Integrand size = 32, antiderivative size = 206 \[ \int \sec (e+f x) (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^5 \, dx=\frac {45 a^3 c^5 \text {arctanh}(\sin (e+f x))}{128 f}-\frac {35 a^3 c^5 \sec (e+f x) \tan (e+f x)}{128 f}-\frac {5 a^3 c^5 \sec ^3(e+f x) \tan (e+f x)}{64 f}+\frac {5 a^3 c^5 \sec (e+f x) \tan ^3(e+f x)}{24 f}+\frac {5 a^3 c^5 \sec ^3(e+f x) \tan ^3(e+f x)}{48 f}-\frac {a^3 c^5 \sec (e+f x) \tan ^5(e+f x)}{6 f}-\frac {a^3 c^5 \sec ^3(e+f x) \tan ^5(e+f x)}{8 f}+\frac {2 a^3 c^5 \tan ^7(e+f x)}{7 f} \] Output:
45/128*a^3*c^5*arctanh(sin(f*x+e))/f-35/128*a^3*c^5*sec(f*x+e)*tan(f*x+e)/ f-5/64*a^3*c^5*sec(f*x+e)^3*tan(f*x+e)/f+5/24*a^3*c^5*sec(f*x+e)*tan(f*x+e )^3/f+5/48*a^3*c^5*sec(f*x+e)^3*tan(f*x+e)^3/f-1/6*a^3*c^5*sec(f*x+e)*tan( f*x+e)^5/f-1/8*a^3*c^5*sec(f*x+e)^3*tan(f*x+e)^5/f+2/7*a^3*c^5*tan(f*x+e)^ 7/f
Time = 1.89 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.54 \[ \int \sec (e+f x) (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^5 \, dx=-\frac {a^3 c^5 \left (-20160 \text {arctanh}(\sin (e+f x))+\sec ^8(e+f x) (5705 \sin (e+f x)-1792 \sin (2 (e+f x))+21 \sin (3 (e+f x))+1792 \sin (4 (e+f x))+2065 \sin (5 (e+f x))-768 \sin (6 (e+f x))+581 \sin (7 (e+f x))+128 \sin (8 (e+f x)))\right )}{57344 f} \] Input:
Integrate[Sec[e + f*x]*(a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^5,x]
Output:
-1/57344*(a^3*c^5*(-20160*ArcTanh[Sin[e + f*x]] + Sec[e + f*x]^8*(5705*Sin [e + f*x] - 1792*Sin[2*(e + f*x)] + 21*Sin[3*(e + f*x)] + 1792*Sin[4*(e + f*x)] + 2065*Sin[5*(e + f*x)] - 768*Sin[6*(e + f*x)] + 581*Sin[7*(e + f*x) ] + 128*Sin[8*(e + f*x)])))/f
Time = 0.52 (sec) , antiderivative size = 190, 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.
| \(\displaystyle \int \sec (e+f x) (a \sec (e+f x)+a)^3 (c-c \sec (e+f x))^5 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \csc \left (e+f x+\frac {\pi }{2}\right ) \left (a \csc \left (e+f x+\frac {\pi }{2}\right )+a\right )^3 \left (c-c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^5dx\) |
| \(\Big \downarrow \) 4446 |
| \(\displaystyle -a^3 c^3 \int \left (c^2 \sec ^3(e+f x) \tan ^6(e+f x)-2 c^2 \sec ^2(e+f x) \tan ^6(e+f x)+c^2 \sec (e+f x) \tan ^6(e+f x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -a^3 c^3 \left (-\frac {45 c^2 \text {arctanh}(\sin (e+f x))}{128 f}-\frac {2 c^2 \tan ^7(e+f x)}{7 f}+\frac {c^2 \tan ^5(e+f x) \sec ^3(e+f x)}{8 f}-\frac {5 c^2 \tan ^3(e+f x) \sec ^3(e+f x)}{48 f}+\frac {5 c^2 \tan (e+f x) \sec ^3(e+f x)}{64 f}+\frac {c^2 \tan ^5(e+f x) \sec (e+f x)}{6 f}-\frac {5 c^2 \tan ^3(e+f x) \sec (e+f x)}{24 f}+\frac {35 c^2 \tan (e+f x) \sec (e+f x)}{128 f}\right )\) |
Input:
Int[Sec[e + f*x]*(a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^5,x]
Output:
-(a^3*c^3*((-45*c^2*ArcTanh[Sin[e + f*x]])/(128*f) + (35*c^2*Sec[e + f*x]* Tan[e + f*x])/(128*f) + (5*c^2*Sec[e + f*x]^3*Tan[e + f*x])/(64*f) - (5*c^ 2*Sec[e + f*x]*Tan[e + f*x]^3)/(24*f) - (5*c^2*Sec[e + f*x]^3*Tan[e + f*x] ^3)/(48*f) + (c^2*Sec[e + f*x]*Tan[e + f*x]^5)/(6*f) + (c^2*Sec[e + f*x]^3 *Tan[e + f*x]^5)/(8*f) - (2*c^2*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]
Time = 0.78 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.16
| method | result | size |
| norman | \(\frac {-\frac {45 a^{3} c^{5} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{64 f}+\frac {345 a^{3} c^{5} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{64 f}-\frac {1149 a^{3} c^{5} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{64 f}+\frac {15159 a^{3} c^{5} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}{448 f}-\frac {17609 a^{3} c^{5} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}{448 f}-\frac {1149 a^{3} c^{5} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{11}}{64 f}+\frac {345 a^{3} c^{5} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{13}}{64 f}-\frac {45 a^{3} c^{5} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{15}}{64 f}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-1\right )^{8}}-\frac {45 a^{3} c^{5} \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{128 f}+\frac {45 a^{3} c^{5} \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{128 f}\) | \(239\) |
| risch | \(\frac {i c^{5} a^{3} \left (581 \,{\mathrm e}^{15 i \left (f x +e \right )}-1792 \,{\mathrm e}^{14 i \left (f x +e \right )}+2065 \,{\mathrm e}^{13 i \left (f x +e \right )}-1792 \,{\mathrm e}^{12 i \left (f x +e \right )}+21 \,{\mathrm e}^{11 i \left (f x +e \right )}-8960 \,{\mathrm e}^{10 i \left (f x +e \right )}+5705 \,{\mathrm e}^{9 i \left (f x +e \right )}-8960 \,{\mathrm e}^{8 i \left (f x +e \right )}-5705 \,{\mathrm e}^{7 i \left (f x +e \right )}-5376 \,{\mathrm e}^{6 i \left (f x +e \right )}-21 \,{\mathrm e}^{5 i \left (f x +e \right )}-5376 \,{\mathrm e}^{4 i \left (f x +e \right )}-2065 \,{\mathrm e}^{3 i \left (f x +e \right )}-256 \,{\mathrm e}^{2 i \left (f x +e \right )}-581 \,{\mathrm e}^{i \left (f x +e \right )}-256\right )}{448 f \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{8}}+\frac {45 a^{3} c^{5} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )}{128 f}-\frac {45 a^{3} c^{5} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )}{128 f}\) | \(242\) |
| parallelrisch | \(-\frac {815 a^{3} \left (\frac {9 \left (\frac {35}{2}+28 \cos \left (2 f x +2 e \right )+14 \cos \left (4 f x +4 e \right )+4 \cos \left (6 f x +6 e \right )+\frac {\cos \left (8 f x +8 e \right )}{2}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{163}+\frac {9 \left (-\frac {35}{2}-28 \cos \left (2 f x +2 e \right )-14 \cos \left (4 f x +4 e \right )-4 \cos \left (6 f x +6 e \right )-\frac {\cos \left (8 f x +8 e \right )}{2}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{163}+\sin \left (f x +e \right )-\frac {256 \sin \left (2 f x +2 e \right )}{815}+\frac {3 \sin \left (3 f x +3 e \right )}{815}+\frac {256 \sin \left (4 f x +4 e \right )}{815}+\frac {59 \sin \left (5 f x +5 e \right )}{163}-\frac {768 \sin \left (6 f x +6 e \right )}{5705}+\frac {83 \sin \left (7 f x +7 e \right )}{815}+\frac {128 \sin \left (8 f x +8 e \right )}{5705}\right ) c^{5}}{64 f \left (35+\cos \left (8 f x +8 e \right )+8 \cos \left (6 f x +6 e \right )+28 \cos \left (4 f x +4 e \right )+56 \cos \left (2 f x +2 e \right )\right )}\) | \(262\) |
| parts | \(-\frac {2 a^{3} c^{5} \tan \left (f x +e \right )}{f}-\frac {a^{3} c^{5} \sec \left (f x +e \right ) \tan \left (f x +e \right )}{f}-\frac {6 a^{3} c^{5} \left (-\frac {2}{3}-\frac {\sec \left (f x +e \right )^{2}}{3}\right ) \tan \left (f x +e \right )}{f}+\frac {6 a^{3} c^{5} \left (-\frac {8}{15}-\frac {\sec \left (f x +e \right )^{4}}{5}-\frac {4 \sec \left (f x +e \right )^{2}}{15}\right ) \tan \left (f x +e \right )}{f}+\frac {2 a^{3} c^{5} \left (-\left (-\frac {\sec \left (f x +e \right )^{5}}{6}-\frac {5 \sec \left (f x +e \right )^{3}}{24}-\frac {5 \sec \left (f x +e \right )}{16}\right ) \tan \left (f x +e \right )+\frac {5 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{16}\right )}{f}-\frac {2 a^{3} c^{5} \left (-\frac {16}{35}-\frac {\sec \left (f x +e \right )^{6}}{7}-\frac {6 \sec \left (f x +e \right )^{4}}{35}-\frac {8 \sec \left (f x +e \right )^{2}}{35}\right ) \tan \left (f x +e \right )}{f}-\frac {a^{3} c^{5} \left (-\left (-\frac {\sec \left (f x +e \right )^{7}}{8}-\frac {7 \sec \left (f x +e \right )^{5}}{48}-\frac {35 \sec \left (f x +e \right )^{3}}{192}-\frac {35 \sec \left (f x +e \right )}{128}\right ) \tan \left (f x +e \right )+\frac {35 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{128}\right )}{f}\) | \(299\) |
| derivativedivides | \(\frac {a^{3} c^{5} \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )-2 a^{3} c^{5} \tan \left (f x +e \right )-2 a^{3} c^{5} \left (\frac {\sec \left (f x +e \right ) \tan \left (f x +e \right )}{2}+\frac {\ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{2}\right )-6 a^{3} c^{5} \left (-\frac {2}{3}-\frac {\sec \left (f x +e \right )^{2}}{3}\right ) \tan \left (f x +e \right )+6 a^{3} c^{5} \left (-\frac {8}{15}-\frac {\sec \left (f x +e \right )^{4}}{5}-\frac {4 \sec \left (f x +e \right )^{2}}{15}\right ) \tan \left (f x +e \right )+2 a^{3} c^{5} \left (-\left (-\frac {\sec \left (f x +e \right )^{5}}{6}-\frac {5 \sec \left (f x +e \right )^{3}}{24}-\frac {5 \sec \left (f x +e \right )}{16}\right ) \tan \left (f x +e \right )+\frac {5 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{16}\right )-2 a^{3} c^{5} \left (-\frac {16}{35}-\frac {\sec \left (f x +e \right )^{6}}{7}-\frac {6 \sec \left (f x +e \right )^{4}}{35}-\frac {8 \sec \left (f x +e \right )^{2}}{35}\right ) \tan \left (f x +e \right )-a^{3} c^{5} \left (-\left (-\frac {\sec \left (f x +e \right )^{7}}{8}-\frac {7 \sec \left (f x +e \right )^{5}}{48}-\frac {35 \sec \left (f x +e \right )^{3}}{192}-\frac {35 \sec \left (f x +e \right )}{128}\right ) \tan \left (f x +e \right )+\frac {35 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{128}\right )}{f}\) | \(322\) |
| default | \(\frac {a^{3} c^{5} \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )-2 a^{3} c^{5} \tan \left (f x +e \right )-2 a^{3} c^{5} \left (\frac {\sec \left (f x +e \right ) \tan \left (f x +e \right )}{2}+\frac {\ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{2}\right )-6 a^{3} c^{5} \left (-\frac {2}{3}-\frac {\sec \left (f x +e \right )^{2}}{3}\right ) \tan \left (f x +e \right )+6 a^{3} c^{5} \left (-\frac {8}{15}-\frac {\sec \left (f x +e \right )^{4}}{5}-\frac {4 \sec \left (f x +e \right )^{2}}{15}\right ) \tan \left (f x +e \right )+2 a^{3} c^{5} \left (-\left (-\frac {\sec \left (f x +e \right )^{5}}{6}-\frac {5 \sec \left (f x +e \right )^{3}}{24}-\frac {5 \sec \left (f x +e \right )}{16}\right ) \tan \left (f x +e \right )+\frac {5 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{16}\right )-2 a^{3} c^{5} \left (-\frac {16}{35}-\frac {\sec \left (f x +e \right )^{6}}{7}-\frac {6 \sec \left (f x +e \right )^{4}}{35}-\frac {8 \sec \left (f x +e \right )^{2}}{35}\right ) \tan \left (f x +e \right )-a^{3} c^{5} \left (-\left (-\frac {\sec \left (f x +e \right )^{7}}{8}-\frac {7 \sec \left (f x +e \right )^{5}}{48}-\frac {35 \sec \left (f x +e \right )^{3}}{192}-\frac {35 \sec \left (f x +e \right )}{128}\right ) \tan \left (f x +e \right )+\frac {35 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{128}\right )}{f}\) | \(322\) |
Input:
int(sec(f*x+e)*(a+a*sec(f*x+e))^3*(c-c*sec(f*x+e))^5,x,method=_RETURNVERBO SE)
Output:
(-45/64*a^3*c^5/f*tan(1/2*f*x+1/2*e)+345/64*a^3*c^5/f*tan(1/2*f*x+1/2*e)^3 -1149/64*a^3*c^5/f*tan(1/2*f*x+1/2*e)^5+15159/448*a^3*c^5/f*tan(1/2*f*x+1/ 2*e)^7-17609/448*a^3*c^5/f*tan(1/2*f*x+1/2*e)^9-1149/64*a^3*c^5/f*tan(1/2* f*x+1/2*e)^11+345/64*a^3*c^5/f*tan(1/2*f*x+1/2*e)^13-45/64*a^3*c^5/f*tan(1 /2*f*x+1/2*e)^15)/(tan(1/2*f*x+1/2*e)^2-1)^8-45/128*a^3*c^5/f*ln(tan(1/2*f *x+1/2*e)-1)+45/128*a^3*c^5/f*ln(tan(1/2*f*x+1/2*e)+1)
Time = 0.14 (sec) , antiderivative size = 193, normalized size of antiderivative = 0.94 \[ \int \sec (e+f x) (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^5 \, dx=\frac {315 \, a^{3} c^{5} \cos \left (f x + e\right )^{8} \log \left (\sin \left (f x + e\right ) + 1\right ) - 315 \, a^{3} c^{5} \cos \left (f x + e\right )^{8} \log \left (-\sin \left (f x + e\right ) + 1\right ) - 2 \, {\left (256 \, a^{3} c^{5} \cos \left (f x + e\right )^{7} + 581 \, a^{3} c^{5} \cos \left (f x + e\right )^{6} - 768 \, a^{3} c^{5} \cos \left (f x + e\right )^{5} - 210 \, a^{3} c^{5} \cos \left (f x + e\right )^{4} + 768 \, a^{3} c^{5} \cos \left (f x + e\right )^{3} - 168 \, a^{3} c^{5} \cos \left (f x + e\right )^{2} - 256 \, a^{3} c^{5} \cos \left (f x + e\right ) + 112 \, a^{3} c^{5}\right )} \sin \left (f x + e\right )}{1792 \, f \cos \left (f x + e\right )^{8}} \] Input:
integrate(sec(f*x+e)*(a+a*sec(f*x+e))^3*(c-c*sec(f*x+e))^5,x, algorithm="f ricas")
Output:
1/1792*(315*a^3*c^5*cos(f*x + e)^8*log(sin(f*x + e) + 1) - 315*a^3*c^5*cos (f*x + e)^8*log(-sin(f*x + e) + 1) - 2*(256*a^3*c^5*cos(f*x + e)^7 + 581*a ^3*c^5*cos(f*x + e)^6 - 768*a^3*c^5*cos(f*x + e)^5 - 210*a^3*c^5*cos(f*x + e)^4 + 768*a^3*c^5*cos(f*x + e)^3 - 168*a^3*c^5*cos(f*x + e)^2 - 256*a^3* c^5*cos(f*x + e) + 112*a^3*c^5)*sin(f*x + e))/(f*cos(f*x + e)^8)
\[ \int \sec (e+f x) (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^5 \, dx=- a^{3} c^{5} \left (\int \left (- \sec {\left (e + f x \right )}\right )\, dx + \int 2 \sec ^{2}{\left (e + f x \right )}\, dx + \int 2 \sec ^{3}{\left (e + f x \right )}\, dx + \int \left (- 6 \sec ^{4}{\left (e + f x \right )}\right )\, dx + \int 6 \sec ^{6}{\left (e + f x \right )}\, dx + \int \left (- 2 \sec ^{7}{\left (e + f x \right )}\right )\, dx + \int \left (- 2 \sec ^{8}{\left (e + f x \right )}\right )\, dx + \int \sec ^{9}{\left (e + f x \right )}\, dx\right ) \] Input:
integrate(sec(f*x+e)*(a+a*sec(f*x+e))**3*(c-c*sec(f*x+e))**5,x)
Output:
-a**3*c**5*(Integral(-sec(e + f*x), x) + Integral(2*sec(e + f*x)**2, x) + Integral(2*sec(e + f*x)**3, x) + Integral(-6*sec(e + f*x)**4, x) + Integra l(6*sec(e + f*x)**6, x) + Integral(-2*sec(e + f*x)**7, x) + Integral(-2*se c(e + f*x)**8, x) + Integral(sec(e + f*x)**9, x))
Leaf count of result is larger than twice the leaf count of optimal. 408 vs. \(2 (190) = 380\).
Time = 0.04 (sec) , antiderivative size = 408, normalized size of antiderivative = 1.98 \[ \int \sec (e+f x) (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^5 \, dx=\frac {1536 \, {\left (5 \, \tan \left (f x + e\right )^{7} + 21 \, \tan \left (f x + e\right )^{5} + 35 \, \tan \left (f x + e\right )^{3} + 35 \, \tan \left (f x + e\right )\right )} a^{3} c^{5} - 10752 \, {\left (3 \, \tan \left (f x + e\right )^{5} + 10 \, \tan \left (f x + e\right )^{3} + 15 \, \tan \left (f x + e\right )\right )} a^{3} c^{5} + 53760 \, {\left (\tan \left (f x + e\right )^{3} + 3 \, \tan \left (f x + e\right )\right )} a^{3} c^{5} + 35 \, a^{3} c^{5} {\left (\frac {2 \, {\left (105 \, \sin \left (f x + e\right )^{7} - 385 \, \sin \left (f x + e\right )^{5} + 511 \, \sin \left (f x + e\right )^{3} - 279 \, \sin \left (f x + e\right )\right )}}{\sin \left (f x + e\right )^{8} - 4 \, \sin \left (f x + e\right )^{6} + 6 \, \sin \left (f x + e\right )^{4} - 4 \, \sin \left (f x + e\right )^{2} + 1} - 105 \, \log \left (\sin \left (f x + e\right ) + 1\right ) + 105 \, \log \left (\sin \left (f x + e\right ) - 1\right )\right )} - 560 \, a^{3} c^{5} {\left (\frac {2 \, {\left (15 \, \sin \left (f x + e\right )^{5} - 40 \, \sin \left (f x + e\right )^{3} + 33 \, \sin \left (f x + e\right )\right )}}{\sin \left (f x + e\right )^{6} - 3 \, \sin \left (f x + e\right )^{4} + 3 \, \sin \left (f x + e\right )^{2} - 1} - 15 \, \log \left (\sin \left (f x + e\right ) + 1\right ) + 15 \, \log \left (\sin \left (f x + e\right ) - 1\right )\right )} + 13440 \, a^{3} c^{5} {\left (\frac {2 \, \sin \left (f x + e\right )}{\sin \left (f x + e\right )^{2} - 1} - \log \left (\sin \left (f x + e\right ) + 1\right ) + \log \left (\sin \left (f x + e\right ) - 1\right )\right )} + 26880 \, a^{3} c^{5} \log \left (\sec \left (f x + e\right ) + \tan \left (f x + e\right )\right ) - 53760 \, a^{3} c^{5} \tan \left (f x + e\right )}{26880 \, f} \] Input:
integrate(sec(f*x+e)*(a+a*sec(f*x+e))^3*(c-c*sec(f*x+e))^5,x, algorithm="m axima")
Output:
1/26880*(1536*(5*tan(f*x + e)^7 + 21*tan(f*x + e)^5 + 35*tan(f*x + e)^3 + 35*tan(f*x + e))*a^3*c^5 - 10752*(3*tan(f*x + e)^5 + 10*tan(f*x + e)^3 + 1 5*tan(f*x + e))*a^3*c^5 + 53760*(tan(f*x + e)^3 + 3*tan(f*x + e))*a^3*c^5 + 35*a^3*c^5*(2*(105*sin(f*x + e)^7 - 385*sin(f*x + e)^5 + 511*sin(f*x + e )^3 - 279*sin(f*x + e))/(sin(f*x + e)^8 - 4*sin(f*x + e)^6 + 6*sin(f*x + e )^4 - 4*sin(f*x + e)^2 + 1) - 105*log(sin(f*x + e) + 1) + 105*log(sin(f*x + e) - 1)) - 560*a^3*c^5*(2*(15*sin(f*x + e)^5 - 40*sin(f*x + e)^3 + 33*si n(f*x + e))/(sin(f*x + e)^6 - 3*sin(f*x + e)^4 + 3*sin(f*x + e)^2 - 1) - 1 5*log(sin(f*x + e) + 1) + 15*log(sin(f*x + e) - 1)) + 13440*a^3*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)) + 26880*a^3*c^5*log(sec(f*x + e) + tan(f*x + e)) - 53760*a^3*c^5*tan (f*x + e))/f
Time = 0.26 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.05 \[ \int \sec (e+f x) (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^5 \, dx=\frac {315 \, a^{3} c^{5} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1 \right |}\right ) - 315 \, a^{3} c^{5} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1 \right |}\right ) - \frac {2 \, {\left (315 \, a^{3} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{15} - 2415 \, a^{3} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{13} + 8043 \, a^{3} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{11} + 17609 \, a^{3} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{9} - 15159 \, a^{3} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 8043 \, a^{3} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 2415 \, a^{3} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 315 \, a^{3} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )}^{8}}}{896 \, f} \] Input:
integrate(sec(f*x+e)*(a+a*sec(f*x+e))^3*(c-c*sec(f*x+e))^5,x, algorithm="g iac")
Output:
1/896*(315*a^3*c^5*log(abs(tan(1/2*f*x + 1/2*e) + 1)) - 315*a^3*c^5*log(ab s(tan(1/2*f*x + 1/2*e) - 1)) - 2*(315*a^3*c^5*tan(1/2*f*x + 1/2*e)^15 - 24 15*a^3*c^5*tan(1/2*f*x + 1/2*e)^13 + 8043*a^3*c^5*tan(1/2*f*x + 1/2*e)^11 + 17609*a^3*c^5*tan(1/2*f*x + 1/2*e)^9 - 15159*a^3*c^5*tan(1/2*f*x + 1/2*e )^7 + 8043*a^3*c^5*tan(1/2*f*x + 1/2*e)^5 - 2415*a^3*c^5*tan(1/2*f*x + 1/2 *e)^3 + 315*a^3*c^5*tan(1/2*f*x + 1/2*e))/(tan(1/2*f*x + 1/2*e)^2 - 1)^8)/ f
Time = 14.10 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.38 \[ \int \sec (e+f x) (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^5 \, dx=\frac {45\,a^3\,c^5\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )}{64\,f}-\frac {\frac {45\,a^3\,c^5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{15}}{64}-\frac {345\,a^3\,c^5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{13}}{64}+\frac {1149\,a^3\,c^5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{11}}{64}+\frac {17609\,a^3\,c^5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9}{448}-\frac {15159\,a^3\,c^5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7}{448}+\frac {1149\,a^3\,c^5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5}{64}-\frac {345\,a^3\,c^5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3}{64}+\frac {45\,a^3\,c^5\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{64}}{f\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{16}-8\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{14}+28\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{12}-56\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}+70\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8-56\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+28\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4-8\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1\right )} \] Input:
int(((a + a/cos(e + f*x))^3*(c - c/cos(e + f*x))^5)/cos(e + f*x),x)
Output:
(45*a^3*c^5*atanh(tan(e/2 + (f*x)/2)))/(64*f) - ((1149*a^3*c^5*tan(e/2 + ( f*x)/2)^5)/64 - (345*a^3*c^5*tan(e/2 + (f*x)/2)^3)/64 - (15159*a^3*c^5*tan (e/2 + (f*x)/2)^7)/448 + (17609*a^3*c^5*tan(e/2 + (f*x)/2)^9)/448 + (1149* a^3*c^5*tan(e/2 + (f*x)/2)^11)/64 - (345*a^3*c^5*tan(e/2 + (f*x)/2)^13)/64 + (45*a^3*c^5*tan(e/2 + (f*x)/2)^15)/64 + (45*a^3*c^5*tan(e/2 + (f*x)/2)) /64)/(f*(28*tan(e/2 + (f*x)/2)^4 - 8*tan(e/2 + (f*x)/2)^2 - 56*tan(e/2 + ( f*x)/2)^6 + 70*tan(e/2 + (f*x)/2)^8 - 56*tan(e/2 + (f*x)/2)^10 + 28*tan(e/ 2 + (f*x)/2)^12 - 8*tan(e/2 + (f*x)/2)^14 + tan(e/2 + (f*x)/2)^16 + 1))
Time = 0.16 (sec) , antiderivative size = 312, normalized size of antiderivative = 1.51 \[ \int \sec (e+f x) (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^5 \, dx=\frac {a^{3} c^{5} \left (256 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{7}-315 \,\mathrm {log}\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right ) \sin \left (f x +e \right )^{8}+1260 \,\mathrm {log}\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right ) \sin \left (f x +e \right )^{6}-1890 \,\mathrm {log}\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right ) \sin \left (f x +e \right )^{4}+1260 \,\mathrm {log}\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right ) \sin \left (f x +e \right )^{2}-315 \,\mathrm {log}\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )+315 \,\mathrm {log}\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right ) \sin \left (f x +e \right )^{8}-1260 \,\mathrm {log}\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right ) \sin \left (f x +e \right )^{6}+1890 \,\mathrm {log}\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right ) \sin \left (f x +e \right )^{4}-1260 \,\mathrm {log}\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right ) \sin \left (f x +e \right )^{2}+315 \,\mathrm {log}\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )+581 \sin \left (f x +e \right )^{7}-1533 \sin \left (f x +e \right )^{5}+1155 \sin \left (f x +e \right )^{3}-315 \sin \left (f x +e \right )\right )}{896 f \left (\sin \left (f x +e \right )^{8}-4 \sin \left (f x +e \right )^{6}+6 \sin \left (f x +e \right )^{4}-4 \sin \left (f x +e \right )^{2}+1\right )} \] Input:
int(sec(f*x+e)*(a+a*sec(f*x+e))^3*(c-c*sec(f*x+e))^5,x)
Output:
(a**3*c**5*(256*cos(e + f*x)*sin(e + f*x)**7 - 315*log(tan((e + f*x)/2) - 1)*sin(e + f*x)**8 + 1260*log(tan((e + f*x)/2) - 1)*sin(e + f*x)**6 - 1890 *log(tan((e + f*x)/2) - 1)*sin(e + f*x)**4 + 1260*log(tan((e + f*x)/2) - 1 )*sin(e + f*x)**2 - 315*log(tan((e + f*x)/2) - 1) + 315*log(tan((e + f*x)/ 2) + 1)*sin(e + f*x)**8 - 1260*log(tan((e + f*x)/2) + 1)*sin(e + f*x)**6 + 1890*log(tan((e + f*x)/2) + 1)*sin(e + f*x)**4 - 1260*log(tan((e + f*x)/2 ) + 1)*sin(e + f*x)**2 + 315*log(tan((e + f*x)/2) + 1) + 581*sin(e + f*x)* *7 - 1533*sin(e + f*x)**5 + 1155*sin(e + f*x)**3 - 315*sin(e + f*x)))/(896 *f*(sin(e + f*x)**8 - 4*sin(e + f*x)**6 + 6*sin(e + f*x)**4 - 4*sin(e + f* x)**2 + 1))