Integrand size = 33, antiderivative size = 246 \[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {1}{8} \left (8 A b^4+24 a^2 b^2 (A+2 C)+a^4 (3 A+4 C)\right ) x+\frac {4 a b^3 C \text {arctanh}(\sin (c+d x))}{d}+\frac {a b \left (12 A b^2+a^2 (23 A+36 C)\right ) \sin (c+d x)}{12 d}+\frac {\left (4 A b^2+a^2 (3 A+4 C)\right ) \cos (c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{8 d}+\frac {A b \cos ^2(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{3 d}+\frac {A \cos ^3(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{4 d}-\frac {b^2 \left (2 b^2 (13 A-12 C)+3 a^2 (3 A+4 C)\right ) \tan (c+d x)}{24 d} \]
1/8*(8*A*b^4+24*a^2*b^2*(A+2*C)+a^4*(3*A+4*C))*x+4*a*b^3*C*arctanh(sin(d*x +c))/d+1/12*a*b*(12*A*b^2+a^2*(23*A+36*C))*sin(d*x+c)/d+1/8*(4*A*b^2+a^2*( 3*A+4*C))*cos(d*x+c)*(a+b*sec(d*x+c))^2*sin(d*x+c)/d+1/3*A*b*cos(d*x+c)^2* (a+b*sec(d*x+c))^3*sin(d*x+c)/d+1/4*A*cos(d*x+c)^3*(a+b*sec(d*x+c))^4*sin( d*x+c)/d-1/24*b^2*(2*b^2*(13*A-12*C)+3*a^2*(3*A+4*C))*tan(d*x+c)/d
Time = 3.40 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.10 \[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {12 \left (8 A b^4+24 a^2 b^2 (A+2 C)+a^4 (3 A+4 C)\right ) (c+d x)-384 a b^3 C \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+384 a b^3 C \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+\frac {96 b^4 C \sin \left (\frac {1}{2} (c+d x)\right )}{\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )}+\frac {96 b^4 C \sin \left (\frac {1}{2} (c+d x)\right )}{\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )}+96 a b \left (4 A b^2+a^2 (3 A+4 C)\right ) \sin (c+d x)+24 a^2 \left (6 A b^2+a^2 (A+C)\right ) \sin (2 (c+d x))+32 a^3 A b \sin (3 (c+d x))+3 a^4 A \sin (4 (c+d x))}{96 d} \]
(12*(8*A*b^4 + 24*a^2*b^2*(A + 2*C) + a^4*(3*A + 4*C))*(c + d*x) - 384*a*b ^3*C*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 384*a*b^3*C*Log[Cos[(c + d *x)/2] + Sin[(c + d*x)/2]] + (96*b^4*C*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) + (96*b^4*C*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin [(c + d*x)/2]) + 96*a*b*(4*A*b^2 + a^2*(3*A + 4*C))*Sin[c + d*x] + 24*a^2* (6*A*b^2 + a^2*(A + C))*Sin[2*(c + d*x)] + 32*a^3*A*b*Sin[3*(c + d*x)] + 3 *a^4*A*Sin[4*(c + d*x)])/(96*d)
Time = 1.95 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.04, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {3042, 4583, 3042, 4582, 3042, 4582, 3042, 4564, 3042, 4535, 24, 3042, 4533, 3042, 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.
| \(\displaystyle \int \cos ^4(c+d x) (a+b \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^4 \left (A+C \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^4}dx\) |
| \(\Big \downarrow \) 4583 |
| \(\displaystyle \frac {1}{4} \int \cos ^3(c+d x) (a+b \sec (c+d x))^3 \left (-b (A-4 C) \sec ^2(c+d x)+a (3 A+4 C) \sec (c+d x)+4 A b\right )dx+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^4}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left (-b (A-4 C) \csc \left (c+d x+\frac {\pi }{2}\right )^2+a (3 A+4 C) \csc \left (c+d x+\frac {\pi }{2}\right )+4 A b\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^3}dx+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^4}{4 d}\) |
| \(\Big \downarrow \) 4582 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \int \cos ^2(c+d x) (a+b \sec (c+d x))^2 \left (-b^2 (7 A-12 C) \sec ^2(c+d x)+2 a b (7 A+12 C) \sec (c+d x)+3 \left ((3 A+4 C) a^2+4 A b^2\right )\right )dx+\frac {4 A b \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^3}{3 d}\right )+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^4}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left (-b^2 (7 A-12 C) \csc \left (c+d x+\frac {\pi }{2}\right )^2+2 a b (7 A+12 C) \csc \left (c+d x+\frac {\pi }{2}\right )+3 \left ((3 A+4 C) a^2+4 A b^2\right )\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^2}dx+\frac {4 A b \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^3}{3 d}\right )+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^4}{4 d}\) |
| \(\Big \downarrow \) 4582 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \int \cos (c+d x) (a+b \sec (c+d x)) \left (-b \left (3 (3 A+4 C) a^2+2 b^2 (13 A-12 C)\right ) \sec ^2(c+d x)+a \left (3 (3 A+4 C) a^2+2 b^2 (13 A+36 C)\right ) \sec (c+d x)+2 \left (12 A b^3+\frac {1}{2} a^2 (46 A b+72 C b)\right )\right )dx+\frac {3 \left (a^2 (3 A+4 C)+4 A b^2\right ) \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^2}{2 d}\right )+\frac {4 A b \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^3}{3 d}\right )+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^4}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right ) \left (-b \left (3 (3 A+4 C) a^2+2 b^2 (13 A-12 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+a \left (3 (3 A+4 C) a^2+2 b^2 (13 A+36 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+2 \left (12 A b^3+\frac {1}{2} a^2 (46 A b+72 C b)\right )\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {3 \left (a^2 (3 A+4 C)+4 A b^2\right ) \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^2}{2 d}\right )+\frac {4 A b \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^3}{3 d}\right )+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^4}{4 d}\) |
| \(\Big \downarrow \) 4564 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (\int \cos (c+d x) \left (96 a C \sec ^2(c+d x) b^3+2 a \left (12 A b^3+\frac {1}{2} a^2 (46 A b+72 C b)\right )+3 \left ((3 A+4 C) a^4+24 b^2 (A+2 C) a^2+8 A b^4\right ) \sec (c+d x)\right )dx-\frac {b^2 \left (3 a^2 (3 A+4 C)+2 b^2 (13 A-12 C)\right ) \tan (c+d x)}{d}\right )+\frac {3 \left (a^2 (3 A+4 C)+4 A b^2\right ) \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^2}{2 d}\right )+\frac {4 A b \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^3}{3 d}\right )+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^4}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (\int \frac {96 a C \csc \left (c+d x+\frac {\pi }{2}\right )^2 b^3+2 a \left (12 A b^3+\frac {1}{2} a^2 (46 A b+72 C b)\right )+3 \left ((3 A+4 C) a^4+24 b^2 (A+2 C) a^2+8 A b^4\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )}dx-\frac {b^2 \left (3 a^2 (3 A+4 C)+2 b^2 (13 A-12 C)\right ) \tan (c+d x)}{d}\right )+\frac {3 \left (a^2 (3 A+4 C)+4 A b^2\right ) \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^2}{2 d}\right )+\frac {4 A b \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^3}{3 d}\right )+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^4}{4 d}\) |
| \(\Big \downarrow \) 4535 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (\int \cos (c+d x) \left (96 a C \sec ^2(c+d x) b^3+2 a \left (12 A b^3+\frac {1}{2} a^2 (46 A b+72 C b)\right )\right )dx+3 \left (a^4 (3 A+4 C)+24 a^2 b^2 (A+2 C)+8 A b^4\right ) \int 1dx-\frac {b^2 \left (3 a^2 (3 A+4 C)+2 b^2 (13 A-12 C)\right ) \tan (c+d x)}{d}\right )+\frac {3 \left (a^2 (3 A+4 C)+4 A b^2\right ) \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^2}{2 d}\right )+\frac {4 A b \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^3}{3 d}\right )+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^4}{4 d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (\int \cos (c+d x) \left (96 a C \sec ^2(c+d x) b^3+2 a \left (12 A b^3+\frac {1}{2} a^2 (46 A b+72 C b)\right )\right )dx-\frac {b^2 \left (3 a^2 (3 A+4 C)+2 b^2 (13 A-12 C)\right ) \tan (c+d x)}{d}+3 x \left (a^4 (3 A+4 C)+24 a^2 b^2 (A+2 C)+8 A b^4\right )\right )+\frac {3 \left (a^2 (3 A+4 C)+4 A b^2\right ) \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^2}{2 d}\right )+\frac {4 A b \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^3}{3 d}\right )+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^4}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (\int \frac {96 a C \csc \left (c+d x+\frac {\pi }{2}\right )^2 b^3+2 a \left (12 A b^3+\frac {1}{2} a^2 (46 A b+72 C b)\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )}dx-\frac {b^2 \left (3 a^2 (3 A+4 C)+2 b^2 (13 A-12 C)\right ) \tan (c+d x)}{d}+3 x \left (a^4 (3 A+4 C)+24 a^2 b^2 (A+2 C)+8 A b^4\right )\right )+\frac {3 \left (a^2 (3 A+4 C)+4 A b^2\right ) \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^2}{2 d}\right )+\frac {4 A b \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^3}{3 d}\right )+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^4}{4 d}\) |
| \(\Big \downarrow \) 4533 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (96 a b^3 C \int \sec (c+d x)dx+\frac {2 a b \left (a^2 (23 A+36 C)+12 A b^2\right ) \sin (c+d x)}{d}-\frac {b^2 \left (3 a^2 (3 A+4 C)+2 b^2 (13 A-12 C)\right ) \tan (c+d x)}{d}+3 x \left (a^4 (3 A+4 C)+24 a^2 b^2 (A+2 C)+8 A b^4\right )\right )+\frac {3 \left (a^2 (3 A+4 C)+4 A b^2\right ) \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^2}{2 d}\right )+\frac {4 A b \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^3}{3 d}\right )+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^4}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (96 a b^3 C \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+\frac {2 a b \left (a^2 (23 A+36 C)+12 A b^2\right ) \sin (c+d x)}{d}-\frac {b^2 \left (3 a^2 (3 A+4 C)+2 b^2 (13 A-12 C)\right ) \tan (c+d x)}{d}+3 x \left (a^4 (3 A+4 C)+24 a^2 b^2 (A+2 C)+8 A b^4\right )\right )+\frac {3 \left (a^2 (3 A+4 C)+4 A b^2\right ) \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^2}{2 d}\right )+\frac {4 A b \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^3}{3 d}\right )+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^4}{4 d}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {3 \left (a^2 (3 A+4 C)+4 A b^2\right ) \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^2}{2 d}+\frac {1}{2} \left (\frac {2 a b \left (a^2 (23 A+36 C)+12 A b^2\right ) \sin (c+d x)}{d}-\frac {b^2 \left (3 a^2 (3 A+4 C)+2 b^2 (13 A-12 C)\right ) \tan (c+d x)}{d}+3 x \left (a^4 (3 A+4 C)+24 a^2 b^2 (A+2 C)+8 A b^4\right )+\frac {96 a b^3 C \text {arctanh}(\sin (c+d x))}{d}\right )\right )+\frac {4 A b \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^3}{3 d}\right )+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^4}{4 d}\) |
(A*Cos[c + d*x]^3*(a + b*Sec[c + d*x])^4*Sin[c + d*x])/(4*d) + ((4*A*b*Cos [c + d*x]^2*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(3*d) + ((3*(4*A*b^2 + a^ 2*(3*A + 4*C))*Cos[c + d*x]*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(2*d) + ( 3*(8*A*b^4 + 24*a^2*b^2*(A + 2*C) + a^4*(3*A + 4*C))*x + (96*a*b^3*C*ArcTa nh[Sin[c + d*x]])/d + (2*a*b*(12*A*b^2 + a^2*(23*A + 36*C))*Sin[c + d*x])/ d - (b^2*(2*b^2*(13*A - 12*C) + 3*a^2*(3*A + 4*C))*Tan[c + d*x])/d)/2)/3)/ 4
3.7.69.3.1 Defintions of rubi rules used
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_.))^(m_.)*(csc[(e_.) + (f_.)*(x_)]^2*(C_.) + (A_)), x_Symbol] :> Simp[A*Cot[e + f*x]*((b*Csc[e + f*x])^m/(f*m)), x] + Simp[(C*m + A*(m + 1))/(b^2*m) Int[(b*Csc[e + f*x])^(m + 2), x], x] /; Fr eeQ[{b, e, f, A, C}, x] && NeQ[C*m + A*(m + 1), 0] && LeQ[m, -1]
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*((A_.) + csc[(e_.) + (f_.)*(x_)]* (B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.)), x_Symbol] :> Simp[B/b Int[(b*Cs c[e + f*x])^(m + 1), x], x] + Int[(b*Csc[e + f*x])^m*(A + C*Csc[e + f*x]^2) , x] /; FreeQ[{b, e, f, A, B, C, m}, x]
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_)), x_Symbol] :> Simp[(-b)*C*Csc[e + f*x]*Cot[e + f*x]*((d*Csc[e + f*x])^ n/(f*(n + 2))), x] + Simp[1/(n + 2) Int[(d*Csc[e + f*x])^n*Simp[A*a*(n + 2) + (B*a*(n + 2) + b*(C*(n + 1) + A*(n + 2)))*Csc[e + f*x] + (a*C + B*b)*( n + 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C, n}, x] && !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/(d*n) Int[(a + b*Csc[e + f*x])^(m - 1)*(d* Csc[e + f*x])^(n + 1)*Simp[A*b*m - a*B*n - (b*B*n + a*(C*n + A*(n + 1)))*Cs c[e + f*x] - b*(C*n + A*(m + n + 1))*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a , b, d, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && LeQ[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/( d*n) Int[(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^(n + 1)*Simp[A*b*m - a*(C*n + A*(n + 1))*Csc[e + f*x] - b*(C*n + A*(m + n + 1))*Csc[e + f*x]^ 2, x], x], x] /; FreeQ[{a, b, d, e, f, A, C}, x] && NeQ[a^2 - b^2, 0] && Gt Q[m, 0] && LeQ[n, -1]
Time = 0.89 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.84
| method | result | size |
| derivativedivides | \(\frac {a^{4} A \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+a^{4} C \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {4 A \,a^{3} b \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+4 a^{3} b C \sin \left (d x +c \right )+6 A \,a^{2} b^{2} \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+6 C \,a^{2} b^{2} \left (d x +c \right )+4 a A \,b^{3} \sin \left (d x +c \right )+4 C a \,b^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+A \,b^{4} \left (d x +c \right )+C \tan \left (d x +c \right ) b^{4}}{d}\) | \(207\) |
| default | \(\frac {a^{4} A \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+a^{4} C \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {4 A \,a^{3} b \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+4 a^{3} b C \sin \left (d x +c \right )+6 A \,a^{2} b^{2} \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+6 C \,a^{2} b^{2} \left (d x +c \right )+4 a A \,b^{3} \sin \left (d x +c \right )+4 C a \,b^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+A \,b^{4} \left (d x +c \right )+C \tan \left (d x +c \right ) b^{4}}{d}\) | \(207\) |
| parallelrisch | \(\frac {-768 C \cos \left (d x +c \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) a \,b^{3}+768 C \cos \left (d x +c \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) a \,b^{3}+320 a b \left (a^{2} \left (A +\frac {6 C}{5}\right )+\frac {6 A \,b^{2}}{5}\right ) \sin \left (2 d x +2 c \right )+\left (\left (27 A +24 C \right ) a^{4}+144 A \,a^{2} b^{2}\right ) \sin \left (3 d x +3 c \right )+32 A \,a^{3} b \sin \left (4 d x +4 c \right )+3 a^{4} A \sin \left (5 d x +5 c \right )+72 x d \left (\left (A +\frac {4 C}{3}\right ) a^{4}+8 a^{2} b^{2} \left (A +2 C \right )+\frac {8 A \,b^{4}}{3}\right ) \cos \left (d x +c \right )+24 \sin \left (d x +c \right ) \left (\left (A +C \right ) a^{4}+6 A \,a^{2} b^{2}+8 C \,b^{4}\right )}{192 d \cos \left (d x +c \right )}\) | \(226\) |
| risch | \(\frac {3 a^{4} A x}{8}+3 A \,a^{2} b^{2} x +x A \,b^{4}+\frac {a^{4} x C}{2}+6 x C \,a^{2} b^{2}+\frac {i {\mathrm e}^{-2 i \left (d x +c \right )} a^{4} C}{8 d}+\frac {2 i {\mathrm e}^{-i \left (d x +c \right )} a A \,b^{3}}{d}-\frac {2 i {\mathrm e}^{i \left (d x +c \right )} a A \,b^{3}}{d}-\frac {3 i A \,a^{3} b \,{\mathrm e}^{i \left (d x +c \right )}}{2 d}+\frac {3 i {\mathrm e}^{-2 i \left (d x +c \right )} A \,a^{2} b^{2}}{4 d}-\frac {i a^{4} A \,{\mathrm e}^{2 i \left (d x +c \right )}}{8 d}+\frac {3 i A \,a^{3} b \,{\mathrm e}^{-i \left (d x +c \right )}}{2 d}-\frac {2 i {\mathrm e}^{i \left (d x +c \right )} a^{3} b C}{d}-\frac {i {\mathrm e}^{2 i \left (d x +c \right )} a^{4} C}{8 d}+\frac {2 i {\mathrm e}^{-i \left (d x +c \right )} a^{3} b C}{d}-\frac {3 i {\mathrm e}^{2 i \left (d x +c \right )} A \,a^{2} b^{2}}{4 d}+\frac {i a^{4} A \,{\mathrm e}^{-2 i \left (d x +c \right )}}{8 d}+\frac {2 i C \,b^{4}}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}-\frac {4 a \,b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{d}+\frac {4 a \,b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{d}+\frac {a^{4} A \sin \left (4 d x +4 c \right )}{32 d}+\frac {A \,a^{3} b \sin \left (3 d x +3 c \right )}{3 d}\) | \(388\) |
1/d*(a^4*A*(1/4*(cos(d*x+c)^3+3/2*cos(d*x+c))*sin(d*x+c)+3/8*d*x+3/8*c)+a^ 4*C*(1/2*sin(d*x+c)*cos(d*x+c)+1/2*d*x+1/2*c)+4/3*A*a^3*b*(2+cos(d*x+c)^2) *sin(d*x+c)+4*a^3*b*C*sin(d*x+c)+6*A*a^2*b^2*(1/2*sin(d*x+c)*cos(d*x+c)+1/ 2*d*x+1/2*c)+6*C*a^2*b^2*(d*x+c)+4*a*A*b^3*sin(d*x+c)+4*C*a*b^3*ln(sec(d*x +c)+tan(d*x+c))+A*b^4*(d*x+c)+C*tan(d*x+c)*b^4)
Time = 0.29 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.83 \[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {48 \, C a b^{3} \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) - 48 \, C a b^{3} \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) + 3 \, {\left ({\left (3 \, A + 4 \, C\right )} a^{4} + 24 \, {\left (A + 2 \, C\right )} a^{2} b^{2} + 8 \, A b^{4}\right )} d x \cos \left (d x + c\right ) + {\left (6 \, A a^{4} \cos \left (d x + c\right )^{4} + 32 \, A a^{3} b \cos \left (d x + c\right )^{3} + 24 \, C b^{4} + 3 \, {\left ({\left (3 \, A + 4 \, C\right )} a^{4} + 24 \, A a^{2} b^{2}\right )} \cos \left (d x + c\right )^{2} + 32 \, {\left ({\left (2 \, A + 3 \, C\right )} a^{3} b + 3 \, A a b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{24 \, d \cos \left (d x + c\right )} \]
1/24*(48*C*a*b^3*cos(d*x + c)*log(sin(d*x + c) + 1) - 48*C*a*b^3*cos(d*x + c)*log(-sin(d*x + c) + 1) + 3*((3*A + 4*C)*a^4 + 24*(A + 2*C)*a^2*b^2 + 8 *A*b^4)*d*x*cos(d*x + c) + (6*A*a^4*cos(d*x + c)^4 + 32*A*a^3*b*cos(d*x + c)^3 + 24*C*b^4 + 3*((3*A + 4*C)*a^4 + 24*A*a^2*b^2)*cos(d*x + c)^2 + 32*( (2*A + 3*C)*a^3*b + 3*A*a*b^3)*cos(d*x + c))*sin(d*x + c))/(d*cos(d*x + c) )
Timed out. \[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]
Time = 0.20 (sec) , antiderivative size = 204, normalized size of antiderivative = 0.83 \[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {3 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} + 24 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{4} - 128 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{3} b + 144 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{2} b^{2} + 576 \, {\left (d x + c\right )} C a^{2} b^{2} + 96 \, {\left (d x + c\right )} A b^{4} + 192 \, C a b^{3} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 384 \, C a^{3} b \sin \left (d x + c\right ) + 384 \, A a b^{3} \sin \left (d x + c\right ) + 96 \, C b^{4} \tan \left (d x + c\right )}{96 \, d} \]
1/96*(3*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*A*a^4 + 24 *(2*d*x + 2*c + sin(2*d*x + 2*c))*C*a^4 - 128*(sin(d*x + c)^3 - 3*sin(d*x + c))*A*a^3*b + 144*(2*d*x + 2*c + sin(2*d*x + 2*c))*A*a^2*b^2 + 576*(d*x + c)*C*a^2*b^2 + 96*(d*x + c)*A*b^4 + 192*C*a*b^3*(log(sin(d*x + c) + 1) - log(sin(d*x + c) - 1)) + 384*C*a^3*b*sin(d*x + c) + 384*A*a*b^3*sin(d*x + c) + 96*C*b^4*tan(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 558 vs. \(2 (234) = 468\).
Time = 0.39 (sec) , antiderivative size = 558, normalized size of antiderivative = 2.27 \[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {96 \, C a b^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 96 \, C a b^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {48 \, C b^{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 (3 \, A a^{4} + 4 \, C a^{4} + 24 \, A a^{2} b^{2} + 48 \, C a^{2} b^{2} + 8 \, A b^{4}\right )} {\left (d x + c\right )} - \frac {2 \, {\left (15 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 12 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 96 \, A a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 96 \, C a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 72 \, A a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 96 \, A a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 9 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 12 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 160 \, A a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 288 \, C a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 72 \, A a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 288 \, A a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 9 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 12 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 160 \, A a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 288 \, C a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 72 \, A a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 288 \, A a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 15 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 12 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 96 \, A a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 96 \, C a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 72 \, A a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 96 \, A a b^{3} \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} \]
1/24*(96*C*a*b^3*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 96*C*a*b^3*log(abs(t an(1/2*d*x + 1/2*c) - 1)) - 48*C*b^4*tan(1/2*d*x + 1/2*c)/(tan(1/2*d*x + 1 /2*c)^2 - 1) + 3*(3*A*a^4 + 4*C*a^4 + 24*A*a^2*b^2 + 48*C*a^2*b^2 + 8*A*b^ 4)*(d*x + c) - 2*(15*A*a^4*tan(1/2*d*x + 1/2*c)^7 + 12*C*a^4*tan(1/2*d*x + 1/2*c)^7 - 96*A*a^3*b*tan(1/2*d*x + 1/2*c)^7 - 96*C*a^3*b*tan(1/2*d*x + 1 /2*c)^7 + 72*A*a^2*b^2*tan(1/2*d*x + 1/2*c)^7 - 96*A*a*b^3*tan(1/2*d*x + 1 /2*c)^7 - 9*A*a^4*tan(1/2*d*x + 1/2*c)^5 + 12*C*a^4*tan(1/2*d*x + 1/2*c)^5 - 160*A*a^3*b*tan(1/2*d*x + 1/2*c)^5 - 288*C*a^3*b*tan(1/2*d*x + 1/2*c)^5 + 72*A*a^2*b^2*tan(1/2*d*x + 1/2*c)^5 - 288*A*a*b^3*tan(1/2*d*x + 1/2*c)^ 5 + 9*A*a^4*tan(1/2*d*x + 1/2*c)^3 - 12*C*a^4*tan(1/2*d*x + 1/2*c)^3 - 160 *A*a^3*b*tan(1/2*d*x + 1/2*c)^3 - 288*C*a^3*b*tan(1/2*d*x + 1/2*c)^3 - 72* A*a^2*b^2*tan(1/2*d*x + 1/2*c)^3 - 288*A*a*b^3*tan(1/2*d*x + 1/2*c)^3 - 15 *A*a^4*tan(1/2*d*x + 1/2*c) - 12*C*a^4*tan(1/2*d*x + 1/2*c) - 96*A*a^3*b*t an(1/2*d*x + 1/2*c) - 96*C*a^3*b*tan(1/2*d*x + 1/2*c) - 72*A*a^2*b^2*tan(1 /2*d*x + 1/2*c) - 96*A*a*b^3*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 + 1)^4)/d
Time = 16.91 (sec) , antiderivative size = 395, normalized size of antiderivative = 1.61 \[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {A\,a^4\,{\cos \left (c+d\,x\right )}^3\,\sin \left (c+d\,x\right )}{4\,d}+\frac {C\,b^4\,\sin \left (c+d\,x\right )}{d\,\cos \left (c+d\,x\right )}+\frac {3\,A\,a^4\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )}{8\,d}+\frac {4\,A\,a\,b^3\,\sin \left (c+d\,x\right )}{d}+\frac {8\,A\,a^3\,b\,\sin \left (c+d\,x\right )}{3\,d}+\frac {C\,a^4\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )}{2\,d}+\frac {4\,C\,a^3\,b\,\sin \left (c+d\,x\right )}{d}+\frac {3\,A\,a^2\,b^2\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )}{d}+\frac {4\,A\,a^3\,b\,{\cos \left (c+d\,x\right )}^2\,\sin \left (c+d\,x\right )}{3\,d}-\frac {A\,a^4\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,3{}\mathrm {i}}{4\,d}-\frac {A\,b^4\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,2{}\mathrm {i}}{d}-\frac {C\,a^4\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,1{}\mathrm {i}}{d}-\frac {C\,a\,b^3\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,8{}\mathrm {i}}{d}-\frac {A\,a^2\,b^2\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,6{}\mathrm {i}}{d}-\frac {C\,a^2\,b^2\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,12{}\mathrm {i}}{d} \]
(A*a^4*cos(c + d*x)^3*sin(c + d*x))/(4*d) - (A*b^4*atanh((sin(c/2 + (d*x)/ 2)*1i)/cos(c/2 + (d*x)/2))*2i)/d - (C*a^4*atanh((sin(c/2 + (d*x)/2)*1i)/co s(c/2 + (d*x)/2))*1i)/d - (C*a*b^3*atan((sin(c/2 + (d*x)/2)*1i)/cos(c/2 + (d*x)/2))*8i)/d - (A*a^4*atanh((sin(c/2 + (d*x)/2)*1i)/cos(c/2 + (d*x)/2)) *3i)/(4*d) + (C*b^4*sin(c + d*x))/(d*cos(c + d*x)) - (A*a^2*b^2*atanh((sin (c/2 + (d*x)/2)*1i)/cos(c/2 + (d*x)/2))*6i)/d - (C*a^2*b^2*atanh((sin(c/2 + (d*x)/2)*1i)/cos(c/2 + (d*x)/2))*12i)/d + (3*A*a^4*cos(c + d*x)*sin(c + d*x))/(8*d) + (4*A*a*b^3*sin(c + d*x))/d + (8*A*a^3*b*sin(c + d*x))/(3*d) + (C*a^4*cos(c + d*x)*sin(c + d*x))/(2*d) + (4*C*a^3*b*sin(c + d*x))/d + ( 3*A*a^2*b^2*cos(c + d*x)*sin(c + d*x))/d + (4*A*a^3*b*cos(c + d*x)^2*sin(c + d*x))/(3*d)