Integrand size = 41, antiderivative size = 209 \[ \int \cos ^2(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {1}{2} a^4 (13 A+8 B+2 C) x+\frac {a^4 (8 A+13 B+12 C) \text {arctanh}(\sin (c+d x))}{2 d}+\frac {5 a^4 (A-B-2 C) \sin (c+d x)}{2 d}-\frac {a (3 A-2 C) (a+a \sec (c+d x))^3 \sin (c+d x)}{6 d}+\frac {A \cos (c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{2 d}-\frac {(A-B-2 C) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{2 d}+\frac {(3 A+18 B+22 C) \left (a^4+a^4 \sec (c+d x)\right ) \sin (c+d x)}{6 d} \]
1/2*a^4*(13*A+8*B+2*C)*x+1/2*a^4*(8*A+13*B+12*C)*arctanh(sin(d*x+c))/d+5/2 *a^4*(A-B-2*C)*sin(d*x+c)/d-1/6*a*(3*A-2*C)*(a+a*sec(d*x+c))^3*sin(d*x+c)/ d+1/2*A*cos(d*x+c)*(a+a*sec(d*x+c))^4*sin(d*x+c)/d-1/2*(A-B-2*C)*(a^2+a^2* sec(d*x+c))^2*sin(d*x+c)/d+1/6*(3*A+18*B+22*C)*(a^4+a^4*sec(d*x+c))*sin(d* x+c)/d
Leaf count is larger than twice the leaf count of optimal. \(524\) vs. \(2(209)=418\).
Time = 9.82 (sec) , antiderivative size = 524, normalized size of antiderivative = 2.51 \[ \int \cos ^2(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {a^4 (1+\cos (c+d x))^4 \left (C+B \cos (c+d x)+A \cos ^2(c+d x)\right ) \sec ^8\left (\frac {1}{2} (c+d x)\right ) \sec ^3(c+d x) \left (-96 (8 A+13 B+12 C) \cos ^3(c+d x) \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )+\sec (c) (36 (13 A+8 B+2 C) d x \cos (d x)+36 (13 A+8 B+2 C) d x \cos (2 c+d x)+156 A d x \cos (2 c+3 d x)+96 B d x \cos (2 c+3 d x)+24 C d x \cos (2 c+3 d x)+156 A d x \cos (4 c+3 d x)+96 B d x \cos (4 c+3 d x)+24 C d x \cos (4 c+3 d x)+102 A \sin (d x)+384 B \sin (d x)+672 C \sin (d x)-42 A \sin (2 c+d x)-192 B \sin (2 c+d x)-288 C \sin (2 c+d x)+96 A \sin (c+2 d x)+48 B \sin (c+2 d x)+96 C \sin (c+2 d x)+96 A \sin (3 c+2 d x)+48 B \sin (3 c+2 d x)+96 C \sin (3 c+2 d x)+57 A \sin (2 c+3 d x)+192 B \sin (2 c+3 d x)+320 C \sin (2 c+3 d x)+9 A \sin (4 c+3 d x)+48 A \sin (3 c+4 d x)+12 B \sin (3 c+4 d x)+48 A \sin (5 c+4 d x)+12 B \sin (5 c+4 d x)+3 A \sin (4 c+5 d x)+3 A \sin (6 c+5 d x))\right )}{1536 d (A+2 C+2 B \cos (c+d x)+A \cos (2 (c+d x)))} \]
(a^4*(1 + Cos[c + d*x])^4*(C + B*Cos[c + d*x] + A*Cos[c + d*x]^2)*Sec[(c + d*x)/2]^8*Sec[c + d*x]^3*(-96*(8*A + 13*B + 12*C)*Cos[c + d*x]^3*(Log[Cos [(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2 ]]) + Sec[c]*(36*(13*A + 8*B + 2*C)*d*x*Cos[d*x] + 36*(13*A + 8*B + 2*C)*d *x*Cos[2*c + d*x] + 156*A*d*x*Cos[2*c + 3*d*x] + 96*B*d*x*Cos[2*c + 3*d*x] + 24*C*d*x*Cos[2*c + 3*d*x] + 156*A*d*x*Cos[4*c + 3*d*x] + 96*B*d*x*Cos[4 *c + 3*d*x] + 24*C*d*x*Cos[4*c + 3*d*x] + 102*A*Sin[d*x] + 384*B*Sin[d*x] + 672*C*Sin[d*x] - 42*A*Sin[2*c + d*x] - 192*B*Sin[2*c + d*x] - 288*C*Sin[ 2*c + d*x] + 96*A*Sin[c + 2*d*x] + 48*B*Sin[c + 2*d*x] + 96*C*Sin[c + 2*d* x] + 96*A*Sin[3*c + 2*d*x] + 48*B*Sin[3*c + 2*d*x] + 96*C*Sin[3*c + 2*d*x] + 57*A*Sin[2*c + 3*d*x] + 192*B*Sin[2*c + 3*d*x] + 320*C*Sin[2*c + 3*d*x] + 9*A*Sin[4*c + 3*d*x] + 48*A*Sin[3*c + 4*d*x] + 12*B*Sin[3*c + 4*d*x] + 48*A*Sin[5*c + 4*d*x] + 12*B*Sin[5*c + 4*d*x] + 3*A*Sin[4*c + 5*d*x] + 3*A *Sin[6*c + 5*d*x])))/(1536*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d* x)]))
Time = 1.27 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.02, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.317, Rules used = {3042, 4574, 3042, 4506, 3042, 4506, 27, 3042, 4506, 27, 3042, 4484, 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.
| \(\displaystyle \int \cos ^2(c+d x) (a \sec (c+d x)+a)^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^4 \left (A+B \csc \left (c+d x+\frac {\pi }{2}\right )+C \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^2}dx\) |
| \(\Big \downarrow \) 4574 |
| \(\displaystyle \frac {\int \cos (c+d x) (\sec (c+d x) a+a)^4 (2 a (2 A+B)-a (3 A-2 C) \sec (c+d x))dx}{2 a}+\frac {A \sin (c+d x) \cos (c+d x) (a \sec (c+d x)+a)^4}{2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^4 \left (2 a (2 A+B)-a (3 A-2 C) \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )}dx}{2 a}+\frac {A \sin (c+d x) \cos (c+d x) (a \sec (c+d x)+a)^4}{2 d}\) |
| \(\Big \downarrow \) 4506 |
| \(\displaystyle \frac {\frac {1}{3} \int \cos (c+d x) (\sec (c+d x) a+a)^3 \left (a^2 (15 A+6 B-2 C)-6 a^2 (A-B-2 C) \sec (c+d x)\right )dx-\frac {a^2 (3 A-2 C) \sin (c+d x) (a \sec (c+d x)+a)^3}{3 d}}{2 a}+\frac {A \sin (c+d x) \cos (c+d x) (a \sec (c+d x)+a)^4}{2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{3} \int \frac {\left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3 \left (a^2 (15 A+6 B-2 C)-6 a^2 (A-B-2 C) \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )}dx-\frac {a^2 (3 A-2 C) \sin (c+d x) (a \sec (c+d x)+a)^3}{3 d}}{2 a}+\frac {A \sin (c+d x) \cos (c+d x) (a \sec (c+d x)+a)^4}{2 d}\) |
| \(\Big \downarrow \) 4506 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {1}{2} \int 2 \cos (c+d x) (\sec (c+d x) a+a)^2 \left ((18 A+3 B-8 C) a^3+(3 A+18 B+22 C) \sec (c+d x) a^3\right )dx-\frac {3 a^3 (A-B-2 C) \sin (c+d x) (a \sec (c+d x)+a)^2}{d}\right )-\frac {a^2 (3 A-2 C) \sin (c+d x) (a \sec (c+d x)+a)^3}{3 d}}{2 a}+\frac {A \sin (c+d x) \cos (c+d x) (a \sec (c+d x)+a)^4}{2 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{3} \left (\int \cos (c+d x) (\sec (c+d x) a+a)^2 \left ((18 A+3 B-8 C) a^3+(3 A+18 B+22 C) \sec (c+d x) a^3\right )dx-\frac {3 a^3 (A-B-2 C) \sin (c+d x) (a \sec (c+d x)+a)^2}{d}\right )-\frac {a^2 (3 A-2 C) \sin (c+d x) (a \sec (c+d x)+a)^3}{3 d}}{2 a}+\frac {A \sin (c+d x) \cos (c+d x) (a \sec (c+d x)+a)^4}{2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{3} \left (\int \frac {\left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2 \left ((18 A+3 B-8 C) a^3+(3 A+18 B+22 C) \csc \left (c+d x+\frac {\pi }{2}\right ) a^3\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )}dx-\frac {3 a^3 (A-B-2 C) \sin (c+d x) (a \sec (c+d x)+a)^2}{d}\right )-\frac {a^2 (3 A-2 C) \sin (c+d x) (a \sec (c+d x)+a)^3}{3 d}}{2 a}+\frac {A \sin (c+d x) \cos (c+d x) (a \sec (c+d x)+a)^4}{2 d}\) |
| \(\Big \downarrow \) 4506 |
| \(\displaystyle \frac {\frac {1}{3} \left (\int 3 \cos (c+d x) (\sec (c+d x) a+a) \left (5 (A-B-2 C) a^4+(8 A+13 B+12 C) \sec (c+d x) a^4\right )dx+\frac {(3 A+18 B+22 C) \sin (c+d x) \left (a^5 \sec (c+d x)+a^5\right )}{d}-\frac {3 a^3 (A-B-2 C) \sin (c+d x) (a \sec (c+d x)+a)^2}{d}\right )-\frac {a^2 (3 A-2 C) \sin (c+d x) (a \sec (c+d x)+a)^3}{3 d}}{2 a}+\frac {A \sin (c+d x) \cos (c+d x) (a \sec (c+d x)+a)^4}{2 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{3} \left (3 \int \cos (c+d x) (\sec (c+d x) a+a) \left (5 (A-B-2 C) a^4+(8 A+13 B+12 C) \sec (c+d x) a^4\right )dx+\frac {(3 A+18 B+22 C) \sin (c+d x) \left (a^5 \sec (c+d x)+a^5\right )}{d}-\frac {3 a^3 (A-B-2 C) \sin (c+d x) (a \sec (c+d x)+a)^2}{d}\right )-\frac {a^2 (3 A-2 C) \sin (c+d x) (a \sec (c+d x)+a)^3}{3 d}}{2 a}+\frac {A \sin (c+d x) \cos (c+d x) (a \sec (c+d x)+a)^4}{2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{3} \left (3 \int \frac {\left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right ) \left (5 (A-B-2 C) a^4+(8 A+13 B+12 C) \csc \left (c+d x+\frac {\pi }{2}\right ) a^4\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {(3 A+18 B+22 C) \sin (c+d x) \left (a^5 \sec (c+d x)+a^5\right )}{d}-\frac {3 a^3 (A-B-2 C) \sin (c+d x) (a \sec (c+d x)+a)^2}{d}\right )-\frac {a^2 (3 A-2 C) \sin (c+d x) (a \sec (c+d x)+a)^3}{3 d}}{2 a}+\frac {A \sin (c+d x) \cos (c+d x) (a \sec (c+d x)+a)^4}{2 d}\) |
| \(\Big \downarrow \) 4484 |
| \(\displaystyle \frac {\frac {1}{3} \left (3 \left (\frac {5 a^5 (A-B-2 C) \sin (c+d x)}{d}-\int \left (-\left ((13 A+8 B+2 C) a^5\right )-(8 A+13 B+12 C) \sec (c+d x) a^5\right )dx\right )+\frac {(3 A+18 B+22 C) \sin (c+d x) \left (a^5 \sec (c+d x)+a^5\right )}{d}-\frac {3 a^3 (A-B-2 C) \sin (c+d x) (a \sec (c+d x)+a)^2}{d}\right )-\frac {a^2 (3 A-2 C) \sin (c+d x) (a \sec (c+d x)+a)^3}{3 d}}{2 a}+\frac {A \sin (c+d x) \cos (c+d x) (a \sec (c+d x)+a)^4}{2 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {1}{3} \left (3 \left (\frac {a^5 (8 A+13 B+12 C) \text {arctanh}(\sin (c+d x))}{d}+\frac {5 a^5 (A-B-2 C) \sin (c+d x)}{d}+a^5 x (13 A+8 B+2 C)\right )+\frac {(3 A+18 B+22 C) \sin (c+d x) \left (a^5 \sec (c+d x)+a^5\right )}{d}-\frac {3 a^3 (A-B-2 C) \sin (c+d x) (a \sec (c+d x)+a)^2}{d}\right )-\frac {a^2 (3 A-2 C) \sin (c+d x) (a \sec (c+d x)+a)^3}{3 d}}{2 a}+\frac {A \sin (c+d x) \cos (c+d x) (a \sec (c+d x)+a)^4}{2 d}\) |
(A*Cos[c + d*x]*(a + a*Sec[c + d*x])^4*Sin[c + d*x])/(2*d) + (-1/3*(a^2*(3 *A - 2*C)*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/d + ((-3*a^3*(A - B - 2*C)* (a + a*Sec[c + d*x])^2*Sin[c + d*x])/d + ((3*A + 18*B + 22*C)*(a^5 + a^5*S ec[c + d*x])*Sin[c + d*x])/d + 3*(a^5*(13*A + 8*B + 2*C)*x + (a^5*(8*A + 1 3*B + 12*C)*ArcTanh[Sin[c + d*x]])/d + (5*a^5*(A - B - 2*C)*Sin[c + d*x])/ d))/3)/(2*a)
3.5.42.3.1 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[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + ( a_))*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)), x_Symbol] :> Simp[A*a*Cot[e + f*x]*((d*Csc[e + f*x])^n/(f*n)), x] + Simp[1/(d*n) Int[(d*Csc[e + f*x])^( n + 1)*Simp[n*(B*a + A*b) + (B*b*n + A*a*(n + 1))*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && LeQ[n, -1]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + ( a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)), x_Symbol] :> Simp[(-b)*B* Cot[e + f*x]*(a + b*Csc[e + f*x])^(m - 1)*((d*Csc[e + f*x])^n/(f*(m + n))), x] + Simp[1/(d*(m + n)) Int[(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x] )^n*Simp[a*A*d*(m + n) + B*(b*d*n) + (A*b*d*(m + n) + a*B*d*(2*m + n - 1))* Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, B, n}, x] && NeQ[A*b - a*B, 0] && EqQ[a^2 - b^2, 0] && GtQ[m, 1/2] && !LtQ[n, -1]
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. ))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a _))^(m_), x_Symbol] :> Simp[A*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*((d*Csc[e + f*x])^n/(f*n)), x] - Simp[1/(b*d*n) Int[(a + b*Csc[e + f*x])^m*(d*Csc[ e + f*x])^(n + 1)*Simp[a*A*m - b*B*n - b*(A*(m + n + 1) + C*n)*Csc[e + f*x] , x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C, m}, x] && EqQ[a^2 - b^2, 0] && !LtQ[m, -2^(-1)] && (LtQ[n, -2^(-1)] || EqQ[m + n + 1, 0])
Time = 0.67 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.11
| method | result | size |
| parallelrisch | \(\frac {5 \left (-\frac {48 \left (A +\frac {13 B}{8}+\frac {3 C}{2}\right ) \left (\frac {\cos \left (3 d x +3 c \right )}{3}+\cos \left (d x +c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{5}+\frac {48 \left (A +\frac {13 B}{8}+\frac {3 C}{2}\right ) \left (\frac {\cos \left (3 d x +3 c \right )}{3}+\cos \left (d x +c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{5}+\frac {26 x d \left (A +\frac {8 B}{13}+\frac {2 C}{13}\right ) \cos \left (3 d x +3 c \right )}{5}+\frac {8 \left (2 A +B +2 C \right ) \sin \left (2 d x +2 c \right )}{5}+\left (\frac {11 A}{10}+\frac {16 B}{5}+\frac {16 C}{3}\right ) \sin \left (3 d x +3 c \right )+\frac {2 \left (4 A +B \right ) \sin \left (4 d x +4 c \right )}{5}+\frac {A \sin \left (5 d x +5 c \right )}{10}+\frac {78 x d \left (A +\frac {8 B}{13}+\frac {2 C}{13}\right ) \cos \left (d x +c \right )}{5}+\sin \left (d x +c \right ) \left (\frac {16 B}{5}+\frac {32 C}{5}+A \right )\right ) a^{4}}{4 d \left (\cos \left (3 d x +3 c \right )+3 \cos \left (d x +c \right )\right )}\) | \(231\) |
| derivativedivides | \(\frac {a^{4} A \tan \left (d x +c \right )+B \,a^{4} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-a^{4} C \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+4 a^{4} A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+4 B \,a^{4} \tan \left (d x +c \right )+4 a^{4} C \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+6 a^{4} A \left (d x +c \right )+6 B \,a^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+6 a^{4} C \tan \left (d x +c \right )+4 a^{4} A \sin \left (d x +c \right )+4 B \,a^{4} \left (d x +c \right )+4 a^{4} C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+a^{4} A \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+B \,a^{4} \sin \left (d x +c \right )+a^{4} C \left (d x +c \right )}{d}\) | \(280\) |
| default | \(\frac {a^{4} A \tan \left (d x +c \right )+B \,a^{4} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-a^{4} C \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+4 a^{4} A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+4 B \,a^{4} \tan \left (d x +c \right )+4 a^{4} C \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+6 a^{4} A \left (d x +c \right )+6 B \,a^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+6 a^{4} C \tan \left (d x +c \right )+4 a^{4} A \sin \left (d x +c \right )+4 B \,a^{4} \left (d x +c \right )+4 a^{4} C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+a^{4} A \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+B \,a^{4} \sin \left (d x +c \right )+a^{4} C \left (d x +c \right )}{d}\) | \(280\) |
| risch | \(\frac {13 a^{4} A x}{2}+4 a^{4} x B +a^{4} x C -\frac {i {\mathrm e}^{i \left (d x +c \right )} B \,a^{4}}{2 d}-\frac {i a^{4} \left (3 B \,{\mathrm e}^{5 i \left (d x +c \right )}+12 C \,{\mathrm e}^{5 i \left (d x +c \right )}-6 A \,{\mathrm e}^{4 i \left (d x +c \right )}-24 B \,{\mathrm e}^{4 i \left (d x +c \right )}-36 C \,{\mathrm e}^{4 i \left (d x +c \right )}-12 A \,{\mathrm e}^{2 i \left (d x +c \right )}-48 B \,{\mathrm e}^{2 i \left (d x +c \right )}-84 C \,{\mathrm e}^{2 i \left (d x +c \right )}-3 B \,{\mathrm e}^{i \left (d x +c \right )}-12 C \,{\mathrm e}^{i \left (d x +c \right )}-6 A -24 B -40 C \right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3}}+\frac {2 i a^{4} A \,{\mathrm e}^{-i \left (d x +c \right )}}{d}+\frac {i {\mathrm e}^{-i \left (d x +c \right )} B \,a^{4}}{2 d}-\frac {2 i a^{4} A \,{\mathrm e}^{i \left (d x +c \right )}}{d}+\frac {i a^{4} A \,{\mathrm e}^{-2 i \left (d x +c \right )}}{8 d}-\frac {i a^{4} A \,{\mathrm e}^{2 i \left (d x +c \right )}}{8 d}-\frac {4 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) A}{d}-\frac {13 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B}{2 d}-\frac {6 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{d}+\frac {4 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) A}{d}+\frac {13 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B}{2 d}+\frac {6 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{d}\) | \(420\) |
| norman | \(\frac {\left (-\frac {13}{2} a^{4} A -4 B \,a^{4}-a^{4} C \right ) x +\left (-\frac {65}{2} a^{4} A -20 B \,a^{4}-5 a^{4} C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\left (-\frac {39}{2} a^{4} A -12 B \,a^{4}-3 a^{4} C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}+\left (-\frac {13}{2} a^{4} A -4 B \,a^{4}-a^{4} C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (\frac {13}{2} a^{4} A +4 B \,a^{4}+a^{4} C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+\left (\frac {13}{2} a^{4} A +4 B \,a^{4}+a^{4} C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}+\left (\frac {39}{2} a^{4} A +12 B \,a^{4}+3 a^{4} C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\left (\frac {65}{2} a^{4} A +20 B \,a^{4}+5 a^{4} C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+\frac {a^{4} \left (27 A +15 B +2 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{d}+\frac {5 a^{4} \left (A -B -2 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{d}+\frac {4 a^{4} \left (9 A -24 B -38 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{3 d}-\frac {a^{4} \left (11 A +11 B +18 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}-\frac {2 a^{4} \left (33 A -12 B -38 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{3 d}-\frac {a^{4} \left (53 A -B -26 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{d}+\frac {2 a^{4} \left (63 A +36 B +38 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{5}}-\frac {a^{4} \left (8 A +13 B +12 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 d}+\frac {a^{4} \left (8 A +13 B +12 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 d}\) | \(535\) |
5/4*(-48/5*(A+13/8*B+3/2*C)*(1/3*cos(3*d*x+3*c)+cos(d*x+c))*ln(tan(1/2*d*x +1/2*c)-1)+48/5*(A+13/8*B+3/2*C)*(1/3*cos(3*d*x+3*c)+cos(d*x+c))*ln(tan(1/ 2*d*x+1/2*c)+1)+26/5*x*d*(A+8/13*B+2/13*C)*cos(3*d*x+3*c)+8/5*(2*A+B+2*C)* sin(2*d*x+2*c)+(11/10*A+16/5*B+16/3*C)*sin(3*d*x+3*c)+2/5*(4*A+B)*sin(4*d* x+4*c)+1/10*A*sin(5*d*x+5*c)+78/5*x*d*(A+8/13*B+2/13*C)*cos(d*x+c)+sin(d*x +c)*(16/5*B+32/5*C+A))*a^4/d/(cos(3*d*x+3*c)+3*cos(d*x+c))
Time = 0.28 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.91 \[ \int \cos ^2(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {6 \, {\left (13 \, A + 8 \, B + 2 \, C\right )} a^{4} d x \cos \left (d x + c\right )^{3} + 3 \, {\left (8 \, A + 13 \, B + 12 \, C\right )} a^{4} \cos \left (d x + c\right )^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (8 \, A + 13 \, B + 12 \, C\right )} a^{4} \cos \left (d x + c\right )^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (3 \, A a^{4} \cos \left (d x + c\right )^{4} + 6 \, {\left (4 \, A + B\right )} a^{4} \cos \left (d x + c\right )^{3} + 2 \, {\left (3 \, A + 12 \, B + 20 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} + 3 \, {\left (B + 4 \, C\right )} a^{4} \cos \left (d x + c\right ) + 2 \, C a^{4}\right )} \sin \left (d x + c\right )}{12 \, d \cos \left (d x + c\right )^{3}} \]
integrate(cos(d*x+c)^2*(a+a*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2), x, algorithm="fricas")
1/12*(6*(13*A + 8*B + 2*C)*a^4*d*x*cos(d*x + c)^3 + 3*(8*A + 13*B + 12*C)* a^4*cos(d*x + c)^3*log(sin(d*x + c) + 1) - 3*(8*A + 13*B + 12*C)*a^4*cos(d *x + c)^3*log(-sin(d*x + c) + 1) + 2*(3*A*a^4*cos(d*x + c)^4 + 6*(4*A + B) *a^4*cos(d*x + c)^3 + 2*(3*A + 12*B + 20*C)*a^4*cos(d*x + c)^2 + 3*(B + 4* C)*a^4*cos(d*x + c) + 2*C*a^4)*sin(d*x + c))/(d*cos(d*x + c)^3)
Timed out. \[ \int \cos ^2(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]
Time = 0.25 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.53 \[ \int \cos ^2(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {3 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} + 72 \, {\left (d x + c\right )} A a^{4} + 48 \, {\left (d x + c\right )} B a^{4} + 4 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C a^{4} + 12 \, {\left (d x + c\right )} C a^{4} - 3 \, B a^{4} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 12 \, C a^{4} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 24 \, A a^{4} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 36 \, B a^{4} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 24 \, C a^{4} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 48 \, A a^{4} \sin \left (d x + c\right ) + 12 \, B a^{4} \sin \left (d x + c\right ) + 12 \, A a^{4} \tan \left (d x + c\right ) + 48 \, B a^{4} \tan \left (d x + c\right ) + 72 \, C a^{4} \tan \left (d x + c\right )}{12 \, d} \]
integrate(cos(d*x+c)^2*(a+a*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2), x, algorithm="maxima")
1/12*(3*(2*d*x + 2*c + sin(2*d*x + 2*c))*A*a^4 + 72*(d*x + c)*A*a^4 + 48*( d*x + c)*B*a^4 + 4*(tan(d*x + c)^3 + 3*tan(d*x + c))*C*a^4 + 12*(d*x + c)* C*a^4 - 3*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)) - 12*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)) + 24*A*a^4*(log(sin(d*x + c) + 1) - log(sin(d*x + c) - 1)) + 36*B*a^4*(log(sin(d*x + c) + 1) - lo g(sin(d*x + c) - 1)) + 24*C*a^4*(log(sin(d*x + c) + 1) - log(sin(d*x + c) - 1)) + 48*A*a^4*sin(d*x + c) + 12*B*a^4*sin(d*x + c) + 12*A*a^4*tan(d*x + c) + 48*B*a^4*tan(d*x + c) + 72*C*a^4*tan(d*x + c))/d
Time = 0.38 (sec) , antiderivative size = 347, normalized size of antiderivative = 1.66 \[ \int \cos ^2(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {3 \, {\left (13 \, A a^{4} + 8 \, B a^{4} + 2 \, C a^{4}\right )} {\left (d x + c\right )} + 3 \, {\left (8 \, A a^{4} + 13 \, B a^{4} + 12 \, C a^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 3 \, {\left (8 \, A a^{4} + 13 \, B a^{4} + 12 \, C a^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + \frac {6 \, {\left (7 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 9 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2}} - \frac {2 \, {\left (6 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 21 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 30 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 12 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 48 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 76 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 6 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 27 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 54 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{3}}}{6 \, d} \]
integrate(cos(d*x+c)^2*(a+a*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2), x, algorithm="giac")
1/6*(3*(13*A*a^4 + 8*B*a^4 + 2*C*a^4)*(d*x + c) + 3*(8*A*a^4 + 13*B*a^4 + 12*C*a^4)*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 3*(8*A*a^4 + 13*B*a^4 + 12* C*a^4)*log(abs(tan(1/2*d*x + 1/2*c) - 1)) + 6*(7*A*a^4*tan(1/2*d*x + 1/2*c )^3 + 2*B*a^4*tan(1/2*d*x + 1/2*c)^3 + 9*A*a^4*tan(1/2*d*x + 1/2*c) + 2*B* a^4*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 + 1)^2 - 2*(6*A*a^4*tan( 1/2*d*x + 1/2*c)^5 + 21*B*a^4*tan(1/2*d*x + 1/2*c)^5 + 30*C*a^4*tan(1/2*d* x + 1/2*c)^5 - 12*A*a^4*tan(1/2*d*x + 1/2*c)^3 - 48*B*a^4*tan(1/2*d*x + 1/ 2*c)^3 - 76*C*a^4*tan(1/2*d*x + 1/2*c)^3 + 6*A*a^4*tan(1/2*d*x + 1/2*c) + 27*B*a^4*tan(1/2*d*x + 1/2*c) + 54*C*a^4*tan(1/2*d*x + 1/2*c))/(tan(1/2*d* x + 1/2*c)^2 - 1)^3)/d
Time = 18.28 (sec) , antiderivative size = 625, normalized size of antiderivative = 2.99 \[ \int \cos ^2(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 {3\,A\,a^4\,\sin \left (2\,c+2\,d\,x\right )+\frac {33\,A\,a^4\,\sin \left (3\,c+3\,d\,x\right )}{32}+\frac {3\,A\,a^4\,\sin \left (4\,c+4\,d\,x\right )}{2}+\frac {3\,A\,a^4\,\sin \left (5\,c+5\,d\,x\right )}{32}+\frac {3\,B\,a^4\,\sin \left (2\,c+2\,d\,x\right )}{2}+3\,B\,a^4\,\sin \left (3\,c+3\,d\,x\right )+\frac {3\,B\,a^4\,\sin \left (4\,c+4\,d\,x\right )}{8}+3\,C\,a^4\,\sin \left (2\,c+2\,d\,x\right )+5\,C\,a^4\,\sin \left (3\,c+3\,d\,x\right )+\frac {15\,A\,a^4\,\sin \left (c+d\,x\right )}{16}+3\,B\,a^4\,\sin \left (c+d\,x\right )+6\,C\,a^4\,\sin \left (c+d\,x\right )+\frac {117\,A\,a^4\,\cos \left (c+d\,x\right )\,\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 )}{4}+18\,A\,a^4\,\cos \left (c+d\,x\right )\,\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 )+18\,B\,a^4\,\cos \left (c+d\,x\right )\,\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 )+\frac {117\,B\,a^4\,\cos \left (c+d\,x\right )\,\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 )}{4}+\frac {9\,C\,a^4\,\cos \left (c+d\,x\right )\,\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 )}{2}+27\,C\,a^4\,\cos \left (c+d\,x\right )\,\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 )+\frac {39\,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 )\,\cos \left (3\,c+3\,d\,x\right )}{4}+6\,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 )\,\cos \left (3\,c+3\,d\,x\right )+6\,B\,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 )\,\cos \left (3\,c+3\,d\,x\right )+\frac {39\,B\,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 )\,\cos \left (3\,c+3\,d\,x\right )}{4}+\frac {3\,C\,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 )\,\cos \left (3\,c+3\,d\,x\right )}{2}+9\,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 )\,\cos \left (3\,c+3\,d\,x\right )}{3\,d\,\left (\frac {3\,\cos \left (c+d\,x\right )}{4}+\frac {\cos \left (3\,c+3\,d\,x\right )}{4}\right )} \]
(3*A*a^4*sin(2*c + 2*d*x) + (33*A*a^4*sin(3*c + 3*d*x))/32 + (3*A*a^4*sin( 4*c + 4*d*x))/2 + (3*A*a^4*sin(5*c + 5*d*x))/32 + (3*B*a^4*sin(2*c + 2*d*x ))/2 + 3*B*a^4*sin(3*c + 3*d*x) + (3*B*a^4*sin(4*c + 4*d*x))/8 + 3*C*a^4*s in(2*c + 2*d*x) + 5*C*a^4*sin(3*c + 3*d*x) + (15*A*a^4*sin(c + d*x))/16 + 3*B*a^4*sin(c + d*x) + 6*C*a^4*sin(c + d*x) + (117*A*a^4*cos(c + d*x)*atan (sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)))/4 + 18*A*a^4*cos(c + d*x)*atanh(s in(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)) + 18*B*a^4*cos(c + d*x)*atan(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)) + (117*B*a^4*cos(c + d*x)*atanh(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)))/4 + (9*C*a^4*cos(c + d*x)*atan(sin(c/2 + (d* x)/2)/cos(c/2 + (d*x)/2)))/2 + 27*C*a^4*cos(c + d*x)*atanh(sin(c/2 + (d*x) /2)/cos(c/2 + (d*x)/2)) + (39*A*a^4*atan(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x )/2))*cos(3*c + 3*d*x))/4 + 6*A*a^4*atanh(sin(c/2 + (d*x)/2)/cos(c/2 + (d* x)/2))*cos(3*c + 3*d*x) + 6*B*a^4*atan(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/ 2))*cos(3*c + 3*d*x) + (39*B*a^4*atanh(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/ 2))*cos(3*c + 3*d*x))/4 + (3*C*a^4*atan(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x) /2))*cos(3*c + 3*d*x))/2 + 9*C*a^4*atanh(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x )/2))*cos(3*c + 3*d*x))/(3*d*((3*cos(c + d*x))/4 + cos(3*c + 3*d*x)/4))