Integrand size = 21, antiderivative size = 279 \[ \int \frac {\sec ^4(c+d x)}{(a+a \sin (c+d x))^8} \, dx=-\frac {\sec ^3(c+d x)}{19 d (a+a \sin (c+d x))^8}-\frac {11 \sec ^3(c+d x)}{323 a d (a+a \sin (c+d x))^7}-\frac {22 \sec ^3(c+d x)}{969 a^2 d (a+a \sin (c+d x))^6}-\frac {66 \sec ^3(c+d x)}{4199 a^3 d (a+a \sin (c+d x))^5}-\frac {48 \sec ^3(c+d x)}{4199 d \left (a^2+a^2 \sin (c+d x)\right )^4}-\frac {112 \sec ^3(c+d x)}{12597 a^2 d \left (a^2+a^2 \sin (c+d x)\right )^3}-\frac {32 \sec ^3(c+d x)}{4199 d \left (a^4+a^4 \sin (c+d x)\right )^2}-\frac {32 \sec ^3(c+d x)}{4199 d \left (a^8+a^8 \sin (c+d x)\right )}+\frac {128 \tan (c+d x)}{4199 a^8 d}+\frac {128 \tan ^3(c+d x)}{12597 a^8 d} \]
-1/19*sec(d*x+c)^3/d/(a+a*sin(d*x+c))^8-11/323*sec(d*x+c)^3/a/d/(a+a*sin(d *x+c))^7-22/969*sec(d*x+c)^3/a^2/d/(a+a*sin(d*x+c))^6-66/4199*sec(d*x+c)^3 /a^3/d/(a+a*sin(d*x+c))^5-48/4199*sec(d*x+c)^3/d/(a^2+a^2*sin(d*x+c))^4-11 2/12597*sec(d*x+c)^3/a^2/d/(a^2+a^2*sin(d*x+c))^3-32/4199*sec(d*x+c)^3/d/( a^4+a^4*sin(d*x+c))^2-32/4199*sec(d*x+c)^3/d/(a^8+a^8*sin(d*x+c))+128/4199 *tan(d*x+c)/a^8/d+128/12597*tan(d*x+c)^3/a^8/d
Time = 0.31 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.45 \[ \int \frac {\sec ^4(c+d x)}{(a+a \sin (c+d x))^8} \, dx=\frac {\sec ^3(c+d x) (-10336 \cos (2 (c+d x))+2736 \cos (6 (c+d x))-512 \cos (8 (c+d x))+16 \cos (10 (c+d x))+8398 \sin (c+d x)-5814 \sin (3 (c+d x))-2907 \sin (5 (c+d x))+1463 \sin (7 (c+d x))-117 \sin (9 (c+d x))+\sin (11 (c+d x)))}{50388 a^8 d (1+\sin (c+d x))^8} \]
(Sec[c + d*x]^3*(-10336*Cos[2*(c + d*x)] + 2736*Cos[6*(c + d*x)] - 512*Cos [8*(c + d*x)] + 16*Cos[10*(c + d*x)] + 8398*Sin[c + d*x] - 5814*Sin[3*(c + d*x)] - 2907*Sin[5*(c + d*x)] + 1463*Sin[7*(c + d*x)] - 117*Sin[9*(c + d* x)] + Sin[11*(c + d*x)]))/(50388*a^8*d*(1 + Sin[c + d*x])^8)
Time = 1.52 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.09, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.905, Rules used = {3042, 3151, 3042, 3151, 3042, 3151, 3042, 3151, 3042, 3151, 3042, 3151, 3042, 3151, 3042, 3151, 3042, 4254, 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.
| \(\displaystyle \int \frac {\sec ^4(c+d x)}{(a \sin (c+d x)+a)^8} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\cos (c+d x)^4 (a \sin (c+d x)+a)^8}dx\) |
| \(\Big \downarrow \) 3151 |
| \(\displaystyle \frac {11 \int \frac {\sec ^4(c+d x)}{(\sin (c+d x) a+a)^7}dx}{19 a}-\frac {\sec ^3(c+d x)}{19 d (a \sin (c+d x)+a)^8}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {11 \int \frac {1}{\cos (c+d x)^4 (\sin (c+d x) a+a)^7}dx}{19 a}-\frac {\sec ^3(c+d x)}{19 d (a \sin (c+d x)+a)^8}\) |
| \(\Big \downarrow \) 3151 |
| \(\displaystyle \frac {11 \left (\frac {10 \int \frac {\sec ^4(c+d x)}{(\sin (c+d x) a+a)^6}dx}{17 a}-\frac {\sec ^3(c+d x)}{17 d (a \sin (c+d x)+a)^7}\right )}{19 a}-\frac {\sec ^3(c+d x)}{19 d (a \sin (c+d x)+a)^8}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {11 \left (\frac {10 \int \frac {1}{\cos (c+d x)^4 (\sin (c+d x) a+a)^6}dx}{17 a}-\frac {\sec ^3(c+d x)}{17 d (a \sin (c+d x)+a)^7}\right )}{19 a}-\frac {\sec ^3(c+d x)}{19 d (a \sin (c+d x)+a)^8}\) |
| \(\Big \downarrow \) 3151 |
| \(\displaystyle \frac {11 \left (\frac {10 \left (\frac {3 \int \frac {\sec ^4(c+d x)}{(\sin (c+d x) a+a)^5}dx}{5 a}-\frac {\sec ^3(c+d x)}{15 d (a \sin (c+d x)+a)^6}\right )}{17 a}-\frac {\sec ^3(c+d x)}{17 d (a \sin (c+d x)+a)^7}\right )}{19 a}-\frac {\sec ^3(c+d x)}{19 d (a \sin (c+d x)+a)^8}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {11 \left (\frac {10 \left (\frac {3 \int \frac {1}{\cos (c+d x)^4 (\sin (c+d x) a+a)^5}dx}{5 a}-\frac {\sec ^3(c+d x)}{15 d (a \sin (c+d x)+a)^6}\right )}{17 a}-\frac {\sec ^3(c+d x)}{17 d (a \sin (c+d x)+a)^7}\right )}{19 a}-\frac {\sec ^3(c+d x)}{19 d (a \sin (c+d x)+a)^8}\) |
| \(\Big \downarrow \) 3151 |
| \(\displaystyle \frac {11 \left (\frac {10 \left (\frac {3 \left (\frac {8 \int \frac {\sec ^4(c+d x)}{(\sin (c+d x) a+a)^4}dx}{13 a}-\frac {\sec ^3(c+d x)}{13 d (a \sin (c+d x)+a)^5}\right )}{5 a}-\frac {\sec ^3(c+d x)}{15 d (a \sin (c+d x)+a)^6}\right )}{17 a}-\frac {\sec ^3(c+d x)}{17 d (a \sin (c+d x)+a)^7}\right )}{19 a}-\frac {\sec ^3(c+d x)}{19 d (a \sin (c+d x)+a)^8}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {11 \left (\frac {10 \left (\frac {3 \left (\frac {8 \int \frac {1}{\cos (c+d x)^4 (\sin (c+d x) a+a)^4}dx}{13 a}-\frac {\sec ^3(c+d x)}{13 d (a \sin (c+d x)+a)^5}\right )}{5 a}-\frac {\sec ^3(c+d x)}{15 d (a \sin (c+d x)+a)^6}\right )}{17 a}-\frac {\sec ^3(c+d x)}{17 d (a \sin (c+d x)+a)^7}\right )}{19 a}-\frac {\sec ^3(c+d x)}{19 d (a \sin (c+d x)+a)^8}\) |
| \(\Big \downarrow \) 3151 |
| \(\displaystyle \frac {11 \left (\frac {10 \left (\frac {3 \left (\frac {8 \left (\frac {7 \int \frac {\sec ^4(c+d x)}{(\sin (c+d x) a+a)^3}dx}{11 a}-\frac {\sec ^3(c+d x)}{11 d (a \sin (c+d x)+a)^4}\right )}{13 a}-\frac {\sec ^3(c+d x)}{13 d (a \sin (c+d x)+a)^5}\right )}{5 a}-\frac {\sec ^3(c+d x)}{15 d (a \sin (c+d x)+a)^6}\right )}{17 a}-\frac {\sec ^3(c+d x)}{17 d (a \sin (c+d x)+a)^7}\right )}{19 a}-\frac {\sec ^3(c+d x)}{19 d (a \sin (c+d x)+a)^8}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {11 \left (\frac {10 \left (\frac {3 \left (\frac {8 \left (\frac {7 \int \frac {1}{\cos (c+d x)^4 (\sin (c+d x) a+a)^3}dx}{11 a}-\frac {\sec ^3(c+d x)}{11 d (a \sin (c+d x)+a)^4}\right )}{13 a}-\frac {\sec ^3(c+d x)}{13 d (a \sin (c+d x)+a)^5}\right )}{5 a}-\frac {\sec ^3(c+d x)}{15 d (a \sin (c+d x)+a)^6}\right )}{17 a}-\frac {\sec ^3(c+d x)}{17 d (a \sin (c+d x)+a)^7}\right )}{19 a}-\frac {\sec ^3(c+d x)}{19 d (a \sin (c+d x)+a)^8}\) |
| \(\Big \downarrow \) 3151 |
| \(\displaystyle \frac {11 \left (\frac {10 \left (\frac {3 \left (\frac {8 \left (\frac {7 \left (\frac {2 \int \frac {\sec ^4(c+d x)}{(\sin (c+d x) a+a)^2}dx}{3 a}-\frac {\sec ^3(c+d x)}{9 d (a \sin (c+d x)+a)^3}\right )}{11 a}-\frac {\sec ^3(c+d x)}{11 d (a \sin (c+d x)+a)^4}\right )}{13 a}-\frac {\sec ^3(c+d x)}{13 d (a \sin (c+d x)+a)^5}\right )}{5 a}-\frac {\sec ^3(c+d x)}{15 d (a \sin (c+d x)+a)^6}\right )}{17 a}-\frac {\sec ^3(c+d x)}{17 d (a \sin (c+d x)+a)^7}\right )}{19 a}-\frac {\sec ^3(c+d x)}{19 d (a \sin (c+d x)+a)^8}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {11 \left (\frac {10 \left (\frac {3 \left (\frac {8 \left (\frac {7 \left (\frac {2 \int \frac {1}{\cos (c+d x)^4 (\sin (c+d x) a+a)^2}dx}{3 a}-\frac {\sec ^3(c+d x)}{9 d (a \sin (c+d x)+a)^3}\right )}{11 a}-\frac {\sec ^3(c+d x)}{11 d (a \sin (c+d x)+a)^4}\right )}{13 a}-\frac {\sec ^3(c+d x)}{13 d (a \sin (c+d x)+a)^5}\right )}{5 a}-\frac {\sec ^3(c+d x)}{15 d (a \sin (c+d x)+a)^6}\right )}{17 a}-\frac {\sec ^3(c+d x)}{17 d (a \sin (c+d x)+a)^7}\right )}{19 a}-\frac {\sec ^3(c+d x)}{19 d (a \sin (c+d x)+a)^8}\) |
| \(\Big \downarrow \) 3151 |
| \(\displaystyle \frac {11 \left (\frac {10 \left (\frac {3 \left (\frac {8 \left (\frac {7 \left (\frac {2 \left (\frac {5 \int \frac {\sec ^4(c+d x)}{\sin (c+d x) a+a}dx}{7 a}-\frac {\sec ^3(c+d x)}{7 d (a \sin (c+d x)+a)^2}\right )}{3 a}-\frac {\sec ^3(c+d x)}{9 d (a \sin (c+d x)+a)^3}\right )}{11 a}-\frac {\sec ^3(c+d x)}{11 d (a \sin (c+d x)+a)^4}\right )}{13 a}-\frac {\sec ^3(c+d x)}{13 d (a \sin (c+d x)+a)^5}\right )}{5 a}-\frac {\sec ^3(c+d x)}{15 d (a \sin (c+d x)+a)^6}\right )}{17 a}-\frac {\sec ^3(c+d x)}{17 d (a \sin (c+d x)+a)^7}\right )}{19 a}-\frac {\sec ^3(c+d x)}{19 d (a \sin (c+d x)+a)^8}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {11 \left (\frac {10 \left (\frac {3 \left (\frac {8 \left (\frac {7 \left (\frac {2 \left (\frac {5 \int \frac {1}{\cos (c+d x)^4 (\sin (c+d x) a+a)}dx}{7 a}-\frac {\sec ^3(c+d x)}{7 d (a \sin (c+d x)+a)^2}\right )}{3 a}-\frac {\sec ^3(c+d x)}{9 d (a \sin (c+d x)+a)^3}\right )}{11 a}-\frac {\sec ^3(c+d x)}{11 d (a \sin (c+d x)+a)^4}\right )}{13 a}-\frac {\sec ^3(c+d x)}{13 d (a \sin (c+d x)+a)^5}\right )}{5 a}-\frac {\sec ^3(c+d x)}{15 d (a \sin (c+d x)+a)^6}\right )}{17 a}-\frac {\sec ^3(c+d x)}{17 d (a \sin (c+d x)+a)^7}\right )}{19 a}-\frac {\sec ^3(c+d x)}{19 d (a \sin (c+d x)+a)^8}\) |
| \(\Big \downarrow \) 3151 |
| \(\displaystyle \frac {11 \left (\frac {10 \left (\frac {3 \left (\frac {8 \left (\frac {7 \left (\frac {2 \left (\frac {5 \left (\frac {4 \int \sec ^4(c+d x)dx}{5 a}-\frac {\sec ^3(c+d x)}{5 d (a \sin (c+d x)+a)}\right )}{7 a}-\frac {\sec ^3(c+d x)}{7 d (a \sin (c+d x)+a)^2}\right )}{3 a}-\frac {\sec ^3(c+d x)}{9 d (a \sin (c+d x)+a)^3}\right )}{11 a}-\frac {\sec ^3(c+d x)}{11 d (a \sin (c+d x)+a)^4}\right )}{13 a}-\frac {\sec ^3(c+d x)}{13 d (a \sin (c+d x)+a)^5}\right )}{5 a}-\frac {\sec ^3(c+d x)}{15 d (a \sin (c+d x)+a)^6}\right )}{17 a}-\frac {\sec ^3(c+d x)}{17 d (a \sin (c+d x)+a)^7}\right )}{19 a}-\frac {\sec ^3(c+d x)}{19 d (a \sin (c+d x)+a)^8}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {11 \left (\frac {10 \left (\frac {3 \left (\frac {8 \left (\frac {7 \left (\frac {2 \left (\frac {5 \left (\frac {4 \int \csc \left (c+d x+\frac {\pi }{2}\right )^4dx}{5 a}-\frac {\sec ^3(c+d x)}{5 d (a \sin (c+d x)+a)}\right )}{7 a}-\frac {\sec ^3(c+d x)}{7 d (a \sin (c+d x)+a)^2}\right )}{3 a}-\frac {\sec ^3(c+d x)}{9 d (a \sin (c+d x)+a)^3}\right )}{11 a}-\frac {\sec ^3(c+d x)}{11 d (a \sin (c+d x)+a)^4}\right )}{13 a}-\frac {\sec ^3(c+d x)}{13 d (a \sin (c+d x)+a)^5}\right )}{5 a}-\frac {\sec ^3(c+d x)}{15 d (a \sin (c+d x)+a)^6}\right )}{17 a}-\frac {\sec ^3(c+d x)}{17 d (a \sin (c+d x)+a)^7}\right )}{19 a}-\frac {\sec ^3(c+d x)}{19 d (a \sin (c+d x)+a)^8}\) |
| \(\Big \downarrow \) 4254 |
| \(\displaystyle \frac {11 \left (\frac {10 \left (\frac {3 \left (\frac {8 \left (\frac {7 \left (\frac {2 \left (\frac {5 \left (-\frac {4 \int \left (\tan ^2(c+d x)+1\right )d(-\tan (c+d x))}{5 a d}-\frac {\sec ^3(c+d x)}{5 d (a \sin (c+d x)+a)}\right )}{7 a}-\frac {\sec ^3(c+d x)}{7 d (a \sin (c+d x)+a)^2}\right )}{3 a}-\frac {\sec ^3(c+d x)}{9 d (a \sin (c+d x)+a)^3}\right )}{11 a}-\frac {\sec ^3(c+d x)}{11 d (a \sin (c+d x)+a)^4}\right )}{13 a}-\frac {\sec ^3(c+d x)}{13 d (a \sin (c+d x)+a)^5}\right )}{5 a}-\frac {\sec ^3(c+d x)}{15 d (a \sin (c+d x)+a)^6}\right )}{17 a}-\frac {\sec ^3(c+d x)}{17 d (a \sin (c+d x)+a)^7}\right )}{19 a}-\frac {\sec ^3(c+d x)}{19 d (a \sin (c+d x)+a)^8}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {11 \left (\frac {10 \left (\frac {3 \left (\frac {8 \left (\frac {7 \left (\frac {2 \left (\frac {5 \left (-\frac {4 \left (-\frac {1}{3} \tan ^3(c+d x)-\tan (c+d x)\right )}{5 a d}-\frac {\sec ^3(c+d x)}{5 d (a \sin (c+d x)+a)}\right )}{7 a}-\frac {\sec ^3(c+d x)}{7 d (a \sin (c+d x)+a)^2}\right )}{3 a}-\frac {\sec ^3(c+d x)}{9 d (a \sin (c+d x)+a)^3}\right )}{11 a}-\frac {\sec ^3(c+d x)}{11 d (a \sin (c+d x)+a)^4}\right )}{13 a}-\frac {\sec ^3(c+d x)}{13 d (a \sin (c+d x)+a)^5}\right )}{5 a}-\frac {\sec ^3(c+d x)}{15 d (a \sin (c+d x)+a)^6}\right )}{17 a}-\frac {\sec ^3(c+d x)}{17 d (a \sin (c+d x)+a)^7}\right )}{19 a}-\frac {\sec ^3(c+d x)}{19 d (a \sin (c+d x)+a)^8}\) |
-1/19*Sec[c + d*x]^3/(d*(a + a*Sin[c + d*x])^8) + (11*(-1/17*Sec[c + d*x]^ 3/(d*(a + a*Sin[c + d*x])^7) + (10*(-1/15*Sec[c + d*x]^3/(d*(a + a*Sin[c + d*x])^6) + (3*(-1/13*Sec[c + d*x]^3/(d*(a + a*Sin[c + d*x])^5) + (8*(-1/1 1*Sec[c + d*x]^3/(d*(a + a*Sin[c + d*x])^4) + (7*(-1/9*Sec[c + d*x]^3/(d*( a + a*Sin[c + d*x])^3) + (2*(-1/7*Sec[c + d*x]^3/(d*(a + a*Sin[c + d*x])^2 ) + (5*(-1/5*Sec[c + d*x]^3/(d*(a + a*Sin[c + d*x])) - (4*(-Tan[c + d*x] - Tan[c + d*x]^3/3))/(5*a*d)))/(7*a)))/(3*a)))/(11*a)))/(13*a)))/(5*a)))/(1 7*a)))/(19*a)
3.1.99.3.1 Defintions of rubi rules used
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_), x_Symbol] :> Simp[b*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x ])^m/(a*f*g*Simplify[2*m + p + 1])), x] + Simp[Simplify[m + p + 1]/(a*Simpl ify[2*m + p + 1]) Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m + 1), x] , x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && ILtQ[Simpli fy[m + p + 1], 0] && NeQ[2*m + p + 1, 0] && !IGtQ[m, 0]
Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp andIntegrand[(1 + x^2)^(n/2 - 1), x], x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]
Result contains complex when optimal does not.
Time = 6.42 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.51
| method | result | size |
| risch | \(\frac {512 i \left (10336 i {\mathrm e}^{9 i \left (d x +c \right )}+8398 \,{\mathrm e}^{10 i \left (d x +c \right )}-5814 \,{\mathrm e}^{8 i \left (d x +c \right )}-2736 i {\mathrm e}^{5 i \left (d x +c \right )}-2907 \,{\mathrm e}^{6 i \left (d x +c \right )}+512 i {\mathrm e}^{3 i \left (d x +c \right )}+1463 \,{\mathrm e}^{4 i \left (d x +c \right )}-16 i {\mathrm e}^{i \left (d x +c \right )}-117 \,{\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{12597 \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{19} \left (-i+{\mathrm e}^{i \left (d x +c \right )}\right )^{3} a^{8} d}\) | \(143\) |
| parallelrisch | \(\frac {\frac {4048}{12597}-176 \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+768 \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {14304 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{13}-\frac {67708 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{39}-\frac {3712 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+168 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {4384 \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+\frac {4756 \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+\frac {89550 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{221}+\frac {156112 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{663}-2 \left (\tan ^{21}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {4864 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{13}-176 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-16 \left (\tan ^{20}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {6976 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{323}-560 \left (\tan ^{16}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-474 \left (\tan ^{17}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {1081612 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12597}+\frac {39574 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{12597}-\frac {236 \left (\tan ^{19}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-\frac {704 \left (\tan ^{18}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}}{d \,a^{8} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{19}}\) | \(308\) |
| derivativedivides | \(\frac {-\frac {1}{768 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{512 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {3}{256 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {256}{19 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{19}}+\frac {128}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{18}}-\frac {10496}{17 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{17}}+\frac {1984}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{16}}-\frac {14192}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{15}}+\frac {8856}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{14}}-\frac {175016}{13 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{13}}+\frac {50936}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{12}}-\frac {18011}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{11}}+\frac {32417}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{10}}-\frac {12430}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{9}}+\frac {32525}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{8}}-\frac {72425}{16 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7}}+\frac {204605}{96 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{6}}-\frac {26871}{32 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}+\frac {2177}{8 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}-\frac {54229}{768 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {7181}{512 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {509}{256 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}}{a^{8} d}\) | \(340\) |
| default | \(\frac {-\frac {1}{768 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{512 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {3}{256 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {256}{19 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{19}}+\frac {128}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{18}}-\frac {10496}{17 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{17}}+\frac {1984}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{16}}-\frac {14192}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{15}}+\frac {8856}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{14}}-\frac {175016}{13 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{13}}+\frac {50936}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{12}}-\frac {18011}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{11}}+\frac {32417}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{10}}-\frac {12430}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{9}}+\frac {32525}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{8}}-\frac {72425}{16 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7}}+\frac {204605}{96 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{6}}-\frac {26871}{32 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}+\frac {2177}{8 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}-\frac {54229}{768 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {7181}{512 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {509}{256 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}}{a^{8} d}\) | \(340\) |
512/12597*I*(10336*I*exp(9*I*(d*x+c))+8398*exp(10*I*(d*x+c))-5814*exp(8*I* (d*x+c))-2736*I*exp(5*I*(d*x+c))-2907*exp(6*I*(d*x+c))+512*I*exp(3*I*(d*x+ c))+1463*exp(4*I*(d*x+c))-16*I*exp(I*(d*x+c))-117*exp(2*I*(d*x+c))+1)/(exp (I*(d*x+c))+I)^19/(-I+exp(I*(d*x+c)))^3/a^8/d
Time = 0.32 (sec) , antiderivative size = 249, normalized size of antiderivative = 0.89 \[ \int \frac {\sec ^4(c+d x)}{(a+a \sin (c+d x))^8} \, dx=\frac {2048 \, \cos \left (d x + c\right )^{10} - 21504 \, \cos \left (d x + c\right )^{8} + 59136 \, \cos \left (d x + c\right )^{6} - 54912 \, \cos \left (d x + c\right )^{4} + 11440 \, \cos \left (d x + c\right )^{2} + {\left (256 \, \cos \left (d x + c\right )^{10} - 8064 \, \cos \left (d x + c\right )^{8} + 36960 \, \cos \left (d x + c\right )^{6} - 48048 \, \cos \left (d x + c\right )^{4} + 12870 \, \cos \left (d x + c\right )^{2} + 2431\right )} \sin \left (d x + c\right ) + 1768}{12597 \, {\left (a^{8} d \cos \left (d x + c\right )^{11} - 32 \, a^{8} d \cos \left (d x + c\right )^{9} + 160 \, a^{8} d \cos \left (d x + c\right )^{7} - 256 \, a^{8} d \cos \left (d x + c\right )^{5} + 128 \, a^{8} d \cos \left (d x + c\right )^{3} - 8 \, {\left (a^{8} d \cos \left (d x + c\right )^{9} - 10 \, a^{8} d \cos \left (d x + c\right )^{7} + 24 \, a^{8} d \cos \left (d x + c\right )^{5} - 16 \, a^{8} d \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )\right )}} \]
1/12597*(2048*cos(d*x + c)^10 - 21504*cos(d*x + c)^8 + 59136*cos(d*x + c)^ 6 - 54912*cos(d*x + c)^4 + 11440*cos(d*x + c)^2 + (256*cos(d*x + c)^10 - 8 064*cos(d*x + c)^8 + 36960*cos(d*x + c)^6 - 48048*cos(d*x + c)^4 + 12870*c os(d*x + c)^2 + 2431)*sin(d*x + c) + 1768)/(a^8*d*cos(d*x + c)^11 - 32*a^8 *d*cos(d*x + c)^9 + 160*a^8*d*cos(d*x + c)^7 - 256*a^8*d*cos(d*x + c)^5 + 128*a^8*d*cos(d*x + c)^3 - 8*(a^8*d*cos(d*x + c)^9 - 10*a^8*d*cos(d*x + c) ^7 + 24*a^8*d*cos(d*x + c)^5 - 16*a^8*d*cos(d*x + c)^3)*sin(d*x + c))
Timed out. \[ \int \frac {\sec ^4(c+d x)}{(a+a \sin (c+d x))^8} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 866 vs. \(2 (259) = 518\).
Time = 0.23 (sec) , antiderivative size = 866, normalized size of antiderivative = 3.10 \[ \int \frac {\sec ^4(c+d x)}{(a+a \sin (c+d x))^8} \, dx=\text {Too large to display} \]
-2/12597*(19787*sin(d*x + c)/(cos(d*x + c) + 1) + 136032*sin(d*x + c)^2/(c os(d*x + c) + 1)^2 + 540806*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 1483064* sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 2552175*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 2356608*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 - 1108536*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 - 6930288*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 - 10934842*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 - 7793344*sin(d*x + c)^10/(co s(d*x + c) + 1)^10 + 1058148*sin(d*x + c)^11/(cos(d*x + c) + 1)^11 + 92042 08*sin(d*x + c)^12/(cos(d*x + c) + 1)^12 + 9985222*sin(d*x + c)^13/(cos(d* x + c) + 1)^13 + 4837248*sin(d*x + c)^14/(cos(d*x + c) + 1)^14 - 1108536*s in(d*x + c)^15/(cos(d*x + c) + 1)^15 - 3527160*sin(d*x + c)^16/(cos(d*x + c) + 1)^16 - 2985489*sin(d*x + c)^17/(cos(d*x + c) + 1)^17 - 1478048*sin(d *x + c)^18/(cos(d*x + c) + 1)^18 - 495482*sin(d*x + c)^19/(cos(d*x + c) + 1)^19 - 100776*sin(d*x + c)^20/(cos(d*x + c) + 1)^20 - 12597*sin(d*x + c)^ 21/(cos(d*x + c) + 1)^21 + 2024)/((a^8 + 16*a^8*sin(d*x + c)/(cos(d*x + c) + 1) + 117*a^8*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 512*a^8*sin(d*x + c) ^3/(cos(d*x + c) + 1)^3 + 1463*a^8*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 2 736*a^8*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 2907*a^8*sin(d*x + c)^6/(cos (d*x + c) + 1)^6 - 5814*a^8*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 - 10336*a^ 8*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 - 8398*a^8*sin(d*x + c)^10/(cos(d*x + c) + 1)^10 + 8398*a^8*sin(d*x + c)^12/(cos(d*x + c) + 1)^12 + 10336*a...
Time = 0.43 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.08 \[ \int \frac {\sec ^4(c+d x)}{(a+a \sin (c+d x))^8} \, dx=-\frac {\frac {4199 \, {\left (18 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 33 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 17\right )}}{a^{8} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{3}} + \frac {12823746 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{18} + 140368371 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{17} + 879644311 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{16} + 3693272440 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{15} + 11467502592 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{14} + 27403194676 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 51919375300 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{12} + 79183835016 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 98304418212 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 99750226290 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 82860874122 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 56110430792 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 30766700912 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 13462452660 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 4616712644 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 1197851960 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 226248618 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 27911475 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2143959}{a^{8} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{19}}}{6449664 \, d} \]
-1/6449664*(4199*(18*tan(1/2*d*x + 1/2*c)^2 - 33*tan(1/2*d*x + 1/2*c) + 17 )/(a^8*(tan(1/2*d*x + 1/2*c) - 1)^3) + (12823746*tan(1/2*d*x + 1/2*c)^18 + 140368371*tan(1/2*d*x + 1/2*c)^17 + 879644311*tan(1/2*d*x + 1/2*c)^16 + 3 693272440*tan(1/2*d*x + 1/2*c)^15 + 11467502592*tan(1/2*d*x + 1/2*c)^14 + 27403194676*tan(1/2*d*x + 1/2*c)^13 + 51919375300*tan(1/2*d*x + 1/2*c)^12 + 79183835016*tan(1/2*d*x + 1/2*c)^11 + 98304418212*tan(1/2*d*x + 1/2*c)^1 0 + 99750226290*tan(1/2*d*x + 1/2*c)^9 + 82860874122*tan(1/2*d*x + 1/2*c)^ 8 + 56110430792*tan(1/2*d*x + 1/2*c)^7 + 30766700912*tan(1/2*d*x + 1/2*c)^ 6 + 13462452660*tan(1/2*d*x + 1/2*c)^5 + 4616712644*tan(1/2*d*x + 1/2*c)^4 + 1197851960*tan(1/2*d*x + 1/2*c)^3 + 226248618*tan(1/2*d*x + 1/2*c)^2 + 27911475*tan(1/2*d*x + 1/2*c) + 2143959)/(a^8*(tan(1/2*d*x + 1/2*c) + 1)^1 9))/d
Time = 11.48 (sec) , antiderivative size = 277, normalized size of antiderivative = 0.99 \[ \int \frac {\sec ^4(c+d x)}{(a+a \sin (c+d x))^8} \, dx=\frac {\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {896971\,\cos \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )}{64}-\frac {1062347\,\cos \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )}{64}-\frac {40375\,\cos \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )}{16}+\frac {40375\,\cos \left (\frac {9\,c}{2}+\frac {9\,d\,x}{2}\right )}{16}+\frac {412471\,\cos \left (\frac {11\,c}{2}+\frac {11\,d\,x}{2}\right )}{128}-\frac {324919\,\cos \left (\frac {13\,c}{2}+\frac {13\,d\,x}{2}\right )}{128}-\frac {11305\,\cos \left (\frac {15\,c}{2}+\frac {15\,d\,x}{2}\right )}{32}+\frac {7209\,\cos \left (\frac {17\,c}{2}+\frac {17\,d\,x}{2}\right )}{32}+\frac {765\,\cos \left (\frac {19\,c}{2}+\frac {19\,d\,x}{2}\right )}{128}-\frac {253\,\cos \left (\frac {21\,c}{2}+\frac {21\,d\,x}{2}\right )}{128}+\frac {65033\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}-\frac {56635\,\sin \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )}{4}-6271\,\sin \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )+\frac {9635\,\sin \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )}{2}-\frac {9635\,\sin \left (\frac {9\,c}{2}+\frac {9\,d\,x}{2}\right )}{2}+\frac {16363\,\sin \left (\frac {11\,c}{2}+\frac {11\,d\,x}{2}\right )}{4}+\frac {10537\,\sin \left (\frac {13\,c}{2}+\frac {13\,d\,x}{2}\right )}{8}-\frac {7611\,\sin \left (\frac {15\,c}{2}+\frac {15\,d\,x}{2}\right )}{8}-\frac {485\,\sin \left (\frac {17\,c}{2}+\frac {17\,d\,x}{2}\right )}{8}+\frac {251\,\sin \left (\frac {19\,c}{2}+\frac {19\,d\,x}{2}\right )}{8}+\frac {\sin \left (\frac {21\,c}{2}+\frac {21\,d\,x}{2}\right )}{4}\right )}{12899328\,a^8\,d\,{\cos \left (\frac {c}{2}-\frac {\pi }{4}+\frac {d\,x}{2}\right )}^{19}\,{\cos \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d\,x}{2}\right )}^3} \]
(cos(c/2 + (d*x)/2)*((896971*cos((5*c)/2 + (5*d*x)/2))/64 - (1062347*cos(( 3*c)/2 + (3*d*x)/2))/64 - (40375*cos((7*c)/2 + (7*d*x)/2))/16 + (40375*cos ((9*c)/2 + (9*d*x)/2))/16 + (412471*cos((11*c)/2 + (11*d*x)/2))/128 - (324 919*cos((13*c)/2 + (13*d*x)/2))/128 - (11305*cos((15*c)/2 + (15*d*x)/2))/3 2 + (7209*cos((17*c)/2 + (17*d*x)/2))/32 + (765*cos((19*c)/2 + (19*d*x)/2) )/128 - (253*cos((21*c)/2 + (21*d*x)/2))/128 + (65033*sin(c/2 + (d*x)/2))/ 4 - (56635*sin((3*c)/2 + (3*d*x)/2))/4 - 6271*sin((5*c)/2 + (5*d*x)/2) + ( 9635*sin((7*c)/2 + (7*d*x)/2))/2 - (9635*sin((9*c)/2 + (9*d*x)/2))/2 + (16 363*sin((11*c)/2 + (11*d*x)/2))/4 + (10537*sin((13*c)/2 + (13*d*x)/2))/8 - (7611*sin((15*c)/2 + (15*d*x)/2))/8 - (485*sin((17*c)/2 + (17*d*x)/2))/8 + (251*sin((19*c)/2 + (19*d*x)/2))/8 + sin((21*c)/2 + (21*d*x)/2)/4))/(128 99328*a^8*d*cos(c/2 - pi/4 + (d*x)/2)^19*cos(c/2 + pi/4 + (d*x)/2)^3)