Integrand size = 21, antiderivative size = 222 \[ \int (a+b \cos (c+d x))^4 \sec ^7(c+d x) \, dx=\frac {\left (5 a^4+36 a^2 b^2+8 b^4\right ) \text {arctanh}(\sin (c+d x))}{16 d}+\frac {4 a b \left (4 a^2+5 b^2\right ) \tan (c+d x)}{5 d}+\frac {\left (5 a^4+36 a^2 b^2+8 b^4\right ) \sec (c+d x) \tan (c+d x)}{16 d}+\frac {a^2 \left (5 a^2+32 b^2\right ) \sec ^3(c+d x) \tan (c+d x)}{24 d}+\frac {7 a^3 b \sec ^4(c+d x) \tan (c+d x)}{15 d}+\frac {a^2 (a+b \cos (c+d x))^2 \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac {4 a b \left (4 a^2+5 b^2\right ) \tan ^3(c+d x)}{15 d} \] Output:
1/16*(5*a^4+36*a^2*b^2+8*b^4)*arctanh(sin(d*x+c))/d+4/5*a*b*(4*a^2+5*b^2)* tan(d*x+c)/d+1/16*(5*a^4+36*a^2*b^2+8*b^4)*sec(d*x+c)*tan(d*x+c)/d+1/24*a^ 2*(5*a^2+32*b^2)*sec(d*x+c)^3*tan(d*x+c)/d+7/15*a^3*b*sec(d*x+c)^4*tan(d*x +c)/d+1/6*a^2*(a+b*cos(d*x+c))^2*sec(d*x+c)^5*tan(d*x+c)/d+4/15*a*b*(4*a^2 +5*b^2)*tan(d*x+c)^3/d
Time = 0.96 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.69 \[ \int (a+b \cos (c+d x))^4 \sec ^7(c+d x) \, dx=\frac {15 \left (5 a^4+36 a^2 b^2+8 b^4\right ) \text {arctanh}(\sin (c+d x))+\tan (c+d x) \left (15 \left (5 a^4+36 a^2 b^2+8 b^4\right ) \sec (c+d x)+10 a^2 \left (5 a^2+36 b^2\right ) \sec ^3(c+d x)+40 a^4 \sec ^5(c+d x)+64 a b \left (15 \left (a^2+b^2\right )+5 \left (2 a^2+b^2\right ) \tan ^2(c+d x)+3 a^2 \tan ^4(c+d x)\right )\right )}{240 d} \] Input:
Integrate[(a + b*Cos[c + d*x])^4*Sec[c + d*x]^7,x]
Output:
(15*(5*a^4 + 36*a^2*b^2 + 8*b^4)*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(15* (5*a^4 + 36*a^2*b^2 + 8*b^4)*Sec[c + d*x] + 10*a^2*(5*a^2 + 36*b^2)*Sec[c + d*x]^3 + 40*a^4*Sec[c + d*x]^5 + 64*a*b*(15*(a^2 + b^2) + 5*(2*a^2 + b^2 )*Tan[c + d*x]^2 + 3*a^2*Tan[c + d*x]^4)))/(240*d)
Time = 1.28 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.93, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.762, Rules used = {3042, 3271, 3042, 3510, 25, 3042, 3500, 27, 3042, 3227, 3042, 4254, 2009, 4255, 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.
| \(\displaystyle \int \sec ^7(c+d x) (a+b \cos (c+d x))^4 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^4}{\sin \left (c+d x+\frac {\pi }{2}\right )^7}dx\) |
| \(\Big \downarrow \) 3271 |
| \(\displaystyle \frac {1}{6} \int (a+b \cos (c+d x)) \left (14 b a^2+\left (5 a^2+18 b^2\right ) \cos (c+d x) a+3 b \left (a^2+2 b^2\right ) \cos ^2(c+d x)\right ) \sec ^6(c+d x)dx+\frac {a^2 \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^2}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right ) \left (14 b a^2+\left (5 a^2+18 b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) a+3 b \left (a^2+2 b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^6}dx+\frac {a^2 \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^2}{6 d}\) |
| \(\Big \downarrow \) 3510 |
| \(\displaystyle \frac {1}{6} \left (\frac {14 a^3 b \tan (c+d x) \sec ^4(c+d x)}{5 d}-\frac {1}{5} \int -\left (\left (5 \left (5 a^2+32 b^2\right ) a^2+24 b \left (4 a^2+5 b^2\right ) \cos (c+d x) a+15 b^2 \left (a^2+2 b^2\right ) \cos ^2(c+d x)\right ) \sec ^5(c+d x)\right )dx\right )+\frac {a^2 \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^2}{6 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \int \left (5 \left (5 a^2+32 b^2\right ) a^2+24 b \left (4 a^2+5 b^2\right ) \cos (c+d x) a+15 b^2 \left (a^2+2 b^2\right ) \cos ^2(c+d x)\right ) \sec ^5(c+d x)dx+\frac {14 a^3 b \tan (c+d x) \sec ^4(c+d x)}{5 d}\right )+\frac {a^2 \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^2}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \int \frac {5 \left (5 a^2+32 b^2\right ) a^2+24 b \left (4 a^2+5 b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) a+15 b^2 \left (a^2+2 b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\sin \left (c+d x+\frac {\pi }{2}\right )^5}dx+\frac {14 a^3 b \tan (c+d x) \sec ^4(c+d x)}{5 d}\right )+\frac {a^2 \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^2}{6 d}\) |
| \(\Big \downarrow \) 3500 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \int 3 \left (32 a b \left (4 a^2+5 b^2\right )+5 \left (5 a^4+36 b^2 a^2+8 b^4\right ) \cos (c+d x)\right ) \sec ^4(c+d x)dx+\frac {5 a^2 \left (5 a^2+32 b^2\right ) \tan (c+d x) \sec ^3(c+d x)}{4 d}\right )+\frac {14 a^3 b \tan (c+d x) \sec ^4(c+d x)}{5 d}\right )+\frac {a^2 \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^2}{6 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {3}{4} \int \left (32 a b \left (4 a^2+5 b^2\right )+5 \left (5 a^4+36 b^2 a^2+8 b^4\right ) \cos (c+d x)\right ) \sec ^4(c+d x)dx+\frac {5 a^2 \left (5 a^2+32 b^2\right ) \tan (c+d x) \sec ^3(c+d x)}{4 d}\right )+\frac {14 a^3 b \tan (c+d x) \sec ^4(c+d x)}{5 d}\right )+\frac {a^2 \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^2}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {3}{4} \int \frac {32 a b \left (4 a^2+5 b^2\right )+5 \left (5 a^4+36 b^2 a^2+8 b^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^4}dx+\frac {5 a^2 \left (5 a^2+32 b^2\right ) \tan (c+d x) \sec ^3(c+d x)}{4 d}\right )+\frac {14 a^3 b \tan (c+d x) \sec ^4(c+d x)}{5 d}\right )+\frac {a^2 \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^2}{6 d}\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {3}{4} \left (32 a b \left (4 a^2+5 b^2\right ) \int \sec ^4(c+d x)dx+5 \left (5 a^4+36 a^2 b^2+8 b^4\right ) \int \sec ^3(c+d x)dx\right )+\frac {5 a^2 \left (5 a^2+32 b^2\right ) \tan (c+d x) \sec ^3(c+d x)}{4 d}\right )+\frac {14 a^3 b \tan (c+d x) \sec ^4(c+d x)}{5 d}\right )+\frac {a^2 \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^2}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {3}{4} \left (32 a b \left (4 a^2+5 b^2\right ) \int \csc \left (c+d x+\frac {\pi }{2}\right )^4dx+5 \left (5 a^4+36 a^2 b^2+8 b^4\right ) \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx\right )+\frac {5 a^2 \left (5 a^2+32 b^2\right ) \tan (c+d x) \sec ^3(c+d x)}{4 d}\right )+\frac {14 a^3 b \tan (c+d x) \sec ^4(c+d x)}{5 d}\right )+\frac {a^2 \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^2}{6 d}\) |
| \(\Big \downarrow \) 4254 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {3}{4} \left (5 \left (5 a^4+36 a^2 b^2+8 b^4\right ) \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx-\frac {32 a b \left (4 a^2+5 b^2\right ) \int \left (\tan ^2(c+d x)+1\right )d(-\tan (c+d x))}{d}\right )+\frac {5 a^2 \left (5 a^2+32 b^2\right ) \tan (c+d x) \sec ^3(c+d x)}{4 d}\right )+\frac {14 a^3 b \tan (c+d x) \sec ^4(c+d x)}{5 d}\right )+\frac {a^2 \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^2}{6 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {3}{4} \left (5 \left (5 a^4+36 a^2 b^2+8 b^4\right ) \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx-\frac {32 a b \left (4 a^2+5 b^2\right ) \left (-\frac {1}{3} \tan ^3(c+d x)-\tan (c+d x)\right )}{d}\right )+\frac {5 a^2 \left (5 a^2+32 b^2\right ) \tan (c+d x) \sec ^3(c+d x)}{4 d}\right )+\frac {14 a^3 b \tan (c+d x) \sec ^4(c+d x)}{5 d}\right )+\frac {a^2 \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^2}{6 d}\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {3}{4} \left (5 \left (5 a^4+36 a^2 b^2+8 b^4\right ) \left (\frac {1}{2} \int \sec (c+d x)dx+\frac {\tan (c+d x) \sec (c+d x)}{2 d}\right )-\frac {32 a b \left (4 a^2+5 b^2\right ) \left (-\frac {1}{3} \tan ^3(c+d x)-\tan (c+d x)\right )}{d}\right )+\frac {5 a^2 \left (5 a^2+32 b^2\right ) \tan (c+d x) \sec ^3(c+d x)}{4 d}\right )+\frac {14 a^3 b \tan (c+d x) \sec ^4(c+d x)}{5 d}\right )+\frac {a^2 \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^2}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {3}{4} \left (5 \left (5 a^4+36 a^2 b^2+8 b^4\right ) \left (\frac {1}{2} \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+\frac {\tan (c+d x) \sec (c+d x)}{2 d}\right )-\frac {32 a b \left (4 a^2+5 b^2\right ) \left (-\frac {1}{3} \tan ^3(c+d x)-\tan (c+d x)\right )}{d}\right )+\frac {5 a^2 \left (5 a^2+32 b^2\right ) \tan (c+d x) \sec ^3(c+d x)}{4 d}\right )+\frac {14 a^3 b \tan (c+d x) \sec ^4(c+d x)}{5 d}\right )+\frac {a^2 \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^2}{6 d}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {a^2 \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^2}{6 d}+\frac {1}{6} \left (\frac {14 a^3 b \tan (c+d x) \sec ^4(c+d x)}{5 d}+\frac {1}{5} \left (\frac {5 a^2 \left (5 a^2+32 b^2\right ) \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac {3}{4} \left (5 \left (5 a^4+36 a^2 b^2+8 b^4\right ) \left (\frac {\text {arctanh}(\sin (c+d x))}{2 d}+\frac {\tan (c+d x) \sec (c+d x)}{2 d}\right )-\frac {32 a b \left (4 a^2+5 b^2\right ) \left (-\frac {1}{3} \tan ^3(c+d x)-\tan (c+d x)\right )}{d}\right )\right )\right )\) |
Input:
Int[(a + b*Cos[c + d*x])^4*Sec[c + d*x]^7,x]
Output:
(a^2*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^5*Tan[c + d*x])/(6*d) + ((14*a^3* b*Sec[c + d*x]^4*Tan[c + d*x])/(5*d) + ((5*a^2*(5*a^2 + 32*b^2)*Sec[c + d* x]^3*Tan[c + d*x])/(4*d) + (3*(5*(5*a^4 + 36*a^2*b^2 + 8*b^4)*(ArcTanh[Sin [c + d*x]]/(2*d) + (Sec[c + d*x]*Tan[c + d*x])/(2*d)) - (32*a*b*(4*a^2 + 5 *b^2)*(-Tan[c + d*x] - Tan[c + d*x]^3/3))/d))/4)/5)/6
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[c Int[(b*Sin[e + f*x])^m, x], x] + Simp[d/b Int [(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-(b^2*c^2 - 2*a*b*c*d + a^2*d^2))*Co s[e + f*x]*(a + b*Sin[e + f*x])^(m - 2)*((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 - 3)*(c + d*Sin[e + f*x])^(n + 1)*Simp[b*(m - 2)*(b*c - a*d)^ 2 + a*d*(n + 1)*(c*(a^2 + b^2) - 2*a*b*d) + (b*(n + 1)*(a*b*c^2 + c*d*(a^2 + b^2) - 3*a*b*d^2) - a*(n + 2)*(b*c - a*d)^2)*Sin[e + f*x] + b*(b^2*(c^2 - d^2) - m*(b*c - a*d)^2 + d*n*(2*a*b*c - d*(a^2 + b^2)))*Sin[e + f*x]^2, x] , x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 2] && LtQ[n, -1] && (IntegerQ[m] || IntegersQ[2*m, 2*n])
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[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp andIntegrand[(1 + x^2)^(n/2 - 1), x], x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Csc[c + d*x])^(n - 1)/(d*(n - 1))), x] + Simp[b^2*((n - 2)/(n - 1)) Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[2*n]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Time = 18.26 (sec) , antiderivative size = 209, normalized size of antiderivative = 0.94
| method | result | size |
| derivativedivides | \(\frac {a^{4} \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )-4 a^{3} b \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )+6 a^{2} b^{2} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-4 a \,b^{3} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+b^{4} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}\) | \(209\) |
| default | \(\frac {a^{4} \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )-4 a^{3} b \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )+6 a^{2} b^{2} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-4 a \,b^{3} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+b^{4} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}\) | \(209\) |
| parts | \(\frac {a^{4} \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )}{d}+\frac {b^{4} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}-\frac {4 a^{3} b \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )}{d}+\frac {6 a^{2} b^{2} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}-\frac {4 a \,b^{3} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}\) | \(220\) |
| parallelrisch | \(\frac {-1125 \left (\frac {\cos \left (6 d x +6 c \right )}{15}+\frac {2 \cos \left (4 d x +4 c \right )}{5}+\cos \left (2 d x +2 c \right )+\frac {2}{3}\right ) \left (a^{4}+\frac {36}{5} a^{2} b^{2}+\frac {8}{5} b^{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+1125 \left (\frac {\cos \left (6 d x +6 c \right )}{15}+\frac {2 \cos \left (4 d x +4 c \right )}{5}+\cos \left (2 d x +2 c \right )+\frac {2}{3}\right ) \left (a^{4}+\frac {36}{5} a^{2} b^{2}+\frac {8}{5} b^{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\left (850 a^{4}+6120 a^{2} b^{2}+720 b^{4}\right ) \sin \left (3 d x +3 c \right )+\left (150 a^{4}+1080 a^{2} b^{2}+240 b^{4}\right ) \sin \left (5 d x +5 c \right )+\left (7680 a^{3} b +5760 a \,b^{3}\right ) \sin \left (2 d x +2 c \right )+\left (3072 a^{3} b +3840 a \,b^{3}\right ) \sin \left (4 d x +4 c \right )+\left (512 a^{3} b +640 a \,b^{3}\right ) \sin \left (6 d x +6 c \right )+1980 \sin \left (d x +c \right ) \left (a^{4}+\frac {28}{11} a^{2} b^{2}+\frac {8}{33} b^{4}\right )}{240 d \left (\cos \left (6 d x +6 c \right )+6 \cos \left (4 d x +4 c \right )+15 \cos \left (2 d x +2 c \right )+10\right )}\) | \(322\) |
| risch | \(-\frac {i \left (-640 a \,b^{3}-512 a^{3} b -7680 a^{3} b \,{\mathrm e}^{4 i \left (d x +c \right )}-540 b^{2} a^{2} {\mathrm e}^{i \left (d x +c \right )}-3072 a^{3} b \,{\mathrm e}^{2 i \left (d x +c \right )}-3060 a^{2} b^{2} {\mathrm e}^{3 i \left (d x +c \right )}+540 a^{2} b^{2} {\mathrm e}^{11 i \left (d x +c \right )}-5120 a^{3} b \,{\mathrm e}^{6 i \left (d x +c \right )}-6400 a \,b^{3} {\mathrm e}^{6 i \left (d x +c \right )}-7680 a \,b^{3} {\mathrm e}^{4 i \left (d x +c \right )}+3060 a^{2} b^{2} {\mathrm e}^{9 i \left (d x +c \right )}-1920 a \,b^{3} {\mathrm e}^{8 i \left (d x +c \right )}-2520 a^{2} b^{2} {\mathrm e}^{5 i \left (d x +c \right )}+2520 \,{\mathrm e}^{7 i \left (d x +c \right )} a^{2} b^{2}-3840 a \,b^{3} {\mathrm e}^{2 i \left (d x +c \right )}+75 a^{4} {\mathrm e}^{11 i \left (d x +c \right )}-75 a^{4} {\mathrm e}^{i \left (d x +c \right )}+240 \,{\mathrm e}^{7 i \left (d x +c \right )} b^{4}-360 b^{4} {\mathrm e}^{3 i \left (d x +c \right )}-990 \,{\mathrm e}^{5 i \left (d x +c \right )} a^{4}-120 b^{4} {\mathrm e}^{i \left (d x +c \right )}+360 b^{4} {\mathrm e}^{9 i \left (d x +c \right )}+120 b^{4} {\mathrm e}^{11 i \left (d x +c \right )}+425 a^{4} {\mathrm e}^{9 i \left (d x +c \right )}-425 a^{4} {\mathrm e}^{3 i \left (d x +c \right )}-240 b^{4} {\mathrm e}^{5 i \left (d x +c \right )}+990 a^{4} {\mathrm e}^{7 i \left (d x +c \right )}\right )}{120 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{6}}-\frac {5 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{16 d}-\frac {9 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) b^{2}}{4 d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) b^{4}}{2 d}+\frac {5 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{16 d}+\frac {9 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) b^{2}}{4 d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) b^{4}}{2 d}\) | \(541\) |
Input:
int((a+cos(d*x+c)*b)^4*sec(d*x+c)^7,x,method=_RETURNVERBOSE)
Output:
1/d*(a^4*(-(-1/6*sec(d*x+c)^5-5/24*sec(d*x+c)^3-5/16*sec(d*x+c))*tan(d*x+c )+5/16*ln(sec(d*x+c)+tan(d*x+c)))-4*a^3*b*(-8/15-1/5*sec(d*x+c)^4-4/15*sec (d*x+c)^2)*tan(d*x+c)+6*a^2*b^2*(-(-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)))-4*a*b^3*(-2/3-1/3*sec(d*x+c)^2)*tan(d *x+c)+b^4*(1/2*sec(d*x+c)*tan(d*x+c)+1/2*ln(sec(d*x+c)+tan(d*x+c))))
Time = 0.09 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.98 \[ \int (a+b \cos (c+d x))^4 \sec ^7(c+d x) \, dx=\frac {15 \, {\left (5 \, a^{4} + 36 \, a^{2} b^{2} + 8 \, b^{4}\right )} \cos \left (d x + c\right )^{6} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \, {\left (5 \, a^{4} + 36 \, a^{2} b^{2} + 8 \, b^{4}\right )} \cos \left (d x + c\right )^{6} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (128 \, {\left (4 \, a^{3} b + 5 \, a b^{3}\right )} \cos \left (d x + c\right )^{5} + 192 \, a^{3} b \cos \left (d x + c\right ) + 15 \, {\left (5 \, a^{4} + 36 \, a^{2} b^{2} + 8 \, b^{4}\right )} \cos \left (d x + c\right )^{4} + 40 \, a^{4} + 64 \, {\left (4 \, a^{3} b + 5 \, a b^{3}\right )} \cos \left (d x + c\right )^{3} + 10 \, {\left (5 \, a^{4} + 36 \, a^{2} b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{480 \, d \cos \left (d x + c\right )^{6}} \] Input:
integrate((a+b*cos(d*x+c))^4*sec(d*x+c)^7,x, algorithm="fricas")
Output:
1/480*(15*(5*a^4 + 36*a^2*b^2 + 8*b^4)*cos(d*x + c)^6*log(sin(d*x + c) + 1 ) - 15*(5*a^4 + 36*a^2*b^2 + 8*b^4)*cos(d*x + c)^6*log(-sin(d*x + c) + 1) + 2*(128*(4*a^3*b + 5*a*b^3)*cos(d*x + c)^5 + 192*a^3*b*cos(d*x + c) + 15* (5*a^4 + 36*a^2*b^2 + 8*b^4)*cos(d*x + c)^4 + 40*a^4 + 64*(4*a^3*b + 5*a*b ^3)*cos(d*x + c)^3 + 10*(5*a^4 + 36*a^2*b^2)*cos(d*x + c)^2)*sin(d*x + c)) /(d*cos(d*x + c)^6)
Timed out. \[ \int (a+b \cos (c+d x))^4 \sec ^7(c+d x) \, dx=\text {Timed out} \] Input:
integrate((a+b*cos(d*x+c))**4*sec(d*x+c)**7,x)
Output:
Timed out
Time = 0.06 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.24 \[ \int (a+b \cos (c+d x))^4 \sec ^7(c+d x) \, dx=\frac {128 \, {\left (3 \, \tan \left (d x + c\right )^{5} + 10 \, \tan \left (d x + c\right )^{3} + 15 \, \tan \left (d x + c\right )\right )} a^{3} b + 640 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} a b^{3} - 5 \, a^{4} {\left (\frac {2 \, {\left (15 \, \sin \left (d x + c\right )^{5} - 40 \, \sin \left (d x + c\right )^{3} + 33 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{6} - 3 \, \sin \left (d x + c\right )^{4} + 3 \, \sin \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 180 \, a^{2} b^{2} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 120 \, b^{4} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )}}{480 \, d} \] Input:
integrate((a+b*cos(d*x+c))^4*sec(d*x+c)^7,x, algorithm="maxima")
Output:
1/480*(128*(3*tan(d*x + c)^5 + 10*tan(d*x + c)^3 + 15*tan(d*x + c))*a^3*b + 640*(tan(d*x + c)^3 + 3*tan(d*x + c))*a*b^3 - 5*a^4*(2*(15*sin(d*x + c)^ 5 - 40*sin(d*x + c)^3 + 33*sin(d*x + c))/(sin(d*x + c)^6 - 3*sin(d*x + c)^ 4 + 3*sin(d*x + c)^2 - 1) - 15*log(sin(d*x + c) + 1) + 15*log(sin(d*x + c) - 1)) - 180*a^2*b^2*(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)) - 120*b^4*(2*sin(d*x + c)/(sin(d*x + c)^2 - 1) - log(sin(d*x + c) + 1 ) + log(sin(d*x + c) - 1)))/d
Leaf count of result is larger than twice the leaf count of optimal. 592 vs. \(2 (208) = 416\).
Time = 0.56 (sec) , antiderivative size = 592, normalized size of antiderivative = 2.67 \[ \int (a+b \cos (c+d x))^4 \sec ^7(c+d x) \, dx =\text {Too large to display} \] Input:
integrate((a+b*cos(d*x+c))^4*sec(d*x+c)^7,x, algorithm="giac")
Output:
1/240*(15*(5*a^4 + 36*a^2*b^2 + 8*b^4)*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 15*(5*a^4 + 36*a^2*b^2 + 8*b^4)*log(abs(tan(1/2*d*x + 1/2*c) - 1)) + 2*( 165*a^4*tan(1/2*d*x + 1/2*c)^11 - 960*a^3*b*tan(1/2*d*x + 1/2*c)^11 + 900* a^2*b^2*tan(1/2*d*x + 1/2*c)^11 - 960*a*b^3*tan(1/2*d*x + 1/2*c)^11 + 120* b^4*tan(1/2*d*x + 1/2*c)^11 + 25*a^4*tan(1/2*d*x + 1/2*c)^9 + 2240*a^3*b*t an(1/2*d*x + 1/2*c)^9 - 1260*a^2*b^2*tan(1/2*d*x + 1/2*c)^9 + 3520*a*b^3*t an(1/2*d*x + 1/2*c)^9 - 360*b^4*tan(1/2*d*x + 1/2*c)^9 + 450*a^4*tan(1/2*d *x + 1/2*c)^7 - 4992*a^3*b*tan(1/2*d*x + 1/2*c)^7 + 360*a^2*b^2*tan(1/2*d* x + 1/2*c)^7 - 5760*a*b^3*tan(1/2*d*x + 1/2*c)^7 + 240*b^4*tan(1/2*d*x + 1 /2*c)^7 + 450*a^4*tan(1/2*d*x + 1/2*c)^5 + 4992*a^3*b*tan(1/2*d*x + 1/2*c) ^5 + 360*a^2*b^2*tan(1/2*d*x + 1/2*c)^5 + 5760*a*b^3*tan(1/2*d*x + 1/2*c)^ 5 + 240*b^4*tan(1/2*d*x + 1/2*c)^5 + 25*a^4*tan(1/2*d*x + 1/2*c)^3 - 2240* a^3*b*tan(1/2*d*x + 1/2*c)^3 - 1260*a^2*b^2*tan(1/2*d*x + 1/2*c)^3 - 3520* a*b^3*tan(1/2*d*x + 1/2*c)^3 - 360*b^4*tan(1/2*d*x + 1/2*c)^3 + 165*a^4*ta n(1/2*d*x + 1/2*c) + 960*a^3*b*tan(1/2*d*x + 1/2*c) + 900*a^2*b^2*tan(1/2* d*x + 1/2*c) + 960*a*b^3*tan(1/2*d*x + 1/2*c) + 120*b^4*tan(1/2*d*x + 1/2* c))/(tan(1/2*d*x + 1/2*c)^2 - 1)^6)/d
Time = 45.09 (sec) , antiderivative size = 370, normalized size of antiderivative = 1.67 \[ \int (a+b \cos (c+d x))^4 \sec ^7(c+d x) \, dx=\frac {\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (\frac {5\,a^4}{8}+\frac {9\,a^2\,b^2}{2}+b^4\right )}{d}+\frac {\left (\frac {11\,a^4}{8}-8\,a^3\,b+\frac {15\,a^2\,b^2}{2}-8\,a\,b^3+b^4\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (\frac {5\,a^4}{24}+\frac {56\,a^3\,b}{3}-\frac {21\,a^2\,b^2}{2}+\frac {88\,a\,b^3}{3}-3\,b^4\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (\frac {15\,a^4}{4}-\frac {208\,a^3\,b}{5}+3\,a^2\,b^2-48\,a\,b^3+2\,b^4\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {15\,a^4}{4}+\frac {208\,a^3\,b}{5}+3\,a^2\,b^2+48\,a\,b^3+2\,b^4\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {5\,a^4}{24}-\frac {56\,a^3\,b}{3}-\frac {21\,a^2\,b^2}{2}-\frac {88\,a\,b^3}{3}-3\,b^4\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {11\,a^4}{8}+8\,a^3\,b+\frac {15\,a^2\,b^2}{2}+8\,a\,b^3+b^4\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )} \] Input:
int((a + b*cos(c + d*x))^4/cos(c + d*x)^7,x)
Output:
(atanh(tan(c/2 + (d*x)/2))*((5*a^4)/8 + b^4 + (9*a^2*b^2)/2))/d + (tan(c/2 + (d*x)/2)^9*((88*a*b^3)/3 + (56*a^3*b)/3 + (5*a^4)/24 - 3*b^4 - (21*a^2* b^2)/2) - tan(c/2 + (d*x)/2)^3*((88*a*b^3)/3 + (56*a^3*b)/3 - (5*a^4)/24 + 3*b^4 + (21*a^2*b^2)/2) + tan(c/2 + (d*x)/2)^5*(48*a*b^3 + (208*a^3*b)/5 + (15*a^4)/4 + 2*b^4 + 3*a^2*b^2) + tan(c/2 + (d*x)/2)^7*((15*a^4)/4 - (20 8*a^3*b)/5 - 48*a*b^3 + 2*b^4 + 3*a^2*b^2) + tan(c/2 + (d*x)/2)*(8*a*b^3 + 8*a^3*b + (11*a^4)/8 + b^4 + (15*a^2*b^2)/2) + tan(c/2 + (d*x)/2)^11*((11 *a^4)/8 - 8*a^3*b - 8*a*b^3 + b^4 + (15*a^2*b^2)/2))/(d*(15*tan(c/2 + (d*x )/2)^4 - 6*tan(c/2 + (d*x)/2)^2 - 20*tan(c/2 + (d*x)/2)^6 + 15*tan(c/2 + ( d*x)/2)^8 - 6*tan(c/2 + (d*x)/2)^10 + tan(c/2 + (d*x)/2)^12 + 1))
Time = 0.19 (sec) , antiderivative size = 1048, normalized size of antiderivative = 4.72 \[ \int (a+b \cos (c+d x))^4 \sec ^7(c+d x) \, dx =\text {Too large to display} \] Input:
int((a+b*cos(d*x+c))^4*sec(d*x+c)^7,x)
Output:
( - 75*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**6*a**4 - 540*c os(c + d*x)*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**6*a**2*b**2 - 120*cos( c + d*x)*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**6*b**4 + 225*cos(c + d*x) *log(tan((c + d*x)/2) - 1)*sin(c + d*x)**4*a**4 + 1620*cos(c + d*x)*log(ta n((c + d*x)/2) - 1)*sin(c + d*x)**4*a**2*b**2 + 360*cos(c + d*x)*log(tan(( c + d*x)/2) - 1)*sin(c + d*x)**4*b**4 - 225*cos(c + d*x)*log(tan((c + d*x) /2) - 1)*sin(c + d*x)**2*a**4 - 1620*cos(c + d*x)*log(tan((c + d*x)/2) - 1 )*sin(c + d*x)**2*a**2*b**2 - 360*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*s in(c + d*x)**2*b**4 + 75*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*a**4 + 540 *cos(c + d*x)*log(tan((c + d*x)/2) - 1)*a**2*b**2 + 120*cos(c + d*x)*log(t an((c + d*x)/2) - 1)*b**4 + 75*cos(c + d*x)*log(tan((c + d*x)/2) + 1)*sin( c + d*x)**6*a**4 + 540*cos(c + d*x)*log(tan((c + d*x)/2) + 1)*sin(c + d*x) **6*a**2*b**2 + 120*cos(c + d*x)*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**6 *b**4 - 225*cos(c + d*x)*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**4*a**4 - 1620*cos(c + d*x)*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**4*a**2*b**2 - 36 0*cos(c + d*x)*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**4*b**4 + 225*cos(c + d*x)*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**2*a**4 + 1620*cos(c + d*x)* log(tan((c + d*x)/2) + 1)*sin(c + d*x)**2*a**2*b**2 + 360*cos(c + d*x)*log (tan((c + d*x)/2) + 1)*sin(c + d*x)**2*b**4 - 75*cos(c + d*x)*log(tan((c + d*x)/2) + 1)*a**4 - 540*cos(c + d*x)*log(tan((c + d*x)/2) + 1)*a**2*b*...