Integrand size = 23, antiderivative size = 211 \[ \int \frac {\sec ^5(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=\frac {99 \text {arctanh}\left (\frac {\sqrt {a+a \sin (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{256 \sqrt {2} a^{3/2} d}-\frac {33}{128 d (a+a \sin (c+d x))^{3/2}}-\frac {99 \sec ^2(c+d x)}{560 d (a+a \sin (c+d x))^{3/2}}-\frac {\sec ^4(c+d x)}{7 d (a+a \sin (c+d x))^{3/2}}-\frac {99}{256 a d \sqrt {a+a \sin (c+d x)}}+\frac {99 \sec ^2(c+d x)}{320 a d \sqrt {a+a \sin (c+d x)}}+\frac {11 \sec ^4(c+d x)}{56 a d \sqrt {a+a \sin (c+d x)}} \]
-33/128/d/(a+a*sin(d*x+c))^(3/2)-99/560*sec(d*x+c)^2/d/(a+a*sin(d*x+c))^(3 /2)-1/7*sec(d*x+c)^4/d/(a+a*sin(d*x+c))^(3/2)+99/512*arctanh(1/2*(a+a*sin( d*x+c))^(1/2)*2^(1/2)/a^(1/2))/a^(3/2)/d*2^(1/2)-99/256/a/d/(a+a*sin(d*x+c ))^(1/2)+99/320*sec(d*x+c)^2/a/d/(a+a*sin(d*x+c))^(1/2)+11/56*sec(d*x+c)^4 /a/d/(a+a*sin(d*x+c))^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.06 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.21 \[ \int \frac {\sec ^5(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=-\frac {a^2 \operatorname {Hypergeometric2F1}\left (-\frac {7}{2},3,-\frac {5}{2},\frac {1}{2} (1+\sin (c+d x))\right )}{28 d (a+a \sin (c+d x))^{7/2}} \]
-1/28*(a^2*Hypergeometric2F1[-7/2, 3, -5/2, (1 + Sin[c + d*x])/2])/(d*(a + a*Sin[c + d*x])^(7/2))
Time = 0.84 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.08, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.609, Rules used = {3042, 3160, 3042, 3166, 3042, 3160, 3042, 3166, 3042, 3146, 61, 61, 73, 219}
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.
| \(\displaystyle \int \frac {\sec ^5(c+d x)}{(a \sin (c+d x)+a)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\cos (c+d x)^5 (a \sin (c+d x)+a)^{3/2}}dx\) |
| \(\Big \downarrow \) 3160 |
| \(\displaystyle \frac {11 \int \frac {\sec ^5(c+d x)}{\sqrt {\sin (c+d x) a+a}}dx}{14 a}-\frac {\sec ^4(c+d x)}{7 d (a \sin (c+d x)+a)^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {11 \int \frac {1}{\cos (c+d x)^5 \sqrt {\sin (c+d x) a+a}}dx}{14 a}-\frac {\sec ^4(c+d x)}{7 d (a \sin (c+d x)+a)^{3/2}}\) |
| \(\Big \downarrow \) 3166 |
| \(\displaystyle \frac {11 \left (\frac {9}{8} a \int \frac {\sec ^3(c+d x)}{(\sin (c+d x) a+a)^{3/2}}dx+\frac {\sec ^4(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}\right )}{14 a}-\frac {\sec ^4(c+d x)}{7 d (a \sin (c+d x)+a)^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {11 \left (\frac {9}{8} a \int \frac {1}{\cos (c+d x)^3 (\sin (c+d x) a+a)^{3/2}}dx+\frac {\sec ^4(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}\right )}{14 a}-\frac {\sec ^4(c+d x)}{7 d (a \sin (c+d x)+a)^{3/2}}\) |
| \(\Big \downarrow \) 3160 |
| \(\displaystyle \frac {11 \left (\frac {9}{8} a \left (\frac {7 \int \frac {\sec ^3(c+d x)}{\sqrt {\sin (c+d x) a+a}}dx}{10 a}-\frac {\sec ^2(c+d x)}{5 d (a \sin (c+d x)+a)^{3/2}}\right )+\frac {\sec ^4(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}\right )}{14 a}-\frac {\sec ^4(c+d x)}{7 d (a \sin (c+d x)+a)^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {11 \left (\frac {9}{8} a \left (\frac {7 \int \frac {1}{\cos (c+d x)^3 \sqrt {\sin (c+d x) a+a}}dx}{10 a}-\frac {\sec ^2(c+d x)}{5 d (a \sin (c+d x)+a)^{3/2}}\right )+\frac {\sec ^4(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}\right )}{14 a}-\frac {\sec ^4(c+d x)}{7 d (a \sin (c+d x)+a)^{3/2}}\) |
| \(\Big \downarrow \) 3166 |
| \(\displaystyle \frac {11 \left (\frac {9}{8} a \left (\frac {7 \left (\frac {5}{4} a \int \frac {\sec (c+d x)}{(\sin (c+d x) a+a)^{3/2}}dx+\frac {\sec ^2(c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )}{10 a}-\frac {\sec ^2(c+d x)}{5 d (a \sin (c+d x)+a)^{3/2}}\right )+\frac {\sec ^4(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}\right )}{14 a}-\frac {\sec ^4(c+d x)}{7 d (a \sin (c+d x)+a)^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {11 \left (\frac {9}{8} a \left (\frac {7 \left (\frac {5}{4} a \int \frac {1}{\cos (c+d x) (\sin (c+d x) a+a)^{3/2}}dx+\frac {\sec ^2(c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )}{10 a}-\frac {\sec ^2(c+d x)}{5 d (a \sin (c+d x)+a)^{3/2}}\right )+\frac {\sec ^4(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}\right )}{14 a}-\frac {\sec ^4(c+d x)}{7 d (a \sin (c+d x)+a)^{3/2}}\) |
| \(\Big \downarrow \) 3146 |
| \(\displaystyle \frac {11 \left (\frac {9}{8} a \left (\frac {7 \left (\frac {5 a^2 \int \frac {1}{(a-a \sin (c+d x)) (\sin (c+d x) a+a)^{5/2}}d(a \sin (c+d x))}{4 d}+\frac {\sec ^2(c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )}{10 a}-\frac {\sec ^2(c+d x)}{5 d (a \sin (c+d x)+a)^{3/2}}\right )+\frac {\sec ^4(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}\right )}{14 a}-\frac {\sec ^4(c+d x)}{7 d (a \sin (c+d x)+a)^{3/2}}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {11 \left (\frac {9}{8} a \left (\frac {7 \left (\frac {5 a^2 \left (\frac {\int \frac {1}{(a-a \sin (c+d x)) (\sin (c+d x) a+a)^{3/2}}d(a \sin (c+d x))}{2 a}-\frac {1}{3 a (a \sin (c+d x)+a)^{3/2}}\right )}{4 d}+\frac {\sec ^2(c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )}{10 a}-\frac {\sec ^2(c+d x)}{5 d (a \sin (c+d x)+a)^{3/2}}\right )+\frac {\sec ^4(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}\right )}{14 a}-\frac {\sec ^4(c+d x)}{7 d (a \sin (c+d x)+a)^{3/2}}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {11 \left (\frac {9}{8} a \left (\frac {7 \left (\frac {5 a^2 \left (\frac {\frac {\int \frac {1}{(a-a \sin (c+d x)) \sqrt {\sin (c+d x) a+a}}d(a \sin (c+d x))}{2 a}-\frac {1}{a \sqrt {a \sin (c+d x)+a}}}{2 a}-\frac {1}{3 a (a \sin (c+d x)+a)^{3/2}}\right )}{4 d}+\frac {\sec ^2(c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )}{10 a}-\frac {\sec ^2(c+d x)}{5 d (a \sin (c+d x)+a)^{3/2}}\right )+\frac {\sec ^4(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}\right )}{14 a}-\frac {\sec ^4(c+d x)}{7 d (a \sin (c+d x)+a)^{3/2}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {11 \left (\frac {9}{8} a \left (\frac {7 \left (\frac {5 a^2 \left (\frac {\frac {\int \frac {1}{2 a-a^2 \sin ^2(c+d x)}d\sqrt {\sin (c+d x) a+a}}{a}-\frac {1}{a \sqrt {a \sin (c+d x)+a}}}{2 a}-\frac {1}{3 a (a \sin (c+d x)+a)^{3/2}}\right )}{4 d}+\frac {\sec ^2(c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )}{10 a}-\frac {\sec ^2(c+d x)}{5 d (a \sin (c+d x)+a)^{3/2}}\right )+\frac {\sec ^4(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}\right )}{14 a}-\frac {\sec ^4(c+d x)}{7 d (a \sin (c+d x)+a)^{3/2}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {11 \left (\frac {9}{8} a \left (\frac {7 \left (\frac {5 a^2 \left (\frac {\frac {\text {arctanh}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2}}\right )}{\sqrt {2} a^{3/2}}-\frac {1}{a \sqrt {a \sin (c+d x)+a}}}{2 a}-\frac {1}{3 a (a \sin (c+d x)+a)^{3/2}}\right )}{4 d}+\frac {\sec ^2(c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )}{10 a}-\frac {\sec ^2(c+d x)}{5 d (a \sin (c+d x)+a)^{3/2}}\right )+\frac {\sec ^4(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}\right )}{14 a}-\frac {\sec ^4(c+d x)}{7 d (a \sin (c+d x)+a)^{3/2}}\) |
-1/7*Sec[c + d*x]^4/(d*(a + a*Sin[c + d*x])^(3/2)) + (11*(Sec[c + d*x]^4/( 4*d*Sqrt[a + a*Sin[c + d*x]]) + (9*a*(-1/5*Sec[c + d*x]^2/(d*(a + a*Sin[c + d*x])^(3/2)) + (7*(Sec[c + d*x]^2/(2*d*Sqrt[a + a*Sin[c + d*x]]) + (5*a^ 2*(-1/3*1/(a*(a + a*Sin[c + d*x])^(3/2)) + (ArcTanh[(Sqrt[a]*Sin[c + d*x]) /Sqrt[2]]/(Sqrt[2]*a^(3/2)) - 1/(a*Sqrt[a + a*Sin[c + d*x]]))/(2*a)))/(4*d )))/(10*a)))/8))/(14*a)
3.2.80.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[m, -1] && !(LtQ[n, -1] && (EqQ[a, 0 ] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d , m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m _.), x_Symbol] :> Simp[1/(b^p*f) Subst[Int[(a + x)^(m + (p - 1)/2)*(a - x )^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && I ntegerQ[(p - 1)/2] && EqQ[a^2 - b^2, 0] && (GeQ[p, -1] || !IntegerQ[m + 1/ 2])
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*(2*m + p + 1))), x] + Simp[(m + p + 1)/(a*(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] && LtQ[m, -1] && NeQ[2*m + p + 1, 0] & & IntegersQ[2*m, 2*p]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)/Sqrt[(a_) + (b_.)*sin[(e_.) + (f_. )*(x_)]], x_Symbol] :> Simp[(-b)*((g*Cos[e + f*x])^(p + 1)/(a*f*g*(p + 1)*S qrt[a + b*Sin[e + f*x]])), x] + Simp[a*((2*p + 1)/(2*g^2*(p + 1))) Int[(g *Cos[e + f*x])^(p + 2)/(a + b*Sin[e + f*x])^(3/2), x], x] /; FreeQ[{a, b, e , f, g}, x] && EqQ[a^2 - b^2, 0] && LtQ[p, -1] && IntegerQ[2*p]
Time = 0.86 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.72
| method | result | size |
| default | \(-\frac {2 a^{5} \left (\frac {5}{32 a^{6} \sqrt {a +a \sin \left (d x +c \right )}}+\frac {1}{16 a^{5} \left (a +a \sin \left (d x +c \right )\right )^{\frac {3}{2}}}+\frac {3}{80 a^{4} \left (a +a \sin \left (d x +c \right )\right )^{\frac {5}{2}}}+\frac {1}{56 a^{3} \left (a +a \sin \left (d x +c \right )\right )^{\frac {7}{2}}}+\frac {\frac {\sqrt {a +a \sin \left (d x +c \right )}\, a \left (19 \sin \left (d x +c \right )-23\right )}{16 \left (a \sin \left (d x +c \right )-a \right )^{2}}-\frac {99 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a +a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )}{32 \sqrt {a}}}{32 a^{6}}\right )}{d}\) | \(152\) |
-2*a^5*(5/32/a^6/(a+a*sin(d*x+c))^(1/2)+1/16/a^5/(a+a*sin(d*x+c))^(3/2)+3/ 80/a^4/(a+a*sin(d*x+c))^(5/2)+1/56/a^3/(a+a*sin(d*x+c))^(7/2)+1/32/a^6*(1/ 16*(a+a*sin(d*x+c))^(1/2)*a*(19*sin(d*x+c)-23)/(a*sin(d*x+c)-a)^2-99/32*2^ (1/2)/a^(1/2)*arctanh(1/2*(a+a*sin(d*x+c))^(1/2)*2^(1/2)/a^(1/2))))/d
Time = 0.31 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.98 \[ \int \frac {\sec ^5(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=\frac {3465 \, \sqrt {2} {\left (\cos \left (d x + c\right )^{6} - 2 \, \cos \left (d x + c\right )^{4} \sin \left (d x + c\right ) - 2 \, \cos \left (d x + c\right )^{4}\right )} \sqrt {a} \log \left (-\frac {a \sin \left (d x + c\right ) + 2 \, \sqrt {2} \sqrt {a \sin \left (d x + c\right ) + a} \sqrt {a} + 3 \, a}{\sin \left (d x + c\right ) - 1}\right ) + 4 \, {\left (5775 \, \cos \left (d x + c\right )^{4} - 1188 \, \cos \left (d x + c\right )^{2} + 11 \, {\left (315 \, \cos \left (d x + c\right )^{4} - 252 \, \cos \left (d x + c\right )^{2} - 160\right )} \sin \left (d x + c\right ) - 480\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{35840 \, {\left (a^{2} d \cos \left (d x + c\right )^{6} - 2 \, a^{2} d \cos \left (d x + c\right )^{4} \sin \left (d x + c\right ) - 2 \, a^{2} d \cos \left (d x + c\right )^{4}\right )}} \]
1/35840*(3465*sqrt(2)*(cos(d*x + c)^6 - 2*cos(d*x + c)^4*sin(d*x + c) - 2* cos(d*x + c)^4)*sqrt(a)*log(-(a*sin(d*x + c) + 2*sqrt(2)*sqrt(a*sin(d*x + c) + a)*sqrt(a) + 3*a)/(sin(d*x + c) - 1)) + 4*(5775*cos(d*x + c)^4 - 1188 *cos(d*x + c)^2 + 11*(315*cos(d*x + c)^4 - 252*cos(d*x + c)^2 - 160)*sin(d *x + c) - 480)*sqrt(a*sin(d*x + c) + a))/(a^2*d*cos(d*x + c)^6 - 2*a^2*d*c os(d*x + c)^4*sin(d*x + c) - 2*a^2*d*cos(d*x + c)^4)
\[ \int \frac {\sec ^5(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=\int \frac {\sec ^{5}{\left (c + d x \right )}}{\left (a \left (\sin {\left (c + d x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Time = 0.28 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.93 \[ \int \frac {\sec ^5(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=-\frac {\frac {3465 \, \sqrt {2} \log \left (-\frac {\sqrt {2} \sqrt {a} - \sqrt {a \sin \left (d x + c\right ) + a}}{\sqrt {2} \sqrt {a} + \sqrt {a \sin \left (d x + c\right ) + a}}\right )}{\sqrt {a}} + \frac {4 \, {\left (3465 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{5} - 11550 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{4} a + 7392 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{3} a^{2} + 2112 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{2} a^{3} + 1408 \, {\left (a \sin \left (d x + c\right ) + a\right )} a^{4} + 1280 \, a^{5}\right )}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {11}{2}} - 4 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {9}{2}} a + 4 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {7}{2}} a^{2}}}{35840 \, a d} \]
-1/35840*(3465*sqrt(2)*log(-(sqrt(2)*sqrt(a) - sqrt(a*sin(d*x + c) + a))/( sqrt(2)*sqrt(a) + sqrt(a*sin(d*x + c) + a)))/sqrt(a) + 4*(3465*(a*sin(d*x + c) + a)^5 - 11550*(a*sin(d*x + c) + a)^4*a + 7392*(a*sin(d*x + c) + a)^3 *a^2 + 2112*(a*sin(d*x + c) + a)^2*a^3 + 1408*(a*sin(d*x + c) + a)*a^4 + 1 280*a^5)/((a*sin(d*x + c) + a)^(11/2) - 4*(a*sin(d*x + c) + a)^(9/2)*a + 4 *(a*sin(d*x + c) + a)^(7/2)*a^2))/(a*d)
Time = 0.32 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.18 \[ \int \frac {\sec ^5(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=\frac {\sqrt {a} {\left (\frac {3465 \, \sqrt {2} \log \left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} - \frac {3465 \, \sqrt {2} \log \left (-\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} - \frac {70 \, {\left (19 \, \sqrt {2} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 21 \, \sqrt {2} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{2} a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} - \frac {16 \, \sqrt {2} {\left (350 \, \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 70 \, \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 21 \, \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 5\right )}}{a^{2} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}\right )}}{35840 \, d} \]
1/35840*sqrt(a)*(3465*sqrt(2)*log(cos(-1/4*pi + 1/2*d*x + 1/2*c) + 1)/(a^2 *sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))) - 3465*sqrt(2)*log(-cos(-1/4*pi + 1/ 2*d*x + 1/2*c) + 1)/(a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))) - 70*(19*sqr t(2)*cos(-1/4*pi + 1/2*d*x + 1/2*c)^3 - 21*sqrt(2)*cos(-1/4*pi + 1/2*d*x + 1/2*c))/((cos(-1/4*pi + 1/2*d*x + 1/2*c)^2 - 1)^2*a^2*sgn(cos(-1/4*pi + 1 /2*d*x + 1/2*c))) - 16*sqrt(2)*(350*cos(-1/4*pi + 1/2*d*x + 1/2*c)^6 + 70* cos(-1/4*pi + 1/2*d*x + 1/2*c)^4 + 21*cos(-1/4*pi + 1/2*d*x + 1/2*c)^2 + 5 )/(a^2*cos(-1/4*pi + 1/2*d*x + 1/2*c)^7*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c) )))/d
Timed out. \[ \int \frac {\sec ^5(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=\int \frac {1}{{\cos \left (c+d\,x\right )}^5\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{3/2}} \,d x \]