Integrand size = 23, antiderivative size = 233 \[ \int \sec ^{10}(c+d x) (a+a \sin (c+d x))^{7/2} \, dx=-\frac {11 a^{7/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a+a \sin (c+d x)}}\right )}{64 \sqrt {2} d}-\frac {11 a^5 \cos (c+d x)}{64 d (a+a \sin (c+d x))^{3/2}}+\frac {11 a^4 \sec (c+d x)}{48 d \sqrt {a+a \sin (c+d x)}}+\frac {11 a^3 \sec ^3(c+d x) \sqrt {a+a \sin (c+d x)}}{120 d}+\frac {11 a^2 \sec ^5(c+d x) (a+a \sin (c+d x))^{3/2}}{140 d}+\frac {11 a \sec ^7(c+d x) (a+a \sin (c+d x))^{5/2}}{126 d}+\frac {\sec ^9(c+d x) (a+a \sin (c+d x))^{7/2}}{9 d} \] Output:
-11/128*a^(7/2)*arctanh(1/2*a^(1/2)*cos(d*x+c)*2^(1/2)/(a+a*sin(d*x+c))^(1 /2))*2^(1/2)/d-11/64*a^5*cos(d*x+c)/d/(a+a*sin(d*x+c))^(3/2)+11/48*a^4*sec (d*x+c)/d/(a+a*sin(d*x+c))^(1/2)+11/120*a^3*sec(d*x+c)^3*(a+a*sin(d*x+c))^ (1/2)/d+11/140*a^2*sec(d*x+c)^5*(a+a*sin(d*x+c))^(3/2)/d+11/126*a*sec(d*x+ c)^7*(a+a*sin(d*x+c))^(5/2)/d+1/9*sec(d*x+c)^9*(a+a*sin(d*x+c))^(7/2)/d
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 1.33 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.27 \[ \int \sec ^{10}(c+d x) (a+a \sin (c+d x))^{7/2} \, dx=\frac {a^3 \operatorname {Hypergeometric2F1}\left (-\frac {9}{2},2,-\frac {7}{2},\frac {1}{2} (1-\sin (c+d x))\right ) \sec ^9(c+d x) (1+\sin (c+d x))^4 \sqrt {a (1+\sin (c+d x))}}{18 d} \] Input:
Integrate[Sec[c + d*x]^10*(a + a*Sin[c + d*x])^(7/2),x]
Output:
(a^3*Hypergeometric2F1[-9/2, 2, -7/2, (1 - Sin[c + d*x])/2]*Sec[c + d*x]^9 *(1 + Sin[c + d*x])^4*Sqrt[a*(1 + Sin[c + d*x])])/(18*d)
Time = 1.15 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.06, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.652, Rules used = {3042, 3154, 3042, 3154, 3042, 3154, 3042, 3154, 3042, 3166, 3042, 3129, 3042, 3128, 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.
| \(\displaystyle \int \sec ^{10}(c+d x) (a \sin (c+d x)+a)^{7/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a \sin (c+d x)+a)^{7/2}}{\cos (c+d x)^{10}}dx\) |
| \(\Big \downarrow \) 3154 |
| \(\displaystyle \frac {11}{18} a \int \sec ^8(c+d x) (\sin (c+d x) a+a)^{5/2}dx+\frac {\sec ^9(c+d x) (a \sin (c+d x)+a)^{7/2}}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {11}{18} a \int \frac {(\sin (c+d x) a+a)^{5/2}}{\cos (c+d x)^8}dx+\frac {\sec ^9(c+d x) (a \sin (c+d x)+a)^{7/2}}{9 d}\) |
| \(\Big \downarrow \) 3154 |
| \(\displaystyle \frac {11}{18} a \left (\frac {9}{14} a \int \sec ^6(c+d x) (\sin (c+d x) a+a)^{3/2}dx+\frac {\sec ^7(c+d x) (a \sin (c+d x)+a)^{5/2}}{7 d}\right )+\frac {\sec ^9(c+d x) (a \sin (c+d x)+a)^{7/2}}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {11}{18} a \left (\frac {9}{14} a \int \frac {(\sin (c+d x) a+a)^{3/2}}{\cos (c+d x)^6}dx+\frac {\sec ^7(c+d x) (a \sin (c+d x)+a)^{5/2}}{7 d}\right )+\frac {\sec ^9(c+d x) (a \sin (c+d x)+a)^{7/2}}{9 d}\) |
| \(\Big \downarrow \) 3154 |
| \(\displaystyle \frac {11}{18} a \left (\frac {9}{14} a \left (\frac {7}{10} a \int \sec ^4(c+d x) \sqrt {\sin (c+d x) a+a}dx+\frac {\sec ^5(c+d x) (a \sin (c+d x)+a)^{3/2}}{5 d}\right )+\frac {\sec ^7(c+d x) (a \sin (c+d x)+a)^{5/2}}{7 d}\right )+\frac {\sec ^9(c+d x) (a \sin (c+d x)+a)^{7/2}}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {11}{18} a \left (\frac {9}{14} a \left (\frac {7}{10} a \int \frac {\sqrt {\sin (c+d x) a+a}}{\cos (c+d x)^4}dx+\frac {\sec ^5(c+d x) (a \sin (c+d x)+a)^{3/2}}{5 d}\right )+\frac {\sec ^7(c+d x) (a \sin (c+d x)+a)^{5/2}}{7 d}\right )+\frac {\sec ^9(c+d x) (a \sin (c+d x)+a)^{7/2}}{9 d}\) |
| \(\Big \downarrow \) 3154 |
| \(\displaystyle \frac {11}{18} a \left (\frac {9}{14} a \left (\frac {7}{10} a \left (\frac {5}{6} a \int \frac {\sec ^2(c+d x)}{\sqrt {\sin (c+d x) a+a}}dx+\frac {\sec ^3(c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}\right )+\frac {\sec ^5(c+d x) (a \sin (c+d x)+a)^{3/2}}{5 d}\right )+\frac {\sec ^7(c+d x) (a \sin (c+d x)+a)^{5/2}}{7 d}\right )+\frac {\sec ^9(c+d x) (a \sin (c+d x)+a)^{7/2}}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {11}{18} a \left (\frac {9}{14} a \left (\frac {7}{10} a \left (\frac {5}{6} a \int \frac {1}{\cos (c+d x)^2 \sqrt {\sin (c+d x) a+a}}dx+\frac {\sec ^3(c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}\right )+\frac {\sec ^5(c+d x) (a \sin (c+d x)+a)^{3/2}}{5 d}\right )+\frac {\sec ^7(c+d x) (a \sin (c+d x)+a)^{5/2}}{7 d}\right )+\frac {\sec ^9(c+d x) (a \sin (c+d x)+a)^{7/2}}{9 d}\) |
| \(\Big \downarrow \) 3166 |
| \(\displaystyle \frac {11}{18} a \left (\frac {9}{14} a \left (\frac {7}{10} a \left (\frac {5}{6} a \left (\frac {3}{2} a \int \frac {1}{(\sin (c+d x) a+a)^{3/2}}dx+\frac {\sec (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )+\frac {\sec ^3(c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}\right )+\frac {\sec ^5(c+d x) (a \sin (c+d x)+a)^{3/2}}{5 d}\right )+\frac {\sec ^7(c+d x) (a \sin (c+d x)+a)^{5/2}}{7 d}\right )+\frac {\sec ^9(c+d x) (a \sin (c+d x)+a)^{7/2}}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {11}{18} a \left (\frac {9}{14} a \left (\frac {7}{10} a \left (\frac {5}{6} a \left (\frac {3}{2} a \int \frac {1}{(\sin (c+d x) a+a)^{3/2}}dx+\frac {\sec (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )+\frac {\sec ^3(c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}\right )+\frac {\sec ^5(c+d x) (a \sin (c+d x)+a)^{3/2}}{5 d}\right )+\frac {\sec ^7(c+d x) (a \sin (c+d x)+a)^{5/2}}{7 d}\right )+\frac {\sec ^9(c+d x) (a \sin (c+d x)+a)^{7/2}}{9 d}\) |
| \(\Big \downarrow \) 3129 |
| \(\displaystyle \frac {11}{18} a \left (\frac {9}{14} a \left (\frac {7}{10} a \left (\frac {5}{6} a \left (\frac {3}{2} a \left (\frac {\int \frac {1}{\sqrt {\sin (c+d x) a+a}}dx}{4 a}-\frac {\cos (c+d x)}{2 d (a \sin (c+d x)+a)^{3/2}}\right )+\frac {\sec (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )+\frac {\sec ^3(c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}\right )+\frac {\sec ^5(c+d x) (a \sin (c+d x)+a)^{3/2}}{5 d}\right )+\frac {\sec ^7(c+d x) (a \sin (c+d x)+a)^{5/2}}{7 d}\right )+\frac {\sec ^9(c+d x) (a \sin (c+d x)+a)^{7/2}}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {11}{18} a \left (\frac {9}{14} a \left (\frac {7}{10} a \left (\frac {5}{6} a \left (\frac {3}{2} a \left (\frac {\int \frac {1}{\sqrt {\sin (c+d x) a+a}}dx}{4 a}-\frac {\cos (c+d x)}{2 d (a \sin (c+d x)+a)^{3/2}}\right )+\frac {\sec (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )+\frac {\sec ^3(c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}\right )+\frac {\sec ^5(c+d x) (a \sin (c+d x)+a)^{3/2}}{5 d}\right )+\frac {\sec ^7(c+d x) (a \sin (c+d x)+a)^{5/2}}{7 d}\right )+\frac {\sec ^9(c+d x) (a \sin (c+d x)+a)^{7/2}}{9 d}\) |
| \(\Big \downarrow \) 3128 |
| \(\displaystyle \frac {11}{18} a \left (\frac {9}{14} a \left (\frac {7}{10} a \left (\frac {5}{6} a \left (\frac {3}{2} a \left (-\frac {\int \frac {1}{2 a-\frac {a^2 \cos ^2(c+d x)}{\sin (c+d x) a+a}}d\frac {a \cos (c+d x)}{\sqrt {\sin (c+d x) a+a}}}{2 a d}-\frac {\cos (c+d x)}{2 d (a \sin (c+d x)+a)^{3/2}}\right )+\frac {\sec (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )+\frac {\sec ^3(c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}\right )+\frac {\sec ^5(c+d x) (a \sin (c+d x)+a)^{3/2}}{5 d}\right )+\frac {\sec ^7(c+d x) (a \sin (c+d x)+a)^{5/2}}{7 d}\right )+\frac {\sec ^9(c+d x) (a \sin (c+d x)+a)^{7/2}}{9 d}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {11}{18} a \left (\frac {9}{14} a \left (\frac {7}{10} a \left (\frac {5}{6} a \left (\frac {3}{2} a \left (-\frac {\text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{2 \sqrt {2} a^{3/2} d}-\frac {\cos (c+d x)}{2 d (a \sin (c+d x)+a)^{3/2}}\right )+\frac {\sec (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )+\frac {\sec ^3(c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}\right )+\frac {\sec ^5(c+d x) (a \sin (c+d x)+a)^{3/2}}{5 d}\right )+\frac {\sec ^7(c+d x) (a \sin (c+d x)+a)^{5/2}}{7 d}\right )+\frac {\sec ^9(c+d x) (a \sin (c+d x)+a)^{7/2}}{9 d}\) |
Input:
Int[Sec[c + d*x]^10*(a + a*Sin[c + d*x])^(7/2),x]
Output:
(Sec[c + d*x]^9*(a + a*Sin[c + d*x])^(7/2))/(9*d) + (11*a*((Sec[c + d*x]^7 *(a + a*Sin[c + d*x])^(5/2))/(7*d) + (9*a*((Sec[c + d*x]^5*(a + a*Sin[c + d*x])^(3/2))/(5*d) + (7*a*((Sec[c + d*x]^3*Sqrt[a + a*Sin[c + d*x]])/(3*d) + (5*a*(Sec[c + d*x]/(d*Sqrt[a + a*Sin[c + d*x]]) + (3*a*(-1/2*ArcTanh[(S qrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])]/(Sqrt[2]*a^(3/2)* d) - Cos[c + d*x]/(2*d*(a + a*Sin[c + d*x])^(3/2))))/2))/6))/10))/14))/18
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[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[-2/d Subst[Int[1/(2*a - x^2), x], x, b*(Cos[c + d*x]/Sqrt[a + b*Sin[c + d*x]])], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*Cos[c + d*x]*((a + b*Sin[c + d*x])^n/(a*d*(2*n + 1))), x] + Simp[(n + 1)/(a*(2*n + 1)) Int[(a + b*Sin[c + d*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0] && LtQ[n, -1] && IntegerQ[2*n]
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*(p + 1))), x] + Simp[a*((m + p + 1)/(g^2*(p + 1))) Int[(g* Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^(m - 1), x], x] /; FreeQ[{a, b, e, f, g}, x] && EqQ[a^2 - b^2, 0] && GtQ[m, 0] && LeQ[p, -2*m] && IntegersQ [m + 1/2, 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 = 0.17 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.89
\[-\frac {-6930 a^{\frac {11}{2}} \cos \left (d x +c \right )^{4} \sin \left (d x +c \right )+25410 a^{\frac {11}{2}} \cos \left (d x +c \right )^{4}+42504 a^{\frac {11}{2}} \cos \left (d x +c \right )^{2} \sin \left (d x +c \right )+3465 \sqrt {2}\, \left (a -a \sin \left (d x +c \right )\right )^{\frac {9}{2}} \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) \sin \left (d x +c \right ) a -50424 \cos \left (d x +c \right )^{2} a^{\frac {11}{2}}+3465 \sqrt {2}\, \left (a -a \sin \left (d x +c \right )\right )^{\frac {9}{2}} \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a -12320 \sin \left (d x +c \right ) a^{\frac {11}{2}}+7840 a^{\frac {11}{2}}}{40320 a^{\frac {3}{2}} \left (\sin \left (d x +c \right )-1\right )^{4} \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\]
Input:
int(sec(d*x+c)^10*(a+a*sin(d*x+c))^(7/2),x)
Output:
-1/40320/a^(3/2)*(-6930*a^(11/2)*cos(d*x+c)^4*sin(d*x+c)+25410*a^(11/2)*co s(d*x+c)^4+42504*a^(11/2)*cos(d*x+c)^2*sin(d*x+c)+3465*2^(1/2)*(a-a*sin(d* x+c))^(9/2)*arctanh(1/2*(a-a*sin(d*x+c))^(1/2)*2^(1/2)/a^(1/2))*sin(d*x+c) *a-50424*cos(d*x+c)^2*a^(11/2)+3465*2^(1/2)*(a-a*sin(d*x+c))^(9/2)*arctanh (1/2*(a-a*sin(d*x+c))^(1/2)*2^(1/2)/a^(1/2))*a-12320*sin(d*x+c)*a^(11/2)+7 840*a^(11/2))/(sin(d*x+c)-1)^4/cos(d*x+c)/(a+a*sin(d*x+c))^(1/2)/d
Time = 0.11 (sec) , antiderivative size = 331, normalized size of antiderivative = 1.42 \[ \int \sec ^{10}(c+d x) (a+a \sin (c+d x))^{7/2} \, dx=\frac {3465 \, \sqrt {\frac {1}{2}} {\left (3 \, a^{3} \cos \left (d x + c\right )^{5} - 4 \, a^{3} \cos \left (d x + c\right )^{3} - {\left (a^{3} \cos \left (d x + c\right )^{5} - 4 \, a^{3} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt {a} \log \left (-\frac {a \cos \left (d x + c\right )^{2} - 4 \, \sqrt {\frac {1}{2}} \sqrt {a \sin \left (d x + c\right ) + a} \sqrt {a} {\left (\cos \left (d x + c\right ) - \sin \left (d x + c\right ) + 1\right )} + 3 \, a \cos \left (d x + c\right ) - {\left (a \cos \left (d x + c\right ) - 2 \, a\right )} \sin \left (d x + c\right ) + 2 \, a}{\cos \left (d x + c\right )^{2} - {\left (\cos \left (d x + c\right ) + 2\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 2}\right ) + 2 \, {\left (12705 \, a^{3} \cos \left (d x + c\right )^{4} - 25212 \, a^{3} \cos \left (d x + c\right )^{2} + 3920 \, a^{3} - 77 \, {\left (45 \, a^{3} \cos \left (d x + c\right )^{4} - 276 \, a^{3} \cos \left (d x + c\right )^{2} + 80 \, a^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{40320 \, {\left (3 \, d \cos \left (d x + c\right )^{5} - 4 \, d \cos \left (d x + c\right )^{3} - {\left (d \cos \left (d x + c\right )^{5} - 4 \, d \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )\right )}} \] Input:
integrate(sec(d*x+c)^10*(a+a*sin(d*x+c))^(7/2),x, algorithm="fricas")
Output:
1/40320*(3465*sqrt(1/2)*(3*a^3*cos(d*x + c)^5 - 4*a^3*cos(d*x + c)^3 - (a^ 3*cos(d*x + c)^5 - 4*a^3*cos(d*x + c)^3)*sin(d*x + c))*sqrt(a)*log(-(a*cos (d*x + c)^2 - 4*sqrt(1/2)*sqrt(a*sin(d*x + c) + a)*sqrt(a)*(cos(d*x + c) - sin(d*x + c) + 1) + 3*a*cos(d*x + c) - (a*cos(d*x + c) - 2*a)*sin(d*x + c ) + 2*a)/(cos(d*x + c)^2 - (cos(d*x + c) + 2)*sin(d*x + c) - cos(d*x + c) - 2)) + 2*(12705*a^3*cos(d*x + c)^4 - 25212*a^3*cos(d*x + c)^2 + 3920*a^3 - 77*(45*a^3*cos(d*x + c)^4 - 276*a^3*cos(d*x + c)^2 + 80*a^3)*sin(d*x + c ))*sqrt(a*sin(d*x + c) + a))/(3*d*cos(d*x + c)^5 - 4*d*cos(d*x + c)^3 - (d *cos(d*x + c)^5 - 4*d*cos(d*x + c)^3)*sin(d*x + c))
Timed out. \[ \int \sec ^{10}(c+d x) (a+a \sin (c+d x))^{7/2} \, dx=\text {Timed out} \] Input:
integrate(sec(d*x+c)**10*(a+a*sin(d*x+c))**(7/2),x)
Output:
Timed out
Timed out. \[ \int \sec ^{10}(c+d x) (a+a \sin (c+d x))^{7/2} \, dx=\text {Timed out} \] Input:
integrate(sec(d*x+c)^10*(a+a*sin(d*x+c))^(7/2),x, algorithm="maxima")
Output:
Timed out
Time = 0.14 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.75 \[ \int \sec ^{10}(c+d x) (a+a \sin (c+d x))^{7/2} \, dx=-\frac {\sqrt {2} a^{\frac {7}{2}} {\left (\frac {630 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1} + \frac {4 \, {\left (1575 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 420 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 189 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 90 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 35\right )}}{\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9}} - 3465 \, \log \left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right ) + 3465 \, \log \left (-\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )\right )} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{80640 \, d} \] Input:
integrate(sec(d*x+c)^10*(a+a*sin(d*x+c))^(7/2),x, algorithm="giac")
Output:
-1/80640*sqrt(2)*a^(7/2)*(630*sin(-1/4*pi + 1/2*d*x + 1/2*c)/(sin(-1/4*pi + 1/2*d*x + 1/2*c)^2 - 1) + 4*(1575*sin(-1/4*pi + 1/2*d*x + 1/2*c)^8 + 420 *sin(-1/4*pi + 1/2*d*x + 1/2*c)^6 + 189*sin(-1/4*pi + 1/2*d*x + 1/2*c)^4 + 90*sin(-1/4*pi + 1/2*d*x + 1/2*c)^2 + 35)/sin(-1/4*pi + 1/2*d*x + 1/2*c)^ 9 - 3465*log(sin(-1/4*pi + 1/2*d*x + 1/2*c) + 1) + 3465*log(-sin(-1/4*pi + 1/2*d*x + 1/2*c) + 1))*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))/d
Timed out. \[ \int \sec ^{10}(c+d x) (a+a \sin (c+d x))^{7/2} \, dx=\int \frac {{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{7/2}}{{\cos \left (c+d\,x\right )}^{10}} \,d x \] Input:
int((a + a*sin(c + d*x))^(7/2)/cos(c + d*x)^10,x)
Output:
int((a + a*sin(c + d*x))^(7/2)/cos(c + d*x)^10, x)
\[ \int \sec ^{10}(c+d x) (a+a \sin (c+d x))^{7/2} \, dx=\sqrt {a}\, a^{3} \left (\int \sqrt {\sin \left (d x +c \right )+1}\, \sec \left (d x +c \right )^{10} \sin \left (d x +c \right )^{3}d x +3 \left (\int \sqrt {\sin \left (d x +c \right )+1}\, \sec \left (d x +c \right )^{10} \sin \left (d x +c \right )^{2}d x \right )+3 \left (\int \sqrt {\sin \left (d x +c \right )+1}\, \sec \left (d x +c \right )^{10} \sin \left (d x +c \right )d x \right )+\int \sqrt {\sin \left (d x +c \right )+1}\, \sec \left (d x +c \right )^{10}d x \right ) \] Input:
int(sec(d*x+c)^10*(a+a*sin(d*x+c))^(7/2),x)
Output:
sqrt(a)*a**3*(int(sqrt(sin(c + d*x) + 1)*sec(c + d*x)**10*sin(c + d*x)**3, x) + 3*int(sqrt(sin(c + d*x) + 1)*sec(c + d*x)**10*sin(c + d*x)**2,x) + 3* int(sqrt(sin(c + d*x) + 1)*sec(c + d*x)**10*sin(c + d*x),x) + int(sqrt(sin (c + d*x) + 1)*sec(c + d*x)**10,x))