Integrand size = 24, antiderivative size = 315 \[ \int \frac {\sin ^9(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=-\frac {\left (5 a-14 \sqrt {a} \sqrt {b}+12 b\right ) \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{64 \sqrt {a} \left (\sqrt {a}-\sqrt {b}\right )^{5/2} b^{9/4} d}-\frac {\left (5 a+14 \sqrt {a} \sqrt {b}+12 b\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{64 \sqrt {a} \left (\sqrt {a}+\sqrt {b}\right )^{5/2} b^{9/4} d}-\frac {a \cos (c+d x) \left (a+b-b \cos ^2(c+d x)\right )}{8 (a-b) b^2 d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )^2}+\frac {\cos (c+d x) \left (9 a^2-11 a b-10 b^2-2 (2 a-5 b) b \cos ^2(c+d x)\right )}{32 (a-b)^2 b^2 d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )} \] Output:
-1/64*(5*a-14*a^(1/2)*b^(1/2)+12*b)*arctan(b^(1/4)*cos(d*x+c)/(a^(1/2)-b^( 1/2))^(1/2))/a^(1/2)/(a^(1/2)-b^(1/2))^(5/2)/b^(9/4)/d-1/64*(5*a+14*a^(1/2 )*b^(1/2)+12*b)*arctanh(b^(1/4)*cos(d*x+c)/(a^(1/2)+b^(1/2))^(1/2))/a^(1/2 )/(a^(1/2)+b^(1/2))^(5/2)/b^(9/4)/d-1/8*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)^2+1/32*cos(d*x+c)*(9*a^2 -11*a*b-10*b^2-2*(2*a-5*b)*b*cos(d*x+c)^2)/(a-b)^2/b^2/d/(a-b+2*b*cos(d*x+ c)^2-b*cos(d*x+c)^4)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 8.06 (sec) , antiderivative size = 785, normalized size of antiderivative = 2.49 \[ \int \frac {\sin ^9(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx =\text {Too large to display} \] Input:
Integrate[Sin[c + d*x]^9/(a - b*Sin[c + d*x]^4)^3,x]
Output:
((-32*Cos[c + d*x]*(-9*a^2 + 13*a*b + 5*b^2 + (2*a - 5*b)*b*Cos[2*(c + d*x )]))/(8*a - 3*b + 4*b*Cos[2*(c + d*x)] - b*Cos[4*(c + d*x)]) - (512*a*(a - b)*Cos[c + d*x]*(2*a + b - b*Cos[2*(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 & , (-4*a*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] + 10*b^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] + (2*I)*a*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] - (5*I)*b^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] - 20*a^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 + 56*a*b*ArcTan[Si n[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 - 78*b^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 + (10*I)*a^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 - (2 8*I)*a*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 + (39*I)*b^2*Log[1 - 2*Cos [c + d*x]*#1 + #1^2]*#1^2 + 20*a^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1) ]*#1^4 - 56*a*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 + 78*b^2*Arc Tan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 - (10*I)*a^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4 + (28*I)*a*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4 - (39*I)*b^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4 + 4*a*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^6 - 10*b^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^6 - (2*I)*a*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^6 + (5*I)*b^2 *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) & ])/(128*(a - b)^2*b^2*d)
Time = 0.71 (sec) , antiderivative size = 371, normalized size of antiderivative = 1.18, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {3042, 3694, 1517, 27, 2206, 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.
| \(\displaystyle \int \frac {\sin ^9(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (c+d x)^9}{\left (a-b \sin (c+d x)^4\right )^3}dx\) |
| \(\Big \downarrow \) 3694 |
| \(\displaystyle -\frac {\int \frac {\left (1-\cos ^2(c+d x)\right )^4}{\left (-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)+a-b\right )^3}d\cos (c+d x)}{d}\) |
| \(\Big \downarrow \) 1517 |
| \(\displaystyle -\frac {\frac {a \cos (c+d x) \left (a-b \cos ^2(c+d x)+b\right )}{8 b^2 (a-b) \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )^2}-\frac {\int \frac {2 \left (8 a (a-b) \cos ^4(c+d x)-a (11 a-16 b) \cos ^2(c+d x)+\frac {a \left (a^2+b a-8 b^2\right )}{b}\right )}{\left (-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)+a-b\right )^2}d\cos (c+d x)}{16 a b (a-b)}}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {a \cos (c+d x) \left (a-b \cos ^2(c+d x)+b\right )}{8 b^2 (a-b) \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )^2}-\frac {\int \frac {8 a (a-b) \cos ^4(c+d x)-a (11 a-16 b) \cos ^2(c+d x)+a \left (\frac {a^2}{b}+a-8 b\right )}{\left (-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)+a-b\right )^2}d\cos (c+d x)}{8 a b (a-b)}}{d}\) |
| \(\Big \downarrow \) 2206 |
| \(\displaystyle -\frac {\frac {a \cos (c+d x) \left (a-b \cos ^2(c+d x)+b\right )}{8 b^2 (a-b) \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )^2}-\frac {\frac {a \cos (c+d x) \left (9 a^2-2 b (2 a-5 b) \cos ^2(c+d x)-11 a b-10 b^2\right )}{4 b (a-b) \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )}-\frac {\int \frac {2 a^2 \left (5 a^2-15 b a+22 b^2+2 (2 a-5 b) b \cos ^2(c+d x)\right )}{-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)+a-b}d\cos (c+d x)}{8 a b (a-b)}}{8 a b (a-b)}}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {a \cos (c+d x) \left (a-b \cos ^2(c+d x)+b\right )}{8 b^2 (a-b) \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )^2}-\frac {\frac {a \cos (c+d x) \left (9 a^2-2 b (2 a-5 b) \cos ^2(c+d x)-11 a b-10 b^2\right )}{4 b (a-b) \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )}-\frac {a \int \frac {5 a^2-15 b a+22 b^2+2 (2 a-5 b) b \cos ^2(c+d x)}{-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)+a-b}d\cos (c+d x)}{4 b (a-b)}}{8 a b (a-b)}}{d}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle -\frac {\frac {a \cos (c+d x) \left (a-b \cos ^2(c+d x)+b\right )}{8 b^2 (a-b) \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )^2}-\frac {\frac {a \cos (c+d x) \left (9 a^2-2 b (2 a-5 b) \cos ^2(c+d x)-11 a b-10 b^2\right )}{4 b (a-b) \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )}-\frac {a \left (\frac {\sqrt {b} \left (-2 \sqrt {a} \sqrt {b}+a+b\right ) \left (14 \sqrt {a} \sqrt {b}+5 a+12 b\right ) \int \frac {1}{\left (\sqrt {a}+\sqrt {b}\right ) \sqrt {b}-b \cos ^2(c+d x)}d\cos (c+d x)}{2 \sqrt {a}}-\frac {\sqrt {b} \left (\sqrt {a}+\sqrt {b}\right )^2 \left (-14 \sqrt {a} \sqrt {b}+5 a+12 b\right ) \int \frac {1}{-b \cos ^2(c+d x)-\left (\sqrt {a}-\sqrt {b}\right ) \sqrt {b}}d\cos (c+d x)}{2 \sqrt {a}}\right )}{4 b (a-b)}}{8 a b (a-b)}}{d}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle -\frac {\frac {a \cos (c+d x) \left (a-b \cos ^2(c+d x)+b\right )}{8 b^2 (a-b) \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )^2}-\frac {\frac {a \cos (c+d x) \left (9 a^2-2 b (2 a-5 b) \cos ^2(c+d x)-11 a b-10 b^2\right )}{4 b (a-b) \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )}-\frac {a \left (\frac {\sqrt {b} \left (-2 \sqrt {a} \sqrt {b}+a+b\right ) \left (14 \sqrt {a} \sqrt {b}+5 a+12 b\right ) \int \frac {1}{\left (\sqrt {a}+\sqrt {b}\right ) \sqrt {b}-b \cos ^2(c+d x)}d\cos (c+d x)}{2 \sqrt {a}}+\frac {\left (-14 \sqrt {a} \sqrt {b}+5 a+12 b\right ) \left (\sqrt {a}+\sqrt {b}\right )^2 \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 \sqrt {a} \sqrt [4]{b} \sqrt {\sqrt {a}-\sqrt {b}}}\right )}{4 b (a-b)}}{8 a b (a-b)}}{d}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {\frac {a \cos (c+d x) \left (a-b \cos ^2(c+d x)+b\right )}{8 b^2 (a-b) \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )^2}-\frac {\frac {a \cos (c+d x) \left (9 a^2-2 b (2 a-5 b) \cos ^2(c+d x)-11 a b-10 b^2\right )}{4 b (a-b) \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )}-\frac {a \left (\frac {\left (-14 \sqrt {a} \sqrt {b}+5 a+12 b\right ) \left (\sqrt {a}+\sqrt {b}\right )^2 \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 \sqrt {a} \sqrt [4]{b} \sqrt {\sqrt {a}-\sqrt {b}}}+\frac {\left (-2 \sqrt {a} \sqrt {b}+a+b\right ) \left (14 \sqrt {a} \sqrt {b}+5 a+12 b\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 \sqrt {a} \sqrt [4]{b} \sqrt {\sqrt {a}+\sqrt {b}}}\right )}{4 b (a-b)}}{8 a b (a-b)}}{d}\) |
Input:
Int[Sin[c + d*x]^9/(a - b*Sin[c + d*x]^4)^3,x]
Output:
-(((a*Cos[c + d*x]*(a + b - b*Cos[c + d*x]^2))/(8*(a - b)*b^2*(a - b + 2*b *Cos[c + d*x]^2 - b*Cos[c + d*x]^4)^2) - (-1/4*(a*(((Sqrt[a] + Sqrt[b])^2* (5*a - 14*Sqrt[a]*Sqrt[b] + 12*b)*ArcTan[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[ a] - Sqrt[b]]])/(2*Sqrt[a]*Sqrt[Sqrt[a] - Sqrt[b]]*b^(1/4)) + ((a - 2*Sqrt [a]*Sqrt[b] + b)*(5*a + 14*Sqrt[a]*Sqrt[b] + 12*b)*ArcTanh[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(2*Sqrt[a]*Sqrt[Sqrt[a] + Sqrt[b]]*b^(1/ 4))))/((a - b)*b) + (a*Cos[c + d*x]*(9*a^2 - 11*a*b - 10*b^2 - 2*(2*a - 5* b)*b*Cos[c + d*x]^2))/(4*(a - b)*b*(a - b + 2*b*Cos[c + d*x]^2 - b*Cos[c + d*x]^4)))/(8*a*(a - b)*b))/d)
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)^(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)^(p_), x_Symbol] :> With[{d = Coeff[PolynomialRemainder[Px, a + b*x^2 + c*x^4, x], x, 0], e = Coeff[Poly nomialRemainder[Px, a + b*x^2 + c*x^4, x], x, 2]}, Simp[x*(a + b*x^2 + c*x^ 4)^(p + 1)*((a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*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[Px, a + b*x^2 + c*x^4, x] + b^2*d*(2*p + 3) - 2*a*c*d*(4*p + 5) - a*b*e + c*(4* p + 7)*(b*d - 2*a*e)*x^2, x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Px, x ^2] && Expon[Px, x^2] > 1 && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -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 = 7.67 (sec) , antiderivative size = 342, normalized size of antiderivative = 1.09
| method | result | size |
| derivativedivides | \(\frac {-\frac {-\frac {\left (2 a -5 b \right ) \cos \left (d x +c \right )^{7}}{16 \left (a^{2}-2 a b +b^{2}\right )}+\frac {3 \left (3 a^{2}-a b -10 b^{2}\right ) \cos \left (d x +c \right )^{5}}{32 b \left (a^{2}-2 a b +b^{2}\right )}-\frac {3 \left (3 a^{2}-2 a b -5 b^{2}\right ) \cos \left (d x +c \right )^{3}}{16 b \left (a^{2}-2 a b +b^{2}\right )}-\frac {5 \left (a^{2}-3 a b -2 b^{2}\right ) \cos \left (d x +c \right )}{32 b^{2} \left (a -b \right )}}{\left (a -b +2 b \cos \left (d x +c \right )^{2}-b \cos \left (d x +c \right )^{4}\right )^{2}}-\frac {-\frac {\left (-4 a \sqrt {a b}+10 \sqrt {a b}\, b -5 a^{2}+11 a b -12 b^{2}\right ) \operatorname {arctanh}\left (\frac {b \cos \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+b \right ) b}}\right )}{2 \sqrt {a b}\, \sqrt {\left (\sqrt {a b}+b \right ) b}}+\frac {\left (-4 a \sqrt {a b}+10 \sqrt {a b}\, b +5 a^{2}-11 a b +12 b^{2}\right ) \arctan \left (\frac {b \cos \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-b \right ) b}}\right )}{2 \sqrt {a b}\, \sqrt {\left (\sqrt {a b}-b \right ) b}}}{32 \left (a^{2}-2 a b +b^{2}\right ) b}}{d}\) | \(342\) |
| default | \(\frac {-\frac {-\frac {\left (2 a -5 b \right ) \cos \left (d x +c \right )^{7}}{16 \left (a^{2}-2 a b +b^{2}\right )}+\frac {3 \left (3 a^{2}-a b -10 b^{2}\right ) \cos \left (d x +c \right )^{5}}{32 b \left (a^{2}-2 a b +b^{2}\right )}-\frac {3 \left (3 a^{2}-2 a b -5 b^{2}\right ) \cos \left (d x +c \right )^{3}}{16 b \left (a^{2}-2 a b +b^{2}\right )}-\frac {5 \left (a^{2}-3 a b -2 b^{2}\right ) \cos \left (d x +c \right )}{32 b^{2} \left (a -b \right )}}{\left (a -b +2 b \cos \left (d x +c \right )^{2}-b \cos \left (d x +c \right )^{4}\right )^{2}}-\frac {-\frac {\left (-4 a \sqrt {a b}+10 \sqrt {a b}\, b -5 a^{2}+11 a b -12 b^{2}\right ) \operatorname {arctanh}\left (\frac {b \cos \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+b \right ) b}}\right )}{2 \sqrt {a b}\, \sqrt {\left (\sqrt {a b}+b \right ) b}}+\frac {\left (-4 a \sqrt {a b}+10 \sqrt {a b}\, b +5 a^{2}-11 a b +12 b^{2}\right ) \arctan \left (\frac {b \cos \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-b \right ) b}}\right )}{2 \sqrt {a b}\, \sqrt {\left (\sqrt {a b}-b \right ) b}}}{32 \left (a^{2}-2 a b +b^{2}\right ) b}}{d}\) | \(342\) |
| risch | \(\text {Expression too large to display}\) | \(1583\) |
Input:
int(sin(d*x+c)^9/(a-b*sin(d*x+c)^4)^3,x,method=_RETURNVERBOSE)
Output:
1/d*(-(-1/16*(2*a-5*b)/(a^2-2*a*b+b^2)*cos(d*x+c)^7+3/32*(3*a^2-a*b-10*b^2 )/b/(a^2-2*a*b+b^2)*cos(d*x+c)^5-3/16*(3*a^2-2*a*b-5*b^2)/b/(a^2-2*a*b+b^2 )*cos(d*x+c)^3-5/32*(a^2-3*a*b-2*b^2)/b^2/(a-b)*cos(d*x+c))/(a-b+2*b*cos(d *x+c)^2-b*cos(d*x+c)^4)^2-1/32/(a^2-2*a*b+b^2)/b*(-1/2*(-4*a*(a*b)^(1/2)+1 0*(a*b)^(1/2)*b-5*a^2+11*a*b-12*b^2)/(a*b)^(1/2)/(((a*b)^(1/2)+b)*b)^(1/2) *arctanh(b*cos(d*x+c)/(((a*b)^(1/2)+b)*b)^(1/2))+1/2*(-4*a*(a*b)^(1/2)+10* (a*b)^(1/2)*b+5*a^2-11*a*b+12*b^2)/(a*b)^(1/2)/(((a*b)^(1/2)-b)*b)^(1/2)*a rctan(b*cos(d*x+c)/(((a*b)^(1/2)-b)*b)^(1/2))))
Leaf count of result is larger than twice the leaf count of optimal. 4640 vs. \(2 (264) = 528\).
Time = 0.80 (sec) , antiderivative size = 4640, normalized size of antiderivative = 14.73 \[ \int \frac {\sin ^9(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=\text {Too large to display} \] Input:
integrate(sin(d*x+c)^9/(a-b*sin(d*x+c)^4)^3,x, algorithm="fricas")
Output:
Too large to include
Timed out. \[ \int \frac {\sin ^9(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=\text {Timed out} \] Input:
integrate(sin(d*x+c)**9/(a-b*sin(d*x+c)**4)**3,x)
Output:
Timed out
\[ \int \frac {\sin ^9(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=\int { -\frac {\sin \left (d x + c\right )^{9}}{{\left (b \sin \left (d x + c\right )^{4} - a\right )}^{3}} \,d x } \] Input:
integrate(sin(d*x+c)^9/(a-b*sin(d*x+c)^4)^3,x, algorithm="maxima")
Output:
-1/8*(8*(2*a*b^4 - 5*b^5)*cos(2*d*x + 2*c)*cos(d*x + c) - 8*(18*a^2*b^3 - 20*a*b^4 - 25*b^5)*sin(3*d*x + 3*c)*sin(2*d*x + 2*c) + 8*(2*a*b^4 - 5*b^5) *sin(2*d*x + 2*c)*sin(d*x + c) - ((2*a*b^4 - 5*b^5)*cos(15*d*x + 15*c) - ( 18*a^2*b^3 - 20*a*b^4 - 25*b^5)*cos(13*d*x + 13*c) + 3*(18*a^2*b^3 - 8*a*b ^4 - 15*b^5)*cos(11*d*x + 11*c) + (160*a^3*b^2 - 388*a^2*b^3 + 2*a*b^4 + 2 5*b^5)*cos(9*d*x + 9*c) + (160*a^3*b^2 - 388*a^2*b^3 + 2*a*b^4 + 25*b^5)*c os(7*d*x + 7*c) + 3*(18*a^2*b^3 - 8*a*b^4 - 15*b^5)*cos(5*d*x + 5*c) - (18 *a^2*b^3 - 20*a*b^4 - 25*b^5)*cos(3*d*x + 3*c) + (2*a*b^4 - 5*b^5)*cos(d*x + c))*cos(16*d*x + 16*c) - (2*a*b^4 - 5*b^5 - 8*(2*a*b^4 - 5*b^5)*cos(14* d*x + 14*c) - 4*(16*a^2*b^3 - 54*a*b^4 + 35*b^5)*cos(12*d*x + 12*c) + 8*(3 2*a^2*b^3 - 94*a*b^4 + 35*b^5)*cos(10*d*x + 10*c) + 2*(256*a^3*b^2 - 832*a ^2*b^3 + 550*a*b^4 - 175*b^5)*cos(8*d*x + 8*c) + 8*(32*a^2*b^3 - 94*a*b^4 + 35*b^5)*cos(6*d*x + 6*c) - 4*(16*a^2*b^3 - 54*a*b^4 + 35*b^5)*cos(4*d*x + 4*c) - 8*(2*a*b^4 - 5*b^5)*cos(2*d*x + 2*c))*cos(15*d*x + 15*c) - 8*((18 *a^2*b^3 - 20*a*b^4 - 25*b^5)*cos(13*d*x + 13*c) - 3*(18*a^2*b^3 - 8*a*b^4 - 15*b^5)*cos(11*d*x + 11*c) - (160*a^3*b^2 - 388*a^2*b^3 + 2*a*b^4 + 25* b^5)*cos(9*d*x + 9*c) - (160*a^3*b^2 - 388*a^2*b^3 + 2*a*b^4 + 25*b^5)*cos (7*d*x + 7*c) - 3*(18*a^2*b^3 - 8*a*b^4 - 15*b^5)*cos(5*d*x + 5*c) + (18*a ^2*b^3 - 20*a*b^4 - 25*b^5)*cos(3*d*x + 3*c) - (2*a*b^4 - 5*b^5)*cos(d*x + c))*cos(14*d*x + 14*c) + (18*a^2*b^3 - 20*a*b^4 - 25*b^5 - 4*(144*a^3*...
\[ \int \frac {\sin ^9(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=\int { -\frac {\sin \left (d x + c\right )^{9}}{{\left (b \sin \left (d x + c\right )^{4} - a\right )}^{3}} \,d x } \] Input:
integrate(sin(d*x+c)^9/(a-b*sin(d*x+c)^4)^3,x, algorithm="giac")
Output:
sage0*x
Time = 41.16 (sec) , antiderivative size = 6675, normalized size of antiderivative = 21.19 \[ \int \frac {\sin ^9(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=\text {Too large to display} \] Input:
int(sin(c + d*x)^9/(a - b*sin(c + d*x)^4)^3,x)
Output:
((cos(c + d*x)^7*(2*a - 5*b))/(16*(a^2 - 2*a*b + b^2)) + (3*cos(c + d*x)^5 *(a*b - 3*a^2 + 10*b^2))/(32*b*(a^2 - 2*a*b + b^2)) - (5*cos(c + d*x)*(3*a *b - a^2 + 2*b^2))/(32*b^2*(a - b)) - (3*cos(c + d*x)^3*(2*a*b - 3*a^2 + 5 *b^2))/(16*b*(a - b)^2))/(d*(a^2 - 2*a*b + b^2 + cos(c + d*x)^2*(4*a*b - 4 *b^2) - cos(c + d*x)^4*(2*a*b - 6*b^2) - 4*b^2*cos(c + d*x)^6 + b^2*cos(c + d*x)^8)) + (atan(((((180224*a*b^8 - 483328*a^2*b^7 + 466944*a^3*b^6 - 20 4800*a^4*b^5 + 40960*a^5*b^4)/(16384*(b^7 - 4*a*b^6 + 6*a^2*b^5 - 4*a^3*b^ 4 + a^4*b^3)) - (cos(c + d*x)*(-(25*a^4*(a^3*b^9)^(1/2) + 384*b^4*(a^3*b^9 )^(1/2) - 144*a*b^9 - 76*a^2*b^8 + 155*a^3*b^7 - 94*a^4*b^6 + 15*a^5*b^5 + 349*a^2*b^2*(a^3*b^9)^(1/2) - 480*a*b^3*(a^3*b^9)^(1/2) - 134*a^3*b*(a^3* b^9)^(1/2))/(16384*(a^2*b^14 - 5*a^3*b^13 + 10*a^4*b^12 - 10*a^5*b^11 + 5* a^6*b^10 - a^7*b^9)))^(1/2)*(16384*a*b^9 - 65536*a^2*b^8 + 98304*a^3*b^7 - 65536*a^4*b^6 + 16384*a^5*b^5))/(256*(a^4*b - 4*a*b^4 + b^5 + 6*a^2*b^3 - 4*a^3*b^2)))*(-(25*a^4*(a^3*b^9)^(1/2) + 384*b^4*(a^3*b^9)^(1/2) - 144*a* b^9 - 76*a^2*b^8 + 155*a^3*b^7 - 94*a^4*b^6 + 15*a^5*b^5 + 349*a^2*b^2*(a^ 3*b^9)^(1/2) - 480*a*b^3*(a^3*b^9)^(1/2) - 134*a^3*b*(a^3*b^9)^(1/2))/(163 84*(a^2*b^14 - 5*a^3*b^13 + 10*a^4*b^12 - 10*a^5*b^11 + 5*a^6*b^10 - a^7*b ^9)))^(1/2) + (cos(c + d*x)*(25*a^4 - 94*a^3*b - 164*a*b^3 + 144*b^4 + 161 *a^2*b^2))/(256*(a^4*b - 4*a*b^4 + b^5 + 6*a^2*b^3 - 4*a^3*b^2)))*(-(25*a^ 4*(a^3*b^9)^(1/2) + 384*b^4*(a^3*b^9)^(1/2) - 144*a*b^9 - 76*a^2*b^8 + ...
\[ \int \frac {\sin ^9(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=\text {too large to display} \] Input:
int(sin(d*x+c)^9/(a-b*sin(d*x+c)^4)^3,x)
Output:
( - 339840*cos(c + d*x)*sin(c + d*x)**6*a**3*b**3 + 287744*cos(c + d*x)*si n(c + d*x)**6*a**2*b**4 - 737280*cos(c + d*x)*sin(c + d*x)**6*a*b**5 + 786 432*cos(c + d*x)*sin(c + d*x)**6*b**6 + 1624320*cos(c + d*x)*sin(c + d*x)* *4*a**3*b**3 - 3467264*cos(c + d*x)*sin(c + d*x)**4*a**2*b**4 + 6782976*co s(c + d*x)*sin(c + d*x)**4*a*b**5 - 7864320*cos(c + d*x)*sin(c + d*x)**4*b **6 - 2744320*cos(c + d*x)*sin(c + d*x)**2*a**3*b**3 + 7643136*cos(c + d*x )*sin(c + d*x)**2*a**2*b**4 - 14548992*cos(c + d*x)*sin(c + d*x)**2*a*b**5 + 18874368*cos(c + d*x)*sin(c + d*x)**2*b**6 + 1392640*cos(c + d*x)*a**3* b**3 - 4898816*cos(c + d*x)*a**2*b**4 + 8650752*cos(c + d*x)*a*b**5 - 1258 2912*cos(c + d*x)*b**6 + 215040000*int(tan((c + d*x)/2)**5/(5*tan((c + d*x )/2)**24*a**4 + 8*tan((c + d*x)/2)**24*a**3*b + 60*tan((c + d*x)/2)**22*a* *4 + 96*tan((c + d*x)/2)**22*a**3*b + 330*tan((c + d*x)/2)**20*a**4 + 288* tan((c + d*x)/2)**20*a**3*b - 384*tan((c + d*x)/2)**20*a**2*b**2 + 1100*ta n((c + d*x)/2)**18*a**4 - 160*tan((c + d*x)/2)**18*a**3*b - 3072*tan((c + d*x)/2)**18*a**2*b**2 + 2475*tan((c + d*x)/2)**16*a**4 - 2760*tan((c + d*x )/2)**16*a**3*b - 6912*tan((c + d*x)/2)**16*a**2*b**2 + 6144*tan((c + d*x) /2)**16*a*b**3 + 3960*tan((c + d*x)/2)**14*a**4 - 7104*tan((c + d*x)/2)**1 4*a**3*b - 6144*tan((c + d*x)/2)**14*a**2*b**2 + 24576*tan((c + d*x)/2)**1 4*a*b**3 + 4620*tan((c + d*x)/2)**12*a**4 - 9408*tan((c + d*x)/2)**12*a**3 *b - 3840*tan((c + d*x)/2)**12*a**2*b**2 + 16384*tan((c + d*x)/2)**12*a...