\(\int \frac {\sin ^3(c+d x) \tan ^7(c+d x)}{a+a \sin (c+d x)} \, dx\) [879]

Optimal result

Integrand size = 29, antiderivative size = 220 \[ \int \frac {\sin ^3(c+d x) \tan ^7(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {325 \log (1-\sin (c+d x))}{256 a d}+\frac {955 \log (1+\sin (c+d x))}{256 a d}-\frac {\sin (c+d x)}{a d}+\frac {\sin ^2(c+d x)}{2 a d}+\frac {a^2}{96 d (a-a \sin (c+d x))^3}-\frac {15 a}{128 d (a-a \sin (c+d x))^2}+\frac {95}{128 d (a-a \sin (c+d x))}-\frac {a^3}{64 d (a+a \sin (c+d x))^4}+\frac {a^2}{6 d (a+a \sin (c+d x))^3}-\frac {55 a}{64 d (a+a \sin (c+d x))^2}+\frac {105}{32 d (a+a \sin (c+d x))} \]

[Out]

325/256*ln(1-sin(d*x+c))/a/d+955/256*ln(1+sin(d*x+c))/a/d-sin(d*x+c)/a/d+1/2*sin(d*x+c)^2/a/d+1/96*a^2/d/(a-a*
sin(d*x+c))^3-15/128*a/d/(a-a*sin(d*x+c))^2+95/128/d/(a-a*sin(d*x+c))-1/64*a^3/d/(a+a*sin(d*x+c))^4+1/6*a^2/d/
(a+a*sin(d*x+c))^3-55/64*a/d/(a+a*sin(d*x+c))^2+105/32/d/(a+a*sin(d*x+c))

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {2915, 12, 90} \[ \int \frac {\sin ^3(c+d x) \tan ^7(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {a^3}{64 d (a \sin (c+d x)+a)^4}+\frac {a^2}{96 d (a-a \sin (c+d x))^3}+\frac {a^2}{6 d (a \sin (c+d x)+a)^3}+\frac {\sin ^2(c+d x)}{2 a d}-\frac {15 a}{128 d (a-a \sin (c+d x))^2}-\frac {55 a}{64 d (a \sin (c+d x)+a)^2}+\frac {95}{128 d (a-a \sin (c+d x))}+\frac {105}{32 d (a \sin (c+d x)+a)}-\frac {\sin (c+d x)}{a d}+\frac {325 \log (1-\sin (c+d x))}{256 a d}+\frac {955 \log (\sin (c+d x)+1)}{256 a d} \]

[In]

Int[(Sin[c + d*x]^3*Tan[c + d*x]^7)/(a + a*Sin[c + d*x]),x]

[Out]

(325*Log[1 - Sin[c + d*x]])/(256*a*d) + (955*Log[1 + Sin[c + d*x]])/(256*a*d) - Sin[c + d*x]/(a*d) + Sin[c + d
*x]^2/(2*a*d) + a^2/(96*d*(a - a*Sin[c + d*x])^3) - (15*a)/(128*d*(a - a*Sin[c + d*x])^2) + 95/(128*d*(a - a*S
in[c + d*x])) - a^3/(64*d*(a + a*Sin[c + d*x])^4) + a^2/(6*d*(a + a*Sin[c + d*x])^3) - (55*a)/(64*d*(a + a*Sin
[c + d*x])^2) + 105/(32*d*(a + a*Sin[c + d*x]))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 90

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rule 2915

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

Rubi steps \begin{align*} \text {integral}& = \frac {a^7 \text {Subst}\left (\int \frac {x^{10}}{a^{10} (a-x)^4 (a+x)^5} \, dx,x,a \sin (c+d x)\right )}{d} \\ & = \frac {\text {Subst}\left (\int \frac {x^{10}}{(a-x)^4 (a+x)^5} \, dx,x,a \sin (c+d x)\right )}{a^3 d} \\ & = \frac {\text {Subst}\left (\int \left (-a+\frac {a^5}{32 (a-x)^4}-\frac {15 a^4}{64 (a-x)^3}+\frac {95 a^3}{128 (a-x)^2}-\frac {325 a^2}{256 (a-x)}+x+\frac {a^6}{16 (a+x)^5}-\frac {a^5}{2 (a+x)^4}+\frac {55 a^4}{32 (a+x)^3}-\frac {105 a^3}{32 (a+x)^2}+\frac {955 a^2}{256 (a+x)}\right ) \, dx,x,a \sin (c+d x)\right )}{a^3 d} \\ & = \frac {325 \log (1-\sin (c+d x))}{256 a d}+\frac {955 \log (1+\sin (c+d x))}{256 a d}-\frac {\sin (c+d x)}{a d}+\frac {\sin ^2(c+d x)}{2 a d}+\frac {a^2}{96 d (a-a \sin (c+d x))^3}-\frac {15 a}{128 d (a-a \sin (c+d x))^2}+\frac {95}{128 d (a-a \sin (c+d x))}-\frac {a^3}{64 d (a+a \sin (c+d x))^4}+\frac {a^2}{6 d (a+a \sin (c+d x))^3}-\frac {55 a}{64 d (a+a \sin (c+d x))^2}+\frac {105}{32 d (a+a \sin (c+d x))} \\ \end{align*}

Mathematica [A] (verified)

Time = 6.10 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.65 \[ \int \frac {\sin ^3(c+d x) \tan ^7(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {975 \log (1-\sin (c+d x))+2865 \log (1+\sin (c+d x))+\frac {8}{(1-\sin (c+d x))^3}-\frac {90}{(1-\sin (c+d x))^2}+\frac {570}{1-\sin (c+d x)}-768 \sin (c+d x)+384 \sin ^2(c+d x)-\frac {12}{(1+\sin (c+d x))^4}+\frac {128}{(1+\sin (c+d x))^3}-\frac {660}{(1+\sin (c+d x))^2}+\frac {2520}{1+\sin (c+d x)}}{768 a d} \]

[In]

Integrate[(Sin[c + d*x]^3*Tan[c + d*x]^7)/(a + a*Sin[c + d*x]),x]

[Out]

(975*Log[1 - Sin[c + d*x]] + 2865*Log[1 + Sin[c + d*x]] + 8/(1 - Sin[c + d*x])^3 - 90/(1 - Sin[c + d*x])^2 + 5
70/(1 - Sin[c + d*x]) - 768*Sin[c + d*x] + 384*Sin[c + d*x]^2 - 12/(1 + Sin[c + d*x])^4 + 128/(1 + Sin[c + d*x
])^3 - 660/(1 + Sin[c + d*x])^2 + 2520/(1 + Sin[c + d*x]))/(768*a*d)

Maple [A] (verified)

Time = 2.31 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.60

method	result	size
derivativedivides	\(\frac {\frac {\left (\sin ^{2}\left (d x +c \right )\right )}{2}-\sin \left (d x +c \right )-\frac {1}{96 \left (\sin \left (d x +c \right )-1\right )^{3}}-\frac {15}{128 \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {95}{128 \left (\sin \left (d x +c \right )-1\right )}+\frac {325 \ln \left (\sin \left (d x +c \right )-1\right )}{256}-\frac {1}{64 \left (1+\sin \left (d x +c \right )\right )^{4}}+\frac {1}{6 \left (1+\sin \left (d x +c \right )\right )^{3}}-\frac {55}{64 \left (1+\sin \left (d x +c \right )\right )^{2}}+\frac {105}{32 \left (1+\sin \left (d x +c \right )\right )}+\frac {955 \ln \left (1+\sin \left (d x +c \right )\right )}{256}}{d a}\)	\(133\)
default	\(\frac {\frac {\left (\sin ^{2}\left (d x +c \right )\right )}{2}-\sin \left (d x +c \right )-\frac {1}{96 \left (\sin \left (d x +c \right )-1\right )^{3}}-\frac {15}{128 \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {95}{128 \left (\sin \left (d x +c \right )-1\right )}+\frac {325 \ln \left (\sin \left (d x +c \right )-1\right )}{256}-\frac {1}{64 \left (1+\sin \left (d x +c \right )\right )^{4}}+\frac {1}{6 \left (1+\sin \left (d x +c \right )\right )^{3}}-\frac {55}{64 \left (1+\sin \left (d x +c \right )\right )^{2}}+\frac {105}{32 \left (1+\sin \left (d x +c \right )\right )}+\frac {955 \ln \left (1+\sin \left (d x +c \right )\right )}{256}}{d a}\)	\(133\)
risch	\(-\frac {5 i x}{a}-\frac {{\mathrm e}^{2 i \left (d x +c \right )}}{8 a d}+\frac {i {\mathrm e}^{i \left (d x +c \right )}}{2 a d}-\frac {i {\mathrm e}^{-i \left (d x +c \right )}}{2 a d}-\frac {{\mathrm e}^{-2 i \left (d x +c \right )}}{8 a d}-\frac {10 i c}{d a}+\frac {i \left (-1890 i {\mathrm e}^{12 i \left (d x +c \right )}+975 \,{\mathrm e}^{13 i \left (d x +c \right )}-3030 i {\mathrm e}^{10 i \left (d x +c \right )}+7110 \,{\mathrm e}^{11 i \left (d x +c \right )}-2932 i {\mathrm e}^{8 i \left (d x +c \right )}+18609 \,{\mathrm e}^{9 i \left (d x +c \right )}+2932 i {\mathrm e}^{6 i \left (d x +c \right )}+25460 \,{\mathrm e}^{7 i \left (d x +c \right )}+3030 i {\mathrm e}^{4 i \left (d x +c \right )}+18609 \,{\mathrm e}^{5 i \left (d x +c \right )}+1890 i {\mathrm e}^{2 i \left (d x +c \right )}+7110 \,{\mathrm e}^{3 i \left (d x +c \right )}+975 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{192 \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{8} \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{6} d a}+\frac {325 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{128 d a}+\frac {955 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{128 a d}\)	\(318\)
parallelrisch	\(\frac {340+1920 \left (-20-\sin \left (7 d x +7 c \right )-5 \sin \left (5 d x +5 c \right )-9 \sin \left (3 d x +3 c \right )-5 \sin \left (d x +c \right )-2 \cos \left (6 d x +6 c \right )-12 \cos \left (4 d x +4 c \right )-30 \cos \left (2 d x +2 c \right )\right ) \ln \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+975 \left (20+\sin \left (7 d x +7 c \right )+5 \sin \left (5 d x +5 c \right )+9 \sin \left (3 d x +3 c \right )+5 \sin \left (d x +c \right )+2 \cos \left (6 d x +6 c \right )+12 \cos \left (4 d x +4 c \right )+30 \cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+2865 \left (20+\sin \left (7 d x +7 c \right )+5 \sin \left (5 d x +5 c \right )+9 \sin \left (3 d x +3 c \right )+5 \sin \left (d x +c \right )+2 \cos \left (6 d x +6 c \right )+12 \cos \left (4 d x +4 c \right )+30 \cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-8100 \sin \left (3 d x +3 c \right )-4300 \sin \left (5 d x +5 c \right )-1760 \sin \left (7 d x +7 c \right )-48 \sin \left (9 d x +9 c \right )-126 \cos \left (2 d x +2 c \right )-180 \cos \left (4 d x +4 c \right )-130 \cos \left (6 d x +6 c \right )+96 \cos \left (8 d x +8 c \right )-1928 \sin \left (d x +c \right )}{384 a d \left (20+\sin \left (7 d x +7 c \right )+5 \sin \left (5 d x +5 c \right )+9 \sin \left (3 d x +3 c \right )+5 \sin \left (d x +c \right )+2 \cos \left (6 d x +6 c \right )+12 \cos \left (4 d x +4 c \right )+30 \cos \left (2 d x +2 c \right )\right )}\)	\(449\)

[In]

int(sec(d*x+c)^7*sin(d*x+c)^10/(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

1/d/a*(1/2*sin(d*x+c)^2-sin(d*x+c)-1/96/(sin(d*x+c)-1)^3-15/128/(sin(d*x+c)-1)^2-95/128/(sin(d*x+c)-1)+325/256
*ln(sin(d*x+c)-1)-1/64/(1+sin(d*x+c))^4+1/6/(1+sin(d*x+c))^3-55/64/(1+sin(d*x+c))^2+105/32/(1+sin(d*x+c))+955/
256*ln(1+sin(d*x+c)))

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.90 \[ \int \frac {\sin ^3(c+d x) \tan ^7(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {384 \, \cos \left (d x + c\right )^{8} + 1374 \, \cos \left (d x + c\right )^{6} + 630 \, \cos \left (d x + c\right )^{4} - 132 \, \cos \left (d x + c\right )^{2} + 2865 \, {\left (\cos \left (d x + c\right )^{6} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{6}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + 975 \, {\left (\cos \left (d x + c\right )^{6} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{6}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (192 \, \cos \left (d x + c\right )^{8} + 288 \, \cos \left (d x + c\right )^{6} - 945 \, \cos \left (d x + c\right )^{4} + 330 \, \cos \left (d x + c\right )^{2} - 56\right )} \sin \left (d x + c\right ) + 16}{768 \, {\left (a d \cos \left (d x + c\right )^{6} \sin \left (d x + c\right ) + a d \cos \left (d x + c\right )^{6}\right )}} \]

[In]

integrate(sec(d*x+c)^7*sin(d*x+c)^10/(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

1/768*(384*cos(d*x + c)^8 + 1374*cos(d*x + c)^6 + 630*cos(d*x + c)^4 - 132*cos(d*x + c)^2 + 2865*(cos(d*x + c)
^6*sin(d*x + c) + cos(d*x + c)^6)*log(sin(d*x + c) + 1) + 975*(cos(d*x + c)^6*sin(d*x + c) + cos(d*x + c)^6)*l
og(-sin(d*x + c) + 1) - 2*(192*cos(d*x + c)^8 + 288*cos(d*x + c)^6 - 945*cos(d*x + c)^4 + 330*cos(d*x + c)^2 -
56)*sin(d*x + c) + 16)/(a*d*cos(d*x + c)^6*sin(d*x + c) + a*d*cos(d*x + c)^6)

Sympy [F(-1)]

Timed out. \[ \int \frac {\sin ^3(c+d x) \tan ^7(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Timed out} \]

[In]

integrate(sec(d*x+c)**7*sin(d*x+c)**10/(a+a*sin(d*x+c)),x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.90 \[ \int \frac {\sin ^3(c+d x) \tan ^7(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {2 \, {\left (975 \, \sin \left (d x + c\right )^{6} - 945 \, \sin \left (d x + c\right )^{5} - 3240 \, \sin \left (d x + c\right )^{4} + 1560 \, \sin \left (d x + c\right )^{3} + 3489 \, \sin \left (d x + c\right )^{2} - 671 \, \sin \left (d x + c\right ) - 1232\right )}}{a \sin \left (d x + c\right )^{7} + a \sin \left (d x + c\right )^{6} - 3 \, a \sin \left (d x + c\right )^{5} - 3 \, a \sin \left (d x + c\right )^{4} + 3 \, a \sin \left (d x + c\right )^{3} + 3 \, a \sin \left (d x + c\right )^{2} - a \sin \left (d x + c\right ) - a} + \frac {384 \, {\left (\sin \left (d x + c\right )^{2} - 2 \, \sin \left (d x + c\right )\right )}}{a} + \frac {2865 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a} + \frac {975 \, \log \left (\sin \left (d x + c\right ) - 1\right )}{a}}{768 \, d} \]

[In]

integrate(sec(d*x+c)^7*sin(d*x+c)^10/(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

1/768*(2*(975*sin(d*x + c)^6 - 945*sin(d*x + c)^5 - 3240*sin(d*x + c)^4 + 1560*sin(d*x + c)^3 + 3489*sin(d*x +
c)^2 - 671*sin(d*x + c) - 1232)/(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) + 384*(sin(d*x + c)^2 - 2*sin(d*x + c))/a +
2865*log(sin(d*x + c) + 1)/a + 975*log(sin(d*x + c) - 1)/a)/d

Giac [A] (verification not implemented)

none

Time = 0.45 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.73 \[ \int \frac {\sin ^3(c+d x) \tan ^7(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {11460 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a} + \frac {3900 \, \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a} + \frac {1536 \, {\left (a \sin \left (d x + c\right )^{2} - 2 \, a \sin \left (d x + c\right )\right )}}{a^{2}} - \frac {2 \, {\left (3575 \, \sin \left (d x + c\right )^{3} - 9585 \, \sin \left (d x + c\right )^{2} + 8625 \, \sin \left (d x + c\right ) - 2599\right )}}{a {\left (\sin \left (d x + c\right ) - 1\right )}^{3}} - \frac {23875 \, \sin \left (d x + c\right )^{4} + 85420 \, \sin \left (d x + c\right )^{3} + 115650 \, \sin \left (d x + c\right )^{2} + 70028 \, \sin \left (d x + c\right ) + 15971}{a {\left (\sin \left (d x + c\right ) + 1\right )}^{4}}}{3072 \, d} \]

[In]

integrate(sec(d*x+c)^7*sin(d*x+c)^10/(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

1/3072*(11460*log(abs(sin(d*x + c) + 1))/a + 3900*log(abs(sin(d*x + c) - 1))/a + 1536*(a*sin(d*x + c)^2 - 2*a*
sin(d*x + c))/a^2 - 2*(3575*sin(d*x + c)^3 - 9585*sin(d*x + c)^2 + 8625*sin(d*x + c) - 2599)/(a*(sin(d*x + c)
- 1)^3) - (23875*sin(d*x + c)^4 + 85420*sin(d*x + c)^3 + 115650*sin(d*x + c)^2 + 70028*sin(d*x + c) + 15971)/(
a*(sin(d*x + c) + 1)^4))/d

Mupad [B] (verification not implemented)

Time = 11.07 (sec) , antiderivative size = 512, normalized size of antiderivative = 2.33 \[ \int \frac {\sin ^3(c+d x) \tan ^7(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {325\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )}{128\,a\,d}+\frac {955\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}{128\,a\,d}+\frac {-\frac {315\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}}{64}+\frac {5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}}{32}+\frac {265\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{8}+\frac {195\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}}{32}-\frac {1217\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{16}-\frac {2389\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{96}+\frac {1189\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{24}+\frac {767\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{32}+\frac {6845\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{96}+\frac {767\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{32}+\frac {1189\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{24}-\frac {2389\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{96}-\frac {1217\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{16}+\frac {195\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{32}+\frac {265\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{8}+\frac {5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{32}-\frac {315\,\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 )}^{18}+2\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}-3\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}-8\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}+8\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}+8\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+8\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}-6\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-20\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9-6\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+8\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+8\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+8\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-8\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-3\,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 {5\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{a\,d} \]

[In]

int(sin(c + d*x)^10/(cos(c + d*x)^7*(a + a*sin(c + d*x))),x)

[Out]

(325*log(tan(c/2 + (d*x)/2) - 1))/(128*a*d) + (955*log(tan(c/2 + (d*x)/2) + 1))/(128*a*d) + ((5*tan(c/2 + (d*x
)/2)^2)/32 - (315*tan(c/2 + (d*x)/2))/64 + (265*tan(c/2 + (d*x)/2)^3)/8 + (195*tan(c/2 + (d*x)/2)^4)/32 - (121
7*tan(c/2 + (d*x)/2)^5)/16 - (2389*tan(c/2 + (d*x)/2)^6)/96 + (1189*tan(c/2 + (d*x)/2)^7)/24 + (767*tan(c/2 +
(d*x)/2)^8)/32 + (6845*tan(c/2 + (d*x)/2)^9)/96 + (767*tan(c/2 + (d*x)/2)^10)/32 + (1189*tan(c/2 + (d*x)/2)^11
)/24 - (2389*tan(c/2 + (d*x)/2)^12)/96 - (1217*tan(c/2 + (d*x)/2)^13)/16 + (195*tan(c/2 + (d*x)/2)^14)/32 + (2
65*tan(c/2 + (d*x)/2)^15)/8 + (5*tan(c/2 + (d*x)/2)^16)/32 - (315*tan(c/2 + (d*x)/2)^17)/64)/(d*(a + 2*a*tan(c
/2 + (d*x)/2) - 3*a*tan(c/2 + (d*x)/2)^2 - 8*a*tan(c/2 + (d*x)/2)^3 + 8*a*tan(c/2 + (d*x)/2)^5 + 8*a*tan(c/2 +
(d*x)/2)^6 + 8*a*tan(c/2 + (d*x)/2)^7 - 6*a*tan(c/2 + (d*x)/2)^8 - 20*a*tan(c/2 + (d*x)/2)^9 - 6*a*tan(c/2 +
(d*x)/2)^10 + 8*a*tan(c/2 + (d*x)/2)^11 + 8*a*tan(c/2 + (d*x)/2)^12 + 8*a*tan(c/2 + (d*x)/2)^13 - 8*a*tan(c/2
+ (d*x)/2)^15 - 3*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)) - (5*log(tan
(c/2 + (d*x)/2)^2 + 1))/(a*d)