Integrand size = 21, antiderivative size = 179 \[ \int \sec ^4(c+d x) (a+a \sin (c+d x))^8 \, dx=\frac {1155 a^8 x}{8}-\frac {385 a^8 \cos ^3(c+d x)}{4 d}+\frac {1155 a^8 \cos (c+d x) \sin (c+d x)}{8 d}+\frac {2 a^{15} \cos ^{11}(c+d x)}{3 d (a-a \sin (c+d x))^7}-\frac {22 a^{13} \cos ^9(c+d x)}{3 d (a-a \sin (c+d x))^5}-\frac {66 a^{14} \cos ^7(c+d x)}{d \left (a^2-a^2 \sin (c+d x)\right )^3}-\frac {231 a^{16} \cos ^5(c+d x)}{4 d \left (a^8-a^8 \sin (c+d x)\right )} \] Output:
1155/8*a^8*x-385/4*a^8*cos(d*x+c)^3/d+1155/8*a^8*cos(d*x+c)*sin(d*x+c)/d+2 /3*a^15*cos(d*x+c)^11/d/(a-a*sin(d*x+c))^7-22/3*a^13*cos(d*x+c)^9/d/(a-a*s in(d*x+c))^5-66*a^14*cos(d*x+c)^7/d/(a^2-a^2*sin(d*x+c))^3-231/4*a^16*cos( d*x+c)^5/d/(a^8-a^8*sin(d*x+c))
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.06 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.33 \[ \int \sec ^4(c+d x) (a+a \sin (c+d x))^8 \, dx=\frac {64 \sqrt {2} a^8 \operatorname {Hypergeometric2F1}\left (-\frac {11}{2},-\frac {3}{2},-\frac {1}{2},\frac {1}{2} (1-\sin (c+d x))\right ) \sec ^3(c+d x) (1+\sin (c+d x))^{3/2}}{3 d} \] Input:
Integrate[Sec[c + d*x]^4*(a + a*Sin[c + d*x])^8,x]
Output:
(64*Sqrt[2]*a^8*Hypergeometric2F1[-11/2, -3/2, -1/2, (1 - Sin[c + d*x])/2] *Sec[c + d*x]^3*(1 + Sin[c + d*x])^(3/2))/(3*d)
Time = 0.97 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.12, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {3042, 3149, 3042, 3159, 3042, 3159, 3042, 3159, 3042, 3158, 3042, 3161, 3042, 3115, 24}
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.
| \(\displaystyle \int \sec ^4(c+d x) (a \sin (c+d x)+a)^8 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a \sin (c+d x)+a)^8}{\cos (c+d x)^4}dx\) |
| \(\Big \downarrow \) 3149 |
| \(\displaystyle a^{16} \int \frac {\cos ^{12}(c+d x)}{(a-a \sin (c+d x))^8}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^{16} \int \frac {\cos (c+d x)^{12}}{(a-a \sin (c+d x))^8}dx\) |
| \(\Big \downarrow \) 3159 |
| \(\displaystyle a^{16} \left (\frac {2 \cos ^{11}(c+d x)}{3 a d (a-a \sin (c+d x))^7}-\frac {11 \int \frac {\cos ^{10}(c+d x)}{(a-a \sin (c+d x))^6}dx}{3 a^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^{16} \left (\frac {2 \cos ^{11}(c+d x)}{3 a d (a-a \sin (c+d x))^7}-\frac {11 \int \frac {\cos (c+d x)^{10}}{(a-a \sin (c+d x))^6}dx}{3 a^2}\right )\) |
| \(\Big \downarrow \) 3159 |
| \(\displaystyle a^{16} \left (\frac {2 \cos ^{11}(c+d x)}{3 a d (a-a \sin (c+d x))^7}-\frac {11 \left (\frac {2 \cos ^9(c+d x)}{a d (a-a \sin (c+d x))^5}-\frac {9 \int \frac {\cos ^8(c+d x)}{(a-a \sin (c+d x))^4}dx}{a^2}\right )}{3 a^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^{16} \left (\frac {2 \cos ^{11}(c+d x)}{3 a d (a-a \sin (c+d x))^7}-\frac {11 \left (\frac {2 \cos ^9(c+d x)}{a d (a-a \sin (c+d x))^5}-\frac {9 \int \frac {\cos (c+d x)^8}{(a-a \sin (c+d x))^4}dx}{a^2}\right )}{3 a^2}\right )\) |
| \(\Big \downarrow \) 3159 |
| \(\displaystyle a^{16} \left (\frac {2 \cos ^{11}(c+d x)}{3 a d (a-a \sin (c+d x))^7}-\frac {11 \left (\frac {2 \cos ^9(c+d x)}{a d (a-a \sin (c+d x))^5}-\frac {9 \left (\frac {7 \int \frac {\cos ^6(c+d x)}{(a-a \sin (c+d x))^2}dx}{a^2}-\frac {2 \cos ^7(c+d x)}{a d (a-a \sin (c+d x))^3}\right )}{a^2}\right )}{3 a^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^{16} \left (\frac {2 \cos ^{11}(c+d x)}{3 a d (a-a \sin (c+d x))^7}-\frac {11 \left (\frac {2 \cos ^9(c+d x)}{a d (a-a \sin (c+d x))^5}-\frac {9 \left (\frac {7 \int \frac {\cos (c+d x)^6}{(a-a \sin (c+d x))^2}dx}{a^2}-\frac {2 \cos ^7(c+d x)}{a d (a-a \sin (c+d x))^3}\right )}{a^2}\right )}{3 a^2}\right )\) |
| \(\Big \downarrow \) 3158 |
| \(\displaystyle a^{16} \left (\frac {2 \cos ^{11}(c+d x)}{3 a d (a-a \sin (c+d x))^7}-\frac {11 \left (\frac {2 \cos ^9(c+d x)}{a d (a-a \sin (c+d x))^5}-\frac {9 \left (\frac {7 \left (\frac {5 \int \frac {\cos ^4(c+d x)}{a-a \sin (c+d x)}dx}{4 a}-\frac {\cos ^5(c+d x)}{4 d \left (a^2-a^2 \sin (c+d x)\right )}\right )}{a^2}-\frac {2 \cos ^7(c+d x)}{a d (a-a \sin (c+d x))^3}\right )}{a^2}\right )}{3 a^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^{16} \left (\frac {2 \cos ^{11}(c+d x)}{3 a d (a-a \sin (c+d x))^7}-\frac {11 \left (\frac {2 \cos ^9(c+d x)}{a d (a-a \sin (c+d x))^5}-\frac {9 \left (\frac {7 \left (\frac {5 \int \frac {\cos (c+d x)^4}{a-a \sin (c+d x)}dx}{4 a}-\frac {\cos ^5(c+d x)}{4 d \left (a^2-a^2 \sin (c+d x)\right )}\right )}{a^2}-\frac {2 \cos ^7(c+d x)}{a d (a-a \sin (c+d x))^3}\right )}{a^2}\right )}{3 a^2}\right )\) |
| \(\Big \downarrow \) 3161 |
| \(\displaystyle a^{16} \left (\frac {2 \cos ^{11}(c+d x)}{3 a d (a-a \sin (c+d x))^7}-\frac {11 \left (\frac {2 \cos ^9(c+d x)}{a d (a-a \sin (c+d x))^5}-\frac {9 \left (\frac {7 \left (\frac {5 \left (\frac {\int \cos ^2(c+d x)dx}{a}-\frac {\cos ^3(c+d x)}{3 a d}\right )}{4 a}-\frac {\cos ^5(c+d x)}{4 d \left (a^2-a^2 \sin (c+d x)\right )}\right )}{a^2}-\frac {2 \cos ^7(c+d x)}{a d (a-a \sin (c+d x))^3}\right )}{a^2}\right )}{3 a^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^{16} \left (\frac {2 \cos ^{11}(c+d x)}{3 a d (a-a \sin (c+d x))^7}-\frac {11 \left (\frac {2 \cos ^9(c+d x)}{a d (a-a \sin (c+d x))^5}-\frac {9 \left (\frac {7 \left (\frac {5 \left (\frac {\int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx}{a}-\frac {\cos ^3(c+d x)}{3 a d}\right )}{4 a}-\frac {\cos ^5(c+d x)}{4 d \left (a^2-a^2 \sin (c+d x)\right )}\right )}{a^2}-\frac {2 \cos ^7(c+d x)}{a d (a-a \sin (c+d x))^3}\right )}{a^2}\right )}{3 a^2}\right )\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle a^{16} \left (\frac {2 \cos ^{11}(c+d x)}{3 a d (a-a \sin (c+d x))^7}-\frac {11 \left (\frac {2 \cos ^9(c+d x)}{a d (a-a \sin (c+d x))^5}-\frac {9 \left (\frac {7 \left (\frac {5 \left (\frac {\frac {\int 1dx}{2}+\frac {\sin (c+d x) \cos (c+d x)}{2 d}}{a}-\frac {\cos ^3(c+d x)}{3 a d}\right )}{4 a}-\frac {\cos ^5(c+d x)}{4 d \left (a^2-a^2 \sin (c+d x)\right )}\right )}{a^2}-\frac {2 \cos ^7(c+d x)}{a d (a-a \sin (c+d x))^3}\right )}{a^2}\right )}{3 a^2}\right )\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle a^{16} \left (\frac {2 \cos ^{11}(c+d x)}{3 a d (a-a \sin (c+d x))^7}-\frac {11 \left (\frac {2 \cos ^9(c+d x)}{a d (a-a \sin (c+d x))^5}-\frac {9 \left (\frac {7 \left (\frac {5 \left (\frac {\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}}{a}-\frac {\cos ^3(c+d x)}{3 a d}\right )}{4 a}-\frac {\cos ^5(c+d x)}{4 d \left (a^2-a^2 \sin (c+d x)\right )}\right )}{a^2}-\frac {2 \cos ^7(c+d x)}{a d (a-a \sin (c+d x))^3}\right )}{a^2}\right )}{3 a^2}\right )\) |
Input:
Int[Sec[c + d*x]^4*(a + a*Sin[c + d*x])^8,x]
Output:
a^16*((2*Cos[c + d*x]^11)/(3*a*d*(a - a*Sin[c + d*x])^7) - (11*((2*Cos[c + d*x]^9)/(a*d*(a - a*Sin[c + d*x])^5) - (9*((-2*Cos[c + d*x]^7)/(a*d*(a - a*Sin[c + d*x])^3) + (7*(-1/4*Cos[c + d*x]^5/(d*(a^2 - a^2*Sin[c + d*x])) + (5*(-1/3*Cos[c + d*x]^3/(a*d) + (x/2 + (Cos[c + d*x]*Sin[c + d*x])/(2*d) )/a))/(4*a)))/a^2))/a^2))/(3*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[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_), x_Symbol] :> Simp[(a/g)^(2*m) Int[(g*Cos[e + f*x])^(2*m + p)/( a - b*Sin[e + f*x])^m, x], x] /; FreeQ[{a, b, e, f, g}, x] && EqQ[a^2 - b^2 , 0] && IntegerQ[m] && LtQ[p, -1] && GeQ[2*m + p, 0]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_), x_Symbol] :> Simp[g*(g*Cos[e + f*x])^(p - 1)*((a + b*Sin[e + f*x ])^(m + 1)/(b*f*(m + p))), x] + Simp[g^2*((p - 1)/(a*(m + p))) 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] && LtQ[m, -1] && GtQ[p, 1] && (GtQ[m, -2] || EqQ[2*m + p + 1, 0] || (EqQ[m, -2] && IntegerQ[p])) && NeQ[m + p, 0] && In tegersQ[2*m, 2*p]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_), x_Symbol] :> Simp[2*g*(g*Cos[e + f*x])^(p - 1)*((a + b*Sin[e + f *x])^(m + 1)/(b*f*(2*m + p + 1))), x] + Simp[g^2*((p - 1)/(b^2*(2*m + p + 1 ))) Int[(g*Cos[e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^(m + 2), x], x] /; FreeQ[{a, b, e, f, g}, x] && EqQ[a^2 - b^2, 0] && LeQ[m, -2] && GtQ[p, 1] & & NeQ[2*m + p + 1, 0] && !ILtQ[m + p + 1, 0] && IntegersQ[2*m, 2*p]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[g*((g*Cos[e + f*x])^(p - 1)/(b*f*(p - 1))), x] + Si mp[g^2/a Int[(g*Cos[e + f*x])^(p - 2), x], x] /; FreeQ[{a, b, e, f, g}, x ] && EqQ[a^2 - b^2, 0] && GtQ[p, 1] && IntegerQ[2*p]
Result contains complex when optimal does not.
Time = 0.96 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.93
| method | result | size |
| risch | \(\frac {1155 a^{8} x}{8}+\frac {31 i a^{8} {\mathrm e}^{2 i \left (d x +c \right )}}{8 d}-\frac {39 a^{8} {\mathrm e}^{i \left (d x +c \right )}}{d}-\frac {39 a^{8} {\mathrm e}^{-i \left (d x +c \right )}}{d}-\frac {31 i a^{8} {\mathrm e}^{-2 i \left (d x +c \right )}}{8 d}-\frac {128 \left (-15 i a^{8} {\mathrm e}^{i \left (d x +c \right )}+9 a^{8} {\mathrm e}^{2 i \left (d x +c \right )}-8 a^{8}\right )}{3 \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{3} d}+\frac {a^{8} \sin \left (4 d x +4 c \right )}{32 d}+\frac {2 a^{8} \cos \left (3 d x +3 c \right )}{3 d}\) | \(166\) |
| parallelrisch | \(\frac {\left (194040 a^{8} x d -507848 a^{8}\right ) \sin \left (\frac {3 d x}{2}+\frac {3 c}{2}\right )+\left (582120 a^{8} x d -137544 a^{8}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (194040 a^{8} x d +235048 a^{8}\right ) \cos \left (\frac {3 d x}{2}+\frac {3 c}{2}\right )+a^{8} \left (-582120 d x \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-38115 \sin \left (\frac {5 d x}{2}+\frac {5 c}{2}\right )-3927 \sin \left (\frac {7 d x}{2}+\frac {7 c}{2}\right )+385 \sin \left (\frac {9 d x}{2}+\frac {9 c}{2}\right )-21 \cos \left (\frac {11 d x}{2}+\frac {11 c}{2}\right )+21 \sin \left (\frac {11 d x}{2}+\frac {11 c}{2}\right )+680856 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-38115 \cos \left (\frac {5 d x}{2}+\frac {5 c}{2}\right )+3927 \cos \left (\frac {7 d x}{2}+\frac {7 c}{2}\right )+385 \cos \left (\frac {9 d x}{2}+\frac {9 c}{2}\right )\right )}{1344 d \left (\sin \left (\frac {3 d x}{2}+\frac {3 c}{2}\right )+3 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )-3 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\cos \left (\frac {3 d x}{2}+\frac {3 c}{2}\right )\right )}\) | \(236\) |
| derivativedivides | \(\frac {a^{8} \left (\frac {\sin \left (d x +c \right )^{9}}{3 \cos \left (d x +c \right )^{3}}-\frac {2 \sin \left (d x +c \right )^{9}}{\cos \left (d x +c \right )}-2 \left (\sin \left (d x +c \right )^{7}+\frac {7 \sin \left (d x +c \right )^{5}}{6}+\frac {35 \sin \left (d x +c \right )^{3}}{24}+\frac {35 \sin \left (d x +c \right )}{16}\right ) \cos \left (d x +c \right )+\frac {35 d x}{8}+\frac {35 c}{8}\right )+8 a^{8} \left (\frac {\sin \left (d x +c \right )^{8}}{3 \cos \left (d x +c \right )^{3}}-\frac {5 \sin \left (d x +c \right )^{8}}{3 \cos \left (d x +c \right )}-\frac {5 \left (\frac {16}{5}+\sin \left (d x +c \right )^{6}+\frac {6 \sin \left (d x +c \right )^{4}}{5}+\frac {8 \sin \left (d x +c \right )^{2}}{5}\right ) \cos \left (d x +c \right )}{3}\right )+28 a^{8} \left (\frac {\sin \left (d x +c \right )^{7}}{3 \cos \left (d x +c \right )^{3}}-\frac {4 \sin \left (d x +c \right )^{7}}{3 \cos \left (d x +c \right )}-\frac {4 \left (\sin \left (d x +c \right )^{5}+\frac {5 \sin \left (d x +c \right )^{3}}{4}+\frac {15 \sin \left (d x +c \right )}{8}\right ) \cos \left (d x +c \right )}{3}+\frac {5 d x}{2}+\frac {5 c}{2}\right )+56 a^{8} \left (\frac {\sin \left (d x +c \right )^{6}}{3 \cos \left (d x +c \right )^{3}}-\frac {\sin \left (d x +c \right )^{6}}{\cos \left (d x +c \right )}-\left (\frac {8}{3}+\sin \left (d x +c \right )^{4}+\frac {4 \sin \left (d x +c \right )^{2}}{3}\right ) \cos \left (d x +c \right )\right )+70 a^{8} \left (\frac {\tan \left (d x +c \right )^{3}}{3}-\tan \left (d x +c \right )+d x +c \right )+56 a^{8} \left (\frac {\sin \left (d x +c \right )^{4}}{3 \cos \left (d x +c \right )^{3}}-\frac {\sin \left (d x +c \right )^{4}}{3 \cos \left (d x +c \right )}-\frac {\left (2+\sin \left (d x +c \right )^{2}\right ) \cos \left (d x +c \right )}{3}\right )+\frac {28 a^{8} \sin \left (d x +c \right )^{3}}{3 \cos \left (d x +c \right )^{3}}+\frac {8 a^{8}}{3 \cos \left (d x +c \right )^{3}}-a^{8} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}\) | \(478\) |
| default | \(\frac {a^{8} \left (\frac {\sin \left (d x +c \right )^{9}}{3 \cos \left (d x +c \right )^{3}}-\frac {2 \sin \left (d x +c \right )^{9}}{\cos \left (d x +c \right )}-2 \left (\sin \left (d x +c \right )^{7}+\frac {7 \sin \left (d x +c \right )^{5}}{6}+\frac {35 \sin \left (d x +c \right )^{3}}{24}+\frac {35 \sin \left (d x +c \right )}{16}\right ) \cos \left (d x +c \right )+\frac {35 d x}{8}+\frac {35 c}{8}\right )+8 a^{8} \left (\frac {\sin \left (d x +c \right )^{8}}{3 \cos \left (d x +c \right )^{3}}-\frac {5 \sin \left (d x +c \right )^{8}}{3 \cos \left (d x +c \right )}-\frac {5 \left (\frac {16}{5}+\sin \left (d x +c \right )^{6}+\frac {6 \sin \left (d x +c \right )^{4}}{5}+\frac {8 \sin \left (d x +c \right )^{2}}{5}\right ) \cos \left (d x +c \right )}{3}\right )+28 a^{8} \left (\frac {\sin \left (d x +c \right )^{7}}{3 \cos \left (d x +c \right )^{3}}-\frac {4 \sin \left (d x +c \right )^{7}}{3 \cos \left (d x +c \right )}-\frac {4 \left (\sin \left (d x +c \right )^{5}+\frac {5 \sin \left (d x +c \right )^{3}}{4}+\frac {15 \sin \left (d x +c \right )}{8}\right ) \cos \left (d x +c \right )}{3}+\frac {5 d x}{2}+\frac {5 c}{2}\right )+56 a^{8} \left (\frac {\sin \left (d x +c \right )^{6}}{3 \cos \left (d x +c \right )^{3}}-\frac {\sin \left (d x +c \right )^{6}}{\cos \left (d x +c \right )}-\left (\frac {8}{3}+\sin \left (d x +c \right )^{4}+\frac {4 \sin \left (d x +c \right )^{2}}{3}\right ) \cos \left (d x +c \right )\right )+70 a^{8} \left (\frac {\tan \left (d x +c \right )^{3}}{3}-\tan \left (d x +c \right )+d x +c \right )+56 a^{8} \left (\frac {\sin \left (d x +c \right )^{4}}{3 \cos \left (d x +c \right )^{3}}-\frac {\sin \left (d x +c \right )^{4}}{3 \cos \left (d x +c \right )}-\frac {\left (2+\sin \left (d x +c \right )^{2}\right ) \cos \left (d x +c \right )}{3}\right )+\frac {28 a^{8} \sin \left (d x +c \right )^{3}}{3 \cos \left (d x +c \right )^{3}}+\frac {8 a^{8}}{3 \cos \left (d x +c \right )^{3}}-a^{8} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}\) | \(478\) |
Input:
int(sec(d*x+c)^4*(a+a*sin(d*x+c))^8,x,method=_RETURNVERBOSE)
Output:
1155/8*a^8*x+31/8*I*a^8/d*exp(2*I*(d*x+c))-39*a^8/d*exp(I*(d*x+c))-39*a^8/ d*exp(-I*(d*x+c))-31/8*I*a^8/d*exp(-2*I*(d*x+c))-128/3*(-15*I*a^8*exp(I*(d *x+c))+9*a^8*exp(2*I*(d*x+c))-8*a^8)/(exp(I*(d*x+c))-I)^3/d+1/32*a^8/d*sin (4*d*x+4*c)+2/3*a^8/d*cos(3*d*x+3*c)
Time = 0.12 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.38 \[ \int \sec ^4(c+d x) (a+a \sin (c+d x))^8 \, dx=-\frac {6 \, a^{8} \cos \left (d x + c\right )^{6} - 52 \, a^{8} \cos \left (d x + c\right )^{5} - 317 \, a^{8} \cos \left (d x + c\right )^{4} + 1286 \, a^{8} \cos \left (d x + c\right )^{3} + 6930 \, a^{8} d x + 512 \, a^{8} - {\left (3465 \, a^{8} d x + 5641 \, a^{8}\right )} \cos \left (d x + c\right )^{2} + {\left (3465 \, a^{8} d x - 6674 \, a^{8}\right )} \cos \left (d x + c\right ) - {\left (6 \, a^{8} \cos \left (d x + c\right )^{5} + 58 \, a^{8} \cos \left (d x + c\right )^{4} - 259 \, a^{8} \cos \left (d x + c\right )^{3} + 6930 \, a^{8} d x - 1545 \, a^{8} \cos \left (d x + c\right )^{2} - 512 \, a^{8} + {\left (3465 \, a^{8} d x - 7186 \, a^{8}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{24 \, {\left (d \cos \left (d x + c\right )^{2} - d \cos \left (d x + c\right ) + {\left (d \cos \left (d x + c\right ) + 2 \, d\right )} \sin \left (d x + c\right ) - 2 \, d\right )}} \] Input:
integrate(sec(d*x+c)^4*(a+a*sin(d*x+c))^8,x, algorithm="fricas")
Output:
-1/24*(6*a^8*cos(d*x + c)^6 - 52*a^8*cos(d*x + c)^5 - 317*a^8*cos(d*x + c) ^4 + 1286*a^8*cos(d*x + c)^3 + 6930*a^8*d*x + 512*a^8 - (3465*a^8*d*x + 56 41*a^8)*cos(d*x + c)^2 + (3465*a^8*d*x - 6674*a^8)*cos(d*x + c) - (6*a^8*c os(d*x + c)^5 + 58*a^8*cos(d*x + c)^4 - 259*a^8*cos(d*x + c)^3 + 6930*a^8* d*x - 1545*a^8*cos(d*x + c)^2 - 512*a^8 + (3465*a^8*d*x - 7186*a^8)*cos(d* x + c))*sin(d*x + c))/(d*cos(d*x + c)^2 - d*cos(d*x + c) + (d*cos(d*x + c) + 2*d)*sin(d*x + c) - 2*d)
Timed out. \[ \int \sec ^4(c+d x) (a+a \sin (c+d x))^8 \, dx=\text {Timed out} \] Input:
integrate(sec(d*x+c)**4*(a+a*sin(d*x+c))**8,x)
Output:
Timed out
Time = 0.12 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.74 \[ \int \sec ^4(c+d x) (a+a \sin (c+d x))^8 \, dx=\frac {224 \, a^{8} \tan \left (d x + c\right )^{3} + 64 \, {\left (\cos \left (d x + c\right )^{3} - \frac {9 \, \cos \left (d x + c\right )^{2} - 1}{\cos \left (d x + c\right )^{3}} - 9 \, \cos \left (d x + c\right )\right )} a^{8} + {\left (8 \, \tan \left (d x + c\right )^{3} + 105 \, d x + 105 \, c - \frac {3 \, {\left (13 \, \tan \left (d x + c\right )^{3} + 11 \, \tan \left (d x + c\right )\right )}}{\tan \left (d x + c\right )^{4} + 2 \, \tan \left (d x + c\right )^{2} + 1} - 72 \, \tan \left (d x + c\right )\right )} a^{8} + 112 \, {\left (2 \, \tan \left (d x + c\right )^{3} + 15 \, d x + 15 \, c - \frac {3 \, \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{2} + 1} - 12 \, \tan \left (d x + c\right )\right )} a^{8} + 560 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, d x + 3 \, c - 3 \, \tan \left (d x + c\right )\right )} a^{8} + 8 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} a^{8} - 448 \, a^{8} {\left (\frac {6 \, \cos \left (d x + c\right )^{2} - 1}{\cos \left (d x + c\right )^{3}} + 3 \, \cos \left (d x + c\right )\right )} - \frac {448 \, {\left (3 \, \cos \left (d x + c\right )^{2} - 1\right )} a^{8}}{\cos \left (d x + c\right )^{3}} + \frac {64 \, a^{8}}{\cos \left (d x + c\right )^{3}}}{24 \, d} \] Input:
integrate(sec(d*x+c)^4*(a+a*sin(d*x+c))^8,x, algorithm="maxima")
Output:
1/24*(224*a^8*tan(d*x + c)^3 + 64*(cos(d*x + c)^3 - (9*cos(d*x + c)^2 - 1) /cos(d*x + c)^3 - 9*cos(d*x + c))*a^8 + (8*tan(d*x + c)^3 + 105*d*x + 105* c - 3*(13*tan(d*x + c)^3 + 11*tan(d*x + c))/(tan(d*x + c)^4 + 2*tan(d*x + c)^2 + 1) - 72*tan(d*x + c))*a^8 + 112*(2*tan(d*x + c)^3 + 15*d*x + 15*c - 3*tan(d*x + c)/(tan(d*x + c)^2 + 1) - 12*tan(d*x + c))*a^8 + 560*(tan(d*x + c)^3 + 3*d*x + 3*c - 3*tan(d*x + c))*a^8 + 8*(tan(d*x + c)^3 + 3*tan(d* x + c))*a^8 - 448*a^8*((6*cos(d*x + c)^2 - 1)/cos(d*x + c)^3 + 3*cos(d*x + c)) - 448*(3*cos(d*x + c)^2 - 1)*a^8/cos(d*x + c)^3 + 64*a^8/cos(d*x + c) ^3)/d
Time = 0.20 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.12 \[ \int \sec ^4(c+d x) (a+a \sin (c+d x))^8 \, dx=\frac {3465 \, {\left (d x + c\right )} a^{8} + \frac {1024 \, {\left (6 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 15 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 7 \, a^{8}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{3}} + \frac {2 \, {\left (369 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 1728 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 393 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 5568 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 393 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 5696 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 369 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1856 \, a^{8}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{4}}}{24 \, d} \] Input:
integrate(sec(d*x+c)^4*(a+a*sin(d*x+c))^8,x, algorithm="giac")
Output:
1/24*(3465*(d*x + c)*a^8 + 1024*(6*a^8*tan(1/2*d*x + 1/2*c)^2 - 15*a^8*tan (1/2*d*x + 1/2*c) + 7*a^8)/(tan(1/2*d*x + 1/2*c) - 1)^3 + 2*(369*a^8*tan(1 /2*d*x + 1/2*c)^7 - 1728*a^8*tan(1/2*d*x + 1/2*c)^6 + 393*a^8*tan(1/2*d*x + 1/2*c)^5 - 5568*a^8*tan(1/2*d*x + 1/2*c)^4 - 393*a^8*tan(1/2*d*x + 1/2*c )^3 - 5696*a^8*tan(1/2*d*x + 1/2*c)^2 - 369*a^8*tan(1/2*d*x + 1/2*c) - 185 6*a^8)/(tan(1/2*d*x + 1/2*c)^2 + 1)^4)/d
Time = 30.79 (sec) , antiderivative size = 437, normalized size of antiderivative = 2.44 \[ \int \sec ^4(c+d x) (a+a \sin (c+d x))^8 \, dx=\frac {1155\,a^8\,x}{8}+\frac {\frac {1155\,a^8\,\left (c+d\,x\right )}{8}-\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {3465\,a^8\,\left (c+d\,x\right )}{8}-\frac {a^8\,\left (10395\,c+10395\,d\,x-25758\right )}{24}\right )-\frac {a^8\,\left (3465\,c+3465\,d\,x-10880\right )}{24}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\left (\frac {3465\,a^8\,\left (c+d\,x\right )}{8}-\frac {a^8\,\left (10395\,c+10395\,d\,x-6882\right )}{24}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\left (\frac {8085\,a^8\,\left (c+d\,x\right )}{8}-\frac {a^8\,\left (24255\,c+24255\,d\,x-21030\right )}{24}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {8085\,a^8\,\left (c+d\,x\right )}{8}-\frac {a^8\,\left (24255\,c+24255\,d\,x-55130\right )}{24}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (\frac {15015\,a^8\,\left (c+d\,x\right )}{8}-\frac {a^8\,\left (45045\,c+45045\,d\,x-45112\right )}{24}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {15015\,a^8\,\left (c+d\,x\right )}{8}-\frac {a^8\,\left (45045\,c+45045\,d\,x-96328\right )}{24}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (\frac {10395\,a^8\,\left (c+d\,x\right )}{4}-\frac {a^8\,\left (62370\,c+62370\,d\,x-86040\right )}{24}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {10395\,a^8\,\left (c+d\,x\right )}{4}-\frac {a^8\,\left (62370\,c+62370\,d\,x-109800\right )}{24}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {12705\,a^8\,\left (c+d\,x\right )}{4}-\frac {a^8\,\left (76230\,c+76230\,d\,x-103972\right )}{24}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {12705\,a^8\,\left (c+d\,x\right )}{4}-\frac {a^8\,\left (76230\,c+76230\,d\,x-135388\right )}{24}\right )}{d\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )}^3\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^4} \] Input:
int((a + a*sin(c + d*x))^8/cos(c + d*x)^4,x)
Output:
(1155*a^8*x)/8 + ((1155*a^8*(c + d*x))/8 - tan(c/2 + (d*x)/2)*((3465*a^8*( c + d*x))/8 - (a^8*(10395*c + 10395*d*x - 25758))/24) - (a^8*(3465*c + 346 5*d*x - 10880))/24 + tan(c/2 + (d*x)/2)^10*((3465*a^8*(c + d*x))/8 - (a^8* (10395*c + 10395*d*x - 6882))/24) - tan(c/2 + (d*x)/2)^9*((8085*a^8*(c + d *x))/8 - (a^8*(24255*c + 24255*d*x - 21030))/24) + tan(c/2 + (d*x)/2)^2*(( 8085*a^8*(c + d*x))/8 - (a^8*(24255*c + 24255*d*x - 55130))/24) + tan(c/2 + (d*x)/2)^8*((15015*a^8*(c + d*x))/8 - (a^8*(45045*c + 45045*d*x - 45112) )/24) - tan(c/2 + (d*x)/2)^3*((15015*a^8*(c + d*x))/8 - (a^8*(45045*c + 45 045*d*x - 96328))/24) - tan(c/2 + (d*x)/2)^7*((10395*a^8*(c + d*x))/4 - (a ^8*(62370*c + 62370*d*x - 86040))/24) + tan(c/2 + (d*x)/2)^4*((10395*a^8*( c + d*x))/4 - (a^8*(62370*c + 62370*d*x - 109800))/24) + tan(c/2 + (d*x)/2 )^6*((12705*a^8*(c + d*x))/4 - (a^8*(76230*c + 76230*d*x - 103972))/24) - tan(c/2 + (d*x)/2)^5*((12705*a^8*(c + d*x))/4 - (a^8*(76230*c + 76230*d*x - 135388))/24))/(d*(tan(c/2 + (d*x)/2) - 1)^3*(tan(c/2 + (d*x)/2)^2 + 1)^4 )
Time = 0.18 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.38 \[ \int \sec ^4(c+d x) (a+a \sin (c+d x))^8 \, dx=\frac {a^{8} \left (-6 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5}-52 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4}-247 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}-1182 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}+3465 \cos \left (d x +c \right ) \sin \left (d x +c \right ) d x +4293 \cos \left (d x +c \right ) \sin \left (d x +c \right )-3465 \cos \left (d x +c \right ) d x -2294 \cos \left (d x +c \right )+6 \sin \left (d x +c \right )^{6}+58 \sin \left (d x +c \right )^{5}+299 \sin \left (d x +c \right )^{4}+1429 \sin \left (d x +c \right )^{3}+3465 \sin \left (d x +c \right )^{2} d x -9403 \sin \left (d x +c \right )^{2}-6930 \sin \left (d x +c \right ) d x +4293 \sin \left (d x +c \right )+3465 d x +2294\right )}{24 d \left (\cos \left (d x +c \right ) \sin \left (d x +c \right )-\cos \left (d x +c \right )+\sin \left (d x +c \right )^{2}-2 \sin \left (d x +c \right )+1\right )} \] Input:
int(sec(d*x+c)^4*(a+a*sin(d*x+c))^8,x)
Output:
(a**8*( - 6*cos(c + d*x)*sin(c + d*x)**5 - 52*cos(c + d*x)*sin(c + d*x)**4 - 247*cos(c + d*x)*sin(c + d*x)**3 - 1182*cos(c + d*x)*sin(c + d*x)**2 + 3465*cos(c + d*x)*sin(c + d*x)*d*x + 4293*cos(c + d*x)*sin(c + d*x) - 3465 *cos(c + d*x)*d*x - 2294*cos(c + d*x) + 6*sin(c + d*x)**6 + 58*sin(c + d*x )**5 + 299*sin(c + d*x)**4 + 1429*sin(c + d*x)**3 + 3465*sin(c + d*x)**2*d *x - 9403*sin(c + d*x)**2 - 6930*sin(c + d*x)*d*x + 4293*sin(c + d*x) + 34 65*d*x + 2294))/(24*d*(cos(c + d*x)*sin(c + d*x) - cos(c + d*x) + sin(c + d*x)**2 - 2*sin(c + d*x) + 1))