Integrand size = 20, antiderivative size = 159 \[ \int \frac {(a+a \cos (e+f x))^2}{(c+d x)^2} \, dx=-\frac {4 a^2 \cos ^4\left (\frac {e}{2}+\frac {f x}{2}\right )}{d (c+d x)}-\frac {a^2 f \operatorname {CosIntegral}\left (\frac {2 c f}{d}+2 f x\right ) \sin \left (2 e-\frac {2 c f}{d}\right )}{d^2}-\frac {2 a^2 f \operatorname {CosIntegral}\left (\frac {c f}{d}+f x\right ) \sin \left (e-\frac {c f}{d}\right )}{d^2}-\frac {2 a^2 f \cos \left (e-\frac {c f}{d}\right ) \text {Si}\left (\frac {c f}{d}+f x\right )}{d^2}-\frac {a^2 f \cos \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (\frac {2 c f}{d}+2 f x\right )}{d^2} \] Output:
-4*a^2*cos(1/2*f*x+1/2*e)^4/d/(d*x+c)+a^2*f*Ci(2*c*f/d+2*f*x)*sin(-2*e+2*c *f/d)/d^2+2*a^2*f*Ci(c*f/d+f*x)*sin(-e+c*f/d)/d^2-2*a^2*f*cos(-e+c*f/d)*Si (c*f/d+f*x)/d^2-a^2*f*cos(-2*e+2*c*f/d)*Si(2*c*f/d+2*f*x)/d^2
Time = 0.93 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.30 \[ \int \frac {(a+a \cos (e+f x))^2}{(c+d x)^2} \, dx=-\frac {a^2 \left (3 d+4 d \cos (e+f x)+d \cos (2 (e+f x))+2 f (c+d x) \operatorname {CosIntegral}\left (\frac {2 f (c+d x)}{d}\right ) \sin \left (2 e-\frac {2 c f}{d}\right )+4 f (c+d x) \operatorname {CosIntegral}\left (f \left (\frac {c}{d}+x\right )\right ) \sin \left (e-\frac {c f}{d}\right )+4 c f \cos \left (e-\frac {c f}{d}\right ) \text {Si}\left (f \left (\frac {c}{d}+x\right )\right )+4 d f x \cos \left (e-\frac {c f}{d}\right ) \text {Si}\left (f \left (\frac {c}{d}+x\right )\right )+2 c f \cos \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (\frac {2 f (c+d x)}{d}\right )+2 d f x \cos \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (\frac {2 f (c+d x)}{d}\right )\right )}{2 d^2 (c+d x)} \] Input:
Integrate[(a + a*Cos[e + f*x])^2/(c + d*x)^2,x]
Output:
-1/2*(a^2*(3*d + 4*d*Cos[e + f*x] + d*Cos[2*(e + f*x)] + 2*f*(c + d*x)*Cos Integral[(2*f*(c + d*x))/d]*Sin[2*e - (2*c*f)/d] + 4*f*(c + d*x)*CosIntegr al[f*(c/d + x)]*Sin[e - (c*f)/d] + 4*c*f*Cos[e - (c*f)/d]*SinIntegral[f*(c /d + x)] + 4*d*f*x*Cos[e - (c*f)/d]*SinIntegral[f*(c/d + x)] + 2*c*f*Cos[2 *e - (2*c*f)/d]*SinIntegral[(2*f*(c + d*x))/d] + 2*d*f*x*Cos[2*e - (2*c*f) /d]*SinIntegral[(2*f*(c + d*x))/d]))/(d^2*(c + d*x))
Time = 0.55 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.01, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3042, 3799, 3042, 3794, 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.
| \(\displaystyle \int \frac {(a \cos (e+f x)+a)^2}{(c+d x)^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a \sin \left (e+f x+\frac {\pi }{2}\right )+a\right )^2}{(c+d x)^2}dx\) |
| \(\Big \downarrow \) 3799 |
| \(\displaystyle 4 a^2 \int \frac {\cos ^4\left (\frac {e}{2}+\frac {f x}{2}\right )}{(c+d x)^2}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle 4 a^2 \int \frac {\sin \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{2}\right )^4}{(c+d x)^2}dx\) |
| \(\Big \downarrow \) 3794 |
| \(\displaystyle 4 a^2 \left (\frac {2 f \int \left (-\frac {\sin (e+f x)}{4 (c+d x)}-\frac {\sin (2 e+2 f x)}{8 (c+d x)}\right )dx}{d}-\frac {\cos ^4\left (\frac {e}{2}+\frac {f x}{2}\right )}{d (c+d x)}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 4 a^2 \left (\frac {2 f \left (-\frac {\operatorname {CosIntegral}\left (2 x f+\frac {2 c f}{d}\right ) \sin \left (2 e-\frac {2 c f}{d}\right )}{8 d}-\frac {\operatorname {CosIntegral}\left (x f+\frac {c f}{d}\right ) \sin \left (e-\frac {c f}{d}\right )}{4 d}-\frac {\cos \left (e-\frac {c f}{d}\right ) \text {Si}\left (x f+\frac {c f}{d}\right )}{4 d}-\frac {\cos \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (2 x f+\frac {2 c f}{d}\right )}{8 d}\right )}{d}-\frac {\cos ^4\left (\frac {e}{2}+\frac {f x}{2}\right )}{d (c+d x)}\right )\) |
Input:
Int[(a + a*Cos[e + f*x])^2/(c + d*x)^2,x]
Output:
4*a^2*(-(Cos[e/2 + (f*x)/2]^4/(d*(c + d*x))) + (2*f*(-1/8*(CosIntegral[(2* c*f)/d + 2*f*x]*Sin[2*e - (2*c*f)/d])/d - (CosIntegral[(c*f)/d + f*x]*Sin[ e - (c*f)/d])/(4*d) - (Cos[e - (c*f)/d]*SinIntegral[(c*f)/d + f*x])/(4*d) - (Cos[2*e - (2*c*f)/d]*SinIntegral[(2*c*f)/d + 2*f*x])/(8*d)))/d)
Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Si mp[(c + d*x)^(m + 1)*(Sin[e + f*x]^n/(d*(m + 1))), x] - Simp[f*(n/(d*(m + 1 ))) Int[ExpandTrigReduce[(c + d*x)^(m + 1), Cos[e + f*x]*Sin[e + f*x]^(n - 1), x], x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && GeQ[m, -2] & & LtQ[m, -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])
Time = 2.12 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.74
| method | result | size |
| derivativedivides | \(\frac {\frac {a^{2} f^{2} \left (-\frac {2 \cos \left (2 f x +2 e \right )}{\left (c f -d e +d \left (f x +e \right )\right ) d}-\frac {2 \left (\frac {2 \,\operatorname {Si}\left (2 f x +2 e +\frac {2 c f -2 d e}{d}\right ) \cos \left (\frac {2 c f -2 d e}{d}\right )}{d}-\frac {2 \,\operatorname {Ci}\left (2 f x +2 e +\frac {2 c f -2 d e}{d}\right ) \sin \left (\frac {2 c f -2 d e}{d}\right )}{d}\right )}{d}\right )}{4}-\frac {3 a^{2} f^{2}}{2 \left (c f -d e +d \left (f x +e \right )\right ) d}+2 a^{2} f^{2} \left (-\frac {\cos \left (f x +e \right )}{\left (c f -d e +d \left (f x +e \right )\right ) d}-\frac {\frac {\operatorname {Si}\left (f x +e +\frac {c f -d e}{d}\right ) \cos \left (\frac {c f -d e}{d}\right )}{d}-\frac {\operatorname {Ci}\left (f x +e +\frac {c f -d e}{d}\right ) \sin \left (\frac {c f -d e}{d}\right )}{d}}{d}\right )}{f}\) | \(276\) |
| default | \(\frac {\frac {a^{2} f^{2} \left (-\frac {2 \cos \left (2 f x +2 e \right )}{\left (c f -d e +d \left (f x +e \right )\right ) d}-\frac {2 \left (\frac {2 \,\operatorname {Si}\left (2 f x +2 e +\frac {2 c f -2 d e}{d}\right ) \cos \left (\frac {2 c f -2 d e}{d}\right )}{d}-\frac {2 \,\operatorname {Ci}\left (2 f x +2 e +\frac {2 c f -2 d e}{d}\right ) \sin \left (\frac {2 c f -2 d e}{d}\right )}{d}\right )}{d}\right )}{4}-\frac {3 a^{2} f^{2}}{2 \left (c f -d e +d \left (f x +e \right )\right ) d}+2 a^{2} f^{2} \left (-\frac {\cos \left (f x +e \right )}{\left (c f -d e +d \left (f x +e \right )\right ) d}-\frac {\frac {\operatorname {Si}\left (f x +e +\frac {c f -d e}{d}\right ) \cos \left (\frac {c f -d e}{d}\right )}{d}-\frac {\operatorname {Ci}\left (f x +e +\frac {c f -d e}{d}\right ) \sin \left (\frac {c f -d e}{d}\right )}{d}}{d}\right )}{f}\) | \(276\) |
| parts | \(-\frac {a^{2}}{d \left (d x +c \right )}+\frac {a^{2} \left (\frac {f^{2} \left (-\frac {2 \cos \left (2 f x +2 e \right )}{\left (c f -d e +d \left (f x +e \right )\right ) d}-\frac {2 \left (\frac {2 \,\operatorname {Si}\left (2 f x +2 e +\frac {2 c f -2 d e}{d}\right ) \cos \left (\frac {2 c f -2 d e}{d}\right )}{d}-\frac {2 \,\operatorname {Ci}\left (2 f x +2 e +\frac {2 c f -2 d e}{d}\right ) \sin \left (\frac {2 c f -2 d e}{d}\right )}{d}\right )}{d}\right )}{4}-\frac {f^{2}}{2 \left (c f -d e +d \left (f x +e \right )\right ) d}\right )}{f}+2 a^{2} f \left (-\frac {\cos \left (f x +e \right )}{\left (c f -d e +d \left (f x +e \right )\right ) d}-\frac {\frac {\operatorname {Si}\left (f x +e +\frac {c f -d e}{d}\right ) \cos \left (\frac {c f -d e}{d}\right )}{d}-\frac {\operatorname {Ci}\left (f x +e +\frac {c f -d e}{d}\right ) \sin \left (\frac {c f -d e}{d}\right )}{d}}{d}\right )\) | \(287\) |
| risch | \(\frac {i a^{2} f \,{\mathrm e}^{\frac {i \left (c f -d e \right )}{d}} \operatorname {expIntegral}_{1}\left (i f x +i e +\frac {i \left (c f -d e \right )}{d}\right )}{d^{2}}-\frac {i f \,a^{2} {\mathrm e}^{-\frac {i \left (c f -d e \right )}{d}} \operatorname {expIntegral}_{1}\left (-i f x -i e -\frac {i c f -i d e}{d}\right )}{d^{2}}-\frac {3 a^{2}}{2 d \left (d x +c \right )}+\frac {i a^{2} f \,{\mathrm e}^{\frac {2 i \left (c f -d e \right )}{d}} \operatorname {expIntegral}_{1}\left (2 i f x +2 i e +\frac {2 i \left (c f -d e \right )}{d}\right )}{2 d^{2}}-\frac {i f \,a^{2} {\mathrm e}^{-\frac {2 i \left (c f -d e \right )}{d}} \operatorname {expIntegral}_{1}\left (-2 i f x -2 i e -\frac {2 \left (i c f -i d e \right )}{d}\right )}{2 d^{2}}-\frac {a^{2} \left (-2 d x f -2 c f \right ) \cos \left (f x +e \right )}{d \left (d x +c \right ) \left (-d x f -c f \right )}-\frac {a^{2} \left (-2 d x f -2 c f \right ) \cos \left (2 f x +2 e \right )}{4 d \left (d x +c \right ) \left (-d x f -c f \right )}\) | \(314\) |
Input:
int((a+cos(f*x+e)*a)^2/(d*x+c)^2,x,method=_RETURNVERBOSE)
Output:
1/f*(1/4*a^2*f^2*(-2*cos(2*f*x+2*e)/(c*f-d*e+d*(f*x+e))/d-2*(2*Si(2*f*x+2* e+2*(c*f-d*e)/d)*cos(2*(c*f-d*e)/d)/d-2*Ci(2*f*x+2*e+2*(c*f-d*e)/d)*sin(2* (c*f-d*e)/d)/d)/d)-3/2*a^2*f^2/(c*f-d*e+d*(f*x+e))/d+2*a^2*f^2*(-cos(f*x+e )/(c*f-d*e+d*(f*x+e))/d-(Si(f*x+e+(c*f-d*e)/d)*cos((c*f-d*e)/d)/d-Ci(f*x+e +(c*f-d*e)/d)*sin((c*f-d*e)/d)/d)/d))
Time = 0.12 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.38 \[ \int \frac {(a+a \cos (e+f x))^2}{(c+d x)^2} \, dx=-\frac {a^{2} d \cos \left (f x + e\right )^{2} + 2 \, a^{2} d \cos \left (f x + e\right ) + a^{2} d - 2 \, {\left (a^{2} d f x + a^{2} c f\right )} \operatorname {Ci}\left (\frac {d f x + c f}{d}\right ) \sin \left (-\frac {d e - c f}{d}\right ) - {\left (a^{2} d f x + a^{2} c f\right )} \operatorname {Ci}\left (\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) \sin \left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right ) + {\left (a^{2} d f x + a^{2} c f\right )} \cos \left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right ) \operatorname {Si}\left (\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) + 2 \, {\left (a^{2} d f x + a^{2} c f\right )} \cos \left (-\frac {d e - c f}{d}\right ) \operatorname {Si}\left (\frac {d f x + c f}{d}\right )}{d^{3} x + c d^{2}} \] Input:
integrate((a+a*cos(f*x+e))^2/(d*x+c)^2,x, algorithm="fricas")
Output:
-(a^2*d*cos(f*x + e)^2 + 2*a^2*d*cos(f*x + e) + a^2*d - 2*(a^2*d*f*x + a^2 *c*f)*cos_integral((d*f*x + c*f)/d)*sin(-(d*e - c*f)/d) - (a^2*d*f*x + a^2 *c*f)*cos_integral(2*(d*f*x + c*f)/d)*sin(-2*(d*e - c*f)/d) + (a^2*d*f*x + a^2*c*f)*cos(-2*(d*e - c*f)/d)*sin_integral(2*(d*f*x + c*f)/d) + 2*(a^2*d *f*x + a^2*c*f)*cos(-(d*e - c*f)/d)*sin_integral((d*f*x + c*f)/d))/(d^3*x + c*d^2)
\[ \int \frac {(a+a \cos (e+f x))^2}{(c+d x)^2} \, dx=a^{2} \left (\int \frac {2 \cos {\left (e + f x \right )}}{c^{2} + 2 c d x + d^{2} x^{2}}\, dx + \int \frac {\cos ^{2}{\left (e + f x \right )}}{c^{2} + 2 c d x + d^{2} x^{2}}\, dx + \int \frac {1}{c^{2} + 2 c d x + d^{2} x^{2}}\, dx\right ) \] Input:
integrate((a+a*cos(f*x+e))**2/(d*x+c)**2,x)
Output:
a**2*(Integral(2*cos(e + f*x)/(c**2 + 2*c*d*x + d**2*x**2), x) + Integral( cos(e + f*x)**2/(c**2 + 2*c*d*x + d**2*x**2), x) + Integral(1/(c**2 + 2*c* d*x + d**2*x**2), x))
Result contains complex when optimal does not.
Time = 0.16 (sec) , antiderivative size = 372, normalized size of antiderivative = 2.34 \[ \int \frac {(a+a \cos (e+f x))^2}{(c+d x)^2} \, dx=-\frac {\frac {4 \, a^{2} f^{2}}{{\left (f x + e\right )} d^{2} - d^{2} e + c d f} + \frac {4 \, {\left (f^{2} {\left (E_{2}\left (\frac {i \, {\left (f x + e\right )} d - i \, d e + i \, c f}{d}\right ) + E_{2}\left (-\frac {i \, {\left (f x + e\right )} d - i \, d e + i \, c f}{d}\right )\right )} \cos \left (-\frac {d e - c f}{d}\right ) - f^{2} {\left (-i \, E_{2}\left (\frac {i \, {\left (f x + e\right )} d - i \, d e + i \, c f}{d}\right ) + i \, E_{2}\left (-\frac {i \, {\left (f x + e\right )} d - i \, d e + i \, c f}{d}\right )\right )} \sin \left (-\frac {d e - c f}{d}\right )\right )} a^{2}}{{\left (f x + e\right )} d^{2} - d^{2} e + c d f} + \frac {{\left (f^{2} {\left (E_{2}\left (\frac {2 \, {\left (-i \, {\left (f x + e\right )} d + i \, d e - i \, c f\right )}}{d}\right ) + E_{2}\left (-\frac {2 \, {\left (-i \, {\left (f x + e\right )} d + i \, d e - i \, c f\right )}}{d}\right )\right )} \cos \left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right ) - f^{2} {\left (i \, E_{2}\left (\frac {2 \, {\left (-i \, {\left (f x + e\right )} d + i \, d e - i \, c f\right )}}{d}\right ) - i \, E_{2}\left (-\frac {2 \, {\left (-i \, {\left (f x + e\right )} d + i \, d e - i \, c f\right )}}{d}\right )\right )} \sin \left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right ) + 2 \, f^{2}\right )} a^{2}}{{\left (f x + e\right )} d^{2} - d^{2} e + c d f}}{4 \, f} \] Input:
integrate((a+a*cos(f*x+e))^2/(d*x+c)^2,x, algorithm="maxima")
Output:
-1/4*(4*a^2*f^2/((f*x + e)*d^2 - d^2*e + c*d*f) + 4*(f^2*(exp_integral_e(2 , (I*(f*x + e)*d - I*d*e + I*c*f)/d) + exp_integral_e(2, -(I*(f*x + e)*d - I*d*e + I*c*f)/d))*cos(-(d*e - c*f)/d) - f^2*(-I*exp_integral_e(2, (I*(f* x + e)*d - I*d*e + I*c*f)/d) + I*exp_integral_e(2, -(I*(f*x + e)*d - I*d*e + I*c*f)/d))*sin(-(d*e - c*f)/d))*a^2/((f*x + e)*d^2 - d^2*e + c*d*f) + ( f^2*(exp_integral_e(2, 2*(-I*(f*x + e)*d + I*d*e - I*c*f)/d) + exp_integra l_e(2, -2*(-I*(f*x + e)*d + I*d*e - I*c*f)/d))*cos(-2*(d*e - c*f)/d) - f^2 *(I*exp_integral_e(2, 2*(-I*(f*x + e)*d + I*d*e - I*c*f)/d) - I*exp_integr al_e(2, -2*(-I*(f*x + e)*d + I*d*e - I*c*f)/d))*sin(-2*(d*e - c*f)/d) + 2* f^2)*a^2/((f*x + e)*d^2 - d^2*e + c*d*f))/f
Leaf count of result is larger than twice the leaf count of optimal. 1049 vs. \(2 (156) = 312\).
Time = 0.50 (sec) , antiderivative size = 1049, normalized size of antiderivative = 6.60 \[ \int \frac {(a+a \cos (e+f x))^2}{(c+d x)^2} \, dx=\text {Too large to display} \] Input:
integrate((a+a*cos(f*x+e))^2/(d*x+c)^2,x, algorithm="giac")
Output:
1/2*(4*(d*x + c)*a^2*(d*e/(d*x + c) - c*f/(d*x + c) + f)*f^2*cos_integral( ((d*x + c)*(d*e/(d*x + c) - c*f/(d*x + c) + f) - d*e + c*f)/d)*sin(-(d*e - c*f)/d) - 4*a^2*d*e*f^2*cos_integral(((d*x + c)*(d*e/(d*x + c) - c*f/(d*x + c) + f) - d*e + c*f)/d)*sin(-(d*e - c*f)/d) + 4*a^2*c*f^3*cos_integral( ((d*x + c)*(d*e/(d*x + c) - c*f/(d*x + c) + f) - d*e + c*f)/d)*sin(-(d*e - c*f)/d) + 2*(d*x + c)*a^2*(d*e/(d*x + c) - c*f/(d*x + c) + f)*f^2*cos_int egral(2*((d*x + c)*(d*e/(d*x + c) - c*f/(d*x + c) + f) - d*e + c*f)/d)*sin (-2*(d*e - c*f)/d) - 2*a^2*d*e*f^2*cos_integral(2*((d*x + c)*(d*e/(d*x + c ) - c*f/(d*x + c) + f) - d*e + c*f)/d)*sin(-2*(d*e - c*f)/d) + 2*a^2*c*f^3 *cos_integral(2*((d*x + c)*(d*e/(d*x + c) - c*f/(d*x + c) + f) - d*e + c*f )/d)*sin(-2*(d*e - c*f)/d) - 2*(d*x + c)*a^2*(d*e/(d*x + c) - c*f/(d*x + c ) + f)*f^2*cos(-2*(d*e - c*f)/d)*sin_integral(2*((d*x + c)*(d*e/(d*x + c) - c*f/(d*x + c) + f) - d*e + c*f)/d) + 2*a^2*d*e*f^2*cos(-2*(d*e - c*f)/d) *sin_integral(2*((d*x + c)*(d*e/(d*x + c) - c*f/(d*x + c) + f) - d*e + c*f )/d) - 2*a^2*c*f^3*cos(-2*(d*e - c*f)/d)*sin_integral(2*((d*x + c)*(d*e/(d *x + c) - c*f/(d*x + c) + f) - d*e + c*f)/d) - 4*(d*x + c)*a^2*(d*e/(d*x + c) - c*f/(d*x + c) + f)*f^2*cos(-(d*e - c*f)/d)*sin_integral(((d*x + c)*( d*e/(d*x + c) - c*f/(d*x + c) + f) - d*e + c*f)/d) + 4*a^2*d*e*f^2*cos(-(d *e - c*f)/d)*sin_integral(((d*x + c)*(d*e/(d*x + c) - c*f/(d*x + c) + f) - d*e + c*f)/d) - 4*a^2*c*f^3*cos(-(d*e - c*f)/d)*sin_integral(((d*x + c...
Timed out. \[ \int \frac {(a+a \cos (e+f x))^2}{(c+d x)^2} \, dx=\int \frac {{\left (a+a\,\cos \left (e+f\,x\right )\right )}^2}{{\left (c+d\,x\right )}^2} \,d x \] Input:
int((a + a*cos(e + f*x))^2/(c + d*x)^2,x)
Output:
int((a + a*cos(e + f*x))^2/(c + d*x)^2, x)
\[ \int \frac {(a+a \cos (e+f x))^2}{(c+d x)^2} \, dx=\frac {a^{2} \left (2 \left (\int \frac {\cos \left (f x +e \right )}{d^{2} x^{2}+2 c d x +c^{2}}d x \right ) c^{2}+2 \left (\int \frac {\cos \left (f x +e \right )}{d^{2} x^{2}+2 c d x +c^{2}}d x \right ) c d x +\left (\int \frac {\cos \left (f x +e \right )^{2}}{d^{2} x^{2}+2 c d x +c^{2}}d x \right ) c^{2}+\left (\int \frac {\cos \left (f x +e \right )^{2}}{d^{2} x^{2}+2 c d x +c^{2}}d x \right ) c d x +2 \left (\int \frac {1}{d^{2} x^{2}+2 c d x +c^{2}}d x \right ) c^{2}+2 \left (\int \frac {1}{d^{2} x^{2}+2 c d x +c^{2}}d x \right ) c d x -x \right )}{c \left (d x +c \right )} \] Input:
int((a+a*cos(f*x+e))^2/(d*x+c)^2,x)
Output:
(a**2*(2*int(cos(e + f*x)/(c**2 + 2*c*d*x + d**2*x**2),x)*c**2 + 2*int(cos (e + f*x)/(c**2 + 2*c*d*x + d**2*x**2),x)*c*d*x + int(cos(e + f*x)**2/(c** 2 + 2*c*d*x + d**2*x**2),x)*c**2 + int(cos(e + f*x)**2/(c**2 + 2*c*d*x + d **2*x**2),x)*c*d*x + 2*int(1/(c**2 + 2*c*d*x + d**2*x**2),x)*c**2 + 2*int( 1/(c**2 + 2*c*d*x + d**2*x**2),x)*c*d*x - x))/(c*(c + d*x))