Integrand size = 33, antiderivative size = 198 \[ \int \cos ^3(c+d x) (a+a \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=2 a^4 (3 A+2 C) x+\frac {a^4 (2 A+13 C) \text {arctanh}(\sin (c+d x))}{2 d}+\frac {5 a^4 (2 A-C) \sin (c+d x)}{2 d}+\frac {2 a A \cos (c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{3 d}+\frac {A \cos ^2(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{3 d}-\frac {(2 A-C) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{2 d}-\frac {(4 A-9 C) \left (a^4+a^4 \sec (c+d x)\right ) \sin (c+d x)}{3 d} \] Output:
2*a^4*(3*A+2*C)*x+1/2*a^4*(2*A+13*C)*arctanh(sin(d*x+c))/d+5/2*a^4*(2*A-C) *sin(d*x+c)/d+2/3*a*A*cos(d*x+c)*(a+a*sec(d*x+c))^3*sin(d*x+c)/d+1/3*A*cos (d*x+c)^2*(a+a*sec(d*x+c))^4*sin(d*x+c)/d-1/2*(2*A-C)*(a^2+a^2*sec(d*x+c)) ^2*sin(d*x+c)/d-1/3*(4*A-9*C)*(a^4+a^4*sec(d*x+c))*sin(d*x+c)/d
Leaf count is larger than twice the leaf count of optimal. \(407\) vs. \(2(198)=396\).
Time = 10.78 (sec) , antiderivative size = 407, normalized size of antiderivative = 2.06 \[ \int \cos ^3(c+d x) (a+a \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {a^4 \cos ^6(c+d x) \sec ^8\left (\frac {1}{2} (c+d x)\right ) (1+\sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \left (24 (3 A+2 C) x-\frac {6 (2 A+13 C) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{d}+\frac {6 (2 A+13 C) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}{d}+\frac {3 (27 A+4 C) \cos (d x) \sin (c)}{d}+\frac {12 A \cos (2 d x) \sin (2 c)}{d}+\frac {A \cos (3 d x) \sin (3 c)}{d}+\frac {3 (27 A+4 C) \cos (c) \sin (d x)}{d}+\frac {12 A \cos (2 c) \sin (2 d x)}{d}+\frac {A \cos (3 c) \sin (3 d x)}{d}+\frac {3 C}{d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {48 C \sin \left (\frac {d x}{2}\right )}{d \left (\cos \left (\frac {c}{2}\right )-\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}-\frac {3 C}{d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {48 C \sin \left (\frac {d x}{2}\right )}{d \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}\right )}{96 (A+2 C+A \cos (2 (c+d x)))} \] Input:
Integrate[Cos[c + d*x]^3*(a + a*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2),x]
Output:
(a^4*Cos[c + d*x]^6*Sec[(c + d*x)/2]^8*(1 + Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2)*(24*(3*A + 2*C)*x - (6*(2*A + 13*C)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/d + (6*(2*A + 13*C)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) /d + (3*(27*A + 4*C)*Cos[d*x]*Sin[c])/d + (12*A*Cos[2*d*x]*Sin[2*c])/d + ( A*Cos[3*d*x]*Sin[3*c])/d + (3*(27*A + 4*C)*Cos[c]*Sin[d*x])/d + (12*A*Cos[ 2*c]*Sin[2*d*x])/d + (A*Cos[3*c]*Sin[3*d*x])/d + (3*C)/(d*(Cos[(c + d*x)/2 ] - Sin[(c + d*x)/2])^2) + (48*C*Sin[(d*x)/2])/(d*(Cos[c/2] - Sin[c/2])*(C os[(c + d*x)/2] - Sin[(c + d*x)/2])) - (3*C)/(d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2) + (48*C*Sin[(d*x)/2])/(d*(Cos[c/2] + Sin[c/2])*(Cos[(c + d* x)/2] + Sin[(c + d*x)/2]))))/(96*(A + 2*C + A*Cos[2*(c + d*x)]))
Time = 1.27 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.04, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.394, Rules used = {3042, 4575, 3042, 4505, 27, 3042, 4506, 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.
| \(\displaystyle \int \cos ^3(c+d x) (a \sec (c+d x)+a)^4 \left (A+C \sec ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^4 \left (A+C \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^3}dx\) |
| \(\Big \downarrow \) 4575 |
| \(\displaystyle \frac {\int \cos ^2(c+d x) (\sec (c+d x) a+a)^4 (4 a A-a (2 A-3 C) \sec (c+d x))dx}{3 a}+\frac {A \sin (c+d x) \cos ^2(c+d x) (a \sec (c+d x)+a)^4}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^4 \left (4 a A-a (2 A-3 C) \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^2}dx}{3 a}+\frac {A \sin (c+d x) \cos ^2(c+d x) (a \sec (c+d x)+a)^4}{3 d}\) |
| \(\Big \downarrow \) 4505 |
| \(\displaystyle \frac {\frac {1}{2} \int 2 \cos (c+d x) (\sec (c+d x) a+a)^3 \left (a^2 (8 A+3 C)-3 a^2 (2 A-C) \sec (c+d x)\right )dx+\frac {2 a^2 A \sin (c+d x) \cos (c+d x) (a \sec (c+d x)+a)^3}{d}}{3 a}+\frac {A \sin (c+d x) \cos ^2(c+d x) (a \sec (c+d x)+a)^4}{3 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \cos (c+d x) (\sec (c+d x) a+a)^3 \left (a^2 (8 A+3 C)-3 a^2 (2 A-C) \sec (c+d x)\right )dx+\frac {2 a^2 A \sin (c+d x) \cos (c+d x) (a \sec (c+d x)+a)^3}{d}}{3 a}+\frac {A \sin (c+d x) \cos ^2(c+d x) (a \sec (c+d x)+a)^4}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3 \left (a^2 (8 A+3 C)-3 a^2 (2 A-C) \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {2 a^2 A \sin (c+d x) \cos (c+d x) (a \sec (c+d x)+a)^3}{d}}{3 a}+\frac {A \sin (c+d x) \cos ^2(c+d x) (a \sec (c+d x)+a)^4}{3 d}\) |
| \(\Big \downarrow \) 4506 |
| \(\displaystyle \frac {\frac {1}{2} \int \cos (c+d x) (\sec (c+d x) a+a)^2 \left (a^3 (22 A+3 C)-2 a^3 (4 A-9 C) \sec (c+d x)\right )dx-\frac {3 a^3 (2 A-C) \sin (c+d x) (a \sec (c+d x)+a)^2}{2 d}+\frac {2 a^2 A \sin (c+d x) \cos (c+d x) (a \sec (c+d x)+a)^3}{d}}{3 a}+\frac {A \sin (c+d x) \cos ^2(c+d x) (a \sec (c+d x)+a)^4}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{2} \int \frac {\left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2 \left (a^3 (22 A+3 C)-2 a^3 (4 A-9 C) \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )}dx-\frac {3 a^3 (2 A-C) \sin (c+d x) (a \sec (c+d x)+a)^2}{2 d}+\frac {2 a^2 A \sin (c+d x) \cos (c+d x) (a \sec (c+d x)+a)^3}{d}}{3 a}+\frac {A \sin (c+d x) \cos ^2(c+d x) (a \sec (c+d x)+a)^4}{3 d}\) |
| \(\Big \downarrow \) 4506 |
| \(\displaystyle \frac {\frac {1}{2} \left (\int 3 \cos (c+d x) (\sec (c+d x) a+a) \left (5 (2 A-C) a^4+(2 A+13 C) \sec (c+d x) a^4\right )dx-\frac {2 (4 A-9 C) \sin (c+d x) \left (a^5 \sec (c+d x)+a^5\right )}{d}\right )-\frac {3 a^3 (2 A-C) \sin (c+d x) (a \sec (c+d x)+a)^2}{2 d}+\frac {2 a^2 A \sin (c+d x) \cos (c+d x) (a \sec (c+d x)+a)^3}{d}}{3 a}+\frac {A \sin (c+d x) \cos ^2(c+d x) (a \sec (c+d x)+a)^4}{3 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{2} \left (3 \int \cos (c+d x) (\sec (c+d x) a+a) \left (5 (2 A-C) a^4+(2 A+13 C) \sec (c+d x) a^4\right )dx-\frac {2 (4 A-9 C) \sin (c+d x) \left (a^5 \sec (c+d x)+a^5\right )}{d}\right )-\frac {3 a^3 (2 A-C) \sin (c+d x) (a \sec (c+d x)+a)^2}{2 d}+\frac {2 a^2 A \sin (c+d x) \cos (c+d x) (a \sec (c+d x)+a)^3}{d}}{3 a}+\frac {A \sin (c+d x) \cos ^2(c+d x) (a \sec (c+d x)+a)^4}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{2} \left (3 \int \frac {\left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right ) \left (5 (2 A-C) a^4+(2 A+13 C) \csc \left (c+d x+\frac {\pi }{2}\right ) a^4\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )}dx-\frac {2 (4 A-9 C) \sin (c+d x) \left (a^5 \sec (c+d x)+a^5\right )}{d}\right )-\frac {3 a^3 (2 A-C) \sin (c+d x) (a \sec (c+d x)+a)^2}{2 d}+\frac {2 a^2 A \sin (c+d x) \cos (c+d x) (a \sec (c+d x)+a)^3}{d}}{3 a}+\frac {A \sin (c+d x) \cos ^2(c+d x) (a \sec (c+d x)+a)^4}{3 d}\) |
| \(\Big \downarrow \) 4484 |
| \(\displaystyle \frac {\frac {1}{2} \left (3 \left (\frac {5 a^5 (2 A-C) \sin (c+d x)}{d}-\int \left (-4 (3 A+2 C) a^5-(2 A+13 C) \sec (c+d x) a^5\right )dx\right )-\frac {2 (4 A-9 C) \sin (c+d x) \left (a^5 \sec (c+d x)+a^5\right )}{d}\right )-\frac {3 a^3 (2 A-C) \sin (c+d x) (a \sec (c+d x)+a)^2}{2 d}+\frac {2 a^2 A \sin (c+d x) \cos (c+d x) (a \sec (c+d x)+a)^3}{d}}{3 a}+\frac {A \sin (c+d x) \cos ^2(c+d x) (a \sec (c+d x)+a)^4}{3 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {1}{2} \left (3 \left (\frac {a^5 (2 A+13 C) \text {arctanh}(\sin (c+d x))}{d}+\frac {5 a^5 (2 A-C) \sin (c+d x)}{d}+4 a^5 x (3 A+2 C)\right )-\frac {2 (4 A-9 C) \sin (c+d x) \left (a^5 \sec (c+d x)+a^5\right )}{d}\right )-\frac {3 a^3 (2 A-C) \sin (c+d x) (a \sec (c+d x)+a)^2}{2 d}+\frac {2 a^2 A \sin (c+d x) \cos (c+d x) (a \sec (c+d x)+a)^3}{d}}{3 a}+\frac {A \sin (c+d x) \cos ^2(c+d x) (a \sec (c+d x)+a)^4}{3 d}\) |
Input:
Int[Cos[c + d*x]^3*(a + a*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2),x]
Output:
(A*Cos[c + d*x]^2*(a + a*Sec[c + d*x])^4*Sin[c + d*x])/(3*d) + ((-3*a^3*(2 *A - C)*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(2*d) + (2*a^2*A*Cos[c + d*x] *(a + a*Sec[c + d*x])^3*Sin[c + d*x])/d + ((-2*(4*A - 9*C)*(a^5 + a^5*Sec[ c + d*x])*Sin[c + d*x])/d + 3*(4*a^5*(3*A + 2*C)*x + (a^5*(2*A + 13*C)*Arc Tanh[Sin[c + d*x]])/d + (5*a^5*(2*A - C)*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_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(d_. ))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[A*Co t[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 *(A*(m + n + 1) + C*n)*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, C, m}, x] && EqQ[a^2 - b^2, 0] && !LtQ[m, -2^(-1)] && (LtQ[n, -2^(-1)] || EqQ[m + n + 1, 0])
Time = 0.78 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.91
| method | result | size |
| parallelrisch | \(\frac {83 \left (-\frac {24 \left (\frac {13 C}{2}+A \right ) \left (1+\cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{83}+\frac {24 \left (\frac {13 C}{2}+A \right ) \left (1+\cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{83}+\frac {144 d \left (A +\frac {2 C}{3}\right ) x \cos \left (2 d x +2 c \right )}{83}+\frac {24 \left (A +4 C \right ) \sin \left (2 d x +2 c \right )}{83}+\left (A +\frac {12 C}{83}\right ) \sin \left (3 d x +3 c \right )+\frac {12 A \sin \left (4 d x +4 c \right )}{83}+\frac {A \sin \left (5 d x +5 c \right )}{83}+\frac {2 \left (41 A +18 C \right ) \sin \left (d x +c \right )}{83}+\frac {144 d \left (A +\frac {2 C}{3}\right ) x}{83}\right ) a^{4}}{24 d \left (1+\cos \left (2 d x +2 c \right )\right )}\) | \(180\) |
| derivativedivides | \(\frac {\frac {a^{4} A \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+a^{4} C \sin \left (d x +c \right )+4 a^{4} A \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+4 a^{4} C \left (d x +c \right )+6 a^{4} A \sin \left (d x +c \right )+6 a^{4} C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+4 a^{4} A \left (d x +c \right )+4 a^{4} C \tan \left (d x +c \right )+a^{4} A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+a^{4} C \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}\) | \(188\) |
| default | \(\frac {\frac {a^{4} A \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+a^{4} C \sin \left (d x +c \right )+4 a^{4} A \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+4 a^{4} C \left (d x +c \right )+6 a^{4} A \sin \left (d x +c \right )+6 a^{4} C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+4 a^{4} A \left (d x +c \right )+4 a^{4} C \tan \left (d x +c \right )+a^{4} A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+a^{4} C \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}\) | \(188\) |
| risch | \(6 a^{4} A x +4 a^{4} x C -\frac {i a^{4} A \,{\mathrm e}^{3 i \left (d x +c \right )}}{24 d}-\frac {i a^{4} A \,{\mathrm e}^{2 i \left (d x +c \right )}}{2 d}-\frac {27 i a^{4} A \,{\mathrm e}^{i \left (d x +c \right )}}{8 d}-\frac {i {\mathrm e}^{i \left (d x +c \right )} a^{4} C}{2 d}+\frac {27 i a^{4} A \,{\mathrm e}^{-i \left (d x +c \right )}}{8 d}+\frac {i {\mathrm e}^{-i \left (d x +c \right )} a^{4} C}{2 d}+\frac {i a^{4} A \,{\mathrm e}^{-2 i \left (d x +c \right )}}{2 d}+\frac {i a^{4} A \,{\mathrm e}^{-3 i \left (d x +c \right )}}{24 d}-\frac {i a^{4} C \left ({\mathrm e}^{3 i \left (d x +c \right )}-8 \,{\mathrm e}^{2 i \left (d x +c \right )}-{\mathrm e}^{i \left (d x +c \right )}-8\right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}-\frac {a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) A}{d}-\frac {13 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{2 d}+\frac {a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) A}{d}+\frac {13 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{2 d}\) | \(311\) |
| norman | \(\frac {\left (-6 a^{4} A -4 a^{4} C \right ) x +\left (-36 a^{4} A -24 a^{4} C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\left (-12 a^{4} A -8 a^{4} C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}+\left (-12 a^{4} A -8 a^{4} C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}+\left (6 a^{4} A +4 a^{4} C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{16}+\left (12 a^{4} A +8 a^{4} C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\left (12 a^{4} A +8 a^{4} C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (36 a^{4} A +24 a^{4} C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+\frac {5 a^{4} \left (2 A -C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{15}}{d}-\frac {a^{4} \left (18 A +11 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}-\frac {a^{4} \left (26 A -69 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{3 d}-\frac {5 a^{4} \left (38 A -15 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{3 d}-\frac {a^{4} \left (70 A +93 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{3 d}-\frac {a^{4} \left (74 A -9 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{3 d}+\frac {a^{4} \left (190 A -51 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{3 d}+\frac {a^{4} \left (194 A +39 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{5}}-\frac {a^{4} \left (2 A +13 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 d}+\frac {a^{4} \left (2 A +13 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 d}\) | \(491\) |
Input:
int(cos(d*x+c)^3*(a+a*sec(d*x+c))^4*(A+C*sec(d*x+c)^2),x,method=_RETURNVER BOSE)
Output:
83/24*(-24/83*(13/2*C+A)*(1+cos(2*d*x+2*c))*ln(tan(1/2*d*x+1/2*c)-1)+24/83 *(13/2*C+A)*(1+cos(2*d*x+2*c))*ln(tan(1/2*d*x+1/2*c)+1)+144/83*d*(A+2/3*C) *x*cos(2*d*x+2*c)+24/83*(A+4*C)*sin(2*d*x+2*c)+(A+12/83*C)*sin(3*d*x+3*c)+ 12/83*A*sin(4*d*x+4*c)+1/83*A*sin(5*d*x+5*c)+2/83*(41*A+18*C)*sin(d*x+c)+1 44/83*d*(A+2/3*C)*x)*a^4/d/(1+cos(2*d*x+2*c))
Time = 0.10 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.86 \[ \int \cos ^3(c+d x) (a+a \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {24 \, {\left (3 \, A + 2 \, C\right )} a^{4} d x \cos \left (d x + c\right )^{2} + 3 \, {\left (2 \, A + 13 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (2 \, A + 13 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (2 \, A a^{4} \cos \left (d x + c\right )^{4} + 12 \, A a^{4} \cos \left (d x + c\right )^{3} + 2 \, {\left (20 \, A + 3 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} + 24 \, C a^{4} \cos \left (d x + c\right ) + 3 \, C a^{4}\right )} \sin \left (d x + c\right )}{12 \, d \cos \left (d x + c\right )^{2}} \] Input:
integrate(cos(d*x+c)^3*(a+a*sec(d*x+c))^4*(A+C*sec(d*x+c)^2),x, algorithm= "fricas")
Output:
1/12*(24*(3*A + 2*C)*a^4*d*x*cos(d*x + c)^2 + 3*(2*A + 13*C)*a^4*cos(d*x + c)^2*log(sin(d*x + c) + 1) - 3*(2*A + 13*C)*a^4*cos(d*x + c)^2*log(-sin(d *x + c) + 1) + 2*(2*A*a^4*cos(d*x + c)^4 + 12*A*a^4*cos(d*x + c)^3 + 2*(20 *A + 3*C)*a^4*cos(d*x + c)^2 + 24*C*a^4*cos(d*x + c) + 3*C*a^4)*sin(d*x + c))/(d*cos(d*x + c)^2)
Timed out. \[ \int \cos ^3(c+d x) (a+a \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)**3*(a+a*sec(d*x+c))**4*(A+C*sec(d*x+c)**2),x)
Output:
Timed out
Time = 0.04 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.07 \[ \int \cos ^3(c+d x) (a+a \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=-\frac {4 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{4} - 12 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} - 48 \, {\left (d x + c\right )} A a^{4} - 48 \, {\left (d x + c\right )} C a^{4} + 3 \, C a^{4} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 6 \, A a^{4} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 36 \, C a^{4} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 72 \, A a^{4} \sin \left (d x + c\right ) - 12 \, C a^{4} \sin \left (d x + c\right ) - 48 \, C a^{4} \tan \left (d x + c\right )}{12 \, d} \] Input:
integrate(cos(d*x+c)^3*(a+a*sec(d*x+c))^4*(A+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^4 - 12*(2*d*x + 2*c + sin(2 *d*x + 2*c))*A*a^4 - 48*(d*x + c)*A*a^4 - 48*(d*x + c)*C*a^4 + 3*C*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)) - 6*A*a^4*(log(sin(d*x + c) + 1) - log(sin(d*x + c) - 1)) - 36*C* a^4*(log(sin(d*x + c) + 1) - log(sin(d*x + c) - 1)) - 72*A*a^4*sin(d*x + c ) - 12*C*a^4*sin(d*x + c) - 48*C*a^4*tan(d*x + c))/d
Time = 0.34 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.25 \[ \int \cos ^3(c+d x) (a+a \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {12 \, {\left (3 \, A a^{4} + 2 \, C a^{4}\right )} {\left (d x + c\right )} + 3 \, {\left (2 \, A a^{4} + 13 \, C a^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 3 \, {\left (2 \, A a^{4} + 13 \, C a^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {6 \, {\left (7 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 9 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{2}} + \frac {4 \, {\left (15 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 38 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 6 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 27 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3}}}{6 \, d} \] Input:
integrate(cos(d*x+c)^3*(a+a*sec(d*x+c))^4*(A+C*sec(d*x+c)^2),x, algorithm= "giac")
Output:
1/6*(12*(3*A*a^4 + 2*C*a^4)*(d*x + c) + 3*(2*A*a^4 + 13*C*a^4)*log(abs(tan (1/2*d*x + 1/2*c) + 1)) - 3*(2*A*a^4 + 13*C*a^4)*log(abs(tan(1/2*d*x + 1/2 *c) - 1)) - 6*(7*C*a^4*tan(1/2*d*x + 1/2*c)^3 - 9*C*a^4*tan(1/2*d*x + 1/2* c))/(tan(1/2*d*x + 1/2*c)^2 - 1)^2 + 4*(15*A*a^4*tan(1/2*d*x + 1/2*c)^5 + 3*C*a^4*tan(1/2*d*x + 1/2*c)^5 + 38*A*a^4*tan(1/2*d*x + 1/2*c)^3 + 6*C*a^4 *tan(1/2*d*x + 1/2*c)^3 + 27*A*a^4*tan(1/2*d*x + 1/2*c) + 3*C*a^4*tan(1/2* d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 + 1)^3)/d
Time = 11.86 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.23 \[ \int \cos ^3(c+d x) (a+a \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {20\,A\,a^4\,\sin \left (c+d\,x\right )}{3\,d}+\frac {C\,a^4\,\sin \left (c+d\,x\right )}{d}+\frac {12\,A\,a^4\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {2\,A\,a^4\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {8\,C\,a^4\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {13\,C\,a^4\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {A\,a^4\,{\cos \left (c+d\,x\right )}^2\,\sin \left (c+d\,x\right )}{3\,d}+\frac {4\,C\,a^4\,\sin \left (c+d\,x\right )}{d\,\cos \left (c+d\,x\right )}+\frac {C\,a^4\,\sin \left (c+d\,x\right )}{2\,d\,{\cos \left (c+d\,x\right )}^2}+\frac {2\,A\,a^4\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )}{d} \] Input:
int(cos(c + d*x)^3*(A + C/cos(c + d*x)^2)*(a + a/cos(c + d*x))^4,x)
Output:
(20*A*a^4*sin(c + d*x))/(3*d) + (C*a^4*sin(c + d*x))/d + (12*A*a^4*atan(si n(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)))/d + (2*A*a^4*atanh(sin(c/2 + (d*x)/2 )/cos(c/2 + (d*x)/2)))/d + (8*C*a^4*atan(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x )/2)))/d + (13*C*a^4*atanh(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)))/d + (A* a^4*cos(c + d*x)^2*sin(c + d*x))/(3*d) + (4*C*a^4*sin(c + d*x))/(d*cos(c + d*x)) + (C*a^4*sin(c + d*x))/(2*d*cos(c + d*x)^2) + (2*A*a^4*cos(c + d*x) *sin(c + d*x))/d
Time = 0.17 (sec) , antiderivative size = 341, normalized size of antiderivative = 1.72 \[ \int \cos ^3(c+d x) (a+a \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {a^{4} \left (12 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} a -12 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a -24 \cos \left (d x +c \right ) \sin \left (d x +c \right ) c -6 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \sin \left (d x +c \right )^{2} a -39 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \sin \left (d x +c \right )^{2} c +6 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) a +39 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) c +6 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \sin \left (d x +c \right )^{2} a +39 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \sin \left (d x +c \right )^{2} c -6 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) a -39 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) c -2 \sin \left (d x +c \right )^{5} a +44 \sin \left (d x +c \right )^{3} a +6 \sin \left (d x +c \right )^{3} c +36 \sin \left (d x +c \right )^{2} a c +36 \sin \left (d x +c \right )^{2} a d x +24 \sin \left (d x +c \right )^{2} c^{2}+24 \sin \left (d x +c \right )^{2} c d x -42 \sin \left (d x +c \right ) a -9 \sin \left (d x +c \right ) c -36 a c -36 a d x -24 c^{2}-24 c d x \right )}{6 d \left (\sin \left (d x +c \right )^{2}-1\right )} \] Input:
int(cos(d*x+c)^3*(a+a*sec(d*x+c))^4*(A+C*sec(d*x+c)^2),x)
Output:
(a**4*(12*cos(c + d*x)*sin(c + d*x)**3*a - 12*cos(c + d*x)*sin(c + d*x)*a - 24*cos(c + d*x)*sin(c + d*x)*c - 6*log(tan((c + d*x)/2) - 1)*sin(c + d*x )**2*a - 39*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**2*c + 6*log(tan((c + d *x)/2) - 1)*a + 39*log(tan((c + d*x)/2) - 1)*c + 6*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**2*a + 39*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**2*c - 6* log(tan((c + d*x)/2) + 1)*a - 39*log(tan((c + d*x)/2) + 1)*c - 2*sin(c + d *x)**5*a + 44*sin(c + d*x)**3*a + 6*sin(c + d*x)**3*c + 36*sin(c + d*x)**2 *a*c + 36*sin(c + d*x)**2*a*d*x + 24*sin(c + d*x)**2*c**2 + 24*sin(c + d*x )**2*c*d*x - 42*sin(c + d*x)*a - 9*sin(c + d*x)*c - 36*a*c - 36*a*d*x - 24 *c**2 - 24*c*d*x))/(6*d*(sin(c + d*x)**2 - 1))