\(\int \frac {\sec ^4(c+d x)}{(a+a \sin (c+d x))^8} \, dx\) [99]

Optimal result

Integrand size = 21, antiderivative size = 279 \[ \int \frac {\sec ^4(c+d x)}{(a+a \sin (c+d x))^8} \, dx=-\frac {\sec ^3(c+d x)}{19 d (a+a \sin (c+d x))^8}-\frac {11 \sec ^3(c+d x)}{323 a d (a+a \sin (c+d x))^7}-\frac {22 \sec ^3(c+d x)}{969 a^2 d (a+a \sin (c+d x))^6}-\frac {66 \sec ^3(c+d x)}{4199 a^3 d (a+a \sin (c+d x))^5}-\frac {48 \sec ^3(c+d x)}{4199 d \left (a^2+a^2 \sin (c+d x)\right )^4}-\frac {112 \sec ^3(c+d x)}{12597 a^2 d \left (a^2+a^2 \sin (c+d x)\right )^3}-\frac {32 \sec ^3(c+d x)}{4199 d \left (a^4+a^4 \sin (c+d x)\right )^2}-\frac {32 \sec ^3(c+d x)}{4199 d \left (a^8+a^8 \sin (c+d x)\right )}+\frac {128 \tan (c+d x)}{4199 a^8 d}+\frac {128 \tan ^3(c+d x)}{12597 a^8 d} \]

[Out]

-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-
112/12597*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

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2751, 3852} \[ \int \frac {\sec ^4(c+d x)}{(a+a \sin (c+d x))^8} \, dx=\frac {128 \tan ^3(c+d x)}{12597 a^8 d}+\frac {128 \tan (c+d x)}{4199 a^8 d}-\frac {32 \sec ^3(c+d x)}{4199 d \left (a^8 \sin (c+d x)+a^8\right )}-\frac {32 \sec ^3(c+d x)}{4199 d \left (a^4 \sin (c+d x)+a^4\right )^2}-\frac {66 \sec ^3(c+d x)}{4199 a^3 d (a \sin (c+d x)+a)^5}-\frac {112 \sec ^3(c+d x)}{12597 a^2 d \left (a^2 \sin (c+d x)+a^2\right )^3}-\frac {48 \sec ^3(c+d x)}{4199 d \left (a^2 \sin (c+d x)+a^2\right )^4}-\frac {22 \sec ^3(c+d x)}{969 a^2 d (a \sin (c+d x)+a)^6}-\frac {11 \sec ^3(c+d x)}{323 a d (a \sin (c+d x)+a)^7}-\frac {\sec ^3(c+d x)}{19 d (a \sin (c+d x)+a)^8} \]

[In]

Int[Sec[c + d*x]^4/(a + a*Sin[c + d*x])^8,x]

[Out]

-1/19*Sec[c + d*x]^3/(d*(a + a*Sin[c + d*x])^8) - (11*Sec[c + d*x]^3)/(323*a*d*(a + a*Sin[c + d*x])^7) - (22*S
ec[c + d*x]^3)/(969*a^2*d*(a + a*Sin[c + d*x])^6) - (66*Sec[c + d*x]^3)/(4199*a^3*d*(a + a*Sin[c + d*x])^5) -
(48*Sec[c + d*x]^3)/(4199*d*(a^2 + a^2*Sin[c + d*x])^4) - (112*Sec[c + d*x]^3)/(12597*a^2*d*(a^2 + a^2*Sin[c +
d*x])^3) - (32*Sec[c + d*x]^3)/(4199*d*(a^4 + a^4*Sin[c + d*x])^2) - (32*Sec[c + d*x]^3)/(4199*d*(a^8 + a^8*S
in[c + d*x])) + (128*Tan[c + d*x])/(4199*a^8*d) + (128*Tan[c + d*x]^3)/(12597*a^8*d)

Rule 2751

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[b*(g*C
os[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^m/(a*f*g*Simplify[2*m + p + 1])), x] + Dist[Simplify[m + p + 1]/(a*
Simplify[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[Simplify[m + p + 1], 0] && NeQ[2*m + p + 1, 0] &&  !IGtQ[m, 0]

Rule 3852

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {\sec ^3(c+d x)}{19 d (a+a \sin (c+d x))^8}+\frac {11 \int \frac {\sec ^4(c+d x)}{(a+a \sin (c+d x))^7} \, dx}{19 a} \\ & = -\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 {110 \int \frac {\sec ^4(c+d x)}{(a+a \sin (c+d x))^6} \, dx}{323 a^2} \\ & = -\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 \int \frac {\sec ^4(c+d x)}{(a+a \sin (c+d x))^5} \, dx}{323 a^3} \\ & = -\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 {528 \int \frac {\sec ^4(c+d x)}{(a+a \sin (c+d x))^4} \, dx}{4199 a^4} \\ & = -\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 {336 \int \frac {\sec ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx}{4199 a^5} \\ & = -\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 {112 \sec ^3(c+d x)}{12597 a^5 d (a+a \sin (c+d x))^3}-\frac {48 \sec ^3(c+d x)}{4199 d \left (a^2+a^2 \sin (c+d x)\right )^4}+\frac {224 \int \frac {\sec ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx}{4199 a^6} \\ & = -\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 {112 \sec ^3(c+d x)}{12597 a^5 d (a+a \sin (c+d x))^3}-\frac {48 \sec ^3(c+d x)}{4199 d \left (a^2+a^2 \sin (c+d x)\right )^4}-\frac {32 \sec ^3(c+d x)}{4199 d \left (a^4+a^4 \sin (c+d x)\right )^2}+\frac {160 \int \frac {\sec ^4(c+d x)}{a+a \sin (c+d x)} \, dx}{4199 a^7} \\ & = -\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 {112 \sec ^3(c+d x)}{12597 a^5 d (a+a \sin (c+d x))^3}-\frac {48 \sec ^3(c+d x)}{4199 d \left (a^2+a^2 \sin (c+d x)\right )^4}-\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 \int \sec ^4(c+d x) \, dx}{4199 a^8} \\ & = -\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 {112 \sec ^3(c+d x)}{12597 a^5 d (a+a \sin (c+d x))^3}-\frac {48 \sec ^3(c+d x)}{4199 d \left (a^2+a^2 \sin (c+d x)\right )^4}-\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 \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{4199 a^8 d} \\ & = -\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 {112 \sec ^3(c+d x)}{12597 a^5 d (a+a \sin (c+d x))^3}-\frac {48 \sec ^3(c+d x)}{4199 d \left (a^2+a^2 \sin (c+d x)\right )^4}-\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} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.31 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.45 \[ \int \frac {\sec ^4(c+d x)}{(a+a \sin (c+d x))^8} \, dx=\frac {\sec ^3(c+d x) (-10336 \cos (2 (c+d x))+2736 \cos (6 (c+d x))-512 \cos (8 (c+d x))+16 \cos (10 (c+d x))+8398 \sin (c+d x)-5814 \sin (3 (c+d x))-2907 \sin (5 (c+d x))+1463 \sin (7 (c+d x))-117 \sin (9 (c+d x))+\sin (11 (c+d x)))}{50388 a^8 d (1+\sin (c+d x))^8} \]

[In]

Integrate[Sec[c + d*x]^4/(a + a*Sin[c + d*x])^8,x]

[Out]

(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)

Maple [C] (verified)

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	\(\frac {-\frac {1}{768 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{512 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {3}{256 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {256}{19 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{19}}+\frac {128}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{18}}-\frac {10496}{17 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{17}}+\frac {1984}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{16}}-\frac {14192}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{15}}+\frac {8856}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{14}}-\frac {175016}{13 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{13}}+\frac {50936}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{12}}-\frac {18011}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{11}}+\frac {32417}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{10}}-\frac {12430}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{9}}+\frac {32525}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{8}}-\frac {72425}{16 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7}}+\frac {204605}{96 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{6}}-\frac {26871}{32 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}+\frac {2177}{8 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}-\frac {54229}{768 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {7181}{512 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {509}{256 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}}{a^{8} d}\)	\(340\)

[In]

int(sec(d*x+c)^4/(a+a*sin(d*x+c))^8,x,method=_RETURNVERBOSE)

[Out]

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))-290
7*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)/(e
xp(I*(d*x+c))+I)^19/(-I+exp(I*(d*x+c)))^3/a^8/d

Fricas [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 249, normalized size of antiderivative = 0.89 \[ \int \frac {\sec ^4(c+d x)}{(a+a \sin (c+d x))^8} \, dx=\frac {2048 \, \cos \left (d x + c\right )^{10} - 21504 \, \cos \left (d x + c\right )^{8} + 59136 \, \cos \left (d x + c\right )^{6} - 54912 \, \cos \left (d x + c\right )^{4} + 11440 \, \cos \left (d x + c\right )^{2} + {\left (256 \, \cos \left (d x + c\right )^{10} - 8064 \, \cos \left (d x + c\right )^{8} + 36960 \, \cos \left (d x + c\right )^{6} - 48048 \, \cos \left (d x + c\right )^{4} + 12870 \, \cos \left (d x + c\right )^{2} + 2431\right )} \sin \left (d x + c\right ) + 1768}{12597 \, {\left (a^{8} d \cos \left (d x + c\right )^{11} - 32 \, a^{8} d \cos \left (d x + c\right )^{9} + 160 \, a^{8} d \cos \left (d x + c\right )^{7} - 256 \, a^{8} d \cos \left (d x + c\right )^{5} + 128 \, a^{8} d \cos \left (d x + c\right )^{3} - 8 \, {\left (a^{8} d \cos \left (d x + c\right )^{9} - 10 \, a^{8} d \cos \left (d x + c\right )^{7} + 24 \, a^{8} d \cos \left (d x + c\right )^{5} - 16 \, a^{8} d \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )\right )}} \]

[In]

integrate(sec(d*x+c)^4/(a+a*sin(d*x+c))^8,x, algorithm="fricas")

[Out]

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 - 8064*cos(d*x + c)^8 + 36960*cos(d*x + c)^6 - 48048*cos(d*x + c)^4 + 12870
*cos(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))

Sympy [F(-1)]

Timed out. \[ \int \frac {\sec ^4(c+d x)}{(a+a \sin (c+d x))^8} \, dx=\text {Timed out} \]

[In]

integrate(sec(d*x+c)**4/(a+a*sin(d*x+c))**8,x)

[Out]

Timed out

Maxima [B] (verification not implemented)

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 \[ \int \frac {\sec ^4(c+d x)}{(a+a \sin (c+d x))^8} \, dx=\text {Too large to display} \]

[In]

integrate(sec(d*x+c)^4/(a+a*sin(d*x+c))^8,x, algorithm="maxima")

[Out]

-2/12597*(19787*sin(d*x + c)/(cos(d*x + c) + 1) + 136032*sin(d*x + c)^2/(cos(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 - 693028
8*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
/(cos(d*x + c) + 1)^10 + 1058148*sin(d*x + c)^11/(cos(d*x + c) + 1)^11 + 9204208*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 - 110
8536*sin(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 + 2
024)/((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*si
n(d*x + c)^3/(cos(d*x + c) + 1)^3 + 1463*a^8*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 2736*a^8*sin(d*x + c)^5/(co
s(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^8*sin(d*x + c)^13/(cos(d*x + c) + 1)^13 + 5814*a^8*sin(d*x + c
)^14/(cos(d*x + c) + 1)^14 - 2907*a^8*sin(d*x + c)^16/(cos(d*x + c) + 1)^16 - 2736*a^8*sin(d*x + c)^17/(cos(d*
x + c) + 1)^17 - 1463*a^8*sin(d*x + c)^18/(cos(d*x + c) + 1)^18 - 512*a^8*sin(d*x + c)^19/(cos(d*x + c) + 1)^1
9 - 117*a^8*sin(d*x + c)^20/(cos(d*x + c) + 1)^20 - 16*a^8*sin(d*x + c)^21/(cos(d*x + c) + 1)^21 - a^8*sin(d*x
+ c)^22/(cos(d*x + c) + 1)^22)*d)

Giac [A] (verification not implemented)

none

Time = 0.43 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.08 \[ \int \frac {\sec ^4(c+d x)}{(a+a \sin (c+d x))^8} \, dx=-\frac {\frac {4199 \, {\left (18 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 33 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 17\right )}}{a^{8} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{3}} + \frac {12823746 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{18} + 140368371 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{17} + 879644311 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{16} + 3693272440 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{15} + 11467502592 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{14} + 27403194676 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 51919375300 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{12} + 79183835016 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 98304418212 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 99750226290 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 82860874122 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 56110430792 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 30766700912 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 13462452660 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 4616712644 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 1197851960 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 226248618 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 27911475 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2143959}{a^{8} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{19}}}{6449664 \, d} \]

[In]

integrate(sec(d*x+c)^4/(a+a*sin(d*x+c))^8,x, algorithm="giac")

[Out]

-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 +
3693272440*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)^1
3 + 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)^10 + 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)^19))/d

Mupad [B] (verification not implemented)

Time = 11.48 (sec) , antiderivative size = 277, normalized size of antiderivative = 0.99 \[ \int \frac {\sec ^4(c+d x)}{(a+a \sin (c+d x))^8} \, dx=\frac {\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {896971\,\cos \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )}{64}-\frac {1062347\,\cos \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )}{64}-\frac {40375\,\cos \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )}{16}+\frac {40375\,\cos \left (\frac {9\,c}{2}+\frac {9\,d\,x}{2}\right )}{16}+\frac {412471\,\cos \left (\frac {11\,c}{2}+\frac {11\,d\,x}{2}\right )}{128}-\frac {324919\,\cos \left (\frac {13\,c}{2}+\frac {13\,d\,x}{2}\right )}{128}-\frac {11305\,\cos \left (\frac {15\,c}{2}+\frac {15\,d\,x}{2}\right )}{32}+\frac {7209\,\cos \left (\frac {17\,c}{2}+\frac {17\,d\,x}{2}\right )}{32}+\frac {765\,\cos \left (\frac {19\,c}{2}+\frac {19\,d\,x}{2}\right )}{128}-\frac {253\,\cos \left (\frac {21\,c}{2}+\frac {21\,d\,x}{2}\right )}{128}+\frac {65033\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}-\frac {56635\,\sin \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )}{4}-6271\,\sin \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )+\frac {9635\,\sin \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )}{2}-\frac {9635\,\sin \left (\frac {9\,c}{2}+\frac {9\,d\,x}{2}\right )}{2}+\frac {16363\,\sin \left (\frac {11\,c}{2}+\frac {11\,d\,x}{2}\right )}{4}+\frac {10537\,\sin \left (\frac {13\,c}{2}+\frac {13\,d\,x}{2}\right )}{8}-\frac {7611\,\sin \left (\frac {15\,c}{2}+\frac {15\,d\,x}{2}\right )}{8}-\frac {485\,\sin \left (\frac {17\,c}{2}+\frac {17\,d\,x}{2}\right )}{8}+\frac {251\,\sin \left (\frac {19\,c}{2}+\frac {19\,d\,x}{2}\right )}{8}+\frac {\sin \left (\frac {21\,c}{2}+\frac {21\,d\,x}{2}\right )}{4}\right )}{12899328\,a^8\,d\,{\cos \left (\frac {c}{2}-\frac {\pi }{4}+\frac {d\,x}{2}\right )}^{19}\,{\cos \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d\,x}{2}\right )}^3} \]

[In]

int(1/(cos(c + d*x)^4*(a + a*sin(c + d*x))^8),x)

[Out]

(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 - (3
24919*cos((13*c)/2 + (13*d*x)/2))/128 - (11305*cos((15*c)/2 + (15*d*x)/2))/32 + (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 + (16363*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))/(12899328*a^8*d*cos(c/2 - pi/4 + (d*x)/2)^19*cos(c/2 + pi/4 + (d*x)/2)
^3)