\(\int (d+e x) \cos (a+b x+c x^2) \, dx\) [30]

Optimal result

Integrand size = 17, antiderivative size = 140 \[ \int (d+e x) \cos \left (a+b x+c x^2\right ) \, dx=\frac {(2 c d-b e) \sqrt {\frac {\pi }{2}} \cos \left (a-\frac {b^2}{4 c}\right ) \operatorname {FresnelC}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right )}{2 c^{3/2}}-\frac {(2 c d-b e) \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right ) \sin \left (a-\frac {b^2}{4 c}\right )}{2 c^{3/2}}+\frac {e \sin \left (a+b x+c x^2\right )}{2 c} \] Output:

1/4*(-b*e+2*c*d)*2^(1/2)*Pi^(1/2)*cos(a-1/4*b^2/c)*FresnelC(1/2*(2*c*x+b)/ 
c^(1/2)*2^(1/2)/Pi^(1/2))/c^(3/2)-1/4*(-b*e+2*c*d)*2^(1/2)*Pi^(1/2)*Fresne 
lS(1/2*(2*c*x+b)/c^(1/2)*2^(1/2)/Pi^(1/2))*sin(a-1/4*b^2/c)/c^(3/2)+1/2*e* 
sin(c*x^2+b*x+a)/c
 

Mathematica [A] (verified)

Time = 0.73 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.92 \[ \int (d+e x) \cos \left (a+b x+c x^2\right ) \, dx=\frac {(2 c d-b e) \sqrt {2 \pi } \cos \left (a-\frac {b^2}{4 c}\right ) \operatorname {FresnelC}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right )-(2 c d-b e) \sqrt {2 \pi } \operatorname {FresnelS}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right ) \sin \left (a-\frac {b^2}{4 c}\right )+2 \sqrt {c} e \sin (a+x (b+c x))}{4 c^{3/2}} \] Input:

Integrate[(d + e*x)*Cos[a + b*x + c*x^2],x]
 

Output:

((2*c*d - b*e)*Sqrt[2*Pi]*Cos[a - b^2/(4*c)]*FresnelC[(b + 2*c*x)/(Sqrt[c] 
*Sqrt[2*Pi])] - (2*c*d - b*e)*Sqrt[2*Pi]*FresnelS[(b + 2*c*x)/(Sqrt[c]*Sqr 
t[2*Pi])]*Sin[a - b^2/(4*c)] + 2*Sqrt[c]*e*Sin[a + x*(b + c*x)])/(4*c^(3/2 
))
 

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.96, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {3943, 3929, 3832, 3833}

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.

\(\displaystyle \int (d+e x) \cos \left (a+b x+c x^2\right ) \, dx\)

\(\Big \downarrow \) 3943

\(\displaystyle \frac {(2 c d-b e) \int \cos \left (c x^2+b x+a\right )dx}{2 c}+\frac {e \sin \left (a+b x+c x^2\right )}{2 c}\)

\(\Big \downarrow \) 3929

\(\displaystyle \frac {(2 c d-b e) \left (\cos \left (a-\frac {b^2}{4 c}\right ) \int \cos \left (\frac {(b+2 c x)^2}{4 c}\right )dx-\sin \left (a-\frac {b^2}{4 c}\right ) \int \sin \left (\frac {(b+2 c x)^2}{4 c}\right )dx\right )}{2 c}+\frac {e \sin \left (a+b x+c x^2\right )}{2 c}\)

\(\Big \downarrow \) 3832

\(\displaystyle \frac {(2 c d-b e) \left (\cos \left (a-\frac {b^2}{4 c}\right ) \int \cos \left (\frac {(b+2 c x)^2}{4 c}\right )dx-\frac {\sqrt {\frac {\pi }{2}} \sin \left (a-\frac {b^2}{4 c}\right ) \operatorname {FresnelS}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right )}{\sqrt {c}}\right )}{2 c}+\frac {e \sin \left (a+b x+c x^2\right )}{2 c}\)

\(\Big \downarrow \) 3833

\(\displaystyle \frac {(2 c d-b e) \left (\frac {\sqrt {\frac {\pi }{2}} \cos \left (a-\frac {b^2}{4 c}\right ) \operatorname {FresnelC}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right )}{\sqrt {c}}-\frac {\sqrt {\frac {\pi }{2}} \sin \left (a-\frac {b^2}{4 c}\right ) \operatorname {FresnelS}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right )}{\sqrt {c}}\right )}{2 c}+\frac {e \sin \left (a+b x+c x^2\right )}{2 c}\)

Input:

Int[(d + e*x)*Cos[a + b*x + c*x^2],x]
 

Output:

((2*c*d - b*e)*((Sqrt[Pi/2]*Cos[a - b^2/(4*c)]*FresnelC[(b + 2*c*x)/(Sqrt[ 
c]*Sqrt[2*Pi])])/Sqrt[c] - (Sqrt[Pi/2]*FresnelS[(b + 2*c*x)/(Sqrt[c]*Sqrt[ 
2*Pi])]*Sin[a - b^2/(4*c)])/Sqrt[c]))/(2*c) + (e*Sin[a + b*x + c*x^2])/(2* 
c)
 

Defintions of rubi rules used

rule 3832

Int[Sin[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[ 
d, 2]))*FresnelS[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]
 

rule 3833

Int[Cos[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[ 
d, 2]))*FresnelC[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]
 

rule 3929

Int[Cos[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[Cos[(b^2 - 4* 
a*c)/(4*c)]   Int[Cos[(b + 2*c*x)^2/(4*c)], x], x] + Simp[Sin[(b^2 - 4*a*c) 
/(4*c)]   Int[Sin[(b + 2*c*x)^2/(4*c)], x], x] /; FreeQ[{a, b, c}, x] && Ne 
Q[b^2 - 4*a*c, 0]
 

rule 3943

Int[Cos[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]*((d_.) + (e_.)*(x_)), x_Symbol] 
:> Simp[e*(Sin[a + b*x + c*x^2]/(2*c)), x] + Simp[(2*c*d - b*e)/(2*c)   Int 
[Cos[a + b*x + c*x^2], x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b 
*e, 0]
 

Maple [A] (verified)

Time = 1.24 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.30

method	result	size
default	\(\frac {e \sin \left (c \,x^{2}+b x +a \right )}{2 c}-\frac {e b \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (\frac {\frac {b^{2}}{4}-a c}{c}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \left (c x +\frac {b}{2}\right )}{\sqrt {\pi }\, \sqrt {c}}\right )+\sin \left (\frac {\frac {b^{2}}{4}-a c}{c}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \left (c x +\frac {b}{2}\right )}{\sqrt {\pi }\, \sqrt {c}}\right )\right )}{4 c^{\frac {3}{2}}}+\frac {\sqrt {2}\, \sqrt {\pi }\, d \left (\cos \left (\frac {\frac {b^{2}}{4}-a c}{c}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \left (c x +\frac {b}{2}\right )}{\sqrt {\pi }\, \sqrt {c}}\right )+\sin \left (\frac {\frac {b^{2}}{4}-a c}{c}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \left (c x +\frac {b}{2}\right )}{\sqrt {\pi }\, \sqrt {c}}\right )\right )}{2 \sqrt {c}}\)	\(182\)
risch	\(\frac {\operatorname {erf}\left (\sqrt {i c}\, x +\frac {i b}{2 \sqrt {i c}}\right ) \sqrt {\pi }\, d \,{\mathrm e}^{-\frac {i \left (4 a c -b^{2}\right )}{4 c}}}{4 \sqrt {i c}}-\frac {e \,\operatorname {erf}\left (\sqrt {i c}\, x +\frac {i b}{2 \sqrt {i c}}\right ) \sqrt {\pi }\, b \,{\mathrm e}^{-\frac {i \left (4 a c -b^{2}\right )}{4 c}}}{8 \sqrt {i c}\, c}-\frac {\operatorname {erf}\left (-\sqrt {-i c}\, x +\frac {i b}{2 \sqrt {-i c}}\right ) \sqrt {\pi }\, d \,{\mathrm e}^{\frac {i \left (4 a c -b^{2}\right )}{4 c}}}{4 \sqrt {-i c}}+\frac {e \,\operatorname {erf}\left (-\sqrt {-i c}\, x +\frac {i b}{2 \sqrt {-i c}}\right ) \sqrt {\pi }\, b \,{\mathrm e}^{\frac {i \left (4 a c -b^{2}\right )}{4 c}}}{8 \sqrt {-i c}\, c}+\frac {e \sin \left (c \,x^{2}+b x +a \right )}{2 c}\)	\(225\)
parts	\(\frac {\sqrt {2}\, \sqrt {\pi }\, \cos \left (\frac {\frac {b^{2}}{4}-a c}{c}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \left (c x +\frac {b}{2}\right )}{\sqrt {\pi }\, \sqrt {c}}\right ) e x}{2 \sqrt {c}}+\frac {\sqrt {2}\, \sqrt {\pi }\, \sin \left (\frac {\frac {b^{2}}{4}-a c}{c}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \left (c x +\frac {b}{2}\right )}{\sqrt {\pi }\, \sqrt {c}}\right ) e x}{2 \sqrt {c}}+\frac {\sqrt {2}\, \sqrt {\pi }\, \cos \left (\frac {\frac {b^{2}}{4}-a c}{c}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \left (c x +\frac {b}{2}\right )}{\sqrt {\pi }\, \sqrt {c}}\right ) d}{2 \sqrt {c}}+\frac {\sqrt {2}\, \sqrt {\pi }\, \sin \left (\frac {\frac {b^{2}}{4}-a c}{c}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \left (c x +\frac {b}{2}\right )}{\sqrt {\pi }\, \sqrt {c}}\right ) d}{2 \sqrt {c}}-\frac {\sqrt {2}\, \sqrt {\pi }\, e \left (\frac {\cos \left (\frac {\frac {b^{2}}{4}-a c}{c}\right ) \sqrt {2}\, \sqrt {\pi }\, \left (\operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {c}\, x}{\sqrt {\pi }}+\frac {\sqrt {2}\, b}{2 \sqrt {\pi }\, \sqrt {c}}\right ) \left (\frac {\sqrt {2}\, \sqrt {c}\, x}{\sqrt {\pi }}+\frac {\sqrt {2}\, b}{2 \sqrt {\pi }\, \sqrt {c}}\right )-\frac {\sin \left (\frac {\pi \left (\frac {\sqrt {2}\, \sqrt {c}\, x}{\sqrt {\pi }}+\frac {\sqrt {2}\, b}{2 \sqrt {\pi }\, \sqrt {c}}\right )^{2}}{2}\right )}{\pi }\right )}{2 \sqrt {c}}+\frac {\sin \left (\frac {\frac {b^{2}}{4}-a c}{c}\right ) \sqrt {2}\, \sqrt {\pi }\, \left (\operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {c}\, x}{\sqrt {\pi }}+\frac {\sqrt {2}\, b}{2 \sqrt {\pi }\, \sqrt {c}}\right ) \left (\frac {\sqrt {2}\, \sqrt {c}\, x}{\sqrt {\pi }}+\frac {\sqrt {2}\, b}{2 \sqrt {\pi }\, \sqrt {c}}\right )+\frac {\cos \left (\frac {\pi \left (\frac {\sqrt {2}\, \sqrt {c}\, x}{\sqrt {\pi }}+\frac {\sqrt {2}\, b}{2 \sqrt {\pi }\, \sqrt {c}}\right )^{2}}{2}\right )}{\pi }\right )}{2 \sqrt {c}}\right )}{2 \sqrt {c}}\)	\(420\)

Input:

int((e*x+d)*cos(c*x^2+b*x+a),x,method=_RETURNVERBOSE)
 

Output:

1/2*e*sin(c*x^2+b*x+a)/c-1/4*e*b/c^(3/2)*2^(1/2)*Pi^(1/2)*(cos((1/4*b^2-a* 
c)/c)*FresnelC(2^(1/2)/Pi^(1/2)/c^(1/2)*(c*x+1/2*b))+sin((1/4*b^2-a*c)/c)* 
FresnelS(2^(1/2)/Pi^(1/2)/c^(1/2)*(c*x+1/2*b)))+1/2*2^(1/2)*Pi^(1/2)/c^(1/ 
2)*d*(cos((1/4*b^2-a*c)/c)*FresnelC(2^(1/2)/Pi^(1/2)/c^(1/2)*(c*x+1/2*b))+ 
sin((1/4*b^2-a*c)/c)*FresnelS(2^(1/2)/Pi^(1/2)/c^(1/2)*(c*x+1/2*b)))
 

Fricas [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.97 \[ \int (d+e x) \cos \left (a+b x+c x^2\right ) \, dx=\frac {\sqrt {2} \pi {\left (2 \, c d - b e\right )} \sqrt {\frac {c}{\pi }} \cos \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right ) \operatorname {C}\left (\frac {\sqrt {2} {\left (2 \, c x + b\right )} \sqrt {\frac {c}{\pi }}}{2 \, c}\right ) - \sqrt {2} \pi {\left (2 \, c d - b e\right )} \sqrt {\frac {c}{\pi }} \operatorname {S}\left (\frac {\sqrt {2} {\left (2 \, c x + b\right )} \sqrt {\frac {c}{\pi }}}{2 \, c}\right ) \sin \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right ) + 2 \, c e \sin \left (c x^{2} + b x + a\right )}{4 \, c^{2}} \] Input:

integrate((e*x+d)*cos(c*x^2+b*x+a),x, algorithm="fricas")
 

Output:

1/4*(sqrt(2)*pi*(2*c*d - b*e)*sqrt(c/pi)*cos(-1/4*(b^2 - 4*a*c)/c)*fresnel 
_cos(1/2*sqrt(2)*(2*c*x + b)*sqrt(c/pi)/c) - sqrt(2)*pi*(2*c*d - b*e)*sqrt 
(c/pi)*fresnel_sin(1/2*sqrt(2)*(2*c*x + b)*sqrt(c/pi)/c)*sin(-1/4*(b^2 - 4 
*a*c)/c) + 2*c*e*sin(c*x^2 + b*x + a))/c^2
 

Sympy [F]

\[ \int (d+e x) \cos \left (a+b x+c x^2\right ) \, dx=\int \left (d + e x\right ) \cos {\left (a + b x + c x^{2} \right )}\, dx \] Input:

integrate((e*x+d)*cos(c*x**2+b*x+a),x)
 

Output:

Integral((d + e*x)*cos(a + b*x + c*x**2), x)
 

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.41 (sec) , antiderivative size = 695, normalized size of antiderivative = 4.96 \[ \int (d+e x) \cos \left (a+b x+c x^2\right ) \, dx =\text {Too large to display} \] Input:

integrate((e*x+d)*cos(c*x^2+b*x+a),x, algorithm="maxima")
 

Output:

-1/8*sqrt(2)*sqrt(pi)*(((I - 1)*cos(-1/4*(b^2 - 4*a*c)/c) + (I + 1)*sin(-1 
/4*(b^2 - 4*a*c)/c))*erf(1/2*(2*I*c*x + I*b)/sqrt(I*c)) + ((I + 1)*cos(-1/ 
4*(b^2 - 4*a*c)/c) + (I - 1)*sin(-1/4*(b^2 - 4*a*c)/c))*erf(1/2*(2*I*c*x + 
 I*b)/sqrt(-I*c)))*d/sqrt(c) + 1/16*(((I - 1)*sqrt(2)*sqrt(pi)*(erf(1/2*sq 
rt((4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c)) - 1) - (I + 1)*sqrt(2)*sqrt(pi)*( 
erf(1/2*sqrt(-(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c)) - 1))*b^2*cos(-1/4*(b^ 
2 - 4*a*c)/c) + ((I + 1)*sqrt(2)*sqrt(pi)*(erf(1/2*sqrt((4*I*c^2*x^2 + 4*I 
*b*c*x + I*b^2)/c)) - 1) - (I - 1)*sqrt(2)*sqrt(pi)*(erf(1/2*sqrt(-(4*I*c^ 
2*x^2 + 4*I*b*c*x + I*b^2)/c)) - 1))*b^2*sin(-1/4*(b^2 - 4*a*c)/c) - 2*((- 
(I - 1)*sqrt(2)*sqrt(pi)*(erf(1/2*sqrt((4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c 
)) - 1) + (I + 1)*sqrt(2)*sqrt(pi)*(erf(1/2*sqrt(-(4*I*c^2*x^2 + 4*I*b*c*x 
 + I*b^2)/c)) - 1))*b*c*cos(-1/4*(b^2 - 4*a*c)/c) + (-(I + 1)*sqrt(2)*sqrt 
(pi)*(erf(1/2*sqrt((4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c)) - 1) + (I - 1)*sq 
rt(2)*sqrt(pi)*(erf(1/2*sqrt(-(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c)) - 1))* 
b*c*sin(-1/4*(b^2 - 4*a*c)/c))*x - 4*(c*(I*e^(1/4*(4*I*c^2*x^2 + 4*I*b*c*x 
 + I*b^2)/c) - I*e^(-1/4*(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c))*cos(-1/4*(b 
^2 - 4*a*c)/c) - c*(e^(1/4*(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c) + e^(-1/4* 
(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c))*sin(-1/4*(b^2 - 4*a*c)/c))*sqrt((4*c 
^2*x^2 + 4*b*c*x + b^2)/c))*e/(c^2*sqrt((4*c^2*x^2 + 4*b*c*x + b^2)/c))
 

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.37 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.44 \[ \int (d+e x) \cos \left (a+b x+c x^2\right ) \, dx=-\frac {-\frac {i \, \sqrt {2} \sqrt {\pi } {\left (2 i \, c d - i \, b e\right )} \operatorname {erf}\left (-\frac {1}{4} \, \sqrt {2} {\left (2 \, x + \frac {b}{c}\right )} {\left (-\frac {i \, c}{{\left | c \right |}} + 1\right )} \sqrt {{\left | c \right |}}\right ) e^{\left (-\frac {i \, b^{2} - 4 i \, a c}{4 \, c}\right )}}{{\left (-\frac {i \, c}{{\left | c \right |}} + 1\right )} \sqrt {{\left | c \right |}}} + 2 i \, e e^{\left (i \, c x^{2} + i \, b x + i \, a\right )}}{8 \, c} - \frac {\frac {i \, \sqrt {2} \sqrt {\pi } {\left (-2 i \, c d + i \, b e\right )} \operatorname {erf}\left (-\frac {1}{4} \, \sqrt {2} {\left (2 \, x + \frac {b}{c}\right )} {\left (\frac {i \, c}{{\left | c \right |}} + 1\right )} \sqrt {{\left | c \right |}}\right ) e^{\left (-\frac {-i \, b^{2} + 4 i \, a c}{4 \, c}\right )}}{{\left (\frac {i \, c}{{\left | c \right |}} + 1\right )} \sqrt {{\left | c \right |}}} - 2 i \, e e^{\left (-i \, c x^{2} - i \, b x - i \, a\right )}}{8 \, c} \] Input:

integrate((e*x+d)*cos(c*x^2+b*x+a),x, algorithm="giac")
 

Output:

-1/8*(-I*sqrt(2)*sqrt(pi)*(2*I*c*d - I*b*e)*erf(-1/4*sqrt(2)*(2*x + b/c)*( 
-I*c/abs(c) + 1)*sqrt(abs(c)))*e^(-1/4*(I*b^2 - 4*I*a*c)/c)/((-I*c/abs(c) 
+ 1)*sqrt(abs(c))) + 2*I*e*e^(I*c*x^2 + I*b*x + I*a))/c - 1/8*(I*sqrt(2)*s 
qrt(pi)*(-2*I*c*d + I*b*e)*erf(-1/4*sqrt(2)*(2*x + b/c)*(I*c/abs(c) + 1)*s 
qrt(abs(c)))*e^(-1/4*(-I*b^2 + 4*I*a*c)/c)/((I*c/abs(c) + 1)*sqrt(abs(c))) 
 - 2*I*e*e^(-I*c*x^2 - I*b*x - I*a))/c
 

Mupad [F(-1)]

Timed out. \[ \int (d+e x) \cos \left (a+b x+c x^2\right ) \, dx=\int \cos \left (c\,x^2+b\,x+a\right )\,\left (d+e\,x\right ) \,d x \] Input:

int(cos(a + b*x + c*x^2)*(d + e*x),x)
 

Output:

int(cos(a + b*x + c*x^2)*(d + e*x), x)
 

Reduce [F]

\[ \int (d+e x) \cos \left (a+b x+c x^2\right ) \, dx=\frac {-2 \left (\int \frac {1}{\tan \left (\frac {1}{2} c \,x^{2}+\frac {1}{2} b x +\frac {1}{2} a \right )^{2}+1}d x \right ) b e +4 \left (\int \frac {1}{\tan \left (\frac {1}{2} c \,x^{2}+\frac {1}{2} b x +\frac {1}{2} a \right )^{2}+1}d x \right ) c d +\sin \left (c \,x^{2}+b x +a \right ) e +b e x -2 c d x}{2 c} \] Input:

int((e*x+d)*cos(c*x^2+b*x+a),x)
 

Output:

( - 2*int(1/(tan((a + b*x + c*x**2)/2)**2 + 1),x)*b*e + 4*int(1/(tan((a + 
b*x + c*x**2)/2)**2 + 1),x)*c*d + sin(a + b*x + c*x**2)*e + b*e*x - 2*c*d* 
x)/(2*c)