Integrand size = 25, antiderivative size = 177 \[ \int (a+a \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=a^4 A x+\frac {a^4 (12 A+7 C) \text {arctanh}(\sin (c+d x))}{2 d}+\frac {a^4 (10 A+7 C) \tan (c+d x)}{2 d}+\frac {a C (a+a \sec (c+d x))^3 \tan (c+d x)}{5 d}+\frac {C (a+a \sec (c+d x))^4 \tan (c+d x)}{5 d}+\frac {(5 A+7 C) \left (a^2+a^2 \sec (c+d x)\right )^2 \tan (c+d x)}{15 d}+\frac {(8 A+7 C) \left (a^4+a^4 \sec (c+d x)\right ) \tan (c+d x)}{6 d} \]
a^4*A*x+1/2*a^4*(12*A+7*C)*arctanh(sin(d*x+c))/d+1/2*a^4*(10*A+7*C)*tan(d* x+c)/d+1/5*a*C*(a+a*sec(d*x+c))^3*tan(d*x+c)/d+1/5*C*(a+a*sec(d*x+c))^4*ta n(d*x+c)/d+1/15*(5*A+7*C)*(a^2+a^2*sec(d*x+c))^2*tan(d*x+c)/d+1/6*(8*A+7*C )*(a^4+a^4*sec(d*x+c))*tan(d*x+c)/d
Time = 6.72 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.63 \[ \int (a+a \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {a^4 \left (30 A d x+15 (12 A+7 C) \text {arctanh}(\sin (c+d x))+3 \left (70 A+66 C+5 (4 A+7 C) \sec (c+d x)+12 C \sec ^2(c+d x)+10 C \sec ^3(c+d x)+2 C \sec ^4(c+d x)\right ) \tan (c+d x)+2 (5 A+16 C) \tan ^3(c+d x)\right )}{30 d} \]
(a^4*(30*A*d*x + 15*(12*A + 7*C)*ArcTanh[Sin[c + d*x]] + 3*(70*A + 66*C + 5*(4*A + 7*C)*Sec[c + d*x] + 12*C*Sec[c + d*x]^2 + 10*C*Sec[c + d*x]^3 + 2 *C*Sec[c + d*x]^4)*Tan[c + d*x] + 2*(5*A + 16*C)*Tan[c + d*x]^3))/(30*d)
Time = 1.11 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.06, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.680, Rules used = {3042, 4543, 3042, 4405, 27, 3042, 4405, 27, 3042, 4405, 27, 3042, 4402, 3042, 4254, 24, 4257}
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.
| \(\displaystyle \int (a \sec (c+d x)+a)^4 \left (A+C \sec ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^4 \left (A+C \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )dx\) |
| \(\Big \downarrow \) 4543 |
| \(\displaystyle \frac {\int (\sec (c+d x) a+a)^4 (5 a A+4 a C \sec (c+d x))dx}{5 a}+\frac {C \tan (c+d x) (a \sec (c+d x)+a)^4}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^4 \left (5 a A+4 a C \csc \left (c+d x+\frac {\pi }{2}\right )\right )dx}{5 a}+\frac {C \tan (c+d x) (a \sec (c+d x)+a)^4}{5 d}\) |
| \(\Big \downarrow \) 4405 |
| \(\displaystyle \frac {\frac {1}{4} \int 4 (\sec (c+d x) a+a)^3 \left (5 A a^2+(5 A+7 C) \sec (c+d x) a^2\right )dx+\frac {a^2 C \tan (c+d x) (a \sec (c+d x)+a)^3}{d}}{5 a}+\frac {C \tan (c+d x) (a \sec (c+d x)+a)^4}{5 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int (\sec (c+d x) a+a)^3 \left (5 A a^2+(5 A+7 C) \sec (c+d x) a^2\right )dx+\frac {a^2 C \tan (c+d x) (a \sec (c+d x)+a)^3}{d}}{5 a}+\frac {C \tan (c+d x) (a \sec (c+d x)+a)^4}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3 \left (5 A a^2+(5 A+7 C) \csc \left (c+d x+\frac {\pi }{2}\right ) a^2\right )dx+\frac {a^2 C \tan (c+d x) (a \sec (c+d x)+a)^3}{d}}{5 a}+\frac {C \tan (c+d x) (a \sec (c+d x)+a)^4}{5 d}\) |
| \(\Big \downarrow \) 4405 |
| \(\displaystyle \frac {\frac {1}{3} \int 5 (\sec (c+d x) a+a)^2 \left (3 A a^3+(8 A+7 C) \sec (c+d x) a^3\right )dx+\frac {a^3 (5 A+7 C) \tan (c+d x) (a \sec (c+d x)+a)^2}{3 d}+\frac {a^2 C \tan (c+d x) (a \sec (c+d x)+a)^3}{d}}{5 a}+\frac {C \tan (c+d x) (a \sec (c+d x)+a)^4}{5 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {5}{3} \int (\sec (c+d x) a+a)^2 \left (3 A a^3+(8 A+7 C) \sec (c+d x) a^3\right )dx+\frac {a^3 (5 A+7 C) \tan (c+d x) (a \sec (c+d x)+a)^2}{3 d}+\frac {a^2 C \tan (c+d x) (a \sec (c+d x)+a)^3}{d}}{5 a}+\frac {C \tan (c+d x) (a \sec (c+d x)+a)^4}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {5}{3} \int \left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2 \left (3 A a^3+(8 A+7 C) \csc \left (c+d x+\frac {\pi }{2}\right ) a^3\right )dx+\frac {a^3 (5 A+7 C) \tan (c+d x) (a \sec (c+d x)+a)^2}{3 d}+\frac {a^2 C \tan (c+d x) (a \sec (c+d x)+a)^3}{d}}{5 a}+\frac {C \tan (c+d x) (a \sec (c+d x)+a)^4}{5 d}\) |
| \(\Big \downarrow \) 4405 |
| \(\displaystyle \frac {\frac {5}{3} \left (\frac {1}{2} \int 3 (\sec (c+d x) a+a) \left (2 A a^4+(10 A+7 C) \sec (c+d x) a^4\right )dx+\frac {(8 A+7 C) \tan (c+d x) \left (a^5 \sec (c+d x)+a^5\right )}{2 d}\right )+\frac {a^3 (5 A+7 C) \tan (c+d x) (a \sec (c+d x)+a)^2}{3 d}+\frac {a^2 C \tan (c+d x) (a \sec (c+d x)+a)^3}{d}}{5 a}+\frac {C \tan (c+d x) (a \sec (c+d x)+a)^4}{5 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {5}{3} \left (\frac {3}{2} \int (\sec (c+d x) a+a) \left (2 A a^4+(10 A+7 C) \sec (c+d x) a^4\right )dx+\frac {(8 A+7 C) \tan (c+d x) \left (a^5 \sec (c+d x)+a^5\right )}{2 d}\right )+\frac {a^3 (5 A+7 C) \tan (c+d x) (a \sec (c+d x)+a)^2}{3 d}+\frac {a^2 C \tan (c+d x) (a \sec (c+d x)+a)^3}{d}}{5 a}+\frac {C \tan (c+d x) (a \sec (c+d x)+a)^4}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {5}{3} \left (\frac {3}{2} \int \left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right ) \left (2 A a^4+(10 A+7 C) \csc \left (c+d x+\frac {\pi }{2}\right ) a^4\right )dx+\frac {(8 A+7 C) \tan (c+d x) \left (a^5 \sec (c+d x)+a^5\right )}{2 d}\right )+\frac {a^3 (5 A+7 C) \tan (c+d x) (a \sec (c+d x)+a)^2}{3 d}+\frac {a^2 C \tan (c+d x) (a \sec (c+d x)+a)^3}{d}}{5 a}+\frac {C \tan (c+d x) (a \sec (c+d x)+a)^4}{5 d}\) |
| \(\Big \downarrow \) 4402 |
| \(\displaystyle \frac {\frac {5}{3} \left (\frac {3}{2} \left (a^5 (10 A+7 C) \int \sec ^2(c+d x)dx+a^5 (12 A+7 C) \int \sec (c+d x)dx+2 a^5 A x\right )+\frac {(8 A+7 C) \tan (c+d x) \left (a^5 \sec (c+d x)+a^5\right )}{2 d}\right )+\frac {a^3 (5 A+7 C) \tan (c+d x) (a \sec (c+d x)+a)^2}{3 d}+\frac {a^2 C \tan (c+d x) (a \sec (c+d x)+a)^3}{d}}{5 a}+\frac {C \tan (c+d x) (a \sec (c+d x)+a)^4}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {5}{3} \left (\frac {3}{2} \left (a^5 (12 A+7 C) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+a^5 (10 A+7 C) \int \csc \left (c+d x+\frac {\pi }{2}\right )^2dx+2 a^5 A x\right )+\frac {(8 A+7 C) \tan (c+d x) \left (a^5 \sec (c+d x)+a^5\right )}{2 d}\right )+\frac {a^3 (5 A+7 C) \tan (c+d x) (a \sec (c+d x)+a)^2}{3 d}+\frac {a^2 C \tan (c+d x) (a \sec (c+d x)+a)^3}{d}}{5 a}+\frac {C \tan (c+d x) (a \sec (c+d x)+a)^4}{5 d}\) |
| \(\Big \downarrow \) 4254 |
| \(\displaystyle \frac {\frac {5}{3} \left (\frac {3}{2} \left (-\frac {a^5 (10 A+7 C) \int 1d(-\tan (c+d x))}{d}+a^5 (12 A+7 C) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+2 a^5 A x\right )+\frac {(8 A+7 C) \tan (c+d x) \left (a^5 \sec (c+d x)+a^5\right )}{2 d}\right )+\frac {a^3 (5 A+7 C) \tan (c+d x) (a \sec (c+d x)+a)^2}{3 d}+\frac {a^2 C \tan (c+d x) (a \sec (c+d x)+a)^3}{d}}{5 a}+\frac {C \tan (c+d x) (a \sec (c+d x)+a)^4}{5 d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {\frac {5}{3} \left (\frac {3}{2} \left (a^5 (12 A+7 C) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+\frac {a^5 (10 A+7 C) \tan (c+d x)}{d}+2 a^5 A x\right )+\frac {(8 A+7 C) \tan (c+d x) \left (a^5 \sec (c+d x)+a^5\right )}{2 d}\right )+\frac {a^3 (5 A+7 C) \tan (c+d x) (a \sec (c+d x)+a)^2}{3 d}+\frac {a^2 C \tan (c+d x) (a \sec (c+d x)+a)^3}{d}}{5 a}+\frac {C \tan (c+d x) (a \sec (c+d x)+a)^4}{5 d}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {\frac {5}{3} \left (\frac {3}{2} \left (\frac {a^5 (12 A+7 C) \text {arctanh}(\sin (c+d x))}{d}+\frac {a^5 (10 A+7 C) \tan (c+d x)}{d}+2 a^5 A x\right )+\frac {(8 A+7 C) \tan (c+d x) \left (a^5 \sec (c+d x)+a^5\right )}{2 d}\right )+\frac {a^3 (5 A+7 C) \tan (c+d x) (a \sec (c+d x)+a)^2}{3 d}+\frac {a^2 C \tan (c+d x) (a \sec (c+d x)+a)^3}{d}}{5 a}+\frac {C \tan (c+d x) (a \sec (c+d x)+a)^4}{5 d}\) |
(C*(a + a*Sec[c + d*x])^4*Tan[c + d*x])/(5*d) + ((a^3*(5*A + 7*C)*(a + a*S ec[c + d*x])^2*Tan[c + d*x])/(3*d) + (a^2*C*(a + a*Sec[c + d*x])^3*Tan[c + d*x])/d + (5*(((8*A + 7*C)*(a^5 + a^5*Sec[c + d*x])*Tan[c + d*x])/(2*d) + (3*(2*a^5*A*x + (a^5*(12*A + 7*C)*ArcTanh[Sin[c + d*x]])/d + (a^5*(10*A + 7*C)*Tan[c + d*x])/d))/2))/3)/(5*a)
3.2.13.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp andIntegrand[(1 + x^2)^(n/2 - 1), x], x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))*(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_)), x_Symbol] :> Simp[a*c*x, x] + (Simp[b*d Int[Csc[e + f*x]^2, x], x ] + Simp[(b*c + a*d) Int[Csc[e + f*x], x], x]) /; FreeQ[{a, b, c, d, e, f }, x] && NeQ[b*c - a*d, 0] && NeQ[b*c + a*d, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(d _.) + (c_)), x_Symbol] :> Simp[(-b)*d*Cot[e + f*x]*((a + b*Csc[e + f*x])^(m - 1)/(f*m)), x] + Simp[1/m Int[(a + b*Csc[e + f*x])^(m - 1)*Simp[a*c*m + (b*c*m + a*d*(2*m - 1))*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && GtQ[m, 1] && EqQ[a^2 - b^2, 0] && IntegerQ[2 *m]
Int[((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/(f*(m + 1))), x] + Simp[1/(b*(m + 1)) Int[(a + b*Csc[e + f*x])^m*Simp[A *b*(m + 1) + a*C*m*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, C, m}, x] && EqQ[a^2 - b^2, 0] && !LtQ[m, -2^(-1)]
Time = 0.53 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.28
| method | result | size |
| parts | \(a^{4} A x -\frac {\left (a^{4} A +6 a^{4} C \right ) \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (4 a^{4} A +4 a^{4} C \right ) \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}+\frac {\left (6 a^{4} A +a^{4} C \right ) \tan \left (d x +c \right )}{d}-\frac {a^{4} C \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )}{d}+\frac {4 A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right ) a^{4}}{d}+\frac {4 a^{4} C \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}\) | \(226\) |
| parallelrisch | \(\frac {44 a^{4} \left (-\frac {45 \left (A +\frac {7 C}{12}\right ) \left (\frac {\cos \left (5 d x +5 c \right )}{10}+\frac {\cos \left (3 d x +3 c \right )}{2}+\cos \left (d x +c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{11}+\frac {45 \left (A +\frac {7 C}{12}\right ) \left (\frac {\cos \left (5 d x +5 c \right )}{10}+\frac {\cos \left (3 d x +3 c \right )}{2}+\cos \left (d x +c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{11}+\frac {15 d x A \cos \left (3 d x +3 c \right )}{44}+\frac {3 d x A \cos \left (5 d x +5 c \right )}{44}+\left (\frac {6 A}{11}+\frac {3 C}{2}\right ) \sin \left (2 d x +2 c \right )+\left (\frac {16 A}{11}+\frac {7 C}{4}\right ) \sin \left (3 d x +3 c \right )+\left (\frac {3 A}{11}+\frac {21 C}{44}\right ) \sin \left (4 d x +4 c \right )+\left (\frac {5 A}{11}+\frac {83 C}{220}\right ) \sin \left (5 d x +5 c \right )+\frac {15 d x A \cos \left (d x +c \right )}{22}+\sin \left (d x +c \right ) \left (\frac {35 C}{22}+A \right )\right )}{3 d \left (\cos \left (5 d x +5 c \right )+5 \cos \left (3 d x +3 c \right )+10 \cos \left (d x +c \right )\right )}\) | \(256\) |
| derivativedivides | \(\frac {a^{4} A \left (d x +c \right )+a^{4} C \tan \left (d x +c \right )+4 a^{4} A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+4 a^{4} C \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+6 a^{4} A \tan \left (d x +c \right )-6 a^{4} C \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+4 a^{4} A \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+4 a^{4} C \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-a^{4} A \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )-a^{4} C \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )}{d}\) | \(265\) |
| default | \(\frac {a^{4} A \left (d x +c \right )+a^{4} C \tan \left (d x +c \right )+4 a^{4} A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+4 a^{4} C \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+6 a^{4} A \tan \left (d x +c \right )-6 a^{4} C \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+4 a^{4} A \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+4 a^{4} C \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-a^{4} A \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )-a^{4} C \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )}{d}\) | \(265\) |
| norman | \(\frac {a^{4} A x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}-a^{4} A x +5 a^{4} A x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-10 a^{4} A x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+10 a^{4} A x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}-5 a^{4} A x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}-\frac {a^{4} \left (10 A +7 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{d}-\frac {a^{4} \left (18 A +25 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {2 a^{4} \left (68 A +49 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{3 d}+\frac {2 a^{4} \left (92 A +79 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 d}-\frac {4 a^{4} \left (295 A +224 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{15 d}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{5}}-\frac {a^{4} \left (12 A +7 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 d}+\frac {a^{4} \left (12 A +7 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 d}\) | \(297\) |
| risch | \(a^{4} A x -\frac {i a^{4} \left (60 A \,{\mathrm e}^{9 i \left (d x +c \right )}+105 C \,{\mathrm e}^{9 i \left (d x +c \right )}-180 A \,{\mathrm e}^{8 i \left (d x +c \right )}-30 C \,{\mathrm e}^{8 i \left (d x +c \right )}+120 A \,{\mathrm e}^{7 i \left (d x +c \right )}+330 C \,{\mathrm e}^{7 i \left (d x +c \right )}-780 A \,{\mathrm e}^{6 i \left (d x +c \right )}-480 C \,{\mathrm e}^{6 i \left (d x +c \right )}-1220 A \,{\mathrm e}^{4 i \left (d x +c \right )}-1180 C \,{\mathrm e}^{4 i \left (d x +c \right )}-120 A \,{\mathrm e}^{3 i \left (d x +c \right )}-330 C \,{\mathrm e}^{3 i \left (d x +c \right )}-820 A \,{\mathrm e}^{2 i \left (d x +c \right )}-800 C \,{\mathrm e}^{2 i \left (d x +c \right )}-60 A \,{\mathrm e}^{i \left (d x +c \right )}-105 C \,{\mathrm e}^{i \left (d x +c \right )}-200 A -166 C \right )}{15 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{5}}-\frac {6 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) A}{d}-\frac {7 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{2 d}+\frac {6 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) A}{d}+\frac {7 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{2 d}\) | \(317\) |
a^4*A*x-(A*a^4+6*C*a^4)/d*(-2/3-1/3*sec(d*x+c)^2)*tan(d*x+c)+(4*A*a^4+4*C* a^4)/d*(1/2*sec(d*x+c)*tan(d*x+c)+1/2*ln(sec(d*x+c)+tan(d*x+c)))+(6*A*a^4+ C*a^4)/d*tan(d*x+c)-a^4*C/d*(-8/15-1/5*sec(d*x+c)^4-4/15*sec(d*x+c)^2)*tan (d*x+c)+4/d*A*ln(sec(d*x+c)+tan(d*x+c))*a^4+4*a^4*C/d*(-(-1/4*sec(d*x+c)^3 -3/8*sec(d*x+c))*tan(d*x+c)+3/8*ln(sec(d*x+c)+tan(d*x+c)))
Time = 0.29 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.00 \[ \int (a+a \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {60 \, A a^{4} d x \cos \left (d x + c\right )^{5} + 15 \, {\left (12 \, A + 7 \, C\right )} a^{4} \cos \left (d x + c\right )^{5} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \, {\left (12 \, A + 7 \, C\right )} a^{4} \cos \left (d x + c\right )^{5} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (2 \, {\left (100 \, A + 83 \, C\right )} a^{4} \cos \left (d x + c\right )^{4} + 15 \, {\left (4 \, A + 7 \, C\right )} a^{4} \cos \left (d x + c\right )^{3} + 2 \, {\left (5 \, A + 34 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} + 30 \, C a^{4} \cos \left (d x + c\right ) + 6 \, C a^{4}\right )} \sin \left (d x + c\right )}{60 \, d \cos \left (d x + c\right )^{5}} \]
1/60*(60*A*a^4*d*x*cos(d*x + c)^5 + 15*(12*A + 7*C)*a^4*cos(d*x + c)^5*log (sin(d*x + c) + 1) - 15*(12*A + 7*C)*a^4*cos(d*x + c)^5*log(-sin(d*x + c) + 1) + 2*(2*(100*A + 83*C)*a^4*cos(d*x + c)^4 + 15*(4*A + 7*C)*a^4*cos(d*x + c)^3 + 2*(5*A + 34*C)*a^4*cos(d*x + c)^2 + 30*C*a^4*cos(d*x + c) + 6*C* a^4)*sin(d*x + c))/(d*cos(d*x + c)^5)
\[ \int (a+a \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=a^{4} \left (\int A\, dx + \int 4 A \sec {\left (c + d x \right )}\, dx + \int 6 A \sec ^{2}{\left (c + d x \right )}\, dx + \int 4 A \sec ^{3}{\left (c + d x \right )}\, dx + \int A \sec ^{4}{\left (c + d x \right )}\, dx + \int C \sec ^{2}{\left (c + d x \right )}\, dx + \int 4 C \sec ^{3}{\left (c + d x \right )}\, dx + \int 6 C \sec ^{4}{\left (c + d x \right )}\, dx + \int 4 C \sec ^{5}{\left (c + d x \right )}\, dx + \int C \sec ^{6}{\left (c + d x \right )}\, dx\right ) \]
a**4*(Integral(A, x) + Integral(4*A*sec(c + d*x), x) + Integral(6*A*sec(c + d*x)**2, x) + Integral(4*A*sec(c + d*x)**3, x) + Integral(A*sec(c + d*x) **4, x) + Integral(C*sec(c + d*x)**2, x) + Integral(4*C*sec(c + d*x)**3, x ) + Integral(6*C*sec(c + d*x)**4, x) + Integral(4*C*sec(c + d*x)**5, x) + Integral(C*sec(c + d*x)**6, x))
Time = 0.21 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.74 \[ \int (a+a \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {20 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a^{4} + 60 \, {\left (d x + c\right )} A a^{4} + 4 \, {\left (3 \, \tan \left (d x + c\right )^{5} + 10 \, \tan \left (d x + c\right )^{3} + 15 \, \tan \left (d x + c\right )\right )} C a^{4} + 120 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C a^{4} - 15 \, C a^{4} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 60 \, A a^{4} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 60 \, C a^{4} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 240 \, A a^{4} \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 360 \, A a^{4} \tan \left (d x + c\right ) + 60 \, C a^{4} \tan \left (d x + c\right )}{60 \, d} \]
1/60*(20*(tan(d*x + c)^3 + 3*tan(d*x + c))*A*a^4 + 60*(d*x + c)*A*a^4 + 4* (3*tan(d*x + c)^5 + 10*tan(d*x + c)^3 + 15*tan(d*x + c))*C*a^4 + 120*(tan( d*x + c)^3 + 3*tan(d*x + c))*C*a^4 - 15*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)) - 60*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)) - 60*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) ) + 240*A*a^4*log(sec(d*x + c) + tan(d*x + c)) + 360*A*a^4*tan(d*x + c) + 60*C*a^4*tan(d*x + c))/d
Time = 0.37 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.45 \[ \int (a+a \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {30 \, {\left (d x + c\right )} A a^{4} + 15 \, {\left (12 \, A a^{4} + 7 \, C a^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 15 \, {\left (12 \, A a^{4} + 7 \, C a^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (150 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 105 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 680 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 490 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 1180 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 896 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 920 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 790 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 270 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 375 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{5}}}{30 \, d} \]
1/30*(30*(d*x + c)*A*a^4 + 15*(12*A*a^4 + 7*C*a^4)*log(abs(tan(1/2*d*x + 1 /2*c) + 1)) - 15*(12*A*a^4 + 7*C*a^4)*log(abs(tan(1/2*d*x + 1/2*c) - 1)) - 2*(150*A*a^4*tan(1/2*d*x + 1/2*c)^9 + 105*C*a^4*tan(1/2*d*x + 1/2*c)^9 - 680*A*a^4*tan(1/2*d*x + 1/2*c)^7 - 490*C*a^4*tan(1/2*d*x + 1/2*c)^7 + 1180 *A*a^4*tan(1/2*d*x + 1/2*c)^5 + 896*C*a^4*tan(1/2*d*x + 1/2*c)^5 - 920*A*a ^4*tan(1/2*d*x + 1/2*c)^3 - 790*C*a^4*tan(1/2*d*x + 1/2*c)^3 + 270*A*a^4*t an(1/2*d*x + 1/2*c) + 375*C*a^4*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c )^2 - 1)^5)/d
Time = 15.49 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.56 \[ \int (a+a \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {2\,A\,a^4\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {12\,A\,a^4\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {7\,C\,a^4\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {20\,A\,a^4\,\sin \left (c+d\,x\right )}{3\,d\,\cos \left (c+d\,x\right )}+\frac {2\,A\,a^4\,\sin \left (c+d\,x\right )}{d\,{\cos \left (c+d\,x\right )}^2}+\frac {A\,a^4\,\sin \left (c+d\,x\right )}{3\,d\,{\cos \left (c+d\,x\right )}^3}+\frac {83\,C\,a^4\,\sin \left (c+d\,x\right )}{15\,d\,\cos \left (c+d\,x\right )}+\frac {7\,C\,a^4\,\sin \left (c+d\,x\right )}{2\,d\,{\cos \left (c+d\,x\right )}^2}+\frac {34\,C\,a^4\,\sin \left (c+d\,x\right )}{15\,d\,{\cos \left (c+d\,x\right )}^3}+\frac {C\,a^4\,\sin \left (c+d\,x\right )}{d\,{\cos \left (c+d\,x\right )}^4}+\frac {C\,a^4\,\sin \left (c+d\,x\right )}{5\,d\,{\cos \left (c+d\,x\right )}^5} \]
(2*A*a^4*atan(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)))/d + (12*A*a^4*atanh( sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)))/d + (7*C*a^4*atanh(sin(c/2 + (d*x) /2)/cos(c/2 + (d*x)/2)))/d + (20*A*a^4*sin(c + d*x))/(3*d*cos(c + d*x)) + (2*A*a^4*sin(c + d*x))/(d*cos(c + d*x)^2) + (A*a^4*sin(c + d*x))/(3*d*cos( c + d*x)^3) + (83*C*a^4*sin(c + d*x))/(15*d*cos(c + d*x)) + (7*C*a^4*sin(c + d*x))/(2*d*cos(c + d*x)^2) + (34*C*a^4*sin(c + d*x))/(15*d*cos(c + d*x) ^3) + (C*a^4*sin(c + d*x))/(d*cos(c + d*x)^4) + (C*a^4*sin(c + d*x))/(5*d* cos(c + d*x)^5)