Integrand size = 22, antiderivative size = 313 \[ \int \frac {\sin (c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=-\frac {3 \left (7 a-10 \sqrt {a} \sqrt {b}+4 b\right ) \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{64 a^{5/2} \left (\sqrt {a}-\sqrt {b}\right )^{5/2} \sqrt [4]{b} d}-\frac {3 \left (7 a+10 \sqrt {a} \sqrt {b}+4 b\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{64 a^{5/2} \left (\sqrt {a}+\sqrt {b}\right )^{5/2} \sqrt [4]{b} d}-\frac {\cos (c+d x) \left (a+b-b \cos ^2(c+d x)\right )}{8 a (a-b) d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )^2}-\frac {\cos (c+d x) \left ((7 a-3 b) (a+2 b)-6 (2 a-b) b \cos ^2(c+d x)\right )}{32 a^2 (a-b)^2 d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )} \] Output:
-3/64*(7*a-10*a^(1/2)*b^(1/2)+4*b)*arctan(b^(1/4)*cos(d*x+c)/(a^(1/2)-b^(1 /2))^(1/2))/a^(5/2)/(a^(1/2)-b^(1/2))^(5/2)/b^(1/4)/d-3/64*(7*a+10*a^(1/2) *b^(1/2)+4*b)*arctanh(b^(1/4)*cos(d*x+c)/(a^(1/2)+b^(1/2))^(1/2))/a^(5/2)/ (a^(1/2)+b^(1/2))^(5/2)/b^(1/4)/d-1/8*cos(d*x+c)*(a+b-b*cos(d*x+c)^2)/a/(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)*((7*a-3*b)*( a+2*b)-6*(2*a-b)*b*cos(d*x+c)^2)/a^2/(a-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 = 2.10 (sec) , antiderivative size = 784, normalized size of antiderivative = 2.50 \[ \int \frac {\sin (c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx =\text {Too large to display} \] Input:
Integrate[Sin[c + d*x]/(a - b*Sin[c + d*x]^4)^3,x]
Output:
((-32*Cos[c + d*x]*(7*a^2 + 5*a*b - 3*b^2 + 3*b*(-2*a + 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 + (3*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)] - 2*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] + I*b^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] - 28*a^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 + 24*a*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 - 10*b^2*ArcTan[Sin[c + d*x]/(Cos[c + d* x] - #1)]*#1^2 + (14*I)*a^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 - (12*I )*a*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 + (5*I)*b^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 + 28*a^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1 ^4 - 24*a*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 + 10*b^2*ArcTan[ Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 - (14*I)*a^2*Log[1 - 2*Cos[c + d*x] *#1 + #1^2]*#1^4 + (12*I)*a*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4 - (5* 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 + 2*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 - 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^2*(a - b)^2*d)
Time = 0.56 (sec) , antiderivative size = 364, normalized size of antiderivative = 1.16, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {3042, 3694, 1405, 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.
| \(\displaystyle \int \frac {\sin (c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (c+d x)}{\left (a-b \sin (c+d x)^4\right )^3}dx\) |
| \(\Big \downarrow \) 3694 |
| \(\displaystyle -\frac {\int \frac {1}{\left (-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)+a-b\right )^3}d\cos (c+d x)}{d}\) |
| \(\Big \downarrow \) 1405 |
| \(\displaystyle -\frac {\frac {\cos (c+d x) \left (a-b \cos ^2(c+d x)+b\right )}{8 a (a-b) \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )^2}-\frac {\int -\frac {2 b \left (-5 b \cos ^2(c+d x)+7 a-b\right )}{\left (-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)+a-b\right )^2}d\cos (c+d x)}{16 a b (a-b)}}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\int \frac {-5 b \cos ^2(c+d x)+7 a-b}{\left (-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)+a-b\right )^2}d\cos (c+d x)}{8 a (a-b)}+\frac {\cos (c+d x) \left (a-b \cos ^2(c+d x)+b\right )}{8 a (a-b) \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )^2}}{d}\) |
| \(\Big \downarrow \) 1492 |
| \(\displaystyle -\frac {\frac {\frac {\cos (c+d x) \left ((7 a-3 b) (a+2 b)-6 b (2 a-b) \cos ^2(c+d x)\right )}{4 a (a-b) \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )}-\frac {\int -\frac {6 b \left (7 a^2-5 b a+2 b^2-2 (2 a-b) b \cos ^2(c+d x)\right )}{-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)+a-b}d\cos (c+d x)}{8 a b (a-b)}}{8 a (a-b)}+\frac {\cos (c+d x) \left (a-b \cos ^2(c+d x)+b\right )}{8 a (a-b) \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )^2}}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\frac {3 \int \frac {7 a^2-5 b a+2 b^2-2 (2 a-b) b \cos ^2(c+d x)}{-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)+a-b}d\cos (c+d x)}{4 a (a-b)}+\frac {\cos (c+d x) \left ((7 a-3 b) (a+2 b)-6 b (2 a-b) \cos ^2(c+d x)\right )}{4 a (a-b) \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )}}{8 a (a-b)}+\frac {\cos (c+d x) \left (a-b \cos ^2(c+d x)+b\right )}{8 a (a-b) \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )^2}}{d}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle -\frac {\frac {\frac {3 \left (\frac {\sqrt {b} \left (-2 \sqrt {a} \sqrt {b}+a+b\right ) \left (10 \sqrt {a} \sqrt {b}+7 a+4 b\right ) \int \frac {1}{\left (\sqrt {a}+\sqrt {b}\right ) \sqrt {b}-b \cos ^2(c+d x)}d\cos (c+d x)}{2 \sqrt {a}}-\frac {\sqrt {b} \left (\sqrt {a}+\sqrt {b}\right )^2 \left (-10 \sqrt {a} \sqrt {b}+7 a+4 b\right ) \int \frac {1}{-b \cos ^2(c+d x)-\left (\sqrt {a}-\sqrt {b}\right ) \sqrt {b}}d\cos (c+d x)}{2 \sqrt {a}}\right )}{4 a (a-b)}+\frac {\cos (c+d x) \left ((7 a-3 b) (a+2 b)-6 b (2 a-b) \cos ^2(c+d x)\right )}{4 a (a-b) \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )}}{8 a (a-b)}+\frac {\cos (c+d x) \left (a-b \cos ^2(c+d x)+b\right )}{8 a (a-b) \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )^2}}{d}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle -\frac {\frac {\frac {3 \left (\frac {\sqrt {b} \left (-2 \sqrt {a} \sqrt {b}+a+b\right ) \left (10 \sqrt {a} \sqrt {b}+7 a+4 b\right ) \int \frac {1}{\left (\sqrt {a}+\sqrt {b}\right ) \sqrt {b}-b \cos ^2(c+d x)}d\cos (c+d x)}{2 \sqrt {a}}+\frac {\left (-10 \sqrt {a} \sqrt {b}+7 a+4 b\right ) \left (\sqrt {a}+\sqrt {b}\right )^2 \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 \sqrt {a} \sqrt [4]{b} \sqrt {\sqrt {a}-\sqrt {b}}}\right )}{4 a (a-b)}+\frac {\cos (c+d x) \left ((7 a-3 b) (a+2 b)-6 b (2 a-b) \cos ^2(c+d x)\right )}{4 a (a-b) \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )}}{8 a (a-b)}+\frac {\cos (c+d x) \left (a-b \cos ^2(c+d x)+b\right )}{8 a (a-b) \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )^2}}{d}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {\frac {\frac {3 \left (\frac {\left (-10 \sqrt {a} \sqrt {b}+7 a+4 b\right ) \left (\sqrt {a}+\sqrt {b}\right )^2 \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 \sqrt {a} \sqrt [4]{b} \sqrt {\sqrt {a}-\sqrt {b}}}+\frac {\left (-2 \sqrt {a} \sqrt {b}+a+b\right ) \left (10 \sqrt {a} \sqrt {b}+7 a+4 b\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 \sqrt {a} \sqrt [4]{b} \sqrt {\sqrt {a}+\sqrt {b}}}\right )}{4 a (a-b)}+\frac {\cos (c+d x) \left ((7 a-3 b) (a+2 b)-6 b (2 a-b) \cos ^2(c+d x)\right )}{4 a (a-b) \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )}}{8 a (a-b)}+\frac {\cos (c+d x) \left (a-b \cos ^2(c+d x)+b\right )}{8 a (a-b) \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )^2}}{d}\) |
Input:
Int[Sin[c + d*x]/(a - b*Sin[c + d*x]^4)^3,x]
Output:
-(((Cos[c + d*x]*(a + b - b*Cos[c + d*x]^2))/(8*a*(a - b)*(a - b + 2*b*Cos [c + d*x]^2 - b*Cos[c + d*x]^4)^2) + ((3*(((Sqrt[a] + Sqrt[b])^2*(7*a - 10 *Sqrt[a]*Sqrt[b] + 4*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)*(7*a + 10*Sqrt[a]*Sqrt[b] + 4*b)*ArcTanh[(b^(1/4)*Cos[c + d*x])/Sqr t[Sqrt[a] + Sqrt[b]]])/(2*Sqrt[a]*Sqrt[Sqrt[a] + Sqrt[b]]*b^(1/4))))/(4*a* (a - b)) + (Cos[c + d*x]*((7*a - 3*b)*(a + 2*b) - 6*(2*a - b)*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 *(a - 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[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(-x)*(b^2 - 2*a*c + b*c*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[(b^2 - 2*a*c + 2*(p + 1)*( b^2 - 4*a*c) + b*c*(4*p + 7)*x^2)*(a + b*x^2 + c*x^4)^(p + 1), x], x] /; Fr eeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && IntegerQ[2*p]
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 = 6.26 (sec) , antiderivative size = 430, normalized size of antiderivative = 1.37
| method | result | size |
| derivativedivides | \(\frac {b^{3} \left (-\frac {\frac {\frac {3 \left (4 a \sqrt {a b}-2 \sqrt {a b}\, b +3 a^{2}-a b \right ) \cos \left (d x +c \right )^{3}}{4 b^{2} \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (11 a \sqrt {a b}-6 \sqrt {a b}\, b +5 a b \right ) \cos \left (d x +c \right )}{4 b^{3} \left (a -b \right )}}{\left (\cos \left (d x +c \right )^{2}+\frac {\sqrt {a b}}{b}-1\right )^{2}}+\frac {3 \left (4 a \sqrt {a b}-2 \sqrt {a b}\, b +7 a^{2}-9 a b +4 b^{2}\right ) \arctan \left (\frac {b \cos \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-b \right ) b}}\right )}{4 b \left (a^{2}-2 a b +b^{2}\right ) \sqrt {\left (\sqrt {a b}-b \right ) b}}}{16 b \sqrt {a b}\, a^{2}}+\frac {\frac {\frac {3 \left (-4 a \sqrt {a b}+2 \sqrt {a b}\, b +3 a^{2}-a b \right ) \cos \left (d x +c \right )^{3}}{4 b^{2} \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (-11 a \sqrt {a b}+6 \sqrt {a b}\, b +5 a b \right ) \cos \left (d x +c \right )}{4 b^{3} \left (a -b \right )}}{\left (\cos \left (d x +c \right )^{2}-1-\frac {\sqrt {a b}}{b}\right )^{2}}+\frac {3 \left (4 a \sqrt {a b}-2 \sqrt {a b}\, b -7 a^{2}+9 a b -4 b^{2}\right ) \operatorname {arctanh}\left (\frac {b \cos \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+b \right ) b}}\right )}{4 b \left (a^{2}-2 a b +b^{2}\right ) \sqrt {\left (\sqrt {a b}+b \right ) b}}}{16 b \sqrt {a b}\, a^{2}}\right )}{d}\) | \(430\) |
| default | \(\frac {b^{3} \left (-\frac {\frac {\frac {3 \left (4 a \sqrt {a b}-2 \sqrt {a b}\, b +3 a^{2}-a b \right ) \cos \left (d x +c \right )^{3}}{4 b^{2} \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (11 a \sqrt {a b}-6 \sqrt {a b}\, b +5 a b \right ) \cos \left (d x +c \right )}{4 b^{3} \left (a -b \right )}}{\left (\cos \left (d x +c \right )^{2}+\frac {\sqrt {a b}}{b}-1\right )^{2}}+\frac {3 \left (4 a \sqrt {a b}-2 \sqrt {a b}\, b +7 a^{2}-9 a b +4 b^{2}\right ) \arctan \left (\frac {b \cos \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-b \right ) b}}\right )}{4 b \left (a^{2}-2 a b +b^{2}\right ) \sqrt {\left (\sqrt {a b}-b \right ) b}}}{16 b \sqrt {a b}\, a^{2}}+\frac {\frac {\frac {3 \left (-4 a \sqrt {a b}+2 \sqrt {a b}\, b +3 a^{2}-a b \right ) \cos \left (d x +c \right )^{3}}{4 b^{2} \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (-11 a \sqrt {a b}+6 \sqrt {a b}\, b +5 a b \right ) \cos \left (d x +c \right )}{4 b^{3} \left (a -b \right )}}{\left (\cos \left (d x +c \right )^{2}-1-\frac {\sqrt {a b}}{b}\right )^{2}}+\frac {3 \left (4 a \sqrt {a b}-2 \sqrt {a b}\, b -7 a^{2}+9 a b -4 b^{2}\right ) \operatorname {arctanh}\left (\frac {b \cos \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+b \right ) b}}\right )}{4 b \left (a^{2}-2 a b +b^{2}\right ) \sqrt {\left (\sqrt {a b}+b \right ) b}}}{16 b \sqrt {a b}\, a^{2}}\right )}{d}\) | \(430\) |
| risch | \(\text {Expression too large to display}\) | \(1321\) |
Input:
int(sin(d*x+c)/(a-b*sin(d*x+c)^4)^3,x,method=_RETURNVERBOSE)
Output:
1/d*b^3*(-1/16/b/(a*b)^(1/2)/a^2*((3/4*(4*a*(a*b)^(1/2)-2*(a*b)^(1/2)*b+3* a^2-a*b)/b^2/(a^2-2*a*b+b^2)*cos(d*x+c)^3+1/4*(11*a*(a*b)^(1/2)-6*(a*b)^(1 /2)*b+5*a*b)/b^3/(a-b)*cos(d*x+c))/(cos(d*x+c)^2+(a*b)^(1/2)/b-1)^2+3/4*(4 *a*(a*b)^(1/2)-2*(a*b)^(1/2)*b+7*a^2-9*a*b+4*b^2)/b/(a^2-2*a*b+b^2)/(((a*b )^(1/2)-b)*b)^(1/2)*arctan(b*cos(d*x+c)/(((a*b)^(1/2)-b)*b)^(1/2)))+1/16/b /(a*b)^(1/2)/a^2*((3/4*(-4*a*(a*b)^(1/2)+2*(a*b)^(1/2)*b+3*a^2-a*b)/b^2/(a ^2-2*a*b+b^2)*cos(d*x+c)^3+1/4*(-11*a*(a*b)^(1/2)+6*(a*b)^(1/2)*b+5*a*b)/b ^3/(a-b)*cos(d*x+c))/(cos(d*x+c)^2-1-(a*b)^(1/2)/b)^2+3/4*(4*a*(a*b)^(1/2) -2*(a*b)^(1/2)*b-7*a^2+9*a*b-4*b^2)/b/(a^2-2*a*b+b^2)/(((a*b)^(1/2)+b)*b)^ (1/2)*arctanh(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. 4160 vs. \(2 (263) = 526\).
Time = 0.69 (sec) , antiderivative size = 4160, normalized size of antiderivative = 13.29 \[ \int \frac {\sin (c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=\text {Too large to display} \] Input:
integrate(sin(d*x+c)/(a-b*sin(d*x+c)^4)^3,x, algorithm="fricas")
Output:
Too large to include
Timed out. \[ \int \frac {\sin (c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=\text {Timed out} \] Input:
integrate(sin(d*x+c)/(a-b*sin(d*x+c)**4)**3,x)
Output:
Timed out
\[ \int \frac {\sin (c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=\int { -\frac {\sin \left (d x + c\right )}{{\left (b \sin \left (d x + c\right )^{4} - a\right )}^{3}} \,d x } \] Input:
integrate(sin(d*x+c)/(a-b*sin(d*x+c)^4)^3,x, algorithm="maxima")
Output:
1/8*(24*(2*a*b^4 - b^5)*cos(2*d*x + 2*c)*cos(d*x + c) - 8*(14*a^2*b^3 + 28 *a*b^4 - 15*b^5)*sin(3*d*x + 3*c)*sin(2*d*x + 2*c) + 24*(2*a*b^4 - b^5)*si n(2*d*x + 2*c)*sin(d*x + c) - (3*(2*a*b^4 - b^5)*cos(15*d*x + 15*c) - (14* a^2*b^3 + 28*a*b^4 - 15*b^5)*cos(13*d*x + 13*c) - (86*a^2*b^3 - 128*a*b^4 + 27*b^5)*cos(11*d*x + 11*c) + (352*a^3*b^2 - 60*a^2*b^3 - 106*a*b^4 + 15* b^5)*cos(9*d*x + 9*c) + (352*a^3*b^2 - 60*a^2*b^3 - 106*a*b^4 + 15*b^5)*co s(7*d*x + 7*c) - (86*a^2*b^3 - 128*a*b^4 + 27*b^5)*cos(5*d*x + 5*c) - (14* a^2*b^3 + 28*a*b^4 - 15*b^5)*cos(3*d*x + 3*c) + 3*(2*a*b^4 - b^5)*cos(d*x + c))*cos(16*d*x + 16*c) - 3*(2*a*b^4 - b^5 - 8*(2*a*b^4 - b^5)*cos(14*d*x + 14*c) - 4*(16*a^2*b^3 - 22*a*b^4 + 7*b^5)*cos(12*d*x + 12*c) + 8*(32*a^ 2*b^3 - 30*a*b^4 + 7*b^5)*cos(10*d*x + 10*c) + 2*(256*a^3*b^2 - 320*a^2*b^ 3 + 166*a*b^4 - 35*b^5)*cos(8*d*x + 8*c) + 8*(32*a^2*b^3 - 30*a*b^4 + 7*b^ 5)*cos(6*d*x + 6*c) - 4*(16*a^2*b^3 - 22*a*b^4 + 7*b^5)*cos(4*d*x + 4*c) - 8*(2*a*b^4 - b^5)*cos(2*d*x + 2*c))*cos(15*d*x + 15*c) - 8*((14*a^2*b^3 + 28*a*b^4 - 15*b^5)*cos(13*d*x + 13*c) + (86*a^2*b^3 - 128*a*b^4 + 27*b^5) *cos(11*d*x + 11*c) - (352*a^3*b^2 - 60*a^2*b^3 - 106*a*b^4 + 15*b^5)*cos( 9*d*x + 9*c) - (352*a^3*b^2 - 60*a^2*b^3 - 106*a*b^4 + 15*b^5)*cos(7*d*x + 7*c) + (86*a^2*b^3 - 128*a*b^4 + 27*b^5)*cos(5*d*x + 5*c) + (14*a^2*b^3 + 28*a*b^4 - 15*b^5)*cos(3*d*x + 3*c) - 3*(2*a*b^4 - b^5)*cos(d*x + c))*cos (14*d*x + 14*c) + (14*a^2*b^3 + 28*a*b^4 - 15*b^5 - 4*(112*a^3*b^2 + 12...
Leaf count of result is larger than twice the leaf count of optimal. 793 vs. \(2 (263) = 526\).
Time = 1.25 (sec) , antiderivative size = 793, normalized size of antiderivative = 2.53 \[ \int \frac {\sin (c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx =\text {Too large to display} \] Input:
integrate(sin(d*x+c)/(a-b*sin(d*x+c)^4)^3,x, algorithm="giac")
Output:
3/64*(4*a^2*b - 2*a*b^2 - (7*a^2 - 9*a*b + 4*b^2)*sqrt(a*b))*sqrt(-b^2 - s qrt(a*b)*b)*arctan(cos(d*x + c)/(d*sqrt(-(a^4*b*d^2 - 2*a^3*b^2*d^2 + a^2* b^3*d^2 + sqrt((a^4*b*d^2 - 2*a^3*b^2*d^2 + a^2*b^3*d^2)^2 + (a^4*b*d^4 - 2*a^3*b^2*d^4 + a^2*b^3*d^4)*(a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3)))/(a^4* b*d^4 - 2*a^3*b^2*d^4 + a^2*b^3*d^4))))/((a^5*b - 2*a^4*b^2 + a^3*b^3 + (a ^5 - 2*a^4*b + a^3*b^2)*sqrt(a*b))*d*abs(b)) + 3/64*(4*a^2*b - 2*a*b^2 + ( 7*a^2 - 9*a*b + 4*b^2)*sqrt(a*b))*sqrt(-b^2 + sqrt(a*b)*b)*arctan(cos(d*x + c)/(d*sqrt(-(a^4*b*d^2 - 2*a^3*b^2*d^2 + a^2*b^3*d^2 - sqrt((a^4*b*d^2 - 2*a^3*b^2*d^2 + a^2*b^3*d^2)^2 + (a^4*b*d^4 - 2*a^3*b^2*d^4 + a^2*b^3*d^4 )*(a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3)))/(a^4*b*d^4 - 2*a^3*b^2*d^4 + a^2 *b^3*d^4))))/((a^5*b - 2*a^4*b^2 + a^3*b^3 - (a^5 - 2*a^4*b + a^3*b^2)*sqr t(a*b))*d*abs(b)) - 1/32*(12*a*b^2*cos(d*x + c)^7/d - 6*b^3*cos(d*x + c)^7 /d - 7*a^2*b*cos(d*x + c)^5/d - 35*a*b^2*cos(d*x + c)^5/d + 18*b^3*cos(d*x + c)^5/d - 2*a^2*b*cos(d*x + c)^3/d + 44*a*b^2*cos(d*x + c)^3/d - 18*b^3* cos(d*x + c)^3/d + 11*a^3*cos(d*x + c)/d + 4*a^2*b*cos(d*x + c)/d - 21*a*b ^2*cos(d*x + c)/d + 6*b^3*cos(d*x + c)/d)/((b*cos(d*x + c)^4 - 2*b*cos(d*x + c)^2 - a + b)^2*(a^4 - 2*a^3*b + a^2*b^2))
Time = 40.33 (sec) , antiderivative size = 5753, normalized size of antiderivative = 18.38 \[ \int \frac {\sin (c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=\text {Too large to display} \] Input:
int(sin(c + d*x)/(a - b*sin(c + d*x)^4)^3,x)
Output:
(atan(((((3*(16384*a^5*b^7 - 73728*a^6*b^6 + 155648*a^7*b^5 - 155648*a^8*b ^4 + 57344*a^9*b^3))/(16384*(a^10 - 4*a^9*b + a^6*b^4 - 4*a^7*b^3 + 6*a^8* b^2)) - (cos(c + d*x)*((9*(49*a^2*(a^15*b)^(1/2) + 21*b^2*(a^15*b)^(1/2) - 105*a^9*b - 16*a^5*b^5 + 84*a^6*b^4 - 189*a^7*b^3 + 210*a^8*b^2 - 54*a*b* (a^15*b)^(1/2)))/(16384*(a^15*b - a^10*b^6 + 5*a^11*b^5 - 10*a^12*b^4 + 10 *a^13*b^3 - 5*a^14*b^2)))^(1/2)*(16384*a^5*b^8 - 65536*a^6*b^7 + 98304*a^7 *b^6 - 65536*a^8*b^5 + 16384*a^9*b^4))/(256*(a^8 - 4*a^7*b + a^4*b^4 - 4*a ^5*b^3 + 6*a^6*b^2)))*((9*(49*a^2*(a^15*b)^(1/2) + 21*b^2*(a^15*b)^(1/2) - 105*a^9*b - 16*a^5*b^5 + 84*a^6*b^4 - 189*a^7*b^3 + 210*a^8*b^2 - 54*a*b* (a^15*b)^(1/2)))/(16384*(a^15*b - a^10*b^6 + 5*a^11*b^5 - 10*a^12*b^4 + 10 *a^13*b^3 - 5*a^14*b^2)))^(1/2) + (cos(c + d*x)*(144*b^7 - 612*a*b^6 + 108 9*a^2*b^5 - 990*a^3*b^4 + 441*a^4*b^3))/(256*(a^8 - 4*a^7*b + a^4*b^4 - 4* a^5*b^3 + 6*a^6*b^2)))*((9*(49*a^2*(a^15*b)^(1/2) + 21*b^2*(a^15*b)^(1/2) - 105*a^9*b - 16*a^5*b^5 + 84*a^6*b^4 - 189*a^7*b^3 + 210*a^8*b^2 - 54*a*b *(a^15*b)^(1/2)))/(16384*(a^15*b - a^10*b^6 + 5*a^11*b^5 - 10*a^12*b^4 + 1 0*a^13*b^3 - 5*a^14*b^2)))^(1/2)*1i - (((3*(16384*a^5*b^7 - 73728*a^6*b^6 + 155648*a^7*b^5 - 155648*a^8*b^4 + 57344*a^9*b^3))/(16384*(a^10 - 4*a^9*b + a^6*b^4 - 4*a^7*b^3 + 6*a^8*b^2)) + (cos(c + d*x)*((9*(49*a^2*(a^15*b)^ (1/2) + 21*b^2*(a^15*b)^(1/2) - 105*a^9*b - 16*a^5*b^5 + 84*a^6*b^4 - 189* a^7*b^3 + 210*a^8*b^2 - 54*a*b*(a^15*b)^(1/2)))/(16384*(a^15*b - a^10*b...
\[ \int \frac {\sin (c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=\text {too large to display} \] Input:
int(sin(d*x+c)/(a-b*sin(d*x+c)^4)^3,x)
Output:
( - 429696*cos(c + d*x)*sin(c + d*x)**6*a**5*b**3 + 805888*cos(c + d*x)*si n(c + d*x)**6*a**4*b**4 - 770048*cos(c + d*x)*sin(c + d*x)**6*a**3*b**5 + 262144*cos(c + d*x)*sin(c + d*x)**6*a**2*b**6 + 47040*cos(c + d*x)*sin(c + d*x)**4*a**6*b**2 + 2447616*cos(c + d*x)*sin(c + d*x)**4*a**5*b**3 - 7862 272*cos(c + d*x)*sin(c + d*x)**4*a**4*b**4 + 7503872*cos(c + d*x)*sin(c + d*x)**4*a**3*b**5 - 2621440*cos(c + d*x)*sin(c + d*x)**4*a**2*b**6 - 26880 *cos(c + d*x)*sin(c + d*x)**2*a**6*b**2 - 3897344*cos(c + d*x)*sin(c + d*x )**2*a**5*b**3 + 17489920*cos(c + d*x)*sin(c + d*x)**2*a**4*b**4 - 1743257 6*cos(c + d*x)*sin(c + d*x)**2*a**3*b**5 + 6291456*cos(c + d*x)*sin(c + d* x)**2*a**2*b**6 - 33600*cos(c + d*x)*a**7*b - 107520*cos(c + d*x)*a**6*b** 2 + 1839104*cos(c + d*x)*a**5*b**3 - 10895360*cos(c + d*x)*a**4*b**4 + 112 72192*cos(c + d*x)*a**3*b**5 - 4194304*cos(c + d*x)*a**2*b**6 + 301056000* 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*t an((c + d*x)/2)**20*a**2*b**2 + 1100*tan((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)**14*a**3*b - 6144*tan((c + d*x)/2)...