Optimal. Leaf size=123 \[ -\frac{5 b^3 \csc (e+f x)}{2 f \sqrt{b \sec (e+f x)}}+\frac{5 b^2 \sqrt{\cos (e+f x)} F\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{b \sec (e+f x)}}{2 f}-\frac{b \csc ^3(e+f x) (b \sec (e+f x))^{3/2}}{3 f}+\frac{b \csc (e+f x) (b \sec (e+f x))^{3/2}}{f} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.150323, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {2625, 2626, 3771, 2641} \[ -\frac{5 b^3 \csc (e+f x)}{2 f \sqrt{b \sec (e+f x)}}+\frac{5 b^2 \sqrt{\cos (e+f x)} F\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{b \sec (e+f x)}}{2 f}-\frac{b \csc ^3(e+f x) (b \sec (e+f x))^{3/2}}{3 f}+\frac{b \csc (e+f x) (b \sec (e+f x))^{3/2}}{f} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2625
Rule 2626
Rule 3771
Rule 2641
Rubi steps
\begin{align*} \int \csc ^4(e+f x) (b \sec (e+f x))^{5/2} \, dx &=-\frac{b \csc ^3(e+f x) (b \sec (e+f x))^{3/2}}{3 f}+\frac{3}{2} \int \csc ^2(e+f x) (b \sec (e+f x))^{5/2} \, dx\\ &=\frac{b \csc (e+f x) (b \sec (e+f x))^{3/2}}{f}-\frac{b \csc ^3(e+f x) (b \sec (e+f x))^{3/2}}{3 f}+\frac{1}{2} \left (5 b^2\right ) \int \csc ^2(e+f x) \sqrt{b \sec (e+f x)} \, dx\\ &=-\frac{5 b^3 \csc (e+f x)}{2 f \sqrt{b \sec (e+f x)}}+\frac{b \csc (e+f x) (b \sec (e+f x))^{3/2}}{f}-\frac{b \csc ^3(e+f x) (b \sec (e+f x))^{3/2}}{3 f}+\frac{1}{4} \left (5 b^2\right ) \int \sqrt{b \sec (e+f x)} \, dx\\ &=-\frac{5 b^3 \csc (e+f x)}{2 f \sqrt{b \sec (e+f x)}}+\frac{b \csc (e+f x) (b \sec (e+f x))^{3/2}}{f}-\frac{b \csc ^3(e+f x) (b \sec (e+f x))^{3/2}}{3 f}+\frac{1}{4} \left (5 b^2 \sqrt{\cos (e+f x)} \sqrt{b \sec (e+f x)}\right ) \int \frac{1}{\sqrt{\cos (e+f x)}} \, dx\\ &=-\frac{5 b^3 \csc (e+f x)}{2 f \sqrt{b \sec (e+f x)}}+\frac{5 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)}}{2 f}+\frac{b \csc (e+f x) (b \sec (e+f x))^{3/2}}{f}-\frac{b \csc ^3(e+f x) (b \sec (e+f x))^{3/2}}{3 f}\\ \end{align*}
Mathematica [A] time = 0.405542, size = 79, normalized size = 0.64 \[ \frac{b \sin (e+f x) (b \sec (e+f x))^{3/2} \left (\cot ^2(e+f x) \left (-\left (2 \csc ^2(e+f x)+11\right )\right )+15 \cos ^{\frac{3}{2}}(e+f x) \csc (e+f x) F\left (\left .\frac{1}{2} (e+f x)\right |2\right )+4\right )}{6 f} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.175, size = 352, normalized size = 2.9 \begin{align*} -{\frac{ \left ( -1+\cos \left ( fx+e \right ) \right ) ^{2}\cos \left ( fx+e \right ) \left ( \cos \left ( fx+e \right ) +1 \right ) ^{2}}{6\,f \left ( \sin \left ( fx+e \right ) \right ) ^{7}} \left ( 15\,i{\it EllipticF} \left ({\frac{i \left ( -1+\cos \left ( fx+e \right ) \right ) }{\sin \left ( fx+e \right ) }},i \right ) \sqrt{{\frac{\cos \left ( fx+e \right ) }{\cos \left ( fx+e \right ) +1}}} \left ( \cos \left ( fx+e \right ) \right ) ^{4}\sin \left ( fx+e \right ) \sqrt{ \left ( \cos \left ( fx+e \right ) +1 \right ) ^{-1}}+15\,i\sqrt{ \left ( \cos \left ( fx+e \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( fx+e \right ) }{\cos \left ( fx+e \right ) +1}}}{\it EllipticF} \left ({\frac{i \left ( -1+\cos \left ( fx+e \right ) \right ) }{\sin \left ( fx+e \right ) }},i \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{3}\sin \left ( fx+e \right ) -15\,i\sqrt{ \left ( \cos \left ( fx+e \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( fx+e \right ) }{\cos \left ( fx+e \right ) +1}}}{\it EllipticF} \left ({\frac{i \left ( -1+\cos \left ( fx+e \right ) \right ) }{\sin \left ( fx+e \right ) }},i \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{2}\sin \left ( fx+e \right ) -15\,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 ) -15\, \left ( \cos \left ( fx+e \right ) \right ) ^{4}+21\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}-4 \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}} \csc \left (f x + e\right )^{4}\,{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 )} b^{2} \csc \left (f x + e\right )^{4} \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}} \csc \left (f x + e\right )^{4}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]