Optimal. Leaf size=62 \[ \frac{\sqrt{\cos (e+f x)} F\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{b \sec (e+f x)}}{f}-\frac{b \csc (e+f x)}{f \sqrt{b \sec (e+f x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0572937, antiderivative size = 62, 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 = {2625, 3771, 2641} \[ \frac{\sqrt{\cos (e+f x)} F\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{b \sec (e+f x)}}{f}-\frac{b \csc (e+f x)}{f \sqrt{b \sec (e+f x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2625
Rule 3771
Rule 2641
Rubi steps
\begin{align*} \int \csc ^2(e+f x) \sqrt{b \sec (e+f x)} \, dx &=-\frac{b \csc (e+f x)}{f \sqrt{b \sec (e+f x)}}+\frac{1}{2} \int \sqrt{b \sec (e+f x)} \, dx\\ &=-\frac{b \csc (e+f x)}{f \sqrt{b \sec (e+f x)}}+\frac{1}{2} \left (\sqrt{\cos (e+f x)} \sqrt{b \sec (e+f x)}\right ) \int \frac{1}{\sqrt{\cos (e+f x)}} \, dx\\ &=-\frac{b \csc (e+f x)}{f \sqrt{b \sec (e+f x)}}+\frac{\sqrt{\cos (e+f x)} F\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{b \sec (e+f x)}}{f}\\ \end{align*}
Mathematica [A] time = 0.106407, size = 47, normalized size = 0.76 \[ \frac{\sqrt{b \sec (e+f x)} \left (\sqrt{\cos (e+f x)} F\left (\left .\frac{1}{2} (e+f x)\right |2\right )-\cot (e+f x)\right )}{f} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.139, size = 184, normalized size = 3. \begin{align*}{\frac{ \left ( -1+\cos \left ( fx+e \right ) \right ) ^{2} \left ( \cos \left ( fx+e \right ) +1 \right ) ^{2}}{f \left ( \sin \left ( fx+e \right ) \right ) ^{5}} \left ( 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 ) +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 ) -\cos \left ( fx+e \right ) \right ) \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 )} \csc \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 (\sqrt{b \sec \left (f x + e\right )} \csc \left (f x + e\right )^{2}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{b \sec{\left (e + f x \right )}} \csc ^{2}{\left (e + f x \right )}\, dx \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 )} \csc \left (f x + e\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]