Integrand size = 33, antiderivative size = 182 \[ \int (a+b \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=b^3 C x+\frac {a \left (12 b^2 (A+2 C)+a^2 (3 A+4 C)\right ) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {b \left (A b^2+a^2 (4 A+6 C)\right ) \tan (c+d x)}{2 d}+\frac {a \left (2 A b^2+a^2 (3 A+4 C)\right ) \sec (c+d x) \tan (c+d x)}{8 d}+\frac {A b (a+b \cos (c+d x))^2 \sec ^2(c+d x) \tan (c+d x)}{4 d}+\frac {A (a+b \cos (c+d x))^3 \sec ^3(c+d x) \tan (c+d x)}{4 d} \]
b^3*C*x+1/8*a*(12*b^2*(A+2*C)+a^2*(3*A+4*C))*arctanh(sin(d*x+c))/d+1/2*b*( A*b^2+a^2*(4*A+6*C))*tan(d*x+c)/d+1/8*a*(2*A*b^2+a^2*(3*A+4*C))*sec(d*x+c) *tan(d*x+c)/d+1/4*A*b*(a+b*cos(d*x+c))^2*sec(d*x+c)^2*tan(d*x+c)/d+1/4*A*( a+b*cos(d*x+c))^3*sec(d*x+c)^3*tan(d*x+c)/d
Time = 0.93 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.70 \[ \int (a+b \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\frac {8 b^3 C d x+a \left (12 b^2 (A+2 C)+a^2 (3 A+4 C)\right ) \text {arctanh}(\sin (c+d x))+\left (8 b \left (A b^2+3 a^2 (A+C)\right )+a \left (12 A b^2+a^2 (3 A+4 C)\right ) \sec (c+d x)+2 a^3 A \sec ^3(c+d x)\right ) \tan (c+d x)+8 a^2 A b \tan ^3(c+d x)}{8 d} \]
(8*b^3*C*d*x + a*(12*b^2*(A + 2*C) + a^2*(3*A + 4*C))*ArcTanh[Sin[c + d*x] ] + (8*b*(A*b^2 + 3*a^2*(A + C)) + a*(12*A*b^2 + a^2*(3*A + 4*C))*Sec[c + d*x] + 2*a^3*A*Sec[c + d*x]^3)*Tan[c + d*x] + 8*a^2*A*b*Tan[c + d*x]^3)/(8 *d)
Time = 1.33 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.02, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.424, Rules used = {3042, 3527, 3042, 3526, 27, 3042, 3510, 25, 3042, 3500, 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.
| \(\displaystyle \int \sec ^5(c+d x) (a+b \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left (A+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^5}dx\) |
| \(\Big \downarrow \) 3527 |
| \(\displaystyle \frac {1}{4} \int (a+b \cos (c+d x))^2 \left (4 b C \cos ^2(c+d x)+a (3 A+4 C) \cos (c+d x)+3 A b\right ) \sec ^4(c+d x)dx+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^3}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left (4 b C \sin \left (c+d x+\frac {\pi }{2}\right )^2+a (3 A+4 C) \sin \left (c+d x+\frac {\pi }{2}\right )+3 A b\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^4}dx+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^3}{4 d}\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \int 3 (a+b \cos (c+d x)) \left ((3 A+4 C) a^2+b (5 A+8 C) \cos (c+d x) a+2 A b^2+4 b^2 C \cos ^2(c+d x)\right ) \sec ^3(c+d x)dx+\frac {A b \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^2}{d}\right )+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^3}{4 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{4} \left (\int (a+b \cos (c+d x)) \left ((3 A+4 C) a^2+b (5 A+8 C) \cos (c+d x) a+2 A b^2+4 b^2 C \cos ^2(c+d x)\right ) \sec ^3(c+d x)dx+\frac {A b \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^2}{d}\right )+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^3}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \left (\int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right ) \left ((3 A+4 C) a^2+b (5 A+8 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a+2 A b^2+4 b^2 C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^3}dx+\frac {A b \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^2}{d}\right )+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^3}{4 d}\) |
| \(\Big \downarrow \) 3510 |
| \(\displaystyle \frac {1}{4} \left (-\frac {1}{2} \int -\left (\left (8 C \cos ^2(c+d x) b^3+4 \left ((4 A+6 C) a^2+A b^2\right ) b+a \left ((3 A+4 C) a^2+12 b^2 (A+2 C)\right ) \cos (c+d x)\right ) \sec ^2(c+d x)\right )dx+\frac {a \left (a^2 (3 A+4 C)+2 A b^2\right ) \tan (c+d x) \sec (c+d x)}{2 d}+\frac {A b \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^2}{d}\right )+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^3}{4 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{2} \int \left (8 C \cos ^2(c+d x) b^3+4 \left ((4 A+6 C) a^2+A b^2\right ) b+a \left ((3 A+4 C) a^2+12 b^2 (A+2 C)\right ) \cos (c+d x)\right ) \sec ^2(c+d x)dx+\frac {a \left (a^2 (3 A+4 C)+2 A b^2\right ) \tan (c+d x) \sec (c+d x)}{2 d}+\frac {A b \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^2}{d}\right )+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^3}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{2} \int \frac {8 C \sin \left (c+d x+\frac {\pi }{2}\right )^2 b^3+4 \left ((4 A+6 C) a^2+A b^2\right ) b+a \left ((3 A+4 C) a^2+12 b^2 (A+2 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^2}dx+\frac {a \left (a^2 (3 A+4 C)+2 A b^2\right ) \tan (c+d x) \sec (c+d x)}{2 d}+\frac {A b \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^2}{d}\right )+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^3}{4 d}\) |
| \(\Big \downarrow \) 3500 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (\int \left (8 C \cos (c+d x) b^3+a \left ((3 A+4 C) a^2+12 b^2 (A+2 C)\right )\right ) \sec (c+d x)dx+\frac {4 b \left (a^2 (4 A+6 C)+A b^2\right ) \tan (c+d x)}{d}\right )+\frac {a \left (a^2 (3 A+4 C)+2 A b^2\right ) \tan (c+d x) \sec (c+d x)}{2 d}+\frac {A b \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^2}{d}\right )+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^3}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (\int \frac {8 C \sin \left (c+d x+\frac {\pi }{2}\right ) b^3+a \left ((3 A+4 C) a^2+12 b^2 (A+2 C)\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {4 b \left (a^2 (4 A+6 C)+A b^2\right ) \tan (c+d x)}{d}\right )+\frac {a \left (a^2 (3 A+4 C)+2 A b^2\right ) \tan (c+d x) \sec (c+d x)}{2 d}+\frac {A b \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^2}{d}\right )+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^3}{4 d}\) |
| \(\Big \downarrow \) 3214 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (a \left (a^2 (3 A+4 C)+12 b^2 (A+2 C)\right ) \int \sec (c+d x)dx+\frac {4 b \left (a^2 (4 A+6 C)+A b^2\right ) \tan (c+d x)}{d}+8 b^3 C x\right )+\frac {a \left (a^2 (3 A+4 C)+2 A b^2\right ) \tan (c+d x) \sec (c+d x)}{2 d}+\frac {A b \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^2}{d}\right )+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^3}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (a \left (a^2 (3 A+4 C)+12 b^2 (A+2 C)\right ) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+\frac {4 b \left (a^2 (4 A+6 C)+A b^2\right ) \tan (c+d x)}{d}+8 b^3 C x\right )+\frac {a \left (a^2 (3 A+4 C)+2 A b^2\right ) \tan (c+d x) \sec (c+d x)}{2 d}+\frac {A b \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^2}{d}\right )+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^3}{4 d}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (\frac {a \left (a^2 (3 A+4 C)+12 b^2 (A+2 C)\right ) \text {arctanh}(\sin (c+d x))}{d}+\frac {4 b \left (a^2 (4 A+6 C)+A b^2\right ) \tan (c+d x)}{d}+8 b^3 C x\right )+\frac {a \left (a^2 (3 A+4 C)+2 A b^2\right ) \tan (c+d x) \sec (c+d x)}{2 d}+\frac {A b \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^2}{d}\right )+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^3}{4 d}\) |
(A*(a + b*Cos[c + d*x])^3*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + ((a*(2*A*b^ 2 + a^2*(3*A + 4*C))*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (A*b*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^2*Tan[c + d*x])/d + (8*b^3*C*x + (a*(12*b^2*(A + 2* C) + a^2*(3*A + 4*C))*ArcTanh[Sin[c + d*x]])/d + (4*b*(A*b^2 + a^2*(4*A + 6*C))*Tan[c + d*x])/d)/2)/4
3.6.46.3.1 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_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b^2 - a*b*B + a^2*C))*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 1)* (a^2 - b^2))), x] + Simp[1/(b*(m + 1)*(a^2 - b^2)) Int[(a + b*Sin[e + f*x ])^(m + 1)*Simp[b*(a*A - b*B + a*C)*(m + 1) - (A*b^2 - a*b*B + a^2*C + b*(A *b - a*B + b*C)*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C}, x] && LtQ[m, -1] && NeQ[a^2 - b^2, 0]
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[((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/(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*c*m + a*d*(n + 1)) - (A*d*(a*d*(n + 2) - b *c*(n + 1)) - C*(b*c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x] - b*( A*d^2*(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, 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 = 9.46 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.04
| method | result | size |
| parts | \(\frac {A \,a^{3} \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}+\frac {\left (A \,b^{3}+3 C \,a^{2} b \right ) \tan \left (d x +c \right )}{d}+\frac {\left (3 A a \,b^{2}+C \,a^{3}\right ) \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}+\frac {C \,b^{3} \left (d x +c \right )}{d}-\frac {3 A \,a^{2} b \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )}{d}+\frac {3 C a \,b^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}\) | \(190\) |
| derivativedivides | \(\frac {A \,a^{3} \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+C \,a^{3} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-3 A \,a^{2} b \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+3 C \,a^{2} b \tan \left (d x +c \right )+3 A a \,b^{2} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+3 C a \,b^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+A \,b^{3} \tan \left (d x +c \right )+C \,b^{3} \left (d x +c \right )}{d}\) | \(209\) |
| default | \(\frac {A \,a^{3} \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+C \,a^{3} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-3 A \,a^{2} b \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+3 C \,a^{2} b \tan \left (d x +c \right )+3 A a \,b^{2} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+3 C a \,b^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+A \,b^{3} \tan \left (d x +c \right )+C \,b^{3} \left (d x +c \right )}{d}\) | \(209\) |
| parallelrisch | \(\frac {-6 \left (\left (A +\frac {4 C}{3}\right ) a^{2}+4 b^{2} \left (A +2 C \right )\right ) \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+6 \left (\left (A +\frac {4 C}{3}\right ) a^{2}+4 b^{2} \left (A +2 C \right )\right ) \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+16 C \,b^{3} d x \cos \left (2 d x +2 c \right )+4 C \,b^{3} d x \cos \left (4 d x +4 c \right )+32 b \left (a^{2} \left (A +\frac {3 C}{4}\right )+\frac {A \,b^{2}}{4}\right ) \sin \left (2 d x +2 c \right )+3 \left (\left (A +\frac {4 C}{3}\right ) a^{2}+4 A \,b^{2}\right ) a \sin \left (3 d x +3 c \right )+8 \left (\left (A +\frac {3 C}{2}\right ) a^{2}+\frac {A \,b^{2}}{2}\right ) b \sin \left (4 d x +4 c \right )+\left (\left (11 A +4 C \right ) a^{3}+12 A a \,b^{2}\right ) \sin \left (d x +c \right )+12 C \,b^{3} d x}{4 d \left (\cos \left (4 d x +4 c \right )+4 \cos \left (2 d x +2 c \right )+3\right )}\) | \(297\) |
| risch | \(b^{3} C x -\frac {i \left (3 A \,a^{3} {\mathrm e}^{7 i \left (d x +c \right )}+12 A a \,b^{2} {\mathrm e}^{7 i \left (d x +c \right )}+4 C \,a^{3} {\mathrm e}^{7 i \left (d x +c \right )}-8 A \,b^{3} {\mathrm e}^{6 i \left (d x +c \right )}-24 C \,a^{2} b \,{\mathrm e}^{6 i \left (d x +c \right )}+11 A \,a^{3} {\mathrm e}^{5 i \left (d x +c \right )}+12 A a \,b^{2} {\mathrm e}^{5 i \left (d x +c \right )}+4 C \,a^{3} {\mathrm e}^{5 i \left (d x +c \right )}-48 A \,a^{2} b \,{\mathrm e}^{4 i \left (d x +c \right )}-24 A \,b^{3} {\mathrm e}^{4 i \left (d x +c \right )}-72 C \,a^{2} b \,{\mathrm e}^{4 i \left (d x +c \right )}-11 A \,a^{3} {\mathrm e}^{3 i \left (d x +c \right )}-12 A a \,b^{2} {\mathrm e}^{3 i \left (d x +c \right )}-4 C \,a^{3} {\mathrm e}^{3 i \left (d x +c \right )}-64 A \,a^{2} b \,{\mathrm e}^{2 i \left (d x +c \right )}-24 A \,b^{3} {\mathrm e}^{2 i \left (d x +c \right )}-72 C \,a^{2} b \,{\mathrm e}^{2 i \left (d x +c \right )}-3 A \,a^{3} {\mathrm e}^{i \left (d x +c \right )}-12 A a \,b^{2} {\mathrm e}^{i \left (d x +c \right )}-4 C \,a^{3} {\mathrm e}^{i \left (d x +c \right )}-16 A \,a^{2} b -8 A \,b^{3}-24 C \,a^{2} b \right )}{4 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{4}}+\frac {3 A \,a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{8 d}+\frac {3 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) A \,b^{2}}{2 d}+\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{2 d}+\frac {3 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) b^{2} C}{d}-\frac {3 A \,a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{8 d}-\frac {3 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) A \,b^{2}}{2 d}-\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{2 d}-\frac {3 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) b^{2} C}{d}\) | \(537\) |
A*a^3/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*b^3+3*C*a^2*b)/d*tan(d*x+c)+(3*A*a*b^2+C*a^3)/d*(1/2*sec(d *x+c)*tan(d*x+c)+1/2*ln(sec(d*x+c)+tan(d*x+c)))+C*b^3/d*(d*x+c)-3*A*a^2*b/ d*(-2/3-1/3*sec(d*x+c)^2)*tan(d*x+c)+3*C*a*b^2/d*ln(sec(d*x+c)+tan(d*x+c))
Time = 0.29 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.09 \[ \int (a+b \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\frac {16 \, C b^{3} d x \cos \left (d x + c\right )^{4} + {\left ({\left (3 \, A + 4 \, C\right )} a^{3} + 12 \, {\left (A + 2 \, C\right )} a b^{2}\right )} \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left ({\left (3 \, A + 4 \, C\right )} a^{3} + 12 \, {\left (A + 2 \, C\right )} a b^{2}\right )} \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (8 \, A a^{2} b \cos \left (d x + c\right ) + 2 \, A a^{3} + 8 \, {\left ({\left (2 \, A + 3 \, C\right )} a^{2} b + A b^{3}\right )} \cos \left (d x + c\right )^{3} + {\left ({\left (3 \, A + 4 \, C\right )} a^{3} + 12 \, A a b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{16 \, d \cos \left (d x + c\right )^{4}} \]
1/16*(16*C*b^3*d*x*cos(d*x + c)^4 + ((3*A + 4*C)*a^3 + 12*(A + 2*C)*a*b^2) *cos(d*x + c)^4*log(sin(d*x + c) + 1) - ((3*A + 4*C)*a^3 + 12*(A + 2*C)*a* b^2)*cos(d*x + c)^4*log(-sin(d*x + c) + 1) + 2*(8*A*a^2*b*cos(d*x + c) + 2 *A*a^3 + 8*((2*A + 3*C)*a^2*b + A*b^3)*cos(d*x + c)^3 + ((3*A + 4*C)*a^3 + 12*A*a*b^2)*cos(d*x + c)^2)*sin(d*x + c))/(d*cos(d*x + c)^4)
Timed out. \[ \int (a+b \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\text {Timed out} \]
Time = 0.21 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.43 \[ \int (a+b \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\frac {16 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a^{2} b + 16 \, {\left (d x + c\right )} C b^{3} - A a^{3} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 4 \, C a^{3} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 12 \, A a b^{2} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 24 \, C a b^{2} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 48 \, C a^{2} b \tan \left (d x + c\right ) + 16 \, A b^{3} \tan \left (d x + c\right )}{16 \, d} \]
1/16*(16*(tan(d*x + c)^3 + 3*tan(d*x + c))*A*a^2*b + 16*(d*x + c)*C*b^3 - A*a^3*(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)) - 4*C*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)) - 12*A*a*b^2*(2*sin(d*x + c)/(sin(d*x + c)^2 - 1) - log(sin(d* x + c) + 1) + log(sin(d*x + c) - 1)) + 24*C*a*b^2*(log(sin(d*x + c) + 1) - log(sin(d*x + c) - 1)) + 48*C*a^2*b*tan(d*x + c) + 16*A*b^3*tan(d*x + c)) /d
Leaf count of result is larger than twice the leaf count of optimal. 526 vs. \(2 (173) = 346\).
Time = 0.35 (sec) , antiderivative size = 526, normalized size of antiderivative = 2.89 \[ \int (a+b \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\frac {8 \, {\left (d x + c\right )} C b^{3} + {\left (3 \, A a^{3} + 4 \, C a^{3} + 12 \, A a b^{2} + 24 \, C a b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - {\left (3 \, A a^{3} + 4 \, C a^{3} + 12 \, A a b^{2} + 24 \, C a b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + \frac {2 \, {\left (5 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 4 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 24 \, A a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 24 \, C a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 12 \, A a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 8 \, A b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 3 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 4 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 40 \, A a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 72 \, C a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 12 \, A a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 24 \, A b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 4 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 40 \, A a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 72 \, C a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 12 \, A a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 24 \, A b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 5 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 4 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 24 \, A a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 24 \, C a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 12 \, A a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 8 \, 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}}}{8 \, d} \]
1/8*(8*(d*x + c)*C*b^3 + (3*A*a^3 + 4*C*a^3 + 12*A*a*b^2 + 24*C*a*b^2)*log (abs(tan(1/2*d*x + 1/2*c) + 1)) - (3*A*a^3 + 4*C*a^3 + 12*A*a*b^2 + 24*C*a *b^2)*log(abs(tan(1/2*d*x + 1/2*c) - 1)) + 2*(5*A*a^3*tan(1/2*d*x + 1/2*c) ^7 + 4*C*a^3*tan(1/2*d*x + 1/2*c)^7 - 24*A*a^2*b*tan(1/2*d*x + 1/2*c)^7 - 24*C*a^2*b*tan(1/2*d*x + 1/2*c)^7 + 12*A*a*b^2*tan(1/2*d*x + 1/2*c)^7 - 8* A*b^3*tan(1/2*d*x + 1/2*c)^7 + 3*A*a^3*tan(1/2*d*x + 1/2*c)^5 - 4*C*a^3*ta n(1/2*d*x + 1/2*c)^5 + 40*A*a^2*b*tan(1/2*d*x + 1/2*c)^5 + 72*C*a^2*b*tan( 1/2*d*x + 1/2*c)^5 - 12*A*a*b^2*tan(1/2*d*x + 1/2*c)^5 + 24*A*b^3*tan(1/2* d*x + 1/2*c)^5 + 3*A*a^3*tan(1/2*d*x + 1/2*c)^3 - 4*C*a^3*tan(1/2*d*x + 1/ 2*c)^3 - 40*A*a^2*b*tan(1/2*d*x + 1/2*c)^3 - 72*C*a^2*b*tan(1/2*d*x + 1/2* c)^3 - 12*A*a*b^2*tan(1/2*d*x + 1/2*c)^3 - 24*A*b^3*tan(1/2*d*x + 1/2*c)^3 + 5*A*a^3*tan(1/2*d*x + 1/2*c) + 4*C*a^3*tan(1/2*d*x + 1/2*c) + 24*A*a^2* b*tan(1/2*d*x + 1/2*c) + 24*C*a^2*b*tan(1/2*d*x + 1/2*c) + 12*A*a*b^2*tan( 1/2*d*x + 1/2*c) + 8*A*b^3*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 - 1)^4)/d
Time = 4.11 (sec) , antiderivative size = 1547, normalized size of antiderivative = 8.50 \[ \int (a+b \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\text {Too large to display} \]
((3*C*b^3*atan((9*A^2*a^6*sin(c/2 + (d*x)/2) + 16*C^2*a^6*sin(c/2 + (d*x)/ 2) + 64*C^2*b^6*sin(c/2 + (d*x)/2) + 144*A^2*a^2*b^4*sin(c/2 + (d*x)/2) + 72*A^2*a^4*b^2*sin(c/2 + (d*x)/2) + 576*C^2*a^2*b^4*sin(c/2 + (d*x)/2) + 1 92*C^2*a^4*b^2*sin(c/2 + (d*x)/2) + 24*A*C*a^6*sin(c/2 + (d*x)/2) + 576*A* C*a^2*b^4*sin(c/2 + (d*x)/2) + 240*A*C*a^4*b^2*sin(c/2 + (d*x)/2))/(cos(c/ 2 + (d*x)/2)*(9*A^2*a^6 + 16*C^2*a^6 + 64*C^2*b^6 + 144*A^2*a^2*b^4 + 72*A ^2*a^4*b^2 + 576*C^2*a^2*b^4 + 192*C^2*a^4*b^2 + 24*A*C*a^6 + 576*A*C*a^2* b^4 + 240*A*C*a^4*b^2))))/4 + (9*A*a^3*atanh(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)))/32 + (3*C*a^3*atanh(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)))/8 + (3*A*a^3*sin(3*c + 3*d*x))/32 + (A*b^3*sin(2*c + 2*d*x))/4 + (A*b^3*sin(4 *c + 4*d*x))/8 + (C*a^3*sin(3*c + 3*d*x))/8 + (11*A*a^3*sin(c + d*x))/32 + (C*a^3*sin(c + d*x))/8 + (3*A*a*b^2*sin(c + d*x))/8 + (9*A*a*b^2*atanh(si n(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)))/8 + (9*C*a*b^2*atanh(sin(c/2 + (d*x) /2)/cos(c/2 + (d*x)/2)))/4 + A*a^2*b*sin(2*c + 2*d*x) + (3*A*a*b^2*sin(3*c + 3*d*x))/8 + (A*a^2*b*sin(4*c + 4*d*x))/4 + (3*C*a^2*b*sin(2*c + 2*d*x)) /4 + (3*C*a^2*b*sin(4*c + 4*d*x))/8 + C*b^3*atan((9*A^2*a^6*sin(c/2 + (d*x )/2) + 16*C^2*a^6*sin(c/2 + (d*x)/2) + 64*C^2*b^6*sin(c/2 + (d*x)/2) + 144 *A^2*a^2*b^4*sin(c/2 + (d*x)/2) + 72*A^2*a^4*b^2*sin(c/2 + (d*x)/2) + 576* C^2*a^2*b^4*sin(c/2 + (d*x)/2) + 192*C^2*a^4*b^2*sin(c/2 + (d*x)/2) + 24*A *C*a^6*sin(c/2 + (d*x)/2) + 576*A*C*a^2*b^4*sin(c/2 + (d*x)/2) + 240*A*...