9/2}}{5 a b f \sqrt{b \sec (e+f x)}}+\frac{a^4 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)}}{24 b^2 f \sqrt{a \sin (e+f x)}}\\ \end{align*}
Mathematica [C] time = 0.861685, size = 103, normalized size = 0.6 \[ -\frac{a^5 \left (-20 \left (-\tan ^2(e+f x)\right )^{3/4} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{3}{2};\sec ^2(e+f x)\right )+17 \cos (2 (e+f x))-16 \cos (4 (e+f x))+3 \cos (6 (e+f x))-4\right )}{480 b f (a \sin (e+f x))^{3/2} \sqrt{b \sec (e+f x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.138, size = 246, normalized size = 1.4 \begin{align*} -{\frac{\sqrt{2}}{120\,f \left ( -1+\cos \left ( fx+e \right ) \right ) \left ( \sin \left ( fx+e \right ) \right ) ^{3} \left ( \cos \left ( fx+e \right ) \right ) ^{2}} \left ( -12\, \left ( \cos \left ( fx+e \right ) \right ) ^{6}\sqrt{2}+12\, \left ( \cos \left ( fx+e \right ) \right ) ^{5}\sqrt{2}+5\,\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 ) }}},1/2\,\sqrt{2} \right ) +22\,\sqrt{2} \left ( \cos \left ( fx+e \right ) \right ) ^{4}-22\,\sqrt{2} \left ( \cos \left ( fx+e \right ) \right ) ^{3}-5\,\sqrt{2} \left ( \cos \left ( fx+e \right ) \right ) ^{2}+5\,\sqrt{2}\cos \left ( fx+e \right ) \right ) \left ( a\sin \left ( fx+e \right ) \right ) ^{{\frac{7}{2}}} \left ({\frac{b}{\cos \left ( fx+e \right ) }} \right ) ^{-{\frac{3}{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 \frac{\left (a \sin \left (f x + e\right )\right )^{\frac{7}{2}}}{\left (b \sec \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 (-\frac{{\left (a^{3} \cos \left (f x + e\right )^{2} - a^{3}\right )} \sqrt{b \sec \left (f x + e\right )} \sqrt{a \sin \left (f x + e\right )} \sin \left (f x + e\right )}{b^{2} \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 \frac{\left (a \sin \left (f x + e\right )\right )^{\frac{7}{2}}}{\left (b \sec \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]