\(\int \frac {\sec (c+d x) \tan ^8(c+d x)}{a+a \sin (c+d x)} \, dx\) [899]

Optimal result

Integrand size = 27, antiderivative size = 160 \[ \int \frac {\sec (c+d x) \tan ^8(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {7 \text {arctanh}(\sin (c+d x))}{256 a d}+\frac {7 \sec (c+d x) \tan (c+d x)}{256 a d}-\frac {7 \sec ^3(c+d x) \tan (c+d x)}{128 a d}+\frac {7 \sec ^3(c+d x) \tan ^3(c+d x)}{96 a d}-\frac {7 \sec ^3(c+d x) \tan ^5(c+d x)}{80 a d}+\frac {\sec ^3(c+d x) \tan ^7(c+d x)}{10 a d}-\frac {\tan ^{10}(c+d x)}{10 a d} \]

[Out]

7/256*arctanh(sin(d*x+c))/a/d+7/256*sec(d*x+c)*tan(d*x+c)/a/d-7/128*sec(d*x+c)^3*tan(d*x+c)/a/d+7/96*sec(d*x+c
)^3*tan(d*x+c)^3/a/d-7/80*sec(d*x+c)^3*tan(d*x+c)^5/a/d+1/10*sec(d*x+c)^3*tan(d*x+c)^7/a/d-1/10*tan(d*x+c)^10/
a/d

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2914, 2691, 3853, 3855, 2687, 30} \[ \int \frac {\sec (c+d x) \tan ^8(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {7 \text {arctanh}(\sin (c+d x))}{256 a d}-\frac {\tan ^{10}(c+d x)}{10 a d}+\frac {\tan ^7(c+d x) \sec ^3(c+d x)}{10 a d}-\frac {7 \tan ^5(c+d x) \sec ^3(c+d x)}{80 a d}+\frac {7 \tan ^3(c+d x) \sec ^3(c+d x)}{96 a d}-\frac {7 \tan (c+d x) \sec ^3(c+d x)}{128 a d}+\frac {7 \tan (c+d x) \sec (c+d x)}{256 a d} \]

[In]

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

[Out]

(7*ArcTanh[Sin[c + d*x]])/(256*a*d) + (7*Sec[c + d*x]*Tan[c + d*x])/(256*a*d) - (7*Sec[c + d*x]^3*Tan[c + d*x]
)/(128*a*d) + (7*Sec[c + d*x]^3*Tan[c + d*x]^3)/(96*a*d) - (7*Sec[c + d*x]^3*Tan[c + d*x]^5)/(80*a*d) + (Sec[c
+ d*x]^3*Tan[c + d*x]^7)/(10*a*d) - Tan[c + d*x]^10/(10*a*d)

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2687

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[1/f, Subst[Int[(b*x)
^n*(1 + x^2)^(m/2 - 1), x], x, Tan[e + f*x]], x] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2] &&  !(IntegerQ[(n
- 1)/2] && LtQ[0, n, m - 1])

Rule 2691

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b*(a*Sec[e +
f*x])^m*((b*Tan[e + f*x])^(n - 1)/(f*(m + n - 1))), x] - Dist[b^2*((n - 1)/(m + n - 1)), Int[(a*Sec[e + f*x])
^m*(b*Tan[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && GtQ[n, 1] && NeQ[m + n - 1, 0] && Integers
Q[2*m, 2*n]

Rule 2914

Int[(cos[(e_.) + (f_.)*(x_)]^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.))/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]
), x_Symbol] :> Dist[1/a, Int[Cos[e + f*x]^(p - 2)*(d*Sin[e + f*x])^n, x], x] - Dist[1/(b*d), Int[Cos[e + f*x]
^(p - 2)*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n, p}, x] && IntegerQ[(p - 1)/2] && EqQ[a^2
- b^2, 0] && IntegerQ[n] && (LtQ[0, n, (p + 1)/2] || (LeQ[p, -n] && LtQ[-n, 2*p - 3]) || (GtQ[n, 0] && LeQ[n,
-p]))

Rule 3853

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Csc[c + d*x])^(n - 1)/(d*(n
- 1))), x] + Dist[b^2*((n - 2)/(n - 1)), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n,
1] && IntegerQ[2*n]

Rule 3855

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps \begin{align*} \text {integral}& = \frac {\int \sec ^3(c+d x) \tan ^8(c+d x) \, dx}{a}-\frac {\int \sec ^2(c+d x) \tan ^9(c+d x) \, dx}{a} \\ & = \frac {\sec ^3(c+d x) \tan ^7(c+d x)}{10 a d}-\frac {7 \int \sec ^3(c+d x) \tan ^6(c+d x) \, dx}{10 a}-\frac {\text {Subst}\left (\int x^9 \, dx,x,\tan (c+d x)\right )}{a d} \\ & = -\frac {7 \sec ^3(c+d x) \tan ^5(c+d x)}{80 a d}+\frac {\sec ^3(c+d x) \tan ^7(c+d x)}{10 a d}-\frac {\tan ^{10}(c+d x)}{10 a d}+\frac {7 \int \sec ^3(c+d x) \tan ^4(c+d x) \, dx}{16 a} \\ & = \frac {7 \sec ^3(c+d x) \tan ^3(c+d x)}{96 a d}-\frac {7 \sec ^3(c+d x) \tan ^5(c+d x)}{80 a d}+\frac {\sec ^3(c+d x) \tan ^7(c+d x)}{10 a d}-\frac {\tan ^{10}(c+d x)}{10 a d}-\frac {7 \int \sec ^3(c+d x) \tan ^2(c+d x) \, dx}{32 a} \\ & = -\frac {7 \sec ^3(c+d x) \tan (c+d x)}{128 a d}+\frac {7 \sec ^3(c+d x) \tan ^3(c+d x)}{96 a d}-\frac {7 \sec ^3(c+d x) \tan ^5(c+d x)}{80 a d}+\frac {\sec ^3(c+d x) \tan ^7(c+d x)}{10 a d}-\frac {\tan ^{10}(c+d x)}{10 a d}+\frac {7 \int \sec ^3(c+d x) \, dx}{128 a} \\ & = \frac {7 \sec (c+d x) \tan (c+d x)}{256 a d}-\frac {7 \sec ^3(c+d x) \tan (c+d x)}{128 a d}+\frac {7 \sec ^3(c+d x) \tan ^3(c+d x)}{96 a d}-\frac {7 \sec ^3(c+d x) \tan ^5(c+d x)}{80 a d}+\frac {\sec ^3(c+d x) \tan ^7(c+d x)}{10 a d}-\frac {\tan ^{10}(c+d x)}{10 a d}+\frac {7 \int \sec (c+d x) \, dx}{256 a} \\ & = \frac {7 \text {arctanh}(\sin (c+d x))}{256 a d}+\frac {7 \sec (c+d x) \tan (c+d x)}{256 a d}-\frac {7 \sec ^3(c+d x) \tan (c+d x)}{128 a d}+\frac {7 \sec ^3(c+d x) \tan ^3(c+d x)}{96 a d}-\frac {7 \sec ^3(c+d x) \tan ^5(c+d x)}{80 a d}+\frac {\sec ^3(c+d x) \tan ^7(c+d x)}{10 a d}-\frac {\tan ^{10}(c+d x)}{10 a d} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.58 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.76 \[ \int \frac {\sec (c+d x) \tan ^8(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {210 \text {arctanh}(\sin (c+d x))+\frac {-768-978 \sin (c+d x)+2862 \sin ^2(c+d x)+3842 \sin ^3(c+d x)-3838 \sin ^4(c+d x)-5630 \sin ^5(c+d x)+2050 \sin ^6(c+d x)+3630 \sin ^7(c+d x)-210 \sin ^8(c+d x)}{(-1+\sin (c+d x))^4 (1+\sin (c+d x))^5}}{7680 a d} \]

[In]

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

[Out]

(210*ArcTanh[Sin[c + d*x]] + (-768 - 978*Sin[c + d*x] + 2862*Sin[c + d*x]^2 + 3842*Sin[c + d*x]^3 - 3838*Sin[c
+ d*x]^4 - 5630*Sin[c + d*x]^5 + 2050*Sin[c + d*x]^6 + 3630*Sin[c + d*x]^7 - 210*Sin[c + d*x]^8)/((-1 + Sin[c
+ d*x])^4*(1 + Sin[c + d*x])^5))/(7680*a*d)

Maple [A] (verified)

Time = 1.74 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.87

method	result	size
derivativedivides	\(\frac {\frac {1}{256 \left (\sin \left (d x +c \right )-1\right )^{4}}+\frac {5}{192 \left (\sin \left (d x +c \right )-1\right )^{3}}+\frac {37}{512 \left (\sin \left (d x +c \right )-1\right )^{2}}+\frac {7}{64 \left (\sin \left (d x +c \right )-1\right )}-\frac {7 \ln \left (\sin \left (d x +c \right )-1\right )}{512}-\frac {1}{160 \left (1+\sin \left (d x +c \right )\right )^{5}}+\frac {11}{256 \left (1+\sin \left (d x +c \right )\right )^{4}}-\frac {47}{384 \left (1+\sin \left (d x +c \right )\right )^{3}}+\frac {93}{512 \left (1+\sin \left (d x +c \right )\right )^{2}}-\frac {35}{256 \left (1+\sin \left (d x +c \right )\right )}+\frac {7 \ln \left (1+\sin \left (d x +c \right )\right )}{512}}{d a}\)	\(139\)
default	\(\frac {\frac {1}{256 \left (\sin \left (d x +c \right )-1\right )^{4}}+\frac {5}{192 \left (\sin \left (d x +c \right )-1\right )^{3}}+\frac {37}{512 \left (\sin \left (d x +c \right )-1\right )^{2}}+\frac {7}{64 \left (\sin \left (d x +c \right )-1\right )}-\frac {7 \ln \left (\sin \left (d x +c \right )-1\right )}{512}-\frac {1}{160 \left (1+\sin \left (d x +c \right )\right )^{5}}+\frac {11}{256 \left (1+\sin \left (d x +c \right )\right )^{4}}-\frac {47}{384 \left (1+\sin \left (d x +c \right )\right )^{3}}+\frac {93}{512 \left (1+\sin \left (d x +c \right )\right )^{2}}-\frac {35}{256 \left (1+\sin \left (d x +c \right )\right )}+\frac {7 \ln \left (1+\sin \left (d x +c \right )\right )}{512}}{d a}\)	\(139\)
risch	\(-\frac {i \left (2890 i {\mathrm e}^{14 i \left (d x +c \right )}+105 \,{\mathrm e}^{17 i \left (d x +c \right )}-3630 i {\mathrm e}^{16 i \left (d x +c \right )}+3260 \,{\mathrm e}^{15 i \left (d x +c \right )}-23674 i {\mathrm e}^{8 i \left (d x +c \right )}+9044 \,{\mathrm e}^{13 i \left (d x +c \right )}+25102 i {\mathrm e}^{6 i \left (d x +c \right )}+24388 \,{\mathrm e}^{11 i \left (d x +c \right )}-25102 i {\mathrm e}^{12 i \left (d x +c \right )}+24710 \,{\mathrm e}^{9 i \left (d x +c \right )}+23674 i {\mathrm e}^{10 i \left (d x +c \right )}+24388 \,{\mathrm e}^{7 i \left (d x +c \right )}-2890 i {\mathrm e}^{4 i \left (d x +c \right )}+9044 \,{\mathrm e}^{5 i \left (d x +c \right )}+3630 i {\mathrm e}^{2 i \left (d x +c \right )}+3260 \,{\mathrm e}^{3 i \left (d x +c \right )}+105 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{1920 \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{10} \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{8} d a}-\frac {7 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{256 d a}+\frac {7 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{256 a d}\)	\(277\)
parallelrisch	\(\frac {\left (-105 \cos \left (10 d x +10 c \right )-22050 \cos \left (2 d x +2 c \right )-12600 \cos \left (4 d x +4 c \right )-4725 \cos \left (6 d x +6 c \right )-1050 \cos \left (8 d x +8 c \right )-13230\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+\left (105 \cos \left (10 d x +10 c \right )+22050 \cos \left (2 d x +2 c \right )+12600 \cos \left (4 d x +4 c \right )+4725 \cos \left (6 d x +6 c \right )+1050 \cos \left (8 d x +8 c \right )+13230\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-69720 \sin \left (3 d x +3 c \right )+23128 \sin \left (5 d x +5 c \right )-8210 \sin \left (7 d x +7 c \right )+210 \sin \left (9 d x +9 c \right )+384 \cos \left (10 d x +10 c \right )+80640 \cos \left (2 d x +2 c \right )-46080 \cos \left (4 d x +4 c \right )+17280 \cos \left (6 d x +6 c \right )-3840 \cos \left (8 d x +8 c \right )+95340 \sin \left (d x +c \right )-48384}{3840 a d \left (\cos \left (10 d x +10 c \right )+10 \cos \left (8 d x +8 c \right )+45 \cos \left (6 d x +6 c \right )+120 \cos \left (4 d x +4 c \right )+210 \cos \left (2 d x +2 c \right )+126\right )}\)	\(315\)

[In]

int(sec(d*x+c)^9*sin(d*x+c)^8/(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

1/d/a*(1/256/(sin(d*x+c)-1)^4+5/192/(sin(d*x+c)-1)^3+37/512/(sin(d*x+c)-1)^2+7/64/(sin(d*x+c)-1)-7/512*ln(sin(
d*x+c)-1)-1/160/(1+sin(d*x+c))^5+11/256/(1+sin(d*x+c))^4-47/384/(1+sin(d*x+c))^3+93/512/(1+sin(d*x+c))^2-35/25
6/(1+sin(d*x+c))+7/512*ln(1+sin(d*x+c)))

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.17 \[ \int \frac {\sec (c+d x) \tan ^8(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {210 \, \cos \left (d x + c\right )^{8} + 1210 \, \cos \left (d x + c\right )^{6} - 1052 \, \cos \left (d x + c\right )^{4} + 496 \, \cos \left (d x + c\right )^{2} - 105 \, {\left (\cos \left (d x + c\right )^{8} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{8}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + 105 \, {\left (\cos \left (d x + c\right )^{8} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{8}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (1815 \, \cos \left (d x + c\right )^{6} - 2630 \, \cos \left (d x + c\right )^{4} + 1736 \, \cos \left (d x + c\right )^{2} - 432\right )} \sin \left (d x + c\right ) - 96}{7680 \, {\left (a d \cos \left (d x + c\right )^{8} \sin \left (d x + c\right ) + a d \cos \left (d x + c\right )^{8}\right )}} \]

[In]

integrate(sec(d*x+c)^9*sin(d*x+c)^8/(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

-1/7680*(210*cos(d*x + c)^8 + 1210*cos(d*x + c)^6 - 1052*cos(d*x + c)^4 + 496*cos(d*x + c)^2 - 105*(cos(d*x +
c)^8*sin(d*x + c) + cos(d*x + c)^8)*log(sin(d*x + c) + 1) + 105*(cos(d*x + c)^8*sin(d*x + c) + cos(d*x + c)^8)
*log(-sin(d*x + c) + 1) + 2*(1815*cos(d*x + c)^6 - 2630*cos(d*x + c)^4 + 1736*cos(d*x + c)^2 - 432)*sin(d*x +
c) - 96)/(a*d*cos(d*x + c)^8*sin(d*x + c) + a*d*cos(d*x + c)^8)

Sympy [F(-1)]

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

[In]

integrate(sec(d*x+c)**9*sin(d*x+c)**8/(a+a*sin(d*x+c)),x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.34 \[ \int \frac {\sec (c+d x) \tan ^8(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\frac {2 \, {\left (105 \, \sin \left (d x + c\right )^{8} - 1815 \, \sin \left (d x + c\right )^{7} - 1025 \, \sin \left (d x + c\right )^{6} + 2815 \, \sin \left (d x + c\right )^{5} + 1919 \, \sin \left (d x + c\right )^{4} - 1921 \, \sin \left (d x + c\right )^{3} - 1431 \, \sin \left (d x + c\right )^{2} + 489 \, \sin \left (d x + c\right ) + 384\right )}}{a \sin \left (d x + c\right )^{9} + a \sin \left (d x + c\right )^{8} - 4 \, a \sin \left (d x + c\right )^{7} - 4 \, a \sin \left (d x + c\right )^{6} + 6 \, a \sin \left (d x + c\right )^{5} + 6 \, a \sin \left (d x + c\right )^{4} - 4 \, a \sin \left (d x + c\right )^{3} - 4 \, a \sin \left (d x + c\right )^{2} + a \sin \left (d x + c\right ) + a} - \frac {105 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a} + \frac {105 \, \log \left (\sin \left (d x + c\right ) - 1\right )}{a}}{7680 \, d} \]

[In]

integrate(sec(d*x+c)^9*sin(d*x+c)^8/(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

-1/7680*(2*(105*sin(d*x + c)^8 - 1815*sin(d*x + c)^7 - 1025*sin(d*x + c)^6 + 2815*sin(d*x + c)^5 + 1919*sin(d*
x + c)^4 - 1921*sin(d*x + c)^3 - 1431*sin(d*x + c)^2 + 489*sin(d*x + c) + 384)/(a*sin(d*x + c)^9 + a*sin(d*x +
c)^8 - 4*a*sin(d*x + c)^7 - 4*a*sin(d*x + c)^6 + 6*a*sin(d*x + c)^5 + 6*a*sin(d*x + c)^4 - 4*a*sin(d*x + c)^3
- 4*a*sin(d*x + c)^2 + a*sin(d*x + c) + a) - 105*log(sin(d*x + c) + 1)/a + 105*log(sin(d*x + c) - 1)/a)/d

Giac [A] (verification not implemented)

none

Time = 0.44 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.98 \[ \int \frac {\sec (c+d x) \tan ^8(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {420 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a} - \frac {420 \, \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a} + \frac {5 \, {\left (175 \, \sin \left (d x + c\right )^{4} - 28 \, \sin \left (d x + c\right )^{3} - 522 \, \sin \left (d x + c\right )^{2} + 588 \, \sin \left (d x + c\right ) - 189\right )}}{a {\left (\sin \left (d x + c\right ) - 1\right )}^{4}} - \frac {959 \, \sin \left (d x + c\right )^{5} + 8995 \, \sin \left (d x + c\right )^{4} + 20810 \, \sin \left (d x + c\right )^{3} + 21810 \, \sin \left (d x + c\right )^{2} + 11055 \, \sin \left (d x + c\right ) + 2211}{a {\left (\sin \left (d x + c\right ) + 1\right )}^{5}}}{30720 \, d} \]

[In]

integrate(sec(d*x+c)^9*sin(d*x+c)^8/(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

1/30720*(420*log(abs(sin(d*x + c) + 1))/a - 420*log(abs(sin(d*x + c) - 1))/a + 5*(175*sin(d*x + c)^4 - 28*sin(
d*x + c)^3 - 522*sin(d*x + c)^2 + 588*sin(d*x + c) - 189)/(a*(sin(d*x + c) - 1)^4) - (959*sin(d*x + c)^5 + 899
5*sin(d*x + c)^4 + 20810*sin(d*x + c)^3 + 21810*sin(d*x + c)^2 + 11055*sin(d*x + c) + 2211)/(a*(sin(d*x + c) +
1)^5))/d

Mupad [B] (verification not implemented)

Time = 17.26 (sec) , antiderivative size = 496, normalized size of antiderivative = 3.10 \[ \int \frac {\sec (c+d x) \tan ^8(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {7\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{128\,a\,d}+\frac {-\frac {7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}}{128}-\frac {7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}}{64}+\frac {35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{96}+\frac {161\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}}{192}-\frac {469\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{480}-\frac {2681\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{960}+\frac {593\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{480}+\frac {5053\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{960}+\frac {10841\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{192}+\frac {5053\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{960}+\frac {593\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{480}-\frac {2681\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{960}-\frac {469\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{480}+\frac {161\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{192}+\frac {35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{96}-\frac {7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{64}-\frac {7\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{128}}{d\,\left (a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{18}+2\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}-7\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}-16\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}+20\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+56\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}-28\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-112\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+14\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+140\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+14\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-112\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-28\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+56\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+20\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-16\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-7\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+2\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+a\right )} \]

[In]

int(sin(c + d*x)^8/(cos(c + d*x)^9*(a + a*sin(c + d*x))),x)

[Out]

(7*atanh(tan(c/2 + (d*x)/2)))/(128*a*d) + ((35*tan(c/2 + (d*x)/2)^3)/96 - (7*tan(c/2 + (d*x)/2)^2)/64 - (7*tan
(c/2 + (d*x)/2))/128 + (161*tan(c/2 + (d*x)/2)^4)/192 - (469*tan(c/2 + (d*x)/2)^5)/480 - (2681*tan(c/2 + (d*x)
/2)^6)/960 + (593*tan(c/2 + (d*x)/2)^7)/480 + (5053*tan(c/2 + (d*x)/2)^8)/960 + (10841*tan(c/2 + (d*x)/2)^9)/1
92 + (5053*tan(c/2 + (d*x)/2)^10)/960 + (593*tan(c/2 + (d*x)/2)^11)/480 - (2681*tan(c/2 + (d*x)/2)^12)/960 - (
469*tan(c/2 + (d*x)/2)^13)/480 + (161*tan(c/2 + (d*x)/2)^14)/192 + (35*tan(c/2 + (d*x)/2)^15)/96 - (7*tan(c/2
+ (d*x)/2)^16)/64 - (7*tan(c/2 + (d*x)/2)^17)/128)/(d*(a + 2*a*tan(c/2 + (d*x)/2) - 7*a*tan(c/2 + (d*x)/2)^2 -
16*a*tan(c/2 + (d*x)/2)^3 + 20*a*tan(c/2 + (d*x)/2)^4 + 56*a*tan(c/2 + (d*x)/2)^5 - 28*a*tan(c/2 + (d*x)/2)^6
- 112*a*tan(c/2 + (d*x)/2)^7 + 14*a*tan(c/2 + (d*x)/2)^8 + 140*a*tan(c/2 + (d*x)/2)^9 + 14*a*tan(c/2 + (d*x)/
2)^10 - 112*a*tan(c/2 + (d*x)/2)^11 - 28*a*tan(c/2 + (d*x)/2)^12 + 56*a*tan(c/2 + (d*x)/2)^13 + 20*a*tan(c/2 +
(d*x)/2)^14 - 16*a*tan(c/2 + (d*x)/2)^15 - 7*a*tan(c/2 + (d*x)/2)^16 + 2*a*tan(c/2 + (d*x)/2)^17 + a*tan(c/2
+ (d*x)/2)^18))