Integrand size = 41, antiderivative size = 196 \[ \int (a+a \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx=\frac {1}{8} a^4 (52 A+48 B+35 C) x+\frac {a^4 (4 A+B) \text {arctanh}(\sin (c+d x))}{d}+\frac {5 a^4 (4 A+8 B+7 C) \sin (c+d x)}{8 d}-\frac {a (4 A-C) (a+a \cos (c+d x))^3 \sin (c+d x)}{4 d}-\frac {(12 A-4 B-7 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{12 d}-\frac {(12 A-32 B-35 C) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{24 d}+\frac {A (a+a \cos (c+d x))^4 \tan (c+d x)}{d} \]
1/8*a^4*(52*A+48*B+35*C)*x+a^4*(4*A+B)*arctanh(sin(d*x+c))/d+5/8*a^4*(4*A+ 8*B+7*C)*sin(d*x+c)/d-1/4*a*(4*A-C)*(a+a*cos(d*x+c))^3*sin(d*x+c)/d-1/12*( 12*A-4*B-7*C)*(a^2+a^2*cos(d*x+c))^2*sin(d*x+c)/d-1/24*(12*A-32*B-35*C)*(a ^4+a^4*cos(d*x+c))*sin(d*x+c)/d+A*(a+a*cos(d*x+c))^4*tan(d*x+c)/d
Time = 8.80 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.26 \[ \int (a+a \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx=\frac {a^4 (1+\cos (c+d x))^4 \sec ^8\left (\frac {1}{2} (c+d x)\right ) \left (12 (52 A+48 B+35 C) (c+d x)-96 (4 A+B) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+96 (4 A+B) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+\frac {96 A \sin \left (\frac {1}{2} (c+d x)\right )}{\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )}+\frac {96 A \sin \left (\frac {1}{2} (c+d x)\right )}{\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )}+24 (16 A+27 B+28 C) \sin (c+d x)+24 (A+4 B+7 C) \sin (2 (c+d x))+8 (B+4 C) \sin (3 (c+d x))+3 C \sin (4 (c+d x))\right )}{1536 d} \]
(a^4*(1 + Cos[c + d*x])^4*Sec[(c + d*x)/2]^8*(12*(52*A + 48*B + 35*C)*(c + d*x) - 96*(4*A + B)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 96*(4*A + B)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (96*A*Sin[(c + d*x)/2])/(Cos [(c + d*x)/2] - Sin[(c + d*x)/2]) + (96*A*Sin[(c + d*x)/2])/(Cos[(c + d*x) /2] + Sin[(c + d*x)/2]) + 24*(16*A + 27*B + 28*C)*Sin[c + d*x] + 24*(A + 4 *B + 7*C)*Sin[2*(c + d*x)] + 8*(B + 4*C)*Sin[3*(c + d*x)] + 3*C*Sin[4*(c + d*x)]))/(1536*d)
Time = 1.67 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.09, number of steps used = 17, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.415, Rules used = {3042, 3522, 3042, 3455, 3042, 3455, 3042, 3455, 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.
| \(\displaystyle \int \sec ^2(c+d x) (a \cos (c+d x)+a)^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^4 \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^2}dx\) |
| \(\Big \downarrow \) 3522 |
| \(\displaystyle \frac {\int (\cos (c+d x) a+a)^4 (a (4 A+B)-a (4 A-C) \cos (c+d x)) \sec (c+d x)dx}{a}+\frac {A \tan (c+d x) (a \cos (c+d x)+a)^4}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^4 \left (a (4 A+B)-a (4 A-C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx}{a}+\frac {A \tan (c+d x) (a \cos (c+d x)+a)^4}{d}\) |
| \(\Big \downarrow \) 3455 |
| \(\displaystyle \frac {\frac {1}{4} \int (\cos (c+d x) a+a)^3 \left (4 a^2 (4 A+B)-a^2 (12 A-4 B-7 C) \cos (c+d x)\right ) \sec (c+d x)dx-\frac {a^2 (4 A-C) \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d}}{a}+\frac {A \tan (c+d x) (a \cos (c+d x)+a)^4}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{4} \int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3 \left (4 a^2 (4 A+B)-a^2 (12 A-4 B-7 C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx-\frac {a^2 (4 A-C) \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d}}{a}+\frac {A \tan (c+d x) (a \cos (c+d x)+a)^4}{d}\) |
| \(\Big \downarrow \) 3455 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {1}{3} \int (\cos (c+d x) a+a)^2 \left (12 a^3 (4 A+B)-a^3 (12 A-32 B-35 C) \cos (c+d x)\right ) \sec (c+d x)dx-\frac {a^3 (12 A-4 B-7 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d}\right )-\frac {a^2 (4 A-C) \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d}}{a}+\frac {A \tan (c+d x) (a \cos (c+d x)+a)^4}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {1}{3} \int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2 \left (12 a^3 (4 A+B)-a^3 (12 A-32 B-35 C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx-\frac {a^3 (12 A-4 B-7 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d}\right )-\frac {a^2 (4 A-C) \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d}}{a}+\frac {A \tan (c+d x) (a \cos (c+d x)+a)^4}{d}\) |
| \(\Big \downarrow \) 3455 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \int 3 (\cos (c+d x) a+a) \left (8 (4 A+B) a^4+5 (4 A+8 B+7 C) \cos (c+d x) a^4\right ) \sec (c+d x)dx-\frac {(12 A-32 B-35 C) \sin (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{2 d}\right )-\frac {a^3 (12 A-4 B-7 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d}\right )-\frac {a^2 (4 A-C) \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d}}{a}+\frac {A \tan (c+d x) (a \cos (c+d x)+a)^4}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {1}{3} \left (\frac {3}{2} \int (\cos (c+d x) a+a) \left (8 (4 A+B) a^4+5 (4 A+8 B+7 C) \cos (c+d x) a^4\right ) \sec (c+d x)dx-\frac {(12 A-32 B-35 C) \sin (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{2 d}\right )-\frac {a^3 (12 A-4 B-7 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d}\right )-\frac {a^2 (4 A-C) \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d}}{a}+\frac {A \tan (c+d x) (a \cos (c+d x)+a)^4}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {1}{3} \left (\frac {3}{2} \int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right ) \left (8 (4 A+B) a^4+5 (4 A+8 B+7 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^4\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx-\frac {(12 A-32 B-35 C) \sin (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{2 d}\right )-\frac {a^3 (12 A-4 B-7 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d}\right )-\frac {a^2 (4 A-C) \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d}}{a}+\frac {A \tan (c+d x) (a \cos (c+d x)+a)^4}{d}\) |
| \(\Big \downarrow \) 3447 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {1}{3} \left (\frac {3}{2} \int \left (5 (4 A+8 B+7 C) \cos ^2(c+d x) a^5+8 (4 A+B) a^5+\left (8 (4 A+B) a^5+5 (4 A+8 B+7 C) a^5\right ) \cos (c+d x)\right ) \sec (c+d x)dx-\frac {(12 A-32 B-35 C) \sin (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{2 d}\right )-\frac {a^3 (12 A-4 B-7 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d}\right )-\frac {a^2 (4 A-C) \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d}}{a}+\frac {A \tan (c+d x) (a \cos (c+d x)+a)^4}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {1}{3} \left (\frac {3}{2} \int \frac {5 (4 A+8 B+7 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2 a^5+8 (4 A+B) a^5+\left (8 (4 A+B) a^5+5 (4 A+8 B+7 C) a^5\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx-\frac {(12 A-32 B-35 C) \sin (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{2 d}\right )-\frac {a^3 (12 A-4 B-7 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d}\right )-\frac {a^2 (4 A-C) \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d}}{a}+\frac {A \tan (c+d x) (a \cos (c+d x)+a)^4}{d}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {1}{3} \left (\frac {3}{2} \left (\int \left (8 (4 A+B) a^5+(52 A+48 B+35 C) \cos (c+d x) a^5\right ) \sec (c+d x)dx+\frac {5 a^5 (4 A+8 B+7 C) \sin (c+d x)}{d}\right )-\frac {(12 A-32 B-35 C) \sin (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{2 d}\right )-\frac {a^3 (12 A-4 B-7 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d}\right )-\frac {a^2 (4 A-C) \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d}}{a}+\frac {A \tan (c+d x) (a \cos (c+d x)+a)^4}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {1}{3} \left (\frac {3}{2} \left (\int \frac {8 (4 A+B) a^5+(52 A+48 B+35 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^5}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {5 a^5 (4 A+8 B+7 C) \sin (c+d x)}{d}\right )-\frac {(12 A-32 B-35 C) \sin (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{2 d}\right )-\frac {a^3 (12 A-4 B-7 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d}\right )-\frac {a^2 (4 A-C) \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d}}{a}+\frac {A \tan (c+d x) (a \cos (c+d x)+a)^4}{d}\) |
| \(\Big \downarrow \) 3214 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {1}{3} \left (\frac {3}{2} \left (8 a^5 (4 A+B) \int \sec (c+d x)dx+\frac {5 a^5 (4 A+8 B+7 C) \sin (c+d x)}{d}+a^5 x (52 A+48 B+35 C)\right )-\frac {(12 A-32 B-35 C) \sin (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{2 d}\right )-\frac {a^3 (12 A-4 B-7 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d}\right )-\frac {a^2 (4 A-C) \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d}}{a}+\frac {A \tan (c+d x) (a \cos (c+d x)+a)^4}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {1}{3} \left (\frac {3}{2} \left (8 a^5 (4 A+B) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+\frac {5 a^5 (4 A+8 B+7 C) \sin (c+d x)}{d}+a^5 x (52 A+48 B+35 C)\right )-\frac {(12 A-32 B-35 C) \sin (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{2 d}\right )-\frac {a^3 (12 A-4 B-7 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d}\right )-\frac {a^2 (4 A-C) \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d}}{a}+\frac {A \tan (c+d x) (a \cos (c+d x)+a)^4}{d}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {1}{3} \left (\frac {3}{2} \left (\frac {8 a^5 (4 A+B) \text {arctanh}(\sin (c+d x))}{d}+\frac {5 a^5 (4 A+8 B+7 C) \sin (c+d x)}{d}+a^5 x (52 A+48 B+35 C)\right )-\frac {(12 A-32 B-35 C) \sin (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{2 d}\right )-\frac {a^3 (12 A-4 B-7 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d}\right )-\frac {a^2 (4 A-C) \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d}}{a}+\frac {A \tan (c+d x) (a \cos (c+d x)+a)^4}{d}\) |
(-1/4*(a^2*(4*A - C)*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/d + (-1/3*(a^3*( 12*A - 4*B - 7*C)*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/d + (-1/2*((12*A - 32*B - 35*C)*(a^5 + a^5*Cos[c + d*x])*Sin[c + d*x])/d + (3*(a^5*(52*A + 48 *B + 35*C)*x + (8*a^5*(4*A + B)*ArcTanh[Sin[c + d*x]])/d + (5*a^5*(4*A + 8 *B + 7*C)*Sin[c + d*x])/d))/2)/3)/4)/a + (A*(a + a*Cos[c + d*x])^4*Tan[c + d*x])/d
3.4.32.3.1 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[((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)*B*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 1))), x] + Simp[1/(d*(m + n + 1)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A*d*(m + n + 1) + B*(a*c*(m - 1 ) + b*d*(n + 1)) + (A*b*d*(m + n + 1) - B*(b*c*m - a*d*(2*m + n)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, 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_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(c^2*C - B*c*d + 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 - B*d)*( a*c*m + b*d*(n + 1)) + b*(d*(B*c - A*d)*(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, B, 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 = 8.17 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.85
| method | result | size |
| parallelrisch | \(\frac {2 a^{4} \left (-2 \cos \left (d x +c \right ) \left (A +\frac {B}{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+2 \cos \left (d x +c \right ) \left (A +\frac {B}{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\left (A +\frac {41 B}{24}+\frac {11 C}{6}\right ) \sin \left (2 d x +2 c \right )+\frac {\left (\frac {A}{4}+B +\frac {57 C}{32}\right ) \sin \left (3 d x +3 c \right )}{4}+\frac {\left (\frac {B}{4}+C \right ) \sin \left (4 d x +4 c \right )}{12}+\frac {\sin \left (5 d x +5 c \right ) C}{128}+\frac {13 x d \left (A +\frac {12 B}{13}+\frac {35 C}{52}\right ) \cos \left (d x +c \right )}{4}+\frac {9 \left (A +\frac {4 B}{9}+\frac {7 C}{9}\right ) \sin \left (d x +c \right )}{16}\right )}{d \cos \left (d x +c \right )}\) | \(167\) |
| parts | \(\frac {a^{4} A \tan \left (d x +c \right )}{d}+\frac {\left (4 a^{4} A +B \,a^{4}\right ) \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {\left (B \,a^{4}+4 C \,a^{4}\right ) \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3 d}+\frac {\left (a^{4} A +4 B \,a^{4}+6 C \,a^{4}\right ) \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {\left (4 a^{4} A +6 B \,a^{4}+4 C \,a^{4}\right ) \sin \left (d x +c \right )}{d}+\frac {\left (6 a^{4} A +4 B \,a^{4}+C \,a^{4}\right ) \left (d x +c \right )}{d}+\frac {C \,a^{4} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\) | \(220\) |
| derivativedivides | \(\frac {a^{4} A \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {B \,a^{4} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+C \,a^{4} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+4 a^{4} A \sin \left (d x +c \right )+4 B \,a^{4} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {4 C \,a^{4} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+6 a^{4} A \left (d x +c \right )+6 B \,a^{4} \sin \left (d x +c \right )+6 C \,a^{4} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+4 a^{4} A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+4 B \,a^{4} \left (d x +c \right )+4 C \,a^{4} \sin \left (d x +c \right )+a^{4} A \tan \left (d x +c \right )+B \,a^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+C \,a^{4} \left (d x +c \right )}{d}\) | \(289\) |
| default | \(\frac {a^{4} A \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {B \,a^{4} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+C \,a^{4} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+4 a^{4} A \sin \left (d x +c \right )+4 B \,a^{4} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {4 C \,a^{4} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+6 a^{4} A \left (d x +c \right )+6 B \,a^{4} \sin \left (d x +c \right )+6 C \,a^{4} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+4 a^{4} A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+4 B \,a^{4} \left (d x +c \right )+4 C \,a^{4} \sin \left (d x +c \right )+a^{4} A \tan \left (d x +c \right )+B \,a^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+C \,a^{4} \left (d x +c \right )}{d}\) | \(289\) |
| risch | \(\frac {13 a^{4} x A}{2}+6 a^{4} B x +\frac {35 a^{4} C x}{8}-\frac {7 i {\mathrm e}^{2 i \left (d x +c \right )} C \,a^{4}}{8 d}+\frac {i {\mathrm e}^{-2 i \left (d x +c \right )} a^{4} A}{8 d}+\frac {i {\mathrm e}^{-2 i \left (d x +c \right )} B \,a^{4}}{2 d}+\frac {7 i {\mathrm e}^{-2 i \left (d x +c \right )} C \,a^{4}}{8 d}-\frac {2 i {\mathrm e}^{i \left (d x +c \right )} a^{4} A}{d}+\frac {2 i a^{4} A}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}+\frac {2 i {\mathrm e}^{-i \left (d x +c \right )} a^{4} A}{d}-\frac {7 i {\mathrm e}^{i \left (d x +c \right )} C \,a^{4}}{2 d}+\frac {27 i {\mathrm e}^{-i \left (d x +c \right )} B \,a^{4}}{8 d}+\frac {7 i {\mathrm e}^{-i \left (d x +c \right )} C \,a^{4}}{2 d}-\frac {i {\mathrm e}^{2 i \left (d x +c \right )} a^{4} A}{8 d}-\frac {i {\mathrm e}^{2 i \left (d x +c \right )} B \,a^{4}}{2 d}-\frac {27 i {\mathrm e}^{i \left (d x +c \right )} B \,a^{4}}{8 d}+\frac {4 a^{4} A \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d}+\frac {a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B}{d}-\frac {4 a^{4} A \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}-\frac {a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B}{d}+\frac {\sin \left (4 d x +4 c \right ) C \,a^{4}}{32 d}+\frac {\sin \left (3 d x +3 c \right ) B \,a^{4}}{12 d}+\frac {\sin \left (3 d x +3 c \right ) C \,a^{4}}{3 d}\) | \(415\) |
| norman | \(\frac {\left (-\frac {13}{2} a^{4} A -6 B \,a^{4}-\frac {35}{8} C \,a^{4}\right ) x +\left (-\frac {117}{2} a^{4} A -54 B \,a^{4}-\frac {315}{8} C \,a^{4}\right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {65}{2} a^{4} A -30 B \,a^{4}-\frac {175}{8} C \,a^{4}\right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {65}{2} a^{4} A -30 B \,a^{4}-\frac {175}{8} C \,a^{4}\right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {13}{2} a^{4} A +6 B \,a^{4}+\frac {35}{8} C \,a^{4}\right ) x \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {65}{2} a^{4} A +30 B \,a^{4}+\frac {175}{8} C \,a^{4}\right ) x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {65}{2} a^{4} A +30 B \,a^{4}+\frac {175}{8} C \,a^{4}\right ) x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {117}{2} a^{4} A +54 B \,a^{4}+\frac {315}{8} C \,a^{4}\right ) x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {5 a^{4} \left (4 A +8 B +7 C \right ) \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}-\frac {a^{4} \left (36 A -16 B -25 C \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {a^{4} \left (44 A +72 B +93 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}+\frac {a^{4} \left (108 A +272 B +245 C \right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d}+\frac {a^{4} \left (132 A +824 B +791 C \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d}-\frac {a^{4} \left (276 A +368 B +395 C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d}-\frac {a^{4} \left (828 A +728 B +617 C \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {a^{4} \left (4 A +B \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}-\frac {a^{4} \left (4 A +B \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}\) | \(529\) |
2*a^4*(-2*cos(d*x+c)*(A+1/4*B)*ln(tan(1/2*d*x+1/2*c)-1)+2*cos(d*x+c)*(A+1/ 4*B)*ln(tan(1/2*d*x+1/2*c)+1)+(A+41/24*B+11/6*C)*sin(2*d*x+2*c)+1/4*(1/4*A +B+57/32*C)*sin(3*d*x+3*c)+1/12*(1/4*B+C)*sin(4*d*x+4*c)+1/128*sin(5*d*x+5 *c)*C+13/4*x*d*(A+12/13*B+35/52*C)*cos(d*x+c)+9/16*(A+4/9*B+7/9*C)*sin(d*x +c))/d/cos(d*x+c)
Time = 0.28 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.91 \[ \int (a+a \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx=\frac {3 \, {\left (52 \, A + 48 \, B + 35 \, C\right )} a^{4} d x \cos \left (d x + c\right ) + 12 \, {\left (4 \, A + B\right )} a^{4} \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) - 12 \, {\left (4 \, A + B\right )} a^{4} \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) + {\left (6 \, C a^{4} \cos \left (d x + c\right )^{4} + 8 \, {\left (B + 4 \, C\right )} a^{4} \cos \left (d x + c\right )^{3} + 3 \, {\left (4 \, A + 16 \, B + 27 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} + 32 \, {\left (3 \, A + 5 \, B + 5 \, C\right )} a^{4} \cos \left (d x + c\right ) + 24 \, A a^{4}\right )} \sin \left (d x + c\right )}{24 \, d \cos \left (d x + c\right )} \]
integrate((a+a*cos(d*x+c))^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^2, x, algorithm="fricas")
1/24*(3*(52*A + 48*B + 35*C)*a^4*d*x*cos(d*x + c) + 12*(4*A + B)*a^4*cos(d *x + c)*log(sin(d*x + c) + 1) - 12*(4*A + B)*a^4*cos(d*x + c)*log(-sin(d*x + c) + 1) + (6*C*a^4*cos(d*x + c)^4 + 8*(B + 4*C)*a^4*cos(d*x + c)^3 + 3* (4*A + 16*B + 27*C)*a^4*cos(d*x + c)^2 + 32*(3*A + 5*B + 5*C)*a^4*cos(d*x + c) + 24*A*a^4)*sin(d*x + c))/(d*cos(d*x + c))
Timed out. \[ \int (a+a \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx=\text {Timed out} \]
Time = 0.22 (sec) , antiderivative size = 290, normalized size of antiderivative = 1.48 \[ \int (a+a \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx=\frac {24 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} + 576 \, {\left (d x + c\right )} A a^{4} - 32 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{4} + 96 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{4} + 384 \, {\left (d x + c\right )} B a^{4} - 128 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{4} + 3 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{4} + 144 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{4} + 96 \, {\left (d x + c\right )} C a^{4} + 192 \, A a^{4} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 48 \, B a^{4} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 384 \, A a^{4} \sin \left (d x + c\right ) + 576 \, B a^{4} \sin \left (d x + c\right ) + 384 \, C a^{4} \sin \left (d x + c\right ) + 96 \, A a^{4} \tan \left (d x + c\right )}{96 \, d} \]
integrate((a+a*cos(d*x+c))^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^2, x, algorithm="maxima")
1/96*(24*(2*d*x + 2*c + sin(2*d*x + 2*c))*A*a^4 + 576*(d*x + c)*A*a^4 - 32 *(sin(d*x + c)^3 - 3*sin(d*x + c))*B*a^4 + 96*(2*d*x + 2*c + sin(2*d*x + 2 *c))*B*a^4 + 384*(d*x + c)*B*a^4 - 128*(sin(d*x + c)^3 - 3*sin(d*x + c))*C *a^4 + 3*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*C*a^4 + 1 44*(2*d*x + 2*c + sin(2*d*x + 2*c))*C*a^4 + 96*(d*x + c)*C*a^4 + 192*A*a^4 *(log(sin(d*x + c) + 1) - log(sin(d*x + c) - 1)) + 48*B*a^4*(log(sin(d*x + c) + 1) - log(sin(d*x + c) - 1)) + 384*A*a^4*sin(d*x + c) + 576*B*a^4*sin (d*x + c) + 384*C*a^4*sin(d*x + c) + 96*A*a^4*tan(d*x + c))/d
Time = 0.37 (sec) , antiderivative size = 332, normalized size of antiderivative = 1.69 \[ \int (a+a \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx=-\frac {\frac {48 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1} - 3 \, {\left (52 \, A a^{4} + 48 \, B a^{4} + 35 \, C a^{4}\right )} {\left (d x + c\right )} - 24 \, {\left (4 \, A a^{4} + B a^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) + 24 \, {\left (4 \, A a^{4} + B a^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (84 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 120 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 105 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 276 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 424 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 385 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 300 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 520 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 511 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 108 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 216 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 279 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{4}}}{24 \, d} \]
integrate((a+a*cos(d*x+c))^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^2, x, algorithm="giac")
-1/24*(48*A*a^4*tan(1/2*d*x + 1/2*c)/(tan(1/2*d*x + 1/2*c)^2 - 1) - 3*(52* A*a^4 + 48*B*a^4 + 35*C*a^4)*(d*x + c) - 24*(4*A*a^4 + B*a^4)*log(abs(tan( 1/2*d*x + 1/2*c) + 1)) + 24*(4*A*a^4 + B*a^4)*log(abs(tan(1/2*d*x + 1/2*c) - 1)) - 2*(84*A*a^4*tan(1/2*d*x + 1/2*c)^7 + 120*B*a^4*tan(1/2*d*x + 1/2* c)^7 + 105*C*a^4*tan(1/2*d*x + 1/2*c)^7 + 276*A*a^4*tan(1/2*d*x + 1/2*c)^5 + 424*B*a^4*tan(1/2*d*x + 1/2*c)^5 + 385*C*a^4*tan(1/2*d*x + 1/2*c)^5 + 3 00*A*a^4*tan(1/2*d*x + 1/2*c)^3 + 520*B*a^4*tan(1/2*d*x + 1/2*c)^3 + 511*C *a^4*tan(1/2*d*x + 1/2*c)^3 + 108*A*a^4*tan(1/2*d*x + 1/2*c) + 216*B*a^4*t an(1/2*d*x + 1/2*c) + 279*C*a^4*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c )^2 + 1)^4)/d
Time = 3.12 (sec) , antiderivative size = 1244, normalized size of antiderivative = 6.35 \[ \int (a+a \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx=\text {Too large to display} \]
- (tan(c/2 + (d*x)/2)^5*(8*B*a^4 - 10*A*a^4 + (21*C*a^4)/2) + tan(c/2 + (d *x)/2)^9*(5*A*a^4 + 10*B*a^4 + (35*C*a^4)/4) - tan(c/2 + (d*x)/2)^3*(24*A* a^4 + (76*B*a^4)/3 + (58*C*a^4)/3) + tan(c/2 + (d*x)/2)^7*(8*A*a^4 + (76*B *a^4)/3 + (70*C*a^4)/3) - tan(c/2 + (d*x)/2)*(11*A*a^4 + 18*B*a^4 + (93*C* a^4)/4))/(d*(3*tan(c/2 + (d*x)/2)^2 + 2*tan(c/2 + (d*x)/2)^4 - 2*tan(c/2 + (d*x)/2)^6 - 3*tan(c/2 + (d*x)/2)^8 - tan(c/2 + (d*x)/2)^10 + 1)) - (a^4* atan((a^4*(tan(c/2 + (d*x)/2)*(1864*A^2*a^8 + 1184*B^2*a^8 + (1225*C^2*a^8 )/2 + 2752*A*B*a^8 + 1820*A*C*a^8 + 1680*B*C*a^8) + a^4*(4*A + B)*(336*A*a ^4 + 224*B*a^4 + 140*C*a^4))*(4*A + B)*1i + a^4*(tan(c/2 + (d*x)/2)*(1864* A^2*a^8 + 1184*B^2*a^8 + (1225*C^2*a^8)/2 + 2752*A*B*a^8 + 1820*A*C*a^8 + 1680*B*C*a^8) - a^4*(4*A + B)*(336*A*a^4 + 224*B*a^4 + 140*C*a^4))*(4*A + B)*1i)/(4160*A^3*a^12 + 1920*B^3*a^12 + 10720*A*B^2*a^12 + 13200*A^2*B*a^1 2 + 4900*A*C^2*a^12 + 10080*A^2*C*a^12 + 1225*B*C^2*a^12 + 3080*B^2*C*a^12 + a^4*(tan(c/2 + (d*x)/2)*(1864*A^2*a^8 + 1184*B^2*a^8 + (1225*C^2*a^8)/2 + 2752*A*B*a^8 + 1820*A*C*a^8 + 1680*B*C*a^8) + a^4*(4*A + B)*(336*A*a^4 + 224*B*a^4 + 140*C*a^4))*(4*A + B) - a^4*(tan(c/2 + (d*x)/2)*(1864*A^2*a^ 8 + 1184*B^2*a^8 + (1225*C^2*a^8)/2 + 2752*A*B*a^8 + 1820*A*C*a^8 + 1680*B *C*a^8) - a^4*(4*A + B)*(336*A*a^4 + 224*B*a^4 + 140*C*a^4))*(4*A + B) + 1 4840*A*B*C*a^12))*(4*A + B)*2i)/d - (a^4*atan(((a^4*(tan(c/2 + (d*x)/2)*(1 864*A^2*a^8 + 1184*B^2*a^8 + (1225*C^2*a^8)/2 + 2752*A*B*a^8 + 1820*A*C...