Integrand size = 33, antiderivative size = 200 \[ \int (a+a \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=4 a^4 C x+\frac {a^4 (35 A+52 C) \text {arctanh}(\sin (c+d x))}{8 d}-\frac {5 a^4 (7 A+4 C) \sin (c+d x)}{8 d}+\frac {(35 A+36 C) \left (a^4+a^4 \cos (c+d x)\right ) \tan (c+d x)}{12 d}+\frac {(7 A+4 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec (c+d x) \tan (c+d x)}{8 d}+\frac {a A (a+a \cos (c+d x))^3 \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac {A (a+a \cos (c+d x))^4 \sec ^3(c+d x) \tan (c+d x)}{4 d} \] Output:
4*a^4*C*x+1/8*a^4*(35*A+52*C)*arctanh(sin(d*x+c))/d-5/8*a^4*(7*A+4*C)*sin( d*x+c)/d+1/12*(35*A+36*C)*(a^4+a^4*cos(d*x+c))*tan(d*x+c)/d+1/8*(7*A+4*C)* (a^2+a^2*cos(d*x+c))^2*sec(d*x+c)*tan(d*x+c)/d+1/3*a*A*(a+a*cos(d*x+c))^3* sec(d*x+c)^2*tan(d*x+c)/d+1/4*A*(a+a*cos(d*x+c))^4*sec(d*x+c)^3*tan(d*x+c) /d
Time = 8.26 (sec) , antiderivative size = 350, normalized size of antiderivative = 1.75 \[ \int (a+a \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\frac {a^4 \sec ^8\left (\frac {1}{2} (c+d x)\right ) (1+\sec (c+d x))^4 \left (-24 (35 A+52 C) \cos ^4(c+d x) \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )+\sec (c) (288 C d x \cos (c)+192 C d x \cos (c+2 d x)+192 C d x \cos (3 c+2 d x)+48 C d x \cos (3 c+4 d x)+48 C d x \cos (5 c+4 d x)-480 A \sin (c)-288 C \sin (c)+105 A \sin (d x)+24 C \sin (d x)+105 A \sin (2 c+d x)+24 C \sin (2 c+d x)+544 A \sin (c+2 d x)+288 C \sin (c+2 d x)-96 A \sin (3 c+2 d x)-96 C \sin (3 c+2 d x)+81 A \sin (2 c+3 d x)+30 C \sin (2 c+3 d x)+81 A \sin (4 c+3 d x)+30 C \sin (4 c+3 d x)+160 A \sin (3 c+4 d x)+96 C \sin (3 c+4 d x)+6 C \sin (4 c+5 d x)+6 C \sin (6 c+5 d x))\right )}{3072 d} \] Input:
Integrate[(a + a*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^5,x]
Output:
(a^4*Sec[(c + d*x)/2]^8*(1 + Sec[c + d*x])^4*(-24*(35*A + 52*C)*Cos[c + d* x]^4*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Si n[(c + d*x)/2]]) + Sec[c]*(288*C*d*x*Cos[c] + 192*C*d*x*Cos[c + 2*d*x] + 1 92*C*d*x*Cos[3*c + 2*d*x] + 48*C*d*x*Cos[3*c + 4*d*x] + 48*C*d*x*Cos[5*c + 4*d*x] - 480*A*Sin[c] - 288*C*Sin[c] + 105*A*Sin[d*x] + 24*C*Sin[d*x] + 1 05*A*Sin[2*c + d*x] + 24*C*Sin[2*c + d*x] + 544*A*Sin[c + 2*d*x] + 288*C*S in[c + 2*d*x] - 96*A*Sin[3*c + 2*d*x] - 96*C*Sin[3*c + 2*d*x] + 81*A*Sin[2 *c + 3*d*x] + 30*C*Sin[2*c + 3*d*x] + 81*A*Sin[4*c + 3*d*x] + 30*C*Sin[4*c + 3*d*x] + 160*A*Sin[3*c + 4*d*x] + 96*C*Sin[3*c + 4*d*x] + 6*C*Sin[4*c + 5*d*x] + 6*C*Sin[6*c + 5*d*x])))/(3072*d)
Time = 1.71 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.08, number of steps used = 17, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.515, Rules used = {3042, 3523, 3042, 3454, 3042, 3454, 3042, 3454, 27, 3042, 3447, 3042, 3502, 3042, 3214, 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.
| \(\displaystyle \int \sec ^5(c+d x) (a \cos (c+d x)+a)^4 \left (A+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^4 \left (A+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^5}dx\) |
| \(\Big \downarrow \) 3523 |
| \(\displaystyle \frac {\int (\cos (c+d x) a+a)^4 (4 a A-a (A-4 C) \cos (c+d x)) \sec ^4(c+d x)dx}{4 a}+\frac {A \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^4}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^4 \left (4 a A-a (A-4 C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^4}dx}{4 a}+\frac {A \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^4}{4 d}\) |
| \(\Big \downarrow \) 3454 |
| \(\displaystyle \frac {\frac {1}{3} \int (\cos (c+d x) a+a)^3 \left (3 a^2 (7 A+4 C)-a^2 (7 A-12 C) \cos (c+d x)\right ) \sec ^3(c+d x)dx+\frac {4 a^2 A \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^3}{3 d}}{4 a}+\frac {A \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^4}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{3} \int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3 \left (3 a^2 (7 A+4 C)-a^2 (7 A-12 C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^3}dx+\frac {4 a^2 A \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^3}{3 d}}{4 a}+\frac {A \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^4}{4 d}\) |
| \(\Big \downarrow \) 3454 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {1}{2} \int (\cos (c+d x) a+a)^2 \left (2 a^3 (35 A+36 C)-a^3 (35 A-12 C) \cos (c+d x)\right ) \sec ^2(c+d x)dx+\frac {3 a^3 (7 A+4 C) \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^2}{2 d}\right )+\frac {4 a^2 A \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^3}{3 d}}{4 a}+\frac {A \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^4}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {1}{2} \int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2 \left (2 a^3 (35 A+36 C)-a^3 (35 A-12 C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^2}dx+\frac {3 a^3 (7 A+4 C) \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^2}{2 d}\right )+\frac {4 a^2 A \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^3}{3 d}}{4 a}+\frac {A \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^4}{4 d}\) |
| \(\Big \downarrow \) 3454 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {1}{2} \left (\int 3 (\cos (c+d x) a+a) \left (a^4 (35 A+52 C)-5 a^4 (7 A+4 C) \cos (c+d x)\right ) \sec (c+d x)dx+\frac {2 (35 A+36 C) \tan (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{d}\right )+\frac {3 a^3 (7 A+4 C) \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^2}{2 d}\right )+\frac {4 a^2 A \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^3}{3 d}}{4 a}+\frac {A \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^4}{4 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {1}{2} \left (3 \int (\cos (c+d x) a+a) \left (a^4 (35 A+52 C)-5 a^4 (7 A+4 C) \cos (c+d x)\right ) \sec (c+d x)dx+\frac {2 (35 A+36 C) \tan (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{d}\right )+\frac {3 a^3 (7 A+4 C) \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^2}{2 d}\right )+\frac {4 a^2 A \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^3}{3 d}}{4 a}+\frac {A \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^4}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {1}{2} \left (3 \int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right ) \left (a^4 (35 A+52 C)-5 a^4 (7 A+4 C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {2 (35 A+36 C) \tan (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{d}\right )+\frac {3 a^3 (7 A+4 C) \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^2}{2 d}\right )+\frac {4 a^2 A \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^3}{3 d}}{4 a}+\frac {A \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^4}{4 d}\) |
| \(\Big \downarrow \) 3447 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {1}{2} \left (3 \int \left (-5 (7 A+4 C) \cos ^2(c+d x) a^5+(35 A+52 C) a^5+\left (a^5 (35 A+52 C)-5 a^5 (7 A+4 C)\right ) \cos (c+d x)\right ) \sec (c+d x)dx+\frac {2 (35 A+36 C) \tan (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{d}\right )+\frac {3 a^3 (7 A+4 C) \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^2}{2 d}\right )+\frac {4 a^2 A \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^3}{3 d}}{4 a}+\frac {A \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^4}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {1}{2} \left (3 \int \frac {-5 (7 A+4 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2 a^5+(35 A+52 C) a^5+\left (a^5 (35 A+52 C)-5 a^5 (7 A+4 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {2 (35 A+36 C) \tan (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{d}\right )+\frac {3 a^3 (7 A+4 C) \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^2}{2 d}\right )+\frac {4 a^2 A \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^3}{3 d}}{4 a}+\frac {A \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^4}{4 d}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {1}{2} \left (3 \left (\int \left ((35 A+52 C) a^5+32 C \cos (c+d x) a^5\right ) \sec (c+d x)dx-\frac {5 a^5 (7 A+4 C) \sin (c+d x)}{d}\right )+\frac {2 (35 A+36 C) \tan (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{d}\right )+\frac {3 a^3 (7 A+4 C) \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^2}{2 d}\right )+\frac {4 a^2 A \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^3}{3 d}}{4 a}+\frac {A \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^4}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {1}{2} \left (3 \left (\int \frac {(35 A+52 C) a^5+32 C \sin \left (c+d x+\frac {\pi }{2}\right ) a^5}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx-\frac {5 a^5 (7 A+4 C) \sin (c+d x)}{d}\right )+\frac {2 (35 A+36 C) \tan (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{d}\right )+\frac {3 a^3 (7 A+4 C) \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^2}{2 d}\right )+\frac {4 a^2 A \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^3}{3 d}}{4 a}+\frac {A \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^4}{4 d}\) |
| \(\Big \downarrow \) 3214 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {1}{2} \left (3 \left (a^5 (35 A+52 C) \int \sec (c+d x)dx-\frac {5 a^5 (7 A+4 C) \sin (c+d x)}{d}+32 a^5 C x\right )+\frac {2 (35 A+36 C) \tan (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{d}\right )+\frac {3 a^3 (7 A+4 C) \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^2}{2 d}\right )+\frac {4 a^2 A \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^3}{3 d}}{4 a}+\frac {A \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^4}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {1}{2} \left (3 \left (a^5 (35 A+52 C) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx-\frac {5 a^5 (7 A+4 C) \sin (c+d x)}{d}+32 a^5 C x\right )+\frac {2 (35 A+36 C) \tan (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{d}\right )+\frac {3 a^3 (7 A+4 C) \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^2}{2 d}\right )+\frac {4 a^2 A \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^3}{3 d}}{4 a}+\frac {A \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^4}{4 d}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {\frac {4 a^2 A \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^3}{3 d}+\frac {1}{3} \left (\frac {1}{2} \left (3 \left (\frac {a^5 (35 A+52 C) \text {arctanh}(\sin (c+d x))}{d}-\frac {5 a^5 (7 A+4 C) \sin (c+d x)}{d}+32 a^5 C x\right )+\frac {2 (35 A+36 C) \tan (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{d}\right )+\frac {3 a^3 (7 A+4 C) \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^2}{2 d}\right )}{4 a}+\frac {A \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^4}{4 d}\) |
Input:
Int[(a + a*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^5,x]
Output:
(A*(a + a*Cos[c + d*x])^4*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + ((4*a^2*A*( a + a*Cos[c + d*x])^3*Sec[c + d*x]^2*Tan[c + d*x])/(3*d) + ((3*a^3*(7*A + 4*C)*(a + a*Cos[c + d*x])^2*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (3*(32*a^5* C*x + (a^5*(35*A + 52*C)*ArcTanh[Sin[c + d*x]])/d - (5*a^5*(7*A + 4*C)*Sin [c + d*x])/d) + (2*(35*A + 36*C)*(a^5 + a^5*Cos[c + d*x])*Tan[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[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])/((c_.) + (d_.)*sin[(e_.) + (f_. )*(x_)]), x_Symbol] :> Simp[b*(x/d), x] - Simp[(b*c - a*d)/d Int[1/(c + d *Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Int[(a + b*Sin[e + f*x])^m*(A*c + (B*c + A*d)*Sin[e + f*x] + B*d*Sin[e + f*x]^2), x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim p[(-b^2)*(B*c - A*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*((c + d*Sin[ e + f*x])^(n + 1)/(d*f*(n + 1)*(b*c + a*d))), x] - Simp[b/(d*(n + 1)*(b*c + a*d)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp [a*A*d*(m - n - 2) - B*(a*c*(m - 1) + b*d*(n + 1)) - (A*b*d*(m + n + 1) - B *(b*c*m - a*d*(n + 1)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f , A, B}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1/2] && LtQ[n, -1] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 0 ])
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Co s[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Simp[1/(b*(m + 2)) Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && !LtQ[m, -1]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(c^2*C + A*d^2))*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2))), x] + Simp[1/(b*d*(n + 1)*(c^2 - d^2)) Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(a *d*m + b*c*(n + 1)) + c*C*(a*c*m + b*d*(n + 1)) - b*(A*d^2*(m + n + 2) + C* (c^2*(m + 1) + d^2*(n + 1)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && !LtQ[m, -2^(-1)] && (LtQ[n, -1] || EqQ[m + n + 2, 0])
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Time = 12.69 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.04
| method | result | size |
| parts | \(\frac {a^{4} A \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}+\frac {\left (a^{4} A +6 a^{4} C \right ) \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {\left (4 a^{4} A +4 a^{4} C \right ) \tan \left (d x +c \right )}{d}+\frac {\left (6 a^{4} A +a^{4} C \right ) \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}+\frac {\sin \left (d x +c \right ) a^{4} C}{d}-\frac {4 a^{4} A \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}+\frac {4 a^{4} C \left (d x +c \right )}{d}\) | \(209\) |
| parallelrisch | \(\frac {56 a^{4} \left (-\frac {15 \left (A +\frac {52 C}{35}\right ) \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{16}+\frac {15 \left (A +\frac {52 C}{35}\right ) \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{16}+\frac {6 d x C \cos \left (2 d x +2 c \right )}{7}+\frac {3 d x C \cos \left (4 d x +4 c \right )}{14}+\left (A +\frac {3 C}{7}\right ) \sin \left (2 d x +2 c \right )+\left (\frac {81 A}{224}+\frac {15 C}{112}\right ) \sin \left (3 d x +3 c \right )+\left (\frac {5 A}{14}+\frac {3 C}{14}\right ) \sin \left (4 d x +4 c \right )+\frac {3 C \sin \left (5 d x +5 c \right )}{112}+\left (\frac {15 A}{32}+\frac {3 C}{28}\right ) \sin \left (d x +c \right )+\frac {9 d x C}{14}\right )}{3 d \left (\cos \left (4 d x +4 c \right )+4 \cos \left (2 d x +2 c \right )+3\right )}\) | \(224\) |
| derivativedivides | \(\frac {a^{4} A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+a^{4} C \sin \left (d x +c \right )+4 a^{4} A \tan \left (d x +c \right )+4 a^{4} C \left (d x +c \right )+6 a^{4} A \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+6 a^{4} C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )-4 a^{4} A \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+4 a^{4} C \tan \left (d x +c \right )+a^{4} A \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+a^{4} C \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}\) | \(237\) |
| default | \(\frac {a^{4} A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+a^{4} C \sin \left (d x +c \right )+4 a^{4} A \tan \left (d x +c \right )+4 a^{4} C \left (d x +c \right )+6 a^{4} A \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+6 a^{4} C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )-4 a^{4} A \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+4 a^{4} C \tan \left (d x +c \right )+a^{4} A \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+a^{4} C \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}\) | \(237\) |
| risch | \(4 a^{4} C x -\frac {i {\mathrm e}^{i \left (d x +c \right )} a^{4} C}{2 d}+\frac {i {\mathrm e}^{-i \left (d x +c \right )} a^{4} C}{2 d}-\frac {i a^{4} \left (81 A \,{\mathrm e}^{7 i \left (d x +c \right )}+12 C \,{\mathrm e}^{7 i \left (d x +c \right )}-96 A \,{\mathrm e}^{6 i \left (d x +c \right )}-96 C \,{\mathrm e}^{6 i \left (d x +c \right )}+105 A \,{\mathrm e}^{5 i \left (d x +c \right )}+12 C \,{\mathrm e}^{5 i \left (d x +c \right )}-480 A \,{\mathrm e}^{4 i \left (d x +c \right )}-288 C \,{\mathrm e}^{4 i \left (d x +c \right )}-105 A \,{\mathrm e}^{3 i \left (d x +c \right )}-12 C \,{\mathrm e}^{3 i \left (d x +c \right )}-544 A \,{\mathrm e}^{2 i \left (d x +c \right )}-288 C \,{\mathrm e}^{2 i \left (d x +c \right )}-81 A \,{\mathrm e}^{i \left (d x +c \right )}-12 C \,{\mathrm e}^{i \left (d x +c \right )}-160 A -96 C \right )}{12 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{4}}+\frac {35 a^{4} A \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{8 d}+\frac {13 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{2 d}-\frac {35 a^{4} A \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{8 d}-\frac {13 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{2 d}\) | \(332\) |
Input:
int((a+a*cos(d*x+c))^4*(A+C*cos(d*x+c)^2)*sec(d*x+c)^5,x,method=_RETURNVER BOSE)
Output:
a^4*A/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)))+(A*a^4+6*C*a^4)/d*ln(sec(d*x+c)+tan(d*x+c))+(4*A*a^4+4*C*a^4) /d*tan(d*x+c)+(6*A*a^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)))+1/d*sin(d*x+c)*a^4*C-4*a^4*A/d*(-2/3-1/3*sec(d*x+c)^2)*tan (d*x+c)+4*a^4*C/d*(d*x+c)
Time = 0.11 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.86 \[ \int (a+a \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\frac {192 \, C a^{4} d x \cos \left (d x + c\right )^{4} + 3 \, {\left (35 \, A + 52 \, C\right )} a^{4} \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (35 \, A + 52 \, C\right )} a^{4} \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (24 \, C a^{4} \cos \left (d x + c\right )^{4} + 32 \, {\left (5 \, A + 3 \, C\right )} a^{4} \cos \left (d x + c\right )^{3} + 3 \, {\left (27 \, A + 4 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} + 32 \, A a^{4} \cos \left (d x + c\right ) + 6 \, A a^{4}\right )} \sin \left (d x + c\right )}{48 \, d \cos \left (d x + c\right )^{4}} \] Input:
integrate((a+a*cos(d*x+c))^4*(A+C*cos(d*x+c)^2)*sec(d*x+c)^5,x, algorithm= "fricas")
Output:
1/48*(192*C*a^4*d*x*cos(d*x + c)^4 + 3*(35*A + 52*C)*a^4*cos(d*x + c)^4*lo g(sin(d*x + c) + 1) - 3*(35*A + 52*C)*a^4*cos(d*x + c)^4*log(-sin(d*x + c) + 1) + 2*(24*C*a^4*cos(d*x + c)^4 + 32*(5*A + 3*C)*a^4*cos(d*x + c)^3 + 3 *(27*A + 4*C)*a^4*cos(d*x + c)^2 + 32*A*a^4*cos(d*x + c) + 6*A*a^4)*sin(d* x + c))/(d*cos(d*x + c)^4)
Timed out. \[ \int (a+a \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\text {Timed out} \] Input:
integrate((a+a*cos(d*x+c))**4*(A+C*cos(d*x+c)**2)*sec(d*x+c)**5,x)
Output:
Timed out
Time = 0.04 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.48 \[ \int (a+a \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\frac {64 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a^{4} + 192 \, {\left (d x + c\right )} C a^{4} - 3 \, A a^{4} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 72 \, A a^{4} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 12 \, C a^{4} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 24 \, A a^{4} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 144 \, C a^{4} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 48 \, C a^{4} \sin \left (d x + c\right ) + 192 \, A a^{4} \tan \left (d x + c\right ) + 192 \, C a^{4} \tan \left (d x + c\right )}{48 \, d} \] Input:
integrate((a+a*cos(d*x+c))^4*(A+C*cos(d*x+c)^2)*sec(d*x+c)^5,x, algorithm= "maxima")
Output:
1/48*(64*(tan(d*x + c)^3 + 3*tan(d*x + c))*A*a^4 + 192*(d*x + c)*C*a^4 - 3 *A*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)) - 72*A*a^ 4*(2*sin(d*x + c)/(sin(d*x + c)^2 - 1) - log(sin(d*x + c) + 1) + log(sin(d *x + c) - 1)) - 12*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) - l og(sin(d*x + c) - 1)) + 144*C*a^4*(log(sin(d*x + c) + 1) - log(sin(d*x + c ) - 1)) + 48*C*a^4*sin(d*x + c) + 192*A*a^4*tan(d*x + c) + 192*C*a^4*tan(d *x + c))/d
Time = 0.39 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.26 \[ \int (a+a \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\frac {96 \, {\left (d x + c\right )} C a^{4} + \frac {48 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1} + 3 \, {\left (35 \, A a^{4} + 52 \, C a^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 3 \, {\left (35 \, A a^{4} + 52 \, C a^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (105 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 84 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 385 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 276 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 511 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 300 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 279 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 108 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{4}}}{24 \, d} \] Input:
integrate((a+a*cos(d*x+c))^4*(A+C*cos(d*x+c)^2)*sec(d*x+c)^5,x, algorithm= "giac")
Output:
1/24*(96*(d*x + c)*C*a^4 + 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)*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 3*(35*A*a^4 + 52*C*a^4)*log(abs(tan(1/2*d*x + 1/2*c) - 1)) - 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 = 0.25 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.23 \[ \int (a+a \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\frac {C\,a^4\,\sin \left (c+d\,x\right )}{d}+\frac {35\,A\,a^4\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{4\,d}+\frac {8\,C\,a^4\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {13\,C\,a^4\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {20\,A\,a^4\,\sin \left (c+d\,x\right )}{3\,d\,\cos \left (c+d\,x\right )}+\frac {27\,A\,a^4\,\sin \left (c+d\,x\right )}{8\,d\,{\cos \left (c+d\,x\right )}^2}+\frac {4\,A\,a^4\,\sin \left (c+d\,x\right )}{3\,d\,{\cos \left (c+d\,x\right )}^3}+\frac {A\,a^4\,\sin \left (c+d\,x\right )}{4\,d\,{\cos \left (c+d\,x\right )}^4}+\frac {4\,C\,a^4\,\sin \left (c+d\,x\right )}{d\,\cos \left (c+d\,x\right )}+\frac {C\,a^4\,\sin \left (c+d\,x\right )}{2\,d\,{\cos \left (c+d\,x\right )}^2} \] Input:
int(((A + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^4)/cos(c + d*x)^5,x)
Output:
(C*a^4*sin(c + d*x))/d + (35*A*a^4*atanh(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x )/2)))/(4*d) + (8*C*a^4*atan(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)))/d + ( 13*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)) + (27*A*a^4*sin(c + d*x))/(8*d*cos(c + d*x)^2) + (4*A*a^4*sin(c + d*x))/(3*d*cos(c + d*x)^3) + (A*a^4*sin(c + d*x))/(4*d *cos(c + d*x)^4) + (4*C*a^4*sin(c + d*x))/(d*cos(c + d*x)) + (C*a^4*sin(c + d*x))/(2*d*cos(c + d*x)^2)
Time = 0.28 (sec) , antiderivative size = 547, normalized size of antiderivative = 2.74 \[ \int (a+a \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx =\text {Too large to display} \] Input:
int((a+a*cos(d*x+c))^4*(A+C*cos(d*x+c)^2)*sec(d*x+c)^5,x)
Output:
(a**4*( - 105*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**4*a - 1 56*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**4*c + 210*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**2*a + 312*cos(c + d*x)*log(ta n((c + d*x)/2) - 1)*sin(c + d*x)**2*c - 105*cos(c + d*x)*log(tan((c + d*x) /2) - 1)*a - 156*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*c + 105*cos(c + d* x)*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**4*a + 156*cos(c + d*x)*log(tan( (c + d*x)/2) + 1)*sin(c + d*x)**4*c - 210*cos(c + d*x)*log(tan((c + d*x)/2 ) + 1)*sin(c + d*x)**2*a - 312*cos(c + d*x)*log(tan((c + d*x)/2) + 1)*sin( c + d*x)**2*c + 105*cos(c + d*x)*log(tan((c + d*x)/2) + 1)*a + 156*cos(c + d*x)*log(tan((c + d*x)/2) + 1)*c + 24*cos(c + d*x)*sin(c + d*x)**5*c + 96 *cos(c + d*x)*sin(c + d*x)**4*c*d*x - 81*cos(c + d*x)*sin(c + d*x)**3*a - 60*cos(c + d*x)*sin(c + d*x)**3*c - 192*cos(c + d*x)*sin(c + d*x)**2*c*d*x + 87*cos(c + d*x)*sin(c + d*x)*a + 36*cos(c + d*x)*sin(c + d*x)*c + 96*co s(c + d*x)*c*d*x + 160*sin(c + d*x)**5*a + 96*sin(c + d*x)**5*c - 352*sin( c + d*x)**3*a - 192*sin(c + d*x)**3*c + 192*sin(c + d*x)*a + 96*sin(c + d* x)*c))/(24*cos(c + d*x)*d*(sin(c + d*x)**4 - 2*sin(c + d*x)**2 + 1))