Optimal. Leaf size=98 \[ \frac{4 E\left (\left .\frac{1}{2} (e+f x)\right |2\right )}{15 b^2 f \sqrt{\cos (e+f x)} \sqrt{b \sec (e+f x)}}+\frac{4 \sin (e+f x)}{45 b f (b \sec (e+f x))^{3/2}}-\frac{2 b \sin (e+f x)}{9 f (b \sec (e+f x))^{7/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0818197, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {2627, 3769, 3771, 2639} \[ \frac{4 E\left (\left .\frac{1}{2} (e+f x)\right |2\right )}{15 b^2 f \sqrt{\cos (e+f x)} \sqrt{b \sec (e+f x)}}+\frac{4 \sin (e+f x)}{45 b f (b \sec (e+f x))^{3/2}}-\frac{2 b \sin (e+f x)}{9 f (b \sec (e+f x))^{7/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2627
Rule 3769
Rule 3771
Rule 2639
Rubi steps
\begin{align*} \int \frac{\sin ^2(e+f x)}{(b \sec (e+f x))^{5/2}} \, dx &=-\frac{2 b \sin (e+f x)}{9 f (b \sec (e+f x))^{7/2}}+\frac{2}{9} \int \frac{1}{(b \sec (e+f x))^{5/2}} \, dx\\ &=-\frac{2 b \sin (e+f x)}{9 f (b \sec (e+f x))^{7/2}}+\frac{4 \sin (e+f x)}{45 b f (b \sec (e+f x))^{3/2}}+\frac{2 \int \frac{1}{\sqrt{b \sec (e+f x)}} \, dx}{15 b^2}\\ &=-\frac{2 b \sin (e+f x)}{9 f (b \sec (e+f x))^{7/2}}+\frac{4 \sin (e+f x)}{45 b f (b \sec (e+f x))^{3/2}}+\frac{2 \int \sqrt{\cos (e+f x)} \, dx}{15 b^2 \sqrt{\cos (e+f x)} \sqrt{b \sec (e+f x)}}\\ &=\frac{4 E\left (\left .\frac{1}{2} (e+f x)\right |2\right )}{15 b^2 f \sqrt{\cos (e+f x)} \sqrt{b \sec (e+f x)}}-\frac{2 b \sin (e+f x)}{9 f (b \sec (e+f x))^{7/2}}+\frac{4 \sin (e+f x)}{45 b f (b \sec (e+f x))^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.392435, size = 66, normalized size = 0.67 \[ \frac{-4 \sin (2 (e+f x))-10 \sin (4 (e+f x))+\frac{96 E\left (\left .\frac{1}{2} (e+f x)\right |2\right )}{\sqrt{\cos (e+f x)}}}{360 b^2 f \sqrt{b \sec (e+f x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.129, size = 333, normalized size = 3.4 \begin{align*}{\frac{2}{45\,f \left ( \cos \left ( fx+e \right ) \right ) ^{3}\sin \left ( fx+e \right ) } \left ( 6\,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 ) -6\,i{\it EllipticE} \left ({\frac{i \left ( -1+\cos \left ( fx+e \right ) \right ) }{\sin \left ( fx+e \right ) }},i \right ) \cos \left ( fx+e \right ) \sin \left ( fx+e \right ) \sqrt{ \left ( \cos \left ( fx+e \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( fx+e \right ) }{\cos \left ( fx+e \right ) +1}}}+5\, \left ( \cos \left ( fx+e \right ) \right ) ^{6}+6\,i{\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 ) -6\,i{\it EllipticE} \left ({\frac{i \left ( -1+\cos \left ( fx+e \right ) \right ) }{\sin \left ( fx+e \right ) }},i \right ) \sin \left ( fx+e \right ) \sqrt{ \left ( \cos \left ( fx+e \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( fx+e \right ) }{\cos \left ( fx+e \right ) +1}}}-7\, \left ( \cos \left ( fx+e \right ) \right ) ^{4}-4\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}+6\,\cos \left ( fx+e \right ) \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 \frac{\sin \left (f x + e\right )^{2}}{\left (b \sec \left (f x + e\right )\right )^{\frac{5}{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 (\cos \left (f x + e\right )^{2} - 1\right )} \sqrt{b \sec \left (f x + e\right )}}{b^{3} \sec \left (f x + e\right )^{3}}, 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{\sin \left (f x + e\right )^{2}}{\left (b \sec \left (f x + e\right )\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]