3.100 \(\int \frac{\sec ^5(c+d x)}{(a+a \sin (c+d x))^8} \, dx\)

Optimal. Leaf size=284 \[ -\frac{a^2}{80 d (a \sin (c+d x)+a)^{10}}+\frac{11}{4096 d \left (a^8-a^8 \sin (c+d x)\right )}-\frac{55}{4096 d \left (a^8 \sin (c+d x)+a^8\right )}+\frac{1}{4096 d \left (a^4-a^4 \sin (c+d x)\right )^2}-\frac{45}{4096 d \left (a^4 \sin (c+d x)+a^4\right )^2}-\frac{7}{512 d \left (a^2 \sin (c+d x)+a^2\right )^4}-\frac{5}{256 a^2 d (a \sin (c+d x)+a)^6}-\frac{21}{1280 a^3 d (a \sin (c+d x)+a)^5}-\frac{3}{256 a^5 d (a \sin (c+d x)+a)^3}+\frac{33 \tanh ^{-1}(\sin (c+d x))}{2048 a^8 d}-\frac{a}{48 d (a \sin (c+d x)+a)^9}-\frac{3}{128 d (a \sin (c+d x)+a)^8}-\frac{5}{224 a d (a \sin (c+d x)+a)^7} \]

[Out]

(33*ArcTanh[Sin[c + d*x]])/(2048*a^8*d) - a^2/(80*d*(a + a*Sin[c + d*x])^10) - a/(48*d*(a + a*Sin[c + d*x])^9)
- 3/(128*d*(a + a*Sin[c + d*x])^8) - 5/(224*a*d*(a + a*Sin[c + d*x])^7) - 5/(256*a^2*d*(a + a*Sin[c + d*x])^6
) - 21/(1280*a^3*d*(a + a*Sin[c + d*x])^5) - 3/(256*a^5*d*(a + a*Sin[c + d*x])^3) - 7/(512*d*(a^2 + a^2*Sin[c
+ d*x])^4) + 1/(4096*d*(a^4 - a^4*Sin[c + d*x])^2) - 45/(4096*d*(a^4 + a^4*Sin[c + d*x])^2) + 11/(4096*d*(a^8
- a^8*Sin[c + d*x])) - 55/(4096*d*(a^8 + a^8*Sin[c + d*x]))

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Rubi [A]  time = 0.215633, antiderivative size = 284, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2667, 44, 206} \[ -\frac{a^2}{80 d (a \sin (c+d x)+a)^{10}}+\frac{11}{4096 d \left (a^8-a^8 \sin (c+d x)\right )}-\frac{55}{4096 d \left (a^8 \sin (c+d x)+a^8\right )}+\frac{1}{4096 d \left (a^4-a^4 \sin (c+d x)\right )^2}-\frac{45}{4096 d \left (a^4 \sin (c+d x)+a^4\right )^2}-\frac{7}{512 d \left (a^2 \sin (c+d x)+a^2\right )^4}-\frac{5}{256 a^2 d (a \sin (c+d x)+a)^6}-\frac{21}{1280 a^3 d (a \sin (c+d x)+a)^5}-\frac{3}{256 a^5 d (a \sin (c+d x)+a)^3}+\frac{33 \tanh ^{-1}(\sin (c+d x))}{2048 a^8 d}-\frac{a}{48 d (a \sin (c+d x)+a)^9}-\frac{3}{128 d (a \sin (c+d x)+a)^8}-\frac{5}{224 a d (a \sin (c+d x)+a)^7} \]

Antiderivative was successfully verified.

[In]

Int[Sec[c + d*x]^5/(a + a*Sin[c + d*x])^8,x]

[Out]

(33*ArcTanh[Sin[c + d*x]])/(2048*a^8*d) - a^2/(80*d*(a + a*Sin[c + d*x])^10) - a/(48*d*(a + a*Sin[c + d*x])^9)
- 3/(128*d*(a + a*Sin[c + d*x])^8) - 5/(224*a*d*(a + a*Sin[c + d*x])^7) - 5/(256*a^2*d*(a + a*Sin[c + d*x])^6
) - 21/(1280*a^3*d*(a + a*Sin[c + d*x])^5) - 3/(256*a^5*d*(a + a*Sin[c + d*x])^3) - 7/(512*d*(a^2 + a^2*Sin[c
+ d*x])^4) + 1/(4096*d*(a^4 - a^4*Sin[c + d*x])^2) - 45/(4096*d*(a^4 + a^4*Sin[c + d*x])^2) + 11/(4096*d*(a^8
- a^8*Sin[c + d*x])) - 55/(4096*d*(a^8 + a^8*Sin[c + d*x]))

Rule 2667

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^(m + (p - 1)/2)*(a - x)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x]
&& IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2, 0] && (GeQ[p, -1] ||  !IntegerQ[m + 1/2])

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{\sec ^5(c+d x)}{(a+a \sin (c+d x))^8} \, dx &=\frac{a^5 \operatorname{Subst}\left (\int \frac{1}{(a-x)^3 (a+x)^{11}} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{a^5 \operatorname{Subst}\left (\int \left (\frac{1}{2048 a^{11} (a-x)^3}+\frac{11}{4096 a^{12} (a-x)^2}+\frac{1}{8 a^3 (a+x)^{11}}+\frac{3}{16 a^4 (a+x)^{10}}+\frac{3}{16 a^5 (a+x)^9}+\frac{5}{32 a^6 (a+x)^8}+\frac{15}{128 a^7 (a+x)^7}+\frac{21}{256 a^8 (a+x)^6}+\frac{7}{128 a^9 (a+x)^5}+\frac{9}{256 a^{10} (a+x)^4}+\frac{45}{2048 a^{11} (a+x)^3}+\frac{55}{4096 a^{12} (a+x)^2}+\frac{33}{2048 a^{12} \left (a^2-x^2\right )}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac{a^2}{80 d (a+a \sin (c+d x))^{10}}-\frac{a}{48 d (a+a \sin (c+d x))^9}-\frac{3}{128 d (a+a \sin (c+d x))^8}-\frac{5}{224 a d (a+a \sin (c+d x))^7}-\frac{5}{256 a^2 d (a+a \sin (c+d x))^6}-\frac{21}{1280 a^3 d (a+a \sin (c+d x))^5}-\frac{3}{256 a^5 d (a+a \sin (c+d x))^3}-\frac{7}{512 d \left (a^2+a^2 \sin (c+d x)\right )^4}+\frac{1}{4096 d \left (a^4-a^4 \sin (c+d x)\right )^2}-\frac{45}{4096 d \left (a^4+a^4 \sin (c+d x)\right )^2}+\frac{11}{4096 d \left (a^8-a^8 \sin (c+d x)\right )}-\frac{55}{4096 d \left (a^8+a^8 \sin (c+d x)\right )}+\frac{33 \operatorname{Subst}\left (\int \frac{1}{a^2-x^2} \, dx,x,a \sin (c+d x)\right )}{2048 a^7 d}\\ &=\frac{33 \tanh ^{-1}(\sin (c+d x))}{2048 a^8 d}-\frac{a^2}{80 d (a+a \sin (c+d x))^{10}}-\frac{a}{48 d (a+a \sin (c+d x))^9}-\frac{3}{128 d (a+a \sin (c+d x))^8}-\frac{5}{224 a d (a+a \sin (c+d x))^7}-\frac{5}{256 a^2 d (a+a \sin (c+d x))^6}-\frac{21}{1280 a^3 d (a+a \sin (c+d x))^5}-\frac{3}{256 a^5 d (a+a \sin (c+d x))^3}-\frac{7}{512 d \left (a^2+a^2 \sin (c+d x)\right )^4}+\frac{1}{4096 d \left (a^4-a^4 \sin (c+d x)\right )^2}-\frac{45}{4096 d \left (a^4+a^4 \sin (c+d x)\right )^2}+\frac{11}{4096 d \left (a^8-a^8 \sin (c+d x)\right )}-\frac{55}{4096 d \left (a^8+a^8 \sin (c+d x)\right )}\\ \end{align*}

Mathematica [A]  time = 2.614, size = 195, normalized size = 0.69 \[ \frac{\sec ^4(c+d x) \left (-3465 \sin ^{11}(c+d x)-27720 \sin ^{10}(c+d x)-91245 \sin ^9(c+d x)-147840 \sin ^8(c+d x)-82698 \sin ^7(c+d x)+114576 \sin ^6(c+d x)+255222 \sin ^5(c+d x)+190080 \sin ^4(c+d x)+21395 \sin ^3(c+d x)-72776 \sin ^2(c+d x)-66953 \sin (c+d x)+3465 \tanh ^{-1}(\sin (c+d x)) \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )^4 \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^{20}-34816\right )}{215040 a^8 d (\sin (c+d x)+1)^8} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[c + d*x]^5/(a + a*Sin[c + d*x])^8,x]

[Out]

(Sec[c + d*x]^4*(-34816 + 3465*ArcTanh[Sin[c + d*x]]*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^4*(Cos[(c + d*x)/2]
+ Sin[(c + d*x)/2])^20 - 66953*Sin[c + d*x] - 72776*Sin[c + d*x]^2 + 21395*Sin[c + d*x]^3 + 190080*Sin[c + d*
x]^4 + 255222*Sin[c + d*x]^5 + 114576*Sin[c + d*x]^6 - 82698*Sin[c + d*x]^7 - 147840*Sin[c + d*x]^8 - 91245*Si
n[c + d*x]^9 - 27720*Sin[c + d*x]^10 - 3465*Sin[c + d*x]^11))/(215040*a^8*d*(1 + Sin[c + d*x])^8)

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Maple [A]  time = 0.144, size = 252, normalized size = 0.9 \begin{align*}{\frac{1}{4096\,d{a}^{8} \left ( \sin \left ( dx+c \right ) -1 \right ) ^{2}}}-{\frac{11}{4096\,d{a}^{8} \left ( \sin \left ( dx+c \right ) -1 \right ) }}-{\frac{33\,\ln \left ( \sin \left ( dx+c \right ) -1 \right ) }{4096\,d{a}^{8}}}-{\frac{1}{80\,d{a}^{8} \left ( 1+\sin \left ( dx+c \right ) \right ) ^{10}}}-{\frac{1}{48\,d{a}^{8} \left ( 1+\sin \left ( dx+c \right ) \right ) ^{9}}}-{\frac{3}{128\,d{a}^{8} \left ( 1+\sin \left ( dx+c \right ) \right ) ^{8}}}-{\frac{5}{224\,d{a}^{8} \left ( 1+\sin \left ( dx+c \right ) \right ) ^{7}}}-{\frac{5}{256\,d{a}^{8} \left ( 1+\sin \left ( dx+c \right ) \right ) ^{6}}}-{\frac{21}{1280\,d{a}^{8} \left ( 1+\sin \left ( dx+c \right ) \right ) ^{5}}}-{\frac{7}{512\,d{a}^{8} \left ( 1+\sin \left ( dx+c \right ) \right ) ^{4}}}-{\frac{3}{256\,d{a}^{8} \left ( 1+\sin \left ( dx+c \right ) \right ) ^{3}}}-{\frac{45}{4096\,d{a}^{8} \left ( 1+\sin \left ( dx+c \right ) \right ) ^{2}}}-{\frac{55}{4096\,d{a}^{8} \left ( 1+\sin \left ( dx+c \right ) \right ) }}+{\frac{33\,\ln \left ( 1+\sin \left ( dx+c \right ) \right ) }{4096\,d{a}^{8}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^5/(a+a*sin(d*x+c))^8,x)

[Out]

1/4096/d/a^8/(sin(d*x+c)-1)^2-11/4096/d/a^8/(sin(d*x+c)-1)-33/4096/d/a^8*ln(sin(d*x+c)-1)-1/80/d/a^8/(1+sin(d*
x+c))^10-1/48/d/a^8/(1+sin(d*x+c))^9-3/128/d/a^8/(1+sin(d*x+c))^8-5/224/d/a^8/(1+sin(d*x+c))^7-5/256/d/a^8/(1+
sin(d*x+c))^6-21/1280/d/a^8/(1+sin(d*x+c))^5-7/512/d/a^8/(1+sin(d*x+c))^4-3/256/d/a^8/(1+sin(d*x+c))^3-45/4096
/d/a^8/(1+sin(d*x+c))^2-55/4096/d/a^8/(1+sin(d*x+c))+33/4096/d/a^8*ln(1+sin(d*x+c))

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Maxima [A]  time = 0.989144, size = 412, normalized size = 1.45 \begin{align*} -\frac{\frac{2 \,{\left (3465 \, \sin \left (d x + c\right )^{11} + 27720 \, \sin \left (d x + c\right )^{10} + 91245 \, \sin \left (d x + c\right )^{9} + 147840 \, \sin \left (d x + c\right )^{8} + 82698 \, \sin \left (d x + c\right )^{7} - 114576 \, \sin \left (d x + c\right )^{6} - 255222 \, \sin \left (d x + c\right )^{5} - 190080 \, \sin \left (d x + c\right )^{4} - 21395 \, \sin \left (d x + c\right )^{3} + 72776 \, \sin \left (d x + c\right )^{2} + 66953 \, \sin \left (d x + c\right ) + 34816\right )}}{a^{8} \sin \left (d x + c\right )^{12} + 8 \, a^{8} \sin \left (d x + c\right )^{11} + 26 \, a^{8} \sin \left (d x + c\right )^{10} + 40 \, a^{8} \sin \left (d x + c\right )^{9} + 15 \, a^{8} \sin \left (d x + c\right )^{8} - 48 \, a^{8} \sin \left (d x + c\right )^{7} - 84 \, a^{8} \sin \left (d x + c\right )^{6} - 48 \, a^{8} \sin \left (d x + c\right )^{5} + 15 \, a^{8} \sin \left (d x + c\right )^{4} + 40 \, a^{8} \sin \left (d x + c\right )^{3} + 26 \, a^{8} \sin \left (d x + c\right )^{2} + 8 \, a^{8} \sin \left (d x + c\right ) + a^{8}} - \frac{3465 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{8}} + \frac{3465 \, \log \left (\sin \left (d x + c\right ) - 1\right )}{a^{8}}}{430080 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^5/(a+a*sin(d*x+c))^8,x, algorithm="maxima")

[Out]

-1/430080*(2*(3465*sin(d*x + c)^11 + 27720*sin(d*x + c)^10 + 91245*sin(d*x + c)^9 + 147840*sin(d*x + c)^8 + 82
698*sin(d*x + c)^7 - 114576*sin(d*x + c)^6 - 255222*sin(d*x + c)^5 - 190080*sin(d*x + c)^4 - 21395*sin(d*x + c
)^3 + 72776*sin(d*x + c)^2 + 66953*sin(d*x + c) + 34816)/(a^8*sin(d*x + c)^12 + 8*a^8*sin(d*x + c)^11 + 26*a^8
*sin(d*x + c)^10 + 40*a^8*sin(d*x + c)^9 + 15*a^8*sin(d*x + c)^8 - 48*a^8*sin(d*x + c)^7 - 84*a^8*sin(d*x + c)
^6 - 48*a^8*sin(d*x + c)^5 + 15*a^8*sin(d*x + c)^4 + 40*a^8*sin(d*x + c)^3 + 26*a^8*sin(d*x + c)^2 + 8*a^8*sin
(d*x + c) + a^8) - 3465*log(sin(d*x + c) + 1)/a^8 + 3465*log(sin(d*x + c) - 1)/a^8)/d

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Fricas [A]  time = 2.26353, size = 1332, normalized size = 4.69 \begin{align*} \frac{55440 \, \cos \left (d x + c\right )^{10} - 572880 \, \cos \left (d x + c\right )^{8} + 1507968 \, \cos \left (d x + c\right )^{6} - 1260864 \, \cos \left (d x + c\right )^{4} + 157696 \, \cos \left (d x + c\right )^{2} + 3465 \,{\left (\cos \left (d x + c\right )^{12} - 32 \, \cos \left (d x + c\right )^{10} + 160 \, \cos \left (d x + c\right )^{8} - 256 \, \cos \left (d x + c\right )^{6} + 128 \, \cos \left (d x + c\right )^{4} - 8 \,{\left (\cos \left (d x + c\right )^{10} - 10 \, \cos \left (d x + c\right )^{8} + 24 \, \cos \left (d x + c\right )^{6} - 16 \, \cos \left (d x + c\right )^{4}\right )} \sin \left (d x + c\right )\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3465 \,{\left (\cos \left (d x + c\right )^{12} - 32 \, \cos \left (d x + c\right )^{10} + 160 \, \cos \left (d x + c\right )^{8} - 256 \, \cos \left (d x + c\right )^{6} + 128 \, \cos \left (d x + c\right )^{4} - 8 \,{\left (\cos \left (d x + c\right )^{10} - 10 \, \cos \left (d x + c\right )^{8} + 24 \, \cos \left (d x + c\right )^{6} - 16 \, \cos \left (d x + c\right )^{4}\right )} \sin \left (d x + c\right )\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (3465 \, \cos \left (d x + c\right )^{10} - 108570 \, \cos \left (d x + c\right )^{8} + 482328 \, \cos \left (d x + c\right )^{6} - 574992 \, \cos \left (d x + c\right )^{4} + 98560 \, \cos \left (d x + c\right )^{2} + 32256\right )} \sin \left (d x + c\right ) + 43008}{430080 \,{\left (a^{8} d \cos \left (d x + c\right )^{12} - 32 \, a^{8} d \cos \left (d x + c\right )^{10} + 160 \, a^{8} d \cos \left (d x + c\right )^{8} - 256 \, a^{8} d \cos \left (d x + c\right )^{6} + 128 \, a^{8} d \cos \left (d x + c\right )^{4} - 8 \,{\left (a^{8} d \cos \left (d x + c\right )^{10} - 10 \, a^{8} d \cos \left (d x + c\right )^{8} + 24 \, a^{8} d \cos \left (d x + c\right )^{6} - 16 \, a^{8} d \cos \left (d x + c\right )^{4}\right )} \sin \left (d x + c\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^5/(a+a*sin(d*x+c))^8,x, algorithm="fricas")

[Out]

1/430080*(55440*cos(d*x + c)^10 - 572880*cos(d*x + c)^8 + 1507968*cos(d*x + c)^6 - 1260864*cos(d*x + c)^4 + 15
7696*cos(d*x + c)^2 + 3465*(cos(d*x + c)^12 - 32*cos(d*x + c)^10 + 160*cos(d*x + c)^8 - 256*cos(d*x + c)^6 + 1
28*cos(d*x + c)^4 - 8*(cos(d*x + c)^10 - 10*cos(d*x + c)^8 + 24*cos(d*x + c)^6 - 16*cos(d*x + c)^4)*sin(d*x +
c))*log(sin(d*x + c) + 1) - 3465*(cos(d*x + c)^12 - 32*cos(d*x + c)^10 + 160*cos(d*x + c)^8 - 256*cos(d*x + c)
^6 + 128*cos(d*x + c)^4 - 8*(cos(d*x + c)^10 - 10*cos(d*x + c)^8 + 24*cos(d*x + c)^6 - 16*cos(d*x + c)^4)*sin(
d*x + c))*log(-sin(d*x + c) + 1) + 2*(3465*cos(d*x + c)^10 - 108570*cos(d*x + c)^8 + 482328*cos(d*x + c)^6 - 5
74992*cos(d*x + c)^4 + 98560*cos(d*x + c)^2 + 32256)*sin(d*x + c) + 43008)/(a^8*d*cos(d*x + c)^12 - 32*a^8*d*c
os(d*x + c)^10 + 160*a^8*d*cos(d*x + c)^8 - 256*a^8*d*cos(d*x + c)^6 + 128*a^8*d*cos(d*x + c)^4 - 8*(a^8*d*cos
(d*x + c)^10 - 10*a^8*d*cos(d*x + c)^8 + 24*a^8*d*cos(d*x + c)^6 - 16*a^8*d*cos(d*x + c)^4)*sin(d*x + c))

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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)**5/(a+a*sin(d*x+c))**8,x)

[Out]

Timed out

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Giac [A]  time = 1.23477, size = 251, normalized size = 0.88 \begin{align*} \frac{\frac{27720 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a^{8}} - \frac{27720 \, \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a^{8}} + \frac{420 \,{\left (99 \, \sin \left (d x + c\right )^{2} - 220 \, \sin \left (d x + c\right ) + 123\right )}}{a^{8}{\left (\sin \left (d x + c\right ) - 1\right )}^{2}} - \frac{81191 \, \sin \left (d x + c\right )^{10} + 858110 \, \sin \left (d x + c\right )^{9} + 4107195 \, \sin \left (d x + c\right )^{8} + 11748840 \, \sin \left (d x + c\right )^{7} + 22318590 \, \sin \left (d x + c\right )^{6} + 29583540 \, \sin \left (d x + c\right )^{5} + 27983550 \, \sin \left (d x + c\right )^{4} + 19002600 \, \sin \left (d x + c\right )^{3} + 9206235 \, \sin \left (d x + c\right )^{2} + 3108990 \, \sin \left (d x + c\right ) + 648327}{a^{8}{\left (\sin \left (d x + c\right ) + 1\right )}^{10}}}{3440640 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^5/(a+a*sin(d*x+c))^8,x, algorithm="giac")

[Out]

1/3440640*(27720*log(abs(sin(d*x + c) + 1))/a^8 - 27720*log(abs(sin(d*x + c) - 1))/a^8 + 420*(99*sin(d*x + c)^
2 - 220*sin(d*x + c) + 123)/(a^8*(sin(d*x + c) - 1)^2) - (81191*sin(d*x + c)^10 + 858110*sin(d*x + c)^9 + 4107
195*sin(d*x + c)^8 + 11748840*sin(d*x + c)^7 + 22318590*sin(d*x + c)^6 + 29583540*sin(d*x + c)^5 + 27983550*si
n(d*x + c)^4 + 19002600*sin(d*x + c)^3 + 9206235*sin(d*x + c)^2 + 3108990*sin(d*x + c) + 648327)/(a^8*(sin(d*x
+ c) + 1)^10))/d