Integrand size = 33, antiderivative size = 200 \[ \int \cos ^4(c+d x) (a+a \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {1}{8} a^4 (35 A+52 C) x+\frac {4 a^4 C \text {arctanh}(\sin (c+d x))}{d}+\frac {5 a^4 (7 A+4 C) \sin (c+d x)}{8 d}+\frac {a A \cos ^2(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{3 d}+\frac {A \cos ^3(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{4 d}+\frac {(7 A+4 C) \cos (c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{8 d}-\frac {(35 A-12 C) \left (a^4+a^4 \sec (c+d x)\right ) \sin (c+d x)}{24 d} \] Output:
1/8*a^4*(35*A+52*C)*x+4*a^4*C*arctanh(sin(d*x+c))/d+5/8*a^4*(7*A+4*C)*sin( d*x+c)/d+1/3*a*A*cos(d*x+c)^2*(a+a*sec(d*x+c))^3*sin(d*x+c)/d+1/4*A*cos(d* x+c)^3*(a+a*sec(d*x+c))^4*sin(d*x+c)/d+1/8*(7*A+4*C)*cos(d*x+c)*(a^2+a^2*s ec(d*x+c))^2*sin(d*x+c)/d-1/24*(35*A-12*C)*(a^4+a^4*sec(d*x+c))*sin(d*x+c) /d
Time = 6.79 (sec) , antiderivative size = 375, normalized size of antiderivative = 1.88 \[ \int \cos ^4(c+d x) (a+a \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {a^4 \cos ^2(c+d x) (1+\cos (c+d x))^4 \sec ^8\left (\frac {1}{2} (c+d x)\right ) \left (A+C \sec ^2(c+d x)\right ) \left (12 (35 A+52 C) x-\frac {384 C \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{d}+\frac {384 C \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}{d}+\frac {96 (7 A+4 C) \cos (d x) \sin (c)}{d}+\frac {24 (7 A+C) \cos (2 d x) \sin (2 c)}{d}+\frac {32 A \cos (3 d x) \sin (3 c)}{d}+\frac {3 A \cos (4 d x) \sin (4 c)}{d}+\frac {96 (7 A+4 C) \cos (c) \sin (d x)}{d}+\frac {24 (7 A+C) \cos (2 c) \sin (2 d x)}{d}+\frac {32 A \cos (3 c) \sin (3 d x)}{d}+\frac {3 A \cos (4 c) \sin (4 d x)}{d}+\frac {96 C \sin \left (\frac {d x}{2}\right )}{d \left (\cos \left (\frac {c}{2}\right )-\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {96 C \sin \left (\frac {d x}{2}\right )}{d \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}\right )}{768 (A+2 C+A \cos (2 (c+d x)))} \] Input:
Integrate[Cos[c + d*x]^4*(a + a*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2),x]
Output:
(a^4*Cos[c + d*x]^2*(1 + Cos[c + d*x])^4*Sec[(c + d*x)/2]^8*(A + C*Sec[c + d*x]^2)*(12*(35*A + 52*C)*x - (384*C*Log[Cos[(c + d*x)/2] - Sin[(c + d*x) /2]])/d + (384*C*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/d + (96*(7*A + 4*C)*Cos[d*x]*Sin[c])/d + (24*(7*A + C)*Cos[2*d*x]*Sin[2*c])/d + (32*A*Cos [3*d*x]*Sin[3*c])/d + (3*A*Cos[4*d*x]*Sin[4*c])/d + (96*(7*A + 4*C)*Cos[c] *Sin[d*x])/d + (24*(7*A + C)*Cos[2*c]*Sin[2*d*x])/d + (32*A*Cos[3*c]*Sin[3 *d*x])/d + (3*A*Cos[4*c]*Sin[4*d*x])/d + (96*C*Sin[(d*x)/2])/(d*(Cos[c/2] - Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + (96*C*Sin[(d*x)/2])/( d*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))))/(768*(A + 2*C + A*Cos[2*(c + d*x)]))
Time = 1.34 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.08, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {3042, 4575, 3042, 4505, 3042, 4505, 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.
| \(\displaystyle \int \cos ^4(c+d x) (a \sec (c+d x)+a)^4 \left (A+C \sec ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^4 \left (A+C \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^4}dx\) |
| \(\Big \downarrow \) 4575 |
| \(\displaystyle \frac {\int \cos ^3(c+d x) (\sec (c+d x) a+a)^4 (4 a A-a (A-4 C) \sec (c+d x))dx}{4 a}+\frac {A \sin (c+d x) \cos ^3(c+d x) (a \sec (c+d x)+a)^4}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^4 \left (4 a A-a (A-4 C) \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^3}dx}{4 a}+\frac {A \sin (c+d x) \cos ^3(c+d x) (a \sec (c+d x)+a)^4}{4 d}\) |
| \(\Big \downarrow \) 4505 |
| \(\displaystyle \frac {\frac {1}{3} \int \cos ^2(c+d x) (\sec (c+d x) a+a)^3 \left (3 a^2 (7 A+4 C)-a^2 (7 A-12 C) \sec (c+d x)\right )dx+\frac {4 a^2 A \sin (c+d x) \cos ^2(c+d x) (a \sec (c+d x)+a)^3}{3 d}}{4 a}+\frac {A \sin (c+d x) \cos ^3(c+d x) (a \sec (c+d x)+a)^4}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{3} \int \frac {\left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3 \left (3 a^2 (7 A+4 C)-a^2 (7 A-12 C) \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^2}dx+\frac {4 a^2 A \sin (c+d x) \cos ^2(c+d x) (a \sec (c+d x)+a)^3}{3 d}}{4 a}+\frac {A \sin (c+d x) \cos ^3(c+d x) (a \sec (c+d x)+a)^4}{4 d}\) |
| \(\Big \downarrow \) 4505 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {1}{2} \int \cos (c+d x) (\sec (c+d x) a+a)^2 \left (2 a^3 (35 A+36 C)-a^3 (35 A-12 C) \sec (c+d x)\right )dx+\frac {3 a^3 (7 A+4 C) \sin (c+d x) \cos (c+d x) (a \sec (c+d x)+a)^2}{2 d}\right )+\frac {4 a^2 A \sin (c+d x) \cos ^2(c+d x) (a \sec (c+d x)+a)^3}{3 d}}{4 a}+\frac {A \sin (c+d x) \cos ^3(c+d x) (a \sec (c+d x)+a)^4}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {1}{2} \int \frac {\left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2 \left (2 a^3 (35 A+36 C)-a^3 (35 A-12 C) \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {3 a^3 (7 A+4 C) \sin (c+d x) \cos (c+d x) (a \sec (c+d x)+a)^2}{2 d}\right )+\frac {4 a^2 A \sin (c+d x) \cos ^2(c+d x) (a \sec (c+d x)+a)^3}{3 d}}{4 a}+\frac {A \sin (c+d x) \cos ^3(c+d x) (a \sec (c+d x)+a)^4}{4 d}\) |
| \(\Big \downarrow \) 4506 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {1}{2} \left (\int 3 \cos (c+d x) (\sec (c+d x) a+a) \left (5 (7 A+4 C) a^4+32 C \sec (c+d x) a^4\right )dx-\frac {(35 A-12 C) \sin (c+d x) \left (a^5 \sec (c+d x)+a^5\right )}{d}\right )+\frac {3 a^3 (7 A+4 C) \sin (c+d x) \cos (c+d x) (a \sec (c+d x)+a)^2}{2 d}\right )+\frac {4 a^2 A \sin (c+d x) \cos ^2(c+d x) (a \sec (c+d x)+a)^3}{3 d}}{4 a}+\frac {A \sin (c+d x) \cos ^3(c+d x) (a \sec (c+d x)+a)^4}{4 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {1}{2} \left (3 \int \cos (c+d x) (\sec (c+d x) a+a) \left (5 (7 A+4 C) a^4+32 C \sec (c+d x) a^4\right )dx-\frac {(35 A-12 C) \sin (c+d x) \left (a^5 \sec (c+d x)+a^5\right )}{d}\right )+\frac {3 a^3 (7 A+4 C) \sin (c+d x) \cos (c+d x) (a \sec (c+d x)+a)^2}{2 d}\right )+\frac {4 a^2 A \sin (c+d x) \cos ^2(c+d x) (a \sec (c+d x)+a)^3}{3 d}}{4 a}+\frac {A \sin (c+d x) \cos ^3(c+d x) (a \sec (c+d x)+a)^4}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {1}{2} \left (3 \int \frac {\left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right ) \left (5 (7 A+4 C) a^4+32 C \csc \left (c+d x+\frac {\pi }{2}\right ) a^4\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )}dx-\frac {(35 A-12 C) \sin (c+d x) \left (a^5 \sec (c+d x)+a^5\right )}{d}\right )+\frac {3 a^3 (7 A+4 C) \sin (c+d x) \cos (c+d x) (a \sec (c+d x)+a)^2}{2 d}\right )+\frac {4 a^2 A \sin (c+d x) \cos ^2(c+d x) (a \sec (c+d x)+a)^3}{3 d}}{4 a}+\frac {A \sin (c+d x) \cos ^3(c+d x) (a \sec (c+d x)+a)^4}{4 d}\) |
| \(\Big \downarrow \) 4484 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {1}{2} \left (3 \left (\frac {5 a^5 (7 A+4 C) \sin (c+d x)}{d}-\int \left (-\left ((35 A+52 C) a^5\right )-32 C \sec (c+d x) a^5\right )dx\right )-\frac {(35 A-12 C) \sin (c+d x) \left (a^5 \sec (c+d x)+a^5\right )}{d}\right )+\frac {3 a^3 (7 A+4 C) \sin (c+d x) \cos (c+d x) (a \sec (c+d x)+a)^2}{2 d}\right )+\frac {4 a^2 A \sin (c+d x) \cos ^2(c+d x) (a \sec (c+d x)+a)^3}{3 d}}{4 a}+\frac {A \sin (c+d x) \cos ^3(c+d x) (a \sec (c+d x)+a)^4}{4 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {4 a^2 A \sin (c+d x) \cos ^2(c+d x) (a \sec (c+d x)+a)^3}{3 d}+\frac {1}{3} \left (\frac {1}{2} \left (3 \left (\frac {5 a^5 (7 A+4 C) \sin (c+d x)}{d}+a^5 x (35 A+52 C)+\frac {32 a^5 C \text {arctanh}(\sin (c+d x))}{d}\right )-\frac {(35 A-12 C) \sin (c+d x) \left (a^5 \sec (c+d x)+a^5\right )}{d}\right )+\frac {3 a^3 (7 A+4 C) \sin (c+d x) \cos (c+d x) (a \sec (c+d x)+a)^2}{2 d}\right )}{4 a}+\frac {A \sin (c+d x) \cos ^3(c+d x) (a \sec (c+d x)+a)^4}{4 d}\) |
Input:
Int[Cos[c + d*x]^4*(a + a*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2),x]
Output:
(A*Cos[c + d*x]^3*(a + a*Sec[c + d*x])^4*Sin[c + d*x])/(4*d) + ((4*a^2*A*C os[c + d*x]^2*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(3*d) + ((3*a^3*(7*A + 4*C)*Cos[c + d*x]*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(2*d) + (-(((35*A - 12*C)*(a^5 + a^5*Sec[c + d*x])*Sin[c + d*x])/d) + 3*(a^5*(35*A + 52*C)*x + (32*a^5*C*ArcTanh[Sin[c + d*x]])/d + (5*a^5*(7*A + 4*C)*Sin[c + d*x])/d) )/2)/3)/(4*a)
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[a*A*Cot [e + f*x]*(a + b*Csc[e + f*x])^(m - 1)*((d*Csc[e + f*x])^n/(f*n)), x] - Sim p[b/(a*d*n) Int[(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^(n + 1)*Sim p[a*A*(m - n - 1) - b*B*n - (a*B*n + A*b*(m + n))*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && EqQ[a^2 - b^2, 0 ] && GtQ[m, 1/2] && LtQ[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_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(d_. ))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[A*Co t[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 *(A*(m + n + 1) + C*n)*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, C, m}, x] && EqQ[a^2 - b^2, 0] && !LtQ[m, -2^(-1)] && (LtQ[n, -2^(-1)] || EqQ[m + n + 1, 0])
Time = 0.64 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.72
| method | result | size |
| parallelrisch | \(\frac {57 a^{4} \left (-\frac {256 C \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \cos \left (d x +c \right )}{57}+\frac {256 C \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \cos \left (d x +c \right )}{57}+\left (\frac {704 A}{171}+\frac {128 C}{57}\right ) \sin \left (2 d x +2 c \right )+\left (A +\frac {8 C}{57}\right ) \sin \left (3 d x +3 c \right )+\frac {32 A \sin \left (4 d x +4 c \right )}{171}+\frac {A \sin \left (5 d x +5 c \right )}{57}+\frac {280 d \left (A +\frac {52 C}{35}\right ) x \cos \left (d x +c \right )}{57}+\frac {56 \sin \left (d x +c \right ) \left (A +\frac {9 C}{7}\right )}{57}\right )}{64 d \cos \left (d x +c \right )}\) | \(144\) |
| derivativedivides | \(\frac {a^{4} A \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+a^{4} C \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {4 a^{4} A \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+4 a^{4} C \sin \left (d x +c \right )+6 a^{4} A \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+6 a^{4} C \left (d x +c \right )+4 a^{4} A \sin \left (d x +c \right )+4 a^{4} C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+a^{4} A \left (d x +c \right )+a^{4} C \tan \left (d x +c \right )}{d}\) | \(197\) |
| default | \(\frac {a^{4} A \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+a^{4} C \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {4 a^{4} A \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+4 a^{4} C \sin \left (d x +c \right )+6 a^{4} A \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+6 a^{4} C \left (d x +c \right )+4 a^{4} A \sin \left (d x +c \right )+4 a^{4} C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+a^{4} A \left (d x +c \right )+a^{4} C \tan \left (d x +c \right )}{d}\) | \(197\) |
| risch | \(\frac {35 a^{4} A x}{8}+\frac {13 a^{4} x C}{2}-\frac {7 i a^{4} A \,{\mathrm e}^{2 i \left (d x +c \right )}}{8 d}-\frac {i {\mathrm e}^{2 i \left (d x +c \right )} a^{4} C}{8 d}-\frac {7 i a^{4} A \,{\mathrm e}^{i \left (d x +c \right )}}{2 d}-\frac {2 i {\mathrm e}^{i \left (d x +c \right )} a^{4} C}{d}+\frac {7 i a^{4} A \,{\mathrm e}^{-i \left (d x +c \right )}}{2 d}+\frac {2 i {\mathrm e}^{-i \left (d x +c \right )} a^{4} C}{d}+\frac {7 i a^{4} A \,{\mathrm e}^{-2 i \left (d x +c \right )}}{8 d}+\frac {i {\mathrm e}^{-2 i \left (d x +c \right )} a^{4} C}{8 d}+\frac {2 i a^{4} C}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}+\frac {4 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{d}-\frac {4 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{d}+\frac {a^{4} A \sin \left (4 d x +4 c \right )}{32 d}+\frac {a^{4} A \sin \left (3 d x +3 c \right )}{3 d}\) | \(271\) |
Input:
int(cos(d*x+c)^4*(a+a*sec(d*x+c))^4*(A+C*sec(d*x+c)^2),x,method=_RETURNVER BOSE)
Output:
57/64*a^4*(-256/57*C*ln(tan(1/2*d*x+1/2*c)-1)*cos(d*x+c)+256/57*C*ln(tan(1 /2*d*x+1/2*c)+1)*cos(d*x+c)+(704/171*A+128/57*C)*sin(2*d*x+2*c)+(A+8/57*C) *sin(3*d*x+3*c)+32/171*A*sin(4*d*x+4*c)+1/57*A*sin(5*d*x+5*c)+280/57*d*(A+ 52/35*C)*x*cos(d*x+c)+56/57*sin(d*x+c)*(A+9/7*C))/d/cos(d*x+c)
Time = 0.09 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.79 \[ \int \cos ^4(c+d x) (a+a \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {3 \, {\left (35 \, A + 52 \, C\right )} a^{4} d x \cos \left (d x + c\right ) + 48 \, C a^{4} \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) - 48 \, C a^{4} \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) + {\left (6 \, A a^{4} \cos \left (d x + c\right )^{4} + 32 \, A a^{4} \cos \left (d x + c\right )^{3} + 3 \, {\left (27 \, A + 4 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} + 32 \, {\left (5 \, A + 3 \, C\right )} a^{4} \cos \left (d x + c\right ) + 24 \, C a^{4}\right )} \sin \left (d x + c\right )}{24 \, d \cos \left (d x + c\right )} \] Input:
integrate(cos(d*x+c)^4*(a+a*sec(d*x+c))^4*(A+C*sec(d*x+c)^2),x, algorithm= "fricas")
Output:
1/24*(3*(35*A + 52*C)*a^4*d*x*cos(d*x + c) + 48*C*a^4*cos(d*x + c)*log(sin (d*x + c) + 1) - 48*C*a^4*cos(d*x + c)*log(-sin(d*x + c) + 1) + (6*A*a^4*c os(d*x + c)^4 + 32*A*a^4*cos(d*x + c)^3 + 3*(27*A + 4*C)*a^4*cos(d*x + c)^ 2 + 32*(5*A + 3*C)*a^4*cos(d*x + c) + 24*C*a^4)*sin(d*x + c))/(d*cos(d*x + c))
Timed out. \[ \int \cos ^4(c+d x) (a+a \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)**4*(a+a*sec(d*x+c))**4*(A+C*sec(d*x+c)**2),x)
Output:
Timed out
Time = 0.04 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.97 \[ \int \cos ^4(c+d x) (a+a \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=-\frac {128 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{4} - 3 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} - 144 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} - 96 \, {\left (d x + c\right )} A a^{4} - 24 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{4} - 576 \, {\left (d x + c\right )} C a^{4} - 192 \, C a^{4} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 384 \, A a^{4} \sin \left (d x + c\right ) - 384 \, C a^{4} \sin \left (d x + c\right ) - 96 \, C a^{4} \tan \left (d x + c\right )}{96 \, d} \] Input:
integrate(cos(d*x+c)^4*(a+a*sec(d*x+c))^4*(A+C*sec(d*x+c)^2),x, algorithm= "maxima")
Output:
-1/96*(128*(sin(d*x + c)^3 - 3*sin(d*x + c))*A*a^4 - 3*(12*d*x + 12*c + si n(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*A*a^4 - 144*(2*d*x + 2*c + sin(2*d*x + 2*c))*A*a^4 - 96*(d*x + c)*A*a^4 - 24*(2*d*x + 2*c + sin(2*d*x + 2*c))*C *a^4 - 576*(d*x + c)*C*a^4 - 192*C*a^4*(log(sin(d*x + c) + 1) - log(sin(d* x + c) - 1)) - 384*A*a^4*sin(d*x + c) - 384*C*a^4*sin(d*x + c) - 96*C*a^4* tan(d*x + c))/d
Time = 0.36 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.22 \[ \int \cos ^4(c+d x) (a+a \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {96 \, C a^{4} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 96 \, C a^{4} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {48 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1} + 3 \, {\left (35 \, A a^{4} + 52 \, C a^{4}\right )} {\left (d x + c\right )} + \frac {2 \, {\left (105 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 84 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 385 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 276 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 511 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 300 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 279 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 108 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{4}}}{24 \, d} \] Input:
integrate(cos(d*x+c)^4*(a+a*sec(d*x+c))^4*(A+C*sec(d*x+c)^2),x, algorithm= "giac")
Output:
1/24*(96*C*a^4*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 96*C*a^4*log(abs(tan(1 /2*d*x + 1/2*c) - 1)) - 48*C*a^4*tan(1/2*d*x + 1/2*c)/(tan(1/2*d*x + 1/2*c )^2 - 1) + 3*(35*A*a^4 + 52*C*a^4)*(d*x + c) + 2*(105*A*a^4*tan(1/2*d*x + 1/2*c)^7 + 84*C*a^4*tan(1/2*d*x + 1/2*c)^7 + 385*A*a^4*tan(1/2*d*x + 1/2*c )^5 + 276*C*a^4*tan(1/2*d*x + 1/2*c)^5 + 511*A*a^4*tan(1/2*d*x + 1/2*c)^3 + 300*C*a^4*tan(1/2*d*x + 1/2*c)^3 + 279*A*a^4*tan(1/2*d*x + 1/2*c) + 108* C*a^4*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 + 1)^4)/d
Time = 11.73 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.17 \[ \int \cos ^4(c+d x) (a+a \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {20\,A\,a^4\,\sin \left (c+d\,x\right )}{3\,d}+\frac {4\,C\,a^4\,\sin \left (c+d\,x\right )}{d}+\frac {35\,A\,a^4\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{4\,d}+\frac {13\,C\,a^4\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {8\,C\,a^4\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {4\,A\,a^4\,{\cos \left (c+d\,x\right )}^2\,\sin \left (c+d\,x\right )}{3\,d}+\frac {A\,a^4\,{\cos \left (c+d\,x\right )}^3\,\sin \left (c+d\,x\right )}{4\,d}+\frac {C\,a^4\,\sin \left (c+d\,x\right )}{d\,\cos \left (c+d\,x\right )}+\frac {27\,A\,a^4\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )}{8\,d}+\frac {C\,a^4\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )}{2\,d} \] Input:
int(cos(c + d*x)^4*(A + C/cos(c + d*x)^2)*(a + a/cos(c + d*x))^4,x)
Output:
(20*A*a^4*sin(c + d*x))/(3*d) + (4*C*a^4*sin(c + d*x))/d + (35*A*a^4*atan( sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)))/(4*d) + (13*C*a^4*atan(sin(c/2 + ( d*x)/2)/cos(c/2 + (d*x)/2)))/d + (8*C*a^4*atanh(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)))/d + (4*A*a^4*cos(c + d*x)^2*sin(c + d*x))/(3*d) + (A*a^4*cos (c + d*x)^3*sin(c + d*x))/(4*d) + (C*a^4*sin(c + d*x))/(d*cos(c + d*x)) + (27*A*a^4*cos(c + d*x)*sin(c + d*x))/(8*d) + (C*a^4*cos(c + d*x)*sin(c + d *x))/(2*d)
Time = 0.16 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.00 \[ \int \cos ^4(c+d x) (a+a \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {a^{4} \left (-96 \cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) c +96 \cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) c -32 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} a +192 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a +96 \cos \left (d x +c \right ) \sin \left (d x +c \right ) c +105 \cos \left (d x +c \right ) a c +105 \cos \left (d x +c \right ) a d x +156 \cos \left (d x +c \right ) c^{2}+156 \cos \left (d x +c \right ) c d x +6 \sin \left (d x +c \right )^{5} a -93 \sin \left (d x +c \right )^{3} a -12 \sin \left (d x +c \right )^{3} c +87 \sin \left (d x +c \right ) a +36 \sin \left (d x +c \right ) c \right )}{24 \cos \left (d x +c \right ) d} \] Input:
int(cos(d*x+c)^4*(a+a*sec(d*x+c))^4*(A+C*sec(d*x+c)^2),x)
Output:
(a**4*( - 96*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*c + 96*cos(c + d*x)*lo g(tan((c + d*x)/2) + 1)*c - 32*cos(c + d*x)*sin(c + d*x)**3*a + 192*cos(c + d*x)*sin(c + d*x)*a + 96*cos(c + d*x)*sin(c + d*x)*c + 105*cos(c + d*x)* a*c + 105*cos(c + d*x)*a*d*x + 156*cos(c + d*x)*c**2 + 156*cos(c + d*x)*c* d*x + 6*sin(c + d*x)**5*a - 93*sin(c + d*x)**3*a - 12*sin(c + d*x)**3*c + 87*sin(c + d*x)*a + 36*sin(c + d*x)*c))/(24*cos(c + d*x)*d)