Integrand size = 21, antiderivative size = 169 \[ \int (a+b \cos (c+d x))^3 \sec ^6(c+d x) \, dx=\frac {b \left (9 a^2+4 b^2\right ) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {a \left (4 a^2+15 b^2\right ) \tan (c+d x)}{5 d}+\frac {b \left (9 a^2+4 b^2\right ) \sec (c+d x) \tan (c+d x)}{8 d}+\frac {11 a^2 b \sec ^3(c+d x) \tan (c+d x)}{20 d}+\frac {a^2 (a+b \cos (c+d x)) \sec ^4(c+d x) \tan (c+d x)}{5 d}+\frac {a \left (4 a^2+15 b^2\right ) \tan ^3(c+d x)}{15 d} \] Output:
1/8*b*(9*a^2+4*b^2)*arctanh(sin(d*x+c))/d+1/5*a*(4*a^2+15*b^2)*tan(d*x+c)/ d+1/8*b*(9*a^2+4*b^2)*sec(d*x+c)*tan(d*x+c)/d+11/20*a^2*b*sec(d*x+c)^3*tan (d*x+c)/d+1/5*a^2*(a+b*cos(d*x+c))*sec(d*x+c)^4*tan(d*x+c)/d+1/15*a*(4*a^2 +15*b^2)*tan(d*x+c)^3/d
Time = 0.91 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.71 \[ \int (a+b \cos (c+d x))^3 \sec ^6(c+d x) \, dx=\frac {15 b \left (9 a^2+4 b^2\right ) \text {arctanh}(\sin (c+d x))+\tan (c+d x) \left (15 b \left (9 a^2+4 b^2\right ) \sec (c+d x)+90 a^2 b \sec ^3(c+d x)+8 a \left (15 \left (a^2+3 b^2\right )+5 \left (2 a^2+3 b^2\right ) \tan ^2(c+d x)+3 a^2 \tan ^4(c+d x)\right )\right )}{120 d} \] Input:
Integrate[(a + b*Cos[c + d*x])^3*Sec[c + d*x]^6,x]
Output:
(15*b*(9*a^2 + 4*b^2)*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(15*b*(9*a^2 + 4*b^2)*Sec[c + d*x] + 90*a^2*b*Sec[c + d*x]^3 + 8*a*(15*(a^2 + 3*b^2) + 5* (2*a^2 + 3*b^2)*Tan[c + d*x]^2 + 3*a^2*Tan[c + d*x]^4)))/(120*d)
Time = 0.90 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.92, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {3042, 3271, 3042, 3500, 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.
| \(\displaystyle \int \sec ^6(c+d x) (a+b \cos (c+d x))^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3}{\sin \left (c+d x+\frac {\pi }{2}\right )^6}dx\) |
| \(\Big \downarrow \) 3271 |
| \(\displaystyle \frac {1}{5} \int \left (11 b a^2+\left (4 a^2+15 b^2\right ) \cos (c+d x) a+b \left (3 a^2+5 b^2\right ) \cos ^2(c+d x)\right ) \sec ^5(c+d x)dx+\frac {a^2 \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \int \frac {11 b a^2+\left (4 a^2+15 b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) a+b \left (3 a^2+5 b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\sin \left (c+d x+\frac {\pi }{2}\right )^5}dx+\frac {a^2 \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))}{5 d}\) |
| \(\Big \downarrow \) 3500 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \int \left (4 a \left (4 a^2+15 b^2\right )+5 b \left (9 a^2+4 b^2\right ) \cos (c+d x)\right ) \sec ^4(c+d x)dx+\frac {11 a^2 b \tan (c+d x) \sec ^3(c+d x)}{4 d}\right )+\frac {a^2 \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \int \frac {4 a \left (4 a^2+15 b^2\right )+5 b \left (9 a^2+4 b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^4}dx+\frac {11 a^2 b \tan (c+d x) \sec ^3(c+d x)}{4 d}\right )+\frac {a^2 \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))}{5 d}\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \left (4 a \left (4 a^2+15 b^2\right ) \int \sec ^4(c+d x)dx+5 b \left (9 a^2+4 b^2\right ) \int \sec ^3(c+d x)dx\right )+\frac {11 a^2 b \tan (c+d x) \sec ^3(c+d x)}{4 d}\right )+\frac {a^2 \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \left (5 b \left (9 a^2+4 b^2\right ) \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx+4 a \left (4 a^2+15 b^2\right ) \int \csc \left (c+d x+\frac {\pi }{2}\right )^4dx\right )+\frac {11 a^2 b \tan (c+d x) \sec ^3(c+d x)}{4 d}\right )+\frac {a^2 \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))}{5 d}\) |
| \(\Big \downarrow \) 4254 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \left (5 b \left (9 a^2+4 b^2\right ) \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx-\frac {4 a \left (4 a^2+15 b^2\right ) \int \left (\tan ^2(c+d x)+1\right )d(-\tan (c+d x))}{d}\right )+\frac {11 a^2 b \tan (c+d x) \sec ^3(c+d x)}{4 d}\right )+\frac {a^2 \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))}{5 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \left (5 b \left (9 a^2+4 b^2\right ) \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx-\frac {4 a \left (4 a^2+15 b^2\right ) \left (-\frac {1}{3} \tan ^3(c+d x)-\tan (c+d x)\right )}{d}\right )+\frac {11 a^2 b \tan (c+d x) \sec ^3(c+d x)}{4 d}\right )+\frac {a^2 \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))}{5 d}\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \left (5 b \left (9 a^2+4 b^2\right ) \left (\frac {1}{2} \int \sec (c+d x)dx+\frac {\tan (c+d x) \sec (c+d x)}{2 d}\right )-\frac {4 a \left (4 a^2+15 b^2\right ) \left (-\frac {1}{3} \tan ^3(c+d x)-\tan (c+d x)\right )}{d}\right )+\frac {11 a^2 b \tan (c+d x) \sec ^3(c+d x)}{4 d}\right )+\frac {a^2 \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \left (5 b \left (9 a^2+4 b^2\right ) \left (\frac {1}{2} \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+\frac {\tan (c+d x) \sec (c+d x)}{2 d}\right )-\frac {4 a \left (4 a^2+15 b^2\right ) \left (-\frac {1}{3} \tan ^3(c+d x)-\tan (c+d x)\right )}{d}\right )+\frac {11 a^2 b \tan (c+d x) \sec ^3(c+d x)}{4 d}\right )+\frac {a^2 \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))}{5 d}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \left (5 b \left (9 a^2+4 b^2\right ) \left (\frac {\text {arctanh}(\sin (c+d x))}{2 d}+\frac {\tan (c+d x) \sec (c+d x)}{2 d}\right )-\frac {4 a \left (4 a^2+15 b^2\right ) \left (-\frac {1}{3} \tan ^3(c+d x)-\tan (c+d x)\right )}{d}\right )+\frac {11 a^2 b \tan (c+d x) \sec ^3(c+d x)}{4 d}\right )+\frac {a^2 \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))}{5 d}\) |
Input:
Int[(a + b*Cos[c + d*x])^3*Sec[c + d*x]^6,x]
Output:
(a^2*(a + b*Cos[c + d*x])*Sec[c + d*x]^4*Tan[c + d*x])/(5*d) + ((11*a^2*b* Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (5*b*(9*a^2 + 4*b^2)*(ArcTanh[Sin[c + d*x]]/(2*d) + (Sec[c + d*x]*Tan[c + d*x])/(2*d)) - (4*a*(4*a^2 + 15*b^2)* (-Tan[c + d*x] - Tan[c + d*x]^3/3))/d)/4)/5
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[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 = 8.43 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.88
| method | result | size |
| derivativedivides | \(\frac {-a^{3} \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )+3 a^{2} b \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-3 b^{2} a \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+b^{3} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}\) | \(148\) |
| default | \(\frac {-a^{3} \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )+3 a^{2} b \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-3 b^{2} a \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+b^{3} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}\) | \(148\) |
| parts | \(-\frac {a^{3} \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )}{d}+\frac {b^{3} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}+\frac {3 a^{2} b \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}-\frac {3 b^{2} a \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}\) | \(156\) |
| parallelrisch | \(\frac {-1350 b \left (a^{2}+\frac {4 b^{2}}{9}\right ) \left (\frac {\cos \left (5 d x +5 c \right )}{10}+\frac {\cos \left (3 d x +3 c \right )}{2}+\cos \left (d x +c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+1350 b \left (a^{2}+\frac {4 b^{2}}{9}\right ) \left (\frac {\cos \left (5 d x +5 c \right )}{10}+\frac {\cos \left (3 d x +3 c \right )}{2}+\cos \left (d x +c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\left (1260 a^{2} b +240 b^{3}\right ) \sin \left (2 d x +2 c \right )+\left (320 a^{3}+1200 b^{2} a \right ) \sin \left (3 d x +3 c \right )+\left (270 a^{2} b +120 b^{3}\right ) \sin \left (4 d x +4 c \right )+64 \left (\left (a^{2}+\frac {15 b^{2}}{4}\right ) \sin \left (5 d x +5 c \right )+\left (10 a^{2}+15 b^{2}\right ) \sin \left (d x +c \right )\right ) a}{120 d \left (\cos \left (5 d x +5 c \right )+5 \cos \left (3 d x +3 c \right )+10 \cos \left (d x +c \right )\right )}\) | \(251\) |
| risch | \(-\frac {i \left (135 a^{2} b \,{\mathrm e}^{9 i \left (d x +c \right )}+60 b^{3} {\mathrm e}^{9 i \left (d x +c \right )}+630 b \,a^{2} {\mathrm e}^{7 i \left (d x +c \right )}+120 b^{3} {\mathrm e}^{7 i \left (d x +c \right )}-720 a \,b^{2} {\mathrm e}^{6 i \left (d x +c \right )}-640 a^{3} {\mathrm e}^{4 i \left (d x +c \right )}-1680 a \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-630 a^{2} b \,{\mathrm e}^{3 i \left (d x +c \right )}-120 b^{3} {\mathrm e}^{3 i \left (d x +c \right )}-320 a^{3} {\mathrm e}^{2 i \left (d x +c \right )}-1200 a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-135 a^{2} b \,{\mathrm e}^{i \left (d x +c \right )}-60 b^{3} {\mathrm e}^{i \left (d x +c \right )}-64 a^{3}-240 b^{2} a \right )}{60 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{5}}+\frac {9 a^{2} b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{8 d}+\frac {b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{2 d}-\frac {9 a^{2} b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{8 d}-\frac {b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{2 d}\) | \(308\) |
| norman | \(\frac {-\frac {\left (8 a^{3}-15 a^{2} b +24 b^{2} a -4 b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{15}}{4 d}-\frac {\left (8 a^{3}+15 a^{2} b +24 b^{2} a +4 b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}-\frac {\left (40 a^{3}-117 a^{2} b +24 b^{2} a -12 b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{12 d}-\frac {\left (40 a^{3}+117 a^{2} b +24 b^{2} a +12 b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{12 d}-\frac {\left (344 a^{3}-405 a^{2} b -600 b^{2} a +180 b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{60 d}-\frac {\left (344 a^{3}+405 a^{2} b -600 b^{2} a -180 b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{60 d}-\frac {\left (872 a^{3}-45 a^{2} b +120 b^{2} a +180 b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{60 d}-\frac {\left (872 a^{3}+45 a^{2} b +120 b^{2} a -180 b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{60 d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{5}}-\frac {b \left (9 a^{2}+4 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 d}+\frac {b \left (9 a^{2}+4 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{8 d}\) | \(402\) |
Input:
int((a+cos(d*x+c)*b)^3*sec(d*x+c)^6,x,method=_RETURNVERBOSE)
Output:
1/d*(-a^3*(-8/15-1/5*sec(d*x+c)^4-4/15*sec(d*x+c)^2)*tan(d*x+c)+3*a^2*b*(- (-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) ))-3*b^2*a*(-2/3-1/3*sec(d*x+c)^2)*tan(d*x+c)+b^3*(1/2*sec(d*x+c)*tan(d*x+ c)+1/2*ln(sec(d*x+c)+tan(d*x+c))))
Time = 0.11 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.01 \[ \int (a+b \cos (c+d x))^3 \sec ^6(c+d x) \, dx=\frac {15 \, {\left (9 \, a^{2} b + 4 \, b^{3}\right )} \cos \left (d x + c\right )^{5} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \, {\left (9 \, a^{2} b + 4 \, b^{3}\right )} \cos \left (d x + c\right )^{5} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (16 \, {\left (4 \, a^{3} + 15 \, a b^{2}\right )} \cos \left (d x + c\right )^{4} + 90 \, a^{2} b \cos \left (d x + c\right ) + 15 \, {\left (9 \, a^{2} b + 4 \, b^{3}\right )} \cos \left (d x + c\right )^{3} + 24 \, a^{3} + 8 \, {\left (4 \, a^{3} + 15 \, a b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{240 \, d \cos \left (d x + c\right )^{5}} \] Input:
integrate((a+b*cos(d*x+c))^3*sec(d*x+c)^6,x, algorithm="fricas")
Output:
1/240*(15*(9*a^2*b + 4*b^3)*cos(d*x + c)^5*log(sin(d*x + c) + 1) - 15*(9*a ^2*b + 4*b^3)*cos(d*x + c)^5*log(-sin(d*x + c) + 1) + 2*(16*(4*a^3 + 15*a* b^2)*cos(d*x + c)^4 + 90*a^2*b*cos(d*x + c) + 15*(9*a^2*b + 4*b^3)*cos(d*x + c)^3 + 24*a^3 + 8*(4*a^3 + 15*a*b^2)*cos(d*x + c)^2)*sin(d*x + c))/(d*c os(d*x + c)^5)
Timed out. \[ \int (a+b \cos (c+d x))^3 \sec ^6(c+d x) \, dx=\text {Timed out} \] Input:
integrate((a+b*cos(d*x+c))**3*sec(d*x+c)**6,x)
Output:
Timed out
Time = 0.05 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.07 \[ \int (a+b \cos (c+d x))^3 \sec ^6(c+d x) \, dx=\frac {16 \, {\left (3 \, \tan \left (d x + c\right )^{5} + 10 \, \tan \left (d x + c\right )^{3} + 15 \, \tan \left (d x + c\right )\right )} a^{3} + 240 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} a b^{2} - 45 \, a^{2} b {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 60 \, b^{3} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )}}{240 \, d} \] Input:
integrate((a+b*cos(d*x+c))^3*sec(d*x+c)^6,x, algorithm="maxima")
Output:
1/240*(16*(3*tan(d*x + c)^5 + 10*tan(d*x + c)^3 + 15*tan(d*x + c))*a^3 + 2 40*(tan(d*x + c)^3 + 3*tan(d*x + c))*a*b^2 - 45*a^2*b*(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)) - 60*b^3*(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. 367 vs. \(2 (157) = 314\).
Time = 0.53 (sec) , antiderivative size = 367, normalized size of antiderivative = 2.17 \[ \int (a+b \cos (c+d x))^3 \sec ^6(c+d x) \, dx=\frac {15 \, {\left (9 \, a^{2} b + 4 \, b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 15 \, {\left (9 \, a^{2} b + 4 \, b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (120 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 225 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 360 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 60 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 160 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 90 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 960 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 120 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 464 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 1200 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 160 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 90 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 960 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 120 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 120 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 225 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 360 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 60 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{5}}}{120 \, d} \] Input:
integrate((a+b*cos(d*x+c))^3*sec(d*x+c)^6,x, algorithm="giac")
Output:
1/120*(15*(9*a^2*b + 4*b^3)*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 15*(9*a^2 *b + 4*b^3)*log(abs(tan(1/2*d*x + 1/2*c) - 1)) - 2*(120*a^3*tan(1/2*d*x + 1/2*c)^9 - 225*a^2*b*tan(1/2*d*x + 1/2*c)^9 + 360*a*b^2*tan(1/2*d*x + 1/2* c)^9 - 60*b^3*tan(1/2*d*x + 1/2*c)^9 - 160*a^3*tan(1/2*d*x + 1/2*c)^7 + 90 *a^2*b*tan(1/2*d*x + 1/2*c)^7 - 960*a*b^2*tan(1/2*d*x + 1/2*c)^7 + 120*b^3 *tan(1/2*d*x + 1/2*c)^7 + 464*a^3*tan(1/2*d*x + 1/2*c)^5 + 1200*a*b^2*tan( 1/2*d*x + 1/2*c)^5 - 160*a^3*tan(1/2*d*x + 1/2*c)^3 - 90*a^2*b*tan(1/2*d*x + 1/2*c)^3 - 960*a*b^2*tan(1/2*d*x + 1/2*c)^3 - 120*b^3*tan(1/2*d*x + 1/2 *c)^3 + 120*a^3*tan(1/2*d*x + 1/2*c) + 225*a^2*b*tan(1/2*d*x + 1/2*c) + 36 0*a*b^2*tan(1/2*d*x + 1/2*c) + 60*b^3*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 - 1)^5)/d
Time = 44.61 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.54 \[ \int (a+b \cos (c+d x))^3 \sec ^6(c+d x) \, dx=\frac {\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (\frac {9\,a^2\,b}{4}+b^3\right )}{d}-\frac {\left (2\,a^3-\frac {15\,a^2\,b}{4}+6\,a\,b^2-b^3\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (-\frac {8\,a^3}{3}+\frac {3\,a^2\,b}{2}-16\,a\,b^2+2\,b^3\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {116\,a^3}{15}+20\,a\,b^2\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (-\frac {8\,a^3}{3}-\frac {3\,a^2\,b}{2}-16\,a\,b^2-2\,b^3\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (2\,a^3+\frac {15\,a^2\,b}{4}+6\,a\,b^2+b^3\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \] Input:
int((a + b*cos(c + d*x))^3/cos(c + d*x)^6,x)
Output:
(atanh(tan(c/2 + (d*x)/2))*((9*a^2*b)/4 + b^3))/d - (tan(c/2 + (d*x)/2)^9* (6*a*b^2 - (15*a^2*b)/4 + 2*a^3 - b^3) - tan(c/2 + (d*x)/2)^3*(16*a*b^2 + (3*a^2*b)/2 + (8*a^3)/3 + 2*b^3) - tan(c/2 + (d*x)/2)^7*(16*a*b^2 - (3*a^2 *b)/2 + (8*a^3)/3 - 2*b^3) + tan(c/2 + (d*x)/2)*(6*a*b^2 + (15*a^2*b)/4 + 2*a^3 + b^3) + tan(c/2 + (d*x)/2)^5*(20*a*b^2 + (116*a^3)/15))/(d*(5*tan(c /2 + (d*x)/2)^2 - 10*tan(c/2 + (d*x)/2)^4 + 10*tan(c/2 + (d*x)/2)^6 - 5*ta n(c/2 + (d*x)/2)^8 + tan(c/2 + (d*x)/2)^10 - 1))
Time = 0.17 (sec) , antiderivative size = 533, normalized size of antiderivative = 3.15 \[ \int (a+b \cos (c+d x))^3 \sec ^6(c+d x) \, dx =\text {Too large to display} \] Input:
int((a+b*cos(d*x+c))^3*sec(d*x+c)^6,x)
Output:
( - 135*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**4*a**2*b - 60 *cos(c + d*x)*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**4*b**3 + 270*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**2*a**2*b + 120*cos(c + d*x)* log(tan((c + d*x)/2) - 1)*sin(c + d*x)**2*b**3 - 135*cos(c + d*x)*log(tan( (c + d*x)/2) - 1)*a**2*b - 60*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*b**3 + 135*cos(c + d*x)*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**4*a**2*b + 60*c os(c + d*x)*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**4*b**3 - 270*cos(c + d *x)*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**2*a**2*b - 120*cos(c + d*x)*lo g(tan((c + d*x)/2) + 1)*sin(c + d*x)**2*b**3 + 135*cos(c + d*x)*log(tan((c + d*x)/2) + 1)*a**2*b + 60*cos(c + d*x)*log(tan((c + d*x)/2) + 1)*b**3 - 135*cos(c + d*x)*sin(c + d*x)**3*a**2*b - 60*cos(c + d*x)*sin(c + d*x)**3* b**3 + 225*cos(c + d*x)*sin(c + d*x)*a**2*b + 60*cos(c + d*x)*sin(c + d*x) *b**3 + 64*sin(c + d*x)**5*a**3 + 240*sin(c + d*x)**5*a*b**2 - 160*sin(c + d*x)**3*a**3 - 600*sin(c + d*x)**3*a*b**2 + 120*sin(c + d*x)*a**3 + 360*s in(c + d*x)*a*b**2)/(120*cos(c + d*x)*d*(sin(c + d*x)**4 - 2*sin(c + d*x)* *2 + 1))