Integrand size = 24, antiderivative size = 315 \[ \int \frac {\sin ^9(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=-\frac {\left (5 a-14 \sqrt {a} \sqrt {b}+12 b\right ) \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{64 \sqrt {a} \left (\sqrt {a}-\sqrt {b}\right )^{5/2} b^{9/4} d}-\frac {\left (5 a+14 \sqrt {a} \sqrt {b}+12 b\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{64 \sqrt {a} \left (\sqrt {a}+\sqrt {b}\right )^{5/2} b^{9/4} d}-\frac {a \cos (c+d x) \left (a+b-b \cos ^2(c+d x)\right )}{8 (a-b) b^2 d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )^2}+\frac {\cos (c+d x) \left (9 a^2-11 a b-10 b^2-2 (2 a-5 b) b \cos ^2(c+d x)\right )}{32 (a-b)^2 b^2 d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )} \]
-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)-1/64*arctan(b^ (1/4)*cos(d*x+c)/(a^(1/2)-b^(1/2))^(1/2))*(5*a+12*b-14*a^(1/2)*b^(1/2))/b^ (9/4)/d/a^(1/2)/(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+12*b+14*a^(1/2)*b^(1/2))/b^(9/4)/d/a^(1/2)/(a^ (1/2)+b^(1/2))^(5/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 7.07 (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=\frac {-\frac {32 \cos (c+d x) \left (-9 a^2+13 a b+5 b^2+(2 a-5 b) b \cos (2 (c+d x))\right )}{8 a-3 b+4 b \cos (2 (c+d x))-b \cos (4 (c+d x))}-\frac {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 \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 {-4 a b \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )+10 b^2 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )+2 i a b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right )-5 i b^2 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right )-20 a^2 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^2+56 a b \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^2-78 b^2 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^2+10 i a^2 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^2-28 i a b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^2+39 i b^2 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^2+20 a^2 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^4-56 a b \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^4+78 b^2 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^4-10 i a^2 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^4+28 i a b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^4-39 i b^2 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^4+4 a b \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^6-10 b^2 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^6-2 i a b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^6+5 i b^2 \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 ]}{128 (a-b)^2 b^2 d} \]
((-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 = 0.68 (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 |
\(\displaystyle -\frac {\frac {a \cos (c+d x) \left (a-b \cos ^2(c+d x)+b\right )}{8 b^2 (a-b) \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )^2}-\frac {\frac {a \cos (c+d x) \left (9 a^2-2 b (2 a-5 b) \cos ^2(c+d x)-11 a b-10 b^2\right )}{4 b (a-b) \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )}-\frac {a \left (\frac {\left (-14 \sqrt {a} \sqrt {b}+5 a+12 b\right ) \left (\sqrt {a}+\sqrt {b}\right )^2 \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 \sqrt {a} \sqrt [4]{b} \sqrt {\sqrt {a}-\sqrt {b}}}+\frac {\left (-2 \sqrt {a} \sqrt {b}+a+b\right ) \left (14 \sqrt {a} \sqrt {b}+5 a+12 b\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 \sqrt {a} \sqrt [4]{b} \sqrt {\sqrt {a}+\sqrt {b}}}\right )}{4 b (a-b)}}{8 a b (a-b)}}{d}\) |
-(((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)
3.3.24.3.1 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)^(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 = 5.59 (sec) , antiderivative size = 342, normalized size of antiderivative = 1.09
method | result | size |
derivativedivides | \(\frac {-\frac {-\frac {\left (2 a -5 b \right ) \left (\cos ^{7}\left (d x +c \right )\right )}{16 \left (a^{2}-2 a b +b^{2}\right )}+\frac {3 \left (3 a^{2}-a b -10 b^{2}\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{32 b \left (a^{2}-2 a b +b^{2}\right )}-\frac {3 \left (3 a^{2}-2 a b -5 b^{2}\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{16 b \left (a^{2}-2 a b +b^{2}\right )}-\frac {5 \left (a^{2}-3 a b -2 b^{2}\right ) \cos \left (d x +c \right )}{32 b^{2} \left (a -b \right )}}{{\left (a -b +2 b \left (\cos ^{2}\left (d x +c \right )\right )-b \left (\cos ^{4}\left (d x +c \right )\right )\right )}^{2}}-\frac {-\frac {\left (-4 a \sqrt {a b}+10 \sqrt {a b}\, b -5 a^{2}+11 a b -12 b^{2}\right ) \operatorname {arctanh}\left (\frac {\cos \left (d x +c \right ) b}{\sqrt {\left (\sqrt {a b}+b \right ) b}}\right )}{2 \sqrt {a b}\, \sqrt {\left (\sqrt {a b}+b \right ) b}}+\frac {\left (-4 a \sqrt {a b}+10 \sqrt {a b}\, b +5 a^{2}-11 a b +12 b^{2}\right ) \arctan \left (\frac {\cos \left (d x +c \right ) b}{\sqrt {\left (\sqrt {a b}-b \right ) b}}\right )}{2 \sqrt {a b}\, \sqrt {\left (\sqrt {a b}-b \right ) b}}}{32 \left (a^{2}-2 a b +b^{2}\right ) b}}{d}\) | \(342\) |
default | \(\frac {-\frac {-\frac {\left (2 a -5 b \right ) \left (\cos ^{7}\left (d x +c \right )\right )}{16 \left (a^{2}-2 a b +b^{2}\right )}+\frac {3 \left (3 a^{2}-a b -10 b^{2}\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{32 b \left (a^{2}-2 a b +b^{2}\right )}-\frac {3 \left (3 a^{2}-2 a b -5 b^{2}\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{16 b \left (a^{2}-2 a b +b^{2}\right )}-\frac {5 \left (a^{2}-3 a b -2 b^{2}\right ) \cos \left (d x +c \right )}{32 b^{2} \left (a -b \right )}}{{\left (a -b +2 b \left (\cos ^{2}\left (d x +c \right )\right )-b \left (\cos ^{4}\left (d x +c \right )\right )\right )}^{2}}-\frac {-\frac {\left (-4 a \sqrt {a b}+10 \sqrt {a b}\, b -5 a^{2}+11 a b -12 b^{2}\right ) \operatorname {arctanh}\left (\frac {\cos \left (d x +c \right ) b}{\sqrt {\left (\sqrt {a b}+b \right ) b}}\right )}{2 \sqrt {a b}\, \sqrt {\left (\sqrt {a b}+b \right ) b}}+\frac {\left (-4 a \sqrt {a b}+10 \sqrt {a b}\, b +5 a^{2}-11 a b +12 b^{2}\right ) \arctan \left (\frac {\cos \left (d x +c \right ) b}{\sqrt {\left (\sqrt {a b}-b \right ) b}}\right )}{2 \sqrt {a b}\, \sqrt {\left (\sqrt {a b}-b \right ) b}}}{32 \left (a^{2}-2 a b +b^{2}\right ) b}}{d}\) | \(342\) |
risch | \(\text {Expression too large to display}\) | \(1583\) |
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(cos(d*x+c)*b/(((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(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. 4640 vs. \(2 (264) = 528\).
Time = 1.07 (sec) , antiderivative size = 4640, normalized size of antiderivative = 14.73 \[ \int \frac {\sin ^9(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=\text {Too large to display} \]
1/128*(8*(2*a*b^2 - 5*b^3)*cos(d*x + c)^7 - 12*(3*a^2*b - a*b^2 - 10*b^3)* cos(d*x + c)^5 + 24*(3*a^2*b - 2*a*b^2 - 5*b^3)*cos(d*x + c)^3 + ((a^2*b^4 - 2*a*b^5 + b^6)*d*cos(d*x + c)^8 - 4*(a^2*b^4 - 2*a*b^5 + b^6)*d*cos(d*x + c)^6 - 2*(a^3*b^3 - 5*a^2*b^4 + 7*a*b^5 - 3*b^6)*d*cos(d*x + c)^4 + 4*( a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 - b^6)*d*cos(d*x + c)^2 + (a^4*b^2 - 4*a^3*b ^3 + 6*a^2*b^4 - 4*a*b^5 + b^6)*d)*sqrt((15*a^4 - 94*a^3*b + 155*a^2*b^2 - 76*a*b^3 - 144*b^4 + (a^6*b^4 - 5*a^5*b^5 + 10*a^4*b^6 - 10*a^3*b^7 + 5*a ^2*b^8 - a*b^9)*d^2*sqrt((625*a^8 - 6700*a^7*b + 35406*a^6*b^2 - 117532*a^ 5*b^3 + 269641*a^4*b^4 - 437952*a^3*b^5 + 498432*a^2*b^6 - 368640*a*b^7 + 147456*b^8)/((a^11*b^9 - 10*a^10*b^10 + 45*a^9*b^11 - 120*a^8*b^12 + 210*a ^7*b^13 - 252*a^6*b^14 + 210*a^5*b^15 - 120*a^4*b^16 + 45*a^3*b^17 - 10*a^ 2*b^18 + a*b^19)*d^4)))/((a^6*b^4 - 5*a^5*b^5 + 10*a^4*b^6 - 10*a^3*b^7 + 5*a^2*b^8 - a*b^9)*d^2))*log((625*a^6 - 5250*a^5*b + 22509*a^4*b^2 - 57820 *a^3*b^3 + 96336*a^2*b^4 - 98304*a*b^5 + 55296*b^6)*cos(d*x + c) - ((a^8*b ^7 - 6*a^7*b^8 + 27*a^6*b^9 - 80*a^5*b^10 + 135*a^4*b^11 - 126*a^3*b^12 + 61*a^2*b^13 - 12*a*b^14)*d^3*sqrt((625*a^8 - 6700*a^7*b + 35406*a^6*b^2 - 117532*a^5*b^3 + 269641*a^4*b^4 - 437952*a^3*b^5 + 498432*a^2*b^6 - 368640 *a*b^7 + 147456*b^8)/((a^11*b^9 - 10*a^10*b^10 + 45*a^9*b^11 - 120*a^8*b^1 2 + 210*a^7*b^13 - 252*a^6*b^14 + 210*a^5*b^15 - 120*a^4*b^16 + 45*a^3*b^1 7 - 10*a^2*b^18 + a*b^19)*d^4)) + (125*a^7*b^2 - 1045*a^6*b^3 + 4305*a^...
Timed out. \[ \int \frac {\sin ^9(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=\text {Timed out} \]
\[ \int \frac {\sin ^9(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=\int { -\frac {\sin \left (d x + c\right )^{9}}{{\left (b \sin \left (d x + c\right )^{4} - a\right )}^{3}} \,d x } \]
-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*...
\[ \int \frac {\sin ^9(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=\int { -\frac {\sin \left (d x + c\right )^{9}}{{\left (b \sin \left (d x + c\right )^{4} - a\right )}^{3}} \,d x } \]
Time = 19.41 (sec) , antiderivative size = 6675, normalized size of antiderivative = 21.19 \[ \int \frac {\sin ^9(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=\text {Too large to display} \]
((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 + ...