Integrand size = 31, antiderivative size = 165 \[ \int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \sec ^4(c+d x) \, dx=a^4 (A+4 B) x+\frac {a^4 (12 A+13 B) \text {arctanh}(\sin (c+d x))}{2 d}-\frac {5 a^4 (2 A+B) \sin (c+d x)}{2 d}+\frac {(11 A+9 B) \left (a^4+a^4 \cos (c+d x)\right ) \tan (c+d x)}{3 d}+\frac {(2 A+B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec (c+d x) \tan (c+d x)}{2 d}+\frac {a A (a+a \cos (c+d x))^3 \sec ^2(c+d x) \tan (c+d x)}{3 d} \]
a^4*(A+4*B)*x+1/2*a^4*(12*A+13*B)*arctanh(sin(d*x+c))/d-5/2*a^4*(2*A+B)*si n(d*x+c)/d+1/3*(11*A+9*B)*(a^4+a^4*cos(d*x+c))*tan(d*x+c)/d+1/2*(2*A+B)*(a ^2+a^2*cos(d*x+c))^2*sec(d*x+c)*tan(d*x+c)/d+1/3*a*A*(a+a*cos(d*x+c))^3*se c(d*x+c)^2*tan(d*x+c)/d
Leaf count is larger than twice the leaf count of optimal. \(380\) vs. \(2(165)=330\).
Time = 9.28 (sec) , antiderivative size = 380, normalized size of antiderivative = 2.30 \[ \int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \sec ^4(c+d x) \, dx=a^4 \left (\frac {(A+4 B) (c+d x)}{d}+\frac {(-12 A-13 B) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{2 d}+\frac {(12 A+13 B) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}{2 d}+\frac {13 A+3 B}{12 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {A \sin \left (\frac {1}{2} (c+d x)\right )}{6 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3}+\frac {A \sin \left (\frac {1}{2} (c+d x)\right )}{6 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^3}+\frac {-13 A-3 B}{12 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {4 \left (5 A \sin \left (\frac {1}{2} (c+d x)\right )+3 B \sin \left (\frac {1}{2} (c+d x)\right )\right )}{3 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {4 \left (5 A \sin \left (\frac {1}{2} (c+d x)\right )+3 B \sin \left (\frac {1}{2} (c+d x)\right )\right )}{3 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {B \sin (c+d x)}{d}\right ) \]
a^4*(((A + 4*B)*(c + d*x))/d + ((-12*A - 13*B)*Log[Cos[(c + d*x)/2] - Sin[ (c + d*x)/2]])/(2*d) + ((12*A + 13*B)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x) /2]])/(2*d) + (13*A + 3*B)/(12*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2) + (A*Sin[(c + d*x)/2])/(6*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3) + (A* Sin[(c + d*x)/2])/(6*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) + (-13*A - 3*B)/(12*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2) + (4*(5*A*Sin[(c + d* x)/2] + 3*B*Sin[(c + d*x)/2]))/(3*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + (4*(5*A*Sin[(c + d*x)/2] + 3*B*Sin[(c + d*x)/2]))/(3*d*(Cos[(c + d*x)/2 ] + Sin[(c + d*x)/2])) + (B*Sin[c + d*x])/d)
Time = 1.40 (sec) , antiderivative size = 172, 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, 3454, 3042, 3454, 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.
| \(\displaystyle \int \sec ^4(c+d x) (a \cos (c+d x)+a)^4 (A+B \cos (c+d x)) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^4 \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^4}dx\) |
| \(\Big \downarrow \) 3454 |
| \(\displaystyle \frac {1}{3} \int (\cos (c+d x) a+a)^3 (3 a (2 A+B)-a (A-3 B) \cos (c+d x)) \sec ^3(c+d x)dx+\frac {a A \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^3}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3 \left (3 a (2 A+B)-a (A-3 B) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^3}dx+\frac {a A \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^3}{3 d}\) |
| \(\Big \downarrow \) 3454 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{2} \int (\cos (c+d x) a+a)^2 \left (2 a^2 (11 A+9 B)-a^2 (8 A-3 B) \cos (c+d x)\right ) \sec ^2(c+d x)dx+\frac {3 (2 A+B) \tan (c+d x) \sec (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{2 d}\right )+\frac {a A \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^3}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{2} \int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2 \left (2 a^2 (11 A+9 B)-a^2 (8 A-3 B) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^2}dx+\frac {3 (2 A+B) \tan (c+d x) \sec (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{2 d}\right )+\frac {a A \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^3}{3 d}\) |
| \(\Big \downarrow \) 3454 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{2} \left (\int 3 (\cos (c+d x) a+a) \left (a^3 (12 A+13 B)-5 a^3 (2 A+B) \cos (c+d x)\right ) \sec (c+d x)dx+\frac {2 (11 A+9 B) \tan (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{d}\right )+\frac {3 (2 A+B) \tan (c+d x) \sec (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{2 d}\right )+\frac {a A \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^3}{3 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{2} \left (3 \int (\cos (c+d x) a+a) \left (a^3 (12 A+13 B)-5 a^3 (2 A+B) \cos (c+d x)\right ) \sec (c+d x)dx+\frac {2 (11 A+9 B) \tan (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{d}\right )+\frac {3 (2 A+B) \tan (c+d x) \sec (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{2 d}\right )+\frac {a A \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^3}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{2} \left (3 \int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right ) \left (a^3 (12 A+13 B)-5 a^3 (2 A+B) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {2 (11 A+9 B) \tan (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{d}\right )+\frac {3 (2 A+B) \tan (c+d x) \sec (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{2 d}\right )+\frac {a A \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^3}{3 d}\) |
| \(\Big \downarrow \) 3447 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{2} \left (3 \int \left (-5 (2 A+B) \cos ^2(c+d x) a^4+(12 A+13 B) a^4+\left (a^4 (12 A+13 B)-5 a^4 (2 A+B)\right ) \cos (c+d x)\right ) \sec (c+d x)dx+\frac {2 (11 A+9 B) \tan (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{d}\right )+\frac {3 (2 A+B) \tan (c+d x) \sec (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{2 d}\right )+\frac {a A \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^3}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{2} \left (3 \int \frac {-5 (2 A+B) \sin \left (c+d x+\frac {\pi }{2}\right )^2 a^4+(12 A+13 B) a^4+\left (a^4 (12 A+13 B)-5 a^4 (2 A+B)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {2 (11 A+9 B) \tan (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{d}\right )+\frac {3 (2 A+B) \tan (c+d x) \sec (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{2 d}\right )+\frac {a A \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^3}{3 d}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{2} \left (3 \left (\int \left ((12 A+13 B) a^4+2 (A+4 B) \cos (c+d x) a^4\right ) \sec (c+d x)dx-\frac {5 a^4 (2 A+B) \sin (c+d x)}{d}\right )+\frac {2 (11 A+9 B) \tan (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{d}\right )+\frac {3 (2 A+B) \tan (c+d x) \sec (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{2 d}\right )+\frac {a A \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^3}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{2} \left (3 \left (\int \frac {(12 A+13 B) a^4+2 (A+4 B) \sin \left (c+d x+\frac {\pi }{2}\right ) a^4}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx-\frac {5 a^4 (2 A+B) \sin (c+d x)}{d}\right )+\frac {2 (11 A+9 B) \tan (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{d}\right )+\frac {3 (2 A+B) \tan (c+d x) \sec (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{2 d}\right )+\frac {a A \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^3}{3 d}\) |
| \(\Big \downarrow \) 3214 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{2} \left (3 \left (a^4 (12 A+13 B) \int \sec (c+d x)dx-\frac {5 a^4 (2 A+B) \sin (c+d x)}{d}+2 a^4 x (A+4 B)\right )+\frac {2 (11 A+9 B) \tan (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{d}\right )+\frac {3 (2 A+B) \tan (c+d x) \sec (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{2 d}\right )+\frac {a A \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^3}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{2} \left (3 \left (a^4 (12 A+13 B) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx-\frac {5 a^4 (2 A+B) \sin (c+d x)}{d}+2 a^4 x (A+4 B)\right )+\frac {2 (11 A+9 B) \tan (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{d}\right )+\frac {3 (2 A+B) \tan (c+d x) \sec (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{2 d}\right )+\frac {a A \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^3}{3 d}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{2} \left (3 \left (\frac {a^4 (12 A+13 B) \text {arctanh}(\sin (c+d x))}{d}-\frac {5 a^4 (2 A+B) \sin (c+d x)}{d}+2 a^4 x (A+4 B)\right )+\frac {2 (11 A+9 B) \tan (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{d}\right )+\frac {3 (2 A+B) \tan (c+d x) \sec (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{2 d}\right )+\frac {a A \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^3}{3 d}\) |
(a*A*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^2*Tan[c + d*x])/(3*d) + ((3*(2*A + B)*(a^2 + a^2*Cos[c + d*x])^2*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (3*(2*a ^4*(A + 4*B)*x + (a^4*(12*A + 13*B)*ArcTanh[Sin[c + d*x]])/d - (5*a^4*(2*A + B)*Sin[c + d*x])/d) + (2*(11*A + 9*B)*(a^4 + a^4*Cos[c + d*x])*Tan[c + d*x])/d)/2)/3
3.1.34.3.1 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_.) + (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_.)*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 = 4.06 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.00
| method | result | size |
| parts | \(-\frac {a^{4} A \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (a^{4} A +4 B \,a^{4}\right ) \left (d x +c \right )}{d}+\frac {\left (4 a^{4} A +B \,a^{4}\right ) \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}+\frac {\left (4 a^{4} A +6 B \,a^{4}\right ) \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {\left (6 a^{4} A +4 B \,a^{4}\right ) \tan \left (d x +c \right )}{d}+\frac {B \,a^{4} \sin \left (d x +c \right )}{d}\) | \(165\) |
| parallelrisch | \(\frac {4 \left (-\frac {9 \left (\frac {\cos \left (3 d x +3 c \right )}{3}+\cos \left (d x +c \right )\right ) \left (A +\frac {13 B}{12}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2}+\frac {9 \left (\frac {\cos \left (3 d x +3 c \right )}{3}+\cos \left (d x +c \right )\right ) \left (A +\frac {13 B}{12}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2}+\frac {d x \left (A +4 B \right ) \cos \left (3 d x +3 c \right )}{4}+\left (A +\frac {B}{2}\right ) \sin \left (2 d x +2 c \right )+\left (\frac {5 A}{3}+B \right ) \sin \left (3 d x +3 c \right )+\frac {\sin \left (4 d x +4 c \right ) B}{8}+\frac {3 d x \left (A +4 B \right ) \cos \left (d x +c \right )}{4}+2 \left (A +\frac {B}{2}\right ) \sin \left (d x +c \right )\right ) a^{4}}{d \left (\cos \left (3 d x +3 c \right )+3 \cos \left (d x +c \right )\right )}\) | \(192\) |
| derivativedivides | \(\frac {a^{4} A \left (d x +c \right )+B \,a^{4} \sin \left (d x +c \right )+4 a^{4} A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+4 B \,a^{4} \left (d x +c \right )+6 a^{4} A \tan \left (d x +c \right )+6 B \,a^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+4 a^{4} A \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+4 B \,a^{4} \tan \left (d x +c \right )-a^{4} A \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+B \,a^{4} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}\) | \(199\) |
| default | \(\frac {a^{4} A \left (d x +c \right )+B \,a^{4} \sin \left (d x +c \right )+4 a^{4} A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+4 B \,a^{4} \left (d x +c \right )+6 a^{4} A \tan \left (d x +c \right )+6 B \,a^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+4 a^{4} A \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+4 B \,a^{4} \tan \left (d x +c \right )-a^{4} A \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+B \,a^{4} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}\) | \(199\) |
| risch | \(a^{4} x A +4 a^{4} B x -\frac {i {\mathrm e}^{i \left (d x +c \right )} B \,a^{4}}{2 d}+\frac {i {\mathrm e}^{-i \left (d x +c \right )} B \,a^{4}}{2 d}-\frac {i a^{4} \left (12 A \,{\mathrm e}^{5 i \left (d x +c \right )}+3 B \,{\mathrm e}^{5 i \left (d x +c \right )}-36 A \,{\mathrm e}^{4 i \left (d x +c \right )}-24 B \,{\mathrm e}^{4 i \left (d x +c \right )}-84 A \,{\mathrm e}^{2 i \left (d x +c \right )}-48 B \,{\mathrm e}^{2 i \left (d x +c \right )}-12 A \,{\mathrm e}^{i \left (d x +c \right )}-3 B \,{\mathrm e}^{i \left (d x +c \right )}-40 A -24 B \right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3}}+\frac {6 a^{4} A \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d}+\frac {13 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B}{2 d}-\frac {6 a^{4} A \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}-\frac {13 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B}{2 d}\) | \(266\) |
| norman | \(\frac {\left (-a^{4} A -4 B \,a^{4}\right ) x +\left (-6 a^{4} A -24 B \,a^{4}\right ) x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-2 a^{4} A -8 B \,a^{4}\right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-2 a^{4} A -8 B \,a^{4}\right ) x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (a^{4} A +4 B \,a^{4}\right ) x \left (\tan ^{16}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (2 a^{4} A +8 B \,a^{4}\right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (2 a^{4} A +8 B \,a^{4}\right ) x \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (6 a^{4} A +24 B \,a^{4}\right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {5 a^{4} \left (2 A +B \right ) \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {a^{4} \left (18 A +11 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {a^{4} \left (26 A -15 B \right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {a^{4} \left (70 A +123 B \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {a^{4} \left (74 A +51 B \right ) \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {a^{4} \left (190 A +33 B \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {a^{4} \left (190 A +117 B \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {a^{4} \left (194 A +93 B \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {a^{4} \left (12 A +13 B \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 d}+\frac {a^{4} \left (12 A +13 B \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 d}\) | \(488\) |
-a^4*A/d*(-2/3-1/3*sec(d*x+c)^2)*tan(d*x+c)+(A*a^4+4*B*a^4)/d*(d*x+c)+(4*A *a^4+B*a^4)/d*(1/2*sec(d*x+c)*tan(d*x+c)+1/2*ln(sec(d*x+c)+tan(d*x+c)))+(4 *A*a^4+6*B*a^4)/d*ln(sec(d*x+c)+tan(d*x+c))+(6*A*a^4+4*B*a^4)/d*tan(d*x+c) +B*a^4/d*sin(d*x+c)
Time = 0.32 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.96 \[ \int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \sec ^4(c+d x) \, dx=\frac {12 \, {\left (A + 4 \, B\right )} a^{4} d x \cos \left (d x + c\right )^{3} + 3 \, {\left (12 \, A + 13 \, B\right )} a^{4} \cos \left (d x + c\right )^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (12 \, A + 13 \, B\right )} a^{4} \cos \left (d x + c\right )^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (6 \, B a^{4} \cos \left (d x + c\right )^{3} + 8 \, {\left (5 \, A + 3 \, B\right )} a^{4} \cos \left (d x + c\right )^{2} + 3 \, {\left (4 \, A + B\right )} a^{4} \cos \left (d x + c\right ) + 2 \, A a^{4}\right )} \sin \left (d x + c\right )}{12 \, d \cos \left (d x + c\right )^{3}} \]
1/12*(12*(A + 4*B)*a^4*d*x*cos(d*x + c)^3 + 3*(12*A + 13*B)*a^4*cos(d*x + c)^3*log(sin(d*x + c) + 1) - 3*(12*A + 13*B)*a^4*cos(d*x + c)^3*log(-sin(d *x + c) + 1) + 2*(6*B*a^4*cos(d*x + c)^3 + 8*(5*A + 3*B)*a^4*cos(d*x + c)^ 2 + 3*(4*A + B)*a^4*cos(d*x + c) + 2*A*a^4)*sin(d*x + c))/(d*cos(d*x + c)^ 3)
Timed out. \[ \int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \sec ^4(c+d x) \, dx=\text {Timed out} \]
Time = 0.21 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.42 \[ \int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \sec ^4(c+d x) \, dx=\frac {4 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a^{4} + 12 \, {\left (d x + c\right )} A a^{4} + 48 \, {\left (d x + c\right )} B a^{4} - 12 \, A a^{4} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 3 \, B a^{4} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 24 \, A a^{4} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 36 \, B a^{4} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 12 \, B a^{4} \sin \left (d x + c\right ) + 72 \, A a^{4} \tan \left (d x + c\right ) + 48 \, B a^{4} \tan \left (d x + c\right )}{12 \, d} \]
1/12*(4*(tan(d*x + c)^3 + 3*tan(d*x + c))*A*a^4 + 12*(d*x + c)*A*a^4 + 48* (d*x + c)*B*a^4 - 12*A*a^4*(2*sin(d*x + c)/(sin(d*x + c)^2 - 1) - log(sin( d*x + c) + 1) + log(sin(d*x + c) - 1)) - 3*B*a^4*(2*sin(d*x + c)/(sin(d*x + c)^2 - 1) - log(sin(d*x + c) + 1) + log(sin(d*x + c) - 1)) + 24*A*a^4*(l og(sin(d*x + c) + 1) - log(sin(d*x + c) - 1)) + 36*B*a^4*(log(sin(d*x + c) + 1) - log(sin(d*x + c) - 1)) + 12*B*a^4*sin(d*x + c) + 72*A*a^4*tan(d*x + c) + 48*B*a^4*tan(d*x + c))/d
Time = 0.36 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.38 \[ \int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \sec ^4(c+d x) \, dx=\frac {\frac {12 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1} + 6 \, {\left (A a^{4} + 4 \, B a^{4}\right )} {\left (d x + c\right )} + 3 \, {\left (12 \, A a^{4} + 13 \, B a^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 3 \, {\left (12 \, A a^{4} + 13 \, B a^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (30 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 21 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 76 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 48 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 54 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 27 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{3}}}{6 \, d} \]
1/6*(12*B*a^4*tan(1/2*d*x + 1/2*c)/(tan(1/2*d*x + 1/2*c)^2 + 1) + 6*(A*a^4 + 4*B*a^4)*(d*x + c) + 3*(12*A*a^4 + 13*B*a^4)*log(abs(tan(1/2*d*x + 1/2* c) + 1)) - 3*(12*A*a^4 + 13*B*a^4)*log(abs(tan(1/2*d*x + 1/2*c) - 1)) - 2* (30*A*a^4*tan(1/2*d*x + 1/2*c)^5 + 21*B*a^4*tan(1/2*d*x + 1/2*c)^5 - 76*A* a^4*tan(1/2*d*x + 1/2*c)^3 - 48*B*a^4*tan(1/2*d*x + 1/2*c)^3 + 54*A*a^4*ta n(1/2*d*x + 1/2*c) + 27*B*a^4*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^ 2 - 1)^3)/d
Time = 0.51 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.54 \[ \int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \sec ^4(c+d x) \, dx=\frac {B\,a^4\,\sin \left (c+d\,x\right )}{d}+\frac {2\,A\,a^4\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {12\,A\,a^4\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {8\,B\,a^4\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {13\,B\,a^4\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {20\,A\,a^4\,\sin \left (c+d\,x\right )}{3\,d\,\cos \left (c+d\,x\right )}+\frac {2\,A\,a^4\,\sin \left (c+d\,x\right )}{d\,{\cos \left (c+d\,x\right )}^2}+\frac {A\,a^4\,\sin \left (c+d\,x\right )}{3\,d\,{\cos \left (c+d\,x\right )}^3}+\frac {4\,B\,a^4\,\sin \left (c+d\,x\right )}{d\,\cos \left (c+d\,x\right )}+\frac {B\,a^4\,\sin \left (c+d\,x\right )}{2\,d\,{\cos \left (c+d\,x\right )}^2} \]
(B*a^4*sin(c + d*x))/d + (2*A*a^4*atan(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/ 2)))/d + (12*A*a^4*atanh(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)))/d + (8*B* a^4*atan(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)))/d + (13*B*a^4*atanh(sin(c /2 + (d*x)/2)/cos(c/2 + (d*x)/2)))/d + (20*A*a^4*sin(c + d*x))/(3*d*cos(c + d*x)) + (2*A*a^4*sin(c + d*x))/(d*cos(c + d*x)^2) + (A*a^4*sin(c + d*x)) /(3*d*cos(c + d*x)^3) + (4*B*a^4*sin(c + d*x))/(d*cos(c + d*x)) + (B*a^4*s in(c + d*x))/(2*d*cos(c + d*x)^2)