Integrand size = 20, antiderivative size = 183 \[ \int \frac {(a+b \sin (e+f x))^2}{(c+d x)^2} \, dx=-\frac {a^2}{d (c+d x)}+\frac {2 a b f \cos \left (e-\frac {c f}{d}\right ) \operatorname {CosIntegral}\left (\frac {c f}{d}+f x\right )}{d^2}+\frac {b^2 f \operatorname {CosIntegral}\left (\frac {2 c f}{d}+2 f x\right ) \sin \left (2 e-\frac {2 c f}{d}\right )}{d^2}-\frac {2 a b \sin (e+f x)}{d (c+d x)}-\frac {b^2 \sin ^2(e+f x)}{d (c+d x)}-\frac {2 a b f \sin \left (e-\frac {c f}{d}\right ) \text {Si}\left (\frac {c f}{d}+f x\right )}{d^2}+\frac {b^2 f \cos \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (\frac {2 c f}{d}+2 f x\right )}{d^2} \] Output:
-a^2/d/(d*x+c)+2*a*b*f*cos(-e+c*f/d)*Ci(c*f/d+f*x)/d^2-b^2*f*Ci(2*c*f/d+2* f*x)*sin(-2*e+2*c*f/d)/d^2-2*a*b*sin(f*x+e)/d/(d*x+c)-b^2*sin(f*x+e)^2/d/( d*x+c)+2*a*b*f*sin(-e+c*f/d)*Si(c*f/d+f*x)/d^2+b^2*f*cos(-2*e+2*c*f/d)*Si( 2*c*f/d+2*f*x)/d^2
Time = 0.69 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.27 \[ \int \frac {(a+b \sin (e+f x))^2}{(c+d x)^2} \, dx=\frac {-2 a^2 d-b^2 d+b^2 d \cos (2 (e+f x))+4 a b f (c+d x) \cos \left (e-\frac {c f}{d}\right ) \operatorname {CosIntegral}\left (f \left (\frac {c}{d}+x\right )\right )+2 b^2 f (c+d x) \operatorname {CosIntegral}\left (\frac {2 f (c+d x)}{d}\right ) \sin \left (2 e-\frac {2 c f}{d}\right )-4 a b d \sin (e+f x)-4 a b c f \sin \left (e-\frac {c f}{d}\right ) \text {Si}\left (f \left (\frac {c}{d}+x\right )\right )-4 a b d f x \sin \left (e-\frac {c f}{d}\right ) \text {Si}\left (f \left (\frac {c}{d}+x\right )\right )+2 b^2 c f \cos \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (\frac {2 f (c+d x)}{d}\right )+2 b^2 d f x \cos \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (\frac {2 f (c+d x)}{d}\right )}{2 d^2 (c+d x)} \] Input:
Integrate[(a + b*Sin[e + f*x])^2/(c + d*x)^2,x]
Output:
(-2*a^2*d - b^2*d + b^2*d*Cos[2*(e + f*x)] + 4*a*b*f*(c + d*x)*Cos[e - (c* f)/d]*CosIntegral[f*(c/d + x)] + 2*b^2*f*(c + d*x)*CosIntegral[(2*f*(c + d *x))/d]*Sin[2*e - (2*c*f)/d] - 4*a*b*d*Sin[e + f*x] - 4*a*b*c*f*Sin[e - (c *f)/d]*SinIntegral[f*(c/d + x)] - 4*a*b*d*f*x*Sin[e - (c*f)/d]*SinIntegral [f*(c/d + x)] + 2*b^2*c*f*Cos[2*e - (2*c*f)/d]*SinIntegral[(2*f*(c + d*x)) /d] + 2*b^2*d*f*x*Cos[2*e - (2*c*f)/d]*SinIntegral[(2*f*(c + d*x))/d])/(2* d^2*(c + d*x))
Time = 0.59 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {3042, 3798, 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.
| \(\displaystyle \int \frac {(a+b \sin (e+f x))^2}{(c+d x)^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a+b \sin (e+f x))^2}{(c+d x)^2}dx\) |
| \(\Big \downarrow \) 3798 |
| \(\displaystyle \int \left (\frac {a^2}{(c+d x)^2}+\frac {2 a b \sin (e+f x)}{(c+d x)^2}+\frac {b^2 \sin ^2(e+f x)}{(c+d x)^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a^2}{d (c+d x)}+\frac {2 a b f \operatorname {CosIntegral}\left (x f+\frac {c f}{d}\right ) \cos \left (e-\frac {c f}{d}\right )}{d^2}-\frac {2 a b f \sin \left (e-\frac {c f}{d}\right ) \text {Si}\left (x f+\frac {c f}{d}\right )}{d^2}-\frac {2 a b \sin (e+f x)}{d (c+d x)}+\frac {b^2 f \operatorname {CosIntegral}\left (2 x f+\frac {2 c f}{d}\right ) \sin \left (2 e-\frac {2 c f}{d}\right )}{d^2}+\frac {b^2 f \cos \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (2 x f+\frac {2 c f}{d}\right )}{d^2}-\frac {b^2 \sin ^2(e+f x)}{d (c+d x)}\) |
Input:
Int[(a + b*Sin[e + f*x])^2/(c + d*x)^2,x]
Output:
-(a^2/(d*(c + d*x))) + (2*a*b*f*Cos[e - (c*f)/d]*CosIntegral[(c*f)/d + f*x ])/d^2 + (b^2*f*CosIntegral[(2*c*f)/d + 2*f*x]*Sin[2*e - (2*c*f)/d])/d^2 - (2*a*b*Sin[e + f*x])/(d*(c + d*x)) - (b^2*Sin[e + f*x]^2)/(d*(c + d*x)) - (2*a*b*f*Sin[e - (c*f)/d]*SinIntegral[(c*f)/d + f*x])/d^2 + (b^2*f*Cos[2* e - (2*c*f)/d]*SinIntegral[(2*c*f)/d + 2*f*x])/d^2
Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_.) , x_Symbol] :> Int[ExpandIntegrand[(c + d*x)^m, (a + b*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[n, 0] && (EqQ[n, 1] || IGtQ[ m, 0] || NeQ[a^2 - b^2, 0])
Time = 1.60 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.55
| method | result | size |
| parts | \(-\frac {a^{2}}{d \left (d x +c \right )}+\frac {b^{2} \left (-\frac {f^{2}}{2 \left (c f -d e +d \left (f x +e \right )\right ) d}-\frac {f^{2} \left (-\frac {2 \cos \left (2 f x +2 e \right )}{\left (c f -d e +d \left (f x +e \right )\right ) d}-\frac {2 \left (\frac {2 \,\operatorname {Si}\left (2 f x +2 e +\frac {2 c f -2 d e}{d}\right ) \cos \left (\frac {2 c f -2 d e}{d}\right )}{d}-\frac {2 \,\operatorname {Ci}\left (2 f x +2 e +\frac {2 c f -2 d e}{d}\right ) \sin \left (\frac {2 c f -2 d e}{d}\right )}{d}\right )}{d}\right )}{4}\right )}{f}+2 a b f \left (-\frac {\sin \left (f x +e \right )}{\left (c f -d e +d \left (f x +e \right )\right ) d}+\frac {\frac {\operatorname {Si}\left (f x +e +\frac {c f -d e}{d}\right ) \sin \left (\frac {c f -d e}{d}\right )}{d}+\frac {\operatorname {Ci}\left (f x +e +\frac {c f -d e}{d}\right ) \cos \left (\frac {c f -d e}{d}\right )}{d}}{d}\right )\) | \(284\) |
| derivativedivides | \(\frac {-\frac {a^{2} f^{2}}{\left (c f -d e +d \left (f x +e \right )\right ) d}+2 f^{2} a b \left (-\frac {\sin \left (f x +e \right )}{\left (c f -d e +d \left (f x +e \right )\right ) d}+\frac {\frac {\operatorname {Si}\left (f x +e +\frac {c f -d e}{d}\right ) \sin \left (\frac {c f -d e}{d}\right )}{d}+\frac {\operatorname {Ci}\left (f x +e +\frac {c f -d e}{d}\right ) \cos \left (\frac {c f -d e}{d}\right )}{d}}{d}\right )-\frac {f^{2} b^{2}}{2 \left (c f -d e +d \left (f x +e \right )\right ) d}-\frac {f^{2} b^{2} \left (-\frac {2 \cos \left (2 f x +2 e \right )}{\left (c f -d e +d \left (f x +e \right )\right ) d}-\frac {2 \left (\frac {2 \,\operatorname {Si}\left (2 f x +2 e +\frac {2 c f -2 d e}{d}\right ) \cos \left (\frac {2 c f -2 d e}{d}\right )}{d}-\frac {2 \,\operatorname {Ci}\left (2 f x +2 e +\frac {2 c f -2 d e}{d}\right ) \sin \left (\frac {2 c f -2 d e}{d}\right )}{d}\right )}{d}\right )}{4}}{f}\) | \(301\) |
| default | \(\frac {-\frac {a^{2} f^{2}}{\left (c f -d e +d \left (f x +e \right )\right ) d}+2 f^{2} a b \left (-\frac {\sin \left (f x +e \right )}{\left (c f -d e +d \left (f x +e \right )\right ) d}+\frac {\frac {\operatorname {Si}\left (f x +e +\frac {c f -d e}{d}\right ) \sin \left (\frac {c f -d e}{d}\right )}{d}+\frac {\operatorname {Ci}\left (f x +e +\frac {c f -d e}{d}\right ) \cos \left (\frac {c f -d e}{d}\right )}{d}}{d}\right )-\frac {f^{2} b^{2}}{2 \left (c f -d e +d \left (f x +e \right )\right ) d}-\frac {f^{2} b^{2} \left (-\frac {2 \cos \left (2 f x +2 e \right )}{\left (c f -d e +d \left (f x +e \right )\right ) d}-\frac {2 \left (\frac {2 \,\operatorname {Si}\left (2 f x +2 e +\frac {2 c f -2 d e}{d}\right ) \cos \left (\frac {2 c f -2 d e}{d}\right )}{d}-\frac {2 \,\operatorname {Ci}\left (2 f x +2 e +\frac {2 c f -2 d e}{d}\right ) \sin \left (\frac {2 c f -2 d e}{d}\right )}{d}\right )}{d}\right )}{4}}{f}\) | \(301\) |
| risch | \(-\frac {f a b \,{\mathrm e}^{\frac {i \left (c f -d e \right )}{d}} \operatorname {expIntegral}_{1}\left (i f x +i e +\frac {i \left (c f -d e \right )}{d}\right )}{d^{2}}-\frac {a^{2}}{d \left (d x +c \right )}-\frac {b^{2}}{2 d \left (d x +c \right )}-\frac {i b^{2} f \,{\mathrm e}^{\frac {2 i \left (c f -d e \right )}{d}} \operatorname {expIntegral}_{1}\left (2 i f x +2 i e +\frac {2 i \left (c f -d e \right )}{d}\right )}{2 d^{2}}+\frac {i f \,b^{2} {\mathrm e}^{-\frac {2 i \left (c f -d e \right )}{d}} \operatorname {expIntegral}_{1}\left (-2 i f x -2 i e -\frac {2 \left (i c f -i d e \right )}{d}\right )}{2 d^{2}}-\frac {a b f \,{\mathrm e}^{-\frac {i \left (c f -d e \right )}{d}} \operatorname {expIntegral}_{1}\left (-i f x -i e -\frac {i c f -i d e}{d}\right )}{d^{2}}-\frac {a b \left (-2 d x f -2 c f \right ) \sin \left (f x +e \right )}{d \left (d x +c \right ) \left (-d x f -c f \right )}+\frac {b^{2} \left (-2 d x f -2 c f \right ) \cos \left (2 f x +2 e \right )}{4 d \left (d x +c \right ) \left (-d x f -c f \right )}\) | \(324\) |
Input:
int((a+b*sin(f*x+e))^2/(d*x+c)^2,x,method=_RETURNVERBOSE)
Output:
-a^2/d/(d*x+c)+b^2/f*(-1/2*f^2/(c*f-d*e+d*(f*x+e))/d-1/4*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))+2*a*b*f*(- sin(f*x+e)/(c*f-d*e+d*(f*x+e))/d+(Si(f*x+e+(c*f-d*e)/d)*sin((c*f-d*e)/d)/d +Ci(f*x+e+(c*f-d*e)/d)*cos((c*f-d*e)/d)/d)/d)
Time = 0.09 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.19 \[ \int \frac {(a+b \sin (e+f x))^2}{(c+d x)^2} \, dx=\frac {b^{2} d \cos \left (f x + e\right )^{2} - 2 \, a b d \sin \left (f x + e\right ) + 2 \, {\left (a b d f x + a b c f\right )} \cos \left (-\frac {d e - c f}{d}\right ) \operatorname {Ci}\left (\frac {d f x + c f}{d}\right ) - {\left (b^{2} d f x + b^{2} c f\right )} \operatorname {Ci}\left (\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) \sin \left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right ) + {\left (b^{2} d f x + b^{2} c f\right )} \cos \left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right ) \operatorname {Si}\left (\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) + 2 \, {\left (a b d f x + a b c f\right )} \sin \left (-\frac {d e - c f}{d}\right ) \operatorname {Si}\left (\frac {d f x + c f}{d}\right ) - {\left (a^{2} + b^{2}\right )} d}{d^{3} x + c d^{2}} \] Input:
integrate((a+b*sin(f*x+e))^2/(d*x+c)^2,x, algorithm="fricas")
Output:
(b^2*d*cos(f*x + e)^2 - 2*a*b*d*sin(f*x + e) + 2*(a*b*d*f*x + a*b*c*f)*cos (-(d*e - c*f)/d)*cos_integral((d*f*x + c*f)/d) - (b^2*d*f*x + b^2*c*f)*cos _integral(2*(d*f*x + c*f)/d)*sin(-2*(d*e - c*f)/d) + (b^2*d*f*x + b^2*c*f) *cos(-2*(d*e - c*f)/d)*sin_integral(2*(d*f*x + c*f)/d) + 2*(a*b*d*f*x + a* b*c*f)*sin(-(d*e - c*f)/d)*sin_integral((d*f*x + c*f)/d) - (a^2 + b^2)*d)/ (d^3*x + c*d^2)
\[ \int \frac {(a+b \sin (e+f x))^2}{(c+d x)^2} \, dx=\int \frac {\left (a + b \sin {\left (e + f x \right )}\right )^{2}}{\left (c + d x\right )^{2}}\, dx \] Input:
integrate((a+b*sin(f*x+e))**2/(d*x+c)**2,x)
Output:
Integral((a + b*sin(e + f*x))**2/(c + d*x)**2, x)
Result contains complex when optimal does not.
Time = 0.17 (sec) , antiderivative size = 371, normalized size of antiderivative = 2.03 \[ \int \frac {(a+b \sin (e+f x))^2}{(c+d x)^2} \, dx=-\frac {\frac {4 \, a^{2} f^{2}}{{\left (f x + e\right )} d^{2} - d^{2} e + c d f} - \frac {4 \, {\left (f^{2} {\left (-i \, E_{2}\left (\frac {i \, {\left (f x + e\right )} d - i \, d e + i \, c f}{d}\right ) + i \, E_{2}\left (-\frac {i \, {\left (f x + e\right )} d - i \, d e + i \, c f}{d}\right )\right )} \cos \left (-\frac {d e - c f}{d}\right ) + f^{2} {\left (E_{2}\left (\frac {i \, {\left (f x + e\right )} d - i \, d e + i \, c f}{d}\right ) + E_{2}\left (-\frac {i \, {\left (f x + e\right )} d - i \, d e + i \, c f}{d}\right )\right )} \sin \left (-\frac {d e - c f}{d}\right )\right )} a b}{{\left (f x + e\right )} d^{2} - d^{2} e + c d f} - \frac {{\left (f^{2} {\left (E_{2}\left (\frac {2 \, {\left (-i \, {\left (f x + e\right )} d + i \, d e - i \, c f\right )}}{d}\right ) + E_{2}\left (-\frac {2 \, {\left (-i \, {\left (f x + e\right )} d + i \, d e - i \, c f\right )}}{d}\right )\right )} \cos \left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right ) - f^{2} {\left (i \, E_{2}\left (\frac {2 \, {\left (-i \, {\left (f x + e\right )} d + i \, d e - i \, c f\right )}}{d}\right ) - i \, E_{2}\left (-\frac {2 \, {\left (-i \, {\left (f x + e\right )} d + i \, d e - i \, c f\right )}}{d}\right )\right )} \sin \left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right ) - 2 \, f^{2}\right )} b^{2}}{{\left (f x + e\right )} d^{2} - d^{2} e + c d f}}{4 \, f} \] Input:
integrate((a+b*sin(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*(-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))*cos(-(d*e - c*f)/d) + 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))*sin(-(d*e - c*f)/d))*a*b/((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)*b^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. 1050 vs. \(2 (186) = 372\).
Time = 0.35 (sec) , antiderivative size = 1050, normalized size of antiderivative = 5.74 \[ \int \frac {(a+b \sin (e+f x))^2}{(c+d x)^2} \, dx=\text {Too large to display} \] Input:
integrate((a+b*sin(f*x+e))^2/(d*x+c)^2,x, algorithm="giac")
Output:
1/2*(4*(d*x + c)*a*b*(d*e/(d*x + c) - c*f/(d*x + c) + f)*f^2*cos(-(d*e - c *f)/d)*cos_integral(((d*x + c)*(d*e/(d*x + c) - c*f/(d*x + c) + f) - d*e + c*f)/d) - 4*a*b*d*e*f^2*cos(-(d*e - c*f)/d)*cos_integral(((d*x + c)*(d*e/ (d*x + c) - c*f/(d*x + c) + f) - d*e + c*f)/d) + 4*a*b*c*f^3*cos(-(d*e - c *f)/d)*cos_integral(((d*x + c)*(d*e/(d*x + c) - c*f/(d*x + c) + f) - d*e + c*f)/d) - 2*(d*x + c)*b^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*b^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*b^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)*b^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*b^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*b^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*b*(d*e/(d*x + c) - c*f/(d*x + c) + f)*f^2*sin(-(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*b*d*e*f^2*sin(-(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*b*c*f^3*sin(-(d*e - c*f)/d)*sin_integral(((d*x + c...
Timed out. \[ \int \frac {(a+b \sin (e+f x))^2}{(c+d x)^2} \, dx=\int \frac {{\left (a+b\,\sin \left (e+f\,x\right )\right )}^2}{{\left (c+d\,x\right )}^2} \,d x \] Input:
int((a + b*sin(e + f*x))^2/(c + d*x)^2,x)
Output:
int((a + b*sin(e + f*x))^2/(c + d*x)^2, x)
\[ \int \frac {(a+b \sin (e+f x))^2}{(c+d x)^2} \, dx=\frac {\left (\int \frac {\sin \left (f x +e \right )^{2}}{d^{2} x^{2}+2 c d x +c^{2}}d x \right ) b^{2} c^{2}+\left (\int \frac {\sin \left (f x +e \right )^{2}}{d^{2} x^{2}+2 c d x +c^{2}}d x \right ) b^{2} c d x +2 \left (\int \frac {\sin \left (f x +e \right )}{d^{2} x^{2}+2 c d x +c^{2}}d x \right ) a b \,c^{2}+2 \left (\int \frac {\sin \left (f x +e \right )}{d^{2} x^{2}+2 c d x +c^{2}}d x \right ) a b c d x +a^{2} x}{c \left (d x +c \right )} \] Input:
int((a+b*sin(f*x+e))^2/(d*x+c)^2,x)
Output:
(int(sin(e + f*x)**2/(c**2 + 2*c*d*x + d**2*x**2),x)*b**2*c**2 + int(sin(e + f*x)**2/(c**2 + 2*c*d*x + d**2*x**2),x)*b**2*c*d*x + 2*int(sin(e + f*x) /(c**2 + 2*c*d*x + d**2*x**2),x)*a*b*c**2 + 2*int(sin(e + f*x)/(c**2 + 2*c *d*x + d**2*x**2),x)*a*b*c*d*x + a**2*x)/(c*(c + d*x))