Integrand size = 21, antiderivative size = 215 \[ \int \frac {\cos ^2(c+d x)}{(a+a \sec (c+d x))^5} \, dx=\frac {31 x}{2 a^5}-\frac {7664 \sin (c+d x)}{315 a^5 d}+\frac {31 \cos (c+d x) \sin (c+d x)}{2 a^5 d}-\frac {\cos (c+d x) \sin (c+d x)}{9 d (a+a \sec (c+d x))^5}-\frac {17 \cos (c+d x) \sin (c+d x)}{63 a d (a+a \sec (c+d x))^4}-\frac {28 \cos (c+d x) \sin (c+d x)}{45 a^2 d (a+a \sec (c+d x))^3}-\frac {577 \cos (c+d x) \sin (c+d x)}{315 a^3 d (a+a \sec (c+d x))^2}-\frac {3832 \cos (c+d x) \sin (c+d x)}{315 d \left (a^5+a^5 \sec (c+d x)\right )} \]
31/2*x/a^5-7664/315*sin(d*x+c)/a^5/d+31/2*cos(d*x+c)*sin(d*x+c)/a^5/d-1/9* cos(d*x+c)*sin(d*x+c)/d/(a+a*sec(d*x+c))^5-17/63*cos(d*x+c)*sin(d*x+c)/a/d /(a+a*sec(d*x+c))^4-28/45*cos(d*x+c)*sin(d*x+c)/a^2/d/(a+a*sec(d*x+c))^3-5 77/315*cos(d*x+c)*sin(d*x+c)/a^3/d/(a+a*sec(d*x+c))^2-3832/315*cos(d*x+c)* sin(d*x+c)/d/(a^5+a^5*sec(d*x+c))
Time = 6.82 (sec) , antiderivative size = 345, normalized size of antiderivative = 1.60 \[ \int \frac {\cos ^2(c+d x)}{(a+a \sec (c+d x))^5} \, dx=\frac {\sec \left (\frac {c}{2}\right ) \sec ^9\left (\frac {1}{2} (c+d x)\right ) \left (4921560 d x \cos \left (\frac {d x}{2}\right )+4921560 d x \cos \left (c+\frac {d x}{2}\right )+3281040 d x \cos \left (c+\frac {3 d x}{2}\right )+3281040 d x \cos \left (2 c+\frac {3 d x}{2}\right )+1406160 d x \cos \left (2 c+\frac {5 d x}{2}\right )+1406160 d x \cos \left (3 c+\frac {5 d x}{2}\right )+351540 d x \cos \left (3 c+\frac {7 d x}{2}\right )+351540 d x \cos \left (4 c+\frac {7 d x}{2}\right )+39060 d x \cos \left (4 c+\frac {9 d x}{2}\right )+39060 d x \cos \left (5 c+\frac {9 d x}{2}\right )-9163224 \sin \left (\frac {d x}{2}\right )+7194600 \sin \left (c+\frac {d x}{2}\right )-7472241 \sin \left (c+\frac {3 d x}{2}\right )+3432975 \sin \left (2 c+\frac {3 d x}{2}\right )-3871989 \sin \left (2 c+\frac {5 d x}{2}\right )+801675 \sin \left (3 c+\frac {5 d x}{2}\right )-1186056 \sin \left (3 c+\frac {7 d x}{2}\right )-17640 \sin \left (4 c+\frac {7 d x}{2}\right )-175184 \sin \left (4 c+\frac {9 d x}{2}\right )-45360 \sin \left (5 c+\frac {9 d x}{2}\right )-3465 \sin \left (5 c+\frac {11 d x}{2}\right )-3465 \sin \left (6 c+\frac {11 d x}{2}\right )+315 \sin \left (6 c+\frac {13 d x}{2}\right )+315 \sin \left (7 c+\frac {13 d x}{2}\right )\right )}{1290240 a^5 d} \]
(Sec[c/2]*Sec[(c + d*x)/2]^9*(4921560*d*x*Cos[(d*x)/2] + 4921560*d*x*Cos[c + (d*x)/2] + 3281040*d*x*Cos[c + (3*d*x)/2] + 3281040*d*x*Cos[2*c + (3*d* x)/2] + 1406160*d*x*Cos[2*c + (5*d*x)/2] + 1406160*d*x*Cos[3*c + (5*d*x)/2 ] + 351540*d*x*Cos[3*c + (7*d*x)/2] + 351540*d*x*Cos[4*c + (7*d*x)/2] + 39 060*d*x*Cos[4*c + (9*d*x)/2] + 39060*d*x*Cos[5*c + (9*d*x)/2] - 9163224*Si n[(d*x)/2] + 7194600*Sin[c + (d*x)/2] - 7472241*Sin[c + (3*d*x)/2] + 34329 75*Sin[2*c + (3*d*x)/2] - 3871989*Sin[2*c + (5*d*x)/2] + 801675*Sin[3*c + (5*d*x)/2] - 1186056*Sin[3*c + (7*d*x)/2] - 17640*Sin[4*c + (7*d*x)/2] - 1 75184*Sin[4*c + (9*d*x)/2] - 45360*Sin[5*c + (9*d*x)/2] - 3465*Sin[5*c + ( 11*d*x)/2] - 3465*Sin[6*c + (11*d*x)/2] + 315*Sin[6*c + (13*d*x)/2] + 315* Sin[7*c + (13*d*x)/2]))/(1290240*a^5*d)
Time = 1.52 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.12, number of steps used = 18, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.857, Rules used = {3042, 4304, 25, 3042, 4508, 3042, 4508, 3042, 4508, 27, 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.
| \(\displaystyle \int \frac {\cos ^2(c+d x)}{(a \sec (c+d x)+a)^5} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\csc \left (c+d x+\frac {\pi }{2}\right )^2 \left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^5}dx\) |
| \(\Big \downarrow \) 4304 |
| \(\displaystyle -\frac {\int -\frac {\cos ^2(c+d x) (11 a-6 a \sec (c+d x))}{(\sec (c+d x) a+a)^4}dx}{9 a^2}-\frac {\sin (c+d x) \cos (c+d x)}{9 d (a \sec (c+d x)+a)^5}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {\cos ^2(c+d x) (11 a-6 a \sec (c+d x))}{(\sec (c+d x) a+a)^4}dx}{9 a^2}-\frac {\sin (c+d x) \cos (c+d x)}{9 d (a \sec (c+d x)+a)^5}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {11 a-6 a \csc \left (c+d x+\frac {\pi }{2}\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^2 \left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^4}dx}{9 a^2}-\frac {\sin (c+d x) \cos (c+d x)}{9 d (a \sec (c+d x)+a)^5}\) |
| \(\Big \downarrow \) 4508 |
| \(\displaystyle \frac {\frac {\int \frac {\cos ^2(c+d x) \left (111 a^2-85 a^2 \sec (c+d x)\right )}{(\sec (c+d x) a+a)^3}dx}{7 a^2}-\frac {17 a \sin (c+d x) \cos (c+d x)}{7 d (a \sec (c+d x)+a)^4}}{9 a^2}-\frac {\sin (c+d x) \cos (c+d x)}{9 d (a \sec (c+d x)+a)^5}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {111 a^2-85 a^2 \csc \left (c+d x+\frac {\pi }{2}\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^2 \left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3}dx}{7 a^2}-\frac {17 a \sin (c+d x) \cos (c+d x)}{7 d (a \sec (c+d x)+a)^4}}{9 a^2}-\frac {\sin (c+d x) \cos (c+d x)}{9 d (a \sec (c+d x)+a)^5}\) |
| \(\Big \downarrow \) 4508 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {\cos ^2(c+d x) \left (947 a^3-784 a^3 \sec (c+d x)\right )}{(\sec (c+d x) a+a)^2}dx}{5 a^2}-\frac {196 a^2 \sin (c+d x) \cos (c+d x)}{5 d (a \sec (c+d x)+a)^3}}{7 a^2}-\frac {17 a \sin (c+d x) \cos (c+d x)}{7 d (a \sec (c+d x)+a)^4}}{9 a^2}-\frac {\sin (c+d x) \cos (c+d x)}{9 d (a \sec (c+d x)+a)^5}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {947 a^3-784 a^3 \csc \left (c+d x+\frac {\pi }{2}\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^2 \left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2}dx}{5 a^2}-\frac {196 a^2 \sin (c+d x) \cos (c+d x)}{5 d (a \sec (c+d x)+a)^3}}{7 a^2}-\frac {17 a \sin (c+d x) \cos (c+d x)}{7 d (a \sec (c+d x)+a)^4}}{9 a^2}-\frac {\sin (c+d x) \cos (c+d x)}{9 d (a \sec (c+d x)+a)^5}\) |
| \(\Big \downarrow \) 4508 |
| \(\displaystyle \frac {\frac {\frac {\frac {\int \frac {3 \cos ^2(c+d x) \left (2101 a^4-1731 a^4 \sec (c+d x)\right )}{\sec (c+d x) a+a}dx}{3 a^2}-\frac {577 a^3 \sin (c+d x) \cos (c+d x)}{d (a \sec (c+d x)+a)^2}}{5 a^2}-\frac {196 a^2 \sin (c+d x) \cos (c+d x)}{5 d (a \sec (c+d x)+a)^3}}{7 a^2}-\frac {17 a \sin (c+d x) \cos (c+d x)}{7 d (a \sec (c+d x)+a)^4}}{9 a^2}-\frac {\sin (c+d x) \cos (c+d x)}{9 d (a \sec (c+d x)+a)^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\frac {\int \frac {\cos ^2(c+d x) \left (2101 a^4-1731 a^4 \sec (c+d x)\right )}{\sec (c+d x) a+a}dx}{a^2}-\frac {577 a^3 \sin (c+d x) \cos (c+d x)}{d (a \sec (c+d x)+a)^2}}{5 a^2}-\frac {196 a^2 \sin (c+d x) \cos (c+d x)}{5 d (a \sec (c+d x)+a)^3}}{7 a^2}-\frac {17 a \sin (c+d x) \cos (c+d x)}{7 d (a \sec (c+d x)+a)^4}}{9 a^2}-\frac {\sin (c+d x) \cos (c+d x)}{9 d (a \sec (c+d x)+a)^5}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\frac {\int \frac {2101 a^4-1731 a^4 \csc \left (c+d x+\frac {\pi }{2}\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^2 \left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )}dx}{a^2}-\frac {577 a^3 \sin (c+d x) \cos (c+d x)}{d (a \sec (c+d x)+a)^2}}{5 a^2}-\frac {196 a^2 \sin (c+d x) \cos (c+d x)}{5 d (a \sec (c+d x)+a)^3}}{7 a^2}-\frac {17 a \sin (c+d x) \cos (c+d x)}{7 d (a \sec (c+d x)+a)^4}}{9 a^2}-\frac {\sin (c+d x) \cos (c+d x)}{9 d (a \sec (c+d x)+a)^5}\) |
| \(\Big \downarrow \) 4508 |
| \(\displaystyle \frac {\frac {\frac {\frac {\frac {\int \cos ^2(c+d x) \left (9765 a^5-7664 a^5 \sec (c+d x)\right )dx}{a^2}-\frac {3832 a^4 \sin (c+d x) \cos (c+d x)}{d (a \sec (c+d x)+a)}}{a^2}-\frac {577 a^3 \sin (c+d x) \cos (c+d x)}{d (a \sec (c+d x)+a)^2}}{5 a^2}-\frac {196 a^2 \sin (c+d x) \cos (c+d x)}{5 d (a \sec (c+d x)+a)^3}}{7 a^2}-\frac {17 a \sin (c+d x) \cos (c+d x)}{7 d (a \sec (c+d x)+a)^4}}{9 a^2}-\frac {\sin (c+d x) \cos (c+d x)}{9 d (a \sec (c+d x)+a)^5}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\frac {\frac {\int \frac {9765 a^5-7664 a^5 \csc \left (c+d x+\frac {\pi }{2}\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^2}dx}{a^2}-\frac {3832 a^4 \sin (c+d x) \cos (c+d x)}{d (a \sec (c+d x)+a)}}{a^2}-\frac {577 a^3 \sin (c+d x) \cos (c+d x)}{d (a \sec (c+d x)+a)^2}}{5 a^2}-\frac {196 a^2 \sin (c+d x) \cos (c+d x)}{5 d (a \sec (c+d x)+a)^3}}{7 a^2}-\frac {17 a \sin (c+d x) \cos (c+d x)}{7 d (a \sec (c+d x)+a)^4}}{9 a^2}-\frac {\sin (c+d x) \cos (c+d x)}{9 d (a \sec (c+d x)+a)^5}\) |
| \(\Big \downarrow \) 4274 |
| \(\displaystyle \frac {\frac {\frac {\frac {\frac {9765 a^5 \int \cos ^2(c+d x)dx-7664 a^5 \int \cos (c+d x)dx}{a^2}-\frac {3832 a^4 \sin (c+d x) \cos (c+d x)}{d (a \sec (c+d x)+a)}}{a^2}-\frac {577 a^3 \sin (c+d x) \cos (c+d x)}{d (a \sec (c+d x)+a)^2}}{5 a^2}-\frac {196 a^2 \sin (c+d x) \cos (c+d x)}{5 d (a \sec (c+d x)+a)^3}}{7 a^2}-\frac {17 a \sin (c+d x) \cos (c+d x)}{7 d (a \sec (c+d x)+a)^4}}{9 a^2}-\frac {\sin (c+d x) \cos (c+d x)}{9 d (a \sec (c+d x)+a)^5}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\frac {\frac {9765 a^5 \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx-7664 a^5 \int \sin \left (c+d x+\frac {\pi }{2}\right )dx}{a^2}-\frac {3832 a^4 \sin (c+d x) \cos (c+d x)}{d (a \sec (c+d x)+a)}}{a^2}-\frac {577 a^3 \sin (c+d x) \cos (c+d x)}{d (a \sec (c+d x)+a)^2}}{5 a^2}-\frac {196 a^2 \sin (c+d x) \cos (c+d x)}{5 d (a \sec (c+d x)+a)^3}}{7 a^2}-\frac {17 a \sin (c+d x) \cos (c+d x)}{7 d (a \sec (c+d x)+a)^4}}{9 a^2}-\frac {\sin (c+d x) \cos (c+d x)}{9 d (a \sec (c+d x)+a)^5}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {\frac {\frac {\frac {\frac {9765 a^5 \left (\frac {\int 1dx}{2}+\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )-7664 a^5 \int \sin \left (c+d x+\frac {\pi }{2}\right )dx}{a^2}-\frac {3832 a^4 \sin (c+d x) \cos (c+d x)}{d (a \sec (c+d x)+a)}}{a^2}-\frac {577 a^3 \sin (c+d x) \cos (c+d x)}{d (a \sec (c+d x)+a)^2}}{5 a^2}-\frac {196 a^2 \sin (c+d x) \cos (c+d x)}{5 d (a \sec (c+d x)+a)^3}}{7 a^2}-\frac {17 a \sin (c+d x) \cos (c+d x)}{7 d (a \sec (c+d x)+a)^4}}{9 a^2}-\frac {\sin (c+d x) \cos (c+d x)}{9 d (a \sec (c+d x)+a)^5}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {\frac {\frac {\frac {\frac {9765 a^5 \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )-7664 a^5 \int \sin \left (c+d x+\frac {\pi }{2}\right )dx}{a^2}-\frac {3832 a^4 \sin (c+d x) \cos (c+d x)}{d (a \sec (c+d x)+a)}}{a^2}-\frac {577 a^3 \sin (c+d x) \cos (c+d x)}{d (a \sec (c+d x)+a)^2}}{5 a^2}-\frac {196 a^2 \sin (c+d x) \cos (c+d x)}{5 d (a \sec (c+d x)+a)^3}}{7 a^2}-\frac {17 a \sin (c+d x) \cos (c+d x)}{7 d (a \sec (c+d x)+a)^4}}{9 a^2}-\frac {\sin (c+d x) \cos (c+d x)}{9 d (a \sec (c+d x)+a)^5}\) |
| \(\Big \downarrow \) 3117 |
| \(\displaystyle \frac {\frac {\frac {\frac {\frac {9765 a^5 \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )-\frac {7664 a^5 \sin (c+d x)}{d}}{a^2}-\frac {3832 a^4 \sin (c+d x) \cos (c+d x)}{d (a \sec (c+d x)+a)}}{a^2}-\frac {577 a^3 \sin (c+d x) \cos (c+d x)}{d (a \sec (c+d x)+a)^2}}{5 a^2}-\frac {196 a^2 \sin (c+d x) \cos (c+d x)}{5 d (a \sec (c+d x)+a)^3}}{7 a^2}-\frac {17 a \sin (c+d x) \cos (c+d x)}{7 d (a \sec (c+d x)+a)^4}}{9 a^2}-\frac {\sin (c+d x) \cos (c+d x)}{9 d (a \sec (c+d x)+a)^5}\) |
-1/9*(Cos[c + d*x]*Sin[c + d*x])/(d*(a + a*Sec[c + d*x])^5) + ((-17*a*Cos[ c + d*x]*Sin[c + d*x])/(7*d*(a + a*Sec[c + d*x])^4) + ((-196*a^2*Cos[c + d *x]*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) + ((-577*a^3*Cos[c + d*x]*S in[c + d*x])/(d*(a + a*Sec[c + d*x])^2) + ((-3832*a^4*Cos[c + d*x]*Sin[c + d*x])/(d*(a + a*Sec[c + d*x])) + ((-7664*a^5*Sin[c + d*x])/d + 9765*a^5*( x/2 + (Cos[c + d*x]*Sin[c + d*x])/(2*d)))/a^2)/a^2)/(5*a^2))/(7*a^2))/(9*a ^2)
3.1.89.3.1 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[((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_), x_Symbol] :> Simp[(-Cot[e + f*x])*(a + b*Csc[e + f*x])^m*((d*Csc [e + f*x])^n/(f*(2*m + 1))), x] + Simp[1/(a^2*(2*m + 1)) Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^n*(a*(2*m + n + 1) - b*(m + n + 1)*Csc[e + f*x]), x], x] /; FreeQ[{a, b, d, e, f, n}, x] && EqQ[a^2 - b^2, 0] && LtQ [m, -1] && (IntegersQ[2*m, 2*n] || IntegerQ[m])
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]
Time = 0.57 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.46
| method | result | size |
| parallelrisch | \(\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\cos \left (6 d x +6 c \right )-\frac {854012 \cos \left (d x +c \right )}{63}-\frac {2250427 \cos \left (2 d x +2 c \right )}{315}-\frac {143054 \cos \left (3 d x +3 c \right )}{63}-\frac {113422 \cos \left (4 d x +4 c \right )}{315}-10 \cos \left (5 d x +5 c \right )-\frac {2627186}{315}\right ) \sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+15872 d x}{1024 d \,a^{5}}\) | \(98\) |
| derivativedivides | \(\frac {-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{9}+\frac {10 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{7}-\frac {48 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{5}+50 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}-351 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {-176 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}-144 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}+496 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d \,a^{5}}\) | \(127\) |
| default | \(\frac {-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{9}+\frac {10 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{7}-\frac {48 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{5}+50 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}-351 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {-176 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}-144 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}+496 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d \,a^{5}}\) | \(127\) |
| norman | \(\frac {\frac {31 x}{2 a}-\frac {495 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{16 a d}-\frac {207 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{4 a d}-\frac {1303 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{80 a d}+\frac {141 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{70 a d}-\frac {2159 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{5040 a d}+\frac {19 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{252 a d}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{144 a d}+\frac {31 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{a}+\frac {31 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{2 a}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2} a^{4}}\) | \(192\) |
| risch | \(\frac {31 x}{2 a^{5}}-\frac {i {\mathrm e}^{2 i \left (d x +c \right )}}{8 d \,a^{5}}+\frac {5 i {\mathrm e}^{i \left (d x +c \right )}}{2 d \,a^{5}}-\frac {5 i {\mathrm e}^{-i \left (d x +c \right )}}{2 d \,a^{5}}+\frac {i {\mathrm e}^{-2 i \left (d x +c \right )}}{8 d \,a^{5}}-\frac {2 i \left (11025 \,{\mathrm e}^{8 i \left (d x +c \right )}+77175 \,{\mathrm e}^{7 i \left (d x +c \right )}+247695 \,{\mathrm e}^{6 i \left (d x +c \right )}+465255 \,{\mathrm e}^{5 i \left (d x +c \right )}+557109 \,{\mathrm e}^{4 i \left (d x +c \right )}+433881 \,{\mathrm e}^{3 i \left (d x +c \right )}+214929 \,{\mathrm e}^{2 i \left (d x +c \right )}+62001 \,{\mathrm e}^{i \left (d x +c \right )}+8114\right )}{315 d \,a^{5} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{9}}\) | \(192\) |
1/1024*(tan(1/2*d*x+1/2*c)*(cos(6*d*x+6*c)-854012/63*cos(d*x+c)-2250427/31 5*cos(2*d*x+2*c)-143054/63*cos(3*d*x+3*c)-113422/315*cos(4*d*x+4*c)-10*cos (5*d*x+5*c)-2627186/315)*sec(1/2*d*x+1/2*c)^8+15872*d*x)/d/a^5
Time = 0.28 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.96 \[ \int \frac {\cos ^2(c+d x)}{(a+a \sec (c+d x))^5} \, dx=\frac {9765 \, d x \cos \left (d x + c\right )^{5} + 48825 \, d x \cos \left (d x + c\right )^{4} + 97650 \, d x \cos \left (d x + c\right )^{3} + 97650 \, d x \cos \left (d x + c\right )^{2} + 48825 \, d x \cos \left (d x + c\right ) + 9765 \, d x + {\left (315 \, \cos \left (d x + c\right )^{6} - 1575 \, \cos \left (d x + c\right )^{5} - 28828 \, \cos \left (d x + c\right )^{4} - 87440 \, \cos \left (d x + c\right )^{3} - 112119 \, \cos \left (d x + c\right )^{2} - 66875 \, \cos \left (d x + c\right ) - 15328\right )} \sin \left (d x + c\right )}{630 \, {\left (a^{5} d \cos \left (d x + c\right )^{5} + 5 \, a^{5} d \cos \left (d x + c\right )^{4} + 10 \, a^{5} d \cos \left (d x + c\right )^{3} + 10 \, a^{5} d \cos \left (d x + c\right )^{2} + 5 \, a^{5} d \cos \left (d x + c\right ) + a^{5} d\right )}} \]
1/630*(9765*d*x*cos(d*x + c)^5 + 48825*d*x*cos(d*x + c)^4 + 97650*d*x*cos( d*x + c)^3 + 97650*d*x*cos(d*x + c)^2 + 48825*d*x*cos(d*x + c) + 9765*d*x + (315*cos(d*x + c)^6 - 1575*cos(d*x + c)^5 - 28828*cos(d*x + c)^4 - 87440 *cos(d*x + c)^3 - 112119*cos(d*x + c)^2 - 66875*cos(d*x + c) - 15328)*sin( d*x + c))/(a^5*d*cos(d*x + c)^5 + 5*a^5*d*cos(d*x + c)^4 + 10*a^5*d*cos(d* x + c)^3 + 10*a^5*d*cos(d*x + c)^2 + 5*a^5*d*cos(d*x + c) + a^5*d)
\[ \int \frac {\cos ^2(c+d x)}{(a+a \sec (c+d x))^5} \, dx=\frac {\int \frac {\cos ^{2}{\left (c + d x \right )}}{\sec ^{5}{\left (c + d x \right )} + 5 \sec ^{4}{\left (c + d x \right )} + 10 \sec ^{3}{\left (c + d x \right )} + 10 \sec ^{2}{\left (c + d x \right )} + 5 \sec {\left (c + d x \right )} + 1}\, dx}{a^{5}} \]
Integral(cos(c + d*x)**2/(sec(c + d*x)**5 + 5*sec(c + d*x)**4 + 10*sec(c + d*x)**3 + 10*sec(c + d*x)**2 + 5*sec(c + d*x) + 1), x)/a**5
Time = 0.32 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.04 \[ \int \frac {\cos ^2(c+d x)}{(a+a \sec (c+d x))^5} \, dx=-\frac {\frac {5040 \, {\left (\frac {9 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {11 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{a^{5} + \frac {2 \, a^{5} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {a^{5} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} + \frac {\frac {110565 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {15750 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {3024 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {450 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {35 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}}}{a^{5}} - \frac {156240 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{5}}}{5040 \, d} \]
-1/5040*(5040*(9*sin(d*x + c)/(cos(d*x + c) + 1) + 11*sin(d*x + c)^3/(cos( d*x + c) + 1)^3)/(a^5 + 2*a^5*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + a^5*si n(d*x + c)^4/(cos(d*x + c) + 1)^4) + (110565*sin(d*x + c)/(cos(d*x + c) + 1) - 15750*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 3024*sin(d*x + c)^5/(cos( d*x + c) + 1)^5 - 450*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 + 35*sin(d*x + c )^9/(cos(d*x + c) + 1)^9)/a^5 - 156240*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a^5)/d
Time = 0.34 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.67 \[ \int \frac {\cos ^2(c+d x)}{(a+a \sec (c+d x))^5} \, dx=\frac {\frac {78120 \, {\left (d x + c\right )}}{a^{5}} - \frac {5040 \, {\left (11 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 9 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2} a^{5}} - \frac {35 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 450 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 3024 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 15750 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 110565 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{45}}}{5040 \, d} \]
1/5040*(78120*(d*x + c)/a^5 - 5040*(11*tan(1/2*d*x + 1/2*c)^3 + 9*tan(1/2* d*x + 1/2*c))/((tan(1/2*d*x + 1/2*c)^2 + 1)^2*a^5) - (35*a^40*tan(1/2*d*x + 1/2*c)^9 - 450*a^40*tan(1/2*d*x + 1/2*c)^7 + 3024*a^40*tan(1/2*d*x + 1/2 *c)^5 - 15750*a^40*tan(1/2*d*x + 1/2*c)^3 + 110565*a^40*tan(1/2*d*x + 1/2* c))/a^45)/d
Time = 13.25 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.84 \[ \int \frac {\cos ^2(c+d x)}{(a+a \sec (c+d x))^5} \, dx=-\frac {35\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-590\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+4584\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-23288\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+129824\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+55440\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-10080\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-78120\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\left (c+d\,x\right )}{5040\,a^5\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9} \]
-(35*sin(c/2 + (d*x)/2) - 590*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2) + 45 84*cos(c/2 + (d*x)/2)^4*sin(c/2 + (d*x)/2) - 23288*cos(c/2 + (d*x)/2)^6*si n(c/2 + (d*x)/2) + 129824*cos(c/2 + (d*x)/2)^8*sin(c/2 + (d*x)/2) + 55440* cos(c/2 + (d*x)/2)^10*sin(c/2 + (d*x)/2) - 10080*cos(c/2 + (d*x)/2)^12*sin (c/2 + (d*x)/2) - 78120*cos(c/2 + (d*x)/2)^9*(c + d*x))/(5040*a^5*d*cos(c/ 2 + (d*x)/2)^9)