Optimal. Leaf size=95 \[ -\frac{4 \sqrt{\sin (e+f x)} F\left (\left .\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{d \sec (e+f x)}}{3 b^2 f \sqrt{b \tan (e+f x)}}-\frac{2 \sqrt{d \sec (e+f x)}}{3 b f (b \tan (e+f x))^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.114301, antiderivative size = 95, 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 = {2609, 2616, 2642, 2641} \[ -\frac{4 \sqrt{\sin (e+f x)} F\left (\left .\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{d \sec (e+f x)}}{3 b^2 f \sqrt{b \tan (e+f x)}}-\frac{2 \sqrt{d \sec (e+f x)}}{3 b f (b \tan (e+f x))^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2609
Rule 2616
Rule 2642
Rule 2641
Rubi steps
\begin{align*} \int \frac{\sqrt{d \sec (e+f x)}}{(b \tan (e+f x))^{5/2}} \, dx &=-\frac{2 \sqrt{d \sec (e+f x)}}{3 b f (b \tan (e+f x))^{3/2}}-\frac{2 \int \frac{\sqrt{d \sec (e+f x)}}{\sqrt{b \tan (e+f x)}} \, dx}{3 b^2}\\ &=-\frac{2 \sqrt{d \sec (e+f x)}}{3 b f (b \tan (e+f x))^{3/2}}-\frac{\left (2 \sqrt{d \sec (e+f x)} \sqrt{b \sin (e+f x)}\right ) \int \frac{1}{\sqrt{b \sin (e+f x)}} \, dx}{3 b^2 \sqrt{b \tan (e+f x)}}\\ &=-\frac{2 \sqrt{d \sec (e+f x)}}{3 b f (b \tan (e+f x))^{3/2}}-\frac{\left (2 \sqrt{d \sec (e+f x)} \sqrt{\sin (e+f x)}\right ) \int \frac{1}{\sqrt{\sin (e+f x)}} \, dx}{3 b^2 \sqrt{b \tan (e+f x)}}\\ &=-\frac{2 \sqrt{d \sec (e+f x)}}{3 b f (b \tan (e+f x))^{3/2}}-\frac{4 F\left (\left .\frac{1}{2} \left (e-\frac{\pi }{2}+f x\right )\right |2\right ) \sqrt{d \sec (e+f x)} \sqrt{\sin (e+f x)}}{3 b^2 f \sqrt{b \tan (e+f x)}}\\ \end{align*}
Mathematica [C] time = 0.705384, size = 70, normalized size = 0.74 \[ -\frac{2 \sqrt{d \sec (e+f x)} \left (2 \left (-\tan ^2(e+f x)\right )^{3/4} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};\sec ^2(e+f x)\right )+1\right )}{3 b f (b \tan (e+f x))^{3/2}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.214, size = 322, normalized size = 3.4 \begin{align*} -{\frac{\sqrt{2}\sin \left ( fx+e \right ) }{3\,f \left ( \cos \left ( fx+e \right ) \right ) ^{2}} \left ( 2\,i\cos \left ( fx+e \right ) \sin \left ( fx+e \right ) \sqrt{{\frac{i\cos \left ( fx+e \right ) -i+\sin \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }}}\sqrt{-{\frac{i\cos \left ( fx+e \right ) -i-\sin \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }}}{\it EllipticF} \left ( \sqrt{{\frac{i\cos \left ( fx+e \right ) -i+\sin \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }}},{\frac{\sqrt{2}}{2}} \right ) \sqrt{{\frac{-i \left ( \cos \left ( fx+e \right ) -1 \right ) }{\sin \left ( fx+e \right ) }}}+2\,i\sin \left ( fx+e \right ) \sqrt{{\frac{i\cos \left ( fx+e \right ) -i+\sin \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }}}\sqrt{-{\frac{i\cos \left ( fx+e \right ) -i-\sin \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }}}{\it EllipticF} \left ( \sqrt{{\frac{i\cos \left ( fx+e \right ) -i+\sin \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }}},{\frac{\sqrt{2}}{2}} \right ) \sqrt{{\frac{-i \left ( \cos \left ( fx+e \right ) -1 \right ) }{\sin \left ( fx+e \right ) }}}+\cos \left ( fx+e \right ) \sqrt{2} \right ) \sqrt{{\frac{d}{\cos \left ( fx+e \right ) }}} \left ({\frac{b\sin \left ( fx+e \right ) }{\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{\sqrt{d \sec \left (f x + e\right )}}{\left (b \tan \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{\sqrt{d \sec \left (f x + e\right )} \sqrt{b \tan \left (f x + e\right )}}{b^{3} \tan \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{\sqrt{d \sec \left (f x + e\right )}}{\left (b \tan \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]