Optimal. Leaf size=171 \[ -\frac{a^2 c^5 \tan ^7(e+f x)}{7 f}-\frac{4 a^2 c^5 \tan ^5(e+f x)}{5 f}+\frac{9 a^2 c^5 \tanh ^{-1}(\sin (e+f x))}{16 f}+\frac{a^2 c^5 \tan ^3(e+f x) \sec ^3(e+f x)}{2 f}-\frac{3 a^2 c^5 \tan (e+f x) \sec ^3(e+f x)}{8 f}+\frac{a^2 c^5 \tan ^3(e+f x) \sec (e+f x)}{4 f}-\frac{3 a^2 c^5 \tan (e+f x) \sec (e+f x)}{16 f} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.273989, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 7, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.219, Rules used = {3958, 2611, 3770, 2607, 30, 3768, 14} \[ -\frac{a^2 c^5 \tan ^7(e+f x)}{7 f}-\frac{4 a^2 c^5 \tan ^5(e+f x)}{5 f}+\frac{9 a^2 c^5 \tanh ^{-1}(\sin (e+f x))}{16 f}+\frac{a^2 c^5 \tan ^3(e+f x) \sec ^3(e+f x)}{2 f}-\frac{3 a^2 c^5 \tan (e+f x) \sec ^3(e+f x)}{8 f}+\frac{a^2 c^5 \tan ^3(e+f x) \sec (e+f x)}{4 f}-\frac{3 a^2 c^5 \tan (e+f x) \sec (e+f x)}{16 f} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3958
Rule 2611
Rule 3770
Rule 2607
Rule 30
Rule 3768
Rule 14
Rubi steps
\begin{align*} \int \sec (e+f x) (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^5 \, dx &=\left (a^2 c^2\right ) \int \left (c^3 \sec (e+f x) \tan ^4(e+f x)-3 c^3 \sec ^2(e+f x) \tan ^4(e+f x)+3 c^3 \sec ^3(e+f x) \tan ^4(e+f x)-c^3 \sec ^4(e+f x) \tan ^4(e+f x)\right ) \, dx\\ &=\left (a^2 c^5\right ) \int \sec (e+f x) \tan ^4(e+f x) \, dx-\left (a^2 c^5\right ) \int \sec ^4(e+f x) \tan ^4(e+f x) \, dx-\left (3 a^2 c^5\right ) \int \sec ^2(e+f x) \tan ^4(e+f x) \, dx+\left (3 a^2 c^5\right ) \int \sec ^3(e+f x) \tan ^4(e+f x) \, dx\\ &=\frac{a^2 c^5 \sec (e+f x) \tan ^3(e+f x)}{4 f}+\frac{a^2 c^5 \sec ^3(e+f x) \tan ^3(e+f x)}{2 f}-\frac{1}{4} \left (3 a^2 c^5\right ) \int \sec (e+f x) \tan ^2(e+f x) \, dx-\frac{1}{2} \left (3 a^2 c^5\right ) \int \sec ^3(e+f x) \tan ^2(e+f x) \, dx-\frac{\left (a^2 c^5\right ) \operatorname{Subst}\left (\int x^4 \left (1+x^2\right ) \, dx,x,\tan (e+f x)\right )}{f}-\frac{\left (3 a^2 c^5\right ) \operatorname{Subst}\left (\int x^4 \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac{3 a^2 c^5 \sec (e+f x) \tan (e+f x)}{8 f}-\frac{3 a^2 c^5 \sec ^3(e+f x) \tan (e+f x)}{8 f}+\frac{a^2 c^5 \sec (e+f x) \tan ^3(e+f x)}{4 f}+\frac{a^2 c^5 \sec ^3(e+f x) \tan ^3(e+f x)}{2 f}-\frac{3 a^2 c^5 \tan ^5(e+f x)}{5 f}+\frac{1}{8} \left (3 a^2 c^5\right ) \int \sec (e+f x) \, dx+\frac{1}{8} \left (3 a^2 c^5\right ) \int \sec ^3(e+f x) \, dx-\frac{\left (a^2 c^5\right ) \operatorname{Subst}\left (\int \left (x^4+x^6\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{3 a^2 c^5 \tanh ^{-1}(\sin (e+f x))}{8 f}-\frac{3 a^2 c^5 \sec (e+f x) \tan (e+f x)}{16 f}-\frac{3 a^2 c^5 \sec ^3(e+f x) \tan (e+f x)}{8 f}+\frac{a^2 c^5 \sec (e+f x) \tan ^3(e+f x)}{4 f}+\frac{a^2 c^5 \sec ^3(e+f x) \tan ^3(e+f x)}{2 f}-\frac{4 a^2 c^5 \tan ^5(e+f x)}{5 f}-\frac{a^2 c^5 \tan ^7(e+f x)}{7 f}+\frac{1}{16} \left (3 a^2 c^5\right ) \int \sec (e+f x) \, dx\\ &=\frac{9 a^2 c^5 \tanh ^{-1}(\sin (e+f x))}{16 f}-\frac{3 a^2 c^5 \sec (e+f x) \tan (e+f x)}{16 f}-\frac{3 a^2 c^5 \sec ^3(e+f x) \tan (e+f x)}{8 f}+\frac{a^2 c^5 \sec (e+f x) \tan ^3(e+f x)}{4 f}+\frac{a^2 c^5 \sec ^3(e+f x) \tan ^3(e+f x)}{2 f}-\frac{4 a^2 c^5 \tan ^5(e+f x)}{5 f}-\frac{a^2 c^5 \tan ^7(e+f x)}{7 f}\\ \end{align*}
Mathematica [A] time = 1.85716, size = 102, normalized size = 0.6 \[ \frac{a^2 c^5 \left (10080 \tanh ^{-1}(\sin (e+f x))-(2520 \sin (e+f x)-455 \sin (2 (e+f x))-616 \sin (3 (e+f x))+2380 \sin (4 (e+f x))-392 \sin (5 (e+f x))+245 \sin (6 (e+f x))+184 \sin (7 (e+f x))) \sec ^7(e+f x)\right )}{17920 f} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.033, size = 192, normalized size = 1.1 \begin{align*} -{\frac{23\,{a}^{2}{c}^{5}\tan \left ( fx+e \right ) }{35\,f}}-{\frac{13\,{a}^{2}{c}^{5}\tan \left ( fx+e \right ) \left ( \sec \left ( fx+e \right ) \right ) ^{4}}{35\,f}}+{\frac{41\,{a}^{2}{c}^{5}\tan \left ( fx+e \right ) \left ( \sec \left ( fx+e \right ) \right ) ^{2}}{35\,f}}-{\frac{5\,{a}^{2}{c}^{5} \left ( \sec \left ( fx+e \right ) \right ) ^{3}\tan \left ( fx+e \right ) }{8\,f}}-{\frac{7\,{a}^{2}{c}^{5}\sec \left ( fx+e \right ) \tan \left ( fx+e \right ) }{16\,f}}+{\frac{9\,{a}^{2}{c}^{5}\ln \left ( \sec \left ( fx+e \right ) +\tan \left ( fx+e \right ) \right ) }{16\,f}}+{\frac{{a}^{2}{c}^{5}\tan \left ( fx+e \right ) \left ( \sec \left ( fx+e \right ) \right ) ^{5}}{2\,f}}-{\frac{{a}^{2}{c}^{5}\tan \left ( fx+e \right ) \left ( \sec \left ( fx+e \right ) \right ) ^{6}}{7\,f}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.02112, size = 497, normalized size = 2.91 \begin{align*} -\frac{96 \,{\left (5 \, \tan \left (f x + e\right )^{7} + 21 \, \tan \left (f x + e\right )^{5} + 35 \, \tan \left (f x + e\right )^{3} + 35 \, \tan \left (f x + e\right )\right )} a^{2} c^{5} + 224 \,{\left (3 \, \tan \left (f x + e\right )^{5} + 10 \, \tan \left (f x + e\right )^{3} + 15 \, \tan \left (f x + e\right )\right )} a^{2} c^{5} - 5600 \,{\left (\tan \left (f x + e\right )^{3} + 3 \, \tan \left (f x + e\right )\right )} a^{2} c^{5} + 105 \, a^{2} c^{5}{\left (\frac{2 \,{\left (15 \, \sin \left (f x + e\right )^{5} - 40 \, \sin \left (f x + e\right )^{3} + 33 \, \sin \left (f x + e\right )\right )}}{\sin \left (f x + e\right )^{6} - 3 \, \sin \left (f x + e\right )^{4} + 3 \, \sin \left (f x + e\right )^{2} - 1} - 15 \, \log \left (\sin \left (f x + e\right ) + 1\right ) + 15 \, \log \left (\sin \left (f x + e\right ) - 1\right )\right )} - 1050 \, a^{2} c^{5}{\left (\frac{2 \,{\left (3 \, \sin \left (f x + e\right )^{3} - 5 \, \sin \left (f x + e\right )\right )}}{\sin \left (f x + e\right )^{4} - 2 \, \sin \left (f x + e\right )^{2} + 1} - 3 \, \log \left (\sin \left (f x + e\right ) + 1\right ) + 3 \, \log \left (\sin \left (f x + e\right ) - 1\right )\right )} + 840 \, a^{2} c^{5}{\left (\frac{2 \, \sin \left (f x + e\right )}{\sin \left (f x + e\right )^{2} - 1} - \log \left (\sin \left (f x + e\right ) + 1\right ) + \log \left (\sin \left (f x + e\right ) - 1\right )\right )} - 3360 \, a^{2} c^{5} \log \left (\sec \left (f x + e\right ) + \tan \left (f x + e\right )\right ) + 10080 \, a^{2} c^{5} \tan \left (f x + e\right )}{3360 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 0.512487, size = 448, normalized size = 2.62 \begin{align*} \frac{315 \, a^{2} c^{5} \cos \left (f x + e\right )^{7} \log \left (\sin \left (f x + e\right ) + 1\right ) - 315 \, a^{2} c^{5} \cos \left (f x + e\right )^{7} \log \left (-\sin \left (f x + e\right ) + 1\right ) - 2 \,{\left (368 \, a^{2} c^{5} \cos \left (f x + e\right )^{6} + 245 \, a^{2} c^{5} \cos \left (f x + e\right )^{5} - 656 \, a^{2} c^{5} \cos \left (f x + e\right )^{4} + 350 \, a^{2} c^{5} \cos \left (f x + e\right )^{3} + 208 \, a^{2} c^{5} \cos \left (f x + e\right )^{2} - 280 \, a^{2} c^{5} \cos \left (f x + e\right ) + 80 \, a^{2} c^{5}\right )} \sin \left (f x + e\right )}{1120 \, f \cos \left (f x + e\right )^{7}} \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*} - a^{2} c^{5} \left (\int - \sec{\left (e + f x \right )}\, dx + \int 3 \sec ^{2}{\left (e + f x \right )}\, dx + \int - \sec ^{3}{\left (e + f x \right )}\, dx + \int - 5 \sec ^{4}{\left (e + f x \right )}\, dx + \int 5 \sec ^{5}{\left (e + f x \right )}\, dx + \int \sec ^{6}{\left (e + f x \right )}\, dx + \int - 3 \sec ^{7}{\left (e + f x \right )}\, dx + \int \sec ^{8}{\left (e + f x \right )}\, dx\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.3115, size = 279, normalized size = 1.63 \begin{align*} \frac{315 \, a^{2} c^{5} \log \left ({\left | \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 1 \right |}\right ) - 315 \, a^{2} c^{5} \log \left ({\left | \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) - 1 \right |}\right ) - \frac{2 \,{\left (315 \, a^{2} c^{5} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{13} - 2100 \, a^{2} c^{5} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{11} - 8393 \, a^{2} c^{5} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{9} + 9216 \, a^{2} c^{5} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{7} - 5943 \, a^{2} c^{5} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{5} + 2100 \, a^{2} c^{5} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} - 315 \, a^{2} c^{5} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )\right )}}{{\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - 1\right )}^{7}}}{560 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]