Integrand size = 33, antiderivative size = 215 \[ \int \frac {\cos ^2(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^4} \, dx=\frac {(21 A+2 C) x}{2 a^4}-\frac {32 (54 A+5 C) \sin (c+d x)}{105 a^4 d}+\frac {(21 A+2 C) \cos (c+d x) \sin (c+d x)}{2 a^4 d}-\frac {(129 A+10 C) \cos (c+d x) \sin (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac {16 (54 A+5 C) \cos (c+d x) \sin (c+d x)}{105 a^4 d (1+\sec (c+d x))}-\frac {(A+C) \cos (c+d x) \sin (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac {2 A \cos (c+d x) \sin (c+d x)}{5 a d (a+a \sec (c+d x))^3} \] Output:
1/2*(21*A+2*C)*x/a^4-32/105*(54*A+5*C)*sin(d*x+c)/a^4/d+1/2*(21*A+2*C)*cos (d*x+c)*sin(d*x+c)/a^4/d-1/105*(129*A+10*C)*cos(d*x+c)*sin(d*x+c)/a^4/d/(1 +sec(d*x+c))^2-16/105*(54*A+5*C)*cos(d*x+c)*sin(d*x+c)/a^4/d/(1+sec(d*x+c) )-1/7*(A+C)*cos(d*x+c)*sin(d*x+c)/d/(a+a*sec(d*x+c))^4-2/5*A*cos(d*x+c)*si n(d*x+c)/a/d/(a+a*sec(d*x+c))^3
Leaf count is larger than twice the leaf count of optimal. \(505\) vs. \(2(215)=430\).
Time = 6.39 (sec) , antiderivative size = 505, normalized size of antiderivative = 2.35 \[ \int \frac {\cos ^2(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^4} \, dx=\frac {\sec \left (\frac {c}{2}\right ) \sec ^7\left (\frac {1}{2} (c+d x)\right ) \left (14700 (21 A+2 C) d x \cos \left (\frac {d x}{2}\right )+14700 (21 A+2 C) d x \cos \left (c+\frac {d x}{2}\right )+185220 A d x \cos \left (c+\frac {3 d x}{2}\right )+17640 C d x \cos \left (c+\frac {3 d x}{2}\right )+185220 A d x \cos \left (2 c+\frac {3 d x}{2}\right )+17640 C d x \cos \left (2 c+\frac {3 d x}{2}\right )+61740 A d x \cos \left (2 c+\frac {5 d x}{2}\right )+5880 C d x \cos \left (2 c+\frac {5 d x}{2}\right )+61740 A d x \cos \left (3 c+\frac {5 d x}{2}\right )+5880 C d x \cos \left (3 c+\frac {5 d x}{2}\right )+8820 A d x \cos \left (3 c+\frac {7 d x}{2}\right )+840 C d x \cos \left (3 c+\frac {7 d x}{2}\right )+8820 A d x \cos \left (4 c+\frac {7 d x}{2}\right )+840 C d x \cos \left (4 c+\frac {7 d x}{2}\right )-539490 A \sin \left (\frac {d x}{2}\right )-79520 C \sin \left (\frac {d x}{2}\right )+386190 A \sin \left (c+\frac {d x}{2}\right )+66080 C \sin \left (c+\frac {d x}{2}\right )-422478 A \sin \left (c+\frac {3 d x}{2}\right )-57120 C \sin \left (c+\frac {3 d x}{2}\right )+132930 A \sin \left (2 c+\frac {3 d x}{2}\right )+30240 C \sin \left (2 c+\frac {3 d x}{2}\right )-181461 A \sin \left (2 c+\frac {5 d x}{2}\right )-22400 C \sin \left (2 c+\frac {5 d x}{2}\right )+3675 A \sin \left (3 c+\frac {5 d x}{2}\right )+6720 C \sin \left (3 c+\frac {5 d x}{2}\right )-36003 A \sin \left (3 c+\frac {7 d x}{2}\right )-4160 C \sin \left (3 c+\frac {7 d x}{2}\right )-9555 A \sin \left (4 c+\frac {7 d x}{2}\right )-945 A \sin \left (4 c+\frac {9 d x}{2}\right )-945 A \sin \left (5 c+\frac {9 d x}{2}\right )+105 A \sin \left (5 c+\frac {11 d x}{2}\right )+105 A \sin \left (6 c+\frac {11 d x}{2}\right )\right )}{107520 a^4 d} \] Input:
Integrate[(Cos[c + d*x]^2*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^4,x ]
Output:
(Sec[c/2]*Sec[(c + d*x)/2]^7*(14700*(21*A + 2*C)*d*x*Cos[(d*x)/2] + 14700* (21*A + 2*C)*d*x*Cos[c + (d*x)/2] + 185220*A*d*x*Cos[c + (3*d*x)/2] + 1764 0*C*d*x*Cos[c + (3*d*x)/2] + 185220*A*d*x*Cos[2*c + (3*d*x)/2] + 17640*C*d *x*Cos[2*c + (3*d*x)/2] + 61740*A*d*x*Cos[2*c + (5*d*x)/2] + 5880*C*d*x*Co s[2*c + (5*d*x)/2] + 61740*A*d*x*Cos[3*c + (5*d*x)/2] + 5880*C*d*x*Cos[3*c + (5*d*x)/2] + 8820*A*d*x*Cos[3*c + (7*d*x)/2] + 840*C*d*x*Cos[3*c + (7*d *x)/2] + 8820*A*d*x*Cos[4*c + (7*d*x)/2] + 840*C*d*x*Cos[4*c + (7*d*x)/2] - 539490*A*Sin[(d*x)/2] - 79520*C*Sin[(d*x)/2] + 386190*A*Sin[c + (d*x)/2] + 66080*C*Sin[c + (d*x)/2] - 422478*A*Sin[c + (3*d*x)/2] - 57120*C*Sin[c + (3*d*x)/2] + 132930*A*Sin[2*c + (3*d*x)/2] + 30240*C*Sin[2*c + (3*d*x)/2 ] - 181461*A*Sin[2*c + (5*d*x)/2] - 22400*C*Sin[2*c + (5*d*x)/2] + 3675*A* Sin[3*c + (5*d*x)/2] + 6720*C*Sin[3*c + (5*d*x)/2] - 36003*A*Sin[3*c + (7* d*x)/2] - 4160*C*Sin[3*c + (7*d*x)/2] - 9555*A*Sin[4*c + (7*d*x)/2] - 945* A*Sin[4*c + (9*d*x)/2] - 945*A*Sin[5*c + (9*d*x)/2] + 105*A*Sin[5*c + (11* d*x)/2] + 105*A*Sin[6*c + (11*d*x)/2]))/(107520*a^4*d)
Time = 1.52 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.07, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {3042, 4573, 25, 3042, 4508, 3042, 4508, 3042, 4508, 3042, 4274, 3042, 3115, 24, 3117}
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.
| \(\displaystyle \int \frac {\cos ^2(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a \sec (c+d x)+a)^4} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+C \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\csc \left (c+d x+\frac {\pi }{2}\right )^2 \left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^4}dx\) |
| \(\Big \downarrow \) 4573 |
| \(\displaystyle -\frac {\int -\frac {\cos ^2(c+d x) (a (9 A+2 C)-a (5 A-2 C) \sec (c+d x))}{(\sec (c+d x) a+a)^3}dx}{7 a^2}-\frac {(A+C) \sin (c+d x) \cos (c+d x)}{7 d (a \sec (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {\cos ^2(c+d x) (a (9 A+2 C)-a (5 A-2 C) \sec (c+d x))}{(\sec (c+d x) a+a)^3}dx}{7 a^2}-\frac {(A+C) \sin (c+d x) \cos (c+d x)}{7 d (a \sec (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {a (9 A+2 C)-a (5 A-2 C) \csc \left (c+d x+\frac {\pi }{2}\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^2 \left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3}dx}{7 a^2}-\frac {(A+C) \sin (c+d x) \cos (c+d x)}{7 d (a \sec (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 4508 |
| \(\displaystyle \frac {\frac {\int \frac {\cos ^2(c+d x) \left (a^2 (73 A+10 C)-56 a^2 A \sec (c+d x)\right )}{(\sec (c+d x) a+a)^2}dx}{5 a^2}-\frac {14 a A \sin (c+d x) \cos (c+d x)}{5 d (a \sec (c+d x)+a)^3}}{7 a^2}-\frac {(A+C) \sin (c+d x) \cos (c+d x)}{7 d (a \sec (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {a^2 (73 A+10 C)-56 a^2 A \csc \left (c+d x+\frac {\pi }{2}\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^2 \left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2}dx}{5 a^2}-\frac {14 a A \sin (c+d x) \cos (c+d x)}{5 d (a \sec (c+d x)+a)^3}}{7 a^2}-\frac {(A+C) \sin (c+d x) \cos (c+d x)}{7 d (a \sec (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 4508 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {\cos ^2(c+d x) \left (a^3 (477 A+50 C)-3 a^3 (129 A+10 C) \sec (c+d x)\right )}{\sec (c+d x) a+a}dx}{3 a^2}-\frac {(129 A+10 C) \sin (c+d x) \cos (c+d x)}{3 d (\sec (c+d x)+1)^2}}{5 a^2}-\frac {14 a A \sin (c+d x) \cos (c+d x)}{5 d (a \sec (c+d x)+a)^3}}{7 a^2}-\frac {(A+C) \sin (c+d x) \cos (c+d x)}{7 d (a \sec (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {a^3 (477 A+50 C)-3 a^3 (129 A+10 C) \csc \left (c+d x+\frac {\pi }{2}\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^2 \left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )}dx}{3 a^2}-\frac {(129 A+10 C) \sin (c+d x) \cos (c+d x)}{3 d (\sec (c+d x)+1)^2}}{5 a^2}-\frac {14 a A \sin (c+d x) \cos (c+d x)}{5 d (a \sec (c+d x)+a)^3}}{7 a^2}-\frac {(A+C) \sin (c+d x) \cos (c+d x)}{7 d (a \sec (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 4508 |
| \(\displaystyle \frac {\frac {\frac {\frac {\int \cos ^2(c+d x) \left (105 a^4 (21 A+2 C)-32 a^4 (54 A+5 C) \sec (c+d x)\right )dx}{a^2}-\frac {16 a^3 (54 A+5 C) \sin (c+d x) \cos (c+d x)}{d (a \sec (c+d x)+a)}}{3 a^2}-\frac {(129 A+10 C) \sin (c+d x) \cos (c+d x)}{3 d (\sec (c+d x)+1)^2}}{5 a^2}-\frac {14 a A \sin (c+d x) \cos (c+d x)}{5 d (a \sec (c+d x)+a)^3}}{7 a^2}-\frac {(A+C) \sin (c+d x) \cos (c+d x)}{7 d (a \sec (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\frac {\int \frac {105 a^4 (21 A+2 C)-32 a^4 (54 A+5 C) \csc \left (c+d x+\frac {\pi }{2}\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^2}dx}{a^2}-\frac {16 a^3 (54 A+5 C) \sin (c+d x) \cos (c+d x)}{d (a \sec (c+d x)+a)}}{3 a^2}-\frac {(129 A+10 C) \sin (c+d x) \cos (c+d x)}{3 d (\sec (c+d x)+1)^2}}{5 a^2}-\frac {14 a A \sin (c+d x) \cos (c+d x)}{5 d (a \sec (c+d x)+a)^3}}{7 a^2}-\frac {(A+C) \sin (c+d x) \cos (c+d x)}{7 d (a \sec (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 4274 |
| \(\displaystyle \frac {\frac {\frac {\frac {105 a^4 (21 A+2 C) \int \cos ^2(c+d x)dx-32 a^4 (54 A+5 C) \int \cos (c+d x)dx}{a^2}-\frac {16 a^3 (54 A+5 C) \sin (c+d x) \cos (c+d x)}{d (a \sec (c+d x)+a)}}{3 a^2}-\frac {(129 A+10 C) \sin (c+d x) \cos (c+d x)}{3 d (\sec (c+d x)+1)^2}}{5 a^2}-\frac {14 a A \sin (c+d x) \cos (c+d x)}{5 d (a \sec (c+d x)+a)^3}}{7 a^2}-\frac {(A+C) \sin (c+d x) \cos (c+d x)}{7 d (a \sec (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\frac {105 a^4 (21 A+2 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx-32 a^4 (54 A+5 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )dx}{a^2}-\frac {16 a^3 (54 A+5 C) \sin (c+d x) \cos (c+d x)}{d (a \sec (c+d x)+a)}}{3 a^2}-\frac {(129 A+10 C) \sin (c+d x) \cos (c+d x)}{3 d (\sec (c+d x)+1)^2}}{5 a^2}-\frac {14 a A \sin (c+d x) \cos (c+d x)}{5 d (a \sec (c+d x)+a)^3}}{7 a^2}-\frac {(A+C) \sin (c+d x) \cos (c+d x)}{7 d (a \sec (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {\frac {\frac {\frac {105 a^4 (21 A+2 C) \left (\frac {\int 1dx}{2}+\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )-32 a^4 (54 A+5 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )dx}{a^2}-\frac {16 a^3 (54 A+5 C) \sin (c+d x) \cos (c+d x)}{d (a \sec (c+d x)+a)}}{3 a^2}-\frac {(129 A+10 C) \sin (c+d x) \cos (c+d x)}{3 d (\sec (c+d x)+1)^2}}{5 a^2}-\frac {14 a A \sin (c+d x) \cos (c+d x)}{5 d (a \sec (c+d x)+a)^3}}{7 a^2}-\frac {(A+C) \sin (c+d x) \cos (c+d x)}{7 d (a \sec (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {\frac {\frac {\frac {105 a^4 (21 A+2 C) \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )-32 a^4 (54 A+5 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )dx}{a^2}-\frac {16 a^3 (54 A+5 C) \sin (c+d x) \cos (c+d x)}{d (a \sec (c+d x)+a)}}{3 a^2}-\frac {(129 A+10 C) \sin (c+d x) \cos (c+d x)}{3 d (\sec (c+d x)+1)^2}}{5 a^2}-\frac {14 a A \sin (c+d x) \cos (c+d x)}{5 d (a \sec (c+d x)+a)^3}}{7 a^2}-\frac {(A+C) \sin (c+d x) \cos (c+d x)}{7 d (a \sec (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3117 |
| \(\displaystyle \frac {\frac {\frac {\frac {105 a^4 (21 A+2 C) \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )-\frac {32 a^4 (54 A+5 C) \sin (c+d x)}{d}}{a^2}-\frac {16 a^3 (54 A+5 C) \sin (c+d x) \cos (c+d x)}{d (a \sec (c+d x)+a)}}{3 a^2}-\frac {(129 A+10 C) \sin (c+d x) \cos (c+d x)}{3 d (\sec (c+d x)+1)^2}}{5 a^2}-\frac {14 a A \sin (c+d x) \cos (c+d x)}{5 d (a \sec (c+d x)+a)^3}}{7 a^2}-\frac {(A+C) \sin (c+d x) \cos (c+d x)}{7 d (a \sec (c+d x)+a)^4}\) |
Input:
Int[(Cos[c + d*x]^2*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^4,x]
Output:
-1/7*((A + C)*Cos[c + d*x]*Sin[c + d*x])/(d*(a + a*Sec[c + d*x])^4) + ((-1 4*a*A*Cos[c + d*x]*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) + (-1/3*((12 9*A + 10*C)*Cos[c + d*x]*Sin[c + d*x])/(d*(1 + Sec[c + d*x])^2) + ((-16*a^ 3*(54*A + 5*C)*Cos[c + d*x]*Sin[c + d*x])/(d*(a + a*Sec[c + d*x])) + ((-32 *a^4*(54*A + 5*C)*Sin[c + d*x])/d + 105*a^4*(21*A + 2*C)*(x/2 + (Cos[c + d *x]*Sin[c + d*x])/(2*d)))/a^2)/(3*a^2))/(5*a^2))/(7*a^2)
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[a Int[(d*Csc[e + f*x])^n, x], x] + Simp[b/d In t[(d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n}, x]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + ( a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)), x_Symbol] :> Simp[(-(A*b - a*B))*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*((d*Csc[e + f*x])^n/(b*f*(2*m + 1))), x] - Simp[1/(a^2*(2*m + 1)) Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Cs c[e + f*x])^n*Simp[b*B*n - a*A*(2*m + n + 1) + (A*b - a*B)*(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] && LtQ[m, -2^(-1)] && !GtQ[n, 0]
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) *(A + C)*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*((d*Csc[e + f*x])^n/(a*f*(2*m + 1))), x] + Simp[1/(a*b*(2*m + 1)) Int[(a + b*Csc[e + f*x])^(m + 1)*(d*C sc[e + f*x])^n*Simp[b*C*n + A*b*(2*m + n + 1) - (a*(A*(m + n + 1) - C*(m - n)))*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, C, n}, x] && EqQ[ a^2 - b^2, 0] && LtQ[m, -2^(-1)]
Time = 0.52 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.55
| method | result | size |
| parallelrisch | \(\frac {-840 \left (\left (\frac {4688 A}{35}+\frac {248 C}{21}\right ) \cos \left (2 d x +2 c \right )+\left (\frac {7873 A}{280}+\frac {52 C}{21}\right ) \cos \left (3 d x +3 c \right )+A \cos \left (4 d x +4 c \right )-\frac {A \cos \left (5 d x +5 c \right )}{8}+\left (\frac {42881 A}{140}+\frac {584 C}{21}\right ) \cos \left (d x +c \right )+\frac {6957 A}{35}+\frac {376 C}{21}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+282240 d \left (A +\frac {2 C}{21}\right ) x}{26880 a^{4} d}\) | \(119\) |
| derivativedivides | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} A}{7}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} C}{7}-\frac {9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} A}{5}-\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} C +13 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} A +\frac {11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} C}{3}-111 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) A -15 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C +\frac {-72 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} A -56 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) A}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}+8 \left (21 A +2 C \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \,a^{4}}\) | \(181\) |
| default | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} A}{7}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} C}{7}-\frac {9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} A}{5}-\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} C +13 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} A +\frac {11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} C}{3}-111 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) A -15 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C +\frac {-72 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} A -56 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) A}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}+8 \left (21 A +2 C \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \,a^{4}}\) | \(181\) |
| risch | \(\frac {21 A x}{2 a^{4}}+\frac {x C}{a^{4}}-\frac {i A \,{\mathrm e}^{2 i \left (d x +c \right )}}{8 a^{4} d}+\frac {2 i A \,{\mathrm e}^{i \left (d x +c \right )}}{a^{4} d}-\frac {2 i A \,{\mathrm e}^{-i \left (d x +c \right )}}{a^{4} d}+\frac {i A \,{\mathrm e}^{-2 i \left (d x +c \right )}}{8 a^{4} d}-\frac {2 i \left (2100 A \,{\mathrm e}^{6 i \left (d x +c \right )}+420 C \,{\mathrm e}^{6 i \left (d x +c \right )}+11025 A \,{\mathrm e}^{5 i \left (d x +c \right )}+1890 C \,{\mathrm e}^{5 i \left (d x +c \right )}+25515 A \,{\mathrm e}^{4 i \left (d x +c \right )}+4130 C \,{\mathrm e}^{4 i \left (d x +c \right )}+32340 A \,{\mathrm e}^{3 i \left (d x +c \right )}+4970 C \,{\mathrm e}^{3 i \left (d x +c \right )}+23688 A \,{\mathrm e}^{2 i \left (d x +c \right )}+3570 C \,{\mathrm e}^{2 i \left (d x +c \right )}+9471 A \,{\mathrm e}^{i \left (d x +c \right )}+1400 C \,{\mathrm e}^{i \left (d x +c \right )}+1653 A +260 C \right )}{105 d \,a^{4} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{7}}\) | \(264\) |
| norman | \(\frac {-\frac {\left (21 A +2 C \right ) x}{2 a}+\frac {\left (A +C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{56 a d}-\frac {\left (21 A +2 C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{2 a}+\frac {\left (21 A +2 C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{2 a}+\frac {\left (21 A +2 C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{2 a}-\frac {\left (29 A +15 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{140 a d}+\frac {\left (167 A +15 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a d}+\frac {\left (171 A +17 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{12 a d}+\frac {\left (1161 A +265 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{840 a d}-\frac {\left (2529 A +275 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{210 a d}-\frac {\left (2913 A +265 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{120 a d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right ) a^{3}}\) | \(297\) |
Input:
int(cos(d*x+c)^2*(A+C*sec(d*x+c)^2)/(a+a*sec(d*x+c))^4,x,method=_RETURNVER BOSE)
Output:
1/26880*(-840*((4688/35*A+248/21*C)*cos(2*d*x+2*c)+(7873/280*A+52/21*C)*co s(3*d*x+3*c)+A*cos(4*d*x+4*c)-1/8*A*cos(5*d*x+5*c)+(42881/140*A+584/21*C)* cos(d*x+c)+6957/35*A+376/21*C)*tan(1/2*d*x+1/2*c)*sec(1/2*d*x+1/2*c)^6+282 240*d*(A+2/21*C)*x)/a^4/d
Time = 0.09 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.09 \[ \int \frac {\cos ^2(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^4} \, dx=\frac {105 \, {\left (21 \, A + 2 \, C\right )} d x \cos \left (d x + c\right )^{4} + 420 \, {\left (21 \, A + 2 \, C\right )} d x \cos \left (d x + c\right )^{3} + 630 \, {\left (21 \, A + 2 \, C\right )} d x \cos \left (d x + c\right )^{2} + 420 \, {\left (21 \, A + 2 \, C\right )} d x \cos \left (d x + c\right ) + 105 \, {\left (21 \, A + 2 \, C\right )} d x + {\left (105 \, A \cos \left (d x + c\right )^{5} - 420 \, A \cos \left (d x + c\right )^{4} - 4 \, {\left (1509 \, A + 130 \, C\right )} \cos \left (d x + c\right )^{3} - 4 \, {\left (3411 \, A + 310 \, C\right )} \cos \left (d x + c\right )^{2} - {\left (11619 \, A + 1070 \, C\right )} \cos \left (d x + c\right ) - 3456 \, A - 320 \, C\right )} \sin \left (d x + c\right )}{210 \, {\left (a^{4} d \cos \left (d x + c\right )^{4} + 4 \, a^{4} d \cos \left (d x + c\right )^{3} + 6 \, a^{4} d \cos \left (d x + c\right )^{2} + 4 \, a^{4} d \cos \left (d x + c\right ) + a^{4} d\right )}} \] Input:
integrate(cos(d*x+c)^2*(A+C*sec(d*x+c)^2)/(a+a*sec(d*x+c))^4,x, algorithm= "fricas")
Output:
1/210*(105*(21*A + 2*C)*d*x*cos(d*x + c)^4 + 420*(21*A + 2*C)*d*x*cos(d*x + c)^3 + 630*(21*A + 2*C)*d*x*cos(d*x + c)^2 + 420*(21*A + 2*C)*d*x*cos(d* x + c) + 105*(21*A + 2*C)*d*x + (105*A*cos(d*x + c)^5 - 420*A*cos(d*x + c) ^4 - 4*(1509*A + 130*C)*cos(d*x + c)^3 - 4*(3411*A + 310*C)*cos(d*x + c)^2 - (11619*A + 1070*C)*cos(d*x + c) - 3456*A - 320*C)*sin(d*x + c))/(a^4*d* cos(d*x + c)^4 + 4*a^4*d*cos(d*x + c)^3 + 6*a^4*d*cos(d*x + c)^2 + 4*a^4*d *cos(d*x + c) + a^4*d)
\[ \int \frac {\cos ^2(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^4} \, dx=\frac {\int \frac {A \cos ^{2}{\left (c + d x \right )}}{\sec ^{4}{\left (c + d x \right )} + 4 \sec ^{3}{\left (c + d x \right )} + 6 \sec ^{2}{\left (c + d x \right )} + 4 \sec {\left (c + d x \right )} + 1}\, dx + \int \frac {C \cos ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}}{\sec ^{4}{\left (c + d x \right )} + 4 \sec ^{3}{\left (c + d x \right )} + 6 \sec ^{2}{\left (c + d x \right )} + 4 \sec {\left (c + d x \right )} + 1}\, dx}{a^{4}} \] Input:
integrate(cos(d*x+c)**2*(A+C*sec(d*x+c)**2)/(a+a*sec(d*x+c))**4,x)
Output:
(Integral(A*cos(c + d*x)**2/(sec(c + d*x)**4 + 4*sec(c + d*x)**3 + 6*sec(c + d*x)**2 + 4*sec(c + d*x) + 1), x) + Integral(C*cos(c + d*x)**2*sec(c + d*x)**2/(sec(c + d*x)**4 + 4*sec(c + d*x)**3 + 6*sec(c + d*x)**2 + 4*sec(c + d*x) + 1), x))/a**4
Time = 0.13 (sec) , antiderivative size = 318, normalized size of antiderivative = 1.48 \[ \int \frac {\cos ^2(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^4} \, dx=-\frac {3 \, A {\left (\frac {280 \, {\left (\frac {7 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {9 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{a^{4} + \frac {2 \, a^{4} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {a^{4} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} + \frac {\frac {3885 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {455 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {63 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {5 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4}} - \frac {5880 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{4}}\right )} + 5 \, C {\left (\frac {\frac {315 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {77 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {21 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {3 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4}} - \frac {336 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{4}}\right )}}{840 \, d} \] Input:
integrate(cos(d*x+c)^2*(A+C*sec(d*x+c)^2)/(a+a*sec(d*x+c))^4,x, algorithm= "maxima")
Output:
-1/840*(3*A*(280*(7*sin(d*x + c)/(cos(d*x + c) + 1) + 9*sin(d*x + c)^3/(co s(d*x + c) + 1)^3)/(a^4 + 2*a^4*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + a^4* sin(d*x + c)^4/(cos(d*x + c) + 1)^4) + (3885*sin(d*x + c)/(cos(d*x + c) + 1) - 455*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 63*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 5*sin(d*x + c)^7/(cos(d*x + c) + 1)^7)/a^4 - 5880*arctan(sin (d*x + c)/(cos(d*x + c) + 1))/a^4) + 5*C*((315*sin(d*x + c)/(cos(d*x + c) + 1) - 77*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 21*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 3*sin(d*x + c)^7/(cos(d*x + c) + 1)^7)/a^4 - 336*arctan(sin (d*x + c)/(cos(d*x + c) + 1))/a^4))/d
Time = 0.32 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.96 \[ \int \frac {\cos ^2(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^4} \, dx=\frac {\frac {420 \, {\left (d x + c\right )} {\left (21 \, A + 2 \, C\right )}}{a^{4}} - \frac {840 \, {\left (9 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 7 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2} a^{4}} + \frac {15 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 15 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 189 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 105 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 1365 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 385 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 11655 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1575 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{28}}}{840 \, d} \] Input:
integrate(cos(d*x+c)^2*(A+C*sec(d*x+c)^2)/(a+a*sec(d*x+c))^4,x, algorithm= "giac")
Output:
1/840*(420*(d*x + c)*(21*A + 2*C)/a^4 - 840*(9*A*tan(1/2*d*x + 1/2*c)^3 + 7*A*tan(1/2*d*x + 1/2*c))/((tan(1/2*d*x + 1/2*c)^2 + 1)^2*a^4) + (15*A*a^2 4*tan(1/2*d*x + 1/2*c)^7 + 15*C*a^24*tan(1/2*d*x + 1/2*c)^7 - 189*A*a^24*t an(1/2*d*x + 1/2*c)^5 - 105*C*a^24*tan(1/2*d*x + 1/2*c)^5 + 1365*A*a^24*ta n(1/2*d*x + 1/2*c)^3 + 385*C*a^24*tan(1/2*d*x + 1/2*c)^3 - 11655*A*a^24*ta n(1/2*d*x + 1/2*c) - 1575*C*a^24*tan(1/2*d*x + 1/2*c))/a^28)/d
Time = 12.27 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.16 \[ \int \frac {\cos ^2(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^4} \, dx=\frac {x\,\left (21\,A+2\,C\right )}{2\,a^4}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {A+C}{4\,a^4}+\frac {6\,A+2\,C}{8\,a^4}+\frac {15\,A-C}{24\,a^4}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {3\,\left (A+C\right )}{40\,a^4}+\frac {6\,A+2\,C}{40\,a^4}\right )}{d}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {5\,\left (A+C\right )}{4\,a^4}+\frac {3\,\left (6\,A+2\,C\right )}{4\,a^4}+\frac {3\,\left (15\,A-C\right )}{8\,a^4}+\frac {20\,A-4\,C}{8\,a^4}\right )}{d}-\frac {9\,A\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+7\,A\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+2\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a^4\right )}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (A+C\right )}{56\,a^4\,d} \] Input:
int((cos(c + d*x)^2*(A + C/cos(c + d*x)^2))/(a + a/cos(c + d*x))^4,x)
Output:
(x*(21*A + 2*C))/(2*a^4) + (tan(c/2 + (d*x)/2)^3*((A + C)/(4*a^4) + (6*A + 2*C)/(8*a^4) + (15*A - C)/(24*a^4)))/d - (tan(c/2 + (d*x)/2)^5*((3*(A + C ))/(40*a^4) + (6*A + 2*C)/(40*a^4)))/d - (tan(c/2 + (d*x)/2)*((5*(A + C))/ (4*a^4) + (3*(6*A + 2*C))/(4*a^4) + (3*(15*A - C))/(8*a^4) + (20*A - 4*C)/ (8*a^4)))/d - (7*A*tan(c/2 + (d*x)/2) + 9*A*tan(c/2 + (d*x)/2)^3)/(d*(2*a^ 4*tan(c/2 + (d*x)/2)^2 + a^4*tan(c/2 + (d*x)/2)^4 + a^4)) + (tan(c/2 + (d* x)/2)^7*(A + C))/(56*a^4*d)
Time = 0.15 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.28 \[ \int \frac {\cos ^2(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^4} \, dx=\frac {15 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11} a +15 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11} c -159 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9} a -75 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9} c +1002 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} a +190 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} c -9114 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} a -910 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} c +8820 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a d x +840 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} c d x -29505 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} a -2765 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} c +17640 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a d x +1680 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} c d x -17535 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a -1575 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) c +8820 a d x +840 c d x}{840 a^{4} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right )} \] Input:
int(cos(d*x+c)^2*(A+C*sec(d*x+c)^2)/(a+a*sec(d*x+c))^4,x)
Output:
(15*tan((c + d*x)/2)**11*a + 15*tan((c + d*x)/2)**11*c - 159*tan((c + d*x) /2)**9*a - 75*tan((c + d*x)/2)**9*c + 1002*tan((c + d*x)/2)**7*a + 190*tan ((c + d*x)/2)**7*c - 9114*tan((c + d*x)/2)**5*a - 910*tan((c + d*x)/2)**5* c + 8820*tan((c + d*x)/2)**4*a*d*x + 840*tan((c + d*x)/2)**4*c*d*x - 29505 *tan((c + d*x)/2)**3*a - 2765*tan((c + d*x)/2)**3*c + 17640*tan((c + d*x)/ 2)**2*a*d*x + 1680*tan((c + d*x)/2)**2*c*d*x - 17535*tan((c + d*x)/2)*a - 1575*tan((c + d*x)/2)*c + 8820*a*d*x + 840*c*d*x)/(840*a**4*d*(tan((c + d* x)/2)**4 + 2*tan((c + d*x)/2)**2 + 1))