3.910 \(\int \frac {\csc ^3(c+d x) \sec ^9(c+d x)}{a+a \sin (c+d x)} \, dx\)

Optimal. Leaf size=279 \[ \frac {a^4}{160 d (a \sin (c+d x)+a)^5}+\frac {a^3}{256 d (a-a \sin (c+d x))^4}+\frac {11 a^3}{256 d (a \sin (c+d x)+a)^4}+\frac {a^2}{32 d (a-a \sin (c+d x))^3}+\frac {23 a^2}{128 d (a \sin (c+d x)+a)^3}+\frac {81 a}{512 d (a-a \sin (c+d x))^2}+\frac {325 a}{512 d (a \sin (c+d x)+a)^2}+\frac {203}{256 d (a-a \sin (c+d x))}+\frac {5}{2 d (a \sin (c+d x)+a)}-\frac {\csc ^2(c+d x)}{2 a d}+\frac {\csc (c+d x)}{a d}-\frac {843 \log (1-\sin (c+d x))}{512 a d}+\frac {6 \log (\sin (c+d x))}{a d}-\frac {2229 \log (\sin (c+d x)+1)}{512 a d} \]

[Out]

csc(d*x+c)/a/d-1/2*csc(d*x+c)^2/a/d-843/512*ln(1-sin(d*x+c))/a/d+6*ln(sin(d*x+c))/a/d-2229/512*ln(1+sin(d*x+c)
)/a/d+1/256*a^3/d/(a-a*sin(d*x+c))^4+1/32*a^2/d/(a-a*sin(d*x+c))^3+81/512*a/d/(a-a*sin(d*x+c))^2+203/256/d/(a-
a*sin(d*x+c))+1/160*a^4/d/(a+a*sin(d*x+c))^5+11/256*a^3/d/(a+a*sin(d*x+c))^4+23/128*a^2/d/(a+a*sin(d*x+c))^3+3
25/512*a/d/(a+a*sin(d*x+c))^2+5/2/d/(a+a*sin(d*x+c))

________________________________________________________________________________________

Rubi [A]  time = 0.30, antiderivative size = 279, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {2836, 12, 88} \[ \frac {a^4}{160 d (a \sin (c+d x)+a)^5}+\frac {a^3}{256 d (a-a \sin (c+d x))^4}+\frac {11 a^3}{256 d (a \sin (c+d x)+a)^4}+\frac {a^2}{32 d (a-a \sin (c+d x))^3}+\frac {23 a^2}{128 d (a \sin (c+d x)+a)^3}+\frac {81 a}{512 d (a-a \sin (c+d x))^2}+\frac {325 a}{512 d (a \sin (c+d x)+a)^2}+\frac {203}{256 d (a-a \sin (c+d x))}+\frac {5}{2 d (a \sin (c+d x)+a)}-\frac {\csc ^2(c+d x)}{2 a d}+\frac {\csc (c+d x)}{a d}-\frac {843 \log (1-\sin (c+d x))}{512 a d}+\frac {6 \log (\sin (c+d x))}{a d}-\frac {2229 \log (\sin (c+d x)+1)}{512 a d} \]

Antiderivative was successfully verified.

[In]

Int[(Csc[c + d*x]^3*Sec[c + d*x]^9)/(a + a*Sin[c + d*x]),x]

[Out]

Csc[c + d*x]/(a*d) - Csc[c + d*x]^2/(2*a*d) - (843*Log[1 - Sin[c + d*x]])/(512*a*d) + (6*Log[Sin[c + d*x]])/(a
*d) - (2229*Log[1 + Sin[c + d*x]])/(512*a*d) + a^3/(256*d*(a - a*Sin[c + d*x])^4) + a^2/(32*d*(a - a*Sin[c + d
*x])^3) + (81*a)/(512*d*(a - a*Sin[c + d*x])^2) + 203/(256*d*(a - a*Sin[c + d*x])) + a^4/(160*d*(a + a*Sin[c +
d*x])^5) + (11*a^3)/(256*d*(a + a*Sin[c + d*x])^4) + (23*a^2)/(128*d*(a + a*Sin[c + d*x])^3) + (325*a)/(512*d
*(a + a*Sin[c + d*x])^2) + 5/(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 88

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 2836

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*x)/b
)^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*} \int \frac {\csc ^3(c+d x) \sec ^9(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac {a^9 \operatorname {Subst}\left (\int \frac {a^3}{(a-x)^5 x^3 (a+x)^6} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a^{12} \operatorname {Subst}\left (\int \frac {1}{(a-x)^5 x^3 (a+x)^6} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a^{12} \operatorname {Subst}\left (\int \left (\frac {1}{64 a^9 (a-x)^5}+\frac {3}{32 a^{10} (a-x)^4}+\frac {81}{256 a^{11} (a-x)^3}+\frac {203}{256 a^{12} (a-x)^2}+\frac {843}{512 a^{13} (a-x)}+\frac {1}{a^{11} x^3}-\frac {1}{a^{12} x^2}+\frac {6}{a^{13} x}-\frac {1}{32 a^8 (a+x)^6}-\frac {11}{64 a^9 (a+x)^5}-\frac {69}{128 a^{10} (a+x)^4}-\frac {325}{256 a^{11} (a+x)^3}-\frac {5}{2 a^{12} (a+x)^2}-\frac {2229}{512 a^{13} (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 {843 \log (1-\sin (c+d x))}{512 a d}+\frac {6 \log (\sin (c+d x))}{a d}-\frac {2229 \log (1+\sin (c+d x))}{512 a d}+\frac {a^3}{256 d (a-a \sin (c+d x))^4}+\frac {a^2}{32 d (a-a \sin (c+d x))^3}+\frac {81 a}{512 d (a-a \sin (c+d x))^2}+\frac {203}{256 d (a-a \sin (c+d x))}+\frac {a^4}{160 d (a+a \sin (c+d x))^5}+\frac {11 a^3}{256 d (a+a \sin (c+d x))^4}+\frac {23 a^2}{128 d (a+a \sin (c+d x))^3}+\frac {325 a}{512 d (a+a \sin (c+d x))^2}+\frac {5}{2 d (a+a \sin (c+d x))}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 6.23, size = 254, normalized size = 0.91 \[ \frac {a^{12} \left (-\frac {\csc ^2(c+d x)}{2 a^{13}}+\frac {\csc (c+d x)}{a^{13}}-\frac {843 \log (1-\sin (c+d x))}{512 a^{13}}+\frac {6 \log (\sin (c+d x))}{a^{13}}-\frac {2229 \log (\sin (c+d x)+1)}{512 a^{13}}+\frac {203}{256 a^{12} (a-a \sin (c+d x))}+\frac {5}{2 a^{12} (a \sin (c+d x)+a)}+\frac {81}{512 a^{11} (a-a \sin (c+d x))^2}+\frac {325}{512 a^{11} (a \sin (c+d x)+a)^2}+\frac {1}{32 a^{10} (a-a \sin (c+d x))^3}+\frac {23}{128 a^{10} (a \sin (c+d x)+a)^3}+\frac {1}{256 a^9 (a-a \sin (c+d x))^4}+\frac {11}{256 a^9 (a \sin (c+d x)+a)^4}+\frac {1}{160 a^8 (a \sin (c+d x)+a)^5}\right )}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[(Csc[c + d*x]^3*Sec[c + d*x]^9)/(a + a*Sin[c + d*x]),x]

[Out]

(a^12*(Csc[c + d*x]/a^13 - Csc[c + d*x]^2/(2*a^13) - (843*Log[1 - Sin[c + d*x]])/(512*a^13) + (6*Log[Sin[c + d
*x]])/a^13 - (2229*Log[1 + Sin[c + d*x]])/(512*a^13) + 1/(256*a^9*(a - a*Sin[c + d*x])^4) + 1/(32*a^10*(a - a*
Sin[c + d*x])^3) + 81/(512*a^11*(a - a*Sin[c + d*x])^2) + 203/(256*a^12*(a - a*Sin[c + d*x])) + 1/(160*a^8*(a
+ a*Sin[c + d*x])^5) + 11/(256*a^9*(a + a*Sin[c + d*x])^4) + 23/(128*a^10*(a + a*Sin[c + d*x])^3) + 325/(512*a
^11*(a + a*Sin[c + d*x])^2) + 5/(2*a^12*(a + a*Sin[c + d*x]))))/d

________________________________________________________________________________________

fricas [A]  time = 0.54, size = 331, normalized size = 1.19 \[ \frac {6930 \, \cos \left (d x + c\right )^{10} - 1560 \, \cos \left (d x + c\right )^{8} - 2454 \, \cos \left (d x + c\right )^{6} - 884 \, \cos \left (d x + c\right )^{4} - 464 \, \cos \left (d x + c\right )^{2} + 15360 \, {\left (\cos \left (d x + c\right )^{10} - \cos \left (d x + c\right )^{8} + {\left (\cos \left (d x + c\right )^{10} - \cos \left (d x + c\right )^{8}\right )} \sin \left (d x + c\right )\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) - 11145 \, {\left (\cos \left (d x + c\right )^{10} - \cos \left (d x + c\right )^{8} + {\left (\cos \left (d x + c\right )^{10} - \cos \left (d x + c\right )^{8}\right )} \sin \left (d x + c\right )\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 4215 \, {\left (\cos \left (d x + c\right )^{10} - \cos \left (d x + c\right )^{8} + {\left (\cos \left (d x + c\right )^{10} - \cos \left (d x + c\right )^{8}\right )} \sin \left (d x + c\right )\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (375 \, \cos \left (d x + c\right )^{8} - 765 \, \cos \left (d x + c\right )^{6} - 178 \, \cos \left (d x + c\right )^{4} - 56 \, \cos \left (d x + c\right )^{2} - 16\right )} \sin \left (d x + c\right ) - 288}{2560 \, {\left (a d \cos \left (d x + c\right )^{10} - a d \cos \left (d x + c\right )^{8} + {\left (a d \cos \left (d x + c\right )^{10} - a d \cos \left (d x + c\right )^{8}\right )} \sin \left (d x + c\right )\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)^3*sec(d*x+c)^9/(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

1/2560*(6930*cos(d*x + c)^10 - 1560*cos(d*x + c)^8 - 2454*cos(d*x + c)^6 - 884*cos(d*x + c)^4 - 464*cos(d*x +
c)^2 + 15360*(cos(d*x + c)^10 - cos(d*x + c)^8 + (cos(d*x + c)^10 - cos(d*x + c)^8)*sin(d*x + c))*log(1/2*sin(
d*x + c)) - 11145*(cos(d*x + c)^10 - cos(d*x + c)^8 + (cos(d*x + c)^10 - cos(d*x + c)^8)*sin(d*x + c))*log(sin
(d*x + c) + 1) - 4215*(cos(d*x + c)^10 - cos(d*x + c)^8 + (cos(d*x + c)^10 - cos(d*x + c)^8)*sin(d*x + c))*log
(-sin(d*x + c) + 1) + 2*(375*cos(d*x + c)^8 - 765*cos(d*x + c)^6 - 178*cos(d*x + c)^4 - 56*cos(d*x + c)^2 - 16
)*sin(d*x + c) - 288)/(a*d*cos(d*x + c)^10 - a*d*cos(d*x + c)^8 + (a*d*cos(d*x + c)^10 - a*d*cos(d*x + c)^8)*s
in(d*x + c))

________________________________________________________________________________________

giac [A]  time = 0.29, size = 202, normalized size = 0.72 \[ -\frac {\frac {44580 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a} + \frac {16860 \, \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a} - \frac {61440 \, \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a} + \frac {5120 \, {\left (18 \, \sin \left (d x + c\right )^{2} - 2 \, \sin \left (d x + c\right ) + 1\right )}}{a \sin \left (d x + c\right )^{2}} - \frac {5 \, {\left (7025 \, \sin \left (d x + c\right )^{4} - 29724 \, \sin \left (d x + c\right )^{3} + 47346 \, \sin \left (d x + c\right )^{2} - 33684 \, \sin \left (d x + c\right ) + 9045\right )}}{a {\left (\sin \left (d x + c\right ) - 1\right )}^{4}} - \frac {101791 \, \sin \left (d x + c\right )^{5} + 534555 \, \sin \left (d x + c\right )^{4} + 1126810 \, \sin \left (d x + c\right )^{3} + 1192850 \, \sin \left (d x + c\right )^{2} + 634975 \, \sin \left (d x + c\right ) + 136235}{a {\left (\sin \left (d x + c\right ) + 1\right )}^{5}}}{10240 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)^3*sec(d*x+c)^9/(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

-1/10240*(44580*log(abs(sin(d*x + c) + 1))/a + 16860*log(abs(sin(d*x + c) - 1))/a - 61440*log(abs(sin(d*x + c)
))/a + 5120*(18*sin(d*x + c)^2 - 2*sin(d*x + c) + 1)/(a*sin(d*x + c)^2) - 5*(7025*sin(d*x + c)^4 - 29724*sin(d
*x + c)^3 + 47346*sin(d*x + c)^2 - 33684*sin(d*x + c) + 9045)/(a*(sin(d*x + c) - 1)^4) - (101791*sin(d*x + c)^
5 + 534555*sin(d*x + c)^4 + 1126810*sin(d*x + c)^3 + 1192850*sin(d*x + c)^2 + 634975*sin(d*x + c) + 136235)/(a
*(sin(d*x + c) + 1)^5))/d

________________________________________________________________________________________

maple [A]  time = 0.52, size = 244, normalized size = 0.87 \[ \frac {1}{256 a d \left (\sin \left (d x +c \right )-1\right )^{4}}-\frac {1}{32 a d \left (\sin \left (d x +c \right )-1\right )^{3}}+\frac {81}{512 a d \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {203}{256 a d \left (\sin \left (d x +c \right )-1\right )}-\frac {843 \ln \left (\sin \left (d x +c \right )-1\right )}{512 a d}-\frac {1}{2 a d \sin \left (d x +c \right )^{2}}+\frac {1}{d a \sin \left (d x +c \right )}+\frac {6 \ln \left (\sin \left (d x +c \right )\right )}{a d}+\frac {1}{160 a d \left (1+\sin \left (d x +c \right )\right )^{5}}+\frac {11}{256 a d \left (1+\sin \left (d x +c \right )\right )^{4}}+\frac {23}{128 a d \left (1+\sin \left (d x +c \right )\right )^{3}}+\frac {325}{512 a d \left (1+\sin \left (d x +c \right )\right )^{2}}+\frac {5}{2 a d \left (1+\sin \left (d x +c \right )\right )}-\frac {2229 \ln \left (1+\sin \left (d x +c \right )\right )}{512 a d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(d*x+c)^3*sec(d*x+c)^9/(a+a*sin(d*x+c)),x)

[Out]

1/256/a/d/(sin(d*x+c)-1)^4-1/32/a/d/(sin(d*x+c)-1)^3+81/512/a/d/(sin(d*x+c)-1)^2-203/256/a/d/(sin(d*x+c)-1)-84
3/512/a/d*ln(sin(d*x+c)-1)-1/2/a/d/sin(d*x+c)^2+1/d/a/sin(d*x+c)+6*ln(sin(d*x+c))/a/d+1/160/a/d/(1+sin(d*x+c))
^5+11/256/a/d/(1+sin(d*x+c))^4+23/128/a/d/(1+sin(d*x+c))^3+325/512/a/d/(1+sin(d*x+c))^2+5/2/a/d/(1+sin(d*x+c))
-2229/512*ln(1+sin(d*x+c))/a/d

________________________________________________________________________________________

maxima [A]  time = 0.33, size = 257, normalized size = 0.92 \[ \frac {\frac {2 \, {\left (3465 \, \sin \left (d x + c\right )^{10} - 375 \, \sin \left (d x + c\right )^{9} - 16545 \, \sin \left (d x + c\right )^{8} + 735 \, \sin \left (d x + c\right )^{7} + 30303 \, \sin \left (d x + c\right )^{6} + 223 \, \sin \left (d x + c\right )^{5} - 25847 \, \sin \left (d x + c\right )^{4} - 1207 \, \sin \left (d x + c\right )^{3} + 9408 \, \sin \left (d x + c\right )^{2} + 640 \, \sin \left (d x + c\right ) - 640\right )}}{a \sin \left (d x + c\right )^{11} + a \sin \left (d x + c\right )^{10} - 4 \, a \sin \left (d x + c\right )^{9} - 4 \, a \sin \left (d x + c\right )^{8} + 6 \, a \sin \left (d x + c\right )^{7} + 6 \, a \sin \left (d x + c\right )^{6} - 4 \, a \sin \left (d x + c\right )^{5} - 4 \, a \sin \left (d x + c\right )^{4} + a \sin \left (d x + c\right )^{3} + a \sin \left (d x + c\right )^{2}} - \frac {11145 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a} - \frac {4215 \, \log \left (\sin \left (d x + c\right ) - 1\right )}{a} + \frac {15360 \, \log \left (\sin \left (d x + c\right )\right )}{a}}{2560 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)^3*sec(d*x+c)^9/(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

1/2560*(2*(3465*sin(d*x + c)^10 - 375*sin(d*x + c)^9 - 16545*sin(d*x + c)^8 + 735*sin(d*x + c)^7 + 30303*sin(d
*x + c)^6 + 223*sin(d*x + c)^5 - 25847*sin(d*x + c)^4 - 1207*sin(d*x + c)^3 + 9408*sin(d*x + c)^2 + 640*sin(d*
x + c) - 640)/(a*sin(d*x + c)^11 + a*sin(d*x + c)^10 - 4*a*sin(d*x + c)^9 - 4*a*sin(d*x + c)^8 + 6*a*sin(d*x +
c)^7 + 6*a*sin(d*x + c)^6 - 4*a*sin(d*x + c)^5 - 4*a*sin(d*x + c)^4 + a*sin(d*x + c)^3 + a*sin(d*x + c)^2) -
11145*log(sin(d*x + c) + 1)/a - 4215*log(sin(d*x + c) - 1)/a + 15360*log(sin(d*x + c))/a)/d

________________________________________________________________________________________

mupad [B]  time = 9.64, size = 263, normalized size = 0.94 \[ \frac {\frac {693\,{\sin \left (c+d\,x\right )}^{10}}{256}-\frac {75\,{\sin \left (c+d\,x\right )}^9}{256}-\frac {3309\,{\sin \left (c+d\,x\right )}^8}{256}+\frac {147\,{\sin \left (c+d\,x\right )}^7}{256}+\frac {30303\,{\sin \left (c+d\,x\right )}^6}{1280}+\frac {223\,{\sin \left (c+d\,x\right )}^5}{1280}-\frac {25847\,{\sin \left (c+d\,x\right )}^4}{1280}-\frac {1207\,{\sin \left (c+d\,x\right )}^3}{1280}+\frac {147\,{\sin \left (c+d\,x\right )}^2}{20}+\frac {\sin \left (c+d\,x\right )}{2}-\frac {1}{2}}{d\,\left (a\,{\sin \left (c+d\,x\right )}^{11}+a\,{\sin \left (c+d\,x\right )}^{10}-4\,a\,{\sin \left (c+d\,x\right )}^9-4\,a\,{\sin \left (c+d\,x\right )}^8+6\,a\,{\sin \left (c+d\,x\right )}^7+6\,a\,{\sin \left (c+d\,x\right )}^6-4\,a\,{\sin \left (c+d\,x\right )}^5-4\,a\,{\sin \left (c+d\,x\right )}^4+a\,{\sin \left (c+d\,x\right )}^3+a\,{\sin \left (c+d\,x\right )}^2\right )}-\frac {2229\,\ln \left (\sin \left (c+d\,x\right )+1\right )}{512\,a\,d}-\frac {843\,\ln \left (\sin \left (c+d\,x\right )-1\right )}{512\,a\,d}+\frac {6\,\ln \left (\sin \left (c+d\,x\right )\right )}{a\,d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cos(c + d*x)^9*sin(c + d*x)^3*(a + a*sin(c + d*x))),x)

[Out]

(sin(c + d*x)/2 + (147*sin(c + d*x)^2)/20 - (1207*sin(c + d*x)^3)/1280 - (25847*sin(c + d*x)^4)/1280 + (223*si
n(c + d*x)^5)/1280 + (30303*sin(c + d*x)^6)/1280 + (147*sin(c + d*x)^7)/256 - (3309*sin(c + d*x)^8)/256 - (75*
sin(c + d*x)^9)/256 + (693*sin(c + d*x)^10)/256 - 1/2)/(d*(a*sin(c + d*x)^2 + a*sin(c + d*x)^3 - 4*a*sin(c + d
*x)^4 - 4*a*sin(c + d*x)^5 + 6*a*sin(c + d*x)^6 + 6*a*sin(c + d*x)^7 - 4*a*sin(c + d*x)^8 - 4*a*sin(c + d*x)^9
+ a*sin(c + d*x)^10 + a*sin(c + d*x)^11)) - (2229*log(sin(c + d*x) + 1))/(512*a*d) - (843*log(sin(c + d*x) -
1))/(512*a*d) + (6*log(sin(c + d*x)))/(a*d)

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)**3*sec(d*x+c)**9/(a+a*sin(d*x+c)),x)

[Out]

Timed out

________________________________________________________________________________________