\(\int \frac {\sec ^6(c+d x) \tan (c+d x)}{a+a \sin (c+d x)} \, dx\) [888]

Optimal result

Integrand size = 27, antiderivative size = 130 \[ \int \frac {\sec ^6(c+d x) \tan (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {5 \text {arctanh}(\sin (c+d x))}{128 a d}+\frac {\sec ^8(c+d x)}{8 a d}+\frac {5 \sec (c+d x) \tan (c+d x)}{128 a d}+\frac {5 \sec ^3(c+d x) \tan (c+d x)}{192 a d}+\frac {\sec ^5(c+d x) \tan (c+d x)}{48 a d}-\frac {\sec ^7(c+d x) \tan (c+d x)}{8 a d} \]

[Out]

5/128*arctanh(sin(d*x+c))/a/d+1/8*sec(d*x+c)^8/a/d+5/128*sec(d*x+c)*tan(d*x+c)/a/d+5/192*sec(d*x+c)^3*tan(d*x+
c)/a/d+1/48*sec(d*x+c)^5*tan(d*x+c)/a/d-1/8*sec(d*x+c)^7*tan(d*x+c)/a/d

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2914, 2686, 30, 2691, 3853, 3855} \[ \int \frac {\sec ^6(c+d x) \tan (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {5 \text {arctanh}(\sin (c+d x))}{128 a d}+\frac {\sec ^8(c+d x)}{8 a d}-\frac {\tan (c+d x) \sec ^7(c+d x)}{8 a d}+\frac {\tan (c+d x) \sec ^5(c+d x)}{48 a d}+\frac {5 \tan (c+d x) \sec ^3(c+d x)}{192 a d}+\frac {5 \tan (c+d x) \sec (c+d x)}{128 a d} \]

[In]

Int[(Sec[c + d*x]^6*Tan[c + d*x])/(a + a*Sin[c + d*x]),x]

[Out]

(5*ArcTanh[Sin[c + d*x]])/(128*a*d) + Sec[c + d*x]^8/(8*a*d) + (5*Sec[c + d*x]*Tan[c + d*x])/(128*a*d) + (5*Se
c[c + d*x]^3*Tan[c + d*x])/(192*a*d) + (Sec[c + d*x]^5*Tan[c + d*x])/(48*a*d) - (Sec[c + d*x]^7*Tan[c + d*x])/
(8*a*d)

Rule 30

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

Rule 2686

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a/f, Subst[
Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n -
1)/2] &&  !(IntegerQ[m/2] && LtQ[0, m, n + 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 ^8(c+d x) \tan (c+d x) \, dx}{a}-\frac {\int \sec ^7(c+d x) \tan ^2(c+d x) \, dx}{a} \\ & = -\frac {\sec ^7(c+d x) \tan (c+d x)}{8 a d}+\frac {\int \sec ^7(c+d x) \, dx}{8 a}+\frac {\text {Subst}\left (\int x^7 \, dx,x,\sec (c+d x)\right )}{a d} \\ & = \frac {\sec ^8(c+d x)}{8 a d}+\frac {\sec ^5(c+d x) \tan (c+d x)}{48 a d}-\frac {\sec ^7(c+d x) \tan (c+d x)}{8 a d}+\frac {5 \int \sec ^5(c+d x) \, dx}{48 a} \\ & = \frac {\sec ^8(c+d x)}{8 a d}+\frac {5 \sec ^3(c+d x) \tan (c+d x)}{192 a d}+\frac {\sec ^5(c+d x) \tan (c+d x)}{48 a d}-\frac {\sec ^7(c+d x) \tan (c+d x)}{8 a d}+\frac {5 \int \sec ^3(c+d x) \, dx}{64 a} \\ & = \frac {\sec ^8(c+d x)}{8 a d}+\frac {5 \sec (c+d x) \tan (c+d x)}{128 a d}+\frac {5 \sec ^3(c+d x) \tan (c+d x)}{192 a d}+\frac {\sec ^5(c+d x) \tan (c+d x)}{48 a d}-\frac {\sec ^7(c+d x) \tan (c+d x)}{8 a d}+\frac {5 \int \sec (c+d x) \, dx}{128 a} \\ & = \frac {5 \text {arctanh}(\sin (c+d x))}{128 a d}+\frac {\sec ^8(c+d x)}{8 a d}+\frac {5 \sec (c+d x) \tan (c+d x)}{128 a d}+\frac {5 \sec ^3(c+d x) \tan (c+d x)}{192 a d}+\frac {\sec ^5(c+d x) \tan (c+d x)}{48 a d}-\frac {\sec ^7(c+d x) \tan (c+d x)}{8 a d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.74 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.71 \[ \int \frac {\sec ^6(c+d x) \tan (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {15 \text {arctanh}(\sin (c+d x))-\frac {4}{(-1+\sin (c+d x))^3}+\frac {9}{(-1+\sin (c+d x))^2}-\frac {15}{-1+\sin (c+d x)}+\frac {6}{(1+\sin (c+d x))^4}+\frac {8}{(1+\sin (c+d x))^3}+\frac {6}{(1+\sin (c+d x))^2}}{384 a d} \]

[In]

Integrate[(Sec[c + d*x]^6*Tan[c + d*x])/(a + a*Sin[c + d*x]),x]

[Out]

(15*ArcTanh[Sin[c + d*x]] - 4/(-1 + Sin[c + d*x])^3 + 9/(-1 + Sin[c + d*x])^2 - 15/(-1 + Sin[c + d*x]) + 6/(1
+ Sin[c + d*x])^4 + 8/(1 + Sin[c + d*x])^3 + 6/(1 + Sin[c + d*x])^2)/(384*a*d)

Maple [A] (verified)

Time = 0.97 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.79

method	result	size
derivativedivides	\(\frac {-\frac {1}{96 \left (\sin \left (d x +c \right )-1\right )^{3}}+\frac {3}{128 \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {5}{128 \left (\sin \left (d x +c \right )-1\right )}-\frac {5 \ln \left (\sin \left (d x +c \right )-1\right )}{256}+\frac {1}{64 \left (1+\sin \left (d x +c \right )\right )^{4}}+\frac {1}{48 \left (1+\sin \left (d x +c \right )\right )^{3}}+\frac {1}{64 \left (1+\sin \left (d x +c \right )\right )^{2}}+\frac {5 \ln \left (1+\sin \left (d x +c \right )\right )}{256}}{d a}\)	\(103\)
default	\(\frac {-\frac {1}{96 \left (\sin \left (d x +c \right )-1\right )^{3}}+\frac {3}{128 \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {5}{128 \left (\sin \left (d x +c \right )-1\right )}-\frac {5 \ln \left (\sin \left (d x +c \right )-1\right )}{256}+\frac {1}{64 \left (1+\sin \left (d x +c \right )\right )^{4}}+\frac {1}{48 \left (1+\sin \left (d x +c \right )\right )^{3}}+\frac {1}{64 \left (1+\sin \left (d x +c \right )\right )^{2}}+\frac {5 \ln \left (1+\sin \left (d x +c \right )\right )}{256}}{d a}\)	\(103\)
risch	\(-\frac {i \left (113 \,{\mathrm e}^{5 i \left (d x +c \right )}-30 i {\mathrm e}^{2 i \left (d x +c \right )}+170 i {\mathrm e}^{10 i \left (d x +c \right )}+70 \,{\mathrm e}^{3 i \left (d x +c \right )}+15 \,{\mathrm e}^{i \left (d x +c \right )}+15 \,{\mathrm e}^{13 i \left (d x +c \right )}+113 \,{\mathrm e}^{9 i \left (d x +c \right )}+70 \,{\mathrm e}^{11 i \left (d x +c \right )}+396 i {\mathrm e}^{8 i \left (d x +c \right )}-396 i {\mathrm e}^{6 i \left (d x +c \right )}-170 i {\mathrm e}^{4 i \left (d x +c \right )}-3468 \,{\mathrm e}^{7 i \left (d x +c \right )}+30 i {\mathrm e}^{12 i \left (d x +c \right )}\right )}{192 \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{8} \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{6} d a}+\frac {5 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{128 a d}-\frac {5 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{128 d a}\)	\(231\)
norman	\(\frac {\frac {95 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d a}+\frac {95 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d a}+\frac {95 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d a}-\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{64 d a}-\frac {5 \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d a}+\frac {59 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d a}+\frac {59 \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d a}+\frac {149 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{96 d a}+\frac {149 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{96 d a}+\frac {163 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{96 d a}+\frac {163 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{96 d a}-\frac {625 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d a}-\frac {625 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d a}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{6} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{8}}-\frac {5 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{128 a d}+\frac {5 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{128 a d}\)	\(315\)
parallelrisch	\(\frac {\left (-75 \sin \left (5 d x +5 c \right )-15 \sin \left (7 d x +7 c \right )-450 \cos \left (2 d x +2 c \right )-180 \cos \left (4 d x +4 c \right )-30 \cos \left (6 d x +6 c \right )-75 \sin \left (d x +c \right )-135 \sin \left (3 d x +3 c \right )-300\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+\left (75 \sin \left (5 d x +5 c \right )+15 \sin \left (7 d x +7 c \right )+450 \cos \left (2 d x +2 c \right )+180 \cos \left (4 d x +4 c \right )+30 \cos \left (6 d x +6 c \right )+75 \sin \left (d x +c \right )+135 \sin \left (3 d x +3 c \right )+300\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-105 \sin \left (5 d x +5 c \right )-33 \sin \left (7 d x +7 c \right )-1216 \cos \left (2 d x +2 c \right )-536 \cos \left (4 d x +4 c \right )-96 \cos \left (6 d x +6 c \right )+627 \sin \left (d x +c \right )+43 \sin \left (3 d x +3 c \right )+2808}{384 a d \left (20+\sin \left (7 d x +7 c \right )+5 \sin \left (5 d x +5 c \right )+9 \sin \left (3 d x +3 c \right )+5 \sin \left (d x +c \right )+2 \cos \left (6 d x +6 c \right )+12 \cos \left (4 d x +4 c \right )+30 \cos \left (2 d x +2 c \right )\right )}\)	\(339\)

[In]

int(sec(d*x+c)^7*sin(d*x+c)/(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

1/d/a*(-1/96/(sin(d*x+c)-1)^3+3/128/(sin(d*x+c)-1)^2-5/128/(sin(d*x+c)-1)-5/256*ln(sin(d*x+c)-1)+1/64/(1+sin(d
*x+c))^4+1/48/(1+sin(d*x+c))^3+1/64/(1+sin(d*x+c))^2+5/256*ln(1+sin(d*x+c)))

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.28 \[ \int \frac {\sec ^6(c+d x) \tan (c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {30 \, \cos \left (d x + c\right )^{6} - 10 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} - 15 \, {\left (\cos \left (d x + c\right )^{6} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{6}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, {\left (\cos \left (d x + c\right )^{6} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{6}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (15 \, \cos \left (d x + c\right )^{4} + 10 \, \cos \left (d x + c\right )^{2} + 8\right )} \sin \left (d x + c\right ) - 112}{768 \, {\left (a d \cos \left (d x + c\right )^{6} \sin \left (d x + c\right ) + a d \cos \left (d x + c\right )^{6}\right )}} \]

[In]

integrate(sec(d*x+c)^7*sin(d*x+c)/(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

-1/768*(30*cos(d*x + c)^6 - 10*cos(d*x + c)^4 - 4*cos(d*x + c)^2 - 15*(cos(d*x + c)^6*sin(d*x + c) + cos(d*x +
c)^6)*log(sin(d*x + c) + 1) + 15*(cos(d*x + c)^6*sin(d*x + c) + cos(d*x + c)^6)*log(-sin(d*x + c) + 1) - 2*(1
5*cos(d*x + c)^4 + 10*cos(d*x + c)^2 + 8)*sin(d*x + c) - 112)/(a*d*cos(d*x + c)^6*sin(d*x + c) + a*d*cos(d*x +
c)^6)

Sympy [F]

\[ \int \frac {\sec ^6(c+d x) \tan (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\int \frac {\sin {\left (c + d x \right )} \sec ^{7}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx}{a} \]

[In]

integrate(sec(d*x+c)**7*sin(d*x+c)/(a+a*sin(d*x+c)),x)

[Out]

Integral(sin(c + d*x)*sec(c + d*x)**7/(sin(c + d*x) + 1), x)/a

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.35 \[ \int \frac {\sec ^6(c+d x) \tan (c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\frac {2 \, {\left (15 \, \sin \left (d x + c\right )^{6} + 15 \, \sin \left (d x + c\right )^{5} - 40 \, \sin \left (d x + c\right )^{4} - 40 \, \sin \left (d x + c\right )^{3} + 33 \, \sin \left (d x + c\right )^{2} + 33 \, \sin \left (d x + c\right ) + 48\right )}}{a \sin \left (d x + c\right )^{7} + a \sin \left (d x + c\right )^{6} - 3 \, a \sin \left (d x + c\right )^{5} - 3 \, a \sin \left (d x + c\right )^{4} + 3 \, a \sin \left (d x + c\right )^{3} + 3 \, a \sin \left (d x + c\right )^{2} - a \sin \left (d x + c\right ) - a} - \frac {15 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a} + \frac {15 \, \log \left (\sin \left (d x + c\right ) - 1\right )}{a}}{768 \, d} \]

[In]

integrate(sec(d*x+c)^7*sin(d*x+c)/(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

-1/768*(2*(15*sin(d*x + c)^6 + 15*sin(d*x + c)^5 - 40*sin(d*x + c)^4 - 40*sin(d*x + c)^3 + 33*sin(d*x + c)^2 +
33*sin(d*x + c) + 48)/(a*sin(d*x + c)^7 + a*sin(d*x + c)^6 - 3*a*sin(d*x + c)^5 - 3*a*sin(d*x + c)^4 + 3*a*si
n(d*x + c)^3 + 3*a*sin(d*x + c)^2 - a*sin(d*x + c) - a) - 15*log(sin(d*x + c) + 1)/a + 15*log(sin(d*x + c) - 1
)/a)/d

Giac [A] (verification not implemented)

none

Time = 0.45 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.05 \[ \int \frac {\sec ^6(c+d x) \tan (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {60 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a} - \frac {60 \, \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a} + \frac {2 \, {\left (55 \, \sin \left (d x + c\right )^{3} - 225 \, \sin \left (d x + c\right )^{2} + 321 \, \sin \left (d x + c\right ) - 167\right )}}{a {\left (\sin \left (d x + c\right ) - 1\right )}^{3}} - \frac {125 \, \sin \left (d x + c\right )^{4} + 500 \, \sin \left (d x + c\right )^{3} + 702 \, \sin \left (d x + c\right )^{2} + 340 \, \sin \left (d x + c\right ) - 35}{a {\left (\sin \left (d x + c\right ) + 1\right )}^{4}}}{3072 \, d} \]

[In]

integrate(sec(d*x+c)^7*sin(d*x+c)/(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

1/3072*(60*log(abs(sin(d*x + c) + 1))/a - 60*log(abs(sin(d*x + c) - 1))/a + 2*(55*sin(d*x + c)^3 - 225*sin(d*x
+ c)^2 + 321*sin(d*x + c) - 167)/(a*(sin(d*x + c) - 1)^3) - (125*sin(d*x + c)^4 + 500*sin(d*x + c)^3 + 702*si
n(d*x + c)^2 + 340*sin(d*x + c) - 35)/(a*(sin(d*x + c) + 1)^4))/d

Mupad [B] (verification not implemented)

Time = 19.51 (sec) , antiderivative size = 388, normalized size of antiderivative = 2.98 \[ \int \frac {\sec ^6(c+d x) \tan (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {5\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{64\,a\,d}+\frac {-\frac {5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{64}+\frac {59\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{32}+\frac {163\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{96}+\frac {149\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{96}-\frac {625\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{192}+\frac {95\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{16}+\frac {95\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{16}+\frac {95\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{16}-\frac {625\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{192}+\frac {149\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{96}+\frac {163\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{96}+\frac {59\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{32}-\frac {5\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64}}{d\,\left (a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+2\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}-5\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-12\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+9\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+30\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9-5\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-40\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-5\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+30\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+9\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-12\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-5\,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)/(cos(c + d*x)^7*(a + a*sin(c + d*x))),x)

[Out]

(5*atanh(tan(c/2 + (d*x)/2)))/(64*a*d) + ((59*tan(c/2 + (d*x)/2)^2)/32 - (5*tan(c/2 + (d*x)/2))/64 + (163*tan(
c/2 + (d*x)/2)^3)/96 + (149*tan(c/2 + (d*x)/2)^4)/96 - (625*tan(c/2 + (d*x)/2)^5)/192 + (95*tan(c/2 + (d*x)/2)
^6)/16 + (95*tan(c/2 + (d*x)/2)^7)/16 + (95*tan(c/2 + (d*x)/2)^8)/16 - (625*tan(c/2 + (d*x)/2)^9)/192 + (149*t
an(c/2 + (d*x)/2)^10)/96 + (163*tan(c/2 + (d*x)/2)^11)/96 + (59*tan(c/2 + (d*x)/2)^12)/32 - (5*tan(c/2 + (d*x)
/2)^13)/64)/(d*(a + 2*a*tan(c/2 + (d*x)/2) - 5*a*tan(c/2 + (d*x)/2)^2 - 12*a*tan(c/2 + (d*x)/2)^3 + 9*a*tan(c/
2 + (d*x)/2)^4 + 30*a*tan(c/2 + (d*x)/2)^5 - 5*a*tan(c/2 + (d*x)/2)^6 - 40*a*tan(c/2 + (d*x)/2)^7 - 5*a*tan(c/
2 + (d*x)/2)^8 + 30*a*tan(c/2 + (d*x)/2)^9 + 9*a*tan(c/2 + (d*x)/2)^10 - 12*a*tan(c/2 + (d*x)/2)^11 - 5*a*tan(
c/2 + (d*x)/2)^12 + 2*a*tan(c/2 + (d*x)/2)^13 + a*tan(c/2 + (d*x)/2)^14))