Integrand size = 18, antiderivative size = 123 \[ \int \frac {c+d x}{(a+a \cos (e+f x))^2} \, dx=\frac {2 d \log \left (\cos \left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{3 a^2 f^2}-\frac {d \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{6 a^2 f^2}+\frac {(c+d x) \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f}+\frac {(c+d x) \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right ) \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{6 a^2 f} \]
2/3*d*ln(cos(1/2*f*x+1/2*e))/a^2/f^2-1/6*d*sec(1/2*f*x+1/2*e)^2/a^2/f^2+1/ 3*(d*x+c)*tan(1/2*f*x+1/2*e)/a^2/f+1/6*(d*x+c)*sec(1/2*f*x+1/2*e)^2*tan(1/ 2*f*x+1/2*e)/a^2/f
Time = 1.13 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.92 \[ \int \frac {c+d x}{(a+a \cos (e+f x))^2} \, dx=\frac {\cos \left (\frac {1}{2} (e+f x)\right ) \left (2 d \cos \left (\frac {3}{2} (e+f x)\right ) \log \left (\cos \left (\frac {1}{2} (e+f x)\right )\right )+2 d \cos \left (\frac {1}{2} (e+f x)\right ) \left (-1+3 \log \left (\cos \left (\frac {1}{2} (e+f x)\right )\right )\right )+f (c+d x) \left (3 \sin \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {3}{2} (e+f x)\right )\right )\right )}{3 a^2 f^2 (1+\cos (e+f x))^2} \]
(Cos[(e + f*x)/2]*(2*d*Cos[(3*(e + f*x))/2]*Log[Cos[(e + f*x)/2]] + 2*d*Co s[(e + f*x)/2]*(-1 + 3*Log[Cos[(e + f*x)/2]]) + f*(c + d*x)*(3*Sin[(e + f* x)/2] + Sin[(3*(e + f*x))/2])))/(3*a^2*f^2*(1 + Cos[e + f*x])^2)
Time = 0.49 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.97, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3042, 3799, 3042, 4673, 3042, 4672, 25, 3042, 3956}
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.
| \(\displaystyle \int \frac {c+d x}{(a \cos (e+f x)+a)^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {c+d x}{\left (a \sin \left (e+f x+\frac {\pi }{2}\right )+a\right )^2}dx\) |
| \(\Big \downarrow \) 3799 |
| \(\displaystyle \frac {\int (c+d x) \sec ^4\left (\frac {e}{2}+\frac {f x}{2}\right )dx}{4 a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int (c+d x) \csc \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{2}\right )^4dx}{4 a^2}\) |
| \(\Big \downarrow \) 4673 |
| \(\displaystyle \frac {\frac {2}{3} \int (c+d x) \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )dx+\frac {2 (c+d x) \tan \left (\frac {e}{2}+\frac {f x}{2}\right ) \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 f}-\frac {2 d \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 f^2}}{4 a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {2}{3} \int (c+d x) \csc \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{2}\right )^2dx+\frac {2 (c+d x) \tan \left (\frac {e}{2}+\frac {f x}{2}\right ) \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 f}-\frac {2 d \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 f^2}}{4 a^2}\) |
| \(\Big \downarrow \) 4672 |
| \(\displaystyle \frac {\frac {2}{3} \left (\frac {2 d \int -\tan \left (\frac {e}{2}+\frac {f x}{2}\right )dx}{f}+\frac {2 (c+d x) \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{f}\right )+\frac {2 (c+d x) \tan \left (\frac {e}{2}+\frac {f x}{2}\right ) \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 f}-\frac {2 d \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 f^2}}{4 a^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {2}{3} \left (\frac {2 (c+d x) \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{f}-\frac {2 d \int \tan \left (\frac {e}{2}+\frac {f x}{2}\right )dx}{f}\right )+\frac {2 (c+d x) \tan \left (\frac {e}{2}+\frac {f x}{2}\right ) \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 f}-\frac {2 d \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 f^2}}{4 a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {2}{3} \left (\frac {2 (c+d x) \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{f}-\frac {2 d \int \tan \left (\frac {e}{2}+\frac {f x}{2}\right )dx}{f}\right )+\frac {2 (c+d x) \tan \left (\frac {e}{2}+\frac {f x}{2}\right ) \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 f}-\frac {2 d \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 f^2}}{4 a^2}\) |
| \(\Big \downarrow \) 3956 |
| \(\displaystyle \frac {\frac {2}{3} \left (\frac {2 (c+d x) \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{f}+\frac {4 d \log \left (\cos \left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{f^2}\right )+\frac {2 (c+d x) \tan \left (\frac {e}{2}+\frac {f x}{2}\right ) \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 f}-\frac {2 d \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 f^2}}{4 a^2}\) |
((-2*d*Sec[e/2 + (f*x)/2]^2)/(3*f^2) + (2*(c + d*x)*Sec[e/2 + (f*x)/2]^2*T an[e/2 + (f*x)/2])/(3*f) + (2*((4*d*Log[Cos[e/2 + (f*x)/2]])/f^2 + (2*(c + d*x)*Tan[e/2 + (f*x)/2])/f))/3)/(4*a^2)
3.2.35.3.1 Defintions of rubi rules used
Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_.) , x_Symbol] :> Simp[(2*a)^n Int[(c + d*x)^m*Sin[(1/2)*(e + Pi*(a/(2*b))) + f*(x/2)]^(2*n), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[a^2 - b^ 2, 0] && IntegerQ[n] && (GtQ[n, 0] || IGtQ[m, 0])
Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d *x], x]]/d, x] /; FreeQ[{c, d}, x]
Int[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp [(-(c + d*x)^m)*(Cot[e + f*x]/f), x] + Simp[d*(m/f) Int[(c + d*x)^(m - 1) *Cot[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(-b^2)*(c + d*x)*Cot[e + f*x]*((b*Csc[e + f*x])^(n - 2)/(f*(n - 1))), x] + (-Simp[b^2*d*((b*Csc[e + f*x])^(n - 2)/(f^2*(n - 1)*(n - 2))), x] + S imp[b^2*((n - 2)/(n - 1)) Int[(c + d*x)*(b*Csc[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && NeQ[n, 2]
Time = 1.68 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.60
| method | result | size |
| parallelrisch | \(\frac {-2 d \ln \left (\sec ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\tan \left (\frac {f x}{2}+\frac {e}{2}\right ) \left (f \left (d x +c \right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+3 \left (d x +c \right ) f \right )}{6 a^{2} f^{2}}\) | \(74\) |
| default | \(\frac {\frac {c \left (\frac {\left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{6}+\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2}\right )}{f}-\frac {d \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{6 f^{2}}+\frac {d x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 f}+\frac {d x \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{6 f}-\frac {d \ln \left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 f^{2}}}{a^{2}}\) | \(109\) |
| norman | \(\frac {\frac {c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 a f}+\frac {c \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{6 a f}-\frac {d \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{6 a \,f^{2}}+\frac {d x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 a f}+\frac {d x \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{6 a f}}{a}-\frac {d \ln \left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 a^{2} f^{2}}\) | \(128\) |
| risch | \(-\frac {2 i d x}{3 a^{2} f}-\frac {2 i d e}{3 a^{2} f^{2}}-\frac {2 \left (-3 i d f x \,{\mathrm e}^{i \left (f x +e \right )}-3 i c f \,{\mathrm e}^{i \left (f x +e \right )}-i d f x -i c f +{\mathrm e}^{2 i \left (f x +e \right )} d +d \,{\mathrm e}^{i \left (f x +e \right )}\right )}{3 f^{2} a^{2} \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{3}}+\frac {2 d \ln \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )}{3 a^{2} f^{2}}\) | \(129\) |
1/6*(-2*d*ln(sec(1/2*f*x+1/2*e)^2)+tan(1/2*f*x+1/2*e)*(f*(d*x+c)*tan(1/2*f *x+1/2*e)^2-d*tan(1/2*f*x+1/2*e)+3*(d*x+c)*f))/a^2/f^2
Time = 0.28 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.96 \[ \int \frac {c+d x}{(a+a \cos (e+f x))^2} \, dx=-\frac {d \cos \left (f x + e\right ) - {\left (d \cos \left (f x + e\right )^{2} + 2 \, d \cos \left (f x + e\right ) + d\right )} \log \left (\frac {1}{2} \, \cos \left (f x + e\right ) + \frac {1}{2}\right ) - {\left (2 \, d f x + 2 \, c f + {\left (d f x + c f\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right ) + d}{3 \, {\left (a^{2} f^{2} \cos \left (f x + e\right )^{2} + 2 \, a^{2} f^{2} \cos \left (f x + e\right ) + a^{2} f^{2}\right )}} \]
-1/3*(d*cos(f*x + e) - (d*cos(f*x + e)^2 + 2*d*cos(f*x + e) + d)*log(1/2*c os(f*x + e) + 1/2) - (2*d*f*x + 2*c*f + (d*f*x + c*f)*cos(f*x + e))*sin(f* x + e) + d)/(a^2*f^2*cos(f*x + e)^2 + 2*a^2*f^2*cos(f*x + e) + a^2*f^2)
Time = 0.45 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.19 \[ \int \frac {c+d x}{(a+a \cos (e+f x))^2} \, dx=\begin {cases} \frac {c \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{6 a^{2} f} + \frac {c \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )}}{2 a^{2} f} + \frac {d x \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{6 a^{2} f} + \frac {d x \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )}}{2 a^{2} f} - \frac {d \log {\left (\tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 1 \right )}}{3 a^{2} f^{2}} - \frac {d \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{6 a^{2} f^{2}} & \text {for}\: f \neq 0 \\\frac {c x + \frac {d x^{2}}{2}}{\left (a \cos {\left (e \right )} + a\right )^{2}} & \text {otherwise} \end {cases} \]
Piecewise((c*tan(e/2 + f*x/2)**3/(6*a**2*f) + c*tan(e/2 + f*x/2)/(2*a**2*f ) + d*x*tan(e/2 + f*x/2)**3/(6*a**2*f) + d*x*tan(e/2 + f*x/2)/(2*a**2*f) - d*log(tan(e/2 + f*x/2)**2 + 1)/(3*a**2*f**2) - d*tan(e/2 + f*x/2)**2/(6*a **2*f**2), Ne(f, 0)), ((c*x + d*x**2/2)/(a*cos(e) + a)**2, True))
Leaf count of result is larger than twice the leaf count of optimal. 763 vs. \(2 (95) = 190\).
Time = 0.35 (sec) , antiderivative size = 763, normalized size of antiderivative = 6.20 \[ \int \frac {c+d x}{(a+a \cos (e+f x))^2} \, dx=-\frac {\frac {2 \, {\left (2 \, {\left (3 \, {\left (f x + e\right )} \sin \left (f x + e\right ) + \cos \left (2 \, f x + 2 \, e\right ) + \cos \left (f x + e\right )\right )} \cos \left (3 \, f x + 3 \, e\right ) + 2 \, {\left (9 \, {\left (f x + e\right )} \sin \left (f x + e\right ) + 6 \, \cos \left (f x + e\right ) + 1\right )} \cos \left (2 \, f x + 2 \, e\right ) + 6 \, \cos \left (2 \, f x + 2 \, e\right )^{2} + 6 \, \cos \left (f x + e\right )^{2} - {\left (2 \, {\left (3 \, \cos \left (2 \, f x + 2 \, e\right ) + 3 \, \cos \left (f x + e\right ) + 1\right )} \cos \left (3 \, f x + 3 \, e\right ) + \cos \left (3 \, f x + 3 \, e\right )^{2} + 6 \, {\left (3 \, \cos \left (f x + e\right ) + 1\right )} \cos \left (2 \, f x + 2 \, e\right ) + 9 \, \cos \left (2 \, f x + 2 \, e\right )^{2} + 9 \, \cos \left (f x + e\right )^{2} + 6 \, {\left (\sin \left (2 \, f x + 2 \, e\right ) + \sin \left (f x + e\right )\right )} \sin \left (3 \, f x + 3 \, e\right ) + \sin \left (3 \, f x + 3 \, e\right )^{2} + 9 \, \sin \left (2 \, f x + 2 \, e\right )^{2} + 18 \, \sin \left (2 \, f x + 2 \, e\right ) \sin \left (f x + e\right ) + 9 \, \sin \left (f x + e\right )^{2} + 6 \, \cos \left (f x + e\right ) + 1\right )} \log \left (\cos \left (f x + e\right )^{2} + \sin \left (f x + e\right )^{2} + 2 \, \cos \left (f x + e\right ) + 1\right ) - 2 \, {\left (f x + 3 \, {\left (f x + e\right )} \cos \left (f x + e\right ) + e - \sin \left (2 \, f x + 2 \, e\right ) - \sin \left (f x + e\right )\right )} \sin \left (3 \, f x + 3 \, e\right ) - 6 \, {\left (f x + 3 \, {\left (f x + e\right )} \cos \left (f x + e\right ) + e - 2 \, \sin \left (f x + e\right )\right )} \sin \left (2 \, f x + 2 \, e\right ) + 6 \, \sin \left (2 \, f x + 2 \, e\right )^{2} + 6 \, \sin \left (f x + e\right )^{2} + 2 \, \cos \left (f x + e\right )\right )} d}{a^{2} f \cos \left (3 \, f x + 3 \, e\right )^{2} + 9 \, a^{2} f \cos \left (2 \, f x + 2 \, e\right )^{2} + 9 \, a^{2} f \cos \left (f x + e\right )^{2} + a^{2} f \sin \left (3 \, f x + 3 \, e\right )^{2} + 9 \, a^{2} f \sin \left (2 \, f x + 2 \, e\right )^{2} + 18 \, a^{2} f \sin \left (2 \, f x + 2 \, e\right ) \sin \left (f x + e\right ) + 9 \, a^{2} f \sin \left (f x + e\right )^{2} + 6 \, a^{2} f \cos \left (f x + e\right ) + a^{2} f + 2 \, {\left (3 \, a^{2} f \cos \left (2 \, f x + 2 \, e\right ) + 3 \, a^{2} f \cos \left (f x + e\right ) + a^{2} f\right )} \cos \left (3 \, f x + 3 \, e\right ) + 6 \, {\left (3 \, a^{2} f \cos \left (f x + e\right ) + a^{2} f\right )} \cos \left (2 \, f x + 2 \, e\right ) + 6 \, {\left (a^{2} f \sin \left (2 \, f x + 2 \, e\right ) + a^{2} f \sin \left (f x + e\right )\right )} \sin \left (3 \, f x + 3 \, e\right )} - \frac {c {\left (\frac {3 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {\sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}\right )}}{a^{2}} + \frac {d e {\left (\frac {3 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {\sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}\right )}}{a^{2} f}}{6 \, f} \]
-1/6*(2*(2*(3*(f*x + e)*sin(f*x + e) + cos(2*f*x + 2*e) + cos(f*x + e))*co s(3*f*x + 3*e) + 2*(9*(f*x + e)*sin(f*x + e) + 6*cos(f*x + e) + 1)*cos(2*f *x + 2*e) + 6*cos(2*f*x + 2*e)^2 + 6*cos(f*x + e)^2 - (2*(3*cos(2*f*x + 2* e) + 3*cos(f*x + e) + 1)*cos(3*f*x + 3*e) + cos(3*f*x + 3*e)^2 + 6*(3*cos( f*x + e) + 1)*cos(2*f*x + 2*e) + 9*cos(2*f*x + 2*e)^2 + 9*cos(f*x + e)^2 + 6*(sin(2*f*x + 2*e) + sin(f*x + e))*sin(3*f*x + 3*e) + sin(3*f*x + 3*e)^2 + 9*sin(2*f*x + 2*e)^2 + 18*sin(2*f*x + 2*e)*sin(f*x + e) + 9*sin(f*x + e )^2 + 6*cos(f*x + e) + 1)*log(cos(f*x + e)^2 + sin(f*x + e)^2 + 2*cos(f*x + e) + 1) - 2*(f*x + 3*(f*x + e)*cos(f*x + e) + e - sin(2*f*x + 2*e) - sin (f*x + e))*sin(3*f*x + 3*e) - 6*(f*x + 3*(f*x + e)*cos(f*x + e) + e - 2*si n(f*x + e))*sin(2*f*x + 2*e) + 6*sin(2*f*x + 2*e)^2 + 6*sin(f*x + e)^2 + 2 *cos(f*x + e))*d/(a^2*f*cos(3*f*x + 3*e)^2 + 9*a^2*f*cos(2*f*x + 2*e)^2 + 9*a^2*f*cos(f*x + e)^2 + a^2*f*sin(3*f*x + 3*e)^2 + 9*a^2*f*sin(2*f*x + 2* e)^2 + 18*a^2*f*sin(2*f*x + 2*e)*sin(f*x + e) + 9*a^2*f*sin(f*x + e)^2 + 6 *a^2*f*cos(f*x + e) + a^2*f + 2*(3*a^2*f*cos(2*f*x + 2*e) + 3*a^2*f*cos(f* x + e) + a^2*f)*cos(3*f*x + 3*e) + 6*(3*a^2*f*cos(f*x + e) + a^2*f)*cos(2* f*x + 2*e) + 6*(a^2*f*sin(2*f*x + 2*e) + a^2*f*sin(f*x + e))*sin(3*f*x + 3 *e)) - c*(3*sin(f*x + e)/(cos(f*x + e) + 1) + sin(f*x + e)^3/(cos(f*x + e) + 1)^3)/a^2 + d*e*(3*sin(f*x + e)/(cos(f*x + e) + 1) + sin(f*x + e)^3/(co s(f*x + e) + 1)^3)/(a^2*f))/f
Leaf count of result is larger than twice the leaf count of optimal. 661 vs. \(2 (95) = 190\).
Time = 0.49 (sec) , antiderivative size = 661, normalized size of antiderivative = 5.37 \[ \int \frac {c+d x}{(a+a \cos (e+f x))^2} \, dx=\text {Too large to display} \]
-1/6*(3*d*f*x*tan(1/2*f*x)^3*tan(1/2*e)^2 + 3*d*f*x*tan(1/2*f*x)^2*tan(1/2 *e)^3 - 2*d*log(4*(tan(1/2*f*x)^2*tan(1/2*e)^2 - 2*tan(1/2*f*x)*tan(1/2*e) + 1)/(tan(1/2*f*x)^2*tan(1/2*e)^2 + tan(1/2*f*x)^2 + tan(1/2*e)^2 + 1))*t an(1/2*f*x)^3*tan(1/2*e)^3 + 3*c*f*tan(1/2*f*x)^3*tan(1/2*e)^2 + 3*c*f*tan (1/2*f*x)^2*tan(1/2*e)^3 + d*tan(1/2*f*x)^3*tan(1/2*e)^3 + d*f*x*tan(1/2*f *x)^3 - 3*d*f*x*tan(1/2*f*x)^2*tan(1/2*e) - 3*d*f*x*tan(1/2*f*x)*tan(1/2*e )^2 + 6*d*log(4*(tan(1/2*f*x)^2*tan(1/2*e)^2 - 2*tan(1/2*f*x)*tan(1/2*e) + 1)/(tan(1/2*f*x)^2*tan(1/2*e)^2 + tan(1/2*f*x)^2 + tan(1/2*e)^2 + 1))*tan (1/2*f*x)^2*tan(1/2*e)^2 + d*f*x*tan(1/2*e)^3 + c*f*tan(1/2*f*x)^3 - 3*c*f *tan(1/2*f*x)^2*tan(1/2*e) + d*tan(1/2*f*x)^3*tan(1/2*e) - 3*c*f*tan(1/2*f *x)*tan(1/2*e)^2 - d*tan(1/2*f*x)^2*tan(1/2*e)^2 + c*f*tan(1/2*e)^3 + d*ta n(1/2*f*x)*tan(1/2*e)^3 + 3*d*f*x*tan(1/2*f*x) + 3*d*f*x*tan(1/2*e) - 6*d* log(4*(tan(1/2*f*x)^2*tan(1/2*e)^2 - 2*tan(1/2*f*x)*tan(1/2*e) + 1)/(tan(1 /2*f*x)^2*tan(1/2*e)^2 + tan(1/2*f*x)^2 + tan(1/2*e)^2 + 1))*tan(1/2*f*x)* tan(1/2*e) + 3*c*f*tan(1/2*f*x) - d*tan(1/2*f*x)^2 + 3*c*f*tan(1/2*e) + d* tan(1/2*f*x)*tan(1/2*e) - d*tan(1/2*e)^2 + 2*d*log(4*(tan(1/2*f*x)^2*tan(1 /2*e)^2 - 2*tan(1/2*f*x)*tan(1/2*e) + 1)/(tan(1/2*f*x)^2*tan(1/2*e)^2 + ta n(1/2*f*x)^2 + tan(1/2*e)^2 + 1)) - d)/(a^2*f^2*tan(1/2*f*x)^3*tan(1/2*e)^ 3 - 3*a^2*f^2*tan(1/2*f*x)^2*tan(1/2*e)^2 + 3*a^2*f^2*tan(1/2*f*x)*tan(1/2 *e) - a^2*f^2)
Time = 18.84 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.42 \[ \int \frac {c+d x}{(a+a \cos (e+f x))^2} \, dx=\frac {2\,d\,\ln \left ({\mathrm {e}}^{e\,1{}\mathrm {i}}\,{\mathrm {e}}^{f\,x\,1{}\mathrm {i}}+1\right )}{3\,a^2\,f^2}+\frac {\left (c\,f+d\,f\,x-d\,1{}\mathrm {i}\right )\,2{}\mathrm {i}}{3\,a^2\,f^2\,\left (2\,{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}+{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1\right )}-\frac {d\,x\,2{}\mathrm {i}}{3\,a^2\,f}-\frac {2\,d}{3\,a^2\,f^2\,\left ({\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}+1\right )}+\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,\left (c+d\,x\right )\,4{}\mathrm {i}}{3\,a^2\,f\,\left (3\,{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}+3\,{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}+1\right )} \]
(2*d*log(exp(e*1i)*exp(f*x*1i) + 1))/(3*a^2*f^2) + ((c*f - d*1i + d*f*x)*2 i)/(3*a^2*f^2*(2*exp(e*1i + f*x*1i) + exp(e*2i + f*x*2i) + 1)) - (d*x*2i)/ (3*a^2*f) - (2*d)/(3*a^2*f^2*(exp(e*1i + f*x*1i) + 1)) + (exp(e*1i + f*x*1 i)*(c + d*x)*4i)/(3*a^2*f*(3*exp(e*1i + f*x*1i) + 3*exp(e*2i + f*x*2i) + e xp(e*3i + f*x*3i) + 1))