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

Optimal. Leaf size=150 \[ \frac {\sqrt {\pi } \cos \left (2 a-\frac {b^2}{2 c}\right ) (2 c d-b e) C\left (\frac {b+2 c x}{\sqrt {c} \sqrt {\pi }}\right )}{8 c^{3/2}}-\frac {\sqrt {\pi } \sin \left (2 a-\frac {b^2}{2 c}\right ) (2 c d-b e) S\left (\frac {b+2 c x}{\sqrt {c} \sqrt {\pi }}\right )}{8 c^{3/2}}+\frac {e \sin \left (2 a+2 b x+2 c x^2\right )}{8 c}+\frac {(d+e x)^2}{4 e} \]

[Out]

1/4*(e*x+d)^2/e+1/8*e*sin(2*c*x^2+2*b*x+2*a)/c+1/8*(-b*e+2*c*d)*cos(2*a-1/2*b^2/c)*FresnelC((2*c*x+b)/c^(1/2)/
Pi^(1/2))*Pi^(1/2)/c^(3/2)-1/8*(-b*e+2*c*d)*FresnelS((2*c*x+b)/c^(1/2)/Pi^(1/2))*sin(2*a-1/2*b^2/c)*Pi^(1/2)/c
^(3/2)

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Rubi [A]  time = 0.09, antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {3468, 3462, 3448, 3352, 3351} \[ \frac {\sqrt {\pi } \cos \left (2 a-\frac {b^2}{2 c}\right ) (2 c d-b e) \text {FresnelC}\left (\frac {b+2 c x}{\sqrt {\pi } \sqrt {c}}\right )}{8 c^{3/2}}-\frac {\sqrt {\pi } \sin \left (2 a-\frac {b^2}{2 c}\right ) (2 c d-b e) S\left (\frac {b+2 c x}{\sqrt {c} \sqrt {\pi }}\right )}{8 c^{3/2}}+\frac {e \sin \left (2 a+2 b x+2 c x^2\right )}{8 c}+\frac {(d+e x)^2}{4 e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(d + e*x)^2/(4*e) + ((2*c*d - b*e)*Sqrt[Pi]*Cos[2*a - b^2/(2*c)]*FresnelC[(b + 2*c*x)/(Sqrt[c]*Sqrt[Pi])])/(8*
c^(3/2)) - ((2*c*d - b*e)*Sqrt[Pi]*FresnelS[(b + 2*c*x)/(Sqrt[c]*Sqrt[Pi])]*Sin[2*a - b^2/(2*c)])/(8*c^(3/2))
+ (e*Sin[2*a + 2*b*x + 2*c*x^2])/(8*c)

Rule 3351

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

Rule 3352

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

Rule 3448

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

Rule 3462

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] + Dist[(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]

Rule 3468

Int[Cos[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]^(n_)*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandTrigReduce[
(d + e*x)^m, Cos[a + b*x + c*x^2]^n, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[n, 1]

Rubi steps

\begin {align*} \int (d+e x) \cos ^2\left (a+b x+c x^2\right ) \, dx &=\int \left (\frac {1}{2} (d+e x)+\frac {1}{2} (d+e x) \cos \left (2 a+2 b x+2 c x^2\right )\right ) \, dx\\ &=\frac {(d+e x)^2}{4 e}+\frac {1}{2} \int (d+e x) \cos \left (2 a+2 b x+2 c x^2\right ) \, dx\\ &=\frac {(d+e x)^2}{4 e}+\frac {e \sin \left (2 a+2 b x+2 c x^2\right )}{8 c}+\frac {(4 c d-2 b e) \int \cos \left (2 a+2 b x+2 c x^2\right ) \, dx}{8 c}\\ &=\frac {(d+e x)^2}{4 e}+\frac {e \sin \left (2 a+2 b x+2 c x^2\right )}{8 c}+\frac {\left ((2 c d-b e) \cos \left (2 a-\frac {b^2}{2 c}\right )\right ) \int \cos \left (\frac {(2 b+4 c x)^2}{8 c}\right ) \, dx}{4 c}-\frac {\left ((2 c d-b e) \sin \left (2 a-\frac {b^2}{2 c}\right )\right ) \int \sin \left (\frac {(2 b+4 c x)^2}{8 c}\right ) \, dx}{4 c}\\ &=\frac {(d+e x)^2}{4 e}+\frac {(2 c d-b e) \sqrt {\pi } \cos \left (2 a-\frac {b^2}{2 c}\right ) C\left (\frac {b+2 c x}{\sqrt {c} \sqrt {\pi }}\right )}{8 c^{3/2}}-\frac {(2 c d-b e) \sqrt {\pi } S\left (\frac {b+2 c x}{\sqrt {c} \sqrt {\pi }}\right ) \sin \left (2 a-\frac {b^2}{2 c}\right )}{8 c^{3/2}}+\frac {e \sin \left (2 a+2 b x+2 c x^2\right )}{8 c}\\ \end {align*}

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Mathematica [A]  time = 0.41, size = 139, normalized size = 0.93 \[ \frac {\sqrt {\pi } \cos \left (2 a-\frac {b^2}{2 c}\right ) (2 c d-b e) C\left (\frac {b+2 c x}{\sqrt {c} \sqrt {\pi }}\right )-\sqrt {\pi } \sin \left (2 a-\frac {b^2}{2 c}\right ) (2 c d-b e) S\left (\frac {b+2 c x}{\sqrt {c} \sqrt {\pi }}\right )+\sqrt {c} (e \sin (2 (a+x (b+c x)))+2 c x (2 d+e x))}{8 c^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((2*c*d - b*e)*Sqrt[Pi]*Cos[2*a - b^2/(2*c)]*FresnelC[(b + 2*c*x)/(Sqrt[c]*Sqrt[Pi])] - (2*c*d - b*e)*Sqrt[Pi]
*FresnelS[(b + 2*c*x)/(Sqrt[c]*Sqrt[Pi])]*Sin[2*a - b^2/(2*c)] + Sqrt[c]*(2*c*x*(2*d + e*x) + e*Sin[2*(a + x*(
b + c*x))]))/(8*c^(3/2))

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fricas [A]  time = 0.91, size = 149, normalized size = 0.99 \[ \frac {2 \, c^{2} e x^{2} + \pi {\left (2 \, c d - b e\right )} \sqrt {\frac {c}{\pi }} \cos \left (-\frac {b^{2} - 4 \, a c}{2 \, c}\right ) \operatorname {C}\left (\frac {{\left (2 \, c x + b\right )} \sqrt {\frac {c}{\pi }}}{c}\right ) - \pi {\left (2 \, c d - b e\right )} \sqrt {\frac {c}{\pi }} \operatorname {S}\left (\frac {{\left (2 \, c x + b\right )} \sqrt {\frac {c}{\pi }}}{c}\right ) \sin \left (-\frac {b^{2} - 4 \, a c}{2 \, c}\right ) + 4 \, c^{2} d x + 2 \, c e \cos \left (c x^{2} + b x + a\right ) \sin \left (c x^{2} + b x + a\right )}{8 \, c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(2*c^2*e*x^2 + pi*(2*c*d - b*e)*sqrt(c/pi)*cos(-1/2*(b^2 - 4*a*c)/c)*fresnel_cos((2*c*x + b)*sqrt(c/pi)/c)
- pi*(2*c*d - b*e)*sqrt(c/pi)*fresnel_sin((2*c*x + b)*sqrt(c/pi)/c)*sin(-1/2*(b^2 - 4*a*c)/c) + 4*c^2*d*x + 2
*c*e*cos(c*x^2 + b*x + a)*sin(c*x^2 + b*x + a))/c^2

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giac [C]  time = 0.74, size = 304, normalized size = 2.03 \[ \frac {1}{4} \, x^{2} e + \frac {1}{2} \, d x - \frac {\sqrt {\pi } d \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {c} {\left (2 \, x + \frac {b}{c}\right )} {\left (-\frac {i \, c}{{\left | c \right |}} + 1\right )}\right ) e^{\left (-\frac {i \, b^{2} - 4 i \, a c}{2 \, c}\right )}}{8 \, \sqrt {c} {\left (-\frac {i \, c}{{\left | c \right |}} + 1\right )}} - \frac {\sqrt {\pi } d \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {c} {\left (2 \, x + \frac {b}{c}\right )} {\left (\frac {i \, c}{{\left | c \right |}} + 1\right )}\right ) e^{\left (-\frac {-i \, b^{2} + 4 i \, a c}{2 \, c}\right )}}{8 \, \sqrt {c} {\left (\frac {i \, c}{{\left | c \right |}} + 1\right )}} + \frac {\frac {\sqrt {\pi } b \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {c} {\left (2 \, x + \frac {b}{c}\right )} {\left (-\frac {i \, c}{{\left | c \right |}} + 1\right )}\right ) e^{\left (-\frac {i \, b^{2} - 4 i \, a c - 2 \, c}{2 \, c}\right )}}{\sqrt {c} {\left (-\frac {i \, c}{{\left | c \right |}} + 1\right )}} - i \, e^{\left (2 i \, c x^{2} + 2 i \, b x + 2 i \, a + 1\right )}}{16 \, c} + \frac {\frac {\sqrt {\pi } b \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {c} {\left (2 \, x + \frac {b}{c}\right )} {\left (\frac {i \, c}{{\left | c \right |}} + 1\right )}\right ) e^{\left (-\frac {-i \, b^{2} + 4 i \, a c - 2 \, c}{2 \, c}\right )}}{\sqrt {c} {\left (\frac {i \, c}{{\left | c \right |}} + 1\right )}} + i \, e^{\left (-2 i \, c x^{2} - 2 i \, b x - 2 i \, a + 1\right )}}{16 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.04, size = 170, normalized size = 1.13 \[ \frac {e \sin \left (2 c \,x^{2}+2 b x +2 a \right )}{8 c}-\frac {e b \sqrt {\pi }\, \left (\cos \left (\frac {-4 c a +b^{2}}{2 c}\right ) \FresnelC \left (\frac {2 c x +b}{\sqrt {c}\, \sqrt {\pi }}\right )+\sin \left (\frac {-4 c a +b^{2}}{2 c}\right ) \mathrm {S}\left (\frac {2 c x +b}{\sqrt {c}\, \sqrt {\pi }}\right )\right )}{8 c^{\frac {3}{2}}}+\frac {\sqrt {\pi }\, d \left (\cos \left (\frac {-4 c a +b^{2}}{2 c}\right ) \FresnelC \left (\frac {2 c x +b}{\sqrt {c}\, \sqrt {\pi }}\right )+\sin \left (\frac {-4 c a +b^{2}}{2 c}\right ) \mathrm {S}\left (\frac {2 c x +b}{\sqrt {c}\, \sqrt {\pi }}\right )\right )}{4 \sqrt {c}}+\frac {d x}{2}+\frac {e \,x^{2}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*e*sin(2*c*x^2+2*b*x+2*a)/c-1/8*e*b/c^(3/2)*Pi^(1/2)*(cos(1/2*(-4*a*c+b^2)/c)*FresnelC((2*c*x+b)/c^(1/2)/Pi
^(1/2))+sin(1/2*(-4*a*c+b^2)/c)*FresnelS((2*c*x+b)/c^(1/2)/Pi^(1/2)))+1/4*Pi^(1/2)/c^(1/2)*d*(cos(1/2*(-4*a*c+
b^2)/c)*FresnelC((2*c*x+b)/c^(1/2)/Pi^(1/2))+sin(1/2*(-4*a*c+b^2)/c)*FresnelS((2*c*x+b)/c^(1/2)/Pi^(1/2)))+1/2
*d*x+1/4*e*x^2

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maxima [C]  time = 1.69, size = 735, normalized size = 4.90 \[ -\frac {{\left (4^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } {\left ({\left (\left (i - 1\right ) \, \cos \left (-\frac {b^{2} - 4 \, a c}{2 \, c}\right ) + \left (i + 1\right ) \, \sin \left (-\frac {b^{2} - 4 \, a c}{2 \, c}\right )\right )} \operatorname {erf}\left (\frac {2 i \, c x + i \, b}{\sqrt {2 i \, c}}\right ) + {\left (\left (i + 1\right ) \, \cos \left (-\frac {b^{2} - 4 \, a c}{2 \, c}\right ) + \left (i - 1\right ) \, \sin \left (-\frac {b^{2} - 4 \, a c}{2 \, c}\right )\right )} \operatorname {erf}\left (\frac {2 i \, c x + i \, b}{\sqrt {-2 i \, c}}\right )\right )} c^{\frac {3}{2}} - 16 \, c^{2} x\right )} d}{32 \, c^{2}} + \frac {\sqrt {2} {\left ({\left (\left (i - 1\right ) \, \sqrt {2} \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {\frac {1}{2}} \sqrt {\frac {4 i \, c^{2} x^{2} + 4 i \, b c x + i \, b^{2}}{c}}\right ) - 1\right )} - \left (i + 1\right ) \, \sqrt {2} \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {\frac {1}{2}} \sqrt {-\frac {4 i \, c^{2} x^{2} + 4 i \, b c x + i \, b^{2}}{c}}\right ) - 1\right )}\right )} b^{2} \cos \left (-\frac {b^{2} - 4 \, a c}{2 \, c}\right ) + {\left (\left (i + 1\right ) \, \sqrt {2} \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {\frac {1}{2}} \sqrt {\frac {4 i \, c^{2} x^{2} + 4 i \, b c x + i \, b^{2}}{c}}\right ) - 1\right )} - \left (i - 1\right ) \, \sqrt {2} \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {\frac {1}{2}} \sqrt {-\frac {4 i \, c^{2} x^{2} + 4 i \, b c x + i \, b^{2}}{c}}\right ) - 1\right )}\right )} b^{2} \sin \left (-\frac {b^{2} - 4 \, a c}{2 \, c}\right ) + {\left ({\left (\left (2 i - 2\right ) \, \sqrt {2} \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {\frac {1}{2}} \sqrt {\frac {4 i \, c^{2} x^{2} + 4 i \, b c x + i \, b^{2}}{c}}\right ) - 1\right )} - \left (2 i + 2\right ) \, \sqrt {2} \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {\frac {1}{2}} \sqrt {-\frac {4 i \, c^{2} x^{2} + 4 i \, b c x + i \, b^{2}}{c}}\right ) - 1\right )}\right )} b c \cos \left (-\frac {b^{2} - 4 \, a c}{2 \, c}\right ) + {\left (\left (2 i + 2\right ) \, \sqrt {2} \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {\frac {1}{2}} \sqrt {\frac {4 i \, c^{2} x^{2} + 4 i \, b c x + i \, b^{2}}{c}}\right ) - 1\right )} - \left (2 i - 2\right ) \, \sqrt {2} \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {\frac {1}{2}} \sqrt {-\frac {4 i \, c^{2} x^{2} + 4 i \, b c x + i \, b^{2}}{c}}\right ) - 1\right )}\right )} b c \sin \left (-\frac {b^{2} - 4 \, a c}{2 \, c}\right )\right )} x + \sqrt {2} {\left (8 \, c^{2} x^{2} + c {\left (-2 i \, e^{\left (\frac {4 i \, c^{2} x^{2} + 4 i \, b c x + i \, b^{2}}{2 \, c}\right )} + 2 i \, e^{\left (-\frac {4 i \, c^{2} x^{2} + 4 i \, b c x + i \, b^{2}}{2 \, c}\right )}\right )} \cos \left (-\frac {b^{2} - 4 \, a c}{2 \, c}\right ) + 2 \, c {\left (e^{\left (\frac {4 i \, c^{2} x^{2} + 4 i \, b c x + i \, b^{2}}{2 \, c}\right )} + e^{\left (-\frac {4 i \, c^{2} x^{2} + 4 i \, b c x + i \, b^{2}}{2 \, c}\right )}\right )} \sin \left (-\frac {b^{2} - 4 \, a c}{2 \, c}\right )\right )} \sqrt {\frac {4 \, c^{2} x^{2} + 4 \, b c x + b^{2}}{c}}\right )} e}{64 \, c^{2} \sqrt {\frac {4 \, c^{2} x^{2} + 4 \, b c x + b^{2}}{c}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/32*(4^(1/4)*sqrt(2)*sqrt(pi)*(((I - 1)*cos(-1/2*(b^2 - 4*a*c)/c) + (I + 1)*sin(-1/2*(b^2 - 4*a*c)/c))*erf((
2*I*c*x + I*b)/sqrt(2*I*c)) + ((I + 1)*cos(-1/2*(b^2 - 4*a*c)/c) + (I - 1)*sin(-1/2*(b^2 - 4*a*c)/c))*erf((2*I
*c*x + I*b)/sqrt(-2*I*c)))*c^(3/2) - 16*c^2*x)*d/c^2 + 1/64*sqrt(2)*(((I - 1)*sqrt(2)*sqrt(pi)*(erf(sqrt(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(sqrt(1/2)*sqrt(-(4*I*c^2*x^2 +
4*I*b*c*x + I*b^2)/c)) - 1))*b^2*cos(-1/2*(b^2 - 4*a*c)/c) + ((I + 1)*sqrt(2)*sqrt(pi)*(erf(sqrt(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(sqrt(1/2)*sqrt(-(4*I*c^2*x^2 + 4*I*b*
c*x + I*b^2)/c)) - 1))*b^2*sin(-1/2*(b^2 - 4*a*c)/c) + (((2*I - 2)*sqrt(2)*sqrt(pi)*(erf(sqrt(1/2)*sqrt((4*I*c
^2*x^2 + 4*I*b*c*x + I*b^2)/c)) - 1) - (2*I + 2)*sqrt(2)*sqrt(pi)*(erf(sqrt(1/2)*sqrt(-(4*I*c^2*x^2 + 4*I*b*c*
x + I*b^2)/c)) - 1))*b*c*cos(-1/2*(b^2 - 4*a*c)/c) + ((2*I + 2)*sqrt(2)*sqrt(pi)*(erf(sqrt(1/2)*sqrt((4*I*c^2*
x^2 + 4*I*b*c*x + I*b^2)/c)) - 1) - (2*I - 2)*sqrt(2)*sqrt(pi)*(erf(sqrt(1/2)*sqrt(-(4*I*c^2*x^2 + 4*I*b*c*x +
I*b^2)/c)) - 1))*b*c*sin(-1/2*(b^2 - 4*a*c)/c))*x + sqrt(2)*(8*c^2*x^2 + c*(-2*I*e^(1/2*(4*I*c^2*x^2 + 4*I*b*
c*x + I*b^2)/c) + 2*I*e^(-1/2*(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c))*cos(-1/2*(b^2 - 4*a*c)/c) + 2*c*(e^(1/2*(4
*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c) + e^(-1/2*(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c))*sin(-1/2*(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))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int {\cos \left (c\,x^2+b\,x+a\right )}^2\,\left (d+e\,x\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (d + e x\right ) \cos ^{2}{\left (a + b x + c x^{2} \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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