Integrand size = 31, antiderivative size = 188 \[ \int \sec (c+d x) (a+a \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {7 a^4 (10 A+7 C) \text {arctanh}(\sin (c+d x))}{16 d}+\frac {4 a^4 (10 A+7 C) \tan (c+d x)}{5 d}+\frac {27 a^4 (10 A+7 C) \sec (c+d x) \tan (c+d x)}{80 d}+\frac {a^4 (10 A+7 C) \sec ^3(c+d x) \tan (c+d x)}{40 d}-\frac {C (a+a \sec (c+d x))^4 \tan (c+d x)}{30 d}+\frac {C (a+a \sec (c+d x))^5 \tan (c+d x)}{6 a d}+\frac {2 a^4 (10 A+7 C) \tan ^3(c+d x)}{15 d} \]
7/16*a^4*(10*A+7*C)*arctanh(sin(d*x+c))/d+4/5*a^4*(10*A+7*C)*tan(d*x+c)/d+ 27/80*a^4*(10*A+7*C)*sec(d*x+c)*tan(d*x+c)/d+1/40*a^4*(10*A+7*C)*sec(d*x+c )^3*tan(d*x+c)/d-1/30*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+7*C)*tan(d*x+c)^3/d
Time = 5.13 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.29 \[ \int \sec (c+d x) (a+a \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {35 a^4 A \text {arctanh}(\sin (c+d x))}{8 d}+\frac {49 a^4 C \text {arctanh}(\sin (c+d x))}{16 d}+\frac {8 a^4 A \tan (c+d x)}{d}+\frac {8 a^4 C \tan (c+d x)}{d}+\frac {27 a^4 A \sec (c+d x) \tan (c+d x)}{8 d}+\frac {49 a^4 C \sec (c+d x) \tan (c+d x)}{16 d}+\frac {a^4 A \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {41 a^4 C \sec ^3(c+d x) \tan (c+d x)}{24 d}+\frac {a^4 C \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac {4 a^4 A \tan ^3(c+d x)}{3 d}+\frac {4 a^4 C \tan ^3(c+d x)}{d}+\frac {4 a^4 C \tan ^5(c+d x)}{5 d} \]
(35*a^4*A*ArcTanh[Sin[c + d*x]])/(8*d) + (49*a^4*C*ArcTanh[Sin[c + d*x]])/ (16*d) + (8*a^4*A*Tan[c + d*x])/d + (8*a^4*C*Tan[c + d*x])/d + (27*a^4*A*S ec[c + d*x]*Tan[c + d*x])/(8*d) + (49*a^4*C*Sec[c + d*x]*Tan[c + d*x])/(16 *d) + (a^4*A*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (41*a^4*C*Sec[c + d*x]^3 *Tan[c + d*x])/(24*d) + (a^4*C*Sec[c + d*x]^5*Tan[c + d*x])/(6*d) + (4*a^4 *A*Tan[c + d*x]^3)/(3*d) + (4*a^4*C*Tan[c + d*x]^3)/d + (4*a^4*C*Tan[c + d *x]^5)/(5*d)
Time = 0.62 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.92, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {3042, 4571, 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.
| \(\displaystyle \int \sec (c+d x) (a \sec (c+d x)+a)^4 \left (A+C \sec ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \csc \left (c+d x+\frac {\pi }{2}\right ) \left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^4 \left (A+C \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )dx\) |
| \(\Big \downarrow \) 4571 |
| \(\displaystyle \frac {\int \sec (c+d x) (\sec (c+d x) a+a)^4 (a (6 A+5 C)-a C \sec (c+d x))dx}{6 a}+\frac {C \tan (c+d x) (a \sec (c+d x)+a)^5}{6 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^4 \left (a (6 A+5 C)-a C \csc \left (c+d x+\frac {\pi }{2}\right )\right )dx}{6 a}+\frac {C \tan (c+d x) (a \sec (c+d x)+a)^5}{6 a d}\) |
| \(\Big \downarrow \) 4489 |
| \(\displaystyle \frac {\frac {3}{5} a (10 A+7 C) \int \sec (c+d x) (\sec (c+d x) a+a)^4dx-\frac {a C \tan (c+d x) (a \sec (c+d x)+a)^4}{5 d}}{6 a}+\frac {C \tan (c+d x) (a \sec (c+d x)+a)^5}{6 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {3}{5} a (10 A+7 C) \int \csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^4dx-\frac {a C \tan (c+d x) (a \sec (c+d x)+a)^4}{5 d}}{6 a}+\frac {C \tan (c+d x) (a \sec (c+d x)+a)^5}{6 a d}\) |
| \(\Big \downarrow \) 4278 |
| \(\displaystyle \frac {\frac {3}{5} a (10 A+7 C) \int \left (a^4 \sec ^5(c+d x)+4 a^4 \sec ^4(c+d x)+6 a^4 \sec ^3(c+d x)+4 a^4 \sec ^2(c+d x)+a^4 \sec (c+d x)\right )dx-\frac {a C \tan (c+d x) (a \sec (c+d x)+a)^4}{5 d}}{6 a}+\frac {C \tan (c+d x) (a \sec (c+d x)+a)^5}{6 a d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {3}{5} a (10 A+7 C) \left (\frac {35 a^4 \text {arctanh}(\sin (c+d x))}{8 d}+\frac {4 a^4 \tan ^3(c+d x)}{3 d}+\frac {8 a^4 \tan (c+d x)}{d}+\frac {a^4 \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac {27 a^4 \tan (c+d x) \sec (c+d x)}{8 d}\right )-\frac {a C \tan (c+d x) (a \sec (c+d x)+a)^4}{5 d}}{6 a}+\frac {C \tan (c+d x) (a \sec (c+d x)+a)^5}{6 a d}\) |
(C*(a + a*Sec[c + d*x])^5*Tan[c + d*x])/(6*a*d) + (-1/5*(a*C*(a + a*Sec[c + d*x])^4*Tan[c + d*x])/d + (3*a*(10*A + 7*C)*((35*a^4*ArcTanh[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)
3.2.12.3.1 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_)]^2*(C_.))*(csc[ (e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[(-C)*Cot[e + f*x] *((a + b*Csc[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Simp[1/(b*(m + 2)) In t[Csc[e + f*x]*(a + b*Csc[e + f*x])^m*Simp[b*A*(m + 2) + b*C*(m + 1) - a*C* Csc[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, C, m}, x] && !LtQ[m, -1]
Time = 0.59 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.21
| method | result | size |
| norman | \(\frac {\frac {281 a^{4} \left (10 A +7 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{20 d}-\frac {231 a^{4} \left (10 A +7 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{20 d}+\frac {119 a^{4} \left (10 A +7 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{24 d}-\frac {7 a^{4} \left (10 A +7 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{8 d}+\frac {3 a^{4} \left (62 A +69 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}-\frac {a^{4} \left (2138 A +1471 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{24 d}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{6}}-\frac {7 a^{4} \left (10 A +7 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{16 d}+\frac {7 a^{4} \left (10 A +7 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{16 d}\) | \(227\) |
| parallelrisch | \(\frac {31 a^{4} \left (-\frac {525 \left (A +\frac {7 C}{10}\right ) \left (\frac {\cos \left (6 d x +6 c \right )}{15}+\frac {2 \cos \left (4 d x +4 c \right )}{5}+\cos \left (2 d x +2 c \right )+\frac {2}{3}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{124}+\frac {525 \left (A +\frac {7 C}{10}\right ) \left (\frac {\cos \left (6 d x +6 c \right )}{15}+\frac {2 \cos \left (4 d x +4 c \right )}{5}+\cos \left (2 d x +2 c \right )+\frac {2}{3}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{124}+\left (\frac {88 A}{31}+\frac {112 C}{31}\right ) \sin \left (2 d x +2 c \right )+\left (\frac {89 A}{62}+\frac {769 C}{372}\right ) \sin \left (3 d x +3 c \right )+\left (\frac {64 A}{31}+\frac {288 C}{155}\right ) \sin \left (4 d x +4 c \right )+\left (\frac {27 A}{62}+\frac {49 C}{124}\right ) \sin \left (5 d x +5 c \right )+\left (\frac {40 A}{93}+\frac {48 C}{155}\right ) \sin \left (6 d x +6 c \right )+\sin \left (d x +c \right ) \left (\frac {125 C}{62}+A \right )\right )}{2 d \left (\cos \left (6 d x +6 c \right )+6 \cos \left (4 d x +4 c \right )+15 \cos \left (2 d x +2 c \right )+10\right )}\) | \(246\) |
| parts | \(\frac {\left (a^{4} A +6 a^{4} C \right ) \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}-\frac {\left (4 a^{4} A +4 a^{4} C \right ) \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (6 a^{4} A +a^{4} C \right ) \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}+\frac {A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right ) a^{4}}{d}+\frac {a^{4} C \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )}{d}+\frac {4 a^{4} A \tan \left (d x +c \right )}{d}-\frac {4 a^{4} C \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )}{d}\) | \(281\) |
| derivativedivides | \(\frac {a^{4} A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+a^{4} C \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+4 a^{4} A \tan \left (d x +c \right )-4 a^{4} C \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+6 a^{4} A \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+6 a^{4} C \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-4 a^{4} A \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )-4 a^{4} C \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )+a^{4} A \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+a^{4} C \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )}{d}\) | \(350\) |
| default | \(\frac {a^{4} A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+a^{4} C \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+4 a^{4} A \tan \left (d x +c \right )-4 a^{4} C \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+6 a^{4} A \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+6 a^{4} C \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-4 a^{4} A \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )-4 a^{4} C \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )+a^{4} A \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+a^{4} C \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )}{d}\) | \(350\) |
| risch | \(-\frac {i a^{4} \left (810 A \,{\mathrm e}^{11 i \left (d x +c \right )}+735 C \,{\mathrm e}^{11 i \left (d x +c \right )}-960 A \,{\mathrm e}^{10 i \left (d x +c \right )}+2670 A \,{\mathrm e}^{9 i \left (d x +c \right )}+3845 C \,{\mathrm e}^{9 i \left (d x +c \right )}-6720 A \,{\mathrm e}^{8 i \left (d x +c \right )}-1920 C \,{\mathrm e}^{8 i \left (d x +c \right )}+1860 A \,{\mathrm e}^{7 i \left (d x +c \right )}+3750 C \,{\mathrm e}^{7 i \left (d x +c \right )}-16000 A \,{\mathrm e}^{6 i \left (d x +c \right )}-11520 C \,{\mathrm e}^{6 i \left (d x +c \right )}-1860 A \,{\mathrm e}^{5 i \left (d x +c \right )}-3750 C \,{\mathrm e}^{5 i \left (d x +c \right )}-17280 A \,{\mathrm e}^{4 i \left (d x +c \right )}-15360 C \,{\mathrm e}^{4 i \left (d x +c \right )}-2670 A \,{\mathrm e}^{3 i \left (d x +c \right )}-3845 C \,{\mathrm e}^{3 i \left (d x +c \right )}-8640 A \,{\mathrm e}^{2 i \left (d x +c \right )}-6912 C \,{\mathrm e}^{2 i \left (d x +c \right )}-810 A \,{\mathrm e}^{i \left (d x +c \right )}-735 C \,{\mathrm e}^{i \left (d x +c \right )}-1600 A -1152 C \right )}{120 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{6}}-\frac {35 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) A}{8 d}-\frac {49 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{16 d}+\frac {35 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) A}{8 d}+\frac {49 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{16 d}\) | \(371\) |
(281/20*a^4*(10*A+7*C)/d*tan(1/2*d*x+1/2*c)^5-231/20*a^4*(10*A+7*C)/d*tan( 1/2*d*x+1/2*c)^7+119/24*a^4*(10*A+7*C)/d*tan(1/2*d*x+1/2*c)^9-7/8*a^4*(10* A+7*C)/d*tan(1/2*d*x+1/2*c)^11+3/8*a^4*(62*A+69*C)/d*tan(1/2*d*x+1/2*c)-1/ 24*a^4*(2138*A+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+7*C)/d*ln(tan(1/2*d*x+1/2*c)-1)+7/16*a^4*(10*A+7*C)/d*ln(ta n(1/2*d*x+1/2*c)+1)
Time = 0.28 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.96 \[ \int \sec (c+d x) (a+a \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {105 \, {\left (10 \, A + 7 \, C\right )} a^{4} \cos \left (d x + c\right )^{6} \log \left (\sin \left (d x + c\right ) + 1\right ) - 105 \, {\left (10 \, A + 7 \, C\right )} a^{4} \cos \left (d x + c\right )^{6} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (64 \, {\left (25 \, A + 18 \, C\right )} a^{4} \cos \left (d x + c\right )^{5} + 15 \, {\left (54 \, A + 49 \, C\right )} a^{4} \cos \left (d x + c\right )^{4} + 64 \, {\left (5 \, A + 9 \, C\right )} a^{4} \cos \left (d x + c\right )^{3} + 10 \, {\left (6 \, A + 41 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} + 192 \, C a^{4} \cos \left (d x + c\right ) + 40 \, C a^{4}\right )} \sin \left (d x + c\right )}{480 \, d \cos \left (d x + c\right )^{6}} \]
1/480*(105*(10*A + 7*C)*a^4*cos(d*x + c)^6*log(sin(d*x + c) + 1) - 105*(10 *A + 7*C)*a^4*cos(d*x + c)^6*log(-sin(d*x + c) + 1) + 2*(64*(25*A + 18*C)* a^4*cos(d*x + c)^5 + 15*(54*A + 49*C)*a^4*cos(d*x + c)^4 + 64*(5*A + 9*C)* a^4*cos(d*x + c)^3 + 10*(6*A + 41*C)*a^4*cos(d*x + c)^2 + 192*C*a^4*cos(d* x + c) + 40*C*a^4)*sin(d*x + c))/(d*cos(d*x + c)^6)
\[ \int \sec (c+d x) (a+a \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=a^{4} \left (\int A \sec {\left (c + d x \right )}\, dx + \int 4 A \sec ^{2}{\left (c + d x \right )}\, dx + \int 6 A \sec ^{3}{\left (c + d x \right )}\, dx + \int 4 A \sec ^{4}{\left (c + d x \right )}\, dx + \int A \sec ^{5}{\left (c + d x \right )}\, dx + \int C \sec ^{3}{\left (c + d x \right )}\, dx + \int 4 C \sec ^{4}{\left (c + d x \right )}\, dx + \int 6 C \sec ^{5}{\left (c + d x \right )}\, dx + \int 4 C \sec ^{6}{\left (c + d x \right )}\, dx + \int C \sec ^{7}{\left (c + d x \right )}\, dx\right ) \]
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(C*sec(c + d*x)**3, x) + Integral(4*C*se c(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. 449 vs. \(2 (174) = 348\).
Time = 0.21 (sec) , antiderivative size = 449, normalized size of antiderivative = 2.39 \[ \int \sec (c+d x) (a+a \sec (c+d x))^4 \left (A+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} + 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 )} - 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 )} - 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 \, d} \]
1/480*(640*(tan(d*x + c)^3 + 3*tan(d*x + c))*A*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*ta n(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*log(sin(d*x + c) - 1)) - 30*A*a^4*(2*(3*si n(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)) - 12 0*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))/d
Time = 0.37 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.49 \[ \int \sec (c+d x) (a+a \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {105 \, {\left (10 \, A a^{4} + 7 \, C a^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 105 \, {\left (10 \, A a^{4} + 7 \, C a^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (1050 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 735 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 5950 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 4165 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 13860 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 9702 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 16860 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 11802 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 10690 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 7355 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2790 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3105 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{6}}}{240 \, d} \]
1/240*(105*(10*A*a^4 + 7*C*a^4)*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 105*( 10*A*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 + 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 - 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 + 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 - 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 + 7355*C*a^4*tan(1/2*d*x + 1/2*c)^3 - 2790*A*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 = 18.20 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.39 \[ \int \sec (c+d x) (a+a \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {\left (-\frac {35\,A\,a^4}{4}-\frac {49\,C\,a^4}{8}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (\frac {595\,A\,a^4}{12}+\frac {833\,C\,a^4}{24}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (-\frac {231\,A\,a^4}{2}-\frac {1617\,C\,a^4}{20}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {281\,A\,a^4}{2}+\frac {1967\,C\,a^4}{20}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (-\frac {1069\,A\,a^4}{12}-\frac {1471\,C\,a^4}{24}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {93\,A\,a^4}{4}+\frac {207\,C\,a^4}{8}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {7\,a^4\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (10\,A+7\,C\right )}{8\,d} \]
(tan(c/2 + (d*x)/2)*((93*A*a^4)/4 + (207*C*a^4)/8) - tan(c/2 + (d*x)/2)^11 *((35*A*a^4)/4 + (49*C*a^4)/8) + tan(c/2 + (d*x)/2)^9*((595*A*a^4)/12 + (8 33*C*a^4)/24) - tan(c/2 + (d*x)/2)^7*((231*A*a^4)/2 + (1617*C*a^4)/20) + t an(c/2 + (d*x)/2)^5*((281*A*a^4)/2 + (1967*C*a^4)/20) - tan(c/2 + (d*x)/2) ^3*((1069*A*a^4)/12 + (1471*C*a^4)/24))/(d*(15*tan(c/2 + (d*x)/2)^4 - 6*ta n(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)) + (7*a^4*atanh(tan(c/ 2 + (d*x)/2))*(10*A + 7*C))/(8*d)