Integrand size = 27, antiderivative size = 70 \[ \int \frac {\cos ^4(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {x}{a^2}-\frac {2 \cos ^3(c+d x)}{3 a^2 d}-\frac {\cos (c+d x) \sin (c+d x)}{a^2 d}-\frac {\cos ^5(c+d x)}{d (a+a \sin (c+d x))^2} \]
Leaf count is larger than twice the leaf count of optimal. \(204\) vs. \(2(70)=140\).
Time = 0.49 (sec) , antiderivative size = 204, normalized size of antiderivative = 2.91 \[ \int \frac {\cos ^4(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {-2 (1+12 d x) \cos \left (\frac {c}{2}\right )-21 \cos \left (\frac {c}{2}+d x\right )-21 \cos \left (\frac {3 c}{2}+d x\right )+6 \cos \left (\frac {3 c}{2}+2 d x\right )-6 \cos \left (\frac {5 c}{2}+2 d x\right )+\cos \left (\frac {5 c}{2}+3 d x\right )+\cos \left (\frac {7 c}{2}+3 d x\right )+2 \sin \left (\frac {c}{2}\right )-24 d x \sin \left (\frac {c}{2}\right )+21 \sin \left (\frac {c}{2}+d x\right )-21 \sin \left (\frac {3 c}{2}+d x\right )+6 \sin \left (\frac {3 c}{2}+2 d x\right )+6 \sin \left (\frac {5 c}{2}+2 d x\right )-\sin \left (\frac {5 c}{2}+3 d x\right )+\sin \left (\frac {7 c}{2}+3 d x\right )}{24 a^2 d \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right )} \]
(-2*(1 + 12*d*x)*Cos[c/2] - 21*Cos[c/2 + d*x] - 21*Cos[(3*c)/2 + d*x] + 6* Cos[(3*c)/2 + 2*d*x] - 6*Cos[(5*c)/2 + 2*d*x] + Cos[(5*c)/2 + 3*d*x] + Cos [(7*c)/2 + 3*d*x] + 2*Sin[c/2] - 24*d*x*Sin[c/2] + 21*Sin[c/2 + d*x] - 21* Sin[(3*c)/2 + d*x] + 6*Sin[(3*c)/2 + 2*d*x] + 6*Sin[(5*c)/2 + 2*d*x] - Sin [(5*c)/2 + 3*d*x] + Sin[(7*c)/2 + 3*d*x])/(24*a^2*d*(Cos[c/2] + Sin[c/2]))
Time = 0.40 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.13, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {3042, 3338, 3042, 3161, 3042, 3115, 24}
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.
| \(\displaystyle \int \frac {\sin (c+d x) \cos ^4(c+d x)}{(a \sin (c+d x)+a)^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (c+d x) \cos (c+d x)^4}{(a \sin (c+d x)+a)^2}dx\) |
| \(\Big \downarrow \) 3338 |
| \(\displaystyle -\frac {2 \int \frac {\cos ^4(c+d x)}{\sin (c+d x) a+a}dx}{a}-\frac {\cos ^5(c+d x)}{d (a \sin (c+d x)+a)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {2 \int \frac {\cos (c+d x)^4}{\sin (c+d x) a+a}dx}{a}-\frac {\cos ^5(c+d x)}{d (a \sin (c+d x)+a)^2}\) |
| \(\Big \downarrow \) 3161 |
| \(\displaystyle -\frac {2 \left (\frac {\int \cos ^2(c+d x)dx}{a}+\frac {\cos ^3(c+d x)}{3 a d}\right )}{a}-\frac {\cos ^5(c+d x)}{d (a \sin (c+d x)+a)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {2 \left (\frac {\int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx}{a}+\frac {\cos ^3(c+d x)}{3 a d}\right )}{a}-\frac {\cos ^5(c+d x)}{d (a \sin (c+d x)+a)^2}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle -\frac {2 \left (\frac {\frac {\int 1dx}{2}+\frac {\sin (c+d x) \cos (c+d x)}{2 d}}{a}+\frac {\cos ^3(c+d x)}{3 a d}\right )}{a}-\frac {\cos ^5(c+d x)}{d (a \sin (c+d x)+a)^2}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle -\frac {\cos ^5(c+d x)}{d (a \sin (c+d x)+a)^2}-\frac {2 \left (\frac {\cos ^3(c+d x)}{3 a d}+\frac {\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}}{a}\right )}{a}\) |
-(Cos[c + d*x]^5/(d*(a + a*Sin[c + d*x])^2)) - (2*(Cos[c + d*x]^3/(3*a*d) + (x/2 + (Cos[c + d*x]*Sin[c + d*x])/(2*d))/a))/a
3.5.24.3.1 Defintions of rubi rules used
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[g*((g*Cos[e + f*x])^(p - 1)/(b*f*(p - 1))), x] + Si mp[g^2/a Int[(g*Cos[e + f*x])^(p - 2), x], x] /; FreeQ[{a, b, e, f, g}, x ] && EqQ[a^2 - b^2, 0] && GtQ[p, 1] && IntegerQ[2*p]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*c - a*d)*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^m/(a*f*g*(2*m + p + 1) )), x] + Simp[(a*d*m + b*c*(m + p + 1))/(a*b*(2*m + p + 1)) Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && (LtQ[m, -1] || ILtQ[Simplify[m + p], 0 ]) && NeQ[2*m + p + 1, 0]
Time = 0.28 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.61
| method | result | size |
| parallelrisch | \(\frac {-12 d x +\cos \left (3 d x +3 c \right )-21 \cos \left (d x +c \right )+6 \sin \left (2 d x +2 c \right )-20}{12 d \,a^{2}}\) | \(43\) |
| risch | \(-\frac {x}{a^{2}}-\frac {7 \cos \left (d x +c \right )}{4 a^{2} d}+\frac {\cos \left (3 d x +3 c \right )}{12 d \,a^{2}}+\frac {\sin \left (2 d x +2 c \right )}{2 d \,a^{2}}\) | \(56\) |
| derivativedivides | \(\frac {\frac {4 \left (-\frac {\left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {\left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-2 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}-\frac {5}{6}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}-2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{2}}\) | \(90\) |
| default | \(\frac {\frac {4 \left (-\frac {\left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {\left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-2 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}-\frac {5}{6}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}-2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{2}}\) | \(90\) |
| norman | \(\frac {-\frac {35 x \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {16 x \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {40 x \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {8 x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {40 x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {25 x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {35 x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {25 x \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {16 x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {8 x \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {10}{3 a d}-\frac {3 x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {x \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {x}{a}-\frac {8 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {56 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}-\frac {8 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d a}-\frac {184 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}-\frac {3 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a}-\frac {32 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {112 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}-\frac {50 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {200 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}-\frac {72 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {154 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}-\frac {16 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {2 \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5} a \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}\) | \(493\) |
Time = 0.25 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.61 \[ \int \frac {\cos ^4(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {\cos \left (d x + c\right )^{3} - 3 \, d x + 3 \, \cos \left (d x + c\right ) \sin \left (d x + c\right ) - 6 \, \cos \left (d x + c\right )}{3 \, a^{2} d} \]
Leaf count of result is larger than twice the leaf count of optimal. 694 vs. \(2 (63) = 126\).
Time = 12.02 (sec) , antiderivative size = 694, normalized size of antiderivative = 9.91 \[ \int \frac {\cos ^4(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^2} \, dx=\begin {cases} - \frac {3 d x \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{3 a^{2} d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{2} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{2} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 3 a^{2} d} - \frac {9 d x \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{3 a^{2} d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{2} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{2} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 3 a^{2} d} - \frac {9 d x \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{3 a^{2} d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{2} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{2} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 3 a^{2} d} - \frac {3 d x}{3 a^{2} d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{2} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{2} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 3 a^{2} d} - \frac {6 \tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{3 a^{2} d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{2} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{2} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 3 a^{2} d} - \frac {6 \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{3 a^{2} d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{2} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{2} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 3 a^{2} d} - \frac {24 \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{3 a^{2} d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{2} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{2} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 3 a^{2} d} + \frac {6 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{3 a^{2} d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{2} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{2} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 3 a^{2} d} - \frac {10}{3 a^{2} d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{2} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{2} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 3 a^{2} d} & \text {for}\: d \neq 0 \\\frac {x \sin {\left (c \right )} \cos ^{4}{\left (c \right )}}{\left (a \sin {\left (c \right )} + a\right )^{2}} & \text {otherwise} \end {cases} \]
Piecewise((-3*d*x*tan(c/2 + d*x/2)**6/(3*a**2*d*tan(c/2 + d*x/2)**6 + 9*a* *2*d*tan(c/2 + d*x/2)**4 + 9*a**2*d*tan(c/2 + d*x/2)**2 + 3*a**2*d) - 9*d* x*tan(c/2 + d*x/2)**4/(3*a**2*d*tan(c/2 + d*x/2)**6 + 9*a**2*d*tan(c/2 + d *x/2)**4 + 9*a**2*d*tan(c/2 + d*x/2)**2 + 3*a**2*d) - 9*d*x*tan(c/2 + d*x/ 2)**2/(3*a**2*d*tan(c/2 + d*x/2)**6 + 9*a**2*d*tan(c/2 + d*x/2)**4 + 9*a** 2*d*tan(c/2 + d*x/2)**2 + 3*a**2*d) - 3*d*x/(3*a**2*d*tan(c/2 + d*x/2)**6 + 9*a**2*d*tan(c/2 + d*x/2)**4 + 9*a**2*d*tan(c/2 + d*x/2)**2 + 3*a**2*d) - 6*tan(c/2 + d*x/2)**5/(3*a**2*d*tan(c/2 + d*x/2)**6 + 9*a**2*d*tan(c/2 + d*x/2)**4 + 9*a**2*d*tan(c/2 + d*x/2)**2 + 3*a**2*d) - 6*tan(c/2 + d*x/2) **4/(3*a**2*d*tan(c/2 + d*x/2)**6 + 9*a**2*d*tan(c/2 + d*x/2)**4 + 9*a**2* d*tan(c/2 + d*x/2)**2 + 3*a**2*d) - 24*tan(c/2 + d*x/2)**2/(3*a**2*d*tan(c /2 + d*x/2)**6 + 9*a**2*d*tan(c/2 + d*x/2)**4 + 9*a**2*d*tan(c/2 + d*x/2)* *2 + 3*a**2*d) + 6*tan(c/2 + d*x/2)/(3*a**2*d*tan(c/2 + d*x/2)**6 + 9*a**2 *d*tan(c/2 + d*x/2)**4 + 9*a**2*d*tan(c/2 + d*x/2)**2 + 3*a**2*d) - 10/(3* a**2*d*tan(c/2 + d*x/2)**6 + 9*a**2*d*tan(c/2 + d*x/2)**4 + 9*a**2*d*tan(c /2 + d*x/2)**2 + 3*a**2*d), Ne(d, 0)), (x*sin(c)*cos(c)**4/(a*sin(c) + a)* *2, True))
Leaf count of result is larger than twice the leaf count of optimal. 184 vs. \(2 (68) = 136\).
Time = 0.31 (sec) , antiderivative size = 184, normalized size of antiderivative = 2.63 \[ \int \frac {\cos ^4(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {2 \, {\left (\frac {\frac {3 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {12 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {3 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {3 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - 5}{a^{2} + \frac {3 \, a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {3 \, a^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {a^{2} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}} - \frac {3 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}}\right )}}{3 \, d} \]
2/3*((3*sin(d*x + c)/(cos(d*x + c) + 1) - 12*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 3*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 3*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 5)/(a^2 + 3*a^2*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 3*a^2 *sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + a^2*sin(d*x + c)^6/(cos(d*x + c) + 1)^6) - 3*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a^2)/d
Time = 0.31 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.26 \[ \int \frac {\cos ^4(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {\frac {3 \, {\left (d x + c\right )}}{a^{2}} + \frac {2 \, {\left (3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 12 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 5\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3} a^{2}}}{3 \, d} \]
-1/3*(3*(d*x + c)/a^2 + 2*(3*tan(1/2*d*x + 1/2*c)^5 + 3*tan(1/2*d*x + 1/2* c)^4 + 12*tan(1/2*d*x + 1/2*c)^2 - 3*tan(1/2*d*x + 1/2*c) + 5)/((tan(1/2*d *x + 1/2*c)^2 + 1)^3*a^2))/d
Time = 9.76 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.79 \[ \int \frac {\cos ^4(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {\cos \left (3\,c+3\,d\,x\right )}{12\,a^2\,d}-\frac {7\,\cos \left (c+d\,x\right )}{4\,a^2\,d}-\frac {x}{a^2}+\frac {\sin \left (2\,c+2\,d\,x\right )}{2\,a^2\,d} \]