3.909 \(\int \frac{\csc ^2(c+d x) \sec ^9(c+d x)}{a+a \sin (c+d x)} \, dx\)

Optimal. Leaf size=262 \[ -\frac{a^4}{160 d (a \sin (c+d x)+a)^5}+\frac{a^3}{256 d (a-a \sin (c+d x))^4}-\frac{9 a^3}{256 d (a \sin (c+d x)+a)^4}+\frac{5 a^2}{192 d (a-a \sin (c+d x))^3}-\frac{47 a^2}{384 d (a \sin (c+d x)+a)^3}+\frac{57 a}{512 d (a-a \sin (c+d x))^2}-\frac{187 a}{512 d (a \sin (c+d x)+a)^2}+\frac{61}{128 d (a-a \sin (c+d x))}-\frac{315}{256 d (a \sin (c+d x)+a)}-\frac{\csc (c+d x)}{a d}-\frac{437 \log (1-\sin (c+d x))}{512 a d}-\frac{\log (\sin (c+d x))}{a d}+\frac{949 \log (\sin (c+d x)+1)}{512 a d} \]

[Out]

-(Csc[c + d*x]/(a*d)) - (437*Log[1 - Sin[c + d*x]])/(512*a*d) - Log[Sin[c + d*x]]/(a*d) + (949*Log[1 + Sin[c +
d*x]])/(512*a*d) + a^3/(256*d*(a - a*Sin[c + d*x])^4) + (5*a^2)/(192*d*(a - a*Sin[c + d*x])^3) + (57*a)/(512*
d*(a - a*Sin[c + d*x])^2) + 61/(128*d*(a - a*Sin[c + d*x])) - a^4/(160*d*(a + a*Sin[c + d*x])^5) - (9*a^3)/(25
6*d*(a + a*Sin[c + d*x])^4) - (47*a^2)/(384*d*(a + a*Sin[c + d*x])^3) - (187*a)/(512*d*(a + a*Sin[c + d*x])^2)
- 315/(256*d*(a + a*Sin[c + d*x]))

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Rubi [A]  time = 0.290825, antiderivative size = 262, 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} \[ -\frac{a^4}{160 d (a \sin (c+d x)+a)^5}+\frac{a^3}{256 d (a-a \sin (c+d x))^4}-\frac{9 a^3}{256 d (a \sin (c+d x)+a)^4}+\frac{5 a^2}{192 d (a-a \sin (c+d x))^3}-\frac{47 a^2}{384 d (a \sin (c+d x)+a)^3}+\frac{57 a}{512 d (a-a \sin (c+d x))^2}-\frac{187 a}{512 d (a \sin (c+d x)+a)^2}+\frac{61}{128 d (a-a \sin (c+d x))}-\frac{315}{256 d (a \sin (c+d x)+a)}-\frac{\csc (c+d x)}{a d}-\frac{437 \log (1-\sin (c+d x))}{512 a d}-\frac{\log (\sin (c+d x))}{a d}+\frac{949 \log (\sin (c+d x)+1)}{512 a d} \]

Antiderivative was successfully verified.

[In]

Int[(Csc[c + d*x]^2*Sec[c + d*x]^9)/(a + a*Sin[c + d*x]),x]

[Out]

-(Csc[c + d*x]/(a*d)) - (437*Log[1 - Sin[c + d*x]])/(512*a*d) - Log[Sin[c + d*x]]/(a*d) + (949*Log[1 + Sin[c +
d*x]])/(512*a*d) + a^3/(256*d*(a - a*Sin[c + d*x])^4) + (5*a^2)/(192*d*(a - a*Sin[c + d*x])^3) + (57*a)/(512*
d*(a - a*Sin[c + d*x])^2) + 61/(128*d*(a - a*Sin[c + d*x])) - a^4/(160*d*(a + a*Sin[c + d*x])^5) - (9*a^3)/(25
6*d*(a + a*Sin[c + d*x])^4) - (47*a^2)/(384*d*(a + a*Sin[c + d*x])^3) - (187*a)/(512*d*(a + a*Sin[c + d*x])^2)
- 315/(256*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 ^2(c+d x) \sec ^9(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac{a^9 \operatorname{Subst}\left (\int \frac{a^2}{(a-x)^5 x^2 (a+x)^6} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{a^{11} \operatorname{Subst}\left (\int \frac{1}{(a-x)^5 x^2 (a+x)^6} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{a^{11} \operatorname{Subst}\left (\int \left (\frac{1}{64 a^8 (a-x)^5}+\frac{5}{64 a^9 (a-x)^4}+\frac{57}{256 a^{10} (a-x)^3}+\frac{61}{128 a^{11} (a-x)^2}+\frac{437}{512 a^{12} (a-x)}+\frac{1}{a^{11} x^2}-\frac{1}{a^{12} x}+\frac{1}{32 a^7 (a+x)^6}+\frac{9}{64 a^8 (a+x)^5}+\frac{47}{128 a^9 (a+x)^4}+\frac{187}{256 a^{10} (a+x)^3}+\frac{315}{256 a^{11} (a+x)^2}+\frac{949}{512 a^{12} (a+x)}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac{\csc (c+d x)}{a d}-\frac{437 \log (1-\sin (c+d x))}{512 a d}-\frac{\log (\sin (c+d x))}{a d}+\frac{949 \log (1+\sin (c+d x))}{512 a d}+\frac{a^3}{256 d (a-a \sin (c+d x))^4}+\frac{5 a^2}{192 d (a-a \sin (c+d x))^3}+\frac{57 a}{512 d (a-a \sin (c+d x))^2}+\frac{61}{128 d (a-a \sin (c+d x))}-\frac{a^4}{160 d (a+a \sin (c+d x))^5}-\frac{9 a^3}{256 d (a+a \sin (c+d x))^4}-\frac{47 a^2}{384 d (a+a \sin (c+d x))^3}-\frac{187 a}{512 d (a+a \sin (c+d x))^2}-\frac{315}{256 d (a+a \sin (c+d x))}\\ \end{align*}

Mathematica [A]  time = 6.20331, size = 240, normalized size = 0.92 \[ \frac{a^{11} \left (\frac{61}{128 a^{11} (a-a \sin (c+d x))}-\frac{315}{256 a^{11} (a \sin (c+d x)+a)}+\frac{57}{512 a^{10} (a-a \sin (c+d x))^2}-\frac{187}{512 a^{10} (a \sin (c+d x)+a)^2}+\frac{5}{192 a^9 (a-a \sin (c+d x))^3}-\frac{47}{384 a^9 (a \sin (c+d x)+a)^3}+\frac{1}{256 a^8 (a-a \sin (c+d x))^4}-\frac{9}{256 a^8 (a \sin (c+d x)+a)^4}-\frac{1}{160 a^7 (a \sin (c+d x)+a)^5}-\frac{\csc (c+d x)}{a^{12}}-\frac{437 \log (1-\sin (c+d x))}{512 a^{12}}-\frac{\log (\sin (c+d x))}{a^{12}}+\frac{949 \log (\sin (c+d x)+1)}{512 a^{12}}\right )}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[(Csc[c + d*x]^2*Sec[c + d*x]^9)/(a + a*Sin[c + d*x]),x]

[Out]

(a^11*(-(Csc[c + d*x]/a^12) - (437*Log[1 - Sin[c + d*x]])/(512*a^12) - Log[Sin[c + d*x]]/a^12 + (949*Log[1 + S
in[c + d*x]])/(512*a^12) + 1/(256*a^8*(a - a*Sin[c + d*x])^4) + 5/(192*a^9*(a - a*Sin[c + d*x])^3) + 57/(512*a
^10*(a - a*Sin[c + d*x])^2) + 61/(128*a^11*(a - a*Sin[c + d*x])) - 1/(160*a^7*(a + a*Sin[c + d*x])^5) - 9/(256
*a^8*(a + a*Sin[c + d*x])^4) - 47/(384*a^9*(a + a*Sin[c + d*x])^3) - 187/(512*a^10*(a + a*Sin[c + d*x])^2) - 3
15/(256*a^11*(a + a*Sin[c + d*x]))))/d

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Maple [A]  time = 0.104, size = 229, normalized size = 0.9 \begin{align*}{\frac{1}{256\,da \left ( \sin \left ( dx+c \right ) -1 \right ) ^{4}}}-{\frac{5}{192\,da \left ( \sin \left ( dx+c \right ) -1 \right ) ^{3}}}+{\frac{57}{512\,da \left ( \sin \left ( dx+c \right ) -1 \right ) ^{2}}}-{\frac{61}{128\,da \left ( \sin \left ( dx+c \right ) -1 \right ) }}-{\frac{437\,\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{9}{256\,da \left ( 1+\sin \left ( dx+c \right ) \right ) ^{4}}}-{\frac{47}{384\,da \left ( 1+\sin \left ( dx+c \right ) \right ) ^{3}}}-{\frac{187}{512\,da \left ( 1+\sin \left ( dx+c \right ) \right ) ^{2}}}-{\frac{315}{256\,da \left ( 1+\sin \left ( dx+c \right ) \right ) }}+{\frac{949\,\ln \left ( 1+\sin \left ( dx+c \right ) \right ) }{512\,da}}-{\frac{1}{da\sin \left ( dx+c \right ) }}-{\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)^2*sec(d*x+c)^9/(a+a*sin(d*x+c)),x)

[Out]

1/256/d/a/(sin(d*x+c)-1)^4-5/192/d/a/(sin(d*x+c)-1)^3+57/512/d/a/(sin(d*x+c)-1)^2-61/128/a/d/(sin(d*x+c)-1)-43
7/512/a/d*ln(sin(d*x+c)-1)-1/160/d/a/(1+sin(d*x+c))^5-9/256/d/a/(1+sin(d*x+c))^4-47/384/d/a/(1+sin(d*x+c))^3-1
87/512/a/d/(1+sin(d*x+c))^2-315/256/a/d/(1+sin(d*x+c))+949/512*ln(1+sin(d*x+c))/a/d-1/d/a/sin(d*x+c)-ln(sin(d*
x+c))/a/d

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Maxima [A]  time = 1.02458, size = 331, normalized size = 1.26 \begin{align*} -\frac{\frac{2 \,{\left (10395 \, \sin \left (d x + c\right )^{9} + 8475 \, \sin \left (d x + c\right )^{8} - 40035 \, \sin \left (d x + c\right )^{7} - 31395 \, \sin \left (d x + c\right )^{6} + 57309 \, \sin \left (d x + c\right )^{5} + 42269 \, \sin \left (d x + c\right )^{4} - 35941 \, \sin \left (d x + c\right )^{3} - 23621 \, \sin \left (d x + c\right )^{2} + 8224 \, \sin \left (d x + c\right ) + 3840\right )}}{a \sin \left (d x + c\right )^{10} + a \sin \left (d x + c\right )^{9} - 4 \, a \sin \left (d x + c\right )^{8} - 4 \, a \sin \left (d x + c\right )^{7} + 6 \, a \sin \left (d x + c\right )^{6} + 6 \, a \sin \left (d x + c\right )^{5} - 4 \, a \sin \left (d x + c\right )^{4} - 4 \, a \sin \left (d x + c\right )^{3} + a \sin \left (d x + c\right )^{2} + a \sin \left (d x + c\right )} - \frac{14235 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a} + \frac{6555 \, \log \left (\sin \left (d x + c\right ) - 1\right )}{a} + \frac{7680 \, \log \left (\sin \left (d x + c\right )\right )}{a}}{7680 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)^2*sec(d*x+c)^9/(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

-1/7680*(2*(10395*sin(d*x + c)^9 + 8475*sin(d*x + c)^8 - 40035*sin(d*x + c)^7 - 31395*sin(d*x + c)^6 + 57309*s
in(d*x + c)^5 + 42269*sin(d*x + c)^4 - 35941*sin(d*x + c)^3 - 23621*sin(d*x + c)^2 + 8224*sin(d*x + c) + 3840)
/(a*sin(d*x + c)^10 + a*sin(d*x + c)^9 - 4*a*sin(d*x + c)^8 - 4*a*sin(d*x + c)^7 + 6*a*sin(d*x + c)^6 + 6*a*si
n(d*x + c)^5 - 4*a*sin(d*x + c)^4 - 4*a*sin(d*x + c)^3 + a*sin(d*x + c)^2 + a*sin(d*x + c)) - 14235*log(sin(d*
x + c) + 1)/a + 6555*log(sin(d*x + c) - 1)/a + 7680*log(sin(d*x + c))/a)/d

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Fricas [A]  time = 2.19109, size = 774, normalized size = 2.95 \begin{align*} \frac{16950 \, \cos \left (d x + c\right )^{8} - 5010 \, \cos \left (d x + c\right )^{6} - 2132 \, \cos \left (d x + c\right )^{4} - 1264 \, \cos \left (d x + c\right )^{2} - 7680 \,{\left (\cos \left (d x + c\right )^{10} - \cos \left (d x + c\right )^{8} \sin \left (d x + c\right ) - \cos \left (d x + c\right )^{8}\right )} \log \left (\frac{1}{2} \, \sin \left (d x + c\right )\right ) + 14235 \,{\left (\cos \left (d x + c\right )^{10} - \cos \left (d x + c\right )^{8} \sin \left (d x + c\right ) - \cos \left (d x + c\right )^{8}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 6555 \,{\left (\cos \left (d x + c\right )^{10} - \cos \left (d x + c\right )^{8} \sin \left (d x + c\right ) - \cos \left (d x + c\right )^{8}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (10395 \, \cos \left (d x + c\right )^{8} - 1545 \, \cos \left (d x + c\right )^{6} - 426 \, \cos \left (d x + c\right )^{4} - 152 \, \cos \left (d x + c\right )^{2} - 48\right )} \sin \left (d x + c\right ) - 864}{7680 \,{\left (a d \cos \left (d x + c\right )^{10} - a d \cos \left (d x + c\right )^{8} \sin \left (d x + c\right ) - a d \cos \left (d x + c\right )^{8}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)^2*sec(d*x+c)^9/(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

1/7680*(16950*cos(d*x + c)^8 - 5010*cos(d*x + c)^6 - 2132*cos(d*x + c)^4 - 1264*cos(d*x + c)^2 - 7680*(cos(d*x
+ c)^10 - cos(d*x + c)^8*sin(d*x + c) - cos(d*x + c)^8)*log(1/2*sin(d*x + c)) + 14235*(cos(d*x + c)^10 - cos(
d*x + c)^8*sin(d*x + c) - cos(d*x + c)^8)*log(sin(d*x + c) + 1) - 6555*(cos(d*x + c)^10 - cos(d*x + c)^8*sin(d
*x + c) - cos(d*x + c)^8)*log(-sin(d*x + c) + 1) + 2*(10395*cos(d*x + c)^8 - 1545*cos(d*x + c)^6 - 426*cos(d*x
+ c)^4 - 152*cos(d*x + c)^2 - 48)*sin(d*x + c) - 864)/(a*d*cos(d*x + c)^10 - a*d*cos(d*x + c)^8*sin(d*x + c)
- a*d*cos(d*x + c)^8)

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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)**2*sec(d*x+c)**9/(a+a*sin(d*x+c)),x)

[Out]

Timed out

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Giac [A]  time = 1.23339, size = 257, normalized size = 0.98 \begin{align*} \frac{\frac{56940 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a} - \frac{26220 \, \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a} - \frac{30720 \, \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a} + \frac{30720 \,{\left (\sin \left (d x + c\right ) - 1\right )}}{a \sin \left (d x + c\right )} + \frac{5 \,{\left (10925 \, \sin \left (d x + c\right )^{4} - 46628 \, \sin \left (d x + c\right )^{3} + 75018 \, \sin \left (d x + c\right )^{2} - 54012 \, \sin \left (d x + c\right ) + 14721\right )}}{a{\left (\sin \left (d x + c\right ) - 1\right )}^{4}} - \frac{130013 \, \sin \left (d x + c\right )^{5} + 687865 \, \sin \left (d x + c\right )^{4} + 1462550 \, \sin \left (d x + c\right )^{3} + 1564350 \, \sin \left (d x + c\right )^{2} + 843525 \, \sin \left (d x + c\right ) + 184065}{a{\left (\sin \left (d x + c\right ) + 1\right )}^{5}}}{30720 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)^2*sec(d*x+c)^9/(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

1/30720*(56940*log(abs(sin(d*x + c) + 1))/a - 26220*log(abs(sin(d*x + c) - 1))/a - 30720*log(abs(sin(d*x + c))
)/a + 30720*(sin(d*x + c) - 1)/(a*sin(d*x + c)) + 5*(10925*sin(d*x + c)^4 - 46628*sin(d*x + c)^3 + 75018*sin(d
*x + c)^2 - 54012*sin(d*x + c) + 14721)/(a*(sin(d*x + c) - 1)^4) - (130013*sin(d*x + c)^5 + 687865*sin(d*x + c
)^4 + 1462550*sin(d*x + c)^3 + 1564350*sin(d*x + c)^2 + 843525*sin(d*x + c) + 184065)/(a*(sin(d*x + c) + 1)^5)
)/d