Integrand size = 39, antiderivative size = 209 \[ \int \sec (c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {7 a^4 (10 A+8 B+7 C) \text {arctanh}(\sin (c+d x))}{16 d}+\frac {4 a^4 (10 A+8 B+7 C) \tan (c+d x)}{5 d}+\frac {27 a^4 (10 A+8 B+7 C) \sec (c+d x) \tan (c+d x)}{80 d}+\frac {a^4 (10 A+8 B+7 C) \sec ^3(c+d x) \tan (c+d x)}{40 d}+\frac {(6 B-C) (a+a \sec (c+d x))^4 \tan (c+d x)}{30 d}+\frac {C (a+a \sec (c+d x))^5 \tan (c+d x)}{6 a d}+\frac {2 a^4 (10 A+8 B+7 C) \tan ^3(c+d x)}{15 d} \]
7/16*a^4*(10*A+8*B+7*C)*arctanh(sin(d*x+c))/d+4/5*a^4*(10*A+8*B+7*C)*tan(d *x+c)/d+27/80*a^4*(10*A+8*B+7*C)*sec(d*x+c)*tan(d*x+c)/d+1/40*a^4*(10*A+8* B+7*C)*sec(d*x+c)^3*tan(d*x+c)/d+1/30*(6*B-C)*(a+a*sec(d*x+c))^4*tan(d*x+c )/d+1/6*C*(a+a*sec(d*x+c))^5*tan(d*x+c)/a/d+2/15*a^4*(10*A+8*B+7*C)*tan(d* x+c)^3/d
Time = 5.97 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.60 \[ \int \sec (c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {a^4 \left (105 (10 A+8 B+7 C) \text {arctanh}(\sin (c+d x))+\tan (c+d x) \left (1920 (A+B+C)+15 (54 A+56 B+49 C) \sec (c+d x)+10 (6 A+24 B+41 C) \sec ^3(c+d x)+40 C \sec ^5(c+d x)+320 (A+2 B+3 C) \tan ^2(c+d x)+48 (B+4 C) \tan ^4(c+d x)\right )\right )}{240 d} \]
(a^4*(105*(10*A + 8*B + 7*C)*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(1920*(A + B + C) + 15*(54*A + 56*B + 49*C)*Sec[c + d*x] + 10*(6*A + 24*B + 41*C)* Sec[c + d*x]^3 + 40*C*Sec[c + d*x]^5 + 320*(A + 2*B + 3*C)*Tan[c + d*x]^2 + 48*(B + 4*C)*Tan[c + d*x]^4)))/(240*d)
Time = 0.66 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.87, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {3042, 4570, 3042, 4489, 3042, 4278, 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.
| \(\displaystyle \int \sec (c+d x) (a \sec (c+d x)+a)^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \csc \left (c+d x+\frac {\pi }{2}\right ) \left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^4 \left (A+B \csc \left (c+d x+\frac {\pi }{2}\right )+C \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )dx\) |
| \(\Big \downarrow \) 4570 |
| \(\displaystyle \frac {\int \sec (c+d x) (\sec (c+d x) a+a)^4 (a (6 A+5 C)+a (6 B-C) \sec (c+d x))dx}{6 a}+\frac {C \tan (c+d x) (a \sec (c+d x)+a)^5}{6 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^4 \left (a (6 A+5 C)+a (6 B-C) \csc \left (c+d x+\frac {\pi }{2}\right )\right )dx}{6 a}+\frac {C \tan (c+d x) (a \sec (c+d x)+a)^5}{6 a d}\) |
| \(\Big \downarrow \) 4489 |
| \(\displaystyle \frac {\frac {3}{5} a (10 A+8 B+7 C) \int \sec (c+d x) (\sec (c+d x) a+a)^4dx+\frac {a (6 B-C) \tan (c+d x) (a \sec (c+d x)+a)^4}{5 d}}{6 a}+\frac {C \tan (c+d x) (a \sec (c+d x)+a)^5}{6 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {3}{5} a (10 A+8 B+7 C) \int \csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^4dx+\frac {a (6 B-C) \tan (c+d x) (a \sec (c+d x)+a)^4}{5 d}}{6 a}+\frac {C \tan (c+d x) (a \sec (c+d x)+a)^5}{6 a d}\) |
| \(\Big \downarrow \) 4278 |
| \(\displaystyle \frac {\frac {3}{5} a (10 A+8 B+7 C) \int \left (a^4 \sec ^5(c+d x)+4 a^4 \sec ^4(c+d x)+6 a^4 \sec ^3(c+d x)+4 a^4 \sec ^2(c+d x)+a^4 \sec (c+d x)\right )dx+\frac {a (6 B-C) \tan (c+d x) (a \sec (c+d x)+a)^4}{5 d}}{6 a}+\frac {C \tan (c+d x) (a \sec (c+d x)+a)^5}{6 a d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {3}{5} a (10 A+8 B+7 C) \left (\frac {35 a^4 \text {arctanh}(\sin (c+d x))}{8 d}+\frac {4 a^4 \tan ^3(c+d x)}{3 d}+\frac {8 a^4 \tan (c+d x)}{d}+\frac {a^4 \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac {27 a^4 \tan (c+d x) \sec (c+d x)}{8 d}\right )+\frac {a (6 B-C) \tan (c+d x) (a \sec (c+d x)+a)^4}{5 d}}{6 a}+\frac {C \tan (c+d x) (a \sec (c+d x)+a)^5}{6 a d}\) |
(C*(a + a*Sec[c + d*x])^5*Tan[c + d*x])/(6*a*d) + ((a*(6*B - C)*(a + a*Sec [c + d*x])^4*Tan[c + d*x])/(5*d) + (3*a*(10*A + 8*B + 7*C)*((35*a^4*ArcTan h[Sin[c + d*x]])/(8*d) + (8*a^4*Tan[c + d*x])/d + (27*a^4*Sec[c + d*x]*Tan [c + d*x])/(8*d) + (a^4*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (4*a^4*Tan[c + d*x]^3)/(3*d)))/5)/(6*a)
3.5.39.3.1 Defintions of rubi rules used
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Int[ExpandTrig[(a + b*csc[e + f*x])^m*(d*csc[e + f *x])^n, x], x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0] && I GtQ[m, 0] && RationalQ[n]
Int[csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(cs c[(e_.) + (f_.)*(x_)]*(B_.) + (A_)), x_Symbol] :> Simp[(-B)*Cot[e + f*x]*(( a + b*Csc[e + f*x])^m/(f*(m + 1))), x] + Simp[(a*B*m + A*b*(m + 1))/(b*(m + 1)) Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^m, x], x] /; FreeQ[{a, b, A, B , e, f, m}, x] && NeQ[A*b - a*B, 0] && EqQ[a^2 - b^2, 0] && NeQ[a*B*m + A*b *(m + 1), 0] && !LtQ[m, -2^(-1)]
Int[csc[(e_.) + (f_.)*(x_)]*((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e _.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_S ymbol] :> Simp[(-C)*Cot[e + f*x]*((a + b*Csc[e + f*x])^(m + 1)/(b*f*(m + 2) )), x] + Simp[1/(b*(m + 2)) Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^m*Simp[ b*A*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && !LtQ[m, -1]
Time = 0.90 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.20
| method | result | size |
| norman | \(\frac {\frac {281 a^{4} \left (10 A +8 B +7 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{20 d}-\frac {231 a^{4} \left (10 A +8 B +7 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{20 d}+\frac {119 a^{4} \left (10 A +8 B +7 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{24 d}-\frac {7 a^{4} \left (10 A +8 B +7 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{8 d}+\frac {a^{4} \left (186 A +200 B +207 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}-\frac {a^{4} \left (2138 A +1864 B +1471 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{24 d}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{6}}-\frac {7 a^{4} \left (10 A +8 B +7 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{16 d}+\frac {7 a^{4} \left (10 A +8 B +7 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{16 d}\) | \(251\) |
| parallelrisch | \(-\frac {35 a^{4} \left (\left (\cos \left (6 d x +6 c \right )+6 \cos \left (4 d x +4 c \right )+15 \cos \left (2 d x +2 c \right )+10\right ) \left (A +\frac {4 B}{5}+\frac {7 C}{10}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-\left (\cos \left (6 d x +6 c \right )+6 \cos \left (4 d x +4 c \right )+15 \cos \left (2 d x +2 c \right )+10\right ) \left (A +\frac {4 B}{5}+\frac {7 C}{10}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\frac {8 \left (-\frac {44 A}{7}-7 B -8 C \right ) \sin \left (2 d x +2 c \right )}{5}+\frac {\left (-\frac {769 C}{3}-178 A -232 B \right ) \sin \left (3 d x +3 c \right )}{35}+\frac {32 \left (-\frac {39 B}{5}-\frac {36 C}{5}-8 A \right ) \sin \left (4 d x +4 c \right )}{35}+\frac {\left (-7 C -\frac {54 A}{7}-8 B \right ) \sin \left (5 d x +5 c \right )}{5}+\frac {8 \left (-\frac {83 B}{75}-\frac {4 A}{3}-\frac {24 C}{25}\right ) \sin \left (6 d x +6 c \right )}{7}-\frac {124 \left (A +\frac {44 B}{31}+\frac {125 C}{62}\right ) \sin \left (d x +c \right )}{35}\right )}{8 d \left (\cos \left (6 d x +6 c \right )+6 \cos \left (4 d x +4 c \right )+15 \cos \left (2 d x +2 c \right )+10\right )}\) | \(275\) |
| parts | \(\frac {\left (4 a^{4} A +B \,a^{4}\right ) \tan \left (d x +c \right )}{d}-\frac {\left (B \,a^{4}+4 a^{4} C \right ) \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (a^{4} A +4 B \,a^{4}+6 a^{4} C \right ) \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}-\frac {\left (4 a^{4} A +6 B \,a^{4}+4 a^{4} C \right ) \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (6 a^{4} A +4 B \,a^{4}+a^{4} C \right ) \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}+\frac {A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right ) a^{4}}{d}+\frac {a^{4} C \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )}{d}\) | \(314\) |
| derivativedivides | \(\frac {a^{4} A \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-B \,a^{4} \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )+a^{4} C \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )-4 a^{4} A \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+4 B \,a^{4} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-4 a^{4} C \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )+6 a^{4} A \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-6 B \,a^{4} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+6 a^{4} C \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+4 a^{4} A \tan \left (d x +c \right )+4 B \,a^{4} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-4 a^{4} C \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+a^{4} A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+B \,a^{4} \tan \left (d x +c \right )+a^{4} C \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}\) | \(506\) |
| default | \(\frac {a^{4} A \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-B \,a^{4} \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )+a^{4} C \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )-4 a^{4} A \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+4 B \,a^{4} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-4 a^{4} C \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )+6 a^{4} A \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-6 B \,a^{4} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+6 a^{4} C \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+4 a^{4} A \tan \left (d x +c \right )+4 B \,a^{4} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-4 a^{4} C \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+a^{4} A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+B \,a^{4} \tan \left (d x +c \right )+a^{4} C \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}\) | \(506\) |
| risch | \(-\frac {i a^{4} \left (-1152 C -1600 A -1328 B -6720 A \,{\mathrm e}^{8 i \left (d x +c \right )}+1860 A \,{\mathrm e}^{7 i \left (d x +c \right )}-1860 A \,{\mathrm e}^{5 i \left (d x +c \right )}-2670 A \,{\mathrm e}^{3 i \left (d x +c \right )}-810 A \,{\mathrm e}^{i \left (d x +c \right )}-15840 B \,{\mathrm e}^{4 i \left (d x +c \right )}-16000 A \,{\mathrm e}^{6 i \left (d x +c \right )}-11520 C \,{\mathrm e}^{6 i \left (d x +c \right )}-17280 A \,{\mathrm e}^{4 i \left (d x +c \right )}-15360 C \,{\mathrm e}^{4 i \left (d x +c \right )}-8640 A \,{\mathrm e}^{2 i \left (d x +c \right )}-6912 C \,{\mathrm e}^{2 i \left (d x +c \right )}+3750 C \,{\mathrm e}^{7 i \left (d x +c \right )}-3750 C \,{\mathrm e}^{5 i \left (d x +c \right )}-3845 C \,{\mathrm e}^{3 i \left (d x +c \right )}-7728 B \,{\mathrm e}^{2 i \left (d x +c \right )}-735 C \,{\mathrm e}^{i \left (d x +c \right )}+810 A \,{\mathrm e}^{11 i \left (d x +c \right )}+735 C \,{\mathrm e}^{11 i \left (d x +c \right )}-960 A \,{\mathrm e}^{10 i \left (d x +c \right )}+2670 A \,{\mathrm e}^{9 i \left (d x +c \right )}+3845 C \,{\mathrm e}^{9 i \left (d x +c \right )}-1920 C \,{\mathrm e}^{8 i \left (d x +c \right )}-840 B \,{\mathrm e}^{i \left (d x +c \right )}-2640 B \,{\mathrm e}^{5 i \left (d x +c \right )}-13280 B \,{\mathrm e}^{6 i \left (d x +c \right )}+2640 B \,{\mathrm e}^{7 i \left (d x +c \right )}+840 B \,{\mathrm e}^{11 i \left (d x +c \right )}-3480 B \,{\mathrm e}^{3 i \left (d x +c \right )}-240 B \,{\mathrm e}^{10 i \left (d x +c \right )}+3480 B \,{\mathrm e}^{9 i \left (d x +c \right )}-4080 B \,{\mathrm e}^{8 i \left (d x +c \right )}\right )}{120 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{6}}+\frac {35 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) A}{8 d}+\frac {7 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B}{2 d}+\frac {49 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{16 d}-\frac {35 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) A}{8 d}-\frac {7 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B}{2 d}-\frac {49 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{16 d}\) | \(550\) |
(281/20*a^4*(10*A+8*B+7*C)/d*tan(1/2*d*x+1/2*c)^5-231/20*a^4*(10*A+8*B+7*C )/d*tan(1/2*d*x+1/2*c)^7+119/24*a^4*(10*A+8*B+7*C)/d*tan(1/2*d*x+1/2*c)^9- 7/8*a^4*(10*A+8*B+7*C)/d*tan(1/2*d*x+1/2*c)^11+1/8*a^4*(186*A+200*B+207*C) /d*tan(1/2*d*x+1/2*c)-1/24*a^4*(2138*A+1864*B+1471*C)/d*tan(1/2*d*x+1/2*c) ^3)/(tan(1/2*d*x+1/2*c)^2-1)^6-7/16*a^4*(10*A+8*B+7*C)/d*ln(tan(1/2*d*x+1/ 2*c)-1)+7/16*a^4*(10*A+8*B+7*C)/d*ln(tan(1/2*d*x+1/2*c)+1)
Time = 0.28 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.97 \[ \int \sec (c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {105 \, {\left (10 \, A + 8 \, B + 7 \, C\right )} a^{4} \cos \left (d x + c\right )^{6} \log \left (\sin \left (d x + c\right ) + 1\right ) - 105 \, {\left (10 \, A + 8 \, B + 7 \, C\right )} a^{4} \cos \left (d x + c\right )^{6} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (16 \, {\left (100 \, A + 83 \, B + 72 \, C\right )} a^{4} \cos \left (d x + c\right )^{5} + 15 \, {\left (54 \, A + 56 \, B + 49 \, C\right )} a^{4} \cos \left (d x + c\right )^{4} + 32 \, {\left (10 \, A + 17 \, B + 18 \, C\right )} a^{4} \cos \left (d x + c\right )^{3} + 10 \, {\left (6 \, A + 24 \, B + 41 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} + 48 \, {\left (B + 4 \, C\right )} a^{4} \cos \left (d x + c\right ) + 40 \, C a^{4}\right )} \sin \left (d x + c\right )}{480 \, d \cos \left (d x + c\right )^{6}} \]
integrate(sec(d*x+c)*(a+a*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="fricas")
1/480*(105*(10*A + 8*B + 7*C)*a^4*cos(d*x + c)^6*log(sin(d*x + c) + 1) - 1 05*(10*A + 8*B + 7*C)*a^4*cos(d*x + c)^6*log(-sin(d*x + c) + 1) + 2*(16*(1 00*A + 83*B + 72*C)*a^4*cos(d*x + c)^5 + 15*(54*A + 56*B + 49*C)*a^4*cos(d *x + c)^4 + 32*(10*A + 17*B + 18*C)*a^4*cos(d*x + c)^3 + 10*(6*A + 24*B + 41*C)*a^4*cos(d*x + c)^2 + 48*(B + 4*C)*a^4*cos(d*x + c) + 40*C*a^4)*sin(d *x + c))/(d*cos(d*x + c)^6)
\[ \int \sec (c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=a^{4} \left (\int A \sec {\left (c + d x \right )}\, dx + \int 4 A \sec ^{2}{\left (c + d x \right )}\, dx + \int 6 A \sec ^{3}{\left (c + d x \right )}\, dx + \int 4 A \sec ^{4}{\left (c + d x \right )}\, dx + \int A \sec ^{5}{\left (c + d x \right )}\, dx + \int B \sec ^{2}{\left (c + d x \right )}\, dx + \int 4 B \sec ^{3}{\left (c + d x \right )}\, dx + \int 6 B \sec ^{4}{\left (c + d x \right )}\, dx + \int 4 B \sec ^{5}{\left (c + d x \right )}\, dx + \int B \sec ^{6}{\left (c + d x \right )}\, dx + \int C \sec ^{3}{\left (c + d x \right )}\, dx + \int 4 C \sec ^{4}{\left (c + d x \right )}\, dx + \int 6 C \sec ^{5}{\left (c + d x \right )}\, dx + \int 4 C \sec ^{6}{\left (c + d x \right )}\, dx + \int C \sec ^{7}{\left (c + d x \right )}\, dx\right ) \]
a**4*(Integral(A*sec(c + d*x), x) + Integral(4*A*sec(c + d*x)**2, x) + Int egral(6*A*sec(c + d*x)**3, x) + Integral(4*A*sec(c + d*x)**4, x) + Integra l(A*sec(c + d*x)**5, x) + Integral(B*sec(c + d*x)**2, x) + Integral(4*B*se c(c + d*x)**3, x) + Integral(6*B*sec(c + d*x)**4, x) + Integral(4*B*sec(c + d*x)**5, x) + Integral(B*sec(c + d*x)**6, x) + Integral(C*sec(c + d*x)** 3, x) + Integral(4*C*sec(c + d*x)**4, x) + Integral(6*C*sec(c + d*x)**5, x ) + Integral(4*C*sec(c + d*x)**6, x) + Integral(C*sec(c + d*x)**7, x))
Leaf count of result is larger than twice the leaf count of optimal. 638 vs. \(2 (195) = 390\).
Time = 0.23 (sec) , antiderivative size = 638, normalized size of antiderivative = 3.05 \[ \int \sec (c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {640 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a^{4} + 32 \, {\left (3 \, \tan \left (d x + c\right )^{5} + 10 \, \tan \left (d x + c\right )^{3} + 15 \, \tan \left (d x + c\right )\right )} B a^{4} + 960 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B a^{4} + 128 \, {\left (3 \, \tan \left (d x + c\right )^{5} + 10 \, \tan \left (d x + c\right )^{3} + 15 \, \tan \left (d x + c\right )\right )} C a^{4} + 640 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C a^{4} - 5 \, C a^{4} {\left (\frac {2 \, {\left (15 \, \sin \left (d x + c\right )^{5} - 40 \, \sin \left (d x + c\right )^{3} + 33 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{6} - 3 \, \sin \left (d x + c\right )^{4} + 3 \, \sin \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 30 \, A a^{4} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 120 \, B a^{4} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 180 \, C a^{4} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 720 \, A a^{4} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 480 \, B a^{4} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 120 \, C a^{4} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 480 \, A a^{4} \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 1920 \, A a^{4} \tan \left (d x + c\right ) + 480 \, B a^{4} \tan \left (d x + c\right )}{480 \, d} \]
integrate(sec(d*x+c)*(a+a*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="maxima")
1/480*(640*(tan(d*x + c)^3 + 3*tan(d*x + c))*A*a^4 + 32*(3*tan(d*x + c)^5 + 10*tan(d*x + c)^3 + 15*tan(d*x + c))*B*a^4 + 960*(tan(d*x + c)^3 + 3*tan (d*x + c))*B*a^4 + 128*(3*tan(d*x + c)^5 + 10*tan(d*x + c)^3 + 15*tan(d*x + c))*C*a^4 + 640*(tan(d*x + c)^3 + 3*tan(d*x + c))*C*a^4 - 5*C*a^4*(2*(15 *sin(d*x + c)^5 - 40*sin(d*x + c)^3 + 33*sin(d*x + c))/(sin(d*x + c)^6 - 3 *sin(d*x + c)^4 + 3*sin(d*x + c)^2 - 1) - 15*log(sin(d*x + c) + 1) + 15*lo g(sin(d*x + c) - 1)) - 30*A*a^4*(2*(3*sin(d*x + c)^3 - 5*sin(d*x + c))/(si n(d*x + c)^4 - 2*sin(d*x + c)^2 + 1) - 3*log(sin(d*x + c) + 1) + 3*log(sin (d*x + c) - 1)) - 120*B*a^4*(2*(3*sin(d*x + c)^3 - 5*sin(d*x + c))/(sin(d* x + c)^4 - 2*sin(d*x + c)^2 + 1) - 3*log(sin(d*x + c) + 1) + 3*log(sin(d*x + c) - 1)) - 180*C*a^4*(2*(3*sin(d*x + c)^3 - 5*sin(d*x + c))/(sin(d*x + c)^4 - 2*sin(d*x + c)^2 + 1) - 3*log(sin(d*x + c) + 1) + 3*log(sin(d*x + c ) - 1)) - 720*A*a^4*(2*sin(d*x + c)/(sin(d*x + c)^2 - 1) - log(sin(d*x + c ) + 1) + log(sin(d*x + c) - 1)) - 480*B*a^4*(2*sin(d*x + c)/(sin(d*x + c)^ 2 - 1) - log(sin(d*x + c) + 1) + log(sin(d*x + c) - 1)) - 120*C*a^4*(2*sin (d*x + c)/(sin(d*x + c)^2 - 1) - log(sin(d*x + c) + 1) + log(sin(d*x + c) - 1)) + 480*A*a^4*log(sec(d*x + c) + tan(d*x + c)) + 1920*A*a^4*tan(d*x + c) + 480*B*a^4*tan(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 392 vs. \(2 (195) = 390\).
Time = 0.41 (sec) , antiderivative size = 392, normalized size of antiderivative = 1.88 \[ \int \sec (c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {105 \, {\left (10 \, A a^{4} + 8 \, B a^{4} + 7 \, C a^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 105 \, {\left (10 \, A a^{4} + 8 \, B a^{4} + 7 \, C a^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (1050 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 840 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 735 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 5950 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 4760 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 4165 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 13860 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 11088 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 9702 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 16860 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 13488 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 11802 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 10690 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 9320 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 7355 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2790 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3000 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3105 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{6}}}{240 \, d} \]
1/240*(105*(10*A*a^4 + 8*B*a^4 + 7*C*a^4)*log(abs(tan(1/2*d*x + 1/2*c) + 1 )) - 105*(10*A*a^4 + 8*B*a^4 + 7*C*a^4)*log(abs(tan(1/2*d*x + 1/2*c) - 1)) - 2*(1050*A*a^4*tan(1/2*d*x + 1/2*c)^11 + 840*B*a^4*tan(1/2*d*x + 1/2*c)^ 11 + 735*C*a^4*tan(1/2*d*x + 1/2*c)^11 - 5950*A*a^4*tan(1/2*d*x + 1/2*c)^9 - 4760*B*a^4*tan(1/2*d*x + 1/2*c)^9 - 4165*C*a^4*tan(1/2*d*x + 1/2*c)^9 + 13860*A*a^4*tan(1/2*d*x + 1/2*c)^7 + 11088*B*a^4*tan(1/2*d*x + 1/2*c)^7 + 9702*C*a^4*tan(1/2*d*x + 1/2*c)^7 - 16860*A*a^4*tan(1/2*d*x + 1/2*c)^5 - 13488*B*a^4*tan(1/2*d*x + 1/2*c)^5 - 11802*C*a^4*tan(1/2*d*x + 1/2*c)^5 + 10690*A*a^4*tan(1/2*d*x + 1/2*c)^3 + 9320*B*a^4*tan(1/2*d*x + 1/2*c)^3 + 7 355*C*a^4*tan(1/2*d*x + 1/2*c)^3 - 2790*A*a^4*tan(1/2*d*x + 1/2*c) - 3000* B*a^4*tan(1/2*d*x + 1/2*c) - 3105*C*a^4*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 - 1)^6)/d
Time = 20.03 (sec) , antiderivative size = 338, normalized size of antiderivative = 1.62 \[ \int \sec (c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {7\,a^4\,\mathrm {atanh}\left (\frac {7\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (10\,A+8\,B+7\,C\right )}{4\,\left (\frac {35\,A\,a^4}{2}+14\,B\,a^4+\frac {49\,C\,a^4}{4}\right )}\right )\,\left (10\,A+8\,B+7\,C\right )}{8\,d}-\frac {\left (\frac {35\,A\,a^4}{4}+7\,B\,a^4+\frac {49\,C\,a^4}{8}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (-\frac {595\,A\,a^4}{12}-\frac {119\,B\,a^4}{3}-\frac {833\,C\,a^4}{24}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (\frac {231\,A\,a^4}{2}+\frac {462\,B\,a^4}{5}+\frac {1617\,C\,a^4}{20}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (-\frac {281\,A\,a^4}{2}-\frac {562\,B\,a^4}{5}-\frac {1967\,C\,a^4}{20}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {1069\,A\,a^4}{12}+\frac {233\,B\,a^4}{3}+\frac {1471\,C\,a^4}{24}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (-\frac {93\,A\,a^4}{4}-25\,B\,a^4-\frac {207\,C\,a^4}{8}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )} \]
(7*a^4*atanh((7*a^4*tan(c/2 + (d*x)/2)*(10*A + 8*B + 7*C))/(4*((35*A*a^4)/ 2 + 14*B*a^4 + (49*C*a^4)/4)))*(10*A + 8*B + 7*C))/(8*d) - (tan(c/2 + (d*x )/2)^11*((35*A*a^4)/4 + 7*B*a^4 + (49*C*a^4)/8) - tan(c/2 + (d*x)/2)^9*((5 95*A*a^4)/12 + (119*B*a^4)/3 + (833*C*a^4)/24) + tan(c/2 + (d*x)/2)^7*((23 1*A*a^4)/2 + (462*B*a^4)/5 + (1617*C*a^4)/20) + tan(c/2 + (d*x)/2)^3*((106 9*A*a^4)/12 + (233*B*a^4)/3 + (1471*C*a^4)/24) - tan(c/2 + (d*x)/2)^5*((28 1*A*a^4)/2 + (562*B*a^4)/5 + (1967*C*a^4)/20) - tan(c/2 + (d*x)/2)*((93*A* a^4)/4 + 25*B*a^4 + (207*C*a^4)/8))/(d*(15*tan(c/2 + (d*x)/2)^4 - 6*tan(c/ 2 + (d*x)/2)^2 - 20*tan(c/2 + (d*x)/2)^6 + 15*tan(c/2 + (d*x)/2)^8 - 6*tan (c/2 + (d*x)/2)^10 + tan(c/2 + (d*x)/2)^12 + 1))