\(\int \frac {\csc ^3(c+d x) \sec ^7(c+d x)}{a+a \sin (c+d x)} \, dx\) [892]

Optimal result

Integrand size = 29, antiderivative size = 232 \[ \int \frac {\csc ^3(c+d x) \sec ^7(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\csc (c+d x)}{a d}-\frac {\csc ^2(c+d x)}{2 a d}-\frac {325 \log (1-\sin (c+d x))}{256 a d}+\frac {5 \log (\sin (c+d x))}{a d}-\frac {955 \log (1+\sin (c+d x))}{256 a d}+\frac {a^2}{96 d (a-a \sin (c+d x))^3}+\frac {11 a}{128 d (a-a \sin (c+d x))^2}+\frac {69}{128 d (a-a \sin (c+d x))}+\frac {a^3}{64 d (a+a \sin (c+d x))^4}+\frac {5 a^2}{48 d (a+a \sin (c+d x))^3}+\frac {29 a}{64 d (a+a \sin (c+d x))^2}+\frac {2}{d (a+a \sin (c+d x))} \]

[Out]

csc(d*x+c)/a/d-1/2*csc(d*x+c)^2/a/d-325/256*ln(1-sin(d*x+c))/a/d+5*ln(sin(d*x+c))/a/d-955/256*ln(1+sin(d*x+c))
/a/d+1/96*a^2/d/(a-a*sin(d*x+c))^3+11/128*a/d/(a-a*sin(d*x+c))^2+69/128/d/(a-a*sin(d*x+c))+1/64*a^3/d/(a+a*sin
(d*x+c))^4+5/48*a^2/d/(a+a*sin(d*x+c))^3+29/64*a/d/(a+a*sin(d*x+c))^2+2/d/(a+a*sin(d*x+c))

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 232, 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} \[ \int \frac {\csc ^3(c+d x) \sec ^7(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {a^3}{64 d (a \sin (c+d x)+a)^4}+\frac {a^2}{96 d (a-a \sin (c+d x))^3}+\frac {5 a^2}{48 d (a \sin (c+d x)+a)^3}+\frac {11 a}{128 d (a-a \sin (c+d x))^2}+\frac {29 a}{64 d (a \sin (c+d x)+a)^2}+\frac {69}{128 d (a-a \sin (c+d x))}+\frac {2}{d (a \sin (c+d x)+a)}-\frac {\csc ^2(c+d x)}{2 a d}+\frac {\csc (c+d x)}{a d}-\frac {325 \log (1-\sin (c+d x))}{256 a d}+\frac {5 \log (\sin (c+d x))}{a d}-\frac {955 \log (\sin (c+d x)+1)}{256 a d} \]

[In]

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

[Out]

Csc[c + d*x]/(a*d) - Csc[c + d*x]^2/(2*a*d) - (325*Log[1 - Sin[c + d*x]])/(256*a*d) + (5*Log[Sin[c + d*x]])/(a
*d) - (955*Log[1 + Sin[c + d*x]])/(256*a*d) + a^2/(96*d*(a - a*Sin[c + d*x])^3) + (11*a)/(128*d*(a - a*Sin[c +
d*x])^2) + 69/(128*d*(a - a*Sin[c + d*x])) + a^3/(64*d*(a + a*Sin[c + d*x])^4) + (5*a^2)/(48*d*(a + a*Sin[c +
d*x])^3) + (29*a)/(64*d*(a + a*Sin[c + d*x])^2) + 2/(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 {a^3}{(a-x)^4 x^3 (a+x)^5} \, dx,x,a \sin (c+d x)\right )}{d} \\ & = \frac {a^{10} \text {Subst}\left (\int \frac {1}{(a-x)^4 x^3 (a+x)^5} \, dx,x,a \sin (c+d x)\right )}{d} \\ & = \frac {a^{10} \text {Subst}\left (\int \left (\frac {1}{32 a^8 (a-x)^4}+\frac {11}{64 a^9 (a-x)^3}+\frac {69}{128 a^{10} (a-x)^2}+\frac {325}{256 a^{11} (a-x)}+\frac {1}{a^9 x^3}-\frac {1}{a^{10} x^2}+\frac {5}{a^{11} x}-\frac {1}{16 a^7 (a+x)^5}-\frac {5}{16 a^8 (a+x)^4}-\frac {29}{32 a^9 (a+x)^3}-\frac {2}{a^{10} (a+x)^2}-\frac {955}{256 a^{11} (a+x)}\right ) \, dx,x,a \sin (c+d x)\right )}{d} \\ & = \frac {\csc (c+d x)}{a d}-\frac {\csc ^2(c+d x)}{2 a d}-\frac {325 \log (1-\sin (c+d x))}{256 a d}+\frac {5 \log (\sin (c+d x))}{a d}-\frac {955 \log (1+\sin (c+d x))}{256 a d}+\frac {a^2}{96 d (a-a \sin (c+d x))^3}+\frac {11 a}{128 d (a-a \sin (c+d x))^2}+\frac {69}{128 d (a-a \sin (c+d x))}+\frac {a^3}{64 d (a+a \sin (c+d x))^4}+\frac {5 a^2}{48 d (a+a \sin (c+d x))^3}+\frac {29 a}{64 d (a+a \sin (c+d x))^2}+\frac {2}{d (a+a \sin (c+d x))} \\ \end{align*}

Mathematica [A] (verified)

Time = 6.11 (sec) , antiderivative size = 213, normalized size of antiderivative = 0.92 \[ \int \frac {\csc ^3(c+d x) \sec ^7(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {a^7 \left (\frac {\csc (c+d x)}{a^8}-\frac {\csc ^2(c+d x)}{2 a^8}-\frac {325 \log (1-\sin (c+d x))}{256 a^8}+\frac {5 \log (\sin (c+d x))}{a^8}-\frac {955 \log (1+\sin (c+d x))}{256 a^8}+\frac {1}{96 a^5 (a-a \sin (c+d x))^3}+\frac {11}{128 a^6 (a-a \sin (c+d x))^2}+\frac {69}{128 a^7 (a-a \sin (c+d x))}+\frac {1}{64 a^4 (a+a \sin (c+d x))^4}+\frac {5}{48 a^5 (a+a \sin (c+d x))^3}+\frac {29}{64 a^6 (a+a \sin (c+d x))^2}+\frac {2}{a^7 (a+a \sin (c+d x))}\right )}{d} \]

[In]

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

[Out]

(a^7*(Csc[c + d*x]/a^8 - Csc[c + d*x]^2/(2*a^8) - (325*Log[1 - Sin[c + d*x]])/(256*a^8) + (5*Log[Sin[c + d*x]]
)/a^8 - (955*Log[1 + Sin[c + d*x]])/(256*a^8) + 1/(96*a^5*(a - a*Sin[c + d*x])^3) + 11/(128*a^6*(a - a*Sin[c +
d*x])^2) + 69/(128*a^7*(a - a*Sin[c + d*x])) + 1/(64*a^4*(a + a*Sin[c + d*x])^4) + 5/(48*a^5*(a + a*Sin[c + d
*x])^3) + 29/(64*a^6*(a + a*Sin[c + d*x])^2) + 2/(a^7*(a + a*Sin[c + d*x]))))/d

Maple [A] (verified)

Time = 1.95 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.61

method	result	size
derivativedivides	\(\frac {-\frac {1}{2 \sin \left (d x +c \right )^{2}}+\frac {1}{\sin \left (d x +c \right )}+5 \ln \left (\sin \left (d x +c \right )\right )-\frac {1}{96 \left (\sin \left (d x +c \right )-1\right )^{3}}+\frac {11}{128 \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {69}{128 \left (\sin \left (d x +c \right )-1\right )}-\frac {325 \ln \left (\sin \left (d x +c \right )-1\right )}{256}+\frac {1}{64 \left (1+\sin \left (d x +c \right )\right )^{4}}+\frac {5}{48 \left (1+\sin \left (d x +c \right )\right )^{3}}+\frac {29}{64 \left (1+\sin \left (d x +c \right )\right )^{2}}+\frac {2}{1+\sin \left (d x +c \right )}-\frac {955 \ln \left (1+\sin \left (d x +c \right )\right )}{256}}{d a}\)	\(142\)
default	\(\frac {-\frac {1}{2 \sin \left (d x +c \right )^{2}}+\frac {1}{\sin \left (d x +c \right )}+5 \ln \left (\sin \left (d x +c \right )\right )-\frac {1}{96 \left (\sin \left (d x +c \right )-1\right )^{3}}+\frac {11}{128 \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {69}{128 \left (\sin \left (d x +c \right )-1\right )}-\frac {325 \ln \left (\sin \left (d x +c \right )-1\right )}{256}+\frac {1}{64 \left (1+\sin \left (d x +c \right )\right )^{4}}+\frac {5}{48 \left (1+\sin \left (d x +c \right )\right )^{3}}+\frac {29}{64 \left (1+\sin \left (d x +c \right )\right )^{2}}+\frac {2}{1+\sin \left (d x +c \right )}-\frac {955 \ln \left (1+\sin \left (d x +c \right )\right )}{256}}{d a}\)	\(142\)
risch	\(\frac {i \left (14604 \,{\mathrm e}^{5 i \left (d x +c \right )}+1170 i {\mathrm e}^{14 i \left (d x +c \right )}+945 \,{\mathrm e}^{i \left (d x +c \right )}+9512 \,{\mathrm e}^{11 i \left (d x +c \right )}-30 i {\mathrm e}^{16 i \left (d x +c \right )}-4602 i {\mathrm e}^{8 i \left (d x +c \right )}-4778 i {\mathrm e}^{6 i \left (d x +c \right )}-13690 \,{\mathrm e}^{9 i \left (d x +c \right )}+14604 \,{\mathrm e}^{13 i \left (d x +c \right )}+6360 \,{\mathrm e}^{3 i \left (d x +c \right )}+9512 \,{\mathrm e}^{7 i \left (d x +c \right )}+6360 \,{\mathrm e}^{15 i \left (d x +c \right )}+945 \,{\mathrm e}^{17 i \left (d x +c \right )}+4778 i {\mathrm e}^{12 i \left (d x +c \right )}+4602 i {\mathrm e}^{10 i \left (d x +c \right )}-1170 i {\mathrm e}^{4 i \left (d x +c \right )}+30 i {\mathrm e}^{2 i \left (d x +c \right )}\right )}{192 \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2} \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{8} \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{6} d a}-\frac {325 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{128 d a}-\frac {955 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{128 a d}+\frac {5 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d a}\)	\(310\)
parallelrisch	\(\frac {5 \left (\csc ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (-\frac {391}{1536}+\frac {65 \left (-\frac {5}{4}+\frac {3 \sin \left (7 d x +7 c \right )}{8}+\cos \left (4 d x +4 c \right )+\cos \left (6 d x +6 c \right )-\frac {3 \sin \left (d x +c \right )}{4}-\sin \left (3 d x +3 c \right )-\cos \left (2 d x +2 c \right )+\frac {\cos \left (8 d x +8 c \right )}{4}+\frac {\sin \left (9 d x +9 c \right )}{8}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{128}+\frac {191 \left (-\frac {5}{4}+\frac {3 \sin \left (7 d x +7 c \right )}{8}+\cos \left (4 d x +4 c \right )+\cos \left (6 d x +6 c \right )-\frac {3 \sin \left (d x +c \right )}{4}-\sin \left (3 d x +3 c \right )-\cos \left (2 d x +2 c \right )+\frac {\cos \left (8 d x +8 c \right )}{4}+\frac {\sin \left (9 d x +9 c \right )}{8}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{128}+\left (\frac {5}{4}+\cos \left (2 d x +2 c \right )-\frac {\sin \left (9 d x +9 c \right )}{8}-\frac {\cos \left (8 d x +8 c \right )}{4}-\frac {3 \sin \left (7 d x +7 c \right )}{8}-\cos \left (6 d x +6 c \right )+\frac {3 \sin \left (d x +c \right )}{4}+\sin \left (3 d x +3 c \right )-\cos \left (4 d x +4 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {2069 \cos \left (2 d x +2 c \right )}{960}-\frac {1891 \cos \left (4 d x +4 c \right )}{1920}-\frac {377 \sin \left (3 d x +3 c \right )}{1280}-\frac {113 \sin \left (d x +c \right )}{1280}+\frac {163 \cos \left (8 d x +8 c \right )}{1536}+\frac {11 \sin \left (9 d x +9 c \right )}{96}+\frac {87 \sin \left (7 d x +7 c \right )}{256}+\frac {39 \sin \left (5 d x +5 c \right )}{256}+\frac {17 \cos \left (6 d x +6 c \right )}{192}\right )}{2 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 )}\)	\(493\)

[In]

int(csc(d*x+c)^3*sec(d*x+c)^7/(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

1/d/a*(-1/2/sin(d*x+c)^2+1/sin(d*x+c)+5*ln(sin(d*x+c))-1/96/(sin(d*x+c)-1)^3+11/128/(sin(d*x+c)-1)^2-69/128/(s
in(d*x+c)-1)-325/256*ln(sin(d*x+c)-1)+1/64/(1+sin(d*x+c))^4+5/48/(1+sin(d*x+c))^3+29/64/(1+sin(d*x+c))^2+2/(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 = 311, normalized size of antiderivative = 1.34 \[ \int \frac {\csc ^3(c+d x) \sec ^7(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {1890 \, \cos \left (d x + c\right )^{8} - 600 \, \cos \left (d x + c\right )^{6} - 582 \, \cos \left (d x + c\right )^{4} - 212 \, \cos \left (d x + c\right )^{2} + 3840 \, {\left (\cos \left (d x + c\right )^{8} - \cos \left (d x + c\right )^{6} + {\left (\cos \left (d x + c\right )^{8} - \cos \left (d x + c\right )^{6}\right )} \sin \left (d x + c\right )\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) - 2865 \, {\left (\cos \left (d x + c\right )^{8} - \cos \left (d x + c\right )^{6} + {\left (\cos \left (d x + c\right )^{8} - \cos \left (d x + c\right )^{6}\right )} \sin \left (d x + c\right )\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 975 \, {\left (\cos \left (d x + c\right )^{8} - \cos \left (d x + c\right )^{6} + {\left (\cos \left (d x + c\right )^{8} - \cos \left (d x + c\right )^{6}\right )} \sin \left (d x + c\right )\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (15 \, \cos \left (d x + c\right )^{6} - 165 \, \cos \left (d x + c\right )^{4} - 34 \, \cos \left (d x + c\right )^{2} - 8\right )} \sin \left (d x + c\right ) - 112}{768 \, {\left (a d \cos \left (d x + c\right )^{8} - a d \cos \left (d x + c\right )^{6} + {\left (a d \cos \left (d x + c\right )^{8} - a d \cos \left (d x + c\right )^{6}\right )} \sin \left (d x + c\right )\right )}} \]

[In]

integrate(csc(d*x+c)^3*sec(d*x+c)^7/(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

1/768*(1890*cos(d*x + c)^8 - 600*cos(d*x + c)^6 - 582*cos(d*x + c)^4 - 212*cos(d*x + c)^2 + 3840*(cos(d*x + c)
^8 - cos(d*x + c)^6 + (cos(d*x + c)^8 - cos(d*x + c)^6)*sin(d*x + c))*log(1/2*sin(d*x + c)) - 2865*(cos(d*x +
c)^8 - cos(d*x + c)^6 + (cos(d*x + c)^8 - cos(d*x + c)^6)*sin(d*x + c))*log(sin(d*x + c) + 1) - 975*(cos(d*x +
c)^8 - cos(d*x + c)^6 + (cos(d*x + c)^8 - cos(d*x + c)^6)*sin(d*x + c))*log(-sin(d*x + c) + 1) + 2*(15*cos(d*
x + c)^6 - 165*cos(d*x + c)^4 - 34*cos(d*x + c)^2 - 8)*sin(d*x + c) - 112)/(a*d*cos(d*x + c)^8 - a*d*cos(d*x +
c)^6 + (a*d*cos(d*x + c)^8 - a*d*cos(d*x + c)^6)*sin(d*x + c))

Sympy [F(-1)]

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

[In]

integrate(csc(d*x+c)**3*sec(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 = 217, normalized size of antiderivative = 0.94 \[ \int \frac {\csc ^3(c+d x) \sec ^7(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {2 \, {\left (945 \, \sin \left (d x + c\right )^{8} - 15 \, \sin \left (d x + c\right )^{7} - 3480 \, \sin \left (d x + c\right )^{6} - 120 \, \sin \left (d x + c\right )^{5} + 4479 \, \sin \left (d x + c\right )^{4} + 319 \, \sin \left (d x + c\right )^{3} - 2192 \, \sin \left (d x + c\right )^{2} - 192 \, \sin \left (d x + c\right ) + 192\right )}}{a \sin \left (d x + c\right )^{9} + a \sin \left (d x + c\right )^{8} - 3 \, a \sin \left (d x + c\right )^{7} - 3 \, a \sin \left (d x + c\right )^{6} + 3 \, a \sin \left (d x + c\right )^{5} + 3 \, a \sin \left (d x + c\right )^{4} - a \sin \left (d x + c\right )^{3} - a \sin \left (d x + c\right )^{2}} - \frac {2865 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a} - \frac {975 \, \log \left (\sin \left (d x + c\right ) - 1\right )}{a} + \frac {3840 \, \log \left (\sin \left (d x + c\right )\right )}{a}}{768 \, d} \]

[In]

integrate(csc(d*x+c)^3*sec(d*x+c)^7/(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

1/768*(2*(945*sin(d*x + c)^8 - 15*sin(d*x + c)^7 - 3480*sin(d*x + c)^6 - 120*sin(d*x + c)^5 + 4479*sin(d*x + c
)^4 + 319*sin(d*x + c)^3 - 2192*sin(d*x + c)^2 - 192*sin(d*x + c) + 192)/(a*sin(d*x + c)^9 + a*sin(d*x + c)^8
- 3*a*sin(d*x + c)^7 - 3*a*sin(d*x + c)^6 + 3*a*sin(d*x + c)^5 + 3*a*sin(d*x + c)^4 - a*sin(d*x + c)^3 - a*sin
(d*x + c)^2) - 2865*log(sin(d*x + c) + 1)/a - 975*log(sin(d*x + c) - 1)/a + 3840*log(sin(d*x + c))/a)/d

Giac [A] (verification not implemented)

none

Time = 0.47 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.78 \[ \int \frac {\csc ^3(c+d x) \sec ^7(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\frac {11460 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a} + \frac {3900 \, \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a} - \frac {15360 \, \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a} + \frac {1536 \, {\left (15 \, \sin \left (d x + c\right )^{2} - 2 \, \sin \left (d x + c\right ) + 1\right )}}{a \sin \left (d x + c\right )^{2}} - \frac {2 \, {\left (3575 \, \sin \left (d x + c\right )^{3} - 11553 \, \sin \left (d x + c\right )^{2} + 12513 \, \sin \left (d x + c\right ) - 4551\right )}}{a {\left (\sin \left (d x + c\right ) - 1\right )}^{3}} - \frac {23875 \, \sin \left (d x + c\right )^{4} + 101644 \, \sin \left (d x + c\right )^{3} + 163074 \, \sin \left (d x + c\right )^{2} + 117036 \, \sin \left (d x + c\right ) + 31779}{a {\left (\sin \left (d x + c\right ) + 1\right )}^{4}}}{3072 \, d} \]

[In]

integrate(csc(d*x+c)^3*sec(d*x+c)^7/(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 - 15360*log(abs(sin(d*x + c)))
/a + 1536*(15*sin(d*x + c)^2 - 2*sin(d*x + c) + 1)/(a*sin(d*x + c)^2) - 2*(3575*sin(d*x + c)^3 - 11553*sin(d*x
+ c)^2 + 12513*sin(d*x + c) - 4551)/(a*(sin(d*x + c) - 1)^3) - (23875*sin(d*x + c)^4 + 101644*sin(d*x + c)^3
+ 163074*sin(d*x + c)^2 + 117036*sin(d*x + c) + 31779)/(a*(sin(d*x + c) + 1)^4))/d

Mupad [B] (verification not implemented)

Time = 10.04 (sec) , antiderivative size = 223, normalized size of antiderivative = 0.96 \[ \int \frac {\csc ^3(c+d x) \sec ^7(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {5\,\ln \left (\sin \left (c+d\,x\right )\right )}{a\,d}-\frac {955\,\ln \left (\sin \left (c+d\,x\right )+1\right )}{256\,a\,d}-\frac {325\,\ln \left (\sin \left (c+d\,x\right )-1\right )}{256\,a\,d}+\frac {-\frac {315\,{\sin \left (c+d\,x\right )}^8}{128}+\frac {5\,{\sin \left (c+d\,x\right )}^7}{128}+\frac {145\,{\sin \left (c+d\,x\right )}^6}{16}+\frac {5\,{\sin \left (c+d\,x\right )}^5}{16}-\frac {1493\,{\sin \left (c+d\,x\right )}^4}{128}-\frac {319\,{\sin \left (c+d\,x\right )}^3}{384}+\frac {137\,{\sin \left (c+d\,x\right )}^2}{24}+\frac {\sin \left (c+d\,x\right )}{2}-\frac {1}{2}}{d\,\left (-a\,{\sin \left (c+d\,x\right )}^9-a\,{\sin \left (c+d\,x\right )}^8+3\,a\,{\sin \left (c+d\,x\right )}^7+3\,a\,{\sin \left (c+d\,x\right )}^6-3\,a\,{\sin \left (c+d\,x\right )}^5-3\,a\,{\sin \left (c+d\,x\right )}^4+a\,{\sin \left (c+d\,x\right )}^3+a\,{\sin \left (c+d\,x\right )}^2\right )} \]

[In]

int(1/(cos(c + d*x)^7*sin(c + d*x)^3*(a + a*sin(c + d*x))),x)

[Out]

(5*log(sin(c + d*x)))/(a*d) - (955*log(sin(c + d*x) + 1))/(256*a*d) - (325*log(sin(c + d*x) - 1))/(256*a*d) +
(sin(c + d*x)/2 + (137*sin(c + d*x)^2)/24 - (319*sin(c + d*x)^3)/384 - (1493*sin(c + d*x)^4)/128 + (5*sin(c +
d*x)^5)/16 + (145*sin(c + d*x)^6)/16 + (5*sin(c + d*x)^7)/128 - (315*sin(c + d*x)^8)/128 - 1/2)/(d*(a*sin(c +
d*x)^2 + a*sin(c + d*x)^3 - 3*a*sin(c + d*x)^4 - 3*a*sin(c + d*x)^5 + 3*a*sin(c + d*x)^6 + 3*a*sin(c + d*x)^7
- a*sin(c + d*x)^8 - a*sin(c + d*x)^9))