Integrand size = 31, antiderivative size = 150 \[ \int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \sec ^2(c+d x) \, dx=\frac {1}{2} a^4 (13 A+12 B) x+\frac {a^4 (4 A+B) \text {arctanh}(\sin (c+d x))}{d}+\frac {5 a^4 (A+2 B) \sin (c+d x)}{2 d}-\frac {(3 A-B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{3 d}-\frac {(3 A-8 B) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{6 d}+\frac {a A (a+a \cos (c+d x))^3 \tan (c+d x)}{d} \] Output:
1/2*a^4*(13*A+12*B)*x+a^4*(4*A+B)*arctanh(sin(d*x+c))/d+5/2*a^4*(A+2*B)*si n(d*x+c)/d-1/3*(3*A-B)*(a^2+a^2*cos(d*x+c))^2*sin(d*x+c)/d-1/6*(3*A-8*B)*( a^4+a^4*cos(d*x+c))*sin(d*x+c)/d+a*A*(a+a*cos(d*x+c))^3*tan(d*x+c)/d
Leaf count is larger than twice the leaf count of optimal. \(312\) vs. \(2(150)=300\).
Time = 6.46 (sec) , antiderivative size = 312, normalized size of antiderivative = 2.08 \[ \int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \sec ^2(c+d x) \, dx=\frac {1}{192} a^4 (1+\cos (c+d x))^4 \sec ^8\left (\frac {1}{2} (c+d x)\right ) \left (78 A x+72 B x-\frac {12 (4 A+B) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{d}+\frac {12 (4 A+B) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}{d}+\frac {3 (16 A+27 B) \cos (d x) \sin (c)}{d}+\frac {3 (A+4 B) \cos (2 d x) \sin (2 c)}{d}+\frac {B \cos (3 d x) \sin (3 c)}{d}+\frac {3 (16 A+27 B) \cos (c) \sin (d x)}{d}+\frac {3 (A+4 B) \cos (2 c) \sin (2 d x)}{d}+\frac {B \cos (3 c) \sin (3 d x)}{d}+\frac {12 A \sin \left (\frac {d x}{2}\right )}{d \left (\cos \left (\frac {c}{2}\right )-\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {12 A \sin \left (\frac {d x}{2}\right )}{d \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}\right ) \] Input:
Integrate[(a + a*Cos[c + d*x])^4*(A + B*Cos[c + d*x])*Sec[c + d*x]^2,x]
Output:
(a^4*(1 + Cos[c + d*x])^4*Sec[(c + d*x)/2]^8*(78*A*x + 72*B*x - (12*(4*A + B)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/d + (12*(4*A + B)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/d + (3*(16*A + 27*B)*Cos[d*x]*Sin[c])/d + (3*(A + 4*B)*Cos[2*d*x]*Sin[2*c])/d + (B*Cos[3*d*x]*Sin[3*c])/d + (3*(16*A + 27*B)*Cos[c]*Sin[d*x])/d + (3*(A + 4*B)*Cos[2*c]*Sin[2*d*x])/d + (B*Cos [3*c]*Sin[3*d*x])/d + (12*A*Sin[(d*x)/2])/(d*(Cos[c/2] - Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + (12*A*Sin[(d*x)/2])/(d*(Cos[c/2] + Sin[c /2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))))/192
Time = 1.24 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.04, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.484, Rules used = {3042, 3454, 3042, 3455, 3042, 3455, 27, 3042, 3447, 3042, 3502, 3042, 3214, 3042, 4257}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \sec ^2(c+d x) (a \cos (c+d x)+a)^4 (A+B \cos (c+d x)) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^4 \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^2}dx\) |
| \(\Big \downarrow \) 3454 |
| \(\displaystyle \int (\cos (c+d x) a+a)^3 (a (4 A+B)-a (3 A-B) \cos (c+d x)) \sec (c+d x)dx+\frac {a A \tan (c+d x) (a \cos (c+d x)+a)^3}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3 \left (a (4 A+B)-a (3 A-B) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {a A \tan (c+d x) (a \cos (c+d x)+a)^3}{d}\) |
| \(\Big \downarrow \) 3455 |
| \(\displaystyle \frac {1}{3} \int (\cos (c+d x) a+a)^2 \left (3 a^2 (4 A+B)-a^2 (3 A-8 B) \cos (c+d x)\right ) \sec (c+d x)dx-\frac {(3 A-B) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{3 d}+\frac {a A \tan (c+d x) (a \cos (c+d x)+a)^3}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2 \left (3 a^2 (4 A+B)-a^2 (3 A-8 B) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx-\frac {(3 A-B) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{3 d}+\frac {a A \tan (c+d x) (a \cos (c+d x)+a)^3}{d}\) |
| \(\Big \downarrow \) 3455 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{2} \int 3 (\cos (c+d x) a+a) \left (2 (4 A+B) a^3+5 (A+2 B) \cos (c+d x) a^3\right ) \sec (c+d x)dx-\frac {(3 A-8 B) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{2 d}\right )-\frac {(3 A-B) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{3 d}+\frac {a A \tan (c+d x) (a \cos (c+d x)+a)^3}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \left (\frac {3}{2} \int (\cos (c+d x) a+a) \left (2 (4 A+B) a^3+5 (A+2 B) \cos (c+d x) a^3\right ) \sec (c+d x)dx-\frac {(3 A-8 B) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{2 d}\right )-\frac {(3 A-B) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{3 d}+\frac {a A \tan (c+d x) (a \cos (c+d x)+a)^3}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \left (\frac {3}{2} \int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right ) \left (2 (4 A+B) a^3+5 (A+2 B) \sin \left (c+d x+\frac {\pi }{2}\right ) a^3\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx-\frac {(3 A-8 B) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{2 d}\right )-\frac {(3 A-B) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{3 d}+\frac {a A \tan (c+d x) (a \cos (c+d x)+a)^3}{d}\) |
| \(\Big \downarrow \) 3447 |
| \(\displaystyle \frac {1}{3} \left (\frac {3}{2} \int \left (5 (A+2 B) \cos ^2(c+d x) a^4+2 (4 A+B) a^4+\left (2 (4 A+B) a^4+5 (A+2 B) a^4\right ) \cos (c+d x)\right ) \sec (c+d x)dx-\frac {(3 A-8 B) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{2 d}\right )-\frac {(3 A-B) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{3 d}+\frac {a A \tan (c+d x) (a \cos (c+d x)+a)^3}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \left (\frac {3}{2} \int \frac {5 (A+2 B) \sin \left (c+d x+\frac {\pi }{2}\right )^2 a^4+2 (4 A+B) a^4+\left (2 (4 A+B) a^4+5 (A+2 B) a^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx-\frac {(3 A-8 B) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{2 d}\right )-\frac {(3 A-B) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{3 d}+\frac {a A \tan (c+d x) (a \cos (c+d x)+a)^3}{d}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {1}{3} \left (\frac {3}{2} \left (\int \left (2 (4 A+B) a^4+(13 A+12 B) \cos (c+d x) a^4\right ) \sec (c+d x)dx+\frac {5 a^4 (A+2 B) \sin (c+d x)}{d}\right )-\frac {(3 A-8 B) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{2 d}\right )-\frac {(3 A-B) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{3 d}+\frac {a A \tan (c+d x) (a \cos (c+d x)+a)^3}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \left (\frac {3}{2} \left (\int \frac {2 (4 A+B) a^4+(13 A+12 B) \sin \left (c+d x+\frac {\pi }{2}\right ) a^4}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {5 a^4 (A+2 B) \sin (c+d x)}{d}\right )-\frac {(3 A-8 B) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{2 d}\right )-\frac {(3 A-B) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{3 d}+\frac {a A \tan (c+d x) (a \cos (c+d x)+a)^3}{d}\) |
| \(\Big \downarrow \) 3214 |
| \(\displaystyle \frac {1}{3} \left (\frac {3}{2} \left (2 a^4 (4 A+B) \int \sec (c+d x)dx+\frac {5 a^4 (A+2 B) \sin (c+d x)}{d}+a^4 x (13 A+12 B)\right )-\frac {(3 A-8 B) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{2 d}\right )-\frac {(3 A-B) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{3 d}+\frac {a A \tan (c+d x) (a \cos (c+d x)+a)^3}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \left (\frac {3}{2} \left (2 a^4 (4 A+B) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+\frac {5 a^4 (A+2 B) \sin (c+d x)}{d}+a^4 x (13 A+12 B)\right )-\frac {(3 A-8 B) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{2 d}\right )-\frac {(3 A-B) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{3 d}+\frac {a A \tan (c+d x) (a \cos (c+d x)+a)^3}{d}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {1}{3} \left (\frac {3}{2} \left (\frac {2 a^4 (4 A+B) \text {arctanh}(\sin (c+d x))}{d}+\frac {5 a^4 (A+2 B) \sin (c+d x)}{d}+a^4 x (13 A+12 B)\right )-\frac {(3 A-8 B) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{2 d}\right )-\frac {(3 A-B) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{3 d}+\frac {a A \tan (c+d x) (a \cos (c+d x)+a)^3}{d}\) |
Input:
Int[(a + a*Cos[c + d*x])^4*(A + B*Cos[c + d*x])*Sec[c + d*x]^2,x]
Output:
-1/3*((3*A - B)*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/d + (-1/2*((3*A - 8*B)*(a^4 + a^4*Cos[c + d*x])*Sin[c + d*x])/d + (3*(a^4*(13*A + 12*B)*x + (2*a^4*(4*A + B)*ArcTanh[Sin[c + d*x]])/d + (5*a^4*(A + 2*B)*Sin[c + d*x] )/d))/2)/3 + (a*A*(a + a*Cos[c + d*x])^3*Tan[c + d*x])/d
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])/((c_.) + (d_.)*sin[(e_.) + (f_. )*(x_)]), x_Symbol] :> Simp[b*(x/d), x] - Simp[(b*c - a*d)/d Int[1/(c + d *Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Int[(a + b*Sin[e + f*x])^m*(A*c + (B*c + A*d)*Sin[e + f*x] + B*d*Sin[e + f*x]^2), x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim p[(-b^2)*(B*c - A*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*((c + d*Sin[ e + f*x])^(n + 1)/(d*f*(n + 1)*(b*c + a*d))), x] - Simp[b/(d*(n + 1)*(b*c + a*d)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp [a*A*d*(m - n - 2) - B*(a*c*(m - 1) + b*d*(n + 1)) - (A*b*d*(m + n + 1) - B *(b*c*m - a*d*(n + 1)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f , A, B}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1/2] && LtQ[n, -1] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 0 ])
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim p[(-b)*B*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 1))), x] + Simp[1/(d*(m + n + 1)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A*d*(m + n + 1) + B*(a*c*(m - 1 ) + b*d*(n + 1)) + (A*b*d*(m + n + 1) - B*(b*c*m - a*d*(2*m + n)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1/2] && !LtQ[n, -1 ] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 0])
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Co s[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Simp[1/(b*(m + 2)) Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && !LtQ[m, -1]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Time = 28.20 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.93
| method | result | size |
| parallelrisch | \(\frac {a^{4} \left (-32 \cos \left (d x +c \right ) \left (A +\frac {B}{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+32 \cos \left (d x +c \right ) \left (A +\frac {B}{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\left (16 A +\frac {82 B}{3}\right ) \sin \left (2 d x +2 c \right )+\left (A +4 B \right ) \sin \left (3 d x +3 c \right )+\frac {\sin \left (4 d x +4 c \right ) B}{3}+52 x \left (A +\frac {12 B}{13}\right ) d \cos \left (d x +c \right )+9 \sin \left (d x +c \right ) \left (A +\frac {4 B}{9}\right )\right )}{8 d \cos \left (d x +c \right )}\) | \(140\) |
| parts | \(\frac {a^{4} A \tan \left (d x +c \right )}{d}+\frac {\left (a^{4} A +4 B \,a^{4}\right ) \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {\left (4 a^{4} A +B \,a^{4}\right ) \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {\left (4 a^{4} A +6 B \,a^{4}\right ) \sin \left (d x +c \right )}{d}+\frac {\left (6 a^{4} A +4 B \,a^{4}\right ) \left (d x +c \right )}{d}+\frac {B \,a^{4} \left (\cos \left (d x +c \right )^{2}+2\right ) \sin \left (d x +c \right )}{3 d}\) | \(154\) |
| derivativedivides | \(\frac {a^{4} A \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {B \,a^{4} \left (\cos \left (d x +c \right )^{2}+2\right ) \sin \left (d x +c \right )}{3}+4 a^{4} A \sin \left (d x +c \right )+4 B \,a^{4} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+6 a^{4} A \left (d x +c \right )+6 B \,a^{4} \sin \left (d x +c \right )+4 a^{4} A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+4 B \,a^{4} \left (d x +c \right )+a^{4} A \tan \left (d x +c \right )+B \,a^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}\) | \(179\) |
| default | \(\frac {a^{4} A \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {B \,a^{4} \left (\cos \left (d x +c \right )^{2}+2\right ) \sin \left (d x +c \right )}{3}+4 a^{4} A \sin \left (d x +c \right )+4 B \,a^{4} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+6 a^{4} A \left (d x +c \right )+6 B \,a^{4} \sin \left (d x +c \right )+4 a^{4} A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+4 B \,a^{4} \left (d x +c \right )+a^{4} A \tan \left (d x +c \right )+B \,a^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}\) | \(179\) |
| risch | \(\frac {13 a^{4} x A}{2}+6 a^{4} B x +\frac {27 i {\mathrm e}^{-i \left (d x +c \right )} B \,a^{4}}{8 d}+\frac {2 i {\mathrm e}^{-i \left (d x +c \right )} a^{4} A}{d}+\frac {2 i a^{4} A}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}-\frac {27 i {\mathrm e}^{i \left (d x +c \right )} B \,a^{4}}{8 d}-\frac {2 i {\mathrm e}^{i \left (d x +c \right )} a^{4} A}{d}-\frac {i {\mathrm e}^{2 i \left (d x +c \right )} B \,a^{4}}{2 d}+\frac {i {\mathrm e}^{-2 i \left (d x +c \right )} B \,a^{4}}{2 d}+\frac {i {\mathrm e}^{-2 i \left (d x +c \right )} a^{4} A}{8 d}-\frac {i {\mathrm e}^{2 i \left (d x +c \right )} a^{4} A}{8 d}+\frac {4 a^{4} A \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d}+\frac {a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B}{d}-\frac {4 a^{4} A \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}-\frac {a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B}{d}+\frac {\sin \left (3 d x +3 c \right ) B \,a^{4}}{12 d}\) | \(296\) |
| norman | \(\frac {\left (-\frac {13}{2} a^{4} A -6 B \,a^{4}\right ) x +\left (-\frac {65}{2} a^{4} A -30 B \,a^{4}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (\frac {13}{2} a^{4} A +6 B \,a^{4}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}+\left (\frac {65}{2} a^{4} A +30 B \,a^{4}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+\left (-26 a^{4} A -24 B \,a^{4}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\left (26 a^{4} A +24 B \,a^{4}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+\frac {5 a^{4} \left (A +2 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{d}-\frac {2 a^{4} \left (3 A -50 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{3 d}-\frac {a^{4} \left (11 A +18 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}-\frac {5 a^{4} \left (21 A +26 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 d}+\frac {a^{4} \left (39 A +106 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{3 d}-\frac {2 a^{4} \left (51 A +26 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{3 d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{5} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )}+\frac {a^{4} \left (4 A +B \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}-\frac {a^{4} \left (4 A +B \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}\) | \(380\) |
Input:
int((a+a*cos(d*x+c))^4*(A+B*cos(d*x+c))*sec(d*x+c)^2,x,method=_RETURNVERBO SE)
Output:
1/8*a^4*(-32*cos(d*x+c)*(A+1/4*B)*ln(tan(1/2*d*x+1/2*c)-1)+32*cos(d*x+c)*( A+1/4*B)*ln(tan(1/2*d*x+1/2*c)+1)+(16*A+82/3*B)*sin(2*d*x+2*c)+(A+4*B)*sin (3*d*x+3*c)+1/3*sin(4*d*x+4*c)*B+52*x*(A+12/13*B)*d*cos(d*x+c)+9*sin(d*x+c )*(A+4/9*B))/d/cos(d*x+c)
Time = 0.09 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.00 \[ \int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \sec ^2(c+d x) \, dx=\frac {3 \, {\left (13 \, A + 12 \, B\right )} a^{4} d x \cos \left (d x + c\right ) + 3 \, {\left (4 \, A + B\right )} a^{4} \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (4 \, A + B\right )} a^{4} \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) + {\left (2 \, B a^{4} \cos \left (d x + c\right )^{3} + 3 \, {\left (A + 4 \, B\right )} a^{4} \cos \left (d x + c\right )^{2} + 8 \, {\left (3 \, A + 5 \, B\right )} a^{4} \cos \left (d x + c\right ) + 6 \, A a^{4}\right )} \sin \left (d x + c\right )}{6 \, d \cos \left (d x + c\right )} \] Input:
integrate((a+a*cos(d*x+c))^4*(A+B*cos(d*x+c))*sec(d*x+c)^2,x, algorithm="f ricas")
Output:
1/6*(3*(13*A + 12*B)*a^4*d*x*cos(d*x + c) + 3*(4*A + B)*a^4*cos(d*x + c)*l og(sin(d*x + c) + 1) - 3*(4*A + B)*a^4*cos(d*x + c)*log(-sin(d*x + c) + 1) + (2*B*a^4*cos(d*x + c)^3 + 3*(A + 4*B)*a^4*cos(d*x + c)^2 + 8*(3*A + 5*B )*a^4*cos(d*x + c) + 6*A*a^4)*sin(d*x + c))/(d*cos(d*x + c))
\[ \int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \sec ^2(c+d x) \, dx=a^{4} \left (\int A \sec ^{2}{\left (c + d x \right )}\, dx + \int 4 A \cos {\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int 6 A \cos ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int 4 A \cos ^{3}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int A \cos ^{4}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int B \cos {\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int 4 B \cos ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int 6 B \cos ^{3}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int 4 B \cos ^{4}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int B \cos ^{5}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx\right ) \] Input:
integrate((a+a*cos(d*x+c))**4*(A+B*cos(d*x+c))*sec(d*x+c)**2,x)
Output:
a**4*(Integral(A*sec(c + d*x)**2, x) + Integral(4*A*cos(c + d*x)*sec(c + d *x)**2, x) + Integral(6*A*cos(c + d*x)**2*sec(c + d*x)**2, x) + Integral(4 *A*cos(c + d*x)**3*sec(c + d*x)**2, x) + Integral(A*cos(c + d*x)**4*sec(c + d*x)**2, x) + Integral(B*cos(c + d*x)*sec(c + d*x)**2, x) + Integral(4*B *cos(c + d*x)**2*sec(c + d*x)**2, x) + Integral(6*B*cos(c + d*x)**3*sec(c + d*x)**2, x) + Integral(4*B*cos(c + d*x)**4*sec(c + d*x)**2, x) + Integra l(B*cos(c + d*x)**5*sec(c + d*x)**2, x))
Time = 0.05 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.25 \[ \int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \sec ^2(c+d x) \, dx=\frac {3 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} + 72 \, {\left (d x + c\right )} A a^{4} - 4 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{4} + 12 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{4} + 48 \, {\left (d x + c\right )} B a^{4} + 24 \, A a^{4} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 6 \, B a^{4} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 48 \, A a^{4} \sin \left (d x + c\right ) + 72 \, B a^{4} \sin \left (d x + c\right ) + 12 \, A a^{4} \tan \left (d x + c\right )}{12 \, d} \] Input:
integrate((a+a*cos(d*x+c))^4*(A+B*cos(d*x+c))*sec(d*x+c)^2,x, algorithm="m axima")
Output:
1/12*(3*(2*d*x + 2*c + sin(2*d*x + 2*c))*A*a^4 + 72*(d*x + c)*A*a^4 - 4*(s in(d*x + c)^3 - 3*sin(d*x + c))*B*a^4 + 12*(2*d*x + 2*c + sin(2*d*x + 2*c) )*B*a^4 + 48*(d*x + c)*B*a^4 + 24*A*a^4*(log(sin(d*x + c) + 1) - log(sin(d *x + c) - 1)) + 6*B*a^4*(log(sin(d*x + c) + 1) - log(sin(d*x + c) - 1)) + 48*A*a^4*sin(d*x + c) + 72*B*a^4*sin(d*x + c) + 12*A*a^4*tan(d*x + c))/d
Time = 0.37 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.51 \[ \int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \sec ^2(c+d x) \, dx=-\frac {\frac {12 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1} - 3 \, {\left (13 \, A a^{4} + 12 \, B a^{4}\right )} {\left (d x + c\right )} - 6 \, {\left (4 \, A a^{4} + B a^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) + 6 \, {\left (4 \, A a^{4} + B a^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (21 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 30 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 48 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 76 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 27 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 54 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3}}}{6 \, d} \] Input:
integrate((a+a*cos(d*x+c))^4*(A+B*cos(d*x+c))*sec(d*x+c)^2,x, algorithm="g iac")
Output:
-1/6*(12*A*a^4*tan(1/2*d*x + 1/2*c)/(tan(1/2*d*x + 1/2*c)^2 - 1) - 3*(13*A *a^4 + 12*B*a^4)*(d*x + c) - 6*(4*A*a^4 + B*a^4)*log(abs(tan(1/2*d*x + 1/2 *c) + 1)) + 6*(4*A*a^4 + B*a^4)*log(abs(tan(1/2*d*x + 1/2*c) - 1)) - 2*(21 *A*a^4*tan(1/2*d*x + 1/2*c)^5 + 30*B*a^4*tan(1/2*d*x + 1/2*c)^5 + 48*A*a^4 *tan(1/2*d*x + 1/2*c)^3 + 76*B*a^4*tan(1/2*d*x + 1/2*c)^3 + 27*A*a^4*tan(1 /2*d*x + 1/2*c) + 54*B*a^4*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 + 1)^3)/d
Time = 42.17 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.61 \[ \int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \sec ^2(c+d x) \, dx=\frac {4\,A\,a^4\,\sin \left (c+d\,x\right )}{d}+\frac {20\,B\,a^4\,\sin \left (c+d\,x\right )}{3\,d}+\frac {13\,A\,a^4\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {8\,A\,a^4\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {12\,B\,a^4\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {2\,B\,a^4\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {A\,a^4\,\sin \left (c+d\,x\right )}{d\,\cos \left (c+d\,x\right )}+\frac {B\,a^4\,{\cos \left (c+d\,x\right )}^2\,\sin \left (c+d\,x\right )}{3\,d}+\frac {A\,a^4\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )}{2\,d}+\frac {2\,B\,a^4\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )}{d} \] Input:
int(((A + B*cos(c + d*x))*(a + a*cos(c + d*x))^4)/cos(c + d*x)^2,x)
Output:
(4*A*a^4*sin(c + d*x))/d + (20*B*a^4*sin(c + d*x))/(3*d) + (13*A*a^4*atan( sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)))/d + (8*A*a^4*atanh(sin(c/2 + (d*x) /2)/cos(c/2 + (d*x)/2)))/d + (12*B*a^4*atan(sin(c/2 + (d*x)/2)/cos(c/2 + ( d*x)/2)))/d + (2*B*a^4*atanh(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)))/d + ( A*a^4*sin(c + d*x))/(d*cos(c + d*x)) + (B*a^4*cos(c + d*x)^2*sin(c + d*x)) /(3*d) + (A*a^4*cos(c + d*x)*sin(c + d*x))/(2*d) + (2*B*a^4*cos(c + d*x)*s in(c + d*x))/d
Time = 0.18 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.42 \[ \int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \sec ^2(c+d x) \, dx=\frac {a^{4} \left (3 \cos \left (d x +c \right )^{2} \sin \left (d x +c \right ) a +12 \cos \left (d x +c \right )^{2} \sin \left (d x +c \right ) b -24 \cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) a -6 \cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) b +24 \cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) a +6 \cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) b -2 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} b +24 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a +42 \cos \left (d x +c \right ) \sin \left (d x +c \right ) b +39 \cos \left (d x +c \right ) a d x +36 \cos \left (d x +c \right ) b d x +6 \sin \left (d x +c \right ) a \right )}{6 \cos \left (d x +c \right ) d} \] Input:
int((a+a*cos(d*x+c))^4*(A+B*cos(d*x+c))*sec(d*x+c)^2,x)
Output:
(a**4*(3*cos(c + d*x)**2*sin(c + d*x)*a + 12*cos(c + d*x)**2*sin(c + d*x)* b - 24*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*a - 6*cos(c + d*x)*log(tan(( c + d*x)/2) - 1)*b + 24*cos(c + d*x)*log(tan((c + d*x)/2) + 1)*a + 6*cos(c + d*x)*log(tan((c + d*x)/2) + 1)*b - 2*cos(c + d*x)*sin(c + d*x)**3*b + 2 4*cos(c + d*x)*sin(c + d*x)*a + 42*cos(c + d*x)*sin(c + d*x)*b + 39*cos(c + d*x)*a*d*x + 36*cos(c + d*x)*b*d*x + 6*sin(c + d*x)*a))/(6*cos(c + d*x)* d)