Integrand size = 24, antiderivative size = 236 \[ \int \frac {\sin ^9(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^2} \, dx=\frac {\sqrt {a} \left (5 \sqrt {a}-6 \sqrt {b}\right ) \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{8 \left (\sqrt {a}-\sqrt {b}\right )^{3/2} b^{9/4} d}+\frac {\sqrt {a} \left (5 \sqrt {a}+6 \sqrt {b}\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{8 \left (\sqrt {a}+\sqrt {b}\right )^{3/2} b^{9/4} d}-\frac {\cos (c+d x)}{b^2 d}-\frac {a \cos (c+d x) \left (a+b-b \cos ^2(c+d x)\right )}{4 (a-b) b^2 d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )} \]
-cos(d*x+c)/b^2/d-1/4*a*cos(d*x+c)*(a+b-b*cos(d*x+c)^2)/(a-b)/b^2/d/(a-b+2 *b*cos(d*x+c)^2-b*cos(d*x+c)^4)+1/8*arctan(b^(1/4)*cos(d*x+c)/(a^(1/2)-b^( 1/2))^(1/2))*a^(1/2)*(5*a^(1/2)-6*b^(1/2))/b^(9/4)/d/(a^(1/2)-b^(1/2))^(3/ 2)+1/8*arctanh(b^(1/4)*cos(d*x+c)/(a^(1/2)+b^(1/2))^(1/2))*a^(1/2)*(5*a^(1 /2)+6*b^(1/2))/b^(9/4)/d/(a^(1/2)+b^(1/2))^(3/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 2.37 (sec) , antiderivative size = 486, normalized size of antiderivative = 2.06 \[ \int \frac {\sin ^9(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^2} \, dx=-\frac {32 \cos (c+d x)+\frac {32 a \cos (c+d x) (2 a+b-b \cos (2 (c+d x)))}{(a-b) (8 a-3 b+4 b \cos (2 (c+d x))-b \cos (4 (c+d x)))}+\frac {i a \text {RootSum}\left [b-4 b \text {$\#$1}^2-16 a \text {$\#$1}^4+6 b \text {$\#$1}^4-4 b \text {$\#$1}^6+b \text {$\#$1}^8\&,\frac {-2 b \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )+i b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right )-40 a \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^2+54 b \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^2+20 i a \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^2-27 i b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^2+40 a \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^4-54 b \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^4-20 i a \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^4+27 i b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^4+2 b \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^6-i b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^6}{-b \text {$\#$1}-8 a \text {$\#$1}^3+3 b \text {$\#$1}^3-3 b \text {$\#$1}^5+b \text {$\#$1}^7}\&\right ]}{a-b}}{32 b^2 d} \]
-1/32*(32*Cos[c + d*x] + (32*a*Cos[c + d*x]*(2*a + b - b*Cos[2*(c + d*x)]) )/((a - b)*(8*a - 3*b + 4*b*Cos[2*(c + d*x)] - b*Cos[4*(c + d*x)])) + (I*a *RootSum[b - 4*b*#1^2 - 16*a*#1^4 + 6*b*#1^4 - 4*b*#1^6 + b*#1^8 & , (-2*b *ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] + I*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] - 40*a*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 + 54*b*ArcTan [Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 + (20*I)*a*Log[1 - 2*Cos[c + d*x]* #1 + #1^2]*#1^2 - (27*I)*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 + 40*a*A rcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 - 54*b*ArcTan[Sin[c + d*x]/(C os[c + d*x] - #1)]*#1^4 - (20*I)*a*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4 + (27*I)*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4 + 2*b*ArcTan[Sin[c + d*x ]/(Cos[c + d*x] - #1)]*#1^6 - I*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^6)/ (-(b*#1) - 8*a*#1^3 + 3*b*#1^3 - 3*b*#1^5 + b*#1^7) & ])/(a - b))/(b^2*d)
Time = 0.62 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.10, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3042, 3694, 1517, 27, 2205, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\sin ^9(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (c+d x)^9}{\left (a-b \sin (c+d x)^4\right )^2}dx\) |
| \(\Big \downarrow \) 3694 |
| \(\displaystyle -\frac {\int \frac {\left (1-\cos ^2(c+d x)\right )^4}{\left (-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)+a-b\right )^2}d\cos (c+d x)}{d}\) |
| \(\Big \downarrow \) 1517 |
| \(\displaystyle -\frac {\frac {a \cos (c+d x) \left (a-b \cos ^2(c+d x)+b\right )}{4 b^2 (a-b) \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )}-\frac {\int \frac {2 \left (4 a (a-b) \cos ^4(c+d x)-a (7 a-8 b) \cos ^2(c+d x)+a \left (\frac {a^2}{b}+a-4 b\right )\right )}{-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)+a-b}d\cos (c+d x)}{8 a b (a-b)}}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {a \cos (c+d x) \left (a-b \cos ^2(c+d x)+b\right )}{4 b^2 (a-b) \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )}-\frac {\int \frac {4 a (a-b) \cos ^4(c+d x)-a (7 a-8 b) \cos ^2(c+d x)+a \left (\frac {a^2}{b}+a-4 b\right )}{-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)+a-b}d\cos (c+d x)}{4 a b (a-b)}}{d}\) |
| \(\Big \downarrow \) 2205 |
| \(\displaystyle -\frac {\frac {a \cos (c+d x) \left (a-b \cos ^2(c+d x)+b\right )}{4 b^2 (a-b) \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )}-\frac {\int \left (\frac {b \cos ^2(c+d x) a^2+(5 a-7 b) a^2}{b \left (-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)+a-b\right )}-\frac {4 a (a-b)}{b}\right )d\cos (c+d x)}{4 a b (a-b)}}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\frac {a \cos (c+d x) \left (a-b \cos ^2(c+d x)+b\right )}{4 b^2 (a-b) \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )}-\frac {\frac {a^{3/2} \left (-\sqrt {a} \sqrt {b}+5 a-6 b\right ) \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 b^{5/4} \sqrt {\sqrt {a}-\sqrt {b}}}+\frac {a^{3/2} \left (\sqrt {a} \sqrt {b}+5 a-6 b\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 b^{5/4} \sqrt {\sqrt {a}+\sqrt {b}}}-\frac {4 a (a-b) \cos (c+d x)}{b}}{4 a b (a-b)}}{d}\) |
-((-1/4*((a^(3/2)*(5*a - Sqrt[a]*Sqrt[b] - 6*b)*ArcTan[(b^(1/4)*Cos[c + d* x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(2*Sqrt[Sqrt[a] - Sqrt[b]]*b^(5/4)) + (a^(3/ 2)*(5*a + Sqrt[a]*Sqrt[b] - 6*b)*ArcTanh[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[ a] + Sqrt[b]]])/(2*Sqrt[Sqrt[a] + Sqrt[b]]*b^(5/4)) - (4*a*(a - b)*Cos[c + d*x])/b)/(a*(a - b)*b) + (a*Cos[c + d*x]*(a + b - b*Cos[c + d*x]^2))/(4*( a - b)*b^2*(a - b + 2*b*Cos[c + d*x]^2 - b*Cos[c + d*x]^4)))/d)
3.3.12.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x _Symbol] :> With[{f = Coeff[PolynomialRemainder[(d + e*x^2)^q, a + b*x^2 + c*x^4, x], x, 0], g = Coeff[PolynomialRemainder[(d + e*x^2)^q, a + b*x^2 + c*x^4, x], x, 2]}, Simp[x*(a + b*x^2 + c*x^4)^(p + 1)*((a*b*g - f*(b^2 - 2* a*c) - c*(b*f - 2*a*g)*x^2)/(2*a*(p + 1)*(b^2 - 4*a*c))), x] + Simp[1/(2*a* (p + 1)*(b^2 - 4*a*c)) Int[(a + b*x^2 + c*x^4)^(p + 1)*ExpandToSum[2*a*(p + 1)*(b^2 - 4*a*c)*PolynomialQuotient[(d + e*x^2)^q, a + b*x^2 + c*x^4, x] + b^2*f*(2*p + 3) - 2*a*c*f*(4*p + 5) - a*b*g + c*(4*p + 7)*(b*f - 2*a*g)* x^2, x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ [c*d^2 - b*d*e + a*e^2, 0] && IGtQ[q, 1] && LtQ[p, -1]
Int[(Px_)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Int[ExpandInte grand[Px/(a + b*x^2 + c*x^4), x], x] /; FreeQ[{a, b, c}, x] && PolyQ[Px, x^ 2] && Expon[Px, x^2] > 1
Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^4)^ (p_.), x_Symbol] :> With[{ff = FreeFactors[Cos[e + f*x], x]}, Simp[-ff/f Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b - 2*b*ff^2*x^2 + b*ff^4*x^4)^p, x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]
Time = 4.12 (sec) , antiderivative size = 209, normalized size of antiderivative = 0.89
| method | result | size |
| derivativedivides | \(\frac {-\frac {\cos \left (d x +c \right )}{b^{2}}+\frac {a \left (\frac {\frac {b \left (\cos ^{3}\left (d x +c \right )\right )}{4 a -4 b}-\frac {\left (a +b \right ) \cos \left (d x +c \right )}{4 \left (a -b \right )}}{a -b +2 b \left (\cos ^{2}\left (d x +c \right )\right )-b \left (\cos ^{4}\left (d x +c \right )\right )}+\frac {b \left (\frac {\left (-\sqrt {a b}-6 b +5 a \right ) \arctan \left (\frac {\cos \left (d x +c \right ) b}{\sqrt {\left (\sqrt {a b}-b \right ) b}}\right )}{2 \sqrt {a b}\, \sqrt {\left (\sqrt {a b}-b \right ) b}}-\frac {\left (-\sqrt {a b}+6 b -5 a \right ) \operatorname {arctanh}\left (\frac {\cos \left (d x +c \right ) b}{\sqrt {\left (\sqrt {a b}+b \right ) b}}\right )}{2 \sqrt {a b}\, \sqrt {\left (\sqrt {a b}+b \right ) b}}\right )}{4 a -4 b}\right )}{b^{2}}}{d}\) | \(209\) |
| default | \(\frac {-\frac {\cos \left (d x +c \right )}{b^{2}}+\frac {a \left (\frac {\frac {b \left (\cos ^{3}\left (d x +c \right )\right )}{4 a -4 b}-\frac {\left (a +b \right ) \cos \left (d x +c \right )}{4 \left (a -b \right )}}{a -b +2 b \left (\cos ^{2}\left (d x +c \right )\right )-b \left (\cos ^{4}\left (d x +c \right )\right )}+\frac {b \left (\frac {\left (-\sqrt {a b}-6 b +5 a \right ) \arctan \left (\frac {\cos \left (d x +c \right ) b}{\sqrt {\left (\sqrt {a b}-b \right ) b}}\right )}{2 \sqrt {a b}\, \sqrt {\left (\sqrt {a b}-b \right ) b}}-\frac {\left (-\sqrt {a b}+6 b -5 a \right ) \operatorname {arctanh}\left (\frac {\cos \left (d x +c \right ) b}{\sqrt {\left (\sqrt {a b}+b \right ) b}}\right )}{2 \sqrt {a b}\, \sqrt {\left (\sqrt {a b}+b \right ) b}}\right )}{4 a -4 b}\right )}{b^{2}}}{d}\) | \(209\) |
| risch | \(-\frac {{\mathrm e}^{i \left (d x +c \right )}}{2 b^{2} d}-\frac {{\mathrm e}^{-i \left (d x +c \right )}}{2 b^{2} d}-\frac {a \left (b \,{\mathrm e}^{7 i \left (d x +c \right )}-4 a \,{\mathrm e}^{5 i \left (d x +c \right )}-b \,{\mathrm e}^{5 i \left (d x +c \right )}-4 a \,{\mathrm e}^{3 i \left (d x +c \right )}-b \,{\mathrm e}^{3 i \left (d x +c \right )}+b \,{\mathrm e}^{i \left (d x +c \right )}\right )}{2 b^{2} \left (a -b \right ) d \left ({\mathrm e}^{8 i \left (d x +c \right )} b -4 b \,{\mathrm e}^{6 i \left (d x +c \right )}-16 a \,{\mathrm e}^{4 i \left (d x +c \right )}+6 b \,{\mathrm e}^{4 i \left (d x +c \right )}-4 b \,{\mathrm e}^{2 i \left (d x +c \right )}+b \right )}-\frac {i \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (a^{3} b^{9} d^{4}-3 a^{2} b^{10} d^{4}+3 a \,b^{11} d^{4}-b^{12} d^{4}\right ) \textit {\_Z}^{4}+\left (-30720 a^{3} b^{5} d^{2}+96256 a^{2} b^{6} d^{2}-73728 a \,b^{7} d^{2}\right ) \textit {\_Z}^{2}-655360000 a^{4}+1887436800 a^{3} b -1358954496 a^{2} b^{2}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\left (\left (\frac {2 i a^{4} b^{7} d^{3}}{5120000 a^{5}-21504000 a^{4} b +30179328 a^{3} b^{2}-14155776 a^{2} b^{3}}-\frac {9 i a^{3} b^{8} d^{3}}{5120000 a^{5}-21504000 a^{4} b +30179328 a^{3} b^{2}-14155776 a^{2} b^{3}}+\frac {15 i a^{2} b^{9} d^{3}}{5120000 a^{5}-21504000 a^{4} b +30179328 a^{3} b^{2}-14155776 a^{2} b^{3}}-\frac {11 i a \,b^{10} d^{3}}{5120000 a^{5}-21504000 a^{4} b +30179328 a^{3} b^{2}-14155776 a^{2} b^{3}}+\frac {3 i b^{11} d^{3}}{5120000 a^{5}-21504000 a^{4} b +30179328 a^{3} b^{2}-14155776 a^{2} b^{3}}\right ) \textit {\_R}^{3}+\left (-\frac {64000 i a^{5} b^{2} d}{5120000 a^{5}-21504000 a^{4} b +30179328 a^{3} b^{2}-14155776 a^{2} b^{3}}+\frac {235520 i a^{4} b^{3} d}{5120000 a^{5}-21504000 a^{4} b +30179328 a^{3} b^{2}-14155776 a^{2} b^{3}}-\frac {227840 i a^{3} b^{4} d}{5120000 a^{5}-21504000 a^{4} b +30179328 a^{3} b^{2}-14155776 a^{2} b^{3}}-\frac {46080 i a^{2} b^{5} d}{5120000 a^{5}-21504000 a^{4} b +30179328 a^{3} b^{2}-14155776 a^{2} b^{3}}+\frac {110592 i a \,b^{6} d}{5120000 a^{5}-21504000 a^{4} b +30179328 a^{3} b^{2}-14155776 a^{2} b^{3}}\right ) \textit {\_R} \right ) {\mathrm e}^{i \left (d x +c \right )}+1\right )\right )}{512}\) | \(728\) |
1/d*(-cos(d*x+c)/b^2+a/b^2*((1/4*b/(a-b)*cos(d*x+c)^3-1/4*(a+b)/(a-b)*cos( d*x+c))/(a-b+2*b*cos(d*x+c)^2-b*cos(d*x+c)^4)+1/4/(a-b)*b*(1/2*(-(a*b)^(1/ 2)-6*b+5*a)/(a*b)^(1/2)/(((a*b)^(1/2)-b)*b)^(1/2)*arctan(cos(d*x+c)*b/(((a *b)^(1/2)-b)*b)^(1/2))-1/2*(-(a*b)^(1/2)+6*b-5*a)/(a*b)^(1/2)/(((a*b)^(1/2 )+b)*b)^(1/2)*arctanh(cos(d*x+c)*b/(((a*b)^(1/2)+b)*b)^(1/2)))))
Leaf count of result is larger than twice the leaf count of optimal. 2649 vs. \(2 (188) = 376\).
Time = 0.60 (sec) , antiderivative size = 2649, normalized size of antiderivative = 11.22 \[ \int \frac {\sin ^9(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^2} \, dx=\text {Too large to display} \]
-1/16*(16*(a*b - b^2)*cos(d*x + c)^5 - 4*(7*a*b - 8*b^2)*cos(d*x + c)^3 + ((a*b^3 - b^4)*d*cos(d*x + c)^4 - 2*(a*b^3 - b^4)*d*cos(d*x + c)^2 - (a^2* b^2 - 2*a*b^3 + b^4)*d)*sqrt(-((a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*d^2*s qrt((625*a^7 - 3450*a^6*b + 7161*a^5*b^2 - 6624*a^4*b^3 + 2304*a^3*b^4)/(( a^6*b^9 - 6*a^5*b^10 + 15*a^4*b^11 - 20*a^3*b^12 + 15*a^2*b^13 - 6*a*b^14 + b^15)*d^4)) + 15*a^3 - 47*a^2*b + 36*a*b^2)/((a^3*b^4 - 3*a^2*b^5 + 3*a* b^6 - b^7)*d^2))*log((625*a^5 - 2625*a^4*b + 3684*a^3*b^2 - 1728*a^2*b^3)* cos(d*x + c) + (2*(2*a^4*b^7 - 9*a^3*b^8 + 15*a^2*b^9 - 11*a*b^10 + 3*b^11 )*d^3*sqrt((625*a^7 - 3450*a^6*b + 7161*a^5*b^2 - 6624*a^4*b^3 + 2304*a^3* b^4)/((a^6*b^9 - 6*a^5*b^10 + 15*a^4*b^11 - 20*a^3*b^12 + 15*a^2*b^13 - 6* a*b^14 + b^15)*d^4)) - (125*a^5*b^2 - 520*a^4*b^3 + 723*a^3*b^4 - 336*a^2* b^5)*d)*sqrt(-((a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*d^2*sqrt((625*a^7 - 3 450*a^6*b + 7161*a^5*b^2 - 6624*a^4*b^3 + 2304*a^3*b^4)/((a^6*b^9 - 6*a^5* b^10 + 15*a^4*b^11 - 20*a^3*b^12 + 15*a^2*b^13 - 6*a*b^14 + b^15)*d^4)) + 15*a^3 - 47*a^2*b + 36*a*b^2)/((a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*d^2)) ) - ((a*b^3 - b^4)*d*cos(d*x + c)^4 - 2*(a*b^3 - b^4)*d*cos(d*x + c)^2 - ( a^2*b^2 - 2*a*b^3 + b^4)*d)*sqrt(((a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*d^ 2*sqrt((625*a^7 - 3450*a^6*b + 7161*a^5*b^2 - 6624*a^4*b^3 + 2304*a^3*b^4) /((a^6*b^9 - 6*a^5*b^10 + 15*a^4*b^11 - 20*a^3*b^12 + 15*a^2*b^13 - 6*a*b^ 14 + b^15)*d^4)) - 15*a^3 + 47*a^2*b - 36*a*b^2)/((a^3*b^4 - 3*a^2*b^5 ...
Timed out. \[ \int \frac {\sin ^9(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^2} \, dx=\text {Timed out} \]
\[ \int \frac {\sin ^9(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^2} \, dx=\int { \frac {\sin \left (d x + c\right )^{9}}{{\left (b \sin \left (d x + c\right )^{4} - a\right )}^{2}} \,d x } \]
1/2*((2*a*b^2 - 3*b^3)*cos(2*d*x + 2*c)*cos(d*x + c) - 4*(2*a*b^2 - 3*b^3) *sin(3*d*x + 3*c)*sin(2*d*x + 2*c) + (2*a*b^2 - 3*b^3)*sin(2*d*x + 2*c)*si n(d*x + c) - ((a*b^2 - b^3)*cos(9*d*x + 9*c) - 4*(a*b^2 - b^3)*cos(7*d*x + 7*c) - 2*(8*a^2*b - 11*a*b^2 + 3*b^3)*cos(5*d*x + 5*c) - 4*(a*b^2 - b^3)* cos(3*d*x + 3*c) + (a*b^2 - b^3)*cos(d*x + c))*cos(10*d*x + 10*c) - (a*b^2 - b^3 - (2*a*b^2 - 3*b^3)*cos(8*d*x + 8*c) - (20*a^2*b - 17*a*b^2 + 2*b^3 )*cos(6*d*x + 6*c) - (20*a^2*b - 17*a*b^2 + 2*b^3)*cos(4*d*x + 4*c) - (2*a *b^2 - 3*b^3)*cos(2*d*x + 2*c))*cos(9*d*x + 9*c) - (4*(2*a*b^2 - 3*b^3)*co s(7*d*x + 7*c) + 2*(16*a^2*b - 30*a*b^2 + 9*b^3)*cos(5*d*x + 5*c) + 4*(2*a *b^2 - 3*b^3)*cos(3*d*x + 3*c) - (2*a*b^2 - 3*b^3)*cos(d*x + c))*cos(8*d*x + 8*c) + 4*(a*b^2 - b^3 - (20*a^2*b - 17*a*b^2 + 2*b^3)*cos(6*d*x + 6*c) - (20*a^2*b - 17*a*b^2 + 2*b^3)*cos(4*d*x + 4*c) - (2*a*b^2 - 3*b^3)*cos(2 *d*x + 2*c))*cos(7*d*x + 7*c) - (2*(160*a^3 - 196*a^2*b + 67*a*b^2 - 6*b^3 )*cos(5*d*x + 5*c) + 4*(20*a^2*b - 17*a*b^2 + 2*b^3)*cos(3*d*x + 3*c) - (2 0*a^2*b - 17*a*b^2 + 2*b^3)*cos(d*x + c))*cos(6*d*x + 6*c) + 2*(8*a^2*b - 11*a*b^2 + 3*b^3 - (160*a^3 - 196*a^2*b + 67*a*b^2 - 6*b^3)*cos(4*d*x + 4* c) - (16*a^2*b - 30*a*b^2 + 9*b^3)*cos(2*d*x + 2*c))*cos(5*d*x + 5*c) - (4 *(20*a^2*b - 17*a*b^2 + 2*b^3)*cos(3*d*x + 3*c) - (20*a^2*b - 17*a*b^2 + 2 *b^3)*cos(d*x + c))*cos(4*d*x + 4*c) + 4*(a*b^2 - b^3 - (2*a*b^2 - 3*b^3)* cos(2*d*x + 2*c))*cos(3*d*x + 3*c) - (a*b^2 - b^3)*cos(d*x + c) + 2*((a...
\[ \int \frac {\sin ^9(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^2} \, dx=\int { \frac {\sin \left (d x + c\right )^{9}}{{\left (b \sin \left (d x + c\right )^{4} - a\right )}^{2}} \,d x } \]
Time = 16.00 (sec) , antiderivative size = 3941, normalized size of antiderivative = 16.70 \[ \int \frac {\sin ^9(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^2} \, dx=\text {Too large to display} \]
- cos(c + d*x)/(b^2*d) - ((cos(c + d*x)*(a*b + a^2))/(4*(a - b)) - (a*b*co s(c + d*x)^3)/(4*(a - b)))/(d*(a*b^2 - b^3 + 2*b^3*cos(c + d*x)^2 - b^3*co s(c + d*x)^4)) - (atan(((((1792*a^2*b^6 - 3072*a^3*b^5 + 1280*a^4*b^4)/(64 *(b^5 - 2*a*b^4 + a^2*b^3)) - (cos(c + d*x)*((25*a^2*(a^3*b^9)^(1/2) + 48* b^2*(a^3*b^9)^(1/2) - 36*a*b^7 + 47*a^2*b^6 - 15*a^3*b^5 - 69*a*b*(a^3*b^9 )^(1/2))/(256*(3*a*b^11 - b^12 - 3*a^2*b^10 + a^3*b^9)))^(1/2)*(256*a*b^7 - 512*a^2*b^6 + 256*a^3*b^5))/(4*(a^2*b - 2*a*b^2 + b^3)))*((25*a^2*(a^3*b ^9)^(1/2) + 48*b^2*(a^3*b^9)^(1/2) - 36*a*b^7 + 47*a^2*b^6 - 15*a^3*b^5 - 69*a*b*(a^3*b^9)^(1/2))/(256*(3*a*b^11 - b^12 - 3*a^2*b^10 + a^3*b^9)))^(1 /2) + (cos(c + d*x)*(25*a^4 - 59*a^3*b + 36*a^2*b^2))/(4*(a^2*b - 2*a*b^2 + b^3)))*((25*a^2*(a^3*b^9)^(1/2) + 48*b^2*(a^3*b^9)^(1/2) - 36*a*b^7 + 47 *a^2*b^6 - 15*a^3*b^5 - 69*a*b*(a^3*b^9)^(1/2))/(256*(3*a*b^11 - b^12 - 3* a^2*b^10 + a^3*b^9)))^(1/2)*1i - (((1792*a^2*b^6 - 3072*a^3*b^5 + 1280*a^4 *b^4)/(64*(b^5 - 2*a*b^4 + a^2*b^3)) + (cos(c + d*x)*((25*a^2*(a^3*b^9)^(1 /2) + 48*b^2*(a^3*b^9)^(1/2) - 36*a*b^7 + 47*a^2*b^6 - 15*a^3*b^5 - 69*a*b *(a^3*b^9)^(1/2))/(256*(3*a*b^11 - b^12 - 3*a^2*b^10 + a^3*b^9)))^(1/2)*(2 56*a*b^7 - 512*a^2*b^6 + 256*a^3*b^5))/(4*(a^2*b - 2*a*b^2 + b^3)))*((25*a ^2*(a^3*b^9)^(1/2) + 48*b^2*(a^3*b^9)^(1/2) - 36*a*b^7 + 47*a^2*b^6 - 15*a ^3*b^5 - 69*a*b*(a^3*b^9)^(1/2))/(256*(3*a*b^11 - b^12 - 3*a^2*b^10 + a^3* b^9)))^(1/2) - (cos(c + d*x)*(25*a^4 - 59*a^3*b + 36*a^2*b^2))/(4*(a^2*...