Integrand size = 13, antiderivative size = 59 \[ \int \frac {\sin ^3(x)}{(a+a \sin (x))^3} \, dx=\frac {x}{a^3}+\frac {\cos (x) \sin ^2(x)}{5 (a+a \sin (x))^3}-\frac {7 \cos (x)}{15 a (a+a \sin (x))^2}+\frac {29 \cos (x)}{15 \left (a^3+a^3 \sin (x)\right )} \] Output:
x/a^3+1/5*cos(x)*sin(x)^2/(a+a*sin(x))^3-7/15*cos(x)/a/(a+a*sin(x))^2+29*c os(x)/(15*a^3+15*a^3*sin(x))
Time = 0.19 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.90 \[ \int \frac {\sin ^3(x)}{(a+a \sin (x))^3} \, dx=\frac {\left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right ) \left (30 (-9+5 x) \cos \left (\frac {x}{2}\right )+(230-75 x) \cos \left (\frac {3 x}{2}\right )-15 x \cos \left (\frac {5 x}{2}\right )-370 \sin \left (\frac {x}{2}\right )+150 x \sin \left (\frac {x}{2}\right )-90 \sin \left (\frac {3 x}{2}\right )+75 x \sin \left (\frac {3 x}{2}\right )+64 \sin \left (\frac {5 x}{2}\right )-15 x \sin \left (\frac {5 x}{2}\right )\right )}{60 a^3 (1+\sin (x))^3} \] Input:
Integrate[Sin[x]^3/(a + a*Sin[x])^3,x]
Output:
((Cos[x/2] + Sin[x/2])*(30*(-9 + 5*x)*Cos[x/2] + (230 - 75*x)*Cos[(3*x)/2] - 15*x*Cos[(5*x)/2] - 370*Sin[x/2] + 150*x*Sin[x/2] - 90*Sin[(3*x)/2] + 7 5*x*Sin[(3*x)/2] + 64*Sin[(5*x)/2] - 15*x*Sin[(5*x)/2]))/(60*a^3*(1 + Sin[ x])^3)
Time = 0.58 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.17, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.846, Rules used = {3042, 3244, 3042, 3447, 3042, 3498, 25, 3042, 3214, 3042, 3127}
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.
| \(\displaystyle \int \frac {\sin ^3(x)}{(a \sin (x)+a)^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (x)^3}{(a \sin (x)+a)^3}dx\) |
| \(\Big \downarrow \) 3244 |
| \(\displaystyle \frac {\sin ^2(x) \cos (x)}{5 (a \sin (x)+a)^3}-\frac {\int \frac {\sin (x) (2 a-5 a \sin (x))}{(\sin (x) a+a)^2}dx}{5 a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sin ^2(x) \cos (x)}{5 (a \sin (x)+a)^3}-\frac {\int \frac {\sin (x) (2 a-5 a \sin (x))}{(\sin (x) a+a)^2}dx}{5 a^2}\) |
| \(\Big \downarrow \) 3447 |
| \(\displaystyle \frac {\sin ^2(x) \cos (x)}{5 (a \sin (x)+a)^3}-\frac {\int \frac {2 a \sin (x)-5 a \sin ^2(x)}{(\sin (x) a+a)^2}dx}{5 a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sin ^2(x) \cos (x)}{5 (a \sin (x)+a)^3}-\frac {\int \frac {2 a \sin (x)-5 a \sin (x)^2}{(\sin (x) a+a)^2}dx}{5 a^2}\) |
| \(\Big \downarrow \) 3498 |
| \(\displaystyle \frac {\sin ^2(x) \cos (x)}{5 (a \sin (x)+a)^3}-\frac {\frac {7 a \cos (x)}{3 (a \sin (x)+a)^2}-\frac {\int -\frac {14 a^2-15 a^2 \sin (x)}{\sin (x) a+a}dx}{3 a^2}}{5 a^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sin ^2(x) \cos (x)}{5 (a \sin (x)+a)^3}-\frac {\frac {\int \frac {14 a^2-15 a^2 \sin (x)}{\sin (x) a+a}dx}{3 a^2}+\frac {7 a \cos (x)}{3 (a \sin (x)+a)^2}}{5 a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sin ^2(x) \cos (x)}{5 (a \sin (x)+a)^3}-\frac {\frac {\int \frac {14 a^2-15 a^2 \sin (x)}{\sin (x) a+a}dx}{3 a^2}+\frac {7 a \cos (x)}{3 (a \sin (x)+a)^2}}{5 a^2}\) |
| \(\Big \downarrow \) 3214 |
| \(\displaystyle \frac {\sin ^2(x) \cos (x)}{5 (a \sin (x)+a)^3}-\frac {\frac {29 a^2 \int \frac {1}{\sin (x) a+a}dx-15 a x}{3 a^2}+\frac {7 a \cos (x)}{3 (a \sin (x)+a)^2}}{5 a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sin ^2(x) \cos (x)}{5 (a \sin (x)+a)^3}-\frac {\frac {29 a^2 \int \frac {1}{\sin (x) a+a}dx-15 a x}{3 a^2}+\frac {7 a \cos (x)}{3 (a \sin (x)+a)^2}}{5 a^2}\) |
| \(\Big \downarrow \) 3127 |
| \(\displaystyle \frac {\sin ^2(x) \cos (x)}{5 (a \sin (x)+a)^3}-\frac {\frac {-\frac {29 a^2 \cos (x)}{a \sin (x)+a}-15 a x}{3 a^2}+\frac {7 a \cos (x)}{3 (a \sin (x)+a)^2}}{5 a^2}\) |
Input:
Int[Sin[x]^3/(a + a*Sin[x])^3,x]
Output:
(Cos[x]*Sin[x]^2)/(5*(a + a*Sin[x])^3) - ((7*a*Cos[x])/(3*(a + a*Sin[x])^2 ) + (-15*a*x - (29*a^2*Cos[x])/(a + a*Sin[x]))/(3*a^2))/(5*a^2)
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> Simp[-Cos[c + d*x]/(d*(b + a*Sin[c + d*x])), x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b ^2, 0]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])/((c_.) + (d_.)*sin[(e_.) + (f_. )*(x_)]), x_Symbol] :> Simp[b*(x/d), x] - Simp[(b*c - a*d)/d Int[1/(c + d *Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*c - a*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n - 1)/(a*f*(2*m + 1))), x] + Simp[1/(a*b* (2*m + 1)) Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n - 2)* Simp[b*(c^2*(m + 1) + d^2*(n - 1)) + a*c*d*(m - n + 1) + d*(a*d*(m - n + 1) + b*c*(m + n))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n] || (IntegerQ[m] && EqQ[c, 0]))
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Int[(a + b*Sin[e + f*x])^m*(A*c + (B*c + A*d)*Sin[e + f*x] + B*d*Sin[e + f*x]^2), x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(A*b - a* B + b*C)*Cos[e + f*x]*((a + b*Sin[e + f*x])^m/(a*f*(2*m + 1))), x] + Simp[1 /(a^2*(2*m + 1)) Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[a*A*(m + 1) + m*(b *B - a*C) + b*C*(2*m + 1)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C}, x] && LtQ[m, -1] && EqQ[a^2 - b^2, 0]
Result contains complex when optimal does not.
Time = 0.37 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.92
| method | result | size |
| risch | \(\frac {x}{a^{3}}+\frac {18 i {\mathrm e}^{3 i x}+6 \,{\mathrm e}^{4 i x}-\frac {46 i {\mathrm e}^{i x}}{3}-\frac {74 \,{\mathrm e}^{2 i x}}{3}+\frac {64}{15}}{a^{3} \left ({\mathrm e}^{i x}+i\right )^{5}}\) | \(54\) |
| default | \(\frac {2 \arctan \left (\tan \left (\frac {x}{2}\right )\right )-\frac {4}{\left (\tan \left (\frac {x}{2}\right )+1\right )^{4}}+\frac {8}{5 \left (\tan \left (\frac {x}{2}\right )+1\right )^{5}}+\frac {4}{3 \left (\tan \left (\frac {x}{2}\right )+1\right )^{3}}+\frac {2}{\left (\tan \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {16}{8 \tan \left (\frac {x}{2}\right )+8}}{a^{3}}\) | \(64\) |
| parallelrisch | \(\frac {15 \tan \left (\frac {x}{2}\right )^{5} x +\left (75 x +30\right ) \tan \left (\frac {x}{2}\right )^{4}+\left (150 x +150\right ) \tan \left (\frac {x}{2}\right )^{3}+\left (150 x +290\right ) \tan \left (\frac {x}{2}\right )^{2}+\left (75 x +190\right ) \tan \left (\frac {x}{2}\right )+15 x +44}{15 a^{3} \left (\tan \left (\frac {x}{2}\right )+1\right )^{5}}\) | \(74\) |
| norman | \(\frac {\frac {x}{a}+\frac {x \tan \left (\frac {x}{2}\right )^{11}}{a}+\frac {2 \tan \left (\frac {x}{2}\right )^{10}}{a}+\frac {10 \tan \left (\frac {x}{2}\right )^{9}}{a}+\frac {48 \tan \left (\frac {x}{2}\right )^{3}}{a}+\frac {44}{15 a}+\frac {5 x \tan \left (\frac {x}{2}\right )}{a}+\frac {13 x \tan \left (\frac {x}{2}\right )^{2}}{a}+\frac {25 x \tan \left (\frac {x}{2}\right )^{3}}{a}+\frac {38 x \tan \left (\frac {x}{2}\right )^{4}}{a}+\frac {46 x \tan \left (\frac {x}{2}\right )^{5}}{a}+\frac {46 x \tan \left (\frac {x}{2}\right )^{6}}{a}+\frac {38 x \tan \left (\frac {x}{2}\right )^{7}}{a}+\frac {25 x \tan \left (\frac {x}{2}\right )^{8}}{a}+\frac {13 x \tan \left (\frac {x}{2}\right )^{9}}{a}+\frac {5 x \tan \left (\frac {x}{2}\right )^{10}}{a}+\frac {38 \tan \left (\frac {x}{2}\right )}{3 a}+\frac {68 \tan \left (\frac {x}{2}\right )^{5}}{a}+\frac {128 \tan \left (\frac {x}{2}\right )^{7}}{3 a}+\frac {76 \tan \left (\frac {x}{2}\right )^{8}}{3 a}+\frac {344 \tan \left (\frac {x}{2}\right )^{4}}{5 a}+\frac {422 \tan \left (\frac {x}{2}\right )^{2}}{15 a}+\frac {1004 \tan \left (\frac {x}{2}\right )^{6}}{15 a}}{\left (\tan \left (\frac {x}{2}\right )^{2}+1\right )^{3} a^{2} \left (\tan \left (\frac {x}{2}\right )+1\right )^{5}}\) | \(271\) |
Input:
int(sin(x)^3/(a+a*sin(x))^3,x,method=_RETURNVERBOSE)
Output:
x/a^3+2/15*(135*I*exp(3*I*x)+45*exp(4*I*x)-115*I*exp(I*x)-185*exp(2*I*x)+3 2)/a^3/(exp(I*x)+I)^5
Leaf count of result is larger than twice the leaf count of optimal. 119 vs. \(2 (53) = 106\).
Time = 0.10 (sec) , antiderivative size = 119, normalized size of antiderivative = 2.02 \[ \int \frac {\sin ^3(x)}{(a+a \sin (x))^3} \, dx=\frac {{\left (15 \, x + 32\right )} \cos \left (x\right )^{3} + {\left (45 \, x - 19\right )} \cos \left (x\right )^{2} - 6 \, {\left (5 \, x + 9\right )} \cos \left (x\right ) + {\left ({\left (15 \, x - 32\right )} \cos \left (x\right )^{2} - 3 \, {\left (10 \, x + 17\right )} \cos \left (x\right ) - 60 \, x + 3\right )} \sin \left (x\right ) - 60 \, x - 3}{15 \, {\left (a^{3} \cos \left (x\right )^{3} + 3 \, a^{3} \cos \left (x\right )^{2} - 2 \, a^{3} \cos \left (x\right ) - 4 \, a^{3} + {\left (a^{3} \cos \left (x\right )^{2} - 2 \, a^{3} \cos \left (x\right ) - 4 \, a^{3}\right )} \sin \left (x\right )\right )}} \] Input:
integrate(sin(x)^3/(a+a*sin(x))^3,x, algorithm="fricas")
Output:
1/15*((15*x + 32)*cos(x)^3 + (45*x - 19)*cos(x)^2 - 6*(5*x + 9)*cos(x) + ( (15*x - 32)*cos(x)^2 - 3*(10*x + 17)*cos(x) - 60*x + 3)*sin(x) - 60*x - 3) /(a^3*cos(x)^3 + 3*a^3*cos(x)^2 - 2*a^3*cos(x) - 4*a^3 + (a^3*cos(x)^2 - 2 *a^3*cos(x) - 4*a^3)*sin(x))
Leaf count of result is larger than twice the leaf count of optimal. 777 vs. \(2 (58) = 116\).
Time = 3.99 (sec) , antiderivative size = 777, normalized size of antiderivative = 13.17 \[ \int \frac {\sin ^3(x)}{(a+a \sin (x))^3} \, dx=\text {Too large to display} \] Input:
integrate(sin(x)**3/(a+a*sin(x))**3,x)
Output:
15*x*tan(x/2)**5/(15*a**3*tan(x/2)**5 + 75*a**3*tan(x/2)**4 + 150*a**3*tan (x/2)**3 + 150*a**3*tan(x/2)**2 + 75*a**3*tan(x/2) + 15*a**3) + 75*x*tan(x /2)**4/(15*a**3*tan(x/2)**5 + 75*a**3*tan(x/2)**4 + 150*a**3*tan(x/2)**3 + 150*a**3*tan(x/2)**2 + 75*a**3*tan(x/2) + 15*a**3) + 150*x*tan(x/2)**3/(1 5*a**3*tan(x/2)**5 + 75*a**3*tan(x/2)**4 + 150*a**3*tan(x/2)**3 + 150*a**3 *tan(x/2)**2 + 75*a**3*tan(x/2) + 15*a**3) + 150*x*tan(x/2)**2/(15*a**3*ta n(x/2)**5 + 75*a**3*tan(x/2)**4 + 150*a**3*tan(x/2)**3 + 150*a**3*tan(x/2) **2 + 75*a**3*tan(x/2) + 15*a**3) + 75*x*tan(x/2)/(15*a**3*tan(x/2)**5 + 7 5*a**3*tan(x/2)**4 + 150*a**3*tan(x/2)**3 + 150*a**3*tan(x/2)**2 + 75*a**3 *tan(x/2) + 15*a**3) + 15*x/(15*a**3*tan(x/2)**5 + 75*a**3*tan(x/2)**4 + 1 50*a**3*tan(x/2)**3 + 150*a**3*tan(x/2)**2 + 75*a**3*tan(x/2) + 15*a**3) + 30*tan(x/2)**4/(15*a**3*tan(x/2)**5 + 75*a**3*tan(x/2)**4 + 150*a**3*tan( x/2)**3 + 150*a**3*tan(x/2)**2 + 75*a**3*tan(x/2) + 15*a**3) + 150*tan(x/2 )**3/(15*a**3*tan(x/2)**5 + 75*a**3*tan(x/2)**4 + 150*a**3*tan(x/2)**3 + 1 50*a**3*tan(x/2)**2 + 75*a**3*tan(x/2) + 15*a**3) + 290*tan(x/2)**2/(15*a* *3*tan(x/2)**5 + 75*a**3*tan(x/2)**4 + 150*a**3*tan(x/2)**3 + 150*a**3*tan (x/2)**2 + 75*a**3*tan(x/2) + 15*a**3) + 190*tan(x/2)/(15*a**3*tan(x/2)**5 + 75*a**3*tan(x/2)**4 + 150*a**3*tan(x/2)**3 + 150*a**3*tan(x/2)**2 + 75* a**3*tan(x/2) + 15*a**3) + 44/(15*a**3*tan(x/2)**5 + 75*a**3*tan(x/2)**4 + 150*a**3*tan(x/2)**3 + 150*a**3*tan(x/2)**2 + 75*a**3*tan(x/2) + 15*a*...
Leaf count of result is larger than twice the leaf count of optimal. 144 vs. \(2 (53) = 106\).
Time = 0.11 (sec) , antiderivative size = 144, normalized size of antiderivative = 2.44 \[ \int \frac {\sin ^3(x)}{(a+a \sin (x))^3} \, dx=\frac {2 \, {\left (\frac {95 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {145 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {75 \, \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac {15 \, \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + 22\right )}}{15 \, {\left (a^{3} + \frac {5 \, a^{3} \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {10 \, a^{3} \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {10 \, a^{3} \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac {5 \, a^{3} \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac {a^{3} \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}}\right )}} + \frac {2 \, \arctan \left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}\right )}{a^{3}} \] Input:
integrate(sin(x)^3/(a+a*sin(x))^3,x, algorithm="maxima")
Output:
2/15*(95*sin(x)/(cos(x) + 1) + 145*sin(x)^2/(cos(x) + 1)^2 + 75*sin(x)^3/( cos(x) + 1)^3 + 15*sin(x)^4/(cos(x) + 1)^4 + 22)/(a^3 + 5*a^3*sin(x)/(cos( x) + 1) + 10*a^3*sin(x)^2/(cos(x) + 1)^2 + 10*a^3*sin(x)^3/(cos(x) + 1)^3 + 5*a^3*sin(x)^4/(cos(x) + 1)^4 + a^3*sin(x)^5/(cos(x) + 1)^5) + 2*arctan( sin(x)/(cos(x) + 1))/a^3
Time = 0.12 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.86 \[ \int \frac {\sin ^3(x)}{(a+a \sin (x))^3} \, dx=\frac {x}{a^{3}} + \frac {2 \, {\left (15 \, \tan \left (\frac {1}{2} \, x\right )^{4} + 75 \, \tan \left (\frac {1}{2} \, x\right )^{3} + 145 \, \tan \left (\frac {1}{2} \, x\right )^{2} + 95 \, \tan \left (\frac {1}{2} \, x\right ) + 22\right )}}{15 \, a^{3} {\left (\tan \left (\frac {1}{2} \, x\right ) + 1\right )}^{5}} \] Input:
integrate(sin(x)^3/(a+a*sin(x))^3,x, algorithm="giac")
Output:
x/a^3 + 2/15*(15*tan(1/2*x)^4 + 75*tan(1/2*x)^3 + 145*tan(1/2*x)^2 + 95*ta n(1/2*x) + 22)/(a^3*(tan(1/2*x) + 1)^5)
Time = 17.25 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.85 \[ \int \frac {\sin ^3(x)}{(a+a \sin (x))^3} \, dx=\frac {x}{a^3}+\frac {2\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4+10\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3+\frac {58\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{3}+\frac {38\,\mathrm {tan}\left (\frac {x}{2}\right )}{3}+\frac {44}{15}}{a^3\,{\left (\mathrm {tan}\left (\frac {x}{2}\right )+1\right )}^5} \] Input:
int(sin(x)^3/(a + a*sin(x))^3,x)
Output:
x/a^3 + ((38*tan(x/2))/3 + (58*tan(x/2)^2)/3 + 10*tan(x/2)^3 + 2*tan(x/2)^ 4 + 44/15)/(a^3*(tan(x/2) + 1)^5)
Time = 0.21 (sec) , antiderivative size = 119, normalized size of antiderivative = 2.02 \[ \int \frac {\sin ^3(x)}{(a+a \sin (x))^3} \, dx=\frac {15 \cos \left (x \right ) \sin \left (x \right )^{2} x -16 \cos \left (x \right ) \sin \left (x \right )^{2}+30 \cos \left (x \right ) \sin \left (x \right ) x -19 \cos \left (x \right ) \sin \left (x \right )+15 \cos \left (x \right ) x -6 \cos \left (x \right )-15 \sin \left (x \right )^{3} x -48 \sin \left (x \right )^{3}-45 \sin \left (x \right )^{2} x -67 \sin \left (x \right )^{2}-45 \sin \left (x \right ) x -19 \sin \left (x \right )-15 x +6}{15 a^{3} \left (\cos \left (x \right ) \sin \left (x \right )^{2}+2 \cos \left (x \right ) \sin \left (x \right )+\cos \left (x \right )-\sin \left (x \right )^{3}-3 \sin \left (x \right )^{2}-3 \sin \left (x \right )-1\right )} \] Input:
int(sin(x)^3/(a+a*sin(x))^3,x)
Output:
(15*cos(x)*sin(x)**2*x - 16*cos(x)*sin(x)**2 + 30*cos(x)*sin(x)*x - 19*cos (x)*sin(x) + 15*cos(x)*x - 6*cos(x) - 15*sin(x)**3*x - 48*sin(x)**3 - 45*s in(x)**2*x - 67*sin(x)**2 - 45*sin(x)*x - 19*sin(x) - 15*x + 6)/(15*a**3*( cos(x)*sin(x)**2 + 2*cos(x)*sin(x) + cos(x) - sin(x)**3 - 3*sin(x)**2 - 3* sin(x) - 1))