\(\int \frac {\sec ^2(c+d x) \tan ^7(c+d x)}{a+a \sin (c+d x)} \, dx\) [900]

Optimal result

Integrand size = 29, antiderivative size = 178 \[ \int \frac {\sec ^2(c+d x) \tan ^7(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {7 \text {arctanh}(\sin (c+d x))}{256 a d}-\frac {7 \sec (c+d x) \tan (c+d x)}{256 a d}+\frac {7 \sec ^3(c+d x) \tan (c+d x)}{128 a d}-\frac {7 \sec ^3(c+d x) \tan ^3(c+d x)}{96 a d}+\frac {7 \sec ^3(c+d x) \tan ^5(c+d x)}{80 a d}-\frac {\sec ^3(c+d x) \tan ^7(c+d x)}{10 a d}+\frac {\tan ^8(c+d x)}{8 a d}+\frac {\tan ^{10}(c+d x)}{10 a d} \]

[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+1/8*tan(d*x+c)^8/a
/d+1/10*tan(d*x+c)^10/a/d

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {2914, 2687, 14, 2691, 3853, 3855} \[ \int \frac {\sec ^2(c+d x) \tan ^7(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {7 \text {arctanh}(\sin (c+d x))}{256 a d}+\frac {\tan ^{10}(c+d x)}{10 a d}+\frac {\tan ^8(c+d x)}{8 a d}-\frac {\tan ^7(c+d x) \sec ^3(c+d x)}{10 a d}+\frac {7 \tan ^5(c+d x) \sec ^3(c+d x)}{80 a d}-\frac {7 \tan ^3(c+d x) \sec ^3(c+d x)}{96 a d}+\frac {7 \tan (c+d x) \sec ^3(c+d x)}{128 a d}-\frac {7 \tan (c+d x) \sec (c+d x)}{256 a d} \]

[In]

Int[(Sec[c + d*x]^2*Tan[c + d*x]^7)/(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]^8/(8*a*d) + Tan[c + d*x]^10/(10*a*d)

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

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 ^4(c+d x) \tan ^7(c+d x) \, dx}{a}-\frac {\int \sec ^3(c+d x) \tan ^8(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^7 \left (1+x^2\right ) \, 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 {7 \int \sec ^3(c+d x) \tan ^4(c+d x) \, dx}{16 a}+\frac {\text {Subst}\left (\int \left (x^7+x^9\right ) \, dx,x,\tan (c+d x)\right )}{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 ^8(c+d x)}{8 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 ^8(c+d x)}{8 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 ^8(c+d x)}{8 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 ^8(c+d x)}{8 a d}+\frac {\tan ^{10}(c+d x)}{10 a d} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.11 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.70 \[ \int \frac {\sec ^2(c+d x) \tan ^7(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {210 \text {arctanh}(\sin (c+d x))-\frac {30}{(1-\sin (c+d x))^4}+\frac {160}{(1-\sin (c+d x))^3}-\frac {315}{(1-\sin (c+d x))^2}+\frac {210}{1-\sin (c+d x)}-\frac {48}{(1+\sin (c+d x))^5}+\frac {270}{(1+\sin (c+d x))^4}-\frac {580}{(1+\sin (c+d x))^3}+\frac {525}{(1+\sin (c+d x))^2}}{7680 a d} \]

[In]

Integrate[(Sec[c + d*x]^2*Tan[c + d*x]^7)/(a + a*Sin[c + d*x]),x]

[Out]

-1/7680*(210*ArcTanh[Sin[c + d*x]] - 30/(1 - Sin[c + d*x])^4 + 160/(1 - Sin[c + d*x])^3 - 315/(1 - Sin[c + d*x
])^2 + 210/(1 - Sin[c + d*x]) - 48/(1 + Sin[c + d*x])^5 + 270/(1 + Sin[c + d*x])^4 - 580/(1 + Sin[c + d*x])^3
+ 525/(1 + Sin[c + d*x])^2)/(a*d)

Maple [A] (verified)

Time = 1.66 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.71

method	result	size
derivativedivides	\(\frac {\frac {1}{256 \left (\sin \left (d x +c \right )-1\right )^{4}}+\frac {1}{48 \left (\sin \left (d x +c \right )-1\right )^{3}}+\frac {21}{512 \left (\sin \left (d x +c \right )-1\right )^{2}}+\frac {7}{256 \left (\sin \left (d x +c \right )-1\right )}+\frac {7 \ln \left (\sin \left (d x +c \right )-1\right )}{512}+\frac {1}{160 \left (1+\sin \left (d x +c \right )\right )^{5}}-\frac {9}{256 \left (1+\sin \left (d x +c \right )\right )^{4}}+\frac {29}{384 \left (1+\sin \left (d x +c \right )\right )^{3}}-\frac {35}{512 \left (1+\sin \left (d x +c \right )\right )^{2}}-\frac {7 \ln \left (1+\sin \left (d x +c \right )\right )}{512}}{d a}\)	\(127\)
default	\(\frac {\frac {1}{256 \left (\sin \left (d x +c \right )-1\right )^{4}}+\frac {1}{48 \left (\sin \left (d x +c \right )-1\right )^{3}}+\frac {21}{512 \left (\sin \left (d x +c \right )-1\right )^{2}}+\frac {7}{256 \left (\sin \left (d x +c \right )-1\right )}+\frac {7 \ln \left (\sin \left (d x +c \right )-1\right )}{512}+\frac {1}{160 \left (1+\sin \left (d x +c \right )\right )^{5}}-\frac {9}{256 \left (1+\sin \left (d x +c \right )\right )^{4}}+\frac {29}{384 \left (1+\sin \left (d x +c \right )\right )^{3}}-\frac {35}{512 \left (1+\sin \left (d x +c \right )\right )^{2}}-\frac {7 \ln \left (1+\sin \left (d x +c \right )\right )}{512}}{d a}\)	\(127\)
risch	\(\frac {i \left (9044 \,{\mathrm e}^{5 i \left (d x +c \right )}+105 \,{\mathrm e}^{17 i \left (d x +c \right )}-4420 \,{\mathrm e}^{15 i \left (d x +c \right )}+9044 \,{\mathrm e}^{13 i \left (d x +c \right )}-29372 \,{\mathrm e}^{11 i \left (d x +c \right )}+24710 \,{\mathrm e}^{9 i \left (d x +c \right )}-950 i {\mathrm e}^{14 i \left (d x +c \right )}+210 i {\mathrm e}^{16 i \left (d x +c \right )}+3206 i {\mathrm e}^{8 i \left (d x +c \right )}-1778 i {\mathrm e}^{6 i \left (d x +c \right )}+1778 i {\mathrm e}^{12 i \left (d x +c \right )}+105 \,{\mathrm e}^{i \left (d x +c \right )}-3206 i {\mathrm e}^{10 i \left (d x +c \right )}+950 i {\mathrm e}^{4 i \left (d x +c \right )}-210 i {\mathrm e}^{2 i \left (d x +c \right )}-4420 \,{\mathrm e}^{3 i \left (d x +c \right )}-29372 \,{\mathrm e}^{7 i \left (d x +c \right )}\right )}{1920 \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{10} \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{8} d a}+\frac {7 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{256 d a}-\frac {7 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{256 a d}\)	\(277\)
parallelrisch	\(\frac {\left (105 \cos \left (10 d x +10 c \right )+22050 \cos \left (2 d x +2 c \right )+12600 \cos \left (4 d x +4 c \right )+4725 \cos \left (6 d x +6 c \right )+1050 \cos \left (8 d x +8 c \right )+13230\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+\left (-105 \cos \left (10 d x +10 c \right )-22050 \cos \left (2 d x +2 c \right )-12600 \cos \left (4 d x +4 c \right )-4725 \cos \left (6 d x +6 c \right )-1050 \cos \left (8 d x +8 c \right )-13230\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+69720 \sin \left (3 d x +3 c \right )-23128 \sin \left (5 d x +5 c \right )+8210 \sin \left (7 d x +7 c \right )-210 \sin \left (9 d x +9 c \right )+96 \cos \left (10 d x +10 c \right )-87360 \cos \left (2 d x +2 c \right )+42240 \cos \left (4 d x +4 c \right )-11040 \cos \left (6 d x +6 c \right )+960 \cos \left (8 d x +8 c \right )-95340 \sin \left (d x +c \right )+55104}{3840 a d \left (\cos \left (10 d x +10 c \right )+10 \cos \left (8 d x +8 c \right )+45 \cos \left (6 d x +6 c \right )+120 \cos \left (4 d x +4 c \right )+210 \cos \left (2 d x +2 c \right )+126\right )}\)	\(315\)

[In]

int(sec(d*x+c)^9*sin(d*x+c)^7/(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

1/d/a*(1/256/(sin(d*x+c)-1)^4+1/48/(sin(d*x+c)-1)^3+21/512/(sin(d*x+c)-1)^2+7/256/(sin(d*x+c)-1)+7/512*ln(sin(
d*x+c)-1)+1/160/(1+sin(d*x+c))^5-9/256/(1+sin(d*x+c))^4+29/384/(1+sin(d*x+c))^3-35/512/(1+sin(d*x+c))^2-7/512*
ln(1+sin(d*x+c)))

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.05 \[ \int \frac {\sec ^2(c+d x) \tan ^7(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {210 \, \cos \left (d x + c\right )^{8} - 2630 \, \cos \left (d x + c\right )^{6} + 4708 \, \cos \left (d x + c\right )^{4} - 3344 \, \cos \left (d x + c\right )^{2} - 105 \, {\left (\cos \left (d x + c\right )^{8} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{8}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + 105 \, {\left (\cos \left (d x + c\right )^{8} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{8}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (105 \, \cos \left (d x + c\right )^{6} - 250 \, \cos \left (d x + c\right )^{4} + 184 \, \cos \left (d x + c\right )^{2} - 48\right )} \sin \left (d x + c\right ) + 864}{7680 \, {\left (a d \cos \left (d x + c\right )^{8} \sin \left (d x + c\right ) + a d \cos \left (d x + c\right )^{8}\right )}} \]

[In]

integrate(sec(d*x+c)^9*sin(d*x+c)^7/(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

1/7680*(210*cos(d*x + c)^8 - 2630*cos(d*x + c)^6 + 4708*cos(d*x + c)^4 - 3344*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*(105*cos(d*x + c)^6 - 250*cos(d*x + c)^4 + 184*cos(d*x + c)^2 - 48)*sin(d*x + c) +
864)/(a*d*cos(d*x + c)^8*sin(d*x + c) + a*d*cos(d*x + c)^8)

Sympy [F(-1)]

Timed out. \[ \int \frac {\sec ^2(c+d x) \tan ^7(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Timed out} \]

[In]

integrate(sec(d*x+c)**9*sin(d*x+c)**7/(a+a*sin(d*x+c)),x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.20 \[ \int \frac {\sec ^2(c+d x) \tan ^7(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {2 \, {\left (105 \, \sin \left (d x + c\right )^{8} + 105 \, \sin \left (d x + c\right )^{7} + 895 \, \sin \left (d x + c\right )^{6} - 65 \, \sin \left (d x + c\right )^{5} - 961 \, \sin \left (d x + c\right )^{4} - \sin \left (d x + c\right )^{3} + 489 \, \sin \left (d x + c\right )^{2} + 9 \, \sin \left (d x + c\right ) - 96\right )}}{a \sin \left (d x + c\right )^{9} + a \sin \left (d x + c\right )^{8} - 4 \, a \sin \left (d x + c\right )^{7} - 4 \, a \sin \left (d x + c\right )^{6} + 6 \, a \sin \left (d x + c\right )^{5} + 6 \, a \sin \left (d x + c\right )^{4} - 4 \, a \sin \left (d x + c\right )^{3} - 4 \, a \sin \left (d x + c\right )^{2} + a \sin \left (d x + c\right ) + a} - \frac {105 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a} + \frac {105 \, \log \left (\sin \left (d x + c\right ) - 1\right )}{a}}{7680 \, d} \]

[In]

integrate(sec(d*x+c)^9*sin(d*x+c)^7/(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

1/7680*(2*(105*sin(d*x + c)^8 + 105*sin(d*x + c)^7 + 895*sin(d*x + c)^6 - 65*sin(d*x + c)^5 - 961*sin(d*x + c)
^4 - sin(d*x + c)^3 + 489*sin(d*x + c)^2 + 9*sin(d*x + c) - 96)/(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 = 0.42 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.88 \[ \int \frac {\sec ^2(c+d x) \tan ^7(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\frac {420 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a} - \frac {420 \, \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a} + \frac {5 \, {\left (175 \, \sin \left (d x + c\right )^{4} - 868 \, \sin \left (d x + c\right )^{3} + 1302 \, \sin \left (d x + c\right )^{2} - 828 \, \sin \left (d x + c\right ) + 195\right )}}{a {\left (\sin \left (d x + c\right ) - 1\right )}^{4}} - \frac {959 \, \sin \left (d x + c\right )^{5} + 4795 \, \sin \left (d x + c\right )^{4} + 7490 \, \sin \left (d x + c\right )^{3} + 5610 \, \sin \left (d x + c\right )^{2} + 2055 \, \sin \left (d x + c\right ) + 291}{a {\left (\sin \left (d x + c\right ) + 1\right )}^{5}}}{30720 \, d} \]

[In]

integrate(sec(d*x+c)^9*sin(d*x+c)^7/(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 - 868*si
n(d*x + c)^3 + 1302*sin(d*x + c)^2 - 828*sin(d*x + c) + 195)/(a*(sin(d*x + c) - 1)^4) - (959*sin(d*x + c)^5 +
4795*sin(d*x + c)^4 + 7490*sin(d*x + c)^3 + 5610*sin(d*x + c)^2 + 2055*sin(d*x + c) + 291)/(a*(sin(d*x + c) +
1)^5))/d

Mupad [B] (verification not implemented)

Time = 18.75 (sec) , antiderivative size = 496, normalized size of antiderivative = 2.79 \[ \int \frac {\sec ^2(c+d x) \tan ^7(c+d x)}{a+a \sin (c+d x)} \, dx=\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 {25667\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{960}+\frac {1447\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{192}+\frac {25667\,{\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 )}-\frac {7\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{128\,a\,d} \]

[In]

int(sin(c + d*x)^7/(cos(c + d*x)^9*(a + a*sin(c + d*x))),x)

[Out]

((7*tan(c/2 + (d*x)/2))/128 + (7*tan(c/2 + (d*x)/2)^2)/64 - (35*tan(c/2 + (d*x)/2)^3)/96 - (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)/48
0 + (25667*tan(c/2 + (d*x)/2)^8)/960 + (1447*tan(c/2 + (d*x)/2)^9)/192 + (25667*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)) - (7*atanh(tan(c/2 + (d*x
)/2)))/(128*a*d)