Integrand size = 20, antiderivative size = 212 \[ \int \frac {(c+d x)^2}{(a+a \cos (e+f x))^2} \, dx=-\frac {i (c+d x)^2}{3 a^2 f}+\frac {4 d (c+d x) \log \left (1+e^{i (e+f x)}\right )}{3 a^2 f^2}-\frac {4 i d^2 \operatorname {PolyLog}\left (2,-e^{i (e+f x)}\right )}{3 a^2 f^3}-\frac {d (c+d x) \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f^2}+\frac {2 d^2 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f^3}+\frac {(c+d x)^2 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f}+\frac {(c+d x)^2 \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right ) \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{6 a^2 f} \]
-1/3*I*(d*x+c)^2/a^2/f+4/3*d*(d*x+c)*ln(1+exp(I*(f*x+e)))/a^2/f^2-4/3*I*d^ 2*polylog(2,-exp(I*(f*x+e)))/a^2/f^3-1/3*d*(d*x+c)*sec(1/2*f*x+1/2*e)^2/a^ 2/f^2+2/3*d^2*tan(1/2*f*x+1/2*e)/a^2/f^3+1/3*(d*x+c)^2*tan(1/2*f*x+1/2*e)/ a^2/f+1/6*(d*x+c)^2*sec(1/2*f*x+1/2*e)^2*tan(1/2*f*x+1/2*e)/a^2/f
Time = 1.17 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.00 \[ \int \frac {(c+d x)^2}{(a+a \cos (e+f x))^2} \, dx=\frac {2 \cos \left (\frac {1}{2} (e+f x)\right ) \left (-2 d f (c+d x) \cos \left (\frac {1}{2} (e+f x)\right )-2 i f (c+d x) \cos ^3\left (\frac {1}{2} (e+f x)\right ) \left (f (c+d x)+4 i d \log \left (1+e^{i (e+f x)}\right )\right )-8 i d^2 \cos ^3\left (\frac {1}{2} (e+f x)\right ) \operatorname {PolyLog}\left (2,-e^{i (e+f x)}\right )+\left (2 \left (c^2 f^2+2 c d f^2 x+d^2 \left (1+f^2 x^2\right )\right )+\left (c^2 f^2+2 c d f^2 x+d^2 \left (2+f^2 x^2\right )\right ) \cos (e+f x)\right ) \sin \left (\frac {1}{2} (e+f x)\right )\right )}{3 a^2 f^3 (1+\cos (e+f x))^2} \]
(2*Cos[(e + f*x)/2]*(-2*d*f*(c + d*x)*Cos[(e + f*x)/2] - (2*I)*f*(c + d*x) *Cos[(e + f*x)/2]^3*(f*(c + d*x) + (4*I)*d*Log[1 + E^(I*(e + f*x))]) - (8* I)*d^2*Cos[(e + f*x)/2]^3*PolyLog[2, -E^(I*(e + f*x))] + (2*(c^2*f^2 + 2*c *d*f^2*x + d^2*(1 + f^2*x^2)) + (c^2*f^2 + 2*c*d*f^2*x + d^2*(2 + f^2*x^2) )*Cos[e + f*x])*Sin[(e + f*x)/2]))/(3*a^2*f^3*(1 + Cos[e + f*x])^2)
Time = 0.92 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.97, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {3042, 3799, 3042, 4674, 3042, 4254, 24, 4672, 25, 3042, 4202, 2620, 2715, 2838}
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.
| \(\displaystyle \int \frac {(c+d x)^2}{(a \cos (e+f x)+a)^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(c+d x)^2}{\left (a \sin \left (e+f x+\frac {\pi }{2}\right )+a\right )^2}dx\) |
| \(\Big \downarrow \) 3799 |
| \(\displaystyle \frac {\int (c+d x)^2 \sec ^4\left (\frac {e}{2}+\frac {f x}{2}\right )dx}{4 a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int (c+d x)^2 \csc \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{2}\right )^4dx}{4 a^2}\) |
| \(\Big \downarrow \) 4674 |
| \(\displaystyle \frac {\frac {2}{3} \int (c+d x)^2 \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )dx+\frac {4 d^2 \int \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )dx}{3 f^2}-\frac {4 d (c+d x) \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 f^2}+\frac {2 (c+d x)^2 \tan \left (\frac {e}{2}+\frac {f x}{2}\right ) \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 f}}{4 a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {2}{3} \int (c+d x)^2 \csc \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{2}\right )^2dx+\frac {4 d^2 \int \csc \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{2}\right )^2dx}{3 f^2}-\frac {4 d (c+d x) \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 f^2}+\frac {2 (c+d x)^2 \tan \left (\frac {e}{2}+\frac {f x}{2}\right ) \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 f}}{4 a^2}\) |
| \(\Big \downarrow \) 4254 |
| \(\displaystyle \frac {\frac {2}{3} \int (c+d x)^2 \csc \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{2}\right )^2dx-\frac {8 d^2 \int 1d\left (-\tan \left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{3 f^3}-\frac {4 d (c+d x) \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 f^2}+\frac {2 (c+d x)^2 \tan \left (\frac {e}{2}+\frac {f x}{2}\right ) \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 f}}{4 a^2}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {\frac {2}{3} \int (c+d x)^2 \csc \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{2}\right )^2dx-\frac {4 d (c+d x) \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 f^2}+\frac {2 (c+d x)^2 \tan \left (\frac {e}{2}+\frac {f x}{2}\right ) \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 f}+\frac {8 d^2 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 f^3}}{4 a^2}\) |
| \(\Big \downarrow \) 4672 |
| \(\displaystyle \frac {\frac {2}{3} \left (\frac {4 d \int -\left ((c+d x) \tan \left (\frac {e}{2}+\frac {f x}{2}\right )\right )dx}{f}+\frac {2 (c+d x)^2 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{f}\right )-\frac {4 d (c+d x) \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 f^2}+\frac {2 (c+d x)^2 \tan \left (\frac {e}{2}+\frac {f x}{2}\right ) \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 f}+\frac {8 d^2 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 f^3}}{4 a^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {2}{3} \left (\frac {2 (c+d x)^2 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{f}-\frac {4 d \int (c+d x) \tan \left (\frac {e}{2}+\frac {f x}{2}\right )dx}{f}\right )-\frac {4 d (c+d x) \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 f^2}+\frac {2 (c+d x)^2 \tan \left (\frac {e}{2}+\frac {f x}{2}\right ) \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 f}+\frac {8 d^2 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 f^3}}{4 a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {2}{3} \left (\frac {2 (c+d x)^2 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{f}-\frac {4 d \int (c+d x) \tan \left (\frac {e}{2}+\frac {f x}{2}\right )dx}{f}\right )-\frac {4 d (c+d x) \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 f^2}+\frac {2 (c+d x)^2 \tan \left (\frac {e}{2}+\frac {f x}{2}\right ) \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 f}+\frac {8 d^2 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 f^3}}{4 a^2}\) |
| \(\Big \downarrow \) 4202 |
| \(\displaystyle \frac {\frac {2}{3} \left (\frac {2 (c+d x)^2 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{f}-\frac {4 d \left (\frac {i (c+d x)^2}{2 d}-2 i \int \frac {e^{i (e+f x)} (c+d x)}{1+e^{i (e+f x)}}dx\right )}{f}\right )-\frac {4 d (c+d x) \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 f^2}+\frac {2 (c+d x)^2 \tan \left (\frac {e}{2}+\frac {f x}{2}\right ) \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 f}+\frac {8 d^2 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 f^3}}{4 a^2}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle \frac {\frac {2}{3} \left (\frac {2 (c+d x)^2 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{f}-\frac {4 d \left (\frac {i (c+d x)^2}{2 d}-2 i \left (\frac {i d \int \log \left (1+e^{i (e+f x)}\right )dx}{f}-\frac {i (c+d x) \log \left (1+e^{i (e+f x)}\right )}{f}\right )\right )}{f}\right )-\frac {4 d (c+d x) \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 f^2}+\frac {2 (c+d x)^2 \tan \left (\frac {e}{2}+\frac {f x}{2}\right ) \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 f}+\frac {8 d^2 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 f^3}}{4 a^2}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {\frac {2}{3} \left (\frac {2 (c+d x)^2 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{f}-\frac {4 d \left (\frac {i (c+d x)^2}{2 d}-2 i \left (\frac {d \int e^{-i (e+f x)} \log \left (1+e^{i (e+f x)}\right )de^{i (e+f x)}}{f^2}-\frac {i (c+d x) \log \left (1+e^{i (e+f x)}\right )}{f}\right )\right )}{f}\right )-\frac {4 d (c+d x) \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 f^2}+\frac {2 (c+d x)^2 \tan \left (\frac {e}{2}+\frac {f x}{2}\right ) \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 f}+\frac {8 d^2 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 f^3}}{4 a^2}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {\frac {2}{3} \left (\frac {2 (c+d x)^2 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{f}-\frac {4 d \left (\frac {i (c+d x)^2}{2 d}-2 i \left (-\frac {i (c+d x) \log \left (1+e^{i (e+f x)}\right )}{f}-\frac {d \operatorname {PolyLog}\left (2,-e^{i (e+f x)}\right )}{f^2}\right )\right )}{f}\right )-\frac {4 d (c+d x) \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 f^2}+\frac {2 (c+d x)^2 \tan \left (\frac {e}{2}+\frac {f x}{2}\right ) \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 f}+\frac {8 d^2 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 f^3}}{4 a^2}\) |
((-4*d*(c + d*x)*Sec[e/2 + (f*x)/2]^2)/(3*f^2) + (8*d^2*Tan[e/2 + (f*x)/2] )/(3*f^3) + (2*(c + d*x)^2*Sec[e/2 + (f*x)/2]^2*Tan[e/2 + (f*x)/2])/(3*f) + (2*((-4*d*(((I/2)*(c + d*x)^2)/d - (2*I)*(((-I)*(c + d*x)*Log[1 + E^(I*( e + f*x))])/f - (d*PolyLog[2, -E^(I*(e + f*x))])/f^2)))/f + (2*(c + d*x)^2 *Tan[e/2 + (f*x)/2])/f))/3)/(4*a^2)
3.2.34.3.1 Defintions of rubi rules used
Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ ((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp [((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si mp[d*(m/(b*f*g*n*Log[F])) Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x )))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Simp[1/(d*e*n*Log[F]) Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) ))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
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[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[I *((c + d*x)^(m + 1)/(d*(m + 1))), x] - Simp[2*I Int[(c + d*x)^m*(E^(2*I*( e + f*x))/(1 + E^(2*I*(e + f*x)))), x], x] /; FreeQ[{c, d, e, f}, x] && IGt Q[m, 0]
Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp andIntegrand[(1 + x^2)^(n/2 - 1), x], x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]
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_))^(m_), x_Symbo l] :> Simp[(-b^2)*(c + d*x)^m*Cot[e + f*x]*((b*Csc[e + f*x])^(n - 2)/(f*(n - 1))), x] + (-Simp[b^2*d*m*(c + d*x)^(m - 1)*((b*Csc[e + f*x])^(n - 2)/(f^ 2*(n - 1)*(n - 2))), x] + Simp[b^2*d^2*m*((m - 1)/(f^2*(n - 1)*(n - 2))) Int[(c + d*x)^(m - 2)*(b*Csc[e + f*x])^(n - 2), x], x] + Simp[b^2*((n - 2)/ (n - 1)) Int[(c + d*x)^m*(b*Csc[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c , d, e, f}, x] && GtQ[n, 1] && NeQ[n, 2] && GtQ[m, 1]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 357 vs. \(2 (172 ) = 344\).
Time = 2.55 (sec) , antiderivative size = 358, normalized size of antiderivative = 1.69
| method | result | size |
| risch | \(\frac {2 i \left (2 i d^{2} f x \,{\mathrm e}^{2 i \left (f x +e \right )}+3 d^{2} f^{2} x^{2} {\mathrm e}^{i \left (f x +e \right )}+2 i c d f \,{\mathrm e}^{2 i \left (f x +e \right )}+2 i d^{2} f x \,{\mathrm e}^{i \left (f x +e \right )}+6 c d \,f^{2} x \,{\mathrm e}^{i \left (f x +e \right )}+d^{2} x^{2} f^{2}+2 i c d f \,{\mathrm e}^{i \left (f x +e \right )}+3 c^{2} f^{2} {\mathrm e}^{i \left (f x +e \right )}+2 c d \,f^{2} x +c^{2} f^{2}+2 d^{2} {\mathrm e}^{2 i \left (f x +e \right )}+4 d^{2} {\mathrm e}^{i \left (f x +e \right )}+2 d^{2}\right )}{3 f^{3} a^{2} \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{3}}-\frac {4 d c \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{3 a^{2} f^{2}}+\frac {4 d c \ln \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )}{3 a^{2} f^{2}}-\frac {2 i d^{2} x^{2}}{3 a^{2} f}-\frac {4 i d^{2} e x}{3 a^{2} f^{2}}-\frac {2 i d^{2} e^{2}}{3 a^{2} f^{3}}+\frac {4 d^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) x}{3 a^{2} f^{2}}-\frac {4 i d^{2} \operatorname {Li}_{2}\left (-{\mathrm e}^{i \left (f x +e \right )}\right )}{3 a^{2} f^{3}}+\frac {4 d^{2} e \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{3 a^{2} f^{3}}\) | \(358\) |
2/3*I*(2*I*d^2*f*x*exp(2*I*(f*x+e))+3*d^2*f^2*x^2*exp(I*(f*x+e))+2*I*c*d*f *exp(2*I*(f*x+e))+2*I*d^2*f*x*exp(I*(f*x+e))+6*c*d*f^2*x*exp(I*(f*x+e))+d^ 2*x^2*f^2+2*I*c*d*f*exp(I*(f*x+e))+3*c^2*f^2*exp(I*(f*x+e))+2*c*d*f^2*x+c^ 2*f^2+2*d^2*exp(2*I*(f*x+e))+4*d^2*exp(I*(f*x+e))+2*d^2)/f^3/a^2/(exp(I*(f *x+e))+1)^3-4/3/a^2*d/f^2*c*ln(exp(I*(f*x+e)))+4/3/a^2*d/f^2*c*ln(exp(I*(f *x+e))+1)-2/3*I/a^2*d^2/f*x^2-4/3*I/a^2*d^2/f^2*e*x-2/3*I/a^2*d^2/f^3*e^2+ 4/3/a^2*d^2/f^2*ln(exp(I*(f*x+e))+1)*x-4/3*I*d^2*polylog(2,-exp(I*(f*x+e)) )/a^2/f^3+4/3/a^2*d^2/f^3*e*ln(exp(I*(f*x+e)))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 390 vs. \(2 (169) = 338\).
Time = 0.27 (sec) , antiderivative size = 390, normalized size of antiderivative = 1.84 \[ \int \frac {(c+d x)^2}{(a+a \cos (e+f x))^2} \, dx=-\frac {2 \, d^{2} f x + 2 \, c d f + 2 \, {\left (d^{2} f x + c d f\right )} \cos \left (f x + e\right ) + 2 \, {\left (-i \, d^{2} \cos \left (f x + e\right )^{2} - 2 i \, d^{2} \cos \left (f x + e\right ) - i \, d^{2}\right )} {\rm Li}_2\left (-\cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right ) + 2 \, {\left (i \, d^{2} \cos \left (f x + e\right )^{2} + 2 i \, d^{2} \cos \left (f x + e\right ) + i \, d^{2}\right )} {\rm Li}_2\left (-\cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right ) - 2 \, {\left (d^{2} f x + c d f + {\left (d^{2} f x + c d f\right )} \cos \left (f x + e\right )^{2} + 2 \, {\left (d^{2} f x + c d f\right )} \cos \left (f x + e\right )\right )} \log \left (\cos \left (f x + e\right ) + i \, \sin \left (f x + e\right ) + 1\right ) - 2 \, {\left (d^{2} f x + c d f + {\left (d^{2} f x + c d f\right )} \cos \left (f x + e\right )^{2} + 2 \, {\left (d^{2} f x + c d f\right )} \cos \left (f x + e\right )\right )} \log \left (\cos \left (f x + e\right ) - i \, \sin \left (f x + e\right ) + 1\right ) - {\left (2 \, d^{2} f^{2} x^{2} + 4 \, c d f^{2} x + 2 \, c^{2} f^{2} + 2 \, d^{2} + {\left (d^{2} f^{2} x^{2} + 2 \, c d f^{2} x + c^{2} f^{2} + 2 \, d^{2}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{3 \, {\left (a^{2} f^{3} \cos \left (f x + e\right )^{2} + 2 \, a^{2} f^{3} \cos \left (f x + e\right ) + a^{2} f^{3}\right )}} \]
-1/3*(2*d^2*f*x + 2*c*d*f + 2*(d^2*f*x + c*d*f)*cos(f*x + e) + 2*(-I*d^2*c os(f*x + e)^2 - 2*I*d^2*cos(f*x + e) - I*d^2)*dilog(-cos(f*x + e) + I*sin( f*x + e)) + 2*(I*d^2*cos(f*x + e)^2 + 2*I*d^2*cos(f*x + e) + I*d^2)*dilog( -cos(f*x + e) - I*sin(f*x + e)) - 2*(d^2*f*x + c*d*f + (d^2*f*x + c*d*f)*c os(f*x + e)^2 + 2*(d^2*f*x + c*d*f)*cos(f*x + e))*log(cos(f*x + e) + I*sin (f*x + e) + 1) - 2*(d^2*f*x + c*d*f + (d^2*f*x + c*d*f)*cos(f*x + e)^2 + 2 *(d^2*f*x + c*d*f)*cos(f*x + e))*log(cos(f*x + e) - I*sin(f*x + e) + 1) - (2*d^2*f^2*x^2 + 4*c*d*f^2*x + 2*c^2*f^2 + 2*d^2 + (d^2*f^2*x^2 + 2*c*d*f^ 2*x + c^2*f^2 + 2*d^2)*cos(f*x + e))*sin(f*x + e))/(a^2*f^3*cos(f*x + e)^2 + 2*a^2*f^3*cos(f*x + e) + a^2*f^3)
\[ \int \frac {(c+d x)^2}{(a+a \cos (e+f x))^2} \, dx=\frac {\int \frac {c^{2}}{\cos ^{2}{\left (e + f x \right )} + 2 \cos {\left (e + f x \right )} + 1}\, dx + \int \frac {d^{2} x^{2}}{\cos ^{2}{\left (e + f x \right )} + 2 \cos {\left (e + f x \right )} + 1}\, dx + \int \frac {2 c d x}{\cos ^{2}{\left (e + f x \right )} + 2 \cos {\left (e + f x \right )} + 1}\, dx}{a^{2}} \]
(Integral(c**2/(cos(e + f*x)**2 + 2*cos(e + f*x) + 1), x) + Integral(d**2* x**2/(cos(e + f*x)**2 + 2*cos(e + f*x) + 1), x) + Integral(2*c*d*x/(cos(e + f*x)**2 + 2*cos(e + f*x) + 1), x))/a**2
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 776 vs. \(2 (169) = 338\).
Time = 0.61 (sec) , antiderivative size = 776, normalized size of antiderivative = 3.66 \[ \int \frac {(c+d x)^2}{(a+a \cos (e+f x))^2} \, dx=\frac {2 \, {\left (c^{2} f^{2} + 2 \, d^{2} + 2 \, {\left (d^{2} f x + c d f + {\left (d^{2} f x + c d f\right )} \cos \left (3 \, f x + 3 \, e\right ) + 3 \, {\left (d^{2} f x + c d f\right )} \cos \left (2 \, f x + 2 \, e\right ) + 3 \, {\left (d^{2} f x + c d f\right )} \cos \left (f x + e\right ) - {\left (-i \, d^{2} f x - i \, c d f\right )} \sin \left (3 \, f x + 3 \, e\right ) - 3 \, {\left (-i \, d^{2} f x - i \, c d f\right )} \sin \left (2 \, f x + 2 \, e\right ) - 3 \, {\left (-i \, d^{2} f x - i \, c d f\right )} \sin \left (f x + e\right )\right )} \arctan \left (\sin \left (f x + e\right ), \cos \left (f x + e\right ) + 1\right ) - {\left (d^{2} f^{2} x^{2} + 2 \, c d f^{2} x\right )} \cos \left (3 \, f x + 3 \, e\right ) - {\left (3 \, d^{2} f^{2} x^{2} - 2 i \, c d f - 2 \, d^{2} + 2 \, {\left (3 \, c d f^{2} - i \, d^{2} f\right )} x\right )} \cos \left (2 \, f x + 2 \, e\right ) + {\left (3 \, c^{2} f^{2} + 2 i \, d^{2} f x + 2 i \, c d f + 4 \, d^{2}\right )} \cos \left (f x + e\right ) - 2 \, {\left (d^{2} \cos \left (3 \, f x + 3 \, e\right ) + 3 \, d^{2} \cos \left (2 \, f x + 2 \, e\right ) + 3 \, d^{2} \cos \left (f x + e\right ) + i \, d^{2} \sin \left (3 \, f x + 3 \, e\right ) + 3 i \, d^{2} \sin \left (2 \, f x + 2 \, e\right ) + 3 i \, d^{2} \sin \left (f x + e\right ) + d^{2}\right )} {\rm Li}_2\left (-e^{\left (i \, f x + i \, e\right )}\right ) - {\left (i \, d^{2} f x + i \, c d f + {\left (i \, d^{2} f x + i \, c d f\right )} \cos \left (3 \, f x + 3 \, e\right ) + 3 \, {\left (i \, d^{2} f x + i \, c d f\right )} \cos \left (2 \, f x + 2 \, e\right ) + 3 \, {\left (i \, d^{2} f x + i \, c d f\right )} \cos \left (f x + e\right ) - {\left (d^{2} f x + c d f\right )} \sin \left (3 \, f x + 3 \, e\right ) - 3 \, {\left (d^{2} f x + c d f\right )} \sin \left (2 \, f x + 2 \, e\right ) - 3 \, {\left (d^{2} f x + c d f\right )} \sin \left (f x + e\right )\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 ) - {\left (i \, d^{2} f^{2} x^{2} + 2 i \, c d f^{2} x\right )} \sin \left (3 \, f x + 3 \, e\right ) - {\left (3 i \, d^{2} f^{2} x^{2} + 2 \, c d f - 2 i \, d^{2} + 2 \, {\left (3 i \, c d f^{2} + d^{2} f\right )} x\right )} \sin \left (2 \, f x + 2 \, e\right ) - {\left (-3 i \, c^{2} f^{2} + 2 \, d^{2} f x + 2 \, c d f - 4 i \, d^{2}\right )} \sin \left (f x + e\right )\right )}}{-3 i \, a^{2} f^{3} \cos \left (3 \, f x + 3 \, e\right ) - 9 i \, a^{2} f^{3} \cos \left (2 \, f x + 2 \, e\right ) - 9 i \, a^{2} f^{3} \cos \left (f x + e\right ) + 3 \, a^{2} f^{3} \sin \left (3 \, f x + 3 \, e\right ) + 9 \, a^{2} f^{3} \sin \left (2 \, f x + 2 \, e\right ) + 9 \, a^{2} f^{3} \sin \left (f x + e\right ) - 3 i \, a^{2} f^{3}} \]
2*(c^2*f^2 + 2*d^2 + 2*(d^2*f*x + c*d*f + (d^2*f*x + c*d*f)*cos(3*f*x + 3* e) + 3*(d^2*f*x + c*d*f)*cos(2*f*x + 2*e) + 3*(d^2*f*x + c*d*f)*cos(f*x + e) - (-I*d^2*f*x - I*c*d*f)*sin(3*f*x + 3*e) - 3*(-I*d^2*f*x - I*c*d*f)*si n(2*f*x + 2*e) - 3*(-I*d^2*f*x - I*c*d*f)*sin(f*x + e))*arctan2(sin(f*x + e), cos(f*x + e) + 1) - (d^2*f^2*x^2 + 2*c*d*f^2*x)*cos(3*f*x + 3*e) - (3* d^2*f^2*x^2 - 2*I*c*d*f - 2*d^2 + 2*(3*c*d*f^2 - I*d^2*f)*x)*cos(2*f*x + 2 *e) + (3*c^2*f^2 + 2*I*d^2*f*x + 2*I*c*d*f + 4*d^2)*cos(f*x + e) - 2*(d^2* cos(3*f*x + 3*e) + 3*d^2*cos(2*f*x + 2*e) + 3*d^2*cos(f*x + e) + I*d^2*sin (3*f*x + 3*e) + 3*I*d^2*sin(2*f*x + 2*e) + 3*I*d^2*sin(f*x + e) + d^2)*dil og(-e^(I*f*x + I*e)) - (I*d^2*f*x + I*c*d*f + (I*d^2*f*x + I*c*d*f)*cos(3* f*x + 3*e) + 3*(I*d^2*f*x + I*c*d*f)*cos(2*f*x + 2*e) + 3*(I*d^2*f*x + I*c *d*f)*cos(f*x + e) - (d^2*f*x + c*d*f)*sin(3*f*x + 3*e) - 3*(d^2*f*x + c*d *f)*sin(2*f*x + 2*e) - 3*(d^2*f*x + c*d*f)*sin(f*x + e))*log(cos(f*x + e)^ 2 + sin(f*x + e)^2 + 2*cos(f*x + e) + 1) - (I*d^2*f^2*x^2 + 2*I*c*d*f^2*x) *sin(3*f*x + 3*e) - (3*I*d^2*f^2*x^2 + 2*c*d*f - 2*I*d^2 + 2*(3*I*c*d*f^2 + d^2*f)*x)*sin(2*f*x + 2*e) - (-3*I*c^2*f^2 + 2*d^2*f*x + 2*c*d*f - 4*I*d ^2)*sin(f*x + e))/(-3*I*a^2*f^3*cos(3*f*x + 3*e) - 9*I*a^2*f^3*cos(2*f*x + 2*e) - 9*I*a^2*f^3*cos(f*x + e) + 3*a^2*f^3*sin(3*f*x + 3*e) + 9*a^2*f^3* sin(2*f*x + 2*e) + 9*a^2*f^3*sin(f*x + e) - 3*I*a^2*f^3)
\[ \int \frac {(c+d x)^2}{(a+a \cos (e+f x))^2} \, dx=\int { \frac {{\left (d x + c\right )}^{2}}{{\left (a \cos \left (f x + e\right ) + a\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {(c+d x)^2}{(a+a \cos (e+f x))^2} \, dx=\text {Hanged} \]