Optimal. Leaf size=113 \[ \frac{a \tan ^{-1}\left (\frac{\sqrt{c} \tan (e+f x)}{\sqrt{2} \sqrt{c-c \sec (e+f x)}}\right )}{8 \sqrt{2} c^{5/2} f}+\frac{a \tan (e+f x)}{8 c f (c-c \sec (e+f x))^{3/2}}-\frac{a \tan (e+f x)}{2 f (c-c \sec (e+f x))^{5/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.154883, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {3957, 3796, 3795, 203} \[ \frac{a \tan ^{-1}\left (\frac{\sqrt{c} \tan (e+f x)}{\sqrt{2} \sqrt{c-c \sec (e+f x)}}\right )}{8 \sqrt{2} c^{5/2} f}+\frac{a \tan (e+f x)}{8 c f (c-c \sec (e+f x))^{3/2}}-\frac{a \tan (e+f x)}{2 f (c-c \sec (e+f x))^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3957
Rule 3796
Rule 3795
Rule 203
Rubi steps
\begin{align*} \int \frac{\sec (e+f x) (a+a \sec (e+f x))}{(c-c \sec (e+f x))^{5/2}} \, dx &=-\frac{a \tan (e+f x)}{2 f (c-c \sec (e+f x))^{5/2}}-\frac{a \int \frac{\sec (e+f x)}{(c-c \sec (e+f x))^{3/2}} \, dx}{4 c}\\ &=-\frac{a \tan (e+f x)}{2 f (c-c \sec (e+f x))^{5/2}}+\frac{a \tan (e+f x)}{8 c f (c-c \sec (e+f x))^{3/2}}-\frac{a \int \frac{\sec (e+f x)}{\sqrt{c-c \sec (e+f x)}} \, dx}{16 c^2}\\ &=-\frac{a \tan (e+f x)}{2 f (c-c \sec (e+f x))^{5/2}}+\frac{a \tan (e+f x)}{8 c f (c-c \sec (e+f x))^{3/2}}+\frac{a \operatorname{Subst}\left (\int \frac{1}{2 c+x^2} \, dx,x,\frac{c \tan (e+f x)}{\sqrt{c-c \sec (e+f x)}}\right )}{8 c^2 f}\\ &=\frac{a \tan ^{-1}\left (\frac{\sqrt{c} \tan (e+f x)}{\sqrt{2} \sqrt{c-c \sec (e+f x)}}\right )}{8 \sqrt{2} c^{5/2} f}-\frac{a \tan (e+f x)}{2 f (c-c \sec (e+f x))^{5/2}}+\frac{a \tan (e+f x)}{8 c f (c-c \sec (e+f x))^{3/2}}\\ \end{align*}
Mathematica [C] time = 1.20918, size = 309, normalized size = 2.73 \[ \frac{a \left (-\frac{i \sqrt{2} \left (-1+e^{i (e+f x)}\right )^5 \tanh ^{-1}\left (\frac{1+e^{i (e+f x)}}{\sqrt{2} \sqrt{1+e^{2 i (e+f x)}}}\right )}{\left (1+e^{2 i (e+f x)}\right )^{5/2}}+48 \sin \left (\frac{e}{2}\right ) \sin \left (\frac{f x}{2}\right ) \sin ^5\left (\frac{1}{2} (e+f x)\right ) \sec ^3(e+f x)-48 \cos \left (\frac{e}{2}\right ) \cos \left (\frac{f x}{2}\right ) \sin ^5\left (\frac{1}{2} (e+f x)\right ) \sec ^3(e+f x)+56 \cot \left (\frac{e}{2}\right ) \sin ^4\left (\frac{1}{2} (e+f x)\right ) \sec ^3(e+f x)-16 \cot \left (\frac{e}{2}\right ) \sin ^2\left (\frac{1}{2} (e+f x)\right ) \sec ^3(e+f x)-56 \csc \left (\frac{e}{2}\right ) \sin \left (\frac{f x}{2}\right ) \sin ^3\left (\frac{1}{2} (e+f x)\right ) \sec ^3(e+f x)+16 \csc \left (\frac{e}{2}\right ) \sin \left (\frac{f x}{2}\right ) \sin \left (\frac{1}{2} (e+f x)\right ) \sec ^3(e+f x)\right )}{16 c^2 f (\sec (e+f x)-1)^2 \sqrt{c-c \sec (e+f x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.214, size = 308, normalized size = 2.7 \begin{align*} -{\frac{a \left ( -1+\cos \left ( fx+e \right ) \right ) ^{3}}{2\,f \left ( \sin \left ( fx+e \right ) \right ) ^{5}} \left ( \left ( \cos \left ( fx+e \right ) \right ) ^{2} \left ( -2\,{\frac{\cos \left ( fx+e \right ) }{1+\cos \left ( fx+e \right ) }} \right ) ^{{\frac{3}{2}}}+4\,\cos \left ( fx+e \right ) \left ( -2\,{\frac{\cos \left ( fx+e \right ) }{1+\cos \left ( fx+e \right ) }} \right ) ^{3/2}- \left ( \cos \left ( fx+e \right ) \right ) ^{2}\sqrt{-2\,{\frac{\cos \left ( fx+e \right ) }{1+\cos \left ( fx+e \right ) }}}- \left ( \cos \left ( fx+e \right ) \right ) ^{2}\arctan \left ({\frac{1}{\sqrt{-2\,{\frac{\cos \left ( fx+e \right ) }{1+\cos \left ( fx+e \right ) }}}}} \right ) +3\, \left ( -2\,{\frac{\cos \left ( fx+e \right ) }{1+\cos \left ( fx+e \right ) }} \right ) ^{3/2}+2\,\sqrt{-2\,{\frac{\cos \left ( fx+e \right ) }{1+\cos \left ( fx+e \right ) }}}\cos \left ( fx+e \right ) +2\,\cos \left ( fx+e \right ) \arctan \left ({\frac{1}{\sqrt{-2\,{\frac{\cos \left ( fx+e \right ) }{1+\cos \left ( fx+e \right ) }}}}} \right ) -\sqrt{-2\,{\frac{\cos \left ( fx+e \right ) }{1+\cos \left ( fx+e \right ) }}}-\arctan \left ({\frac{1}{\sqrt{-2\,{\frac{\cos \left ( fx+e \right ) }{1+\cos \left ( fx+e \right ) }}}}} \right ) \right ) \left ({\frac{c \left ( -1+\cos \left ( fx+e \right ) \right ) }{\cos \left ( fx+e \right ) }} \right ) ^{-{\frac{5}{2}}} \left ( -2\,{\frac{\cos \left ( fx+e \right ) }{1+\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(-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]
________________________________________________________________________________________
Fricas [A] time = 0.619489, size = 1042, normalized size = 9.22 \begin{align*} \left [-\frac{\sqrt{2}{\left (a \cos \left (f x + e\right )^{2} - 2 \, a \cos \left (f x + e\right ) + a\right )} \sqrt{-c} \log \left (-\frac{2 \, \sqrt{2}{\left (\cos \left (f x + e\right )^{2} + \cos \left (f x + e\right )\right )} \sqrt{-c} \sqrt{\frac{c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}} -{\left (3 \, c \cos \left (f x + e\right ) + c\right )} \sin \left (f x + e\right )}{{\left (\cos \left (f x + e\right ) - 1\right )} \sin \left (f x + e\right )}\right ) \sin \left (f x + e\right ) - 4 \,{\left (3 \, a \cos \left (f x + e\right )^{3} + 4 \, a \cos \left (f x + e\right )^{2} + a \cos \left (f x + e\right )\right )} \sqrt{\frac{c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}}}{32 \,{\left (c^{3} f \cos \left (f x + e\right )^{2} - 2 \, c^{3} f \cos \left (f x + e\right ) + c^{3} f\right )} \sin \left (f x + e\right )}, -\frac{\sqrt{2}{\left (a \cos \left (f x + e\right )^{2} - 2 \, a \cos \left (f x + e\right ) + a\right )} \sqrt{c} \arctan \left (\frac{\sqrt{2} \sqrt{\frac{c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{\sqrt{c} \sin \left (f x + e\right )}\right ) \sin \left (f x + e\right ) - 2 \,{\left (3 \, a \cos \left (f x + e\right )^{3} + 4 \, a \cos \left (f x + e\right )^{2} + a \cos \left (f x + e\right )\right )} \sqrt{\frac{c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}}}{16 \,{\left (c^{3} f \cos \left (f x + e\right )^{2} - 2 \, c^{3} f \cos \left (f x + e\right ) + c^{3} f\right )} \sin \left (f x + e\right )}\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 [A] time = 1.88553, size = 178, normalized size = 1.58 \begin{align*} \frac{\sqrt{2} a{\left (\frac{\arctan \left (\frac{\sqrt{c \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - c}}{\sqrt{c}}\right )}{c^{\frac{3}{2}}} + \frac{{\left (c \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - c\right )}^{\frac{3}{2}} - \sqrt{c \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - c} c}{c^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4}}\right )}}{16 \, c f \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - 1\right ) \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]