3.897 \(\int \frac{\sin (c+d x) \tan ^9(c+d x)}{a+a \sin (c+d x)} \, dx\)

Optimal. Leaf size=233 \[ -\frac{a^4}{160 d (a \sin (c+d x)+a)^5}+\frac{a^3}{256 d (a-a \sin (c+d x))^4}+\frac{15 a^3}{256 d (a \sin (c+d x)+a)^4}-\frac{7 a^2}{192 d (a-a \sin (c+d x))^3}-\frac{95 a^2}{384 d (a \sin (c+d x)+a)^3}+\frac{81 a}{512 d (a-a \sin (c+d x))^2}+\frac{325 a}{512 d (a \sin (c+d x)+a)^2}-\frac{61}{128 d (a-a \sin (c+d x))}-\frac{315}{256 d (a \sin (c+d x)+a)}-\frac{193 \log (1-\sin (c+d x))}{512 a d}-\frac{319 \log (\sin (c+d x)+1)}{512 a d} \]

[Out]

(-193*Log[1 - Sin[c + d*x]])/(512*a*d) - (319*Log[1 + Sin[c + d*x]])/(512*a*d) + a^3/(256*d*(a - a*Sin[c + d*x
])^4) - (7*a^2)/(192*d*(a - a*Sin[c + d*x])^3) + (81*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) + (15*a^3)/(256*d*(a + a*Sin[c + d*x])^4) - (95*a^2)/(384*d*(a
+ a*Sin[c + d*x])^3) + (325*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 = 0.234589, antiderivative size = 233, 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} \[ -\frac{a^4}{160 d (a \sin (c+d x)+a)^5}+\frac{a^3}{256 d (a-a \sin (c+d x))^4}+\frac{15 a^3}{256 d (a \sin (c+d x)+a)^4}-\frac{7 a^2}{192 d (a-a \sin (c+d x))^3}-\frac{95 a^2}{384 d (a \sin (c+d x)+a)^3}+\frac{81 a}{512 d (a-a \sin (c+d x))^2}+\frac{325 a}{512 d (a \sin (c+d x)+a)^2}-\frac{61}{128 d (a-a \sin (c+d x))}-\frac{315}{256 d (a \sin (c+d x)+a)}-\frac{193 \log (1-\sin (c+d x))}{512 a d}-\frac{319 \log (\sin (c+d x)+1)}{512 a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-193*Log[1 - Sin[c + d*x]])/(512*a*d) - (319*Log[1 + Sin[c + d*x]])/(512*a*d) + a^3/(256*d*(a - a*Sin[c + d*x
])^4) - (7*a^2)/(192*d*(a - a*Sin[c + d*x])^3) + (81*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) + (15*a^3)/(256*d*(a + a*Sin[c + d*x])^4) - (95*a^2)/(384*d*(a
+ a*Sin[c + d*x])^3) + (325*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{\sin (c+d x) \tan ^9(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac{a^9 \operatorname{Subst}\left (\int \frac{x^{10}}{a^{10} (a-x)^5 (a+x)^6} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{x^{10}}{(a-x)^5 (a+x)^6} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{a^4}{64 (a-x)^5}-\frac{7 a^3}{64 (a-x)^4}+\frac{81 a^2}{256 (a-x)^3}-\frac{61 a}{128 (a-x)^2}+\frac{193}{512 (a-x)}+\frac{a^5}{32 (a+x)^6}-\frac{15 a^4}{64 (a+x)^5}+\frac{95 a^3}{128 (a+x)^4}-\frac{325 a^2}{256 (a+x)^3}+\frac{315 a}{256 (a+x)^2}-\frac{319}{512 (a+x)}\right ) \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=-\frac{193 \log (1-\sin (c+d x))}{512 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{7 a^2}{192 d (a-a \sin (c+d x))^3}+\frac{81 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{15 a^3}{256 d (a+a \sin (c+d x))^4}-\frac{95 a^2}{384 d (a+a \sin (c+d x))^3}+\frac{325 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 = 4.91451, size = 137, normalized size = 0.59 \[ -\frac{\frac{2 \left (2895 \sin ^8(c+d x)-6705 \sin ^7(c+d x)-13815 \sin ^6(c+d x)+14985 \sin ^5(c+d x)+23049 \sin ^4(c+d x)-12151 \sin ^3(c+d x)-16561 \sin ^2(c+d x)+3439 \sin (c+d x)+4384\right )}{(\sin (c+d x)-1)^4 (\sin (c+d x)+1)^5}+2895 \log (1-\sin (c+d x))+4785 \log (\sin (c+d x)+1)}{7680 a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(2895*Log[1 - Sin[c + d*x]] + 4785*Log[1 + Sin[c + d*x]] + (2*(4384 + 3439*Sin[c + d*x] - 16561*Sin[c + d*x]^
2 - 12151*Sin[c + d*x]^3 + 23049*Sin[c + d*x]^4 + 14985*Sin[c + d*x]^5 - 13815*Sin[c + d*x]^6 - 6705*Sin[c + d
*x]^7 + 2895*Sin[c + d*x]^8))/((-1 + Sin[c + d*x])^4*(1 + Sin[c + d*x])^5))/(7680*a*d)

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Maple [A]  time = 0.109, size = 198, normalized size = 0.9 \begin{align*}{\frac{1}{256\,da \left ( \sin \left ( dx+c \right ) -1 \right ) ^{4}}}+{\frac{7}{192\,da \left ( \sin \left ( dx+c \right ) -1 \right ) ^{3}}}+{\frac{81}{512\,da \left ( \sin \left ( dx+c \right ) -1 \right ) ^{2}}}+{\frac{61}{128\,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{15}{256\,da \left ( 1+\sin \left ( dx+c \right ) \right ) ^{4}}}-{\frac{95}{384\,da \left ( 1+\sin \left ( dx+c \right ) \right ) ^{3}}}+{\frac{325}{512\,da \left ( 1+\sin \left ( dx+c \right ) \right ) ^{2}}}-{\frac{315}{256\,da \left ( 1+\sin \left ( dx+c \right ) \right ) }}-{\frac{319\,\ln \left ( 1+\sin \left ( dx+c \right ) \right ) }{512\,da}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^9*sin(d*x+c)^10/(a+a*sin(d*x+c)),x)

[Out]

1/256/d/a/(sin(d*x+c)-1)^4+7/192/d/a/(sin(d*x+c)-1)^3+81/512/d/a/(sin(d*x+c)-1)^2+61/128/a/d/(sin(d*x+c)-1)-19
3/512/a/d*ln(sin(d*x+c)-1)-1/160/d/a/(1+sin(d*x+c))^5+15/256/d/a/(1+sin(d*x+c))^4-95/384/d/a/(1+sin(d*x+c))^3+
325/512/a/d/(1+sin(d*x+c))^2-315/256/a/d/(1+sin(d*x+c))-319/512*ln(1+sin(d*x+c))/a/d

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Maxima [A]  time = 1.05919, size = 289, normalized size = 1.24 \begin{align*} -\frac{\frac{2 \,{\left (2895 \, \sin \left (d x + c\right )^{8} - 6705 \, \sin \left (d x + c\right )^{7} - 13815 \, \sin \left (d x + c\right )^{6} + 14985 \, \sin \left (d x + c\right )^{5} + 23049 \, \sin \left (d x + c\right )^{4} - 12151 \, \sin \left (d x + c\right )^{3} - 16561 \, \sin \left (d x + c\right )^{2} + 3439 \, \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}}{7680 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^9*sin(d*x+c)^10/(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

-1/7680*(2*(2895*sin(d*x + c)^8 - 6705*sin(d*x + c)^7 - 13815*sin(d*x + c)^6 + 14985*sin(d*x + c)^5 + 23049*si
n(d*x + c)^4 - 12151*sin(d*x + c)^3 - 16561*sin(d*x + c)^2 + 3439*sin(d*x + c) + 4384)/(a*sin(d*x + c)^9 + a*s
in(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)/d

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Fricas [A]  time = 2.1346, size = 537, normalized size = 2.3 \begin{align*} -\frac{5790 \, \cos \left (d x + c\right )^{8} + 4470 \, \cos \left (d x + c\right )^{6} - 2052 \, \cos \left (d x + c\right )^{4} + 656 \, \cos \left (d x + c\right )^{2} + 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 (6705 \, \cos \left (d x + c\right )^{6} - 5130 \, \cos \left (d x + c\right )^{4} + 2296 \, \cos \left (d x + c\right )^{2} - 432\right )} \sin \left (d x + c\right ) - 96}{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(sec(d*x+c)^9*sin(d*x+c)^10/(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

-1/7680*(5790*cos(d*x + c)^8 + 4470*cos(d*x + c)^6 - 2052*cos(d*x + c)^4 + 656*cos(d*x + c)^2 + 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*(6705*cos(d*x + c)^6 - 5130*cos(d*x + c)^4 + 2296*cos(d*x + c)^2 - 432)*sin(d*x
+ c) - 96)/(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(sec(d*x+c)**9*sin(d*x+c)**10/(a+a*sin(d*x+c)),x)

[Out]

Timed out

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Giac [A]  time = 1.35615, size = 211, normalized size = 0.91 \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{5 \,{\left (4825 \, \sin \left (d x + c\right )^{4} - 16372 \, \sin \left (d x + c\right )^{3} + 21138 \, \sin \left (d x + c\right )^{2} - 12236 \, \sin \left (d x + c\right ) + 2669\right )}}{a{\left (\sin \left (d x + c\right ) - 1\right )}^{4}} - \frac{43703 \, \sin \left (d x + c\right )^{5} + 180715 \, \sin \left (d x + c\right )^{4} + 305330 \, \sin \left (d x + c\right )^{3} + 261130 \, \sin \left (d x + c\right )^{2} + 112415 \, \sin \left (d x + c\right ) + 19411}{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(sec(d*x+c)^9*sin(d*x+c)^10/(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 - 5*(4825*sin(d*x + c)^4 - 1
6372*sin(d*x + c)^3 + 21138*sin(d*x + c)^2 - 12236*sin(d*x + c) + 2669)/(a*(sin(d*x + c) - 1)^4) - (43703*sin(
d*x + c)^5 + 180715*sin(d*x + c)^4 + 305330*sin(d*x + c)^3 + 261130*sin(d*x + c)^2 + 112415*sin(d*x + c) + 194
11)/(a*(sin(d*x + c) + 1)^5))/d