\(\int \frac {\tan ^7(c+d x)}{a+a \sin (c+d x)} \, dx\) [882]

Optimal result

Integrand size = 21, antiderivative size = 130 \[ \int \frac {\tan ^7(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {35 \text {arctanh}(\sin (c+d x))}{128 a d}+\frac {35 \sec (c+d x) \tan (c+d x)}{128 a d}-\frac {35 \sec (c+d x) \tan ^3(c+d x)}{192 a d}+\frac {7 \sec (c+d x) \tan ^5(c+d x)}{48 a d}-\frac {\sec (c+d x) \tan ^7(c+d x)}{8 a d}+\frac {\tan ^8(c+d x)}{8 a d} \]

[Out]

-35/128*arctanh(sin(d*x+c))/a/d+35/128*sec(d*x+c)*tan(d*x+c)/a/d-35/192*sec(d*x+c)*tan(d*x+c)^3/a/d+7/48*sec(d
*x+c)*tan(d*x+c)^5/a/d-1/8*sec(d*x+c)*tan(d*x+c)^7/a/d+1/8*tan(d*x+c)^8/a/d

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {2785, 2687, 30, 2691, 3855} \[ \int \frac {\tan ^7(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {35 \text {arctanh}(\sin (c+d x))}{128 a d}+\frac {\tan ^8(c+d x)}{8 a d}-\frac {\tan ^7(c+d x) \sec (c+d x)}{8 a d}+\frac {7 \tan ^5(c+d x) \sec (c+d x)}{48 a d}-\frac {35 \tan ^3(c+d x) \sec (c+d x)}{192 a d}+\frac {35 \tan (c+d x) \sec (c+d x)}{128 a d} \]

[In]

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

[Out]

(-35*ArcTanh[Sin[c + d*x]])/(128*a*d) + (35*Sec[c + d*x]*Tan[c + d*x])/(128*a*d) - (35*Sec[c + d*x]*Tan[c + d*
x]^3)/(192*a*d) + (7*Sec[c + d*x]*Tan[c + d*x]^5)/(48*a*d) - (Sec[c + d*x]*Tan[c + d*x]^7)/(8*a*d) + Tan[c + d
*x]^8/(8*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 2785

Int[((g_.)*tan[(e_.) + (f_.)*(x_)])^(p_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[1/a, Int[S
ec[e + f*x]^2*(g*Tan[e + f*x])^p, x], x] - Dist[1/(b*g), Int[Sec[e + f*x]*(g*Tan[e + f*x])^(p + 1), x], x] /;
FreeQ[{a, b, e, f, g, p}, x] && EqQ[a^2 - b^2, 0] && NeQ[p, -1]

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 ^2(c+d x) \tan ^7(c+d x) \, dx}{a}-\frac {\int \sec (c+d x) \tan ^8(c+d x) \, dx}{a} \\ & = -\frac {\sec (c+d x) \tan ^7(c+d x)}{8 a d}+\frac {7 \int \sec (c+d x) \tan ^6(c+d x) \, dx}{8 a}+\frac {\text {Subst}\left (\int x^7 \, dx,x,\tan (c+d x)\right )}{a d} \\ & = \frac {7 \sec (c+d x) \tan ^5(c+d x)}{48 a d}-\frac {\sec (c+d x) \tan ^7(c+d x)}{8 a d}+\frac {\tan ^8(c+d x)}{8 a d}-\frac {35 \int \sec (c+d x) \tan ^4(c+d x) \, dx}{48 a} \\ & = -\frac {35 \sec (c+d x) \tan ^3(c+d x)}{192 a d}+\frac {7 \sec (c+d x) \tan ^5(c+d x)}{48 a d}-\frac {\sec (c+d x) \tan ^7(c+d x)}{8 a d}+\frac {\tan ^8(c+d x)}{8 a d}+\frac {35 \int \sec (c+d x) \tan ^2(c+d x) \, dx}{64 a} \\ & = \frac {35 \sec (c+d x) \tan (c+d x)}{128 a d}-\frac {35 \sec (c+d x) \tan ^3(c+d x)}{192 a d}+\frac {7 \sec (c+d x) \tan ^5(c+d x)}{48 a d}-\frac {\sec (c+d x) \tan ^7(c+d x)}{8 a d}+\frac {\tan ^8(c+d x)}{8 a d}-\frac {35 \int \sec (c+d x) \, dx}{128 a} \\ & = -\frac {35 \text {arctanh}(\sin (c+d x))}{128 a d}+\frac {35 \sec (c+d x) \tan (c+d x)}{128 a d}-\frac {35 \sec (c+d x) \tan ^3(c+d x)}{192 a d}+\frac {7 \sec (c+d x) \tan ^5(c+d x)}{48 a d}-\frac {\sec (c+d x) \tan ^7(c+d x)}{8 a d}+\frac {\tan ^8(c+d x)}{8 a d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.65 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.78 \[ \int \frac {\tan ^7(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {105 \text {arctanh}(\sin (c+d x))+\frac {-48+57 \sin (c+d x)+249 \sin ^2(c+d x)-136 \sin ^3(c+d x)-424 \sin ^4(c+d x)+87 \sin ^5(c+d x)+279 \sin ^6(c+d x)}{(-1+\sin (c+d x))^3 (1+\sin (c+d x))^4}}{384 a d} \]

[In]

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

[Out]

-1/384*(105*ArcTanh[Sin[c + d*x]] + (-48 + 57*Sin[c + d*x] + 249*Sin[c + d*x]^2 - 136*Sin[c + d*x]^3 - 424*Sin
[c + d*x]^4 + 87*Sin[c + d*x]^5 + 279*Sin[c + d*x]^6)/((-1 + Sin[c + d*x])^3*(1 + Sin[c + d*x])^4))/(a*d)

Maple [A] (verified)

Time = 0.93 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.88

method	result	size
derivativedivides	\(\frac {-\frac {1}{96 \left (\sin \left (d x +c \right )-1\right )^{3}}-\frac {9}{128 \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {29}{128 \left (\sin \left (d x +c \right )-1\right )}+\frac {35 \ln \left (\sin \left (d x +c \right )-1\right )}{256}+\frac {1}{64 \left (1+\sin \left (d x +c \right )\right )^{4}}-\frac {5}{48 \left (1+\sin \left (d x +c \right )\right )^{3}}+\frac {19}{64 \left (1+\sin \left (d x +c \right )\right )^{2}}-\frac {1}{2 \left (1+\sin \left (d x +c \right )\right )}-\frac {35 \ln \left (1+\sin \left (d x +c \right )\right )}{256}}{d a}\)	\(115\)
default	\(\frac {-\frac {1}{96 \left (\sin \left (d x +c \right )-1\right )^{3}}-\frac {9}{128 \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {29}{128 \left (\sin \left (d x +c \right )-1\right )}+\frac {35 \ln \left (\sin \left (d x +c \right )-1\right )}{256}+\frac {1}{64 \left (1+\sin \left (d x +c \right )\right )^{4}}-\frac {5}{48 \left (1+\sin \left (d x +c \right )\right )^{3}}+\frac {19}{64 \left (1+\sin \left (d x +c \right )\right )^{2}}-\frac {1}{2 \left (1+\sin \left (d x +c \right )\right )}-\frac {35 \ln \left (1+\sin \left (d x +c \right )\right )}{256}}{d a}\)	\(115\)
risch	\(-\frac {i \left (1385 \,{\mathrm e}^{5 i \left (d x +c \right )}+174 i {\mathrm e}^{12 i \left (d x +c \right )}-174 i {\mathrm e}^{2 i \left (d x +c \right )}+218 i {\mathrm e}^{10 i \left (d x +c \right )}+279 \,{\mathrm e}^{i \left (d x +c \right )}+1385 \,{\mathrm e}^{9 i \left (d x +c \right )}+279 \,{\mathrm e}^{13 i \left (d x +c \right )}+22 \,{\mathrm e}^{11 i \left (d x +c \right )}+22 \,{\mathrm e}^{3 i \left (d x +c \right )}-300 \,{\mathrm e}^{7 i \left (d x +c \right )}+300 i {\mathrm e}^{8 i \left (d x +c \right )}-300 i {\mathrm e}^{6 i \left (d x +c \right )}-218 i {\mathrm e}^{4 i \left (d x +c \right )}\right )}{192 \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{8} \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{6} d a}-\frac {35 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{128 a d}+\frac {35 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{128 d a}\)	\(231\)
parallelrisch	\(\frac {\left (5880 \cos \left (2 d x +2 c \right )+2940 \cos \left (4 d x +4 c \right )+840 \cos \left (6 d x +6 c \right )+105 \cos \left (8 d x +8 c \right )+3675\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+\left (-5880 \cos \left (2 d x +2 c \right )-2940 \cos \left (4 d x +4 c \right )-840 \cos \left (6 d x +6 c \right )-105 \cos \left (8 d x +8 c \right )-3675\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+3598 \sin \left (3 d x +3 c \right )+182 \sin \left (5 d x +5 c \right )+558 \sin \left (7 d x +7 c \right )-2688 \cos \left (2 d x +2 c \right )+1344 \cos \left (4 d x +4 c \right )-384 \cos \left (6 d x +6 c \right )+48 \cos \left (8 d x +8 c \right )-2170 \sin \left (d x +c \right )+1680}{384 a d \left (\cos \left (8 d x +8 c \right )+8 \cos \left (6 d x +6 c \right )+28 \cos \left (4 d x +4 c \right )+56 \cos \left (2 d x +2 c \right )+35\right )}\)	\(260\)
norman	\(\frac {\frac {35 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d a}+\frac {35 \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d a}+\frac {231 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d a}+\frac {231 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d a}-\frac {25 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d a}+\frac {35 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{64 d a}+\frac {35 \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d a}-\frac {595 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{96 d a}-\frac {595 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{96 d a}-\frac {245 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{96 d a}-\frac {245 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{96 d a}+\frac {791 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d a}+\frac {791 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d a}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{8} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{6}}+\frac {35 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{128 a d}-\frac {35 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{128 a d}\)	\(315\)

[In]

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

[Out]

1/d/a*(-1/96/(sin(d*x+c)-1)^3-9/128/(sin(d*x+c)-1)^2-29/128/(sin(d*x+c)-1)+35/256*ln(sin(d*x+c)-1)+1/64/(1+sin
(d*x+c))^4-5/48/(1+sin(d*x+c))^3+19/64/(1+sin(d*x+c))^2-1/2/(1+sin(d*x+c))-35/256*ln(1+sin(d*x+c)))

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.28 \[ \int \frac {\tan ^7(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {558 \, \cos \left (d x + c\right )^{6} - 826 \, \cos \left (d x + c\right )^{4} + 476 \, \cos \left (d x + c\right )^{2} + 105 \, {\left (\cos \left (d x + c\right )^{6} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{6}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 105 \, {\left (\cos \left (d x + c\right )^{6} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{6}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (87 \, \cos \left (d x + c\right )^{4} - 38 \, \cos \left (d x + c\right )^{2} + 8\right )} \sin \left (d x + c\right ) - 112}{768 \, {\left (a d \cos \left (d x + c\right )^{6} \sin \left (d x + c\right ) + a d \cos \left (d x + c\right )^{6}\right )}} \]

[In]

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

[Out]

-1/768*(558*cos(d*x + c)^6 - 826*cos(d*x + c)^4 + 476*cos(d*x + c)^2 + 105*(cos(d*x + c)^6*sin(d*x + c) + cos(
d*x + c)^6)*log(sin(d*x + c) + 1) - 105*(cos(d*x + c)^6*sin(d*x + c) + cos(d*x + c)^6)*log(-sin(d*x + c) + 1)
- 2*(87*cos(d*x + c)^4 - 38*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(-1)]

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

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.35 \[ \int \frac {\tan ^7(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\frac {2 \, {\left (279 \, \sin \left (d x + c\right )^{6} + 87 \, \sin \left (d x + c\right )^{5} - 424 \, \sin \left (d x + c\right )^{4} - 136 \, \sin \left (d x + c\right )^{3} + 249 \, \sin \left (d x + c\right )^{2} + 57 \, \sin \left (d x + c\right ) - 48\right )}}{a \sin \left (d x + c\right )^{7} + a \sin \left (d x + c\right )^{6} - 3 \, a \sin \left (d x + c\right )^{5} - 3 \, a \sin \left (d x + c\right )^{4} + 3 \, a \sin \left (d x + c\right )^{3} + 3 \, a \sin \left (d x + c\right )^{2} - a \sin \left (d x + c\right ) - a} + \frac {105 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a} - \frac {105 \, \log \left (\sin \left (d x + c\right ) - 1\right )}{a}}{768 \, d} \]

[In]

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

[Out]

-1/768*(2*(279*sin(d*x + c)^6 + 87*sin(d*x + c)^5 - 424*sin(d*x + c)^4 - 136*sin(d*x + c)^3 + 249*sin(d*x + c)
^2 + 57*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*sin(d*x + c)^3 + 3*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 = 0.40 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.05 \[ \int \frac {\tan ^7(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\frac {420 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a} - \frac {420 \, \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a} + \frac {2 \, {\left (385 \, \sin \left (d x + c\right )^{3} - 807 \, \sin \left (d x + c\right )^{2} + 567 \, \sin \left (d x + c\right ) - 129\right )}}{a {\left (\sin \left (d x + c\right ) - 1\right )}^{3}} - \frac {875 \, \sin \left (d x + c\right )^{4} + 1964 \, \sin \left (d x + c\right )^{3} + 1554 \, \sin \left (d x + c\right )^{2} + 396 \, \sin \left (d x + c\right ) - 21}{a {\left (\sin \left (d x + c\right ) + 1\right )}^{4}}}{3072 \, d} \]

[In]

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

[Out]

-1/3072*(420*log(abs(sin(d*x + c) + 1))/a - 420*log(abs(sin(d*x + c) - 1))/a + 2*(385*sin(d*x + c)^3 - 807*sin
(d*x + c)^2 + 567*sin(d*x + c) - 129)/(a*(sin(d*x + c) - 1)^3) - (875*sin(d*x + c)^4 + 1964*sin(d*x + c)^3 + 1
554*sin(d*x + c)^2 + 396*sin(d*x + c) - 21)/(a*(sin(d*x + c) + 1)^4))/d

Mupad [B] (verification not implemented)

Time = 18.59 (sec) , antiderivative size = 388, normalized size of antiderivative = 2.98 \[ \int \frac {\tan ^7(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{64}+\frac {35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{32}-\frac {245\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{96}-\frac {595\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{96}+\frac {791\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{192}+\frac {231\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{16}-\frac {25\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{16}+\frac {231\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{16}+\frac {791\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{192}-\frac {595\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{96}-\frac {245\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{96}+\frac {35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{32}+\frac {35\,\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 )}-\frac {35\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{64\,a\,d} \]

[In]

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

[Out]

((35*tan(c/2 + (d*x)/2))/64 + (35*tan(c/2 + (d*x)/2)^2)/32 - (245*tan(c/2 + (d*x)/2)^3)/96 - (595*tan(c/2 + (d
*x)/2)^4)/96 + (791*tan(c/2 + (d*x)/2)^5)/192 + (231*tan(c/2 + (d*x)/2)^6)/16 - (25*tan(c/2 + (d*x)/2)^7)/16 +
(231*tan(c/2 + (d*x)/2)^8)/16 + (791*tan(c/2 + (d*x)/2)^9)/192 - (595*tan(c/2 + (d*x)/2)^10)/96 - (245*tan(c/
2 + (d*x)/2)^11)/96 + (35*tan(c/2 + (d*x)/2)^12)/32 + (35*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)) - (35*atanh(tan(c/2 + (d*x)/2)))/(64*a*d)