Integrand size = 21, antiderivative size = 129 \[ \int \sin ^3(e+f x) (a+a \sin (e+f x))^3 \, dx=\frac {23 a^3 x}{16}-\frac {4 a^3 \cos (e+f x)}{f}+\frac {7 a^3 \cos ^3(e+f x)}{3 f}-\frac {3 a^3 \cos ^5(e+f x)}{5 f}-\frac {23 a^3 \cos (e+f x) \sin (e+f x)}{16 f}-\frac {23 a^3 \cos (e+f x) \sin ^3(e+f x)}{24 f}-\frac {a^3 \cos (e+f x) \sin ^5(e+f x)}{6 f} \] Output:
23/16*a^3*x-4*a^3*cos(f*x+e)/f+7/3*a^3*cos(f*x+e)^3/f-3/5*a^3*cos(f*x+e)^5 /f-23/16*a^3*cos(f*x+e)*sin(f*x+e)/f-23/24*a^3*cos(f*x+e)*sin(f*x+e)^3/f-1 /6*a^3*cos(f*x+e)*sin(f*x+e)^5/f
Time = 0.41 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.89 \[ \int \sin ^3(e+f x) (a+a \sin (e+f x))^3 \, dx=-\frac {a^3 \cos (e+f x) \left (690 \arcsin \left (\frac {\sqrt {1-\sin (e+f x)}}{\sqrt {2}}\right )+\sqrt {\cos ^2(e+f x)} \left (544+345 \sin (e+f x)+272 \sin ^2(e+f x)+230 \sin ^3(e+f x)+144 \sin ^4(e+f x)+40 \sin ^5(e+f x)\right )\right )}{240 f \sqrt {\cos ^2(e+f x)}} \] Input:
Integrate[Sin[e + f*x]^3*(a + a*Sin[e + f*x])^3,x]
Output:
-1/240*(a^3*Cos[e + f*x]*(690*ArcSin[Sqrt[1 - Sin[e + f*x]]/Sqrt[2]] + Sqr t[Cos[e + f*x]^2]*(544 + 345*Sin[e + f*x] + 272*Sin[e + f*x]^2 + 230*Sin[e + f*x]^3 + 144*Sin[e + f*x]^4 + 40*Sin[e + f*x]^5)))/(f*Sqrt[Cos[e + f*x] ^2])
Time = 0.35 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3042, 3236, 2009}
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.
| \(\displaystyle \int \sin ^3(e+f x) (a \sin (e+f x)+a)^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin (e+f x)^3 (a \sin (e+f x)+a)^3dx\) |
| \(\Big \downarrow \) 3236 |
| \(\displaystyle \int \left (a^3 \sin ^6(e+f x)+3 a^3 \sin ^5(e+f x)+3 a^3 \sin ^4(e+f x)+a^3 \sin ^3(e+f x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3 a^3 \cos ^5(e+f x)}{5 f}+\frac {7 a^3 \cos ^3(e+f x)}{3 f}-\frac {4 a^3 \cos (e+f x)}{f}-\frac {a^3 \sin ^5(e+f x) \cos (e+f x)}{6 f}-\frac {23 a^3 \sin ^3(e+f x) \cos (e+f x)}{24 f}-\frac {23 a^3 \sin (e+f x) \cos (e+f x)}{16 f}+\frac {23 a^3 x}{16}\) |
Input:
Int[Sin[e + f*x]^3*(a + a*Sin[e + f*x])^3,x]
Output:
(23*a^3*x)/16 - (4*a^3*Cos[e + f*x])/f + (7*a^3*Cos[e + f*x]^3)/(3*f) - (3 *a^3*Cos[e + f*x]^5)/(5*f) - (23*a^3*Cos[e + f*x]*Sin[e + f*x])/(16*f) - ( 23*a^3*Cos[e + f*x]*Sin[e + f*x]^3)/(24*f) - (a^3*Cos[e + f*x]*Sin[e + f*x ]^5)/(6*f)
Int[((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*( x_)])^(m_.), x_Symbol] :> Int[ExpandTrig[(a + b*sin[e + f*x])^m*(d*sin[e + f*x])^n, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && EqQ[a^2 - b^2, 0] && IGt Q[m, 0] && RationalQ[n]
Time = 43.77 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.59
| method | result | size |
| parallelrisch | \(-\frac {\left (-276 f x +\sin \left (6 f x +6 e \right )+504 \cos \left (f x +e \right )-76 \cos \left (3 f x +3 e \right )+\frac {36 \cos \left (5 f x +5 e \right )}{5}+189 \sin \left (2 f x +2 e \right )-27 \sin \left (4 f x +4 e \right )+\frac {2176}{5}\right ) a^{3}}{192 f}\) | \(76\) |
| risch | \(\frac {23 a^{3} x}{16}-\frac {21 a^{3} \cos \left (f x +e \right )}{8 f}-\frac {a^{3} \sin \left (6 f x +6 e \right )}{192 f}-\frac {3 a^{3} \cos \left (5 f x +5 e \right )}{80 f}+\frac {9 a^{3} \sin \left (4 f x +4 e \right )}{64 f}+\frac {19 a^{3} \cos \left (3 f x +3 e \right )}{48 f}-\frac {63 a^{3} \sin \left (2 f x +2 e \right )}{64 f}\) | \(107\) |
| derivativedivides | \(\frac {a^{3} \left (-\frac {\left (\sin \left (f x +e \right )^{5}+\frac {5 \sin \left (f x +e \right )^{3}}{4}+\frac {15 \sin \left (f x +e \right )}{8}\right ) \cos \left (f x +e \right )}{6}+\frac {5 f x}{16}+\frac {5 e}{16}\right )-\frac {3 a^{3} \left (\frac {8}{3}+\sin \left (f x +e \right )^{4}+\frac {4 \sin \left (f x +e \right )^{2}}{3}\right ) \cos \left (f x +e \right )}{5}+3 a^{3} \left (-\frac {\left (\sin \left (f x +e \right )^{3}+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )-\frac {a^{3} \left (2+\sin \left (f x +e \right )^{2}\right ) \cos \left (f x +e \right )}{3}}{f}\) | \(143\) |
| default | \(\frac {a^{3} \left (-\frac {\left (\sin \left (f x +e \right )^{5}+\frac {5 \sin \left (f x +e \right )^{3}}{4}+\frac {15 \sin \left (f x +e \right )}{8}\right ) \cos \left (f x +e \right )}{6}+\frac {5 f x}{16}+\frac {5 e}{16}\right )-\frac {3 a^{3} \left (\frac {8}{3}+\sin \left (f x +e \right )^{4}+\frac {4 \sin \left (f x +e \right )^{2}}{3}\right ) \cos \left (f x +e \right )}{5}+3 a^{3} \left (-\frac {\left (\sin \left (f x +e \right )^{3}+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )-\frac {a^{3} \left (2+\sin \left (f x +e \right )^{2}\right ) \cos \left (f x +e \right )}{3}}{f}\) | \(143\) |
| parts | \(-\frac {a^{3} \left (2+\sin \left (f x +e \right )^{2}\right ) \cos \left (f x +e \right )}{3 f}+\frac {a^{3} \left (-\frac {\left (\sin \left (f x +e \right )^{5}+\frac {5 \sin \left (f x +e \right )^{3}}{4}+\frac {15 \sin \left (f x +e \right )}{8}\right ) \cos \left (f x +e \right )}{6}+\frac {5 f x}{16}+\frac {5 e}{16}\right )}{f}+\frac {3 a^{3} \left (-\frac {\left (\sin \left (f x +e \right )^{3}+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )}{f}-\frac {3 a^{3} \left (\frac {8}{3}+\sin \left (f x +e \right )^{4}+\frac {4 \sin \left (f x +e \right )^{2}}{3}\right ) \cos \left (f x +e \right )}{5 f}\) | \(151\) |
| norman | \(\frac {\frac {23 a^{3} x}{16}-\frac {68 a^{3}}{15 f}-\frac {23 a^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{8 f}-\frac {391 a^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{24 f}-\frac {75 a^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{4 f}+\frac {75 a^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}{4 f}+\frac {391 a^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}{24 f}+\frac {23 a^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{11}}{8 f}+\frac {69 a^{3} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{8}+\frac {345 a^{3} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}}{16}+\frac {115 a^{3} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}}{4}+\frac {345 a^{3} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}}{16}+\frac {69 a^{3} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{10}}{8}+\frac {23 a^{3} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{12}}{16}-\frac {4 a^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}}{f}-\frac {136 a^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}}{3 f}-\frac {64 a^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}}{f}-\frac {136 a^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{5 f}}{\left (1+\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}\right )^{6}}\) | \(322\) |
| orering | \(\text {Expression too large to display}\) | \(3179\) |
Input:
int(sin(f*x+e)^3*(a+sin(f*x+e)*a)^3,x,method=_RETURNVERBOSE)
Output:
-1/192*(-276*f*x+sin(6*f*x+6*e)+504*cos(f*x+e)-76*cos(3*f*x+3*e)+36/5*cos( 5*f*x+5*e)+189*sin(2*f*x+2*e)-27*sin(4*f*x+4*e)+2176/5)*a^3/f
Time = 0.09 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.74 \[ \int \sin ^3(e+f x) (a+a \sin (e+f x))^3 \, dx=-\frac {144 \, a^{3} \cos \left (f x + e\right )^{5} - 560 \, a^{3} \cos \left (f x + e\right )^{3} - 345 \, a^{3} f x + 960 \, a^{3} \cos \left (f x + e\right ) + 5 \, {\left (8 \, a^{3} \cos \left (f x + e\right )^{5} - 62 \, a^{3} \cos \left (f x + e\right )^{3} + 123 \, a^{3} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{240 \, f} \] Input:
integrate(sin(f*x+e)^3*(a+a*sin(f*x+e))^3,x, algorithm="fricas")
Output:
-1/240*(144*a^3*cos(f*x + e)^5 - 560*a^3*cos(f*x + e)^3 - 345*a^3*f*x + 96 0*a^3*cos(f*x + e) + 5*(8*a^3*cos(f*x + e)^5 - 62*a^3*cos(f*x + e)^3 + 123 *a^3*cos(f*x + e))*sin(f*x + e))/f
Leaf count of result is larger than twice the leaf count of optimal. 379 vs. \(2 (122) = 244\).
Time = 0.37 (sec) , antiderivative size = 379, normalized size of antiderivative = 2.94 \[ \int \sin ^3(e+f x) (a+a \sin (e+f x))^3 \, dx=\begin {cases} \frac {5 a^{3} x \sin ^{6}{\left (e + f x \right )}}{16} + \frac {15 a^{3} x \sin ^{4}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{16} + \frac {9 a^{3} x \sin ^{4}{\left (e + f x \right )}}{8} + \frac {15 a^{3} x \sin ^{2}{\left (e + f x \right )} \cos ^{4}{\left (e + f x \right )}}{16} + \frac {9 a^{3} x \sin ^{2}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{4} + \frac {5 a^{3} x \cos ^{6}{\left (e + f x \right )}}{16} + \frac {9 a^{3} x \cos ^{4}{\left (e + f x \right )}}{8} - \frac {11 a^{3} \sin ^{5}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{16 f} - \frac {3 a^{3} \sin ^{4}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {5 a^{3} \sin ^{3}{\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{6 f} - \frac {15 a^{3} \sin ^{3}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{8 f} - \frac {4 a^{3} \sin ^{2}{\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{f} - \frac {a^{3} \sin ^{2}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {5 a^{3} \sin {\left (e + f x \right )} \cos ^{5}{\left (e + f x \right )}}{16 f} - \frac {9 a^{3} \sin {\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{8 f} - \frac {8 a^{3} \cos ^{5}{\left (e + f x \right )}}{5 f} - \frac {2 a^{3} \cos ^{3}{\left (e + f x \right )}}{3 f} & \text {for}\: f \neq 0 \\x \left (a \sin {\left (e \right )} + a\right )^{3} \sin ^{3}{\left (e \right )} & \text {otherwise} \end {cases} \] Input:
integrate(sin(f*x+e)**3*(a+a*sin(f*x+e))**3,x)
Output:
Piecewise((5*a**3*x*sin(e + f*x)**6/16 + 15*a**3*x*sin(e + f*x)**4*cos(e + f*x)**2/16 + 9*a**3*x*sin(e + f*x)**4/8 + 15*a**3*x*sin(e + f*x)**2*cos(e + f*x)**4/16 + 9*a**3*x*sin(e + f*x)**2*cos(e + f*x)**2/4 + 5*a**3*x*cos( e + f*x)**6/16 + 9*a**3*x*cos(e + f*x)**4/8 - 11*a**3*sin(e + f*x)**5*cos( e + f*x)/(16*f) - 3*a**3*sin(e + f*x)**4*cos(e + f*x)/f - 5*a**3*sin(e + f *x)**3*cos(e + f*x)**3/(6*f) - 15*a**3*sin(e + f*x)**3*cos(e + f*x)/(8*f) - 4*a**3*sin(e + f*x)**2*cos(e + f*x)**3/f - a**3*sin(e + f*x)**2*cos(e + f*x)/f - 5*a**3*sin(e + f*x)*cos(e + f*x)**5/(16*f) - 9*a**3*sin(e + f*x)* cos(e + f*x)**3/(8*f) - 8*a**3*cos(e + f*x)**5/(5*f) - 2*a**3*cos(e + f*x) **3/(3*f), Ne(f, 0)), (x*(a*sin(e) + a)**3*sin(e)**3, True))
Time = 0.04 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.11 \[ \int \sin ^3(e+f x) (a+a \sin (e+f x))^3 \, dx=-\frac {192 \, {\left (3 \, \cos \left (f x + e\right )^{5} - 10 \, \cos \left (f x + e\right )^{3} + 15 \, \cos \left (f x + e\right )\right )} a^{3} - 320 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} a^{3} - 5 \, {\left (4 \, \sin \left (2 \, f x + 2 \, e\right )^{3} + 60 \, f x + 60 \, e + 9 \, \sin \left (4 \, f x + 4 \, e\right ) - 48 \, \sin \left (2 \, f x + 2 \, e\right )\right )} a^{3} - 90 \, {\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} a^{3}}{960 \, f} \] Input:
integrate(sin(f*x+e)^3*(a+a*sin(f*x+e))^3,x, algorithm="maxima")
Output:
-1/960*(192*(3*cos(f*x + e)^5 - 10*cos(f*x + e)^3 + 15*cos(f*x + e))*a^3 - 320*(cos(f*x + e)^3 - 3*cos(f*x + e))*a^3 - 5*(4*sin(2*f*x + 2*e)^3 + 60* f*x + 60*e + 9*sin(4*f*x + 4*e) - 48*sin(2*f*x + 2*e))*a^3 - 90*(12*f*x + 12*e + sin(4*f*x + 4*e) - 8*sin(2*f*x + 2*e))*a^3)/f
Time = 0.13 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.82 \[ \int \sin ^3(e+f x) (a+a \sin (e+f x))^3 \, dx=\frac {23}{16} \, a^{3} x - \frac {3 \, a^{3} \cos \left (5 \, f x + 5 \, e\right )}{80 \, f} + \frac {19 \, a^{3} \cos \left (3 \, f x + 3 \, e\right )}{48 \, f} - \frac {21 \, a^{3} \cos \left (f x + e\right )}{8 \, f} - \frac {a^{3} \sin \left (6 \, f x + 6 \, e\right )}{192 \, f} + \frac {9 \, a^{3} \sin \left (4 \, f x + 4 \, e\right )}{64 \, f} - \frac {63 \, a^{3} \sin \left (2 \, f x + 2 \, e\right )}{64 \, f} \] Input:
integrate(sin(f*x+e)^3*(a+a*sin(f*x+e))^3,x, algorithm="giac")
Output:
23/16*a^3*x - 3/80*a^3*cos(5*f*x + 5*e)/f + 19/48*a^3*cos(3*f*x + 3*e)/f - 21/8*a^3*cos(f*x + e)/f - 1/192*a^3*sin(6*f*x + 6*e)/f + 9/64*a^3*sin(4*f *x + 4*e)/f - 63/64*a^3*sin(2*f*x + 2*e)/f
Time = 20.14 (sec) , antiderivative size = 294, normalized size of antiderivative = 2.28 \[ \int \sin ^3(e+f x) (a+a \sin (e+f x))^3 \, dx=\frac {23\,a^3\,x}{16}-\frac {\frac {23\,a^3\,\left (e+f\,x\right )}{16}+\frac {391\,a^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3}{24}+\frac {75\,a^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5}{4}-\frac {75\,a^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7}{4}-\frac {391\,a^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9}{24}-\frac {23\,a^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{11}}{8}-\frac {a^3\,\left (345\,e+345\,f\,x-1088\right )}{240}+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (\frac {69\,a^3\,\left (e+f\,x\right )}{8}-\frac {a^3\,\left (2070\,e+2070\,f\,x-6528\right )}{240}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8\,\left (\frac {345\,a^3\,\left (e+f\,x\right )}{16}-\frac {a^3\,\left (5175\,e+5175\,f\,x-960\right )}{240}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\left (\frac {115\,a^3\,\left (e+f\,x\right )}{4}-\frac {a^3\,\left (6900\,e+6900\,f\,x-10880\right )}{240}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (\frac {345\,a^3\,\left (e+f\,x\right )}{16}-\frac {a^3\,\left (5175\,e+5175\,f\,x-15360\right )}{240}\right )+\frac {23\,a^3\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{8}}{f\,{\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1\right )}^6} \] Input:
int(sin(e + f*x)^3*(a + a*sin(e + f*x))^3,x)
Output:
(23*a^3*x)/16 - ((23*a^3*(e + f*x))/16 + (391*a^3*tan(e/2 + (f*x)/2)^3)/24 + (75*a^3*tan(e/2 + (f*x)/2)^5)/4 - (75*a^3*tan(e/2 + (f*x)/2)^7)/4 - (39 1*a^3*tan(e/2 + (f*x)/2)^9)/24 - (23*a^3*tan(e/2 + (f*x)/2)^11)/8 - (a^3*( 345*e + 345*f*x - 1088))/240 + tan(e/2 + (f*x)/2)^2*((69*a^3*(e + f*x))/8 - (a^3*(2070*e + 2070*f*x - 6528))/240) + tan(e/2 + (f*x)/2)^8*((345*a^3*( e + f*x))/16 - (a^3*(5175*e + 5175*f*x - 960))/240) + tan(e/2 + (f*x)/2)^6 *((115*a^3*(e + f*x))/4 - (a^3*(6900*e + 6900*f*x - 10880))/240) + tan(e/2 + (f*x)/2)^4*((345*a^3*(e + f*x))/16 - (a^3*(5175*e + 5175*f*x - 15360))/ 240) + (23*a^3*tan(e/2 + (f*x)/2))/8)/(f*(tan(e/2 + (f*x)/2)^2 + 1)^6)
Time = 0.17 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.78 \[ \int \sin ^3(e+f x) (a+a \sin (e+f x))^3 \, dx=\frac {a^{3} \left (-40 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{5}-144 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{4}-230 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{3}-272 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{2}-345 \cos \left (f x +e \right ) \sin \left (f x +e \right )-544 \cos \left (f x +e \right )+345 f x +544\right )}{240 f} \] Input:
int(sin(f*x+e)^3*(a+a*sin(f*x+e))^3,x)
Output:
(a**3*( - 40*cos(e + f*x)*sin(e + f*x)**5 - 144*cos(e + f*x)*sin(e + f*x)* *4 - 230*cos(e + f*x)*sin(e + f*x)**3 - 272*cos(e + f*x)*sin(e + f*x)**2 - 345*cos(e + f*x)*sin(e + f*x) - 544*cos(e + f*x) + 345*f*x + 544))/(240*f )