Integrand size = 41, antiderivative size = 169 \[ \int \cos ^3(c+d x) (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {1}{2} a^3 (5 A+7 B+6 C) x+\frac {a^3 (B+3 C) \text {arctanh}(\sin (c+d x))}{d}+\frac {5 a^3 (A+B) \sin (c+d x)}{2 d}+\frac {A \cos ^2(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{3 d}+\frac {(A+B) \cos (c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{2 a d}-\frac {(5 A+3 B-6 C) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{6 d} \] Output:
1/2*a^3*(5*A+7*B+6*C)*x+a^3*(B+3*C)*arctanh(sin(d*x+c))/d+5/2*a^3*(A+B)*si n(d*x+c)/d+1/3*A*cos(d*x+c)^2*(a+a*sec(d*x+c))^3*sin(d*x+c)/d+1/2*(A+B)*co s(d*x+c)*(a^2+a^2*sec(d*x+c))^2*sin(d*x+c)/a/d-1/6*(5*A+3*B-6*C)*(a^3+a^3* sec(d*x+c))*sin(d*x+c)/d
Leaf count is larger than twice the leaf count of optimal. \(379\) vs. \(2(169)=338\).
Time = 4.65 (sec) , antiderivative size = 379, normalized size of antiderivative = 2.24 \[ \int \cos ^3(c+d x) (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {a^3 \cos ^2(c+d x) (1+\cos (c+d x))^3 \sec ^6\left (\frac {1}{2} (c+d x)\right ) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (6 (5 A+7 B+6 C) x-\frac {12 (B+3 C) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{d}+\frac {12 (B+3 C) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}{d}+\frac {3 (15 A+4 (3 B+C)) \cos (d x) \sin (c)}{d}+\frac {3 (3 A+B) \cos (2 d x) \sin (2 c)}{d}+\frac {A \cos (3 d x) \sin (3 c)}{d}+\frac {3 (15 A+4 (3 B+C)) \cos (c) \sin (d x)}{d}+\frac {3 (3 A+B) \cos (2 c) \sin (2 d x)}{d}+\frac {A \cos (3 c) \sin (3 d x)}{d}+\frac {12 C \sin \left (\frac {d x}{2}\right )}{d \left (\cos \left (\frac {c}{2}\right )-\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {12 C \sin \left (\frac {d x}{2}\right )}{d \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}\right )}{48 (A+2 C+2 B \cos (c+d x)+A \cos (2 (c+d x)))} \] Input:
Integrate[Cos[c + d*x]^3*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Se c[c + d*x]^2),x]
Output:
(a^3*Cos[c + d*x]^2*(1 + Cos[c + d*x])^3*Sec[(c + d*x)/2]^6*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(6*(5*A + 7*B + 6*C)*x - (12*(B + 3*C)*Log[Cos[( c + d*x)/2] - Sin[(c + d*x)/2]])/d + (12*(B + 3*C)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/d + (3*(15*A + 4*(3*B + C))*Cos[d*x]*Sin[c])/d + (3*(3* A + B)*Cos[2*d*x]*Sin[2*c])/d + (A*Cos[3*d*x]*Sin[3*c])/d + (3*(15*A + 4*( 3*B + C))*Cos[c]*Sin[d*x])/d + (3*(3*A + B)*Cos[2*c]*Sin[2*d*x])/d + (A*Co s[3*c]*Sin[3*d*x])/d + (12*C*Sin[(d*x)/2])/(d*(Cos[c/2] - Sin[c/2])*(Cos[( c + d*x)/2] - Sin[(c + d*x)/2])) + (12*C*Sin[(d*x)/2])/(d*(Cos[c/2] + Sin[ c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))))/(48*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)]))
Time = 1.05 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.04, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.244, Rules used = {3042, 4574, 3042, 4505, 3042, 4506, 27, 3042, 4484, 2009}
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.
| \(\displaystyle \int \cos ^3(c+d x) (a \sec (c+d x)+a)^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^3 \left (A+B \csc \left (c+d x+\frac {\pi }{2}\right )+C \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^3}dx\) |
| \(\Big \downarrow \) 4574 |
| \(\displaystyle \frac {\int \cos ^2(c+d x) (\sec (c+d x) a+a)^3 (3 a (A+B)-a (A-3 C) \sec (c+d x))dx}{3 a}+\frac {A \sin (c+d x) \cos ^2(c+d x) (a \sec (c+d x)+a)^3}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3 \left (3 a (A+B)-a (A-3 C) \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^2}dx}{3 a}+\frac {A \sin (c+d x) \cos ^2(c+d x) (a \sec (c+d x)+a)^3}{3 d}\) |
| \(\Big \downarrow \) 4505 |
| \(\displaystyle \frac {\frac {1}{2} \int \cos (c+d x) (\sec (c+d x) a+a)^2 \left (2 a^2 (5 A+6 B+3 C)-a^2 (5 A+3 B-6 C) \sec (c+d x)\right )dx+\frac {3 (A+B) \sin (c+d x) \cos (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{2 d}}{3 a}+\frac {A \sin (c+d x) \cos ^2(c+d x) (a \sec (c+d x)+a)^3}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{2} \int \frac {\left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2 \left (2 a^2 (5 A+6 B+3 C)-a^2 (5 A+3 B-6 C) \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {3 (A+B) \sin (c+d x) \cos (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{2 d}}{3 a}+\frac {A \sin (c+d x) \cos ^2(c+d x) (a \sec (c+d x)+a)^3}{3 d}\) |
| \(\Big \downarrow \) 4506 |
| \(\displaystyle \frac {\frac {1}{2} \left (\int 3 \cos (c+d x) (\sec (c+d x) a+a) \left (5 (A+B) a^3+2 (B+3 C) \sec (c+d x) a^3\right )dx-\frac {(5 A+3 B-6 C) \sin (c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{d}\right )+\frac {3 (A+B) \sin (c+d x) \cos (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{2 d}}{3 a}+\frac {A \sin (c+d x) \cos ^2(c+d x) (a \sec (c+d x)+a)^3}{3 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{2} \left (3 \int \cos (c+d x) (\sec (c+d x) a+a) \left (5 (A+B) a^3+2 (B+3 C) \sec (c+d x) a^3\right )dx-\frac {(5 A+3 B-6 C) \sin (c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{d}\right )+\frac {3 (A+B) \sin (c+d x) \cos (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{2 d}}{3 a}+\frac {A \sin (c+d x) \cos ^2(c+d x) (a \sec (c+d x)+a)^3}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{2} \left (3 \int \frac {\left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right ) \left (5 (A+B) a^3+2 (B+3 C) \csc \left (c+d x+\frac {\pi }{2}\right ) a^3\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )}dx-\frac {(5 A+3 B-6 C) \sin (c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{d}\right )+\frac {3 (A+B) \sin (c+d x) \cos (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{2 d}}{3 a}+\frac {A \sin (c+d x) \cos ^2(c+d x) (a \sec (c+d x)+a)^3}{3 d}\) |
| \(\Big \downarrow \) 4484 |
| \(\displaystyle \frac {\frac {1}{2} \left (3 \left (\frac {5 a^4 (A+B) \sin (c+d x)}{d}-\int \left (-\left ((5 A+7 B+6 C) a^4\right )-2 (B+3 C) \sec (c+d x) a^4\right )dx\right )-\frac {(5 A+3 B-6 C) \sin (c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{d}\right )+\frac {3 (A+B) \sin (c+d x) \cos (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{2 d}}{3 a}+\frac {A \sin (c+d x) \cos ^2(c+d x) (a \sec (c+d x)+a)^3}{3 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {1}{2} \left (3 \left (\frac {5 a^4 (A+B) \sin (c+d x)}{d}+a^4 x (5 A+7 B+6 C)+\frac {2 a^4 (B+3 C) \text {arctanh}(\sin (c+d x))}{d}\right )-\frac {(5 A+3 B-6 C) \sin (c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{d}\right )+\frac {3 (A+B) \sin (c+d x) \cos (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{2 d}}{3 a}+\frac {A \sin (c+d x) \cos ^2(c+d x) (a \sec (c+d x)+a)^3}{3 d}\) |
Input:
Int[Cos[c + d*x]^3*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]
Output:
(A*Cos[c + d*x]^2*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(3*d) + ((3*(A + B) *Cos[c + d*x]*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(2*d) + (-(((5*A + 3*B - 6*C)*(a^4 + a^4*Sec[c + d*x])*Sin[c + d*x])/d) + 3*(a^4*(5*A + 7*B + 6*C)*x + (2*a^4*(B + 3*C)*ArcTanh[Sin[c + d*x]])/d + (5*a^4*(A + B)*Sin[c + d*x])/d))/2)/(3*a)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + ( a_))*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)), x_Symbol] :> Simp[A*a*Cot[e + f*x]*((d*Csc[e + f*x])^n/(f*n)), x] + Simp[1/(d*n) Int[(d*Csc[e + f*x])^( n + 1)*Simp[n*(B*a + A*b) + (B*b*n + A*a*(n + 1))*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && LeQ[n, -1]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + ( a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)), x_Symbol] :> Simp[a*A*Cot [e + f*x]*(a + b*Csc[e + f*x])^(m - 1)*((d*Csc[e + f*x])^n/(f*n)), x] - Sim p[b/(a*d*n) Int[(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^(n + 1)*Sim p[a*A*(m - n - 1) - b*B*n - (a*B*n + A*b*(m + n))*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && EqQ[a^2 - b^2, 0 ] && GtQ[m, 1/2] && LtQ[n, -1]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + ( a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)), x_Symbol] :> Simp[(-b)*B* Cot[e + f*x]*(a + b*Csc[e + f*x])^(m - 1)*((d*Csc[e + f*x])^n/(f*(m + n))), x] + Simp[1/(d*(m + n)) Int[(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x] )^n*Simp[a*A*d*(m + n) + B*(b*d*n) + (A*b*d*(m + n) + a*B*d*(2*m + n - 1))* Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, B, n}, x] && NeQ[A*b - a*B, 0] && EqQ[a^2 - b^2, 0] && GtQ[m, 1/2] && !LtQ[n, -1]
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. ))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a _))^(m_), x_Symbol] :> Simp[A*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*((d*Csc[e + f*x])^n/(f*n)), x] - Simp[1/(b*d*n) Int[(a + b*Csc[e + f*x])^m*(d*Csc[ e + f*x])^(n + 1)*Simp[a*A*m - b*B*n - b*(A*(m + n + 1) + C*n)*Csc[e + f*x] , x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C, m}, x] && EqQ[a^2 - b^2, 0] && !LtQ[m, -2^(-1)] && (LtQ[n, -2^(-1)] || EqQ[m + n + 1, 0])
Time = 0.85 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.88
| method | result | size |
| parallelrisch | \(\frac {23 \left (-\frac {12 \cos \left (d x +c \right ) \left (B +3 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{23}+\frac {12 \cos \left (d x +c \right ) \left (B +3 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{23}+\left (A +\frac {18 B}{23}+\frac {6 C}{23}\right ) \sin \left (2 d x +2 c \right )+\frac {3 \left (3 A +B \right ) \sin \left (3 d x +3 c \right )}{46}+\frac {A \sin \left (4 d x +4 c \right )}{46}+\frac {30 x \left (A +\frac {7 B}{5}+\frac {6 C}{5}\right ) d \cos \left (d x +c \right )}{23}+\frac {9 \left (A +\frac {B}{3}+\frac {8 C}{3}\right ) \sin \left (d x +c \right )}{46}\right ) a^{3}}{12 d \cos \left (d x +c \right )}\) | \(148\) |
| derivativedivides | \(\frac {a^{3} A \left (d x +c \right )+B \,a^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+a^{3} C \tan \left (d x +c \right )+3 a^{3} A \sin \left (d x +c \right )+3 B \,a^{3} \left (d x +c \right )+3 a^{3} C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+3 a^{3} A \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+3 B \,a^{3} \sin \left (d x +c \right )+3 a^{3} C \left (d x +c \right )+\frac {a^{3} A \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+B \,a^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a^{3} C \sin \left (d x +c \right )}{d}\) | \(200\) |
| default | \(\frac {a^{3} A \left (d x +c \right )+B \,a^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+a^{3} C \tan \left (d x +c \right )+3 a^{3} A \sin \left (d x +c \right )+3 B \,a^{3} \left (d x +c \right )+3 a^{3} C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+3 a^{3} A \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+3 B \,a^{3} \sin \left (d x +c \right )+3 a^{3} C \left (d x +c \right )+\frac {a^{3} A \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+B \,a^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a^{3} C \sin \left (d x +c \right )}{d}\) | \(200\) |
| risch | \(\frac {5 a^{3} A x}{2}+\frac {7 a^{3} B x}{2}+3 a^{3} x C -\frac {15 i a^{3} A \,{\mathrm e}^{i \left (d x +c \right )}}{8 d}-\frac {3 i B \,a^{3} {\mathrm e}^{i \left (d x +c \right )}}{2 d}+\frac {i {\mathrm e}^{-2 i \left (d x +c \right )} B \,a^{3}}{8 d}+\frac {3 i a^{3} A \,{\mathrm e}^{-2 i \left (d x +c \right )}}{8 d}-\frac {i {\mathrm e}^{i \left (d x +c \right )} a^{3} C}{2 d}-\frac {3 i a^{3} A \,{\mathrm e}^{2 i \left (d x +c \right )}}{8 d}-\frac {i {\mathrm e}^{2 i \left (d x +c \right )} B \,a^{3}}{8 d}+\frac {i {\mathrm e}^{-i \left (d x +c \right )} a^{3} C}{2 d}+\frac {2 i a^{3} C}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}+\frac {15 i a^{3} A \,{\mathrm e}^{-i \left (d x +c \right )}}{8 d}+\frac {3 i {\mathrm e}^{-i \left (d x +c \right )} B \,a^{3}}{2 d}+\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B}{d}+\frac {3 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{d}-\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B}{d}-\frac {3 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{d}+\frac {a^{3} A \sin \left (3 d x +3 c \right )}{12 d}\) | \(341\) |
| norman | \(\frac {\left (\frac {5}{2} a^{3} A +\frac {7}{2} B \,a^{3}+3 a^{3} C \right ) x +\left (-\frac {15}{2} a^{3} A -\frac {21}{2} B \,a^{3}-9 a^{3} C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (-\frac {15}{2} a^{3} A -\frac {21}{2} B \,a^{3}-9 a^{3} C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+\left (-\frac {5}{2} a^{3} A -\frac {7}{2} B \,a^{3}-3 a^{3} C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\left (-\frac {5}{2} a^{3} A -\frac {7}{2} B \,a^{3}-3 a^{3} C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}+\left (\frac {5}{2} a^{3} A +\frac {7}{2} B \,a^{3}+3 a^{3} C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}+\left (\frac {15}{2} a^{3} A +\frac {21}{2} B \,a^{3}+9 a^{3} C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\left (\frac {15}{2} a^{3} A +\frac {21}{2} B \,a^{3}+9 a^{3} C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+\frac {a^{3} \left (11 A +7 B +4 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {5 a^{3} \left (A +B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{d}+\frac {8 a^{3} \left (2 A +3 B +C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{d}-\frac {4 a^{3} \left (5 A +6 B +3 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{3 d}-\frac {4 a^{3} \left (23 A +12 B +3 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 d}-\frac {a^{3} \left (37 A +33 B -12 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{3 d}+\frac {a^{3} \left (53 A -3 B -24 C \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 )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{4}}+\frac {a^{3} \left (B +3 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}-\frac {a^{3} \left (B +3 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}\) | \(519\) |
Input:
int(cos(d*x+c)^3*(a+a*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x,meth od=_RETURNVERBOSE)
Output:
23/12*(-12/23*cos(d*x+c)*(B+3*C)*ln(tan(1/2*d*x+1/2*c)-1)+12/23*cos(d*x+c) *(B+3*C)*ln(tan(1/2*d*x+1/2*c)+1)+(A+18/23*B+6/23*C)*sin(2*d*x+2*c)+3/46*( 3*A+B)*sin(3*d*x+3*c)+1/46*A*sin(4*d*x+4*c)+30/23*x*(A+7/5*B+6/5*C)*d*cos( d*x+c)+9/46*(A+1/3*B+8/3*C)*sin(d*x+c))*a^3/d/cos(d*x+c)
Time = 0.09 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.92 \[ \int \cos ^3(c+d x) (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {3 \, {\left (5 \, A + 7 \, B + 6 \, C\right )} a^{3} d x \cos \left (d x + c\right ) + 3 \, {\left (B + 3 \, C\right )} a^{3} \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (B + 3 \, C\right )} a^{3} \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) + {\left (2 \, A a^{3} \cos \left (d x + c\right )^{3} + 3 \, {\left (3 \, A + B\right )} a^{3} \cos \left (d x + c\right )^{2} + 2 \, {\left (11 \, A + 9 \, B + 3 \, C\right )} a^{3} \cos \left (d x + c\right ) + 6 \, C a^{3}\right )} \sin \left (d x + c\right )}{6 \, d \cos \left (d x + c\right )} \] Input:
integrate(cos(d*x+c)^3*(a+a*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2), x, algorithm="fricas")
Output:
1/6*(3*(5*A + 7*B + 6*C)*a^3*d*x*cos(d*x + c) + 3*(B + 3*C)*a^3*cos(d*x + c)*log(sin(d*x + c) + 1) - 3*(B + 3*C)*a^3*cos(d*x + c)*log(-sin(d*x + c) + 1) + (2*A*a^3*cos(d*x + c)^3 + 3*(3*A + B)*a^3*cos(d*x + c)^2 + 2*(11*A + 9*B + 3*C)*a^3*cos(d*x + c) + 6*C*a^3)*sin(d*x + c))/(d*cos(d*x + c))
Timed out. \[ \int \cos ^3(c+d x) (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)**3*(a+a*sec(d*x+c))**3*(A+B*sec(d*x+c)+C*sec(d*x+c)** 2),x)
Output:
Timed out
Time = 0.04 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.24 \[ \int \cos ^3(c+d x) (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=-\frac {4 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{3} - 9 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{3} - 12 \, {\left (d x + c\right )} A a^{3} - 3 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{3} - 36 \, {\left (d x + c\right )} B a^{3} - 36 \, {\left (d x + c\right )} C a^{3} - 6 \, B a^{3} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 18 \, C a^{3} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 36 \, A a^{3} \sin \left (d x + c\right ) - 36 \, B a^{3} \sin \left (d x + c\right ) - 12 \, C a^{3} \sin \left (d x + c\right ) - 12 \, C a^{3} \tan \left (d x + c\right )}{12 \, d} \] Input:
integrate(cos(d*x+c)^3*(a+a*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2), x, algorithm="maxima")
Output:
-1/12*(4*(sin(d*x + c)^3 - 3*sin(d*x + c))*A*a^3 - 9*(2*d*x + 2*c + sin(2* d*x + 2*c))*A*a^3 - 12*(d*x + c)*A*a^3 - 3*(2*d*x + 2*c + sin(2*d*x + 2*c) )*B*a^3 - 36*(d*x + c)*B*a^3 - 36*(d*x + c)*C*a^3 - 6*B*a^3*(log(sin(d*x + c) + 1) - log(sin(d*x + c) - 1)) - 18*C*a^3*(log(sin(d*x + c) + 1) - log( sin(d*x + c) - 1)) - 36*A*a^3*sin(d*x + c) - 36*B*a^3*sin(d*x + c) - 12*C* a^3*sin(d*x + c) - 12*C*a^3*tan(d*x + c))/d
Time = 0.32 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.66 \[ \int \cos ^3(c+d x) (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=-\frac {\frac {12 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1} - 3 \, {\left (5 \, A a^{3} + 7 \, B a^{3} + 6 \, C a^{3}\right )} {\left (d x + c\right )} - 6 \, {\left (B a^{3} + 3 \, C a^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) + 6 \, {\left (B a^{3} + 3 \, C a^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (15 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 15 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 40 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 36 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 12 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 33 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 21 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3}}}{6 \, d} \] Input:
integrate(cos(d*x+c)^3*(a+a*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2), x, algorithm="giac")
Output:
-1/6*(12*C*a^3*tan(1/2*d*x + 1/2*c)/(tan(1/2*d*x + 1/2*c)^2 - 1) - 3*(5*A* a^3 + 7*B*a^3 + 6*C*a^3)*(d*x + c) - 6*(B*a^3 + 3*C*a^3)*log(abs(tan(1/2*d *x + 1/2*c) + 1)) + 6*(B*a^3 + 3*C*a^3)*log(abs(tan(1/2*d*x + 1/2*c) - 1)) - 2*(15*A*a^3*tan(1/2*d*x + 1/2*c)^5 + 15*B*a^3*tan(1/2*d*x + 1/2*c)^5 + 6*C*a^3*tan(1/2*d*x + 1/2*c)^5 + 40*A*a^3*tan(1/2*d*x + 1/2*c)^3 + 36*B*a^ 3*tan(1/2*d*x + 1/2*c)^3 + 12*C*a^3*tan(1/2*d*x + 1/2*c)^3 + 33*A*a^3*tan( 1/2*d*x + 1/2*c) + 21*B*a^3*tan(1/2*d*x + 1/2*c) + 6*C*a^3*tan(1/2*d*x + 1 /2*c))/(tan(1/2*d*x + 1/2*c)^2 + 1)^3)/d
Time = 14.48 (sec) , antiderivative size = 290, normalized size of antiderivative = 1.72 \[ \int \cos ^3(c+d x) (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {5\,A\,a^3\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )+7\,B\,a^3\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )-B\,a^3\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,2{}\mathrm {i}+6\,C\,a^3\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )-C\,a^3\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,6{}\mathrm {i}}{d}+\frac {\frac {23\,A\,a^3\,\sin \left (2\,c+2\,d\,x\right )}{12}+\frac {3\,A\,a^3\,\sin \left (3\,c+3\,d\,x\right )}{8}+\frac {A\,a^3\,\sin \left (4\,c+4\,d\,x\right )}{24}+\frac {3\,B\,a^3\,\sin \left (2\,c+2\,d\,x\right )}{2}+\frac {B\,a^3\,\sin \left (3\,c+3\,d\,x\right )}{8}+\frac {C\,a^3\,\sin \left (2\,c+2\,d\,x\right )}{2}+\frac {3\,A\,a^3\,\sin \left (c+d\,x\right )}{8}+\frac {B\,a^3\,\sin \left (c+d\,x\right )}{8}+C\,a^3\,\sin \left (c+d\,x\right )}{d\,\cos \left (c+d\,x\right )} \] Input:
int(cos(c + d*x)^3*(a + a/cos(c + d*x))^3*(A + B/cos(c + d*x) + C/cos(c + d*x)^2),x)
Output:
(5*A*a^3*atan(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)) + 7*B*a^3*atan(sin(c/ 2 + (d*x)/2)/cos(c/2 + (d*x)/2)) - B*a^3*atan((sin(c/2 + (d*x)/2)*1i)/cos( c/2 + (d*x)/2))*2i + 6*C*a^3*atan(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)) - C*a^3*atan((sin(c/2 + (d*x)/2)*1i)/cos(c/2 + (d*x)/2))*6i)/d + ((23*A*a^3 *sin(2*c + 2*d*x))/12 + (3*A*a^3*sin(3*c + 3*d*x))/8 + (A*a^3*sin(4*c + 4* d*x))/24 + (3*B*a^3*sin(2*c + 2*d*x))/2 + (B*a^3*sin(3*c + 3*d*x))/8 + (C* a^3*sin(2*c + 2*d*x))/2 + (3*A*a^3*sin(c + d*x))/8 + (B*a^3*sin(c + d*x))/ 8 + C*a^3*sin(c + d*x))/(d*cos(c + d*x))
Time = 0.18 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.63 \[ \int \cos ^3(c+d x) (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {a^{3} \left (-6 \cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) b -18 \cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) c +6 \cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) b +18 \cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) c -2 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} a +24 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a +18 \cos \left (d x +c \right ) \sin \left (d x +c \right ) b +6 \cos \left (d x +c \right ) \sin \left (d x +c \right ) c +15 \cos \left (d x +c \right ) a c +15 \cos \left (d x +c \right ) a d x +21 \cos \left (d x +c \right ) b c +21 \cos \left (d x +c \right ) b d x +18 \cos \left (d x +c \right ) c^{2}+18 \cos \left (d x +c \right ) c d x -9 \sin \left (d x +c \right )^{3} a -3 \sin \left (d x +c \right )^{3} b +9 \sin \left (d x +c \right ) a +3 \sin \left (d x +c \right ) b +6 \sin \left (d x +c \right ) c \right )}{6 \cos \left (d x +c \right ) d} \] Input:
int(cos(d*x+c)^3*(a+a*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x)
Output:
(a**3*( - 6*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*b - 18*cos(c + d*x)*log (tan((c + d*x)/2) - 1)*c + 6*cos(c + d*x)*log(tan((c + d*x)/2) + 1)*b + 18 *cos(c + d*x)*log(tan((c + d*x)/2) + 1)*c - 2*cos(c + d*x)*sin(c + d*x)**3 *a + 24*cos(c + d*x)*sin(c + d*x)*a + 18*cos(c + d*x)*sin(c + d*x)*b + 6*c os(c + d*x)*sin(c + d*x)*c + 15*cos(c + d*x)*a*c + 15*cos(c + d*x)*a*d*x + 21*cos(c + d*x)*b*c + 21*cos(c + d*x)*b*d*x + 18*cos(c + d*x)*c**2 + 18*c os(c + d*x)*c*d*x - 9*sin(c + d*x)**3*a - 3*sin(c + d*x)**3*b + 9*sin(c + d*x)*a + 3*sin(c + d*x)*b + 6*sin(c + d*x)*c))/(6*cos(c + d*x)*d)