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[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]))

________________________________________________________________________________________

Rubi [A]  time = 0.302406, antiderivative size = 279, normalized size of antiderivative = 1., 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 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]

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]))

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 = 6.22719, size = 254, normalized size = 0.91 \[ \frac{a^{12} \left (\frac{203}{256 a^{12} (a-a \sin (c+d x))}+\frac{5}{2 a^{12} (a \sin (c+d x)+a)}+\frac{81}{512 a^{11} (a-a \sin (c+d x))^2}+\frac{325}{512 a^{11} (a \sin (c+d x)+a)^2}+\frac{1}{32 a^{10} (a-a \sin (c+d x))^3}+\frac{23}{128 a^{10} (a \sin (c+d x)+a)^3}+\frac{1}{256 a^9 (a-a \sin (c+d x))^4}+\frac{11}{256 a^9 (a \sin (c+d x)+a)^4}+\frac{1}{160 a^8 (a \sin (c+d x)+a)^5}-\frac{\csc ^2(c+d x)}{2 a^{13}}+\frac{\csc (c+d x)}{a^{13}}-\frac{843 \log (1-\sin (c+d x))}{512 a^{13}}+\frac{6 \log (\sin (c+d x))}{a^{13}}-\frac{2229 \log (\sin (c+d x)+1)}{512 a^{13}}\right )}{d} \]

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

________________________________________________________________________________________

Maple [A]  time = 0.112, size = 244, normalized size = 0.9 \begin{align*}{\frac{1}{256\,da \left ( \sin \left ( dx+c \right ) -1 \right ) ^{4}}}-{\frac{1}{32\,da \left ( \sin \left ( dx+c \right ) -1 \right ) ^{3}}}+{\frac{81}{512\,da \left ( \sin \left ( dx+c \right ) -1 \right ) ^{2}}}-{\frac{203}{256\,da \left ( \sin \left ( dx+c \right ) -1 \right ) }}-{\frac{843\,\ln \left ( \sin \left ( dx+c \right ) -1 \right ) }{512\,da}}+{\frac{1}{160\,da \left ( 1+\sin \left ( dx+c \right ) \right ) ^{5}}}+{\frac{11}{256\,da \left ( 1+\sin \left ( dx+c \right ) \right ) ^{4}}}+{\frac{23}{128\,da \left ( 1+\sin \left ( dx+c \right ) \right ) ^{3}}}+{\frac{325}{512\,da \left ( 1+\sin \left ( dx+c \right ) \right ) ^{2}}}+{\frac{5}{2\,da \left ( 1+\sin \left ( dx+c \right ) \right ) }}-{\frac{2229\,\ln \left ( 1+\sin \left ( dx+c \right ) \right ) }{512\,da}}-{\frac{1}{2\,da \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}+{\frac{1}{da\sin \left ( dx+c \right ) }}+6\,{\frac{\ln \left ( \sin \left ( dx+c \right ) \right ) }{da}} \end{align*}

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

________________________________________________________________________________________

Maxima [A]  time = 1.03702, size = 347, normalized size = 1.24 \begin{align*} \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} \end{align*}

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

________________________________________________________________________________________

Fricas [A]  time = 2.33834, size = 910, normalized size = 3.26 \begin{align*} \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 )}} \end{align*}

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))

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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

________________________________________________________________________________________

Giac [A]  time = 1.28225, size = 273, normalized size = 0.98 \begin{align*} -\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} \end{align*}

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