Integrand size = 24, antiderivative size = 288 \[ \int \frac {\sin ^3(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=-\frac {\left (5 \sqrt {a}-2 \sqrt {b}\right ) \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{64 a^{3/2} \left (\sqrt {a}-\sqrt {b}\right )^{5/2} b^{3/4} d}+\frac {\left (5 \sqrt {a}+2 \sqrt {b}\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{64 a^{3/2} \left (\sqrt {a}+\sqrt {b}\right )^{5/2} b^{3/4} d}-\frac {\cos (c+d x) \left (2-\cos ^2(c+d x)\right )}{8 (a-b) d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )^2}-\frac {\cos (c+d x) \left (11 a+b-(5 a+b) \cos ^2(c+d x)\right )}{32 a (a-b)^2 d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )} \]
-1/8*cos(d*x+c)*(2-cos(d*x+c)^2)/(a-b)/d/(a-b+2*b*cos(d*x+c)^2-b*cos(d*x+c )^4)^2-1/32*cos(d*x+c)*(11*a+b-(5*a+b)*cos(d*x+c)^2)/a/(a-b)^2/d/(a-b+2*b* cos(d*x+c)^2-b*cos(d*x+c)^4)-1/64*arctan(b^(1/4)*cos(d*x+c)/(a^(1/2)-b^(1/ 2))^(1/2))*(5*a^(1/2)-2*b^(1/2))/a^(3/2)/b^(3/4)/d/(a^(1/2)-b^(1/2))^(5/2) +1/64*arctanh(b^(1/4)*cos(d*x+c)/(a^(1/2)+b^(1/2))^(1/2))*(5*a^(1/2)+2*b^( 1/2))/a^(3/2)/b^(3/4)/d/(a^(1/2)+b^(1/2))^(5/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 1.14 (sec) , antiderivative size = 631, normalized size of antiderivative = 2.19 \[ \int \frac {\sin ^3(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=\frac {\frac {32 \cos (c+d x) (-17 a-b+(5 a+b) \cos (2 (c+d x)))}{a (8 a-3 b+4 b \cos (2 (c+d x))-b \cos (4 (c+d x)))}+\frac {512 (a-b) (-5 \cos (c+d x)+\cos (3 (c+d x)))}{(-8 a+3 b-4 b \cos (2 (c+d x))+b \cos (4 (c+d x)))^2}+\frac {i \text {RootSum}\left [b-4 b \text {$\#$1}^2-16 a \text {$\#$1}^4+6 b \text {$\#$1}^4-4 b \text {$\#$1}^6+b \text {$\#$1}^8\&,\frac {10 a \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )+2 b \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )-5 i a \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right )-i b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right )-94 a \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^2+10 b \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^2+47 i a \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^2-5 i b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^2+94 a \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^4-10 b \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^4-47 i a \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^4+5 i b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^4-10 a \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^6-2 b \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^6+5 i a \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^6+i b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^6}{-b \text {$\#$1}-8 a \text {$\#$1}^3+3 b \text {$\#$1}^3-3 b \text {$\#$1}^5+b \text {$\#$1}^7}\&\right ]}{a}}{256 (a-b)^2 d} \]
((32*Cos[c + d*x]*(-17*a - b + (5*a + b)*Cos[2*(c + d*x)]))/(a*(8*a - 3*b + 4*b*Cos[2*(c + d*x)] - b*Cos[4*(c + d*x)])) + (512*(a - b)*(-5*Cos[c + d *x] + Cos[3*(c + d*x)]))/(-8*a + 3*b - 4*b*Cos[2*(c + d*x)] + b*Cos[4*(c + d*x)])^2 + (I*RootSum[b - 4*b*#1^2 - 16*a*#1^4 + 6*b*#1^4 - 4*b*#1^6 + b* #1^8 & , (10*a*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] + 2*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] - (5*I)*a*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] - I*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] - 94*a*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 + 10*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 + ( 47*I)*a*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 - (5*I)*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 + 94*a*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 - 10*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 - (47*I)*a*Log[1 - 2 *Cos[c + d*x]*#1 + #1^2]*#1^4 + (5*I)*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]* #1^4 - 10*a*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^6 - 2*b*ArcTan[Sin [c + d*x]/(Cos[c + d*x] - #1)]*#1^6 + (5*I)*a*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#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)/(256*(a - b)^2*d)
Time = 0.54 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.10, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {3042, 3694, 1492, 27, 1492, 27, 1480, 218, 221}
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.
| \(\displaystyle \int \frac {\sin ^3(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (c+d x)^3}{\left (a-b \sin (c+d x)^4\right )^3}dx\) |
| \(\Big \downarrow \) 3694 |
| \(\displaystyle -\frac {\int \frac {1-\cos ^2(c+d x)}{\left (-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)+a-b\right )^3}d\cos (c+d x)}{d}\) |
| \(\Big \downarrow \) 1492 |
| \(\displaystyle -\frac {\frac {\cos (c+d x) \left (2-\cos ^2(c+d x)\right )}{8 (a-b) \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )^2}-\frac {\int -\frac {2 a b \left (6-5 \cos ^2(c+d x)\right )}{\left (-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)+a-b\right )^2}d\cos (c+d x)}{16 a b (a-b)}}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\int \frac {6-5 \cos ^2(c+d x)}{\left (-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)+a-b\right )^2}d\cos (c+d x)}{8 (a-b)}+\frac {\cos (c+d x) \left (2-\cos ^2(c+d x)\right )}{8 (a-b) \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )^2}}{d}\) |
| \(\Big \downarrow \) 1492 |
| \(\displaystyle -\frac {\frac {\frac {\cos (c+d x) \left (-\left ((5 a+b) \cos ^2(c+d x)\right )+11 a+b\right )}{4 a (a-b) \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )}-\frac {\int -\frac {2 b \left (-\left ((5 a+b) \cos ^2(c+d x)\right )+13 a-b\right )}{-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)+a-b}d\cos (c+d x)}{8 a b (a-b)}}{8 (a-b)}+\frac {\cos (c+d x) \left (2-\cos ^2(c+d x)\right )}{8 (a-b) \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )^2}}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\frac {\int \frac {-\left ((5 a+b) \cos ^2(c+d x)\right )+13 a-b}{-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)+a-b}d\cos (c+d x)}{4 a (a-b)}+\frac {\cos (c+d x) \left (-\left ((5 a+b) \cos ^2(c+d x)\right )+11 a+b\right )}{4 a (a-b) \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )}}{8 (a-b)}+\frac {\cos (c+d x) \left (2-\cos ^2(c+d x)\right )}{8 (a-b) \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )^2}}{d}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle -\frac {\frac {\frac {-\frac {1}{2} \left (\frac {2 \sqrt {b} (4 a-b)}{\sqrt {a}}+5 a+b\right ) \int \frac {1}{-b \cos ^2(c+d x)-\left (\sqrt {a}-\sqrt {b}\right ) \sqrt {b}}d\cos (c+d x)-\frac {1}{2} \left (-\frac {2 \sqrt {b} (4 a-b)}{\sqrt {a}}+5 a+b\right ) \int \frac {1}{\left (\sqrt {a}+\sqrt {b}\right ) \sqrt {b}-b \cos ^2(c+d x)}d\cos (c+d x)}{4 a (a-b)}+\frac {\cos (c+d x) \left (-\left ((5 a+b) \cos ^2(c+d x)\right )+11 a+b\right )}{4 a (a-b) \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )}}{8 (a-b)}+\frac {\cos (c+d x) \left (2-\cos ^2(c+d x)\right )}{8 (a-b) \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )^2}}{d}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle -\frac {\frac {\frac {\frac {\left (\frac {2 \sqrt {b} (4 a-b)}{\sqrt {a}}+5 a+b\right ) \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 b^{3/4} \sqrt {\sqrt {a}-\sqrt {b}}}-\frac {1}{2} \left (-\frac {2 \sqrt {b} (4 a-b)}{\sqrt {a}}+5 a+b\right ) \int \frac {1}{\left (\sqrt {a}+\sqrt {b}\right ) \sqrt {b}-b \cos ^2(c+d x)}d\cos (c+d x)}{4 a (a-b)}+\frac {\cos (c+d x) \left (-\left ((5 a+b) \cos ^2(c+d x)\right )+11 a+b\right )}{4 a (a-b) \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )}}{8 (a-b)}+\frac {\cos (c+d x) \left (2-\cos ^2(c+d x)\right )}{8 (a-b) \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )^2}}{d}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {\frac {\frac {\frac {\left (\frac {2 \sqrt {b} (4 a-b)}{\sqrt {a}}+5 a+b\right ) \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 b^{3/4} \sqrt {\sqrt {a}-\sqrt {b}}}-\frac {\left (-\frac {2 \sqrt {b} (4 a-b)}{\sqrt {a}}+5 a+b\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 b^{3/4} \sqrt {\sqrt {a}+\sqrt {b}}}}{4 a (a-b)}+\frac {\cos (c+d x) \left (-\left ((5 a+b) \cos ^2(c+d x)\right )+11 a+b\right )}{4 a (a-b) \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )}}{8 (a-b)}+\frac {\cos (c+d x) \left (2-\cos ^2(c+d x)\right )}{8 (a-b) \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )^2}}{d}\) |
-(((Cos[c + d*x]*(2 - Cos[c + d*x]^2))/(8*(a - b)*(a - b + 2*b*Cos[c + d*x ]^2 - b*Cos[c + d*x]^4)^2) + ((((5*a + (2*(4*a - b)*Sqrt[b])/Sqrt[a] + b)* ArcTan[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(2*Sqrt[Sqrt[a] - Sqrt[b]]*b^(3/4)) - ((5*a - (2*(4*a - b)*Sqrt[b])/Sqrt[a] + b)*ArcTanh[(b^ (1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(2*Sqrt[Sqrt[a] + Sqrt[b]]*b ^(3/4)))/(4*a*(a - b)) + (Cos[c + d*x]*(11*a + b - (5*a + b)*Cos[c + d*x]^ 2))/(4*a*(a - b)*(a - b + 2*b*Cos[c + d*x]^2 - b*Cos[c + d*x]^4)))/(8*(a - b)))/d)
3.3.27.3.1 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[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q)) Int[1/( b/2 - q/2 + c*x^2), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q)) Int[1/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^2 - 4*a*c]
Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symb ol] :> Simp[x*(a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2)*((a + b*x^2 + c*x^4)^(p + 1)/(2*a*(p + 1)*(b^2 - 4*a*c))), x] + Simp[1/(2*a*(p + 1)*(b^2 - 4*a*c)) Int[Simp[(2*p + 3)*d*b^2 - a*b*e - 2*a*c*d*(4*p + 5) + (4*p + 7)*(d*b - 2*a*e)*c*x^2, x]*(a + b*x^2 + c*x^4)^(p + 1), 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] && LtQ[p, -1] && IntegerQ[2*p]
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 = 5.86 (sec) , antiderivative size = 403, normalized size of antiderivative = 1.40
| method | result | size |
| derivativedivides | \(\frac {b^{3} \left (\frac {\frac {\frac {\left (-5 a \sqrt {a b}-\sqrt {a b}\, b +8 a b -2 b^{2}\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{4 b^{2} \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (7 a -5 \sqrt {a b}-2 b \right ) \cos \left (d x +c \right )}{4 b^{2} \left (a -b \right )}}{\left (\cos ^{2}\left (d x +c \right )-1-\frac {\sqrt {a b}}{b}\right )^{2}}+\frac {\left (5 a \sqrt {a b}+\sqrt {a b}\, b -8 a b +2 b^{2}\right ) \operatorname {arctanh}\left (\frac {\cos \left (d x +c \right ) b}{\sqrt {\left (\sqrt {a b}+b \right ) b}}\right )}{4 b \left (a^{2}-2 a b +b^{2}\right ) \sqrt {\left (\sqrt {a b}+b \right ) b}}}{16 \sqrt {a b}\, a \,b^{2}}-\frac {\frac {\frac {\left (5 a \sqrt {a b}+\sqrt {a b}\, b +8 a b -2 b^{2}\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{4 b^{2} \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (7 a +5 \sqrt {a b}-2 b \right ) \cos \left (d x +c \right )}{4 b^{2} \left (a -b \right )}}{\left (\cos ^{2}\left (d x +c \right )+\frac {\sqrt {a b}}{b}-1\right )^{2}}+\frac {\left (5 a \sqrt {a b}+\sqrt {a b}\, b +8 a b -2 b^{2}\right ) \arctan \left (\frac {\cos \left (d x +c \right ) b}{\sqrt {\left (\sqrt {a b}-b \right ) b}}\right )}{4 b \left (a^{2}-2 a b +b^{2}\right ) \sqrt {\left (\sqrt {a b}-b \right ) b}}}{16 \sqrt {a b}\, a \,b^{2}}\right )}{d}\) | \(403\) |
| default | \(\frac {b^{3} \left (\frac {\frac {\frac {\left (-5 a \sqrt {a b}-\sqrt {a b}\, b +8 a b -2 b^{2}\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{4 b^{2} \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (7 a -5 \sqrt {a b}-2 b \right ) \cos \left (d x +c \right )}{4 b^{2} \left (a -b \right )}}{\left (\cos ^{2}\left (d x +c \right )-1-\frac {\sqrt {a b}}{b}\right )^{2}}+\frac {\left (5 a \sqrt {a b}+\sqrt {a b}\, b -8 a b +2 b^{2}\right ) \operatorname {arctanh}\left (\frac {\cos \left (d x +c \right ) b}{\sqrt {\left (\sqrt {a b}+b \right ) b}}\right )}{4 b \left (a^{2}-2 a b +b^{2}\right ) \sqrt {\left (\sqrt {a b}+b \right ) b}}}{16 \sqrt {a b}\, a \,b^{2}}-\frac {\frac {\frac {\left (5 a \sqrt {a b}+\sqrt {a b}\, b +8 a b -2 b^{2}\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{4 b^{2} \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (7 a +5 \sqrt {a b}-2 b \right ) \cos \left (d x +c \right )}{4 b^{2} \left (a -b \right )}}{\left (\cos ^{2}\left (d x +c \right )+\frac {\sqrt {a b}}{b}-1\right )^{2}}+\frac {\left (5 a \sqrt {a b}+\sqrt {a b}\, b +8 a b -2 b^{2}\right ) \arctan \left (\frac {\cos \left (d x +c \right ) b}{\sqrt {\left (\sqrt {a b}-b \right ) b}}\right )}{4 b \left (a^{2}-2 a b +b^{2}\right ) \sqrt {\left (\sqrt {a b}-b \right ) b}}}{16 \sqrt {a b}\, a \,b^{2}}\right )}{d}\) | \(403\) |
| risch | \(-\frac {5 a b \,{\mathrm e}^{15 i \left (d x +c \right )}+b^{2} {\mathrm e}^{15 i \left (d x +c \right )}-49 a b \,{\mathrm e}^{13 i \left (d x +c \right )}-5 b^{2} {\mathrm e}^{13 i \left (d x +c \right )}-144 a^{2} {\mathrm e}^{11 i \left (d x +c \right )}+165 a b \,{\mathrm e}^{11 i \left (d x +c \right )}+9 b^{2} {\mathrm e}^{11 i \left (d x +c \right )}+784 a^{2} {\mathrm e}^{9 i \left (d x +c \right )}-377 a b \,{\mathrm e}^{9 i \left (d x +c \right )}-5 b^{2} {\mathrm e}^{9 i \left (d x +c \right )}+784 a^{2} {\mathrm e}^{7 i \left (d x +c \right )}-377 a b \,{\mathrm e}^{7 i \left (d x +c \right )}-5 b^{2} {\mathrm e}^{7 i \left (d x +c \right )}-144 a^{2} {\mathrm e}^{5 i \left (d x +c \right )}+165 a b \,{\mathrm e}^{5 i \left (d x +c \right )}+9 b^{2} {\mathrm e}^{5 i \left (d x +c \right )}-49 a b \,{\mathrm e}^{3 i \left (d x +c \right )}-5 b^{2} {\mathrm e}^{3 i \left (d x +c \right )}+5 a b \,{\mathrm e}^{i \left (d x +c \right )}+b^{2} {\mathrm e}^{i \left (d x +c \right )}}{16 \left (a -b \right )^{2} 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 )^{2} a}+\frac {i \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (65536 a^{11} b^{3} d^{4}-327680 a^{10} b^{4} d^{4}+655360 a^{9} b^{5} d^{4}-655360 a^{8} b^{6} d^{4}+327680 a^{7} b^{7} d^{4}-65536 a^{6} b^{8} d^{4}\right ) \textit {\_Z}^{4}+\left (-53760 a^{6} b^{2} d^{2}-35840 a^{5} b^{3} d^{2}+17920 a^{4} b^{4} d^{2}-2048 a^{3} b^{5} d^{2}\right ) \textit {\_Z}^{2}-625 a^{2}+200 a b -16 b^{2}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\left (\left (-\frac {40960 i a^{10} b^{2} d^{3}}{625 a^{3}+3750 a^{2} b -1491 a \,b^{2}+140 b^{3}}+\frac {131072 i a^{9} b^{3} d^{3}}{625 a^{3}+3750 a^{2} b -1491 a \,b^{2}+140 b^{3}}-\frac {24576 i a^{8} b^{4} d^{3}}{625 a^{3}+3750 a^{2} b -1491 a \,b^{2}+140 b^{3}}-\frac {409600 i a^{7} b^{5} d^{3}}{625 a^{3}+3750 a^{2} b -1491 a \,b^{2}+140 b^{3}}+\frac {696320 i a^{6} b^{6} d^{3}}{625 a^{3}+3750 a^{2} b -1491 a \,b^{2}+140 b^{3}}-\frac {491520 i a^{5} b^{7} d^{3}}{625 a^{3}+3750 a^{2} b -1491 a \,b^{2}+140 b^{3}}+\frac {155648 i a^{4} b^{8} d^{3}}{625 a^{3}+3750 a^{2} b -1491 a \,b^{2}+140 b^{3}}-\frac {16384 i a^{3} b^{9} d^{3}}{625 a^{3}+3750 a^{2} b -1491 a \,b^{2}+140 b^{3}}\right ) \textit {\_R}^{3}+\left (\frac {27200 i a^{5} b d}{625 a^{3}+3750 a^{2} b -1491 a \,b^{2}+140 b^{3}}+\frac {104704 i a^{4} b^{2} d}{625 a^{3}+3750 a^{2} b -1491 a \,b^{2}+140 b^{3}}-\frac {11648 i a^{3} b^{3} d}{625 a^{3}+3750 a^{2} b -1491 a \,b^{2}+140 b^{3}}-\frac {12800 i a^{2} b^{4} d}{625 a^{3}+3750 a^{2} b -1491 a \,b^{2}+140 b^{3}}+\frac {3392 i a \,b^{5} d}{625 a^{3}+3750 a^{2} b -1491 a \,b^{2}+140 b^{3}}-\frac {256 i b^{6} d}{625 a^{3}+3750 a^{2} b -1491 a \,b^{2}+140 b^{3}}\right ) \textit {\_R} \right ) {\mathrm e}^{i \left (d x +c \right )}+1\right )\right )}{8}\) | \(1025\) |
1/d*b^3*(1/16/(a*b)^(1/2)/a/b^2*((1/4*(-5*a*(a*b)^(1/2)-(a*b)^(1/2)*b+8*a* b-2*b^2)/b^2/(a^2-2*a*b+b^2)*cos(d*x+c)^3+1/4*(7*a-5*(a*b)^(1/2)-2*b)/b^2/ (a-b)*cos(d*x+c))/(cos(d*x+c)^2-1-(a*b)^(1/2)/b)^2+1/4*(5*a*(a*b)^(1/2)+(a *b)^(1/2)*b-8*a*b+2*b^2)/b/(a^2-2*a*b+b^2)/(((a*b)^(1/2)+b)*b)^(1/2)*arcta nh(cos(d*x+c)*b/(((a*b)^(1/2)+b)*b)^(1/2)))-1/16/(a*b)^(1/2)/a/b^2*((1/4*( 5*a*(a*b)^(1/2)+(a*b)^(1/2)*b+8*a*b-2*b^2)/b^2/(a^2-2*a*b+b^2)*cos(d*x+c)^ 3+1/4*(7*a+5*(a*b)^(1/2)-2*b)/b^2/(a-b)*cos(d*x+c))/(cos(d*x+c)^2+(a*b)^(1 /2)/b-1)^2+1/4*(5*a*(a*b)^(1/2)+(a*b)^(1/2)*b+8*a*b-2*b^2)/b/(a^2-2*a*b+b^ 2)/(((a*b)^(1/2)-b)*b)^(1/2)*arctan(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. 4050 vs. \(2 (233) = 466\).
Time = 0.75 (sec) , antiderivative size = 4050, normalized size of antiderivative = 14.06 \[ \int \frac {\sin ^3(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=\text {Too large to display} \]
-1/128*(4*(5*a*b + b^2)*cos(d*x + c)^7 - 12*(7*a*b + b^2)*cos(d*x + c)^5 - 12*(3*a^2 - 10*a*b - b^2)*cos(d*x + c)^3 + ((a^3*b^2 - 2*a^2*b^3 + a*b^4) *d*cos(d*x + c)^8 - 4*(a^3*b^2 - 2*a^2*b^3 + a*b^4)*d*cos(d*x + c)^6 - 2*( a^4*b - 5*a^3*b^2 + 7*a^2*b^3 - 3*a*b^4)*d*cos(d*x + c)^4 + 4*(a^4*b - 3*a ^3*b^2 + 3*a^2*b^3 - a*b^4)*d*cos(d*x + c)^2 + (a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d)*sqrt(-((a^8*b - 5*a^7*b^2 + 10*a^6*b^3 - 10*a^5*b^ 4 + 5*a^4*b^5 - a^3*b^6)*d^2*sqrt((625*a^4 + 7700*a^3*b + 21966*a^2*b^2 - 10780*a*b^3 + 1225*b^4)/((a^13*b^3 - 10*a^12*b^4 + 45*a^11*b^5 - 120*a^10* b^6 + 210*a^9*b^7 - 252*a^8*b^8 + 210*a^7*b^9 - 120*a^6*b^10 + 45*a^5*b^11 - 10*a^4*b^12 + a^3*b^13)*d^4)) + 105*a^3 + 70*a^2*b - 35*a*b^2 + 4*b^3)/ ((a^8*b - 5*a^7*b^2 + 10*a^6*b^3 - 10*a^5*b^4 + 5*a^4*b^5 - a^3*b^6)*d^2)) *log((625*a^3 + 3750*a^2*b - 1491*a*b^2 + 140*b^3)*cos(d*x + c) + ((5*a^10 *b^2 - 16*a^9*b^3 + 3*a^8*b^4 + 50*a^7*b^5 - 85*a^6*b^6 + 60*a^5*b^7 - 19* a^4*b^8 + 2*a^3*b^9)*d^3*sqrt((625*a^4 + 7700*a^3*b + 21966*a^2*b^2 - 1078 0*a*b^3 + 1225*b^4)/((a^13*b^3 - 10*a^12*b^4 + 45*a^11*b^5 - 120*a^10*b^6 + 210*a^9*b^7 - 252*a^8*b^8 + 210*a^7*b^9 - 120*a^6*b^10 + 45*a^5*b^11 - 1 0*a^4*b^12 + a^3*b^13)*d^4)) - (325*a^5*b + 1977*a^4*b^2 - 609*a^3*b^3 + 3 5*a^2*b^4)*d)*sqrt(-((a^8*b - 5*a^7*b^2 + 10*a^6*b^3 - 10*a^5*b^4 + 5*a^4* b^5 - a^3*b^6)*d^2*sqrt((625*a^4 + 7700*a^3*b + 21966*a^2*b^2 - 10780*a*b^ 3 + 1225*b^4)/((a^13*b^3 - 10*a^12*b^4 + 45*a^11*b^5 - 120*a^10*b^6 + 2...
Timed out. \[ \int \frac {\sin ^3(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=\text {Timed out} \]
\[ \int \frac {\sin ^3(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=\int { -\frac {\sin \left (d x + c\right )^{3}}{{\left (b \sin \left (d x + c\right )^{4} - a\right )}^{3}} \,d x } \]
1/16*(8*(5*a*b^3 + b^4)*cos(2*d*x + 2*c)*cos(d*x + c) - 8*(49*a*b^3 + 5*b^ 4)*sin(3*d*x + 3*c)*sin(2*d*x + 2*c) + 8*(5*a*b^3 + b^4)*sin(2*d*x + 2*c)* sin(d*x + c) - ((5*a*b^3 + b^4)*cos(15*d*x + 15*c) - (49*a*b^3 + 5*b^4)*co s(13*d*x + 13*c) - 3*(48*a^2*b^2 - 55*a*b^3 - 3*b^4)*cos(11*d*x + 11*c) + (784*a^2*b^2 - 377*a*b^3 - 5*b^4)*cos(9*d*x + 9*c) + (784*a^2*b^2 - 377*a* b^3 - 5*b^4)*cos(7*d*x + 7*c) - 3*(48*a^2*b^2 - 55*a*b^3 - 3*b^4)*cos(5*d* x + 5*c) - (49*a*b^3 + 5*b^4)*cos(3*d*x + 3*c) + (5*a*b^3 + b^4)*cos(d*x + c))*cos(16*d*x + 16*c) - (5*a*b^3 + b^4 - 8*(5*a*b^3 + b^4)*cos(14*d*x + 14*c) - 4*(40*a^2*b^2 - 27*a*b^3 - 7*b^4)*cos(12*d*x + 12*c) + 8*(80*a^2*b ^2 - 19*a*b^3 - 7*b^4)*cos(10*d*x + 10*c) + 2*(640*a^3*b - 352*a^2*b^2 + 7 9*a*b^3 + 35*b^4)*cos(8*d*x + 8*c) + 8*(80*a^2*b^2 - 19*a*b^3 - 7*b^4)*cos (6*d*x + 6*c) - 4*(40*a^2*b^2 - 27*a*b^3 - 7*b^4)*cos(4*d*x + 4*c) - 8*(5* a*b^3 + b^4)*cos(2*d*x + 2*c))*cos(15*d*x + 15*c) - 8*((49*a*b^3 + 5*b^4)* cos(13*d*x + 13*c) + 3*(48*a^2*b^2 - 55*a*b^3 - 3*b^4)*cos(11*d*x + 11*c) - (784*a^2*b^2 - 377*a*b^3 - 5*b^4)*cos(9*d*x + 9*c) - (784*a^2*b^2 - 377* a*b^3 - 5*b^4)*cos(7*d*x + 7*c) + 3*(48*a^2*b^2 - 55*a*b^3 - 3*b^4)*cos(5* d*x + 5*c) + (49*a*b^3 + 5*b^4)*cos(3*d*x + 3*c) - (5*a*b^3 + b^4)*cos(d*x + c))*cos(14*d*x + 14*c) + (49*a*b^3 + 5*b^4 - 4*(392*a^2*b^2 - 303*a*b^3 - 35*b^4)*cos(12*d*x + 12*c) + 8*(784*a^2*b^2 - 263*a*b^3 - 35*b^4)*cos(1 0*d*x + 10*c) + 2*(6272*a^3*b - 4064*a^2*b^2 + 1235*a*b^3 + 175*b^4)*co...
Leaf count of result is larger than twice the leaf count of optimal. 1076 vs. \(2 (233) = 466\).
Time = 1.65 (sec) , antiderivative size = 1076, normalized size of antiderivative = 3.74 \[ \int \frac {\sin ^3(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=-\frac {\frac {5 \, a b \cos \left (d x + c\right )^{7}}{d} + \frac {b^{2} \cos \left (d x + c\right )^{7}}{d} - \frac {21 \, a b \cos \left (d x + c\right )^{5}}{d} - \frac {3 \, b^{2} \cos \left (d x + c\right )^{5}}{d} - \frac {9 \, a^{2} \cos \left (d x + c\right )^{3}}{d} + \frac {30 \, a b \cos \left (d x + c\right )^{3}}{d} + \frac {3 \, b^{2} \cos \left (d x + c\right )^{3}}{d} + \frac {19 \, a^{2} \cos \left (d x + c\right )}{d} - \frac {18 \, a b \cos \left (d x + c\right )}{d} - \frac {b^{2} \cos \left (d x + c\right )}{d}}{32 \, {\left (b \cos \left (d x + c\right )^{4} - 2 \, b \cos \left (d x + c\right )^{2} - a + b\right )}^{2} {\left (a^{3} - 2 \, a^{2} b + a b^{2}\right )}} - \frac {{\left (2 \, {\left (4 \, a^{6} b - 17 \, a^{5} b^{2} + 28 \, a^{4} b^{3} - 22 \, a^{3} b^{4} + 8 \, a^{2} b^{5} - a b^{6}\right )} \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} d^{4} + {\left (13 \, a^{4} b - 27 \, a^{3} b^{2} + 15 \, a^{2} b^{3} - a b^{4}\right )} \sqrt {-b^{2} + \sqrt {a b} b} d^{2} {\left | a^{3} d^{2} - 2 \, a^{2} b d^{2} + a b^{2} d^{2} \right |} + {\left (a^{3} d^{2} - 2 \, a^{2} b d^{2} + a b^{2} d^{2}\right )}^{2} \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} {\left (5 \, a + b\right )}\right )} \arctan \left (\frac {\cos \left (d x + c\right )}{d \sqrt {-\frac {a^{3} b d^{2} - 2 \, a^{2} b^{2} d^{2} + a b^{3} d^{2} - \sqrt {{\left (a^{3} b d^{2} - 2 \, a^{2} b^{2} d^{2} + a b^{3} d^{2}\right )}^{2} + {\left (a^{3} b d^{4} - 2 \, a^{2} b^{2} d^{4} + a b^{3} d^{4}\right )} {\left (a^{4} - 3 \, a^{3} b + 3 \, a^{2} b^{2} - a b^{3}\right )}}}{a^{3} b d^{4} - 2 \, a^{2} b^{2} d^{4} + a b^{3} d^{4}}}}\right )}{64 \, {\left (a^{7} b - 5 \, a^{6} b^{2} + 10 \, a^{5} b^{3} - 10 \, a^{4} b^{4} + 5 \, a^{3} b^{5} - a^{2} b^{6}\right )} d^{3} {\left | a^{3} d^{2} - 2 \, a^{2} b d^{2} + a b^{2} d^{2} \right |} {\left | b \right |}} + \frac {{\left (2 \, {\left (4 \, a^{6} b - 17 \, a^{5} b^{2} + 28 \, a^{4} b^{3} - 22 \, a^{3} b^{4} + 8 \, a^{2} b^{5} - a b^{6}\right )} \sqrt {-b^{2} - \sqrt {a b} b} d^{4} - {\left (13 \, a^{3} - 27 \, a^{2} b + 15 \, a b^{2} - b^{3}\right )} \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} d^{2} {\left | a^{3} d^{2} - 2 \, a^{2} b d^{2} + a b^{2} d^{2} \right |} + {\left (a^{3} d^{2} - 2 \, a^{2} b d^{2} + a b^{2} d^{2}\right )}^{2} \sqrt {-b^{2} - \sqrt {a b} b} {\left (5 \, a + b\right )}\right )} \arctan \left (\frac {\cos \left (d x + c\right )}{d \sqrt {-\frac {a^{3} b d^{2} - 2 \, a^{2} b^{2} d^{2} + a b^{3} d^{2} + \sqrt {{\left (a^{3} b d^{2} - 2 \, a^{2} b^{2} d^{2} + a b^{3} d^{2}\right )}^{2} + {\left (a^{3} b d^{4} - 2 \, a^{2} b^{2} d^{4} + a b^{3} d^{4}\right )} {\left (a^{4} - 3 \, a^{3} b + 3 \, a^{2} b^{2} - a b^{3}\right )}}}{a^{3} b d^{4} - 2 \, a^{2} b^{2} d^{4} + a b^{3} d^{4}}}}\right )}{64 \, {\left (a^{6} - 5 \, a^{5} b + 10 \, a^{4} b^{2} - 10 \, a^{3} b^{3} + 5 \, a^{2} b^{4} - a b^{5}\right )} \sqrt {a b} d^{3} {\left | a^{3} d^{2} - 2 \, a^{2} b d^{2} + a b^{2} d^{2} \right |} {\left | b \right |}} \]
-1/32*(5*a*b*cos(d*x + c)^7/d + b^2*cos(d*x + c)^7/d - 21*a*b*cos(d*x + c) ^5/d - 3*b^2*cos(d*x + c)^5/d - 9*a^2*cos(d*x + c)^3/d + 30*a*b*cos(d*x + c)^3/d + 3*b^2*cos(d*x + c)^3/d + 19*a^2*cos(d*x + c)/d - 18*a*b*cos(d*x + c)/d - b^2*cos(d*x + c)/d)/((b*cos(d*x + c)^4 - 2*b*cos(d*x + c)^2 - a + b)^2*(a^3 - 2*a^2*b + a*b^2)) - 1/64*(2*(4*a^6*b - 17*a^5*b^2 + 28*a^4*b^3 - 22*a^3*b^4 + 8*a^2*b^5 - a*b^6)*sqrt(a*b)*sqrt(-b^2 + sqrt(a*b)*b)*d^4 + (13*a^4*b - 27*a^3*b^2 + 15*a^2*b^3 - a*b^4)*sqrt(-b^2 + sqrt(a*b)*b)*d^ 2*abs(a^3*d^2 - 2*a^2*b*d^2 + a*b^2*d^2) + (a^3*d^2 - 2*a^2*b*d^2 + a*b^2* d^2)^2*sqrt(a*b)*sqrt(-b^2 + sqrt(a*b)*b)*(5*a + b))*arctan(cos(d*x + c)/( d*sqrt(-(a^3*b*d^2 - 2*a^2*b^2*d^2 + a*b^3*d^2 - sqrt((a^3*b*d^2 - 2*a^2*b ^2*d^2 + a*b^3*d^2)^2 + (a^3*b*d^4 - 2*a^2*b^2*d^4 + a*b^3*d^4)*(a^4 - 3*a ^3*b + 3*a^2*b^2 - a*b^3)))/(a^3*b*d^4 - 2*a^2*b^2*d^4 + a*b^3*d^4))))/((a ^7*b - 5*a^6*b^2 + 10*a^5*b^3 - 10*a^4*b^4 + 5*a^3*b^5 - a^2*b^6)*d^3*abs( a^3*d^2 - 2*a^2*b*d^2 + a*b^2*d^2)*abs(b)) + 1/64*(2*(4*a^6*b - 17*a^5*b^2 + 28*a^4*b^3 - 22*a^3*b^4 + 8*a^2*b^5 - a*b^6)*sqrt(-b^2 - sqrt(a*b)*b)*d ^4 - (13*a^3 - 27*a^2*b + 15*a*b^2 - b^3)*sqrt(a*b)*sqrt(-b^2 - sqrt(a*b)* b)*d^2*abs(a^3*d^2 - 2*a^2*b*d^2 + a*b^2*d^2) + (a^3*d^2 - 2*a^2*b*d^2 + a *b^2*d^2)^2*sqrt(-b^2 - sqrt(a*b)*b)*(5*a + b))*arctan(cos(d*x + c)/(d*sqr t(-(a^3*b*d^2 - 2*a^2*b^2*d^2 + a*b^3*d^2 + sqrt((a^3*b*d^2 - 2*a^2*b^2*d^ 2 + a*b^3*d^2)^2 + (a^3*b*d^4 - 2*a^2*b^2*d^4 + a*b^3*d^4)*(a^4 - 3*a^3...
Time = 18.40 (sec) , antiderivative size = 5566, normalized size of antiderivative = 19.33 \[ \int \frac {\sin ^3(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=\text {Too large to display} \]
- (atan(((((16384*a^3*b^6 - 245760*a^4*b^5 + 442368*a^5*b^4 - 212992*a^6*b ^3)/(32768*(a^7 - 4*a^6*b + a^3*b^4 - 4*a^4*b^3 + 6*a^5*b^2)) - (cos(c + d *x)*((35*b^2*(a^9*b^3)^(1/2) - 25*a^2*(a^9*b^3)^(1/2) + 4*a^3*b^5 - 35*a^4 *b^4 + 70*a^5*b^3 + 105*a^6*b^2 - 154*a*b*(a^9*b^3)^(1/2))/(16384*(a^6*b^8 - 5*a^7*b^7 + 10*a^8*b^6 - 10*a^9*b^5 + 5*a^10*b^4 - a^11*b^3)))^(1/2)*(1 6384*a^3*b^8 - 65536*a^4*b^7 + 98304*a^5*b^6 - 65536*a^6*b^5 + 16384*a^7*b ^4))/(256*(a^6 - 4*a^5*b + a^2*b^4 - 4*a^3*b^3 + 6*a^4*b^2)))*((35*b^2*(a^ 9*b^3)^(1/2) - 25*a^2*(a^9*b^3)^(1/2) + 4*a^3*b^5 - 35*a^4*b^4 + 70*a^5*b^ 3 + 105*a^6*b^2 - 154*a*b*(a^9*b^3)^(1/2))/(16384*(a^6*b^8 - 5*a^7*b^7 + 1 0*a^8*b^6 - 10*a^9*b^5 + 5*a^10*b^4 - a^11*b^3)))^(1/2) + (cos(c + d*x)*(4 *b^5 - 31*a*b^4 + 74*a^2*b^3 + 25*a^3*b^2))/(256*(a^6 - 4*a^5*b + a^2*b^4 - 4*a^3*b^3 + 6*a^4*b^2)))*((35*b^2*(a^9*b^3)^(1/2) - 25*a^2*(a^9*b^3)^(1/ 2) + 4*a^3*b^5 - 35*a^4*b^4 + 70*a^5*b^3 + 105*a^6*b^2 - 154*a*b*(a^9*b^3) ^(1/2))/(16384*(a^6*b^8 - 5*a^7*b^7 + 10*a^8*b^6 - 10*a^9*b^5 + 5*a^10*b^4 - a^11*b^3)))^(1/2)*1i - (((16384*a^3*b^6 - 245760*a^4*b^5 + 442368*a^5*b ^4 - 212992*a^6*b^3)/(32768*(a^7 - 4*a^6*b + a^3*b^4 - 4*a^4*b^3 + 6*a^5*b ^2)) + (cos(c + d*x)*((35*b^2*(a^9*b^3)^(1/2) - 25*a^2*(a^9*b^3)^(1/2) + 4 *a^3*b^5 - 35*a^4*b^4 + 70*a^5*b^3 + 105*a^6*b^2 - 154*a*b*(a^9*b^3)^(1/2) )/(16384*(a^6*b^8 - 5*a^7*b^7 + 10*a^8*b^6 - 10*a^9*b^5 + 5*a^10*b^4 - a^1 1*b^3)))^(1/2)*(16384*a^3*b^8 - 65536*a^4*b^7 + 98304*a^5*b^6 - 65536*a...