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

Optimal. Leaf size=247 \[ \frac{a^4}{160 d (a \sin (c+d x)+a)^5}+\frac{a^3}{256 d (a-a \sin (c+d x))^4}+\frac{7 a^3}{256 d (a \sin (c+d x)+a)^4}+\frac{a^2}{48 d (a-a \sin (c+d x))^3}+\frac{29 a^2}{384 d (a \sin (c+d x)+a)^3}+\frac{37 a}{512 d (a-a \sin (c+d x))^2}+\frac{93 a}{512 d (a \sin (c+d x)+a)^2}+\frac{65}{256 d (a-a \sin (c+d x))}+\frac{1}{2 d (a \sin (c+d x)+a)}-\frac{193 \log (1-\sin (c+d x))}{512 a d}+\frac{\log (\sin (c+d x))}{a d}-\frac{319 \log (\sin (c+d x)+1)}{512 a d} \]

[Out]

(-193*Log[1 - Sin[c + d*x]])/(512*a*d) + Log[Sin[c + d*x]]/(a*d) - (319*Log[1 + Sin[c + d*x]])/(512*a*d) + a^3
/(256*d*(a - a*Sin[c + d*x])^4) + a^2/(48*d*(a - a*Sin[c + d*x])^3) + (37*a)/(512*d*(a - a*Sin[c + d*x])^2) +
65/(256*d*(a - a*Sin[c + d*x])) + a^4/(160*d*(a + a*Sin[c + d*x])^5) + (7*a^3)/(256*d*(a + a*Sin[c + d*x])^4)
+ (29*a^2)/(384*d*(a + a*Sin[c + d*x])^3) + (93*a)/(512*d*(a + a*Sin[c + d*x])^2) + 1/(2*d*(a + a*Sin[c + d*x]
))

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Rubi [A]  time = 0.244415, antiderivative size = 247, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2836, 12, 88} \[ \frac{a^4}{160 d (a \sin (c+d x)+a)^5}+\frac{a^3}{256 d (a-a \sin (c+d x))^4}+\frac{7 a^3}{256 d (a \sin (c+d x)+a)^4}+\frac{a^2}{48 d (a-a \sin (c+d x))^3}+\frac{29 a^2}{384 d (a \sin (c+d x)+a)^3}+\frac{37 a}{512 d (a-a \sin (c+d x))^2}+\frac{93 a}{512 d (a \sin (c+d x)+a)^2}+\frac{65}{256 d (a-a \sin (c+d x))}+\frac{1}{2 d (a \sin (c+d x)+a)}-\frac{193 \log (1-\sin (c+d x))}{512 a d}+\frac{\log (\sin (c+d x))}{a d}-\frac{319 \log (\sin (c+d x)+1)}{512 a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-193*Log[1 - Sin[c + d*x]])/(512*a*d) + Log[Sin[c + d*x]]/(a*d) - (319*Log[1 + Sin[c + d*x]])/(512*a*d) + a^3
/(256*d*(a - a*Sin[c + d*x])^4) + a^2/(48*d*(a - a*Sin[c + d*x])^3) + (37*a)/(512*d*(a - a*Sin[c + d*x])^2) +
65/(256*d*(a - a*Sin[c + d*x])) + a^4/(160*d*(a + a*Sin[c + d*x])^5) + (7*a^3)/(256*d*(a + a*Sin[c + d*x])^4)
+ (29*a^2)/(384*d*(a + a*Sin[c + d*x])^3) + (93*a)/(512*d*(a + a*Sin[c + d*x])^2) + 1/(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 (c+d x) \sec ^9(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac{a^9 \operatorname{Subst}\left (\int \frac{a}{(a-x)^5 x (a+x)^6} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{a^{10} \operatorname{Subst}\left (\int \frac{1}{(a-x)^5 x (a+x)^6} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{a^{10} \operatorname{Subst}\left (\int \left (\frac{1}{64 a^7 (a-x)^5}+\frac{1}{16 a^8 (a-x)^4}+\frac{37}{256 a^9 (a-x)^3}+\frac{65}{256 a^{10} (a-x)^2}+\frac{193}{512 a^{11} (a-x)}+\frac{1}{a^{11} x}-\frac{1}{32 a^6 (a+x)^6}-\frac{7}{64 a^7 (a+x)^5}-\frac{29}{128 a^8 (a+x)^4}-\frac{93}{256 a^9 (a+x)^3}-\frac{1}{2 a^{10} (a+x)^2}-\frac{319}{512 a^{11} (a+x)}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac{193 \log (1-\sin (c+d x))}{512 a d}+\frac{\log (\sin (c+d x))}{a d}-\frac{319 \log (1+\sin (c+d x))}{512 a d}+\frac{a^3}{256 d (a-a \sin (c+d x))^4}+\frac{a^2}{48 d (a-a \sin (c+d x))^3}+\frac{37 a}{512 d (a-a \sin (c+d x))^2}+\frac{65}{256 d (a-a \sin (c+d x))}+\frac{a^4}{160 d (a+a \sin (c+d x))^5}+\frac{7 a^3}{256 d (a+a \sin (c+d x))^4}+\frac{29 a^2}{384 d (a+a \sin (c+d x))^3}+\frac{93 a}{512 d (a+a \sin (c+d x))^2}+\frac{1}{2 d (a+a \sin (c+d x))}\\ \end{align*}

Mathematica [A]  time = 6.19715, size = 228, normalized size = 0.92 \[ \frac{a^{10} \left (\frac{65}{256 a^{10} (a-a \sin (c+d x))}+\frac{1}{2 a^{10} (a \sin (c+d x)+a)}+\frac{37}{512 a^9 (a-a \sin (c+d x))^2}+\frac{93}{512 a^9 (a \sin (c+d x)+a)^2}+\frac{1}{48 a^8 (a-a \sin (c+d x))^3}+\frac{29}{384 a^8 (a \sin (c+d x)+a)^3}+\frac{1}{256 a^7 (a-a \sin (c+d x))^4}+\frac{7}{256 a^7 (a \sin (c+d x)+a)^4}+\frac{1}{160 a^6 (a \sin (c+d x)+a)^5}-\frac{193 \log (1-\sin (c+d x))}{512 a^{11}}+\frac{\log (\sin (c+d x))}{a^{11}}-\frac{319 \log (\sin (c+d x)+1)}{512 a^{11}}\right )}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^10*((-193*Log[1 - Sin[c + d*x]])/(512*a^11) + Log[Sin[c + d*x]]/a^11 - (319*Log[1 + Sin[c + d*x]])/(512*a^1
1) + 1/(256*a^7*(a - a*Sin[c + d*x])^4) + 1/(48*a^8*(a - a*Sin[c + d*x])^3) + 37/(512*a^9*(a - a*Sin[c + d*x])
^2) + 65/(256*a^10*(a - a*Sin[c + d*x])) + 1/(160*a^6*(a + a*Sin[c + d*x])^5) + 7/(256*a^7*(a + a*Sin[c + d*x]
)^4) + 29/(384*a^8*(a + a*Sin[c + d*x])^3) + 93/(512*a^9*(a + a*Sin[c + d*x])^2) + 1/(2*a^10*(a + a*Sin[c + d*
x]))))/d

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Maple [A]  time = 0.1, size = 212, normalized size = 0.9 \begin{align*}{\frac{1}{256\,da \left ( \sin \left ( dx+c \right ) -1 \right ) ^{4}}}-{\frac{1}{48\,da \left ( \sin \left ( dx+c \right ) -1 \right ) ^{3}}}+{\frac{37}{512\,da \left ( \sin \left ( dx+c \right ) -1 \right ) ^{2}}}-{\frac{65}{256\,da \left ( \sin \left ( dx+c \right ) -1 \right ) }}-{\frac{193\,\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{7}{256\,da \left ( 1+\sin \left ( dx+c \right ) \right ) ^{4}}}+{\frac{29}{384\,da \left ( 1+\sin \left ( dx+c \right ) \right ) ^{3}}}+{\frac{93}{512\,da \left ( 1+\sin \left ( dx+c \right ) \right ) ^{2}}}+{\frac{1}{2\,da \left ( 1+\sin \left ( dx+c \right ) \right ) }}-{\frac{319\,\ln \left ( 1+\sin \left ( dx+c \right ) \right ) }{512\,da}}+{\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)*sec(d*x+c)^9/(a+a*sin(d*x+c)),x)

[Out]

1/256/d/a/(sin(d*x+c)-1)^4-1/48/d/a/(sin(d*x+c)-1)^3+37/512/d/a/(sin(d*x+c)-1)^2-65/256/a/d/(sin(d*x+c)-1)-193
/512/a/d*ln(sin(d*x+c)-1)+1/160/d/a/(1+sin(d*x+c))^5+7/256/d/a/(1+sin(d*x+c))^4+29/384/d/a/(1+sin(d*x+c))^3+93
/512/a/d/(1+sin(d*x+c))^2+1/2/a/d/(1+sin(d*x+c))-319/512*ln(1+sin(d*x+c))/a/d+ln(sin(d*x+c))/a/d

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Maxima [A]  time = 1.02366, size = 305, normalized size = 1.23 \begin{align*} \frac{\frac{2 \,{\left (945 \, \sin \left (d x + c\right )^{8} - 975 \, \sin \left (d x + c\right )^{7} - 5385 \, \sin \left (d x + c\right )^{6} + 3255 \, \sin \left (d x + c\right )^{5} + 11319 \, \sin \left (d x + c\right )^{4} - 3721 \, \sin \left (d x + c\right )^{3} - 10831 \, \sin \left (d x + c\right )^{2} + 1489 \, \sin \left (d x + c\right ) + 4384\right )}}{a \sin \left (d x + c\right )^{9} + a \sin \left (d x + c\right )^{8} - 4 \, a \sin \left (d x + c\right )^{7} - 4 \, a \sin \left (d x + c\right )^{6} + 6 \, a \sin \left (d x + c\right )^{5} + 6 \, a \sin \left (d x + c\right )^{4} - 4 \, a \sin \left (d x + c\right )^{3} - 4 \, a \sin \left (d x + c\right )^{2} + a \sin \left (d x + c\right ) + a} - \frac{4785 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a} - \frac{2895 \, \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)*sec(d*x+c)^9/(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

1/7680*(2*(945*sin(d*x + c)^8 - 975*sin(d*x + c)^7 - 5385*sin(d*x + c)^6 + 3255*sin(d*x + c)^5 + 11319*sin(d*x
+ c)^4 - 3721*sin(d*x + c)^3 - 10831*sin(d*x + c)^2 + 1489*sin(d*x + c) + 4384)/(a*sin(d*x + c)^9 + a*sin(d*x
+ c)^8 - 4*a*sin(d*x + c)^7 - 4*a*sin(d*x + c)^6 + 6*a*sin(d*x + c)^5 + 6*a*sin(d*x + c)^4 - 4*a*sin(d*x + c)
^3 - 4*a*sin(d*x + c)^2 + a*sin(d*x + c) + a) - 4785*log(sin(d*x + c) + 1)/a - 2895*log(sin(d*x + c) - 1)/a +
7680*log(sin(d*x + c))/a)/d

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Fricas [A]  time = 2.22443, size = 636, normalized size = 2.57 \begin{align*} \frac{1890 \, \cos \left (d x + c\right )^{8} + 3210 \, \cos \left (d x + c\right )^{6} + 1668 \, \cos \left (d x + c\right )^{4} + 1136 \, \cos \left (d x + c\right )^{2} + 7680 \,{\left (\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 ) - 4785 \,{\left (\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 ) - 2895 \,{\left (\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 (975 \, \cos \left (d x + c\right )^{6} + 330 \, \cos \left (d x + c\right )^{4} + 136 \, \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 )^{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)*sec(d*x+c)^9/(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

1/7680*(1890*cos(d*x + c)^8 + 3210*cos(d*x + c)^6 + 1668*cos(d*x + c)^4 + 1136*cos(d*x + c)^2 + 7680*(cos(d*x
+ c)^8*sin(d*x + c) + cos(d*x + c)^8)*log(1/2*sin(d*x + c)) - 4785*(cos(d*x + c)^8*sin(d*x + c) + cos(d*x + c)
^8)*log(sin(d*x + c) + 1) - 2895*(cos(d*x + c)^8*sin(d*x + c) + cos(d*x + c)^8)*log(-sin(d*x + c) + 1) + 2*(97
5*cos(d*x + c)^6 + 330*cos(d*x + c)^4 + 136*cos(d*x + c)^2 + 48)*sin(d*x + c) + 864)/(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)*sec(d*x+c)**9/(a+a*sin(d*x+c)),x)

[Out]

Timed out

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Giac [A]  time = 1.33199, size = 228, normalized size = 0.92 \begin{align*} -\frac{\frac{19140 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a} + \frac{11580 \, \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{5 \,{\left (4825 \, \sin \left (d x + c\right )^{4} - 20860 \, \sin \left (d x + c\right )^{3} + 34074 \, \sin \left (d x + c\right )^{2} - 24996 \, \sin \left (d x + c\right ) + 6981\right )}}{a{\left (\sin \left (d x + c\right ) - 1\right )}^{4}} - \frac{43703 \, \sin \left (d x + c\right )^{5} + 233875 \, \sin \left (d x + c\right )^{4} + 504050 \, \sin \left (d x + c\right )^{3} + 548250 \, \sin \left (d x + c\right )^{2} + 302175 \, \sin \left (d x + c\right ) + 67995}{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)*sec(d*x+c)^9/(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

-1/30720*(19140*log(abs(sin(d*x + c) + 1))/a + 11580*log(abs(sin(d*x + c) - 1))/a - 30720*log(abs(sin(d*x + c)
))/a - 5*(4825*sin(d*x + c)^4 - 20860*sin(d*x + c)^3 + 34074*sin(d*x + c)^2 - 24996*sin(d*x + c) + 6981)/(a*(s
in(d*x + c) - 1)^4) - (43703*sin(d*x + c)^5 + 233875*sin(d*x + c)^4 + 504050*sin(d*x + c)^3 + 548250*sin(d*x +
c)^2 + 302175*sin(d*x + c) + 67995)/(a*(sin(d*x + c) + 1)^5))/d