3.97 \(\int \frac{\sec ^2(c+d x)}{(a+a \sin (c+d x))^8} \, dx\)

Optimal. Leaf size=245 \[ \frac{128 \tan (c+d x)}{12155 a^8 d}-\frac{64 \sec (c+d x)}{12155 d \left (a^8 \sin (c+d x)+a^8\right )}-\frac{64 \sec (c+d x)}{12155 d \left (a^4 \sin (c+d x)+a^4\right )^2}-\frac{16 \sec (c+d x)}{2431 a^2 d \left (a^2 \sin (c+d x)+a^2\right )^3}-\frac{112 \sec (c+d x)}{12155 d \left (a^2 \sin (c+d x)+a^2\right )^4}-\frac{168 \sec (c+d x)}{12155 a^3 d (a \sin (c+d x)+a)^5}-\frac{24 \sec (c+d x)}{1105 a^2 d (a \sin (c+d x)+a)^6}-\frac{3 \sec (c+d x)}{85 a d (a \sin (c+d x)+a)^7}-\frac{\sec (c+d x)}{17 d (a \sin (c+d x)+a)^8} \]

[Out]

-Sec[c + d*x]/(17*d*(a + a*Sin[c + d*x])^8) - (3*Sec[c + d*x])/(85*a*d*(a + a*Sin[c + d*x])^7) - (24*Sec[c + d
*x])/(1105*a^2*d*(a + a*Sin[c + d*x])^6) - (168*Sec[c + d*x])/(12155*a^3*d*(a + a*Sin[c + d*x])^5) - (112*Sec[
c + d*x])/(12155*d*(a^2 + a^2*Sin[c + d*x])^4) - (16*Sec[c + d*x])/(2431*a^2*d*(a^2 + a^2*Sin[c + d*x])^3) - (
64*Sec[c + d*x])/(12155*d*(a^4 + a^4*Sin[c + d*x])^2) - (64*Sec[c + d*x])/(12155*d*(a^8 + a^8*Sin[c + d*x])) +
(128*Tan[c + d*x])/(12155*a^8*d)

________________________________________________________________________________________

Rubi [A]  time = 0.40397, antiderivative size = 245, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2672, 3767, 8} \[ \frac{128 \tan (c+d x)}{12155 a^8 d}-\frac{64 \sec (c+d x)}{12155 d \left (a^8 \sin (c+d x)+a^8\right )}-\frac{64 \sec (c+d x)}{12155 d \left (a^4 \sin (c+d x)+a^4\right )^2}-\frac{16 \sec (c+d x)}{2431 a^2 d \left (a^2 \sin (c+d x)+a^2\right )^3}-\frac{112 \sec (c+d x)}{12155 d \left (a^2 \sin (c+d x)+a^2\right )^4}-\frac{168 \sec (c+d x)}{12155 a^3 d (a \sin (c+d x)+a)^5}-\frac{24 \sec (c+d x)}{1105 a^2 d (a \sin (c+d x)+a)^6}-\frac{3 \sec (c+d x)}{85 a d (a \sin (c+d x)+a)^7}-\frac{\sec (c+d x)}{17 d (a \sin (c+d x)+a)^8} \]

Antiderivative was successfully verified.

[In]

Int[Sec[c + d*x]^2/(a + a*Sin[c + d*x])^8,x]

[Out]

-Sec[c + d*x]/(17*d*(a + a*Sin[c + d*x])^8) - (3*Sec[c + d*x])/(85*a*d*(a + a*Sin[c + d*x])^7) - (24*Sec[c + d
*x])/(1105*a^2*d*(a + a*Sin[c + d*x])^6) - (168*Sec[c + d*x])/(12155*a^3*d*(a + a*Sin[c + d*x])^5) - (112*Sec[
c + d*x])/(12155*d*(a^2 + a^2*Sin[c + d*x])^4) - (16*Sec[c + d*x])/(2431*a^2*d*(a^2 + a^2*Sin[c + d*x])^3) - (
64*Sec[c + d*x])/(12155*d*(a^4 + a^4*Sin[c + d*x])^2) - (64*Sec[c + d*x])/(12155*d*(a^8 + a^8*Sin[c + d*x])) +
(128*Tan[c + d*x])/(12155*a^8*d)

Rule 2672

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] + 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 3767

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]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \frac{\sec ^2(c+d x)}{(a+a \sin (c+d x))^8} \, dx &=-\frac{\sec (c+d x)}{17 d (a+a \sin (c+d x))^8}+\frac{9 \int \frac{\sec ^2(c+d x)}{(a+a \sin (c+d x))^7} \, dx}{17 a}\\ &=-\frac{\sec (c+d x)}{17 d (a+a \sin (c+d x))^8}-\frac{3 \sec (c+d x)}{85 a d (a+a \sin (c+d x))^7}+\frac{24 \int \frac{\sec ^2(c+d x)}{(a+a \sin (c+d x))^6} \, dx}{85 a^2}\\ &=-\frac{\sec (c+d x)}{17 d (a+a \sin (c+d x))^8}-\frac{3 \sec (c+d x)}{85 a d (a+a \sin (c+d x))^7}-\frac{24 \sec (c+d x)}{1105 a^2 d (a+a \sin (c+d x))^6}+\frac{168 \int \frac{\sec ^2(c+d x)}{(a+a \sin (c+d x))^5} \, dx}{1105 a^3}\\ &=-\frac{\sec (c+d x)}{17 d (a+a \sin (c+d x))^8}-\frac{3 \sec (c+d x)}{85 a d (a+a \sin (c+d x))^7}-\frac{24 \sec (c+d x)}{1105 a^2 d (a+a \sin (c+d x))^6}-\frac{168 \sec (c+d x)}{12155 a^3 d (a+a \sin (c+d x))^5}+\frac{1008 \int \frac{\sec ^2(c+d x)}{(a+a \sin (c+d x))^4} \, dx}{12155 a^4}\\ &=-\frac{\sec (c+d x)}{17 d (a+a \sin (c+d x))^8}-\frac{3 \sec (c+d x)}{85 a d (a+a \sin (c+d x))^7}-\frac{24 \sec (c+d x)}{1105 a^2 d (a+a \sin (c+d x))^6}-\frac{168 \sec (c+d x)}{12155 a^3 d (a+a \sin (c+d x))^5}-\frac{112 \sec (c+d x)}{12155 d \left (a^2+a^2 \sin (c+d x)\right )^4}+\frac{112 \int \frac{\sec ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx}{2431 a^5}\\ &=-\frac{\sec (c+d x)}{17 d (a+a \sin (c+d x))^8}-\frac{3 \sec (c+d x)}{85 a d (a+a \sin (c+d x))^7}-\frac{24 \sec (c+d x)}{1105 a^2 d (a+a \sin (c+d x))^6}-\frac{168 \sec (c+d x)}{12155 a^3 d (a+a \sin (c+d x))^5}-\frac{16 \sec (c+d x)}{2431 a^5 d (a+a \sin (c+d x))^3}-\frac{112 \sec (c+d x)}{12155 d \left (a^2+a^2 \sin (c+d x)\right )^4}+\frac{64 \int \frac{\sec ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx}{2431 a^6}\\ &=-\frac{\sec (c+d x)}{17 d (a+a \sin (c+d x))^8}-\frac{3 \sec (c+d x)}{85 a d (a+a \sin (c+d x))^7}-\frac{24 \sec (c+d x)}{1105 a^2 d (a+a \sin (c+d x))^6}-\frac{168 \sec (c+d x)}{12155 a^3 d (a+a \sin (c+d x))^5}-\frac{16 \sec (c+d x)}{2431 a^5 d (a+a \sin (c+d x))^3}-\frac{112 \sec (c+d x)}{12155 d \left (a^2+a^2 \sin (c+d x)\right )^4}-\frac{64 \sec (c+d x)}{12155 d \left (a^4+a^4 \sin (c+d x)\right )^2}+\frac{192 \int \frac{\sec ^2(c+d x)}{a+a \sin (c+d x)} \, dx}{12155 a^7}\\ &=-\frac{\sec (c+d x)}{17 d (a+a \sin (c+d x))^8}-\frac{3 \sec (c+d x)}{85 a d (a+a \sin (c+d x))^7}-\frac{24 \sec (c+d x)}{1105 a^2 d (a+a \sin (c+d x))^6}-\frac{168 \sec (c+d x)}{12155 a^3 d (a+a \sin (c+d x))^5}-\frac{16 \sec (c+d x)}{2431 a^5 d (a+a \sin (c+d x))^3}-\frac{112 \sec (c+d x)}{12155 d \left (a^2+a^2 \sin (c+d x)\right )^4}-\frac{64 \sec (c+d x)}{12155 d \left (a^4+a^4 \sin (c+d x)\right )^2}-\frac{64 \sec (c+d x)}{12155 d \left (a^8+a^8 \sin (c+d x)\right )}+\frac{128 \int \sec ^2(c+d x) \, dx}{12155 a^8}\\ &=-\frac{\sec (c+d x)}{17 d (a+a \sin (c+d x))^8}-\frac{3 \sec (c+d x)}{85 a d (a+a \sin (c+d x))^7}-\frac{24 \sec (c+d x)}{1105 a^2 d (a+a \sin (c+d x))^6}-\frac{168 \sec (c+d x)}{12155 a^3 d (a+a \sin (c+d x))^5}-\frac{16 \sec (c+d x)}{2431 a^5 d (a+a \sin (c+d x))^3}-\frac{112 \sec (c+d x)}{12155 d \left (a^2+a^2 \sin (c+d x)\right )^4}-\frac{64 \sec (c+d x)}{12155 d \left (a^4+a^4 \sin (c+d x)\right )^2}-\frac{64 \sec (c+d x)}{12155 d \left (a^8+a^8 \sin (c+d x)\right )}-\frac{128 \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{12155 a^8 d}\\ &=-\frac{\sec (c+d x)}{17 d (a+a \sin (c+d x))^8}-\frac{3 \sec (c+d x)}{85 a d (a+a \sin (c+d x))^7}-\frac{24 \sec (c+d x)}{1105 a^2 d (a+a \sin (c+d x))^6}-\frac{168 \sec (c+d x)}{12155 a^3 d (a+a \sin (c+d x))^5}-\frac{16 \sec (c+d x)}{2431 a^5 d (a+a \sin (c+d x))^3}-\frac{112 \sec (c+d x)}{12155 d \left (a^2+a^2 \sin (c+d x)\right )^4}-\frac{64 \sec (c+d x)}{12155 d \left (a^4+a^4 \sin (c+d x)\right )^2}-\frac{64 \sec (c+d x)}{12155 d \left (a^8+a^8 \sin (c+d x)\right )}+\frac{128 \tan (c+d x)}{12155 a^8 d}\\ \end{align*}

Mathematica [A]  time = 0.342428, size = 113, normalized size = 0.46 \[ \frac{\sec (c+d x) (4862 \sin (c+d x)-6188 \sin (3 (c+d x))+1700 \sin (5 (c+d x))-119 \sin (7 (c+d x))+\sin (9 (c+d x))-7072 \cos (2 (c+d x))+3808 \cos (4 (c+d x))-544 \cos (6 (c+d x))+16 \cos (8 (c+d x)))}{24310 a^8 d (\sin (c+d x)+1)^8} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[c + d*x]^2/(a + a*Sin[c + d*x])^8,x]

[Out]

(Sec[c + d*x]*(-7072*Cos[2*(c + d*x)] + 3808*Cos[4*(c + d*x)] - 544*Cos[6*(c + d*x)] + 16*Cos[8*(c + d*x)] + 4
862*Sin[c + d*x] - 6188*Sin[3*(c + d*x)] + 1700*Sin[5*(c + d*x)] - 119*Sin[7*(c + d*x)] + Sin[9*(c + d*x)]))/(
24310*a^8*d*(1 + Sin[c + d*x])^8)

________________________________________________________________________________________

Maple [A]  time = 0.098, size = 280, normalized size = 1.1 \begin{align*} 2\,{\frac{1}{d{a}^{8}} \left ( -{\frac{1}{512\,\tan \left ( 1/2\,dx+c/2 \right ) -512}}-{\frac{128}{17\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{17}}}+64\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{-16}-{\frac{1376}{5\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{15}}}+784\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{-14}-{\frac{21400}{13\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{13}}}+2692\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{-12}-{\frac{38954}{11\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{11}}}+{\frac{19109}{5\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{10}}}-{\frac{6847}{2\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{9}}}+{\frac{10241}{4\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{8}}}-{\frac{12799}{8\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{7}}}+{\frac{13313}{16\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{6}}}-{\frac{57083}{160\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{5}}}+{\frac{7937}{64\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{4}}}-{\frac{4351}{128\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{3}}}+{\frac{1793}{256\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{2}}}-{\frac{511}{512\,\tan \left ( 1/2\,dx+c/2 \right ) +512}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^2/(a+a*sin(d*x+c))^8,x)

[Out]

2/d/a^8*(-1/512/(tan(1/2*d*x+1/2*c)-1)-128/17/(tan(1/2*d*x+1/2*c)+1)^17+64/(tan(1/2*d*x+1/2*c)+1)^16-1376/5/(t
an(1/2*d*x+1/2*c)+1)^15+784/(tan(1/2*d*x+1/2*c)+1)^14-21400/13/(tan(1/2*d*x+1/2*c)+1)^13+2692/(tan(1/2*d*x+1/2
*c)+1)^12-38954/11/(tan(1/2*d*x+1/2*c)+1)^11+19109/5/(tan(1/2*d*x+1/2*c)+1)^10-6847/2/(tan(1/2*d*x+1/2*c)+1)^9
+10241/4/(tan(1/2*d*x+1/2*c)+1)^8-12799/8/(tan(1/2*d*x+1/2*c)+1)^7+13313/16/(tan(1/2*d*x+1/2*c)+1)^6-57083/160
/(tan(1/2*d*x+1/2*c)+1)^5+7937/64/(tan(1/2*d*x+1/2*c)+1)^4-4351/128/(tan(1/2*d*x+1/2*c)+1)^3+1793/256/(tan(1/2
*d*x+1/2*c)+1)^2-511/512/(tan(1/2*d*x+1/2*c)+1))

________________________________________________________________________________________

Maxima [B]  time = 1.15014, size = 999, normalized size = 4.08 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^2/(a+a*sin(d*x+c))^8,x, algorithm="maxima")

[Out]

-2/12155*(18181*sin(d*x + c)/(cos(d*x + c) + 1) + 128384*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 545224*sin(d*x
+ c)^3/(cos(d*x + c) + 1)^3 + 1667360*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 3612364*sin(d*x + c)^5/(cos(d*x +
c) + 1)^5 + 5742464*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 6271096*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 + 392849
6*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 - 850850*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 - 5289856*sin(d*x + c)^10/(
cos(d*x + c) + 1)^10 - 7137416*sin(d*x + c)^11/(cos(d*x + c) + 1)^11 - 5989984*sin(d*x + c)^12/(cos(d*x + c) +
1)^12 - 3607604*sin(d*x + c)^13/(cos(d*x + c) + 1)^13 - 1555840*sin(d*x + c)^14/(cos(d*x + c) + 1)^14 - 48620
0*sin(d*x + c)^15/(cos(d*x + c) + 1)^15 - 97240*sin(d*x + c)^16/(cos(d*x + c) + 1)^16 - 12155*sin(d*x + c)^17/
(cos(d*x + c) + 1)^17 + 1896)/((a^8 + 16*a^8*sin(d*x + c)/(cos(d*x + c) + 1) + 119*a^8*sin(d*x + c)^2/(cos(d*x
+ c) + 1)^2 + 544*a^8*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 1700*a^8*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 38
08*a^8*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 6188*a^8*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 7072*a^8*sin(d*x +
c)^7/(cos(d*x + c) + 1)^7 + 4862*a^8*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 - 4862*a^8*sin(d*x + c)^10/(cos(d*x
+ c) + 1)^10 - 7072*a^8*sin(d*x + c)^11/(cos(d*x + c) + 1)^11 - 6188*a^8*sin(d*x + c)^12/(cos(d*x + c) + 1)^12
- 3808*a^8*sin(d*x + c)^13/(cos(d*x + c) + 1)^13 - 1700*a^8*sin(d*x + c)^14/(cos(d*x + c) + 1)^14 - 544*a^8*s
in(d*x + c)^15/(cos(d*x + c) + 1)^15 - 119*a^8*sin(d*x + c)^16/(cos(d*x + c) + 1)^16 - 16*a^8*sin(d*x + c)^17/
(cos(d*x + c) + 1)^17 - a^8*sin(d*x + c)^18/(cos(d*x + c) + 1)^18)*d)

________________________________________________________________________________________

Fricas [A]  time = 1.87013, size = 624, normalized size = 2.55 \begin{align*} \frac{1024 \, \cos \left (d x + c\right )^{8} - 10752 \, \cos \left (d x + c\right )^{6} + 29568 \, \cos \left (d x + c\right )^{4} - 27456 \, \cos \left (d x + c\right )^{2} +{\left (128 \, \cos \left (d x + c\right )^{8} - 4032 \, \cos \left (d x + c\right )^{6} + 18480 \, \cos \left (d x + c\right )^{4} - 24024 \, \cos \left (d x + c\right )^{2} + 6435\right )} \sin \left (d x + c\right ) + 5720}{12155 \,{\left (a^{8} d \cos \left (d x + c\right )^{9} - 32 \, a^{8} d \cos \left (d x + c\right )^{7} + 160 \, a^{8} d \cos \left (d x + c\right )^{5} - 256 \, a^{8} d \cos \left (d x + c\right )^{3} + 128 \, a^{8} d \cos \left (d x + c\right ) - 8 \,{\left (a^{8} d \cos \left (d x + c\right )^{7} - 10 \, a^{8} d \cos \left (d x + c\right )^{5} + 24 \, a^{8} d \cos \left (d x + c\right )^{3} - 16 \, a^{8} d \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^2/(a+a*sin(d*x+c))^8,x, algorithm="fricas")

[Out]

1/12155*(1024*cos(d*x + c)^8 - 10752*cos(d*x + c)^6 + 29568*cos(d*x + c)^4 - 27456*cos(d*x + c)^2 + (128*cos(d
*x + c)^8 - 4032*cos(d*x + c)^6 + 18480*cos(d*x + c)^4 - 24024*cos(d*x + c)^2 + 6435)*sin(d*x + c) + 5720)/(a^
8*d*cos(d*x + c)^9 - 32*a^8*d*cos(d*x + c)^7 + 160*a^8*d*cos(d*x + c)^5 - 256*a^8*d*cos(d*x + c)^3 + 128*a^8*d
*cos(d*x + c) - 8*(a^8*d*cos(d*x + c)^7 - 10*a^8*d*cos(d*x + c)^5 + 24*a^8*d*cos(d*x + c)^3 - 16*a^8*d*cos(d*x
+ c))*sin(d*x + c))

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)**2/(a+a*sin(d*x+c))**8,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.21859, size = 336, normalized size = 1.37 \begin{align*} -\frac{\frac{12155}{a^{8}{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1\right )}} + \frac{6211205 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{16} + 55791450 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{15} + 303072770 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{14} + 1091397450 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{13} + 2909561798 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{12} + 5901218466 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} + 9405145178 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{10} + 11877161010 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 12017308160 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} + 9710430158 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 6263238566 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 3172666718 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 1247921210 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 365303990 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 77883902 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 10498214 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 982907}{a^{8}{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1\right )}^{17}}}{3111680 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^2/(a+a*sin(d*x+c))^8,x, algorithm="giac")

[Out]

-1/3111680*(12155/(a^8*(tan(1/2*d*x + 1/2*c) - 1)) + (6211205*tan(1/2*d*x + 1/2*c)^16 + 55791450*tan(1/2*d*x +
1/2*c)^15 + 303072770*tan(1/2*d*x + 1/2*c)^14 + 1091397450*tan(1/2*d*x + 1/2*c)^13 + 2909561798*tan(1/2*d*x +
1/2*c)^12 + 5901218466*tan(1/2*d*x + 1/2*c)^11 + 9405145178*tan(1/2*d*x + 1/2*c)^10 + 11877161010*tan(1/2*d*x
+ 1/2*c)^9 + 12017308160*tan(1/2*d*x + 1/2*c)^8 + 9710430158*tan(1/2*d*x + 1/2*c)^7 + 6263238566*tan(1/2*d*x
+ 1/2*c)^6 + 3172666718*tan(1/2*d*x + 1/2*c)^5 + 1247921210*tan(1/2*d*x + 1/2*c)^4 + 365303990*tan(1/2*d*x + 1
/2*c)^3 + 77883902*tan(1/2*d*x + 1/2*c)^2 + 10498214*tan(1/2*d*x + 1/2*c) + 982907)/(a^8*(tan(1/2*d*x + 1/2*c)
+ 1)^17))/d