Optimal. Leaf size=76 \[ \frac{\cos (e+f x) \sqrt{d \tan (e+f x)}}{d f}+\frac{\sqrt{\sin (2 e+2 f x)} \sec (e+f x) F\left (\left .e+f x-\frac{\pi }{4}\right |2\right )}{2 f \sqrt{d \tan (e+f x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0932574, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {2612, 2614, 2573, 2641} \[ \frac{\cos (e+f x) \sqrt{d \tan (e+f x)}}{d f}+\frac{\sqrt{\sin (2 e+2 f x)} \sec (e+f x) F\left (\left .e+f x-\frac{\pi }{4}\right |2\right )}{2 f \sqrt{d \tan (e+f x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2612
Rule 2614
Rule 2573
Rule 2641
Rubi steps
\begin{align*} \int \frac{\cos (e+f x)}{\sqrt{d \tan (e+f x)}} \, dx &=\frac{\cos (e+f x) \sqrt{d \tan (e+f x)}}{d f}+\frac{1}{2} \int \frac{\sec (e+f x)}{\sqrt{d \tan (e+f x)}} \, dx\\ &=\frac{\cos (e+f x) \sqrt{d \tan (e+f x)}}{d f}+\frac{\sqrt{\sin (e+f x)} \int \frac{1}{\sqrt{\cos (e+f x)} \sqrt{\sin (e+f x)}} \, dx}{2 \sqrt{\cos (e+f x)} \sqrt{d \tan (e+f x)}}\\ &=\frac{\cos (e+f x) \sqrt{d \tan (e+f x)}}{d f}+\frac{\left (\sec (e+f x) \sqrt{\sin (2 e+2 f x)}\right ) \int \frac{1}{\sqrt{\sin (2 e+2 f x)}} \, dx}{2 \sqrt{d \tan (e+f x)}}\\ &=\frac{F\left (\left .e-\frac{\pi }{4}+f x\right |2\right ) \sec (e+f x) \sqrt{\sin (2 e+2 f x)}}{2 f \sqrt{d \tan (e+f x)}}+\frac{\cos (e+f x) \sqrt{d \tan (e+f x)}}{d f}\\ \end{align*}
Mathematica [C] time = 0.537385, size = 126, normalized size = 1.66 \[ \frac{\cos (2 (e+f x)) \sqrt{\tan (e+f x)} \sec (e+f x) \left (-\sqrt{\tan (e+f x)} \sqrt{\sec ^2(e+f x)}+\sqrt [4]{-1} \sec ^2(e+f x) F\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-1} \sqrt{\tan (e+f x)}\right )\right |-1\right )\right )}{f \left (\tan ^2(e+f x)-1\right ) \sqrt{\sec ^2(e+f x)} \sqrt{d \tan (e+f x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.145, size = 196, normalized size = 2.6 \begin{align*} -{\frac{\sqrt{2} \left ( \cos \left ( fx+e \right ) -1 \right ) \left ( \cos \left ( fx+e \right ) +1 \right ) ^{2}}{2\,f \left ( \sin \left ( fx+e \right ) \right ) ^{3}\cos \left ( fx+e \right ) } \left ( \sin \left ( fx+e \right ) \sqrt{{\frac{\cos \left ( fx+e \right ) -1}{\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{\cos \left ( fx+e \right ) -1+\sin \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 ) }}},{\frac{\sqrt{2}}{2}} \right ) - \left ( \cos \left ( fx+e \right ) \right ) ^{2}\sqrt{2}+\cos \left ( fx+e \right ) \sqrt{2} \right ){\frac{1}{\sqrt{{\frac{d\sin \left ( fx+e \right ) }{\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 \frac{\cos \left (f x + e\right )}{\sqrt{d \tan \left (f x + e\right )}}\,{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 \tan \left (f x + e\right )} \cos \left (f x + e\right )}{d \tan \left (f x + e\right )}, 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 \frac{\cos{\left (e + f x \right )}}{\sqrt{d \tan{\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 \frac{\cos \left (f x + e\right )}{\sqrt{d \tan \left (f x + e\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]