Integrand size = 22, antiderivative size = 313 \[ \int \frac {\sin (c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=-\frac {3 \left (7 a-10 \sqrt {a} \sqrt {b}+4 b\right ) \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{64 a^{5/2} \left (\sqrt {a}-\sqrt {b}\right )^{5/2} \sqrt [4]{b} d}-\frac {3 \left (7 a+10 \sqrt {a} \sqrt {b}+4 b\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{64 a^{5/2} \left (\sqrt {a}+\sqrt {b}\right )^{5/2} \sqrt [4]{b} d}-\frac {\cos (c+d x) \left (a+b-b \cos ^2(c+d x)\right )}{8 a (a-b) d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )^2}-\frac {\cos (c+d x) \left ((7 a-3 b) (a+2 b)-6 (2 a-b) b \cos ^2(c+d x)\right )}{32 a^2 (a-b)^2 d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )} \]
-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)-3/64*arctan(b^(1/4)*co s(d*x+c)/(a^(1/2)-b^(1/2))^(1/2))*(7*a+4*b-10*a^(1/2)*b^(1/2))/a^(5/2)/b^( 1/4)/d/(a^(1/2)-b^(1/2))^(5/2)-3/64*arctanh(b^(1/4)*cos(d*x+c)/(a^(1/2)+b^ (1/2))^(1/2))*(7*a+4*b+10*a^(1/2)*b^(1/2))/a^(5/2)/b^(1/4)/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 = 1.65 (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=\frac {-\frac {32 \cos (c+d x) \left (7 a^2+5 a b-3 b^2+3 b (-2 a+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}+3 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 )-2 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 )+i b^2 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right )-28 a^2 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^2+24 a b \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^2-10 b^2 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^2+14 i a^2 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^2-12 i a b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^2+5 i b^2 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^2+28 a^2 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^4-24 a b \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^4+10 b^2 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^4-14 i a^2 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^4+12 i a b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^4-5 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+2 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-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^2 (a-b)^2 d} \]
((-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 = 0.54 (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 |
| \(\displaystyle -\frac {\frac {\frac {3 \left (\frac {\left (-10 \sqrt {a} \sqrt {b}+7 a+4 b\right ) \left (\sqrt {a}+\sqrt {b}\right )^2 \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 \sqrt {a} \sqrt [4]{b} \sqrt {\sqrt {a}-\sqrt {b}}}+\frac {\left (-2 \sqrt {a} \sqrt {b}+a+b\right ) \left (10 \sqrt {a} \sqrt {b}+7 a+4 b\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 \sqrt {a} \sqrt [4]{b} \sqrt {\sqrt {a}+\sqrt {b}}}\right )}{4 a (a-b)}+\frac {\cos (c+d x) \left ((7 a-3 b) (a+2 b)-6 b (2 a-b) \cos ^2(c+d x)\right )}{4 a (a-b) \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )}}{8 a (a-b)}+\frac {\cos (c+d x) \left (a-b \cos ^2(c+d x)+b\right )}{8 a (a-b) \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )^2}}{d}\) |
-(((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)
3.3.28.3.1 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[((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 = 6.15 (sec) , antiderivative size = 430, normalized size of antiderivative = 1.37
| method | result | size |
| derivativedivides | \(\frac {b^{3} \left (\frac {\frac {\frac {3 \left (-4 a \sqrt {a b}+2 \sqrt {a b}\, b +3 a^{2}-a b \right ) \left (\cos ^{3}\left (d x +c \right )\right )}{4 b^{2} \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (-11 a \sqrt {a b}+6 \sqrt {a b}\, b +5 a b \right ) \cos \left (d x +c \right )}{4 b^{3} \left (a -b \right )}}{\left (\cos ^{2}\left (d x +c \right )-1-\frac {\sqrt {a b}}{b}\right )^{2}}+\frac {3 \left (4 a \sqrt {a b}-2 \sqrt {a b}\, b -7 a^{2}+9 a b -4 b^{2}\right ) \operatorname {arctanh}\left (\frac {\cos \left (d x +c \right ) b}{\sqrt {\left (\sqrt {a b}+b \right ) b}}\right )}{4 b \left (a^{2}-2 a b +b^{2}\right ) \sqrt {\left (\sqrt {a b}+b \right ) b}}}{16 b \,a^{2} \sqrt {a b}}-\frac {\frac {\frac {3 \left (4 a \sqrt {a b}-2 \sqrt {a b}\, b +3 a^{2}-a b \right ) \left (\cos ^{3}\left (d x +c \right )\right )}{4 b^{2} \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (11 a \sqrt {a b}-6 \sqrt {a b}\, b +5 a b \right ) \cos \left (d x +c \right )}{4 b^{3} \left (a -b \right )}}{\left (\cos ^{2}\left (d x +c \right )+\frac {\sqrt {a b}}{b}-1\right )^{2}}+\frac {3 \left (4 a \sqrt {a b}-2 \sqrt {a b}\, b +7 a^{2}-9 a b +4 b^{2}\right ) \arctan \left (\frac {\cos \left (d x +c \right ) b}{\sqrt {\left (\sqrt {a b}-b \right ) b}}\right )}{4 b \left (a^{2}-2 a b +b^{2}\right ) \sqrt {\left (\sqrt {a b}-b \right ) b}}}{16 b \,a^{2} \sqrt {a b}}\right )}{d}\) | \(430\) |
| default | \(\frac {b^{3} \left (\frac {\frac {\frac {3 \left (-4 a \sqrt {a b}+2 \sqrt {a b}\, b +3 a^{2}-a b \right ) \left (\cos ^{3}\left (d x +c \right )\right )}{4 b^{2} \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (-11 a \sqrt {a b}+6 \sqrt {a b}\, b +5 a b \right ) \cos \left (d x +c \right )}{4 b^{3} \left (a -b \right )}}{\left (\cos ^{2}\left (d x +c \right )-1-\frac {\sqrt {a b}}{b}\right )^{2}}+\frac {3 \left (4 a \sqrt {a b}-2 \sqrt {a b}\, b -7 a^{2}+9 a b -4 b^{2}\right ) \operatorname {arctanh}\left (\frac {\cos \left (d x +c \right ) b}{\sqrt {\left (\sqrt {a b}+b \right ) b}}\right )}{4 b \left (a^{2}-2 a b +b^{2}\right ) \sqrt {\left (\sqrt {a b}+b \right ) b}}}{16 b \,a^{2} \sqrt {a b}}-\frac {\frac {\frac {3 \left (4 a \sqrt {a b}-2 \sqrt {a b}\, b +3 a^{2}-a b \right ) \left (\cos ^{3}\left (d x +c \right )\right )}{4 b^{2} \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (11 a \sqrt {a b}-6 \sqrt {a b}\, b +5 a b \right ) \cos \left (d x +c \right )}{4 b^{3} \left (a -b \right )}}{\left (\cos ^{2}\left (d x +c \right )+\frac {\sqrt {a b}}{b}-1\right )^{2}}+\frac {3 \left (4 a \sqrt {a b}-2 \sqrt {a b}\, b +7 a^{2}-9 a b +4 b^{2}\right ) \arctan \left (\frac {\cos \left (d x +c \right ) b}{\sqrt {\left (\sqrt {a b}-b \right ) b}}\right )}{4 b \left (a^{2}-2 a b +b^{2}\right ) \sqrt {\left (\sqrt {a b}-b \right ) b}}}{16 b \,a^{2} \sqrt {a b}}\right )}{d}\) | \(430\) |
| risch | \(\text {Expression too large to display}\) | \(1321\) |
1/d*b^3*(1/16/b/a^2/(a*b)^(1/2)*((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(cos(d*x+c)*b/(((a*b)^(1/2)+b)*b)^(1/2)))-1/16 /b/a^2/(a*b)^(1/2)*((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(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. 4160 vs. \(2 (263) = 526\).
Time = 0.89 (sec) , antiderivative size = 4160, normalized size of antiderivative = 13.29 \[ \int \frac {\sin (c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=\text {Too large to display} \]
-1/128*(24*(2*a*b^2 - b^3)*cos(d*x + c)^7 - 4*(7*a^2*b + 35*a*b^2 - 18*b^3 )*cos(d*x + c)^5 - 8*(a^2*b - 22*a*b^2 + 9*b^3)*cos(d*x + c)^3 + 3*((a^4*b ^2 - 2*a^3*b^3 + a^2*b^4)*d*cos(d*x + c)^8 - 4*(a^4*b^2 - 2*a^3*b^3 + a^2* b^4)*d*cos(d*x + c)^6 - 2*(a^5*b - 5*a^4*b^2 + 7*a^3*b^3 - 3*a^2*b^4)*d*co s(d*x + c)^4 + 4*(a^5*b - 3*a^4*b^2 + 3*a^3*b^3 - a^2*b^4)*d*cos(d*x + c)^ 2 + (a^6 - 4*a^5*b + 6*a^4*b^2 - 4*a^3*b^3 + a^2*b^4)*d)*sqrt(-(105*a^4 - 210*a^3*b + 189*a^2*b^2 - 84*a*b^3 + 16*b^4 + (a^10 - 5*a^9*b + 10*a^8*b^2 - 10*a^7*b^3 + 5*a^6*b^4 - a^5*b^5)*d^2*sqrt((2401*a^4 - 5292*a^3*b + 497 4*a^2*b^2 - 2268*a*b^3 + 441*b^4)/((a^15*b - 10*a^14*b^2 + 45*a^13*b^3 - 1 20*a^12*b^4 + 210*a^11*b^5 - 252*a^10*b^6 + 210*a^9*b^7 - 120*a^8*b^8 + 45 *a^7*b^9 - 10*a^6*b^10 + a^5*b^11)*d^4)))/((a^10 - 5*a^9*b + 10*a^8*b^2 - 10*a^7*b^3 + 5*a^6*b^4 - a^5*b^5)*d^2))*log(27*(2401*a^4 - 4802*a^3*b + 41 89*a^2*b^2 - 1788*a*b^3 + 336*b^4)*cos(d*x + c) - 27*((11*a^12*b - 66*a^11 *b^2 + 169*a^10*b^3 - 240*a^9*b^4 + 205*a^8*b^5 - 106*a^7*b^6 + 31*a^6*b^7 - 4*a^5*b^8)*d^3*sqrt((2401*a^4 - 5292*a^3*b + 4974*a^2*b^2 - 2268*a*b^3 + 441*b^4)/((a^15*b - 10*a^14*b^2 + 45*a^13*b^3 - 120*a^12*b^4 + 210*a^11* b^5 - 252*a^10*b^6 + 210*a^9*b^7 - 120*a^8*b^8 + 45*a^7*b^9 - 10*a^6*b^10 + a^5*b^11)*d^4)) - (343*a^7 - 623*a^6*b + 515*a^5*b^2 - 213*a^4*b^3 + 42* a^3*b^4)*d)*sqrt(-(105*a^4 - 210*a^3*b + 189*a^2*b^2 - 84*a*b^3 + 16*b^4 + (a^10 - 5*a^9*b + 10*a^8*b^2 - 10*a^7*b^3 + 5*a^6*b^4 - a^5*b^5)*d^2*s...
Timed out. \[ \int \frac {\sin (c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=\text {Timed out} \]
\[ \int \frac {\sin (c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=\int { -\frac {\sin \left (d x + c\right )}{{\left (b \sin \left (d x + c\right )^{4} - a\right )}^{3}} \,d x } \]
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 = 1.96 (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=\frac {3 \, {\left (4 \, a^{2} b - 2 \, a b^{2} - {\left (7 \, a^{2} - 9 \, a b + 4 \, b^{2}\right )} \sqrt {a b}\right )} \sqrt {-b^{2} - \sqrt {a b} b} \arctan \left (\frac {\cos \left (d x + c\right )}{d \sqrt {-\frac {a^{4} b d^{2} - 2 \, a^{3} b^{2} d^{2} + a^{2} b^{3} d^{2} + \sqrt {{\left (a^{4} b d^{2} - 2 \, a^{3} b^{2} d^{2} + a^{2} b^{3} d^{2}\right )}^{2} + {\left (a^{4} b d^{4} - 2 \, a^{3} b^{2} d^{4} + a^{2} b^{3} d^{4}\right )} {\left (a^{5} - 3 \, a^{4} b + 3 \, a^{3} b^{2} - a^{2} b^{3}\right )}}}{a^{4} b d^{4} - 2 \, a^{3} b^{2} d^{4} + a^{2} b^{3} d^{4}}}}\right )}{64 \, {\left (a^{5} b - 2 \, a^{4} b^{2} + a^{3} b^{3} + {\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} \sqrt {a b}\right )} d {\left | b \right |}} + \frac {3 \, {\left (4 \, a^{2} b - 2 \, a b^{2} + {\left (7 \, a^{2} - 9 \, a b + 4 \, b^{2}\right )} \sqrt {a b}\right )} \sqrt {-b^{2} + \sqrt {a b} b} \arctan \left (\frac {\cos \left (d x + c\right )}{d \sqrt {-\frac {a^{4} b d^{2} - 2 \, a^{3} b^{2} d^{2} + a^{2} b^{3} d^{2} - \sqrt {{\left (a^{4} b d^{2} - 2 \, a^{3} b^{2} d^{2} + a^{2} b^{3} d^{2}\right )}^{2} + {\left (a^{4} b d^{4} - 2 \, a^{3} b^{2} d^{4} + a^{2} b^{3} d^{4}\right )} {\left (a^{5} - 3 \, a^{4} b + 3 \, a^{3} b^{2} - a^{2} b^{3}\right )}}}{a^{4} b d^{4} - 2 \, a^{3} b^{2} d^{4} + a^{2} b^{3} d^{4}}}}\right )}{64 \, {\left (a^{5} b - 2 \, a^{4} b^{2} + a^{3} b^{3} - {\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} \sqrt {a b}\right )} d {\left | b \right |}} - \frac {\frac {12 \, a b^{2} \cos \left (d x + c\right )^{7}}{d} - \frac {6 \, b^{3} \cos \left (d x + c\right )^{7}}{d} - \frac {7 \, a^{2} b \cos \left (d x + c\right )^{5}}{d} - \frac {35 \, a b^{2} \cos \left (d x + c\right )^{5}}{d} + \frac {18 \, b^{3} \cos \left (d x + c\right )^{5}}{d} - \frac {2 \, a^{2} b \cos \left (d x + c\right )^{3}}{d} + \frac {44 \, a b^{2} \cos \left (d x + c\right )^{3}}{d} - \frac {18 \, b^{3} \cos \left (d x + c\right )^{3}}{d} + \frac {11 \, a^{3} \cos \left (d x + c\right )}{d} + \frac {4 \, a^{2} b \cos \left (d x + c\right )}{d} - \frac {21 \, a b^{2} \cos \left (d x + c\right )}{d} + \frac {6 \, b^{3} \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^{4} - 2 \, a^{3} b + a^{2} b^{2}\right )}} \]
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 = 18.28 (sec) , antiderivative size = 5753, normalized size of antiderivative = 18.38 \[ \int \frac {\sin (c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=\text {Too large to display} \]
(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...