Integrand size = 41, antiderivative size = 196 \[ \int (a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx=b^2 (b B+3 a C) x+\frac {\left (2 A b^3+a^3 B+6 a b^2 B+3 a^2 b (A+2 C)\right ) \text {arctanh}(\sin (c+d x))}{2 d}-\frac {b^2 (5 A b+3 a B-6 b C) \sin (c+d x)}{6 d}+\frac {a \left (3 A b^2+6 a b B+a^2 (2 A+3 C)\right ) \tan (c+d x)}{3 d}+\frac {(A b+a B) (a+b \cos (c+d x))^2 \sec (c+d x) \tan (c+d x)}{2 d}+\frac {A (a+b \cos (c+d x))^3 \sec ^2(c+d x) \tan (c+d x)}{3 d} \] Output:
b^2*(B*b+3*C*a)*x+1/2*(2*A*b^3+B*a^3+6*B*a*b^2+3*a^2*b*(A+2*C))*arctanh(si n(d*x+c))/d-1/6*b^2*(5*A*b+3*B*a-6*C*b)*sin(d*x+c)/d+1/3*a*(3*A*b^2+6*B*a* b+a^2*(2*A+3*C))*tan(d*x+c)/d+1/2*(A*b+B*a)*(a+b*cos(d*x+c))^2*sec(d*x+c)* tan(d*x+c)/d+1/3*A*(a+b*cos(d*x+c))^3*sec(d*x+c)^2*tan(d*x+c)/d
Leaf count is larger than twice the leaf count of optimal. \(429\) vs. \(2(196)=392\).
Time = 8.76 (sec) , antiderivative size = 429, normalized size of antiderivative = 2.19 \[ \int (a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx=\frac {12 b^2 (b B+3 a C) (c+d x)-6 \left (2 A b^3+a^3 B+6 a b^2 B+3 a^2 b (A+2 C)\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+6 \left (2 A b^3+a^3 B+6 a b^2 B+3 a^2 b (A+2 C)\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+\frac {a^2 (9 A b+a (A+3 B))}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {2 a^3 A \sin \left (\frac {1}{2} (c+d x)\right )}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3}+\frac {4 a \left (9 A b^2+9 a b B+a^2 (2 A+3 C)\right ) \sin \left (\frac {1}{2} (c+d x)\right )}{\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )}+\frac {2 a^3 A \sin \left (\frac {1}{2} (c+d x)\right )}{\left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^3}-\frac {a^2 (9 A b+a (A+3 B))}{\left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {4 a \left (9 A b^2+9 a b B+a^2 (2 A+3 C)\right ) \sin \left (\frac {1}{2} (c+d x)\right )}{\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )}+12 b^3 C \sin (c+d x)}{12 d} \] Input:
Integrate[(a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*S ec[c + d*x]^4,x]
Output:
(12*b^2*(b*B + 3*a*C)*(c + d*x) - 6*(2*A*b^3 + a^3*B + 6*a*b^2*B + 3*a^2*b *(A + 2*C))*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 6*(2*A*b^3 + a^3*B + 6*a*b^2*B + 3*a^2*b*(A + 2*C))*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (a^2*(9*A*b + a*(A + 3*B)))/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 + (2 *a^3*A*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3 + (4*a*(9 *A*b^2 + 9*a*b*B + a^2*(2*A + 3*C))*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) + (2*a^3*A*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3 - (a^2*(9*A*b + a*(A + 3*B)))/(Cos[(c + d*x)/2] + Sin[(c + d* x)/2])^2 + (4*a*(9*A*b^2 + 9*a*b*B + a^2*(2*A + 3*C))*Sin[(c + d*x)/2])/(C os[(c + d*x)/2] + Sin[(c + d*x)/2]) + 12*b^3*C*Sin[c + d*x])/(12*d)
Time = 1.52 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.04, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.341, Rules used = {3042, 3526, 3042, 3526, 3042, 3510, 25, 3042, 3502, 27, 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.
| \(\displaystyle \int \sec ^4(c+d x) (a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^4}dx\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \frac {1}{3} \int (a+b \cos (c+d x))^2 \left (-b (A-3 C) \cos ^2(c+d x)+(2 a A+3 b B+3 a C) \cos (c+d x)+3 (A b+a B)\right ) \sec ^3(c+d x)dx+\frac {A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^3}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left (-b (A-3 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+(2 a A+3 b B+3 a C) \sin \left (c+d x+\frac {\pi }{2}\right )+3 (A b+a B)\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^3}dx+\frac {A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^3}{3 d}\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{2} \int (a+b \cos (c+d x)) \left (-b (5 A b-6 C b+3 a B) \cos ^2(c+d x)+\left (3 B a^2+b (5 A+12 C) a+6 b^2 B\right ) \cos (c+d x)+2 \left (\frac {1}{2} (4 A+6 C) a^2+6 b B a+3 A b^2\right )\right ) \sec ^2(c+d x)dx+\frac {3 (a B+A b) \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^2}{2 d}\right )+\frac {A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^3}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{2} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right ) \left (-b (5 A b-6 C b+3 a B) \sin \left (c+d x+\frac {\pi }{2}\right )^2+\left (3 B a^2+b (5 A+12 C) a+6 b^2 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+2 \left (\frac {1}{2} (4 A+6 C) a^2+6 b B a+3 A b^2\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^2}dx+\frac {3 (a B+A b) \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^2}{2 d}\right )+\frac {A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^3}{3 d}\) |
| \(\Big \downarrow \) 3510 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{2} \left (\frac {2 a \tan (c+d x) \left (a^2 (2 A+3 C)+6 a b B+3 A b^2\right )}{d}-\int -\left (\left (-\left ((5 A b-6 C b+3 a B) \cos ^2(c+d x) b^2\right )+6 (b B+3 a C) \cos (c+d x) b^2+3 \left (B a^3+3 b (A+2 C) a^2+6 b^2 B a+2 A b^3\right )\right ) \sec (c+d x)\right )dx\right )+\frac {3 (a B+A b) \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^2}{2 d}\right )+\frac {A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^3}{3 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{2} \left (\int \left (-\left ((5 A b-6 C b+3 a B) \cos ^2(c+d x) b^2\right )+6 (b B+3 a C) \cos (c+d x) b^2+3 \left (B a^3+3 b (A+2 C) a^2+6 b^2 B a+2 A b^3\right )\right ) \sec (c+d x)dx+\frac {2 a \tan (c+d x) \left (a^2 (2 A+3 C)+6 a b B+3 A b^2\right )}{d}\right )+\frac {3 (a B+A b) \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^2}{2 d}\right )+\frac {A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^3}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{2} \left (\int \frac {-\left ((5 A b-6 C b+3 a B) \sin \left (c+d x+\frac {\pi }{2}\right )^2 b^2\right )+6 (b B+3 a C) \sin \left (c+d x+\frac {\pi }{2}\right ) b^2+3 \left (B a^3+3 b (A+2 C) a^2+6 b^2 B a+2 A b^3\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {2 a \tan (c+d x) \left (a^2 (2 A+3 C)+6 a b B+3 A b^2\right )}{d}\right )+\frac {3 (a B+A b) \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^2}{2 d}\right )+\frac {A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^3}{3 d}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{2} \left (\int 3 \left (B a^3+3 b (A+2 C) a^2+6 b^2 B a+2 A b^3+2 b^2 (b B+3 a C) \cos (c+d x)\right ) \sec (c+d x)dx+\frac {2 a \tan (c+d x) \left (a^2 (2 A+3 C)+6 a b B+3 A b^2\right )}{d}-\frac {b^2 \sin (c+d x) (3 a B+5 A b-6 b C)}{d}\right )+\frac {3 (a B+A b) \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^2}{2 d}\right )+\frac {A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^3}{3 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{2} \left (3 \int \left (B a^3+3 b (A+2 C) a^2+6 b^2 B a+2 A b^3+2 b^2 (b B+3 a C) \cos (c+d x)\right ) \sec (c+d x)dx+\frac {2 a \tan (c+d x) \left (a^2 (2 A+3 C)+6 a b B+3 A b^2\right )}{d}-\frac {b^2 \sin (c+d x) (3 a B+5 A b-6 b C)}{d}\right )+\frac {3 (a B+A b) \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^2}{2 d}\right )+\frac {A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^3}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{2} \left (3 \int \frac {B a^3+3 b (A+2 C) a^2+6 b^2 B a+2 A b^3+2 b^2 (b B+3 a C) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {2 a \tan (c+d x) \left (a^2 (2 A+3 C)+6 a b B+3 A b^2\right )}{d}-\frac {b^2 \sin (c+d x) (3 a B+5 A b-6 b C)}{d}\right )+\frac {3 (a B+A b) \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^2}{2 d}\right )+\frac {A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^3}{3 d}\) |
| \(\Big \downarrow \) 3214 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{2} \left (3 \left (\left (a^3 B+3 a^2 b (A+2 C)+6 a b^2 B+2 A b^3\right ) \int \sec (c+d x)dx+2 b^2 x (3 a C+b B)\right )+\frac {2 a \tan (c+d x) \left (a^2 (2 A+3 C)+6 a b B+3 A b^2\right )}{d}-\frac {b^2 \sin (c+d x) (3 a B+5 A b-6 b C)}{d}\right )+\frac {3 (a B+A b) \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^2}{2 d}\right )+\frac {A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^3}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{2} \left (3 \left (\left (a^3 B+3 a^2 b (A+2 C)+6 a b^2 B+2 A b^3\right ) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+2 b^2 x (3 a C+b B)\right )+\frac {2 a \tan (c+d x) \left (a^2 (2 A+3 C)+6 a b B+3 A b^2\right )}{d}-\frac {b^2 \sin (c+d x) (3 a B+5 A b-6 b C)}{d}\right )+\frac {3 (a B+A b) \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^2}{2 d}\right )+\frac {A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^3}{3 d}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{2} \left (\frac {2 a \tan (c+d x) \left (a^2 (2 A+3 C)+6 a b B+3 A b^2\right )}{d}+3 \left (\frac {\left (a^3 B+3 a^2 b (A+2 C)+6 a b^2 B+2 A b^3\right ) \text {arctanh}(\sin (c+d x))}{d}+2 b^2 x (3 a C+b B)\right )-\frac {b^2 \sin (c+d x) (3 a B+5 A b-6 b C)}{d}\right )+\frac {3 (a B+A b) \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^2}{2 d}\right )+\frac {A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^3}{3 d}\) |
Input:
Int[(a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4,x]
Output:
(A*(a + b*Cos[c + d*x])^3*Sec[c + d*x]^2*Tan[c + d*x])/(3*d) + ((3*(A*b + a*B)*(a + b*Cos[c + d*x])^2*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (3*(2*b^2*( b*B + 3*a*C)*x + ((2*A*b^3 + a^3*B + 6*a*b^2*B + 3*a^2*b*(A + 2*C))*ArcTan h[Sin[c + d*x]])/d) - (b^2*(5*A*b + 3*a*B - 6*b*C)*Sin[c + d*x])/d + (2*a* (3*A*b^2 + 6*a*b*B + a^2*(2*A + 3*C))*Tan[c + d*x])/d)/2)/3
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_.)*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_)])*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f _.)*(x_)]^2), x_Symbol] :> Simp[(-(b*c - a*d))*(A*b^2 - a*b*B + a^2*C)*Cos[ e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b^2*f*(m + 1)*(a^2 - b^2))), x] - S imp[1/(b^2*(m + 1)*(a^2 - b^2)) Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[b*( m + 1)*((b*B - a*C)*(b*c - a*d) - A*b*(a*c - b*d)) + (b*B*(a^2*d + b^2*d*(m + 1) - a*b*c*(m + 2)) + (b*c - a*d)*(A*b^2*(m + 2) + C*(a^2 + b^2*(m + 1)) ))*Sin[e + f*x] - b*C*d*(m + 1)*(a^2 - b^2)*Sin[e + f*x]^2, x], x], x] /; F reeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && 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/(d*(n + 1)*(c^2 - d^2)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(b*d*m + a*c*(n + 1)) + (c*C - B* d)*(b*c*m + a*d*(n + 1)) - (d*(A*(a*d*(n + 2) - b*c*(n + 1)) + B*(b*d*(n + 1) - a*c*(n + 2))) - C*(b*c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x ] + b*(d*(B*c - A*d)*(m + n + 2) - C*(c^2*(m + 1) + d^2*(n + 1)))*Sin[e + f *x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d , 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && LtQ[n, -1]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Time = 2.25 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.92
| method | result | size |
| parts | \(-\frac {A \,a^{3} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (3 A \,a^{2} b +B \,a^{3}\right ) \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}+\frac {\left (B \,b^{3}+3 C a \,b^{2}\right ) \left (d x +c \right )}{d}+\frac {\left (A \,b^{3}+3 B a \,b^{2}+3 a^{2} b C \right ) \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {\left (3 a A \,b^{2}+3 B \,a^{2} b +a^{3} C \right ) \tan \left (d x +c \right )}{d}+\frac {\sin \left (d x +c \right ) C \,b^{3}}{d}\) | \(181\) |
| derivativedivides | \(\frac {-A \,a^{3} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+B \,a^{3} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+C \tan \left (d x +c \right ) a^{3}+3 A \,a^{2} b \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+3 B \tan \left (d x +c \right ) a^{2} b +3 a^{2} b C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+3 A \tan \left (d x +c \right ) a \,b^{2}+3 B a \,b^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+3 C a \,b^{2} \left (d x +c \right )+A \,b^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+B \,b^{3} \left (d x +c \right )+C \sin \left (d x +c \right ) b^{3}}{d}\) | \(235\) |
| default | \(\frac {-A \,a^{3} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+B \,a^{3} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+C \tan \left (d x +c \right ) a^{3}+3 A \,a^{2} b \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+3 B \tan \left (d x +c \right ) a^{2} b +3 a^{2} b C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+3 A \tan \left (d x +c \right ) a \,b^{2}+3 B a \,b^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+3 C a \,b^{2} \left (d x +c \right )+A \,b^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+B \,b^{3} \left (d x +c \right )+C \sin \left (d x +c \right ) b^{3}}{d}\) | \(235\) |
| parallelrisch | \(\frac {-9 \left (\cos \left (3 d x +3 c \right )+3 \cos \left (d x +c \right )\right ) \left (\frac {B \,a^{3}}{3}+a^{2} b \left (A +2 C \right )+2 B a \,b^{2}+\frac {2 A \,b^{3}}{3}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+9 \left (\cos \left (3 d x +3 c \right )+3 \cos \left (d x +c \right )\right ) \left (\frac {B \,a^{3}}{3}+a^{2} b \left (A +2 C \right )+2 B a \,b^{2}+\frac {2 A \,b^{3}}{3}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+6 b^{2} d x \left (B b +3 C a \right ) \cos \left (3 d x +3 c \right )+2 \left (a^{3} \left (2 A +3 C \right )+9 B \,a^{2} b +9 a A \,b^{2}\right ) \sin \left (3 d x +3 c \right )+6 \left (3 A \,a^{2} b +B \,a^{3}+C \,b^{3}\right ) \sin \left (2 d x +2 c \right )+3 C \sin \left (4 d x +4 c \right ) b^{3}+18 b^{2} d x \left (B b +3 C a \right ) \cos \left (d x +c \right )+12 \left (a^{2} \left (A +\frac {C}{2}\right )+\frac {3 B a b}{2}+\frac {3 A \,b^{2}}{2}\right ) \sin \left (d x +c \right ) a}{6 d \left (\cos \left (3 d x +3 c \right )+3 \cos \left (d x +c \right )\right )}\) | \(307\) |
| risch | \(x B \,b^{3}+3 a \,b^{2} C x -\frac {i {\mathrm e}^{i \left (d x +c \right )} C \,b^{3}}{2 d}+\frac {i {\mathrm e}^{-i \left (d x +c \right )} C \,b^{3}}{2 d}-\frac {i a \left (9 A a b \,{\mathrm e}^{5 i \left (d x +c \right )}+3 B \,a^{2} {\mathrm e}^{5 i \left (d x +c \right )}-18 A \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-18 B a b \,{\mathrm e}^{4 i \left (d x +c \right )}-6 C \,a^{2} {\mathrm e}^{4 i \left (d x +c \right )}-12 A \,a^{2} {\mathrm e}^{2 i \left (d x +c \right )}-36 A \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-36 B a b \,{\mathrm e}^{2 i \left (d x +c \right )}-12 C \,a^{2} {\mathrm e}^{2 i \left (d x +c \right )}-9 A a b \,{\mathrm e}^{i \left (d x +c \right )}-3 B \,a^{2} {\mathrm e}^{i \left (d x +c \right )}-4 A \,a^{2}-18 A \,b^{2}-18 B a b -6 a^{2} C \right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3}}-\frac {3 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) A b}{2 d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) A \,b^{3}}{d}-\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B}{2 d}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B a \,b^{2}}{d}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) a^{2} b C}{d}+\frac {3 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) A b}{2 d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) A \,b^{3}}{d}+\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B}{2 d}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B a \,b^{2}}{d}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) a^{2} b C}{d}\) | \(484\) |
| norman | \(\frac {\left (-B \,b^{3}-3 C a \,b^{2}\right ) x +\left (-6 B \,b^{3}-18 C a \,b^{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+\left (-2 B \,b^{3}-6 C a \,b^{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\left (-2 B \,b^{3}-6 C a \,b^{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}+\left (B \,b^{3}+3 C a \,b^{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{16}+\left (2 B \,b^{3}+6 C a \,b^{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (2 B \,b^{3}+6 C a \,b^{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}+\left (6 B \,b^{3}+18 C a \,b^{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}-\frac {\left (2 A \,a^{3}-3 A \,a^{2} b +6 a A \,b^{2}-B \,a^{3}+6 B \,a^{2} b +2 a^{3} C -2 C \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{15}}{d}-\frac {\left (2 A \,a^{3}+3 A \,a^{2} b +6 a A \,b^{2}+B \,a^{3}+6 B \,a^{2} b +2 a^{3} C +2 C \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}-\frac {\left (26 A \,a^{3}-45 A \,a^{2} b +54 a A \,b^{2}-15 B \,a^{3}+54 B \,a^{2} b +18 a^{3} C -6 C \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{3 d}-\frac {\left (26 A \,a^{3}+45 A \,a^{2} b +54 a A \,b^{2}+15 B \,a^{3}+54 B \,a^{2} b +18 a^{3} C +6 C \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 d}-\frac {\left (46 A \,a^{3}-81 A \,a^{2} b +18 a A \,b^{2}-27 B \,a^{3}+18 B \,a^{2} b +6 a^{3} C +18 C \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{3 d}-\frac {\left (46 A \,a^{3}+81 A \,a^{2} b +18 a A \,b^{2}+27 B \,a^{3}+18 B \,a^{2} b +6 a^{3} C -18 C \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{3 d}-\frac {\left (50 A \,a^{3}-45 A \,a^{2} b -90 a A \,b^{2}-15 B \,a^{3}-90 B \,a^{2} b -30 a^{3} C +18 C \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{3 d}-\frac {\left (50 A \,a^{3}+45 A \,a^{2} b -90 a A \,b^{2}+15 B \,a^{3}-90 B \,a^{2} b -30 a^{3} C -18 C \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{3 d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{5} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{3}}-\frac {\left (3 A \,a^{2} b +2 A \,b^{3}+B \,a^{3}+6 B a \,b^{2}+6 a^{2} b C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 d}+\frac {\left (3 A \,a^{2} b +2 A \,b^{3}+B \,a^{3}+6 B a \,b^{2}+6 a^{2} b C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 d}\) | \(831\) |
Input:
int((a+b*cos(d*x+c))^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^4,x,meth od=_RETURNVERBOSE)
Output:
-A*a^3/d*(-2/3-1/3*sec(d*x+c)^2)*tan(d*x+c)+(3*A*a^2*b+B*a^3)/d*(1/2*sec(d *x+c)*tan(d*x+c)+1/2*ln(sec(d*x+c)+tan(d*x+c)))+(B*b^3+3*C*a*b^2)/d*(d*x+c )+(A*b^3+3*B*a*b^2+3*C*a^2*b)/d*ln(sec(d*x+c)+tan(d*x+c))+(3*A*a*b^2+3*B*a ^2*b+C*a^3)/d*tan(d*x+c)+1/d*sin(d*x+c)*C*b^3
Time = 0.10 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.15 \[ \int (a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx=\frac {12 \, {\left (3 \, C a b^{2} + B b^{3}\right )} d x \cos \left (d x + c\right )^{3} + 3 \, {\left (B a^{3} + 3 \, {\left (A + 2 \, C\right )} a^{2} b + 6 \, B a b^{2} + 2 \, A b^{3}\right )} \cos \left (d x + c\right )^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (B a^{3} + 3 \, {\left (A + 2 \, C\right )} a^{2} b + 6 \, B a b^{2} + 2 \, A b^{3}\right )} \cos \left (d x + c\right )^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (6 \, C b^{3} \cos \left (d x + c\right )^{3} + 2 \, A a^{3} + 2 \, {\left ({\left (2 \, A + 3 \, C\right )} a^{3} + 9 \, B a^{2} b + 9 \, A a b^{2}\right )} \cos \left (d x + c\right )^{2} + 3 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{12 \, d \cos \left (d x + c\right )^{3}} \] Input:
integrate((a+b*cos(d*x+c))^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^4, x, algorithm="fricas")
Output:
1/12*(12*(3*C*a*b^2 + B*b^3)*d*x*cos(d*x + c)^3 + 3*(B*a^3 + 3*(A + 2*C)*a ^2*b + 6*B*a*b^2 + 2*A*b^3)*cos(d*x + c)^3*log(sin(d*x + c) + 1) - 3*(B*a^ 3 + 3*(A + 2*C)*a^2*b + 6*B*a*b^2 + 2*A*b^3)*cos(d*x + c)^3*log(-sin(d*x + c) + 1) + 2*(6*C*b^3*cos(d*x + c)^3 + 2*A*a^3 + 2*((2*A + 3*C)*a^3 + 9*B* a^2*b + 9*A*a*b^2)*cos(d*x + c)^2 + 3*(B*a^3 + 3*A*a^2*b)*cos(d*x + c))*si n(d*x + c))/(d*cos(d*x + c)^3)
Timed out. \[ \int (a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx=\text {Timed out} \] Input:
integrate((a+b*cos(d*x+c))**3*(A+B*cos(d*x+c)+C*cos(d*x+c)**2)*sec(d*x+c)* *4,x)
Output:
Timed out
Time = 0.05 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.43 \[ \int (a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx=\frac {4 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a^{3} + 36 \, {\left (d x + c\right )} C a b^{2} + 12 \, {\left (d x + c\right )} B b^{3} - 3 \, B a^{3} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 9 \, A a^{2} b {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 18 \, C a^{2} b {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 18 \, B a b^{2} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 6 \, A b^{3} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 12 \, C b^{3} \sin \left (d x + c\right ) + 12 \, C a^{3} \tan \left (d x + c\right ) + 36 \, B a^{2} b \tan \left (d x + c\right ) + 36 \, A a b^{2} \tan \left (d x + c\right )}{12 \, d} \] Input:
integrate((a+b*cos(d*x+c))^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^4, x, algorithm="maxima")
Output:
1/12*(4*(tan(d*x + c)^3 + 3*tan(d*x + c))*A*a^3 + 36*(d*x + c)*C*a*b^2 + 1 2*(d*x + c)*B*b^3 - 3*B*a^3*(2*sin(d*x + c)/(sin(d*x + c)^2 - 1) - log(sin (d*x + c) + 1) + log(sin(d*x + c) - 1)) - 9*A*a^2*b*(2*sin(d*x + c)/(sin(d *x + c)^2 - 1) - log(sin(d*x + c) + 1) + log(sin(d*x + c) - 1)) + 18*C*a^2 *b*(log(sin(d*x + c) + 1) - log(sin(d*x + c) - 1)) + 18*B*a*b^2*(log(sin(d *x + c) + 1) - log(sin(d*x + c) - 1)) + 6*A*b^3*(log(sin(d*x + c) + 1) - l og(sin(d*x + c) - 1)) + 12*C*b^3*sin(d*x + c) + 12*C*a^3*tan(d*x + c) + 36 *B*a^2*b*tan(d*x + c) + 36*A*a*b^2*tan(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 438 vs. \(2 (186) = 372\).
Time = 0.18 (sec) , antiderivative size = 438, normalized size of antiderivative = 2.23 \[ \int (a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx=\frac {\frac {12 \, C b^{3} \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} + 6 \, {\left (3 \, C a b^{2} + B b^{3}\right )} {\left (d x + c\right )} + 3 \, {\left (B a^{3} + 3 \, A a^{2} b + 6 \, C a^{2} b + 6 \, B a b^{2} + 2 \, A b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 3 \, {\left (B a^{3} + 3 \, A a^{2} b + 6 \, C a^{2} b + 6 \, B a b^{2} + 2 \, A b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (6 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 3 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 9 \, A a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 18 \, B a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 18 \, A a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 4 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 12 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 36 \, B a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 36 \, A a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 6 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 9 \, A a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 18 \, B a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 18 \, A a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{3}}}{6 \, d} \] Input:
integrate((a+b*cos(d*x+c))^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^4, x, algorithm="giac")
Output:
1/6*(12*C*b^3*tan(1/2*d*x + 1/2*c)/(tan(1/2*d*x + 1/2*c)^2 + 1) + 6*(3*C*a *b^2 + B*b^3)*(d*x + c) + 3*(B*a^3 + 3*A*a^2*b + 6*C*a^2*b + 6*B*a*b^2 + 2 *A*b^3)*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 3*(B*a^3 + 3*A*a^2*b + 6*C*a^ 2*b + 6*B*a*b^2 + 2*A*b^3)*log(abs(tan(1/2*d*x + 1/2*c) - 1)) - 2*(6*A*a^3 *tan(1/2*d*x + 1/2*c)^5 - 3*B*a^3*tan(1/2*d*x + 1/2*c)^5 + 6*C*a^3*tan(1/2 *d*x + 1/2*c)^5 - 9*A*a^2*b*tan(1/2*d*x + 1/2*c)^5 + 18*B*a^2*b*tan(1/2*d* x + 1/2*c)^5 + 18*A*a*b^2*tan(1/2*d*x + 1/2*c)^5 - 4*A*a^3*tan(1/2*d*x + 1 /2*c)^3 - 12*C*a^3*tan(1/2*d*x + 1/2*c)^3 - 36*B*a^2*b*tan(1/2*d*x + 1/2*c )^3 - 36*A*a*b^2*tan(1/2*d*x + 1/2*c)^3 + 6*A*a^3*tan(1/2*d*x + 1/2*c) + 3 *B*a^3*tan(1/2*d*x + 1/2*c) + 6*C*a^3*tan(1/2*d*x + 1/2*c) + 9*A*a^2*b*tan (1/2*d*x + 1/2*c) + 18*B*a^2*b*tan(1/2*d*x + 1/2*c) + 18*A*a*b^2*tan(1/2*d *x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 - 1)^3)/d
Time = 2.01 (sec) , antiderivative size = 2437, normalized size of antiderivative = 12.43 \[ \int (a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx=\text {Too large to display} \] Input:
int(((a + b*cos(c + d*x))^3*(A + B*cos(c + d*x) + C*cos(c + d*x)^2))/cos(c + d*x)^4,x)
Output:
(2*b^2*atan((b^2*(tan(c/2 + (d*x)/2)*(32*A^2*b^6 + 8*B^2*a^6 + 32*B^2*b^6 + 96*A^2*a^2*b^4 + 72*A^2*a^4*b^2 + 288*B^2*a^2*b^4 + 96*B^2*a^4*b^2 + 288 *C^2*a^2*b^4 + 288*C^2*a^4*b^2 + 192*A*B*a*b^5 + 48*A*B*a^5*b + 192*B*C*a* b^5 + 96*B*C*a^5*b + 320*A*B*a^3*b^3 + 192*A*C*a^2*b^4 + 288*A*C*a^4*b^2 + 576*B*C*a^3*b^3) - b^2*(B*b + 3*C*a)*(32*A*b^3 + 16*B*a^3 + 32*B*b^3 + 48 *A*a^2*b + 96*B*a*b^2 + 96*C*a*b^2 + 96*C*a^2*b)*1i)*(B*b + 3*C*a) + b^2*( tan(c/2 + (d*x)/2)*(32*A^2*b^6 + 8*B^2*a^6 + 32*B^2*b^6 + 96*A^2*a^2*b^4 + 72*A^2*a^4*b^2 + 288*B^2*a^2*b^4 + 96*B^2*a^4*b^2 + 288*C^2*a^2*b^4 + 288 *C^2*a^4*b^2 + 192*A*B*a*b^5 + 48*A*B*a^5*b + 192*B*C*a*b^5 + 96*B*C*a^5*b + 320*A*B*a^3*b^3 + 192*A*C*a^2*b^4 + 288*A*C*a^4*b^2 + 576*B*C*a^3*b^3) + b^2*(B*b + 3*C*a)*(32*A*b^3 + 16*B*a^3 + 32*B*b^3 + 48*A*a^2*b + 96*B*a* b^2 + 96*C*a*b^2 + 96*C*a^2*b)*1i)*(B*b + 3*C*a))/(64*A^2*B*b^9 - 64*A*B^2 *b^9 - 192*B^3*a*b^8 + 576*B^3*a^2*b^7 - 32*B^3*a^3*b^6 + 192*B^3*a^4*b^5 + 16*B^3*a^6*b^3 - 1728*C^3*a^4*b^5 + 1728*C^3*a^5*b^4 + b^2*(tan(c/2 + (d *x)/2)*(32*A^2*b^6 + 8*B^2*a^6 + 32*B^2*b^6 + 96*A^2*a^2*b^4 + 72*A^2*a^4* b^2 + 288*B^2*a^2*b^4 + 96*B^2*a^4*b^2 + 288*C^2*a^2*b^4 + 288*C^2*a^4*b^2 + 192*A*B*a*b^5 + 48*A*B*a^5*b + 192*B*C*a*b^5 + 96*B*C*a^5*b + 320*A*B*a ^3*b^3 + 192*A*C*a^2*b^4 + 288*A*C*a^4*b^2 + 576*B*C*a^3*b^3) - b^2*(B*b + 3*C*a)*(32*A*b^3 + 16*B*a^3 + 32*B*b^3 + 48*A*a^2*b + 96*B*a*b^2 + 96*C*a *b^2 + 96*C*a^2*b)*1i)*(B*b + 3*C*a)*1i - b^2*(tan(c/2 + (d*x)/2)*(32*A...
Time = 0.20 (sec) , antiderivative size = 574, normalized size of antiderivative = 2.93 \[ \int (a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx =\text {Too large to display} \] Input:
int((a+b*cos(d*x+c))^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^4,x)
Output:
( - 6*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**2*a**3*b - 9*co s(c + d*x)*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**2*a**2*b*c - 12*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**2*a*b**3 + 6*cos(c + d*x)*lo g(tan((c + d*x)/2) - 1)*a**3*b + 9*cos(c + d*x)*log(tan((c + d*x)/2) - 1)* a**2*b*c + 12*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*a*b**3 + 6*cos(c + d* x)*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**2*a**3*b + 9*cos(c + d*x)*log(t an((c + d*x)/2) + 1)*sin(c + d*x)**2*a**2*b*c + 12*cos(c + d*x)*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**2*a*b**3 - 6*cos(c + d*x)*log(tan((c + d*x)/ 2) + 1)*a**3*b - 9*cos(c + d*x)*log(tan((c + d*x)/2) + 1)*a**2*b*c - 12*co s(c + d*x)*log(tan((c + d*x)/2) + 1)*a*b**3 + 3*cos(c + d*x)*sin(c + d*x)* *3*b**3*c + 9*cos(c + d*x)*sin(c + d*x)**2*a*b**2*c*d*x + 3*cos(c + d*x)*s in(c + d*x)**2*b**4*d*x - 6*cos(c + d*x)*sin(c + d*x)*a**3*b - 3*cos(c + d *x)*sin(c + d*x)*b**3*c - 9*cos(c + d*x)*a*b**2*c*d*x - 3*cos(c + d*x)*b** 4*d*x + 2*sin(c + d*x)**3*a**4 + 3*sin(c + d*x)**3*a**3*c + 18*sin(c + d*x )**3*a**2*b**2 - 3*sin(c + d*x)*a**4 - 3*sin(c + d*x)*a**3*c - 18*sin(c + d*x)*a**2*b**2)/(3*cos(c + d*x)*d*(sin(c + d*x)**2 - 1))