Integrand size = 32, antiderivative size = 171 \[ \int \sec (e+f x) (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^5 \, dx=\frac {9 a^2 c^5 \text {arctanh}(\sin (e+f x))}{16 f}-\frac {3 a^2 c^5 \sec (e+f x) \tan (e+f x)}{16 f}-\frac {3 a^2 c^5 \sec ^3(e+f x) \tan (e+f x)}{8 f}+\frac {a^2 c^5 \sec (e+f x) \tan ^3(e+f x)}{4 f}+\frac {a^2 c^5 \sec ^3(e+f x) \tan ^3(e+f x)}{2 f}-\frac {4 a^2 c^5 \tan ^5(e+f x)}{5 f}-\frac {a^2 c^5 \tan ^7(e+f x)}{7 f} \] Output:
9/16*a^2*c^5*arctanh(sin(f*x+e))/f-3/16*a^2*c^5*sec(f*x+e)*tan(f*x+e)/f-3/ 8*a^2*c^5*sec(f*x+e)^3*tan(f*x+e)/f+1/4*a^2*c^5*sec(f*x+e)*tan(f*x+e)^3/f+ 1/2*a^2*c^5*sec(f*x+e)^3*tan(f*x+e)^3/f-4/5*a^2*c^5*tan(f*x+e)^5/f-1/7*a^2 *c^5*tan(f*x+e)^7/f
Time = 1.09 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.60 \[ \int \sec (e+f x) (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^5 \, dx=\frac {a^2 c^5 \left (10080 \text {arctanh}(\sin (e+f x))-\sec ^7(e+f x) (2520 \sin (e+f x)-455 \sin (2 (e+f x))-616 \sin (3 (e+f x))+2380 \sin (4 (e+f x))-392 \sin (5 (e+f x))+245 \sin (6 (e+f x))+184 \sin (7 (e+f x)))\right )}{17920 f} \] Input:
Integrate[Sec[e + f*x]*(a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x])^5,x]
Output:
(a^2*c^5*(10080*ArcTanh[Sin[e + f*x]] - Sec[e + f*x]^7*(2520*Sin[e + f*x] - 455*Sin[2*(e + f*x)] - 616*Sin[3*(e + f*x)] + 2380*Sin[4*(e + f*x)] - 39 2*Sin[5*(e + f*x)] + 245*Sin[6*(e + f*x)] + 184*Sin[7*(e + f*x)])))/(17920 *f)
Time = 0.48 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.92, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used = {3042, 4446, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \sec (e+f x) (a \sec (e+f x)+a)^2 (c-c \sec (e+f x))^5 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \csc \left (e+f x+\frac {\pi }{2}\right ) \left (a \csc \left (e+f x+\frac {\pi }{2}\right )+a\right )^2 \left (c-c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^5dx\) |
| \(\Big \downarrow \) 4446 |
| \(\displaystyle a^2 c^2 \int \left (-c^3 \sec ^4(e+f x) \tan ^4(e+f x)+3 c^3 \sec ^3(e+f x) \tan ^4(e+f x)-3 c^3 \sec ^2(e+f x) \tan ^4(e+f x)+c^3 \sec (e+f x) \tan ^4(e+f x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle a^2 c^2 \left (\frac {9 c^3 \text {arctanh}(\sin (e+f x))}{16 f}-\frac {c^3 \tan ^7(e+f x)}{7 f}-\frac {4 c^3 \tan ^5(e+f x)}{5 f}+\frac {c^3 \tan ^3(e+f x) \sec ^3(e+f x)}{2 f}-\frac {3 c^3 \tan (e+f x) \sec ^3(e+f x)}{8 f}+\frac {c^3 \tan ^3(e+f x) \sec (e+f x)}{4 f}-\frac {3 c^3 \tan (e+f x) \sec (e+f x)}{16 f}\right )\) |
Input:
Int[Sec[e + f*x]*(a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x])^5,x]
Output:
a^2*c^2*((9*c^3*ArcTanh[Sin[e + f*x]])/(16*f) - (3*c^3*Sec[e + f*x]*Tan[e + f*x])/(16*f) - (3*c^3*Sec[e + f*x]^3*Tan[e + f*x])/(8*f) + (c^3*Sec[e + f*x]*Tan[e + f*x]^3)/(4*f) + (c^3*Sec[e + f*x]^3*Tan[e + f*x]^3)/(2*f) - ( 4*c^3*Tan[e + f*x]^5)/(5*f) - (c^3*Tan[e + f*x]^7)/(7*f))
Int[csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.)*(c sc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))^(n_.), x_Symbol] :> Simp[((-a)*c)^m Int[ExpandTrig[csc[e + f*x]*cot[e + f*x]^(2*m), (c + d*csc[e + f*x])^(n - m ), x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && EqQ[b*c + a*d, 0] && Eq Q[a^2 - b^2, 0] && IntegersQ[m, n] && GeQ[n - m, 0] && GtQ[m*n, 0]
Result contains complex when optimal does not.
Time = 0.72 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.22
| method | result | size |
| risch | \(\frac {i a^{2} c^{5} \left (245 \,{\mathrm e}^{13 i \left (f x +e \right )}-1680 \,{\mathrm e}^{12 i \left (f x +e \right )}+2380 \,{\mathrm e}^{11 i \left (f x +e \right )}-4480 \,{\mathrm e}^{10 i \left (f x +e \right )}-455 \,{\mathrm e}^{9 i \left (f x +e \right )}-3920 \,{\mathrm e}^{8 i \left (f x +e \right )}-8960 \,{\mathrm e}^{6 i \left (f x +e \right )}+455 \,{\mathrm e}^{5 i \left (f x +e \right )}-3248 \,{\mathrm e}^{4 i \left (f x +e \right )}-2380 \,{\mathrm e}^{3 i \left (f x +e \right )}-896 \,{\mathrm e}^{2 i \left (f x +e \right )}-245 \,{\mathrm e}^{i \left (f x +e \right )}-368\right )}{280 f \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{7}}+\frac {9 a^{2} c^{5} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )}{16 f}-\frac {9 a^{2} c^{5} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )}{16 f}\) | \(209\) |
| norman | \(\frac {\frac {9 a^{2} c^{5} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{8 f}-\frac {15 a^{2} c^{5} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{2 f}+\frac {849 a^{2} c^{5} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{40 f}-\frac {1152 a^{2} c^{5} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}{35 f}+\frac {1199 a^{2} c^{5} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}{40 f}+\frac {15 a^{2} c^{5} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{11}}{2 f}-\frac {9 a^{2} c^{5} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{13}}{8 f}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-1\right )^{7}}-\frac {9 a^{2} c^{5} \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{16 f}+\frac {9 a^{2} c^{5} \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{16 f}\) | \(217\) |
| parallelrisch | \(-\frac {9 a^{2} c^{5} \left (\frac {\left (\cos \left (7 f x +7 e \right )+21 \cos \left (3 f x +3 e \right )+35 \cos \left (f x +e \right )+7 \cos \left (5 f x +5 e \right )\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{16}+\frac {\left (-\cos \left (7 f x +7 e \right )-7 \cos \left (5 f x +5 e \right )-21 \cos \left (3 f x +3 e \right )-35 \cos \left (f x +e \right )\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{16}+\sin \left (f x +e \right )-\frac {13 \sin \left (2 f x +2 e \right )}{72}-\frac {11 \sin \left (3 f x +3 e \right )}{45}+\frac {17 \sin \left (4 f x +4 e \right )}{18}-\frac {7 \sin \left (5 f x +5 e \right )}{45}+\frac {7 \sin \left (6 f x +6 e \right )}{72}+\frac {23 \sin \left (7 f x +7 e \right )}{315}\right )}{f \left (\cos \left (7 f x +7 e \right )+21 \cos \left (3 f x +3 e \right )+35 \cos \left (f x +e \right )+7 \cos \left (5 f x +5 e \right )\right )}\) | \(237\) |
| derivativedivides | \(\frac {a^{2} c^{5} \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )-3 a^{2} c^{5} \tan \left (f x +e \right )+a^{2} c^{5} \left (\frac {\sec \left (f x +e \right ) \tan \left (f x +e \right )}{2}+\frac {\ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{2}\right )-5 a^{2} c^{5} \left (-\frac {2}{3}-\frac {\sec \left (f x +e \right )^{2}}{3}\right ) \tan \left (f x +e \right )-5 a^{2} c^{5} \left (-\left (-\frac {\sec \left (f x +e \right )^{3}}{4}-\frac {3 \sec \left (f x +e \right )}{8}\right ) \tan \left (f x +e \right )+\frac {3 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{8}\right )+a^{2} c^{5} \left (-\frac {8}{15}-\frac {\sec \left (f x +e \right )^{4}}{5}-\frac {4 \sec \left (f x +e \right )^{2}}{15}\right ) \tan \left (f x +e \right )+3 a^{2} c^{5} \left (-\left (-\frac {\sec \left (f x +e \right )^{5}}{6}-\frac {5 \sec \left (f x +e \right )^{3}}{24}-\frac {5 \sec \left (f x +e \right )}{16}\right ) \tan \left (f x +e \right )+\frac {5 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{16}\right )+a^{2} c^{5} \left (-\frac {16}{35}-\frac {\sec \left (f x +e \right )^{6}}{7}-\frac {6 \sec \left (f x +e \right )^{4}}{35}-\frac {8 \sec \left (f x +e \right )^{2}}{35}\right ) \tan \left (f x +e \right )}{f}\) | \(299\) |
| default | \(\frac {a^{2} c^{5} \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )-3 a^{2} c^{5} \tan \left (f x +e \right )+a^{2} c^{5} \left (\frac {\sec \left (f x +e \right ) \tan \left (f x +e \right )}{2}+\frac {\ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{2}\right )-5 a^{2} c^{5} \left (-\frac {2}{3}-\frac {\sec \left (f x +e \right )^{2}}{3}\right ) \tan \left (f x +e \right )-5 a^{2} c^{5} \left (-\left (-\frac {\sec \left (f x +e \right )^{3}}{4}-\frac {3 \sec \left (f x +e \right )}{8}\right ) \tan \left (f x +e \right )+\frac {3 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{8}\right )+a^{2} c^{5} \left (-\frac {8}{15}-\frac {\sec \left (f x +e \right )^{4}}{5}-\frac {4 \sec \left (f x +e \right )^{2}}{15}\right ) \tan \left (f x +e \right )+3 a^{2} c^{5} \left (-\left (-\frac {\sec \left (f x +e \right )^{5}}{6}-\frac {5 \sec \left (f x +e \right )^{3}}{24}-\frac {5 \sec \left (f x +e \right )}{16}\right ) \tan \left (f x +e \right )+\frac {5 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{16}\right )+a^{2} c^{5} \left (-\frac {16}{35}-\frac {\sec \left (f x +e \right )^{6}}{7}-\frac {6 \sec \left (f x +e \right )^{4}}{35}-\frac {8 \sec \left (f x +e \right )^{2}}{35}\right ) \tan \left (f x +e \right )}{f}\) | \(299\) |
| parts | \(\frac {a^{2} c^{5} \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{f}+\frac {a^{2} c^{5} \left (\frac {\sec \left (f x +e \right ) \tan \left (f x +e \right )}{2}+\frac {\ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{2}\right )}{f}-\frac {3 a^{2} c^{5} \tan \left (f x +e \right )}{f}-\frac {5 a^{2} c^{5} \left (-\frac {2}{3}-\frac {\sec \left (f x +e \right )^{2}}{3}\right ) \tan \left (f x +e \right )}{f}-\frac {5 a^{2} c^{5} \left (-\left (-\frac {\sec \left (f x +e \right )^{3}}{4}-\frac {3 \sec \left (f x +e \right )}{8}\right ) \tan \left (f x +e \right )+\frac {3 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{8}\right )}{f}+\frac {a^{2} c^{5} \left (-\frac {8}{15}-\frac {\sec \left (f x +e \right )^{4}}{5}-\frac {4 \sec \left (f x +e \right )^{2}}{15}\right ) \tan \left (f x +e \right )}{f}+\frac {3 a^{2} c^{5} \left (-\left (-\frac {\sec \left (f x +e \right )^{5}}{6}-\frac {5 \sec \left (f x +e \right )^{3}}{24}-\frac {5 \sec \left (f x +e \right )}{16}\right ) \tan \left (f x +e \right )+\frac {5 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{16}\right )}{f}+\frac {a^{2} c^{5} \left (-\frac {16}{35}-\frac {\sec \left (f x +e \right )^{6}}{7}-\frac {6 \sec \left (f x +e \right )^{4}}{35}-\frac {8 \sec \left (f x +e \right )^{2}}{35}\right ) \tan \left (f x +e \right )}{f}\) | \(319\) |
Input:
int(sec(f*x+e)*(a+a*sec(f*x+e))^2*(c-c*sec(f*x+e))^5,x,method=_RETURNVERBO SE)
Output:
1/280*I*a^2*c^5*(245*exp(13*I*(f*x+e))-1680*exp(12*I*(f*x+e))+2380*exp(11* I*(f*x+e))-4480*exp(10*I*(f*x+e))-455*exp(9*I*(f*x+e))-3920*exp(8*I*(f*x+e ))-8960*exp(6*I*(f*x+e))+455*exp(5*I*(f*x+e))-3248*exp(4*I*(f*x+e))-2380*e xp(3*I*(f*x+e))-896*exp(2*I*(f*x+e))-245*exp(I*(f*x+e))-368)/f/(exp(2*I*(f *x+e))+1)^7+9/16*a^2*c^5/f*ln(exp(I*(f*x+e))+I)-9/16*a^2*c^5/f*ln(exp(I*(f *x+e))-I)
Time = 0.09 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.04 \[ \int \sec (e+f x) (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^5 \, dx=\frac {315 \, a^{2} c^{5} \cos \left (f x + e\right )^{7} \log \left (\sin \left (f x + e\right ) + 1\right ) - 315 \, a^{2} c^{5} \cos \left (f x + e\right )^{7} \log \left (-\sin \left (f x + e\right ) + 1\right ) - 2 \, {\left (368 \, a^{2} c^{5} \cos \left (f x + e\right )^{6} + 245 \, a^{2} c^{5} \cos \left (f x + e\right )^{5} - 656 \, a^{2} c^{5} \cos \left (f x + e\right )^{4} + 350 \, a^{2} c^{5} \cos \left (f x + e\right )^{3} + 208 \, a^{2} c^{5} \cos \left (f x + e\right )^{2} - 280 \, a^{2} c^{5} \cos \left (f x + e\right ) + 80 \, a^{2} c^{5}\right )} \sin \left (f x + e\right )}{1120 \, f \cos \left (f x + e\right )^{7}} \] Input:
integrate(sec(f*x+e)*(a+a*sec(f*x+e))^2*(c-c*sec(f*x+e))^5,x, algorithm="f ricas")
Output:
1/1120*(315*a^2*c^5*cos(f*x + e)^7*log(sin(f*x + e) + 1) - 315*a^2*c^5*cos (f*x + e)^7*log(-sin(f*x + e) + 1) - 2*(368*a^2*c^5*cos(f*x + e)^6 + 245*a ^2*c^5*cos(f*x + e)^5 - 656*a^2*c^5*cos(f*x + e)^4 + 350*a^2*c^5*cos(f*x + e)^3 + 208*a^2*c^5*cos(f*x + e)^2 - 280*a^2*c^5*cos(f*x + e) + 80*a^2*c^5 )*sin(f*x + e))/(f*cos(f*x + e)^7)
\[ \int \sec (e+f x) (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^5 \, dx=- a^{2} c^{5} \left (\int \left (- \sec {\left (e + f x \right )}\right )\, dx + \int 3 \sec ^{2}{\left (e + f x \right )}\, dx + \int \left (- \sec ^{3}{\left (e + f x \right )}\right )\, dx + \int \left (- 5 \sec ^{4}{\left (e + f x \right )}\right )\, dx + \int 5 \sec ^{5}{\left (e + f x \right )}\, dx + \int \sec ^{6}{\left (e + f x \right )}\, dx + \int \left (- 3 \sec ^{7}{\left (e + f x \right )}\right )\, dx + \int \sec ^{8}{\left (e + f x \right )}\, dx\right ) \] Input:
integrate(sec(f*x+e)*(a+a*sec(f*x+e))**2*(c-c*sec(f*x+e))**5,x)
Output:
-a**2*c**5*(Integral(-sec(e + f*x), x) + Integral(3*sec(e + f*x)**2, x) + Integral(-sec(e + f*x)**3, x) + Integral(-5*sec(e + f*x)**4, x) + Integral (5*sec(e + f*x)**5, x) + Integral(sec(e + f*x)**6, x) + Integral(-3*sec(e + f*x)**7, x) + Integral(sec(e + f*x)**8, x))
Leaf count of result is larger than twice the leaf count of optimal. 368 vs. \(2 (157) = 314\).
Time = 0.04 (sec) , antiderivative size = 368, normalized size of antiderivative = 2.15 \[ \int \sec (e+f x) (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^5 \, dx=-\frac {96 \, {\left (5 \, \tan \left (f x + e\right )^{7} + 21 \, \tan \left (f x + e\right )^{5} + 35 \, \tan \left (f x + e\right )^{3} + 35 \, \tan \left (f x + e\right )\right )} a^{2} c^{5} + 224 \, {\left (3 \, \tan \left (f x + e\right )^{5} + 10 \, \tan \left (f x + e\right )^{3} + 15 \, \tan \left (f x + e\right )\right )} a^{2} c^{5} - 5600 \, {\left (\tan \left (f x + e\right )^{3} + 3 \, \tan \left (f x + e\right )\right )} a^{2} c^{5} + 105 \, a^{2} c^{5} {\left (\frac {2 \, {\left (15 \, \sin \left (f x + e\right )^{5} - 40 \, \sin \left (f x + e\right )^{3} + 33 \, \sin \left (f x + e\right )\right )}}{\sin \left (f x + e\right )^{6} - 3 \, \sin \left (f x + e\right )^{4} + 3 \, \sin \left (f x + e\right )^{2} - 1} - 15 \, \log \left (\sin \left (f x + e\right ) + 1\right ) + 15 \, \log \left (\sin \left (f x + e\right ) - 1\right )\right )} - 1050 \, a^{2} c^{5} {\left (\frac {2 \, {\left (3 \, \sin \left (f x + e\right )^{3} - 5 \, \sin \left (f x + e\right )\right )}}{\sin \left (f x + e\right )^{4} - 2 \, \sin \left (f x + e\right )^{2} + 1} - 3 \, \log \left (\sin \left (f x + e\right ) + 1\right ) + 3 \, \log \left (\sin \left (f x + e\right ) - 1\right )\right )} + 840 \, a^{2} c^{5} {\left (\frac {2 \, \sin \left (f x + e\right )}{\sin \left (f x + e\right )^{2} - 1} - \log \left (\sin \left (f x + e\right ) + 1\right ) + \log \left (\sin \left (f x + e\right ) - 1\right )\right )} - 3360 \, a^{2} c^{5} \log \left (\sec \left (f x + e\right ) + \tan \left (f x + e\right )\right ) + 10080 \, a^{2} c^{5} \tan \left (f x + e\right )}{3360 \, f} \] Input:
integrate(sec(f*x+e)*(a+a*sec(f*x+e))^2*(c-c*sec(f*x+e))^5,x, algorithm="m axima")
Output:
-1/3360*(96*(5*tan(f*x + e)^7 + 21*tan(f*x + e)^5 + 35*tan(f*x + e)^3 + 35 *tan(f*x + e))*a^2*c^5 + 224*(3*tan(f*x + e)^5 + 10*tan(f*x + e)^3 + 15*ta n(f*x + e))*a^2*c^5 - 5600*(tan(f*x + e)^3 + 3*tan(f*x + e))*a^2*c^5 + 105 *a^2*c^5*(2*(15*sin(f*x + e)^5 - 40*sin(f*x + e)^3 + 33*sin(f*x + e))/(sin (f*x + e)^6 - 3*sin(f*x + e)^4 + 3*sin(f*x + e)^2 - 1) - 15*log(sin(f*x + e) + 1) + 15*log(sin(f*x + e) - 1)) - 1050*a^2*c^5*(2*(3*sin(f*x + e)^3 - 5*sin(f*x + e))/(sin(f*x + e)^4 - 2*sin(f*x + e)^2 + 1) - 3*log(sin(f*x + e) + 1) + 3*log(sin(f*x + e) - 1)) + 840*a^2*c^5*(2*sin(f*x + e)/(sin(f*x + e)^2 - 1) - log(sin(f*x + e) + 1) + log(sin(f*x + e) - 1)) - 3360*a^2*c^ 5*log(sec(f*x + e) + tan(f*x + e)) + 10080*a^2*c^5*tan(f*x + e))/f
Time = 0.23 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.15 \[ \int \sec (e+f x) (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^5 \, dx=\frac {315 \, a^{2} c^{5} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1 \right |}\right ) - 315 \, a^{2} c^{5} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1 \right |}\right ) - \frac {2 \, {\left (315 \, a^{2} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{13} - 2100 \, a^{2} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{11} - 8393 \, a^{2} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{9} + 9216 \, a^{2} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} - 5943 \, a^{2} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 2100 \, a^{2} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 315 \, a^{2} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )}^{7}}}{560 \, f} \] Input:
integrate(sec(f*x+e)*(a+a*sec(f*x+e))^2*(c-c*sec(f*x+e))^5,x, algorithm="g iac")
Output:
1/560*(315*a^2*c^5*log(abs(tan(1/2*f*x + 1/2*e) + 1)) - 315*a^2*c^5*log(ab s(tan(1/2*f*x + 1/2*e) - 1)) - 2*(315*a^2*c^5*tan(1/2*f*x + 1/2*e)^13 - 21 00*a^2*c^5*tan(1/2*f*x + 1/2*e)^11 - 8393*a^2*c^5*tan(1/2*f*x + 1/2*e)^9 + 9216*a^2*c^5*tan(1/2*f*x + 1/2*e)^7 - 5943*a^2*c^5*tan(1/2*f*x + 1/2*e)^5 + 2100*a^2*c^5*tan(1/2*f*x + 1/2*e)^3 - 315*a^2*c^5*tan(1/2*f*x + 1/2*e)) /(tan(1/2*f*x + 1/2*e)^2 - 1)^7)/f
Time = 14.14 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.47 \[ \int \sec (e+f x) (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^5 \, dx=\frac {-\frac {9\,a^2\,c^5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{13}}{8}+\frac {15\,a^2\,c^5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{11}}{2}+\frac {1199\,a^2\,c^5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9}{40}-\frac {1152\,a^2\,c^5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7}{35}+\frac {849\,a^2\,c^5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5}{40}-\frac {15\,a^2\,c^5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3}{2}+\frac {9\,a^2\,c^5\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{8}}{f\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{14}-7\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{12}+21\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}-35\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8+35\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6-21\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+7\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2-1\right )}+\frac {9\,a^2\,c^5\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )}{8\,f} \] Input:
int(((a + a/cos(e + f*x))^2*(c - c/cos(e + f*x))^5)/cos(e + f*x),x)
Output:
((849*a^2*c^5*tan(e/2 + (f*x)/2)^5)/40 - (15*a^2*c^5*tan(e/2 + (f*x)/2)^3) /2 - (1152*a^2*c^5*tan(e/2 + (f*x)/2)^7)/35 + (1199*a^2*c^5*tan(e/2 + (f*x )/2)^9)/40 + (15*a^2*c^5*tan(e/2 + (f*x)/2)^11)/2 - (9*a^2*c^5*tan(e/2 + ( f*x)/2)^13)/8 + (9*a^2*c^5*tan(e/2 + (f*x)/2))/8)/(f*(7*tan(e/2 + (f*x)/2) ^2 - 21*tan(e/2 + (f*x)/2)^4 + 35*tan(e/2 + (f*x)/2)^6 - 35*tan(e/2 + (f*x )/2)^8 + 21*tan(e/2 + (f*x)/2)^10 - 7*tan(e/2 + (f*x)/2)^12 + tan(e/2 + (f *x)/2)^14 - 1)) + (9*a^2*c^5*atanh(tan(e/2 + (f*x)/2)))/(8*f)
Time = 0.18 (sec) , antiderivative size = 326, normalized size of antiderivative = 1.91 \[ \int \sec (e+f x) (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^5 \, dx=\frac {a^{2} c^{5} \left (-315 \cos \left (f x +e \right ) \mathrm {log}\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right ) \sin \left (f x +e \right )^{6}+945 \cos \left (f x +e \right ) \mathrm {log}\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right ) \sin \left (f x +e \right )^{4}-945 \cos \left (f x +e \right ) \mathrm {log}\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right ) \sin \left (f x +e \right )^{2}+315 \cos \left (f x +e \right ) \mathrm {log}\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )+315 \cos \left (f x +e \right ) \mathrm {log}\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right ) \sin \left (f x +e \right )^{6}-945 \cos \left (f x +e \right ) \mathrm {log}\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right ) \sin \left (f x +e \right )^{4}+945 \cos \left (f x +e \right ) \mathrm {log}\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right ) \sin \left (f x +e \right )^{2}-315 \cos \left (f x +e \right ) \mathrm {log}\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )+245 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{5}-840 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{3}+315 \cos \left (f x +e \right ) \sin \left (f x +e \right )-368 \sin \left (f x +e \right )^{7}+448 \sin \left (f x +e \right )^{5}\right )}{560 \cos \left (f x +e \right ) f \left (\sin \left (f x +e \right )^{6}-3 \sin \left (f x +e \right )^{4}+3 \sin \left (f x +e \right )^{2}-1\right )} \] Input:
int(sec(f*x+e)*(a+a*sec(f*x+e))^2*(c-c*sec(f*x+e))^5,x)
Output:
(a**2*c**5*( - 315*cos(e + f*x)*log(tan((e + f*x)/2) - 1)*sin(e + f*x)**6 + 945*cos(e + f*x)*log(tan((e + f*x)/2) - 1)*sin(e + f*x)**4 - 945*cos(e + f*x)*log(tan((e + f*x)/2) - 1)*sin(e + f*x)**2 + 315*cos(e + f*x)*log(tan ((e + f*x)/2) - 1) + 315*cos(e + f*x)*log(tan((e + f*x)/2) + 1)*sin(e + f* x)**6 - 945*cos(e + f*x)*log(tan((e + f*x)/2) + 1)*sin(e + f*x)**4 + 945*c os(e + f*x)*log(tan((e + f*x)/2) + 1)*sin(e + f*x)**2 - 315*cos(e + f*x)*l og(tan((e + f*x)/2) + 1) + 245*cos(e + f*x)*sin(e + f*x)**5 - 840*cos(e + f*x)*sin(e + f*x)**3 + 315*cos(e + f*x)*sin(e + f*x) - 368*sin(e + f*x)**7 + 448*sin(e + f*x)**5))/(560*cos(e + f*x)*f*(sin(e + f*x)**6 - 3*sin(e + f*x)**4 + 3*sin(e + f*x)**2 - 1))