\(\int \frac {\csc ^4(c+d x) \sec ^7(c+d x)}{a+a \sin (c+d x)} \, dx\) [893]

Optimal result

Integrand size = 29, antiderivative size = 253 \[ \int \frac {\csc ^4(c+d x) \sec ^7(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {5 \csc (c+d x)}{a d}+\frac {\csc ^2(c+d x)}{2 a d}-\frac {\csc ^3(c+d x)}{3 a d}-\frac {515 \log (1-\sin (c+d x))}{256 a d}-\frac {5 \log (\sin (c+d x))}{a d}+\frac {1795 \log (1+\sin (c+d x))}{256 a d}+\frac {a^2}{96 d (a-a \sin (c+d x))^3}+\frac {13 a}{128 d (a-a \sin (c+d x))^2}+\frac {95}{128 d (a-a \sin (c+d x))}-\frac {a^3}{64 d (a+a \sin (c+d x))^4}-\frac {a^2}{8 d (a+a \sin (c+d x))^3}-\frac {41 a}{64 d (a+a \sin (c+d x))^2}-\frac {105}{32 d (a+a \sin (c+d x))} \]

[Out]

-5*csc(d*x+c)/a/d+1/2*csc(d*x+c)^2/a/d-1/3*csc(d*x+c)^3/a/d-515/256*ln(1-sin(d*x+c))/a/d-5*ln(sin(d*x+c))/a/d+
1795/256*ln(1+sin(d*x+c))/a/d+1/96*a^2/d/(a-a*sin(d*x+c))^3+13/128*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))^4-1/8*a^2/d/(a+a*sin(d*x+c))^3-41/64*a/d/(a+a*sin(d*x+c))^2-105/32/d/(a+a*si
n(d*x+c))

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 253, 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} \[ \int \frac {\csc ^4(c+d x) \sec ^7(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {a^3}{64 d (a \sin (c+d x)+a)^4}+\frac {a^2}{96 d (a-a \sin (c+d x))^3}-\frac {a^2}{8 d (a \sin (c+d x)+a)^3}+\frac {13 a}{128 d (a-a \sin (c+d x))^2}-\frac {41 a}{64 d (a \sin (c+d x)+a)^2}+\frac {95}{128 d (a-a \sin (c+d x))}-\frac {105}{32 d (a \sin (c+d x)+a)}-\frac {\csc ^3(c+d x)}{3 a d}+\frac {\csc ^2(c+d x)}{2 a d}-\frac {5 \csc (c+d x)}{a d}-\frac {515 \log (1-\sin (c+d x))}{256 a d}-\frac {5 \log (\sin (c+d x))}{a d}+\frac {1795 \log (\sin (c+d x)+1)}{256 a d} \]

[In]

Int[(Csc[c + d*x]^4*Sec[c + d*x]^7)/(a + a*Sin[c + d*x]),x]

[Out]

(-5*Csc[c + d*x])/(a*d) + Csc[c + d*x]^2/(2*a*d) - Csc[c + d*x]^3/(3*a*d) - (515*Log[1 - Sin[c + d*x]])/(256*a
*d) - (5*Log[Sin[c + d*x]])/(a*d) + (1795*Log[1 + Sin[c + d*x]])/(256*a*d) + a^2/(96*d*(a - a*Sin[c + d*x])^3)
+ (13*a)/(128*d*(a - a*Sin[c + d*x])^2) + 95/(128*d*(a - a*Sin[c + d*x])) - a^3/(64*d*(a + a*Sin[c + d*x])^4)
- a^2/(8*d*(a + a*Sin[c + d*x])^3) - (41*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 {a^4}{(a-x)^4 x^4 (a+x)^5} \, dx,x,a \sin (c+d x)\right )}{d} \\ & = \frac {a^{11} \text {Subst}\left (\int \frac {1}{(a-x)^4 x^4 (a+x)^5} \, dx,x,a \sin (c+d x)\right )}{d} \\ & = \frac {a^{11} \text {Subst}\left (\int \left (\frac {1}{32 a^9 (a-x)^4}+\frac {13}{64 a^{10} (a-x)^3}+\frac {95}{128 a^{11} (a-x)^2}+\frac {515}{256 a^{12} (a-x)}+\frac {1}{a^9 x^4}-\frac {1}{a^{10} x^3}+\frac {5}{a^{11} x^2}-\frac {5}{a^{12} x}+\frac {1}{16 a^8 (a+x)^5}+\frac {3}{8 a^9 (a+x)^4}+\frac {41}{32 a^{10} (a+x)^3}+\frac {105}{32 a^{11} (a+x)^2}+\frac {1795}{256 a^{12} (a+x)}\right ) \, dx,x,a \sin (c+d x)\right )}{d} \\ & = -\frac {5 \csc (c+d x)}{a d}+\frac {\csc ^2(c+d x)}{2 a d}-\frac {\csc ^3(c+d x)}{3 a d}-\frac {515 \log (1-\sin (c+d x))}{256 a d}-\frac {5 \log (\sin (c+d x))}{a d}+\frac {1795 \log (1+\sin (c+d x))}{256 a d}+\frac {a^2}{96 d (a-a \sin (c+d x))^3}+\frac {13 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}{8 d (a+a \sin (c+d x))^3}-\frac {41 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 = 231, normalized size of antiderivative = 0.91 \[ \int \frac {\csc ^4(c+d x) \sec ^7(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {a^7 \left (-\frac {5 \csc (c+d x)}{a^8}+\frac {\csc ^2(c+d x)}{2 a^8}-\frac {\csc ^3(c+d x)}{3 a^8}-\frac {515 \log (1-\sin (c+d x))}{256 a^8}-\frac {5 \log (\sin (c+d x))}{a^8}+\frac {1795 \log (1+\sin (c+d x))}{256 a^8}+\frac {1}{96 a^5 (a-a \sin (c+d x))^3}+\frac {13}{128 a^6 (a-a \sin (c+d x))^2}+\frac {95}{128 a^7 (a-a \sin (c+d x))}-\frac {1}{64 a^4 (a+a \sin (c+d x))^4}-\frac {1}{8 a^5 (a+a \sin (c+d x))^3}-\frac {41}{64 a^6 (a+a \sin (c+d x))^2}-\frac {105}{32 a^7 (a+a \sin (c+d x))}\right )}{d} \]

[In]

Integrate[(Csc[c + d*x]^4*Sec[c + d*x]^7)/(a + a*Sin[c + d*x]),x]

[Out]

(a^7*((-5*Csc[c + d*x])/a^8 + Csc[c + d*x]^2/(2*a^8) - Csc[c + d*x]^3/(3*a^8) - (515*Log[1 - Sin[c + d*x]])/(2
56*a^8) - (5*Log[Sin[c + d*x]])/a^8 + (1795*Log[1 + Sin[c + d*x]])/(256*a^8) + 1/(96*a^5*(a - a*Sin[c + d*x])^
3) + 13/(128*a^6*(a - a*Sin[c + d*x])^2) + 95/(128*a^7*(a - a*Sin[c + d*x])) - 1/(64*a^4*(a + a*Sin[c + d*x])^
4) - 1/(8*a^5*(a + a*Sin[c + d*x])^3) - 41/(64*a^6*(a + a*Sin[c + d*x])^2) - 105/(32*a^7*(a + a*Sin[c + d*x]))
))/d

Maple [A] (verified)

Time = 3.13 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.61

method	result	size
derivativedivides	\(\frac {-\frac {1}{3 \sin \left (d x +c \right )^{3}}+\frac {1}{2 \sin \left (d x +c \right )^{2}}-\frac {5}{\sin \left (d x +c \right )}-5 \ln \left (\sin \left (d x +c \right )\right )-\frac {1}{96 \left (\sin \left (d x +c \right )-1\right )^{3}}+\frac {13}{128 \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {95}{128 \left (\sin \left (d x +c \right )-1\right )}-\frac {515 \ln \left (\sin \left (d x +c \right )-1\right )}{256}-\frac {1}{64 \left (1+\sin \left (d x +c \right )\right )^{4}}-\frac {1}{8 \left (1+\sin \left (d x +c \right )\right )^{3}}-\frac {41}{64 \left (1+\sin \left (d x +c \right )\right )^{2}}-\frac {105}{32 \left (1+\sin \left (d x +c \right )\right )}+\frac {1795 \ln \left (1+\sin \left (d x +c \right )\right )}{256}}{d a}\)	\(154\)
default	\(\frac {-\frac {1}{3 \sin \left (d x +c \right )^{3}}+\frac {1}{2 \sin \left (d x +c \right )^{2}}-\frac {5}{\sin \left (d x +c \right )}-5 \ln \left (\sin \left (d x +c \right )\right )-\frac {1}{96 \left (\sin \left (d x +c \right )-1\right )^{3}}+\frac {13}{128 \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {95}{128 \left (\sin \left (d x +c \right )-1\right )}-\frac {515 \ln \left (\sin \left (d x +c \right )-1\right )}{256}-\frac {1}{64 \left (1+\sin \left (d x +c \right )\right )^{4}}-\frac {1}{8 \left (1+\sin \left (d x +c \right )\right )^{3}}-\frac {41}{64 \left (1+\sin \left (d x +c \right )\right )^{2}}-\frac {105}{32 \left (1+\sin \left (d x +c \right )\right )}+\frac {1795 \ln \left (1+\sin \left (d x +c \right )\right )}{256}}{d a}\)	\(154\)
risch	\(-\frac {i \left (-424 i {\mathrm e}^{14 i \left (d x +c \right )}+3465 \,{\mathrm e}^{19 i \left (d x +c \right )}+14640 i {\mathrm e}^{16 i \left (d x +c \right )}+9615 \,{\mathrm e}^{17 i \left (d x +c \right )}-38192 i {\mathrm e}^{8 i \left (d x +c \right )}-492 \,{\mathrm e}^{15 i \left (d x +c \right )}+5010 i {\mathrm e}^{18 i \left (d x +c \right )}-22084 \,{\mathrm e}^{13 i \left (d x +c \right )}-424 i {\mathrm e}^{6 i \left (d x +c \right )}-13394 \,{\mathrm e}^{11 i \left (d x +c \right )}-38192 i {\mathrm e}^{12 i \left (d x +c \right )}+13394 \,{\mathrm e}^{9 i \left (d x +c \right )}-27604 i {\mathrm e}^{10 i \left (d x +c \right )}+22084 \,{\mathrm e}^{7 i \left (d x +c \right )}+14640 i {\mathrm e}^{4 i \left (d x +c \right )}+492 \,{\mathrm e}^{5 i \left (d x +c \right )}+5010 i {\mathrm e}^{2 i \left (d x +c \right )}-9615 \,{\mathrm e}^{3 i \left (d x +c \right )}-3465 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{192 \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{3} \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{8} \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{6} d a}-\frac {515 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{128 d a}+\frac {1795 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{128 a d}-\frac {5 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d a}\)	\(333\)
parallelrisch	\(-\frac {5 \left (\csc ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\frac {2101}{7680}+\frac {103 \left (-\frac {\cos \left (4 d x +4 c \right )}{2}+\frac {3 \sin \left (d x +c \right )}{4}+\sin \left (3 d x +3 c \right )+\frac {\cos \left (2 d x +2 c \right )}{8}+\frac {\cos \left (10 d x +10 c \right )}{16}-\frac {3 \sin \left (7 d x +7 c \right )}{8}-\frac {3 \cos \left (6 d x +6 c \right )}{16}-\frac {\sin \left (9 d x +9 c \right )}{8}+\frac {\cos \left (8 d x +8 c \right )}{8}+\frac {3}{8}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{128}+\frac {359 \left (-\frac {3}{8}-\frac {\cos \left (2 d x +2 c \right )}{8}+\frac {\sin \left (9 d x +9 c \right )}{8}-\frac {\cos \left (8 d x +8 c \right )}{8}+\frac {3 \sin \left (7 d x +7 c \right )}{8}+\frac {3 \cos \left (6 d x +6 c \right )}{16}-\frac {3 \sin \left (d x +c \right )}{4}-\sin \left (3 d x +3 c \right )+\frac {\cos \left (4 d x +4 c \right )}{2}-\frac {\cos \left (10 d x +10 c \right )}{16}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{128}+\left (-\frac {\cos \left (4 d x +4 c \right )}{2}+\frac {3 \sin \left (d x +c \right )}{4}+\sin \left (3 d x +3 c \right )+\frac {\cos \left (2 d x +2 c \right )}{8}+\frac {\cos \left (10 d x +10 c \right )}{16}-\frac {3 \sin \left (7 d x +7 c \right )}{8}-\frac {3 \cos \left (6 d x +6 c \right )}{16}-\frac {\sin \left (9 d x +9 c \right )}{8}+\frac {\cos \left (8 d x +8 c \right )}{8}+\frac {3}{8}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {729 \cos \left (2 d x +2 c \right )}{320}+\frac {551 \cos \left (4 d x +4 c \right )}{640}-\frac {293 \sin \left (3 d x +3 c \right )}{1280}-\frac {2903 \sin \left (d x +c \right )}{7680}-\frac {821 \cos \left (8 d x +8 c \right )}{1536}-\frac {53 \sin \left (9 d x +9 c \right )}{3072}-\frac {\sin \left (7 d x +7 c \right )}{1024}-\frac {5 \cos \left (10 d x +10 c \right )}{48}+\frac {41 \sin \left (5 d x +5 c \right )}{1280}-\frac {41 \cos \left (6 d x +6 c \right )}{64}\right ) \left (\sec ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 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 )}\)	\(545\)

[In]

int(csc(d*x+c)^4*sec(d*x+c)^7/(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

1/d/a*(-1/3/sin(d*x+c)^3+1/2/sin(d*x+c)^2-5/sin(d*x+c)-5*ln(sin(d*x+c))-1/96/(sin(d*x+c)-1)^3+13/128/(sin(d*x+
c)-1)^2-95/128/(sin(d*x+c)-1)-515/256*ln(sin(d*x+c)-1)-1/64/(1+sin(d*x+c))^4-1/8/(1+sin(d*x+c))^3-41/64/(1+sin
(d*x+c))^2-105/32/(1+sin(d*x+c))+1795/256*ln(1+sin(d*x+c)))

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 360, normalized size of antiderivative = 1.42 \[ \int \frac {\csc ^4(c+d x) \sec ^7(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {5010 \, \cos \left (d x + c\right )^{8} - 6360 \, \cos \left (d x + c\right )^{6} + 746 \, \cos \left (d x + c\right )^{4} + 236 \, \cos \left (d x + c\right )^{2} - 3840 \, {\left (\cos \left (d x + c\right )^{10} - 2 \, \cos \left (d x + c\right )^{8} + \cos \left (d x + c\right )^{6} - {\left (\cos \left (d x + c\right )^{8} - \cos \left (d x + c\right )^{6}\right )} \sin \left (d x + c\right )\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) + 5385 \, {\left (\cos \left (d x + c\right )^{10} - 2 \, \cos \left (d x + c\right )^{8} + \cos \left (d x + c\right )^{6} - {\left (\cos \left (d x + c\right )^{8} - \cos \left (d x + c\right )^{6}\right )} \sin \left (d x + c\right )\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 1545 \, {\left (\cos \left (d x + c\right )^{10} - 2 \, \cos \left (d x + c\right )^{8} + \cos \left (d x + c\right )^{6} - {\left (\cos \left (d x + c\right )^{8} - \cos \left (d x + c\right )^{6}\right )} \sin \left (d x + c\right )\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (3465 \, \cos \left (d x + c\right )^{8} - 3660 \, \cos \left (d x + c\right )^{6} + 213 \, \cos \left (d x + c\right )^{4} + 38 \, \cos \left (d x + c\right )^{2} + 8\right )} \sin \left (d x + c\right ) + 112}{768 \, {\left (a d \cos \left (d x + c\right )^{10} - 2 \, a d \cos \left (d x + c\right )^{8} + a d \cos \left (d x + c\right )^{6} - {\left (a d \cos \left (d x + c\right )^{8} - a d \cos \left (d x + c\right )^{6}\right )} \sin \left (d x + c\right )\right )}} \]

[In]

integrate(csc(d*x+c)^4*sec(d*x+c)^7/(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

1/768*(5010*cos(d*x + c)^8 - 6360*cos(d*x + c)^6 + 746*cos(d*x + c)^4 + 236*cos(d*x + c)^2 - 3840*(cos(d*x + c
)^10 - 2*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
)) + 5385*(cos(d*x + c)^10 - 2*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) - 1545*(cos(d*x + c)^10 - 2*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*(3465*cos(d*x + c)^8 - 3660*cos(d*x + c)^6 + 213*cos(d*x +
c)^4 + 38*cos(d*x + c)^2 + 8)*sin(d*x + c) + 112)/(a*d*cos(d*x + c)^10 - 2*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. \[ \int \frac {\csc ^4(c+d x) \sec ^7(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Timed out} \]

[In]

integrate(csc(d*x+c)**4*sec(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 = 227, normalized size of antiderivative = 0.90 \[ \int \frac {\csc ^4(c+d x) \sec ^7(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\frac {2 \, {\left (3465 \, \sin \left (d x + c\right )^{9} + 2505 \, \sin \left (d x + c\right )^{8} - 10200 \, \sin \left (d x + c\right )^{7} - 6840 \, \sin \left (d x + c\right )^{6} + 10023 \, \sin \left (d x + c\right )^{5} + 5863 \, \sin \left (d x + c\right )^{4} - 3344 \, \sin \left (d x + c\right )^{3} - 1344 \, \sin \left (d x + c\right )^{2} + 64 \, \sin \left (d x + c\right ) - 128\right )}}{a \sin \left (d x + c\right )^{10} + a \sin \left (d x + c\right )^{9} - 3 \, a \sin \left (d x + c\right )^{8} - 3 \, a \sin \left (d x + c\right )^{7} + 3 \, a \sin \left (d x + c\right )^{6} + 3 \, a \sin \left (d x + c\right )^{5} - a \sin \left (d x + c\right )^{4} - a \sin \left (d x + c\right )^{3}} - \frac {5385 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a} + \frac {1545 \, \log \left (\sin \left (d x + c\right ) - 1\right )}{a} + \frac {3840 \, \log \left (\sin \left (d x + c\right )\right )}{a}}{768 \, d} \]

[In]

integrate(csc(d*x+c)^4*sec(d*x+c)^7/(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

-1/768*(2*(3465*sin(d*x + c)^9 + 2505*sin(d*x + c)^8 - 10200*sin(d*x + c)^7 - 6840*sin(d*x + c)^6 + 10023*sin(
d*x + c)^5 + 5863*sin(d*x + c)^4 - 3344*sin(d*x + c)^3 - 1344*sin(d*x + c)^2 + 64*sin(d*x + c) - 128)/(a*sin(d
*x + c)^10 + a*sin(d*x + c)^9 - 3*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 - a*sin(d*x + c)^4 - a*sin(d*x + c)^3) - 5385*log(sin(d*x + c) + 1)/a + 1545*log(sin(d*x + c) - 1)/a + 384
0*log(sin(d*x + c))/a)/d

Giac [A] (verification not implemented)

none

Time = 0.58 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.74 \[ \int \frac {\csc ^4(c+d x) \sec ^7(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {21540 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a} - \frac {6180 \, \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a} - \frac {15360 \, \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a} + \frac {19745 \, \sin \left (d x + c\right )^{6} - 76875 \, \sin \left (d x + c\right )^{5} + 111723 \, \sin \left (d x + c\right )^{4} - 74081 \, \sin \left (d x + c\right )^{3} + 23040 \, \sin \left (d x + c\right )^{2} - 4608 \, \sin \left (d x + c\right ) + 1024}{{\left (\sin \left (d x + c\right )^{2} - \sin \left (d x + c\right )\right )}^{3} a} - \frac {44875 \, \sin \left (d x + c\right )^{4} + 189580 \, \sin \left (d x + c\right )^{3} + 301458 \, \sin \left (d x + c\right )^{2} + 214060 \, \sin \left (d x + c\right ) + 57355}{a {\left (\sin \left (d x + c\right ) + 1\right )}^{4}}}{3072 \, d} \]

[In]

integrate(csc(d*x+c)^4*sec(d*x+c)^7/(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

1/3072*(21540*log(abs(sin(d*x + c) + 1))/a - 6180*log(abs(sin(d*x + c) - 1))/a - 15360*log(abs(sin(d*x + c)))/
a + (19745*sin(d*x + c)^6 - 76875*sin(d*x + c)^5 + 111723*sin(d*x + c)^4 - 74081*sin(d*x + c)^3 + 23040*sin(d*
x + c)^2 - 4608*sin(d*x + c) + 1024)/((sin(d*x + c)^2 - sin(d*x + c))^3*a) - (44875*sin(d*x + c)^4 + 189580*si
n(d*x + c)^3 + 301458*sin(d*x + c)^2 + 214060*sin(d*x + c) + 57355)/(a*(sin(d*x + c) + 1)^4))/d

Mupad [B] (verification not implemented)

Time = 9.57 (sec) , antiderivative size = 233, normalized size of antiderivative = 0.92 \[ \int \frac {\csc ^4(c+d x) \sec ^7(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {1155\,{\sin \left (c+d\,x\right )}^9}{128}+\frac {835\,{\sin \left (c+d\,x\right )}^8}{128}-\frac {425\,{\sin \left (c+d\,x\right )}^7}{16}-\frac {285\,{\sin \left (c+d\,x\right )}^6}{16}+\frac {3341\,{\sin \left (c+d\,x\right )}^5}{128}+\frac {5863\,{\sin \left (c+d\,x\right )}^4}{384}-\frac {209\,{\sin \left (c+d\,x\right )}^3}{24}-\frac {7\,{\sin \left (c+d\,x\right )}^2}{2}+\frac {\sin \left (c+d\,x\right )}{6}-\frac {1}{3}}{d\,\left (-a\,{\sin \left (c+d\,x\right )}^{10}-a\,{\sin \left (c+d\,x\right )}^9+3\,a\,{\sin \left (c+d\,x\right )}^8+3\,a\,{\sin \left (c+d\,x\right )}^7-3\,a\,{\sin \left (c+d\,x\right )}^6-3\,a\,{\sin \left (c+d\,x\right )}^5+a\,{\sin \left (c+d\,x\right )}^4+a\,{\sin \left (c+d\,x\right )}^3\right )}-\frac {515\,\ln \left (\sin \left (c+d\,x\right )-1\right )}{256\,a\,d}+\frac {1795\,\ln \left (\sin \left (c+d\,x\right )+1\right )}{256\,a\,d}-\frac {5\,\ln \left (\sin \left (c+d\,x\right )\right )}{a\,d} \]

[In]

int(1/(cos(c + d*x)^7*sin(c + d*x)^4*(a + a*sin(c + d*x))),x)

[Out]

(sin(c + d*x)/6 - (7*sin(c + d*x)^2)/2 - (209*sin(c + d*x)^3)/24 + (5863*sin(c + d*x)^4)/384 + (3341*sin(c + d
*x)^5)/128 - (285*sin(c + d*x)^6)/16 - (425*sin(c + d*x)^7)/16 + (835*sin(c + d*x)^8)/128 + (1155*sin(c + d*x)
^9)/128 - 1/3)/(d*(a*sin(c + d*x)^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 + 3*a*sin(c + d*x)^8 - a*sin(c + d*x)^9 - a*sin(c + d*x)^10)) - (515*log(sin(c + d*x) - 1))/(256*a*d)
+ (1795*log(sin(c + d*x) + 1))/(256*a*d) - (5*log(sin(c + d*x)))/(a*d)