3.32 \(\int (d+e x)^2 \cos ^2(a+b x+c x^2) \, dx\)
Optimal. Leaf size=291 \[ \frac{\sqrt{\pi } \cos \left (2 a-\frac{b^2}{2 c}\right ) (2 c d-b e)^2 \text{FresnelC}\left (\frac{b+2 c x}{\sqrt{\pi } \sqrt{c}}\right )}{16 c^{5/2}}-\frac{\sqrt{\pi } \sin \left (2 a-\frac{b^2}{2 c}\right ) (2 c d-b e)^2 S\left (\frac{b+2 c x}{\sqrt{c} \sqrt{\pi }}\right )}{16 c^{5/2}}-\frac{\sqrt{\pi } e^2 \sin \left (2 a-\frac{b^2}{2 c}\right ) \text{FresnelC}\left (\frac{b+2 c x}{\sqrt{\pi } \sqrt{c}}\right )}{16 c^{3/2}}-\frac{\sqrt{\pi } e^2 \cos \left (2 a-\frac{b^2}{2 c}\right ) S\left (\frac{b+2 c x}{\sqrt{c} \sqrt{\pi }}\right )}{16 c^{3/2}}+\frac{e (2 c d-b e) \sin \left (2 a+2 b x+2 c x^2\right )}{16 c^2}+\frac{e (d+e x) \sin \left (2 a+2 b x+2 c x^2\right )}{8 c}+\frac{(d+e x)^3}{6 e} \]
[Out]
(d + e*x)^3/(6*e) + ((2*c*d - b*e)^2*Sqrt[Pi]*Cos[2*a - b^2/(2*c)]*FresnelC[(b + 2*c*x)/(Sqrt[c]*Sqrt[Pi])])/(
16*c^(5/2)) - (e^2*Sqrt[Pi]*Cos[2*a - b^2/(2*c)]*FresnelS[(b + 2*c*x)/(Sqrt[c]*Sqrt[Pi])])/(16*c^(3/2)) - (e^2
*Sqrt[Pi]*FresnelC[(b + 2*c*x)/(Sqrt[c]*Sqrt[Pi])]*Sin[2*a - b^2/(2*c)])/(16*c^(3/2)) - ((2*c*d - b*e)^2*Sqrt[
Pi]*FresnelS[(b + 2*c*x)/(Sqrt[c]*Sqrt[Pi])]*Sin[2*a - b^2/(2*c)])/(16*c^(5/2)) + (e*(2*c*d - b*e)*Sin[2*a + 2
*b*x + 2*c*x^2])/(16*c^2) + (e*(d + e*x)*Sin[2*a + 2*b*x + 2*c*x^2])/(8*c)
________________________________________________________________________________________
Rubi [A] time = 0.363667, antiderivative size = 291, normalized size of antiderivative = 1.,
number of steps used = 10, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used
= {3468, 3464, 3447, 3351, 3352, 3462, 3448} \[ \frac{\sqrt{\pi } \cos \left (2 a-\frac{b^2}{2 c}\right ) (2 c d-b e)^2 \text{FresnelC}\left (\frac{b+2 c x}{\sqrt{\pi } \sqrt{c}}\right )}{16 c^{5/2}}-\frac{\sqrt{\pi } \sin \left (2 a-\frac{b^2}{2 c}\right ) (2 c d-b e)^2 S\left (\frac{b+2 c x}{\sqrt{c} \sqrt{\pi }}\right )}{16 c^{5/2}}-\frac{\sqrt{\pi } e^2 \sin \left (2 a-\frac{b^2}{2 c}\right ) \text{FresnelC}\left (\frac{b+2 c x}{\sqrt{\pi } \sqrt{c}}\right )}{16 c^{3/2}}-\frac{\sqrt{\pi } e^2 \cos \left (2 a-\frac{b^2}{2 c}\right ) S\left (\frac{b+2 c x}{\sqrt{c} \sqrt{\pi }}\right )}{16 c^{3/2}}+\frac{e (2 c d-b e) \sin \left (2 a+2 b x+2 c x^2\right )}{16 c^2}+\frac{e (d+e x) \sin \left (2 a+2 b x+2 c x^2\right )}{8 c}+\frac{(d+e x)^3}{6 e} \]
Antiderivative was successfully verified.
[In]
Int[(d + e*x)^2*Cos[a + b*x + c*x^2]^2,x]
[Out]
(d + e*x)^3/(6*e) + ((2*c*d - b*e)^2*Sqrt[Pi]*Cos[2*a - b^2/(2*c)]*FresnelC[(b + 2*c*x)/(Sqrt[c]*Sqrt[Pi])])/(
16*c^(5/2)) - (e^2*Sqrt[Pi]*Cos[2*a - b^2/(2*c)]*FresnelS[(b + 2*c*x)/(Sqrt[c]*Sqrt[Pi])])/(16*c^(3/2)) - (e^2
*Sqrt[Pi]*FresnelC[(b + 2*c*x)/(Sqrt[c]*Sqrt[Pi])]*Sin[2*a - b^2/(2*c)])/(16*c^(3/2)) - ((2*c*d - b*e)^2*Sqrt[
Pi]*FresnelS[(b + 2*c*x)/(Sqrt[c]*Sqrt[Pi])]*Sin[2*a - b^2/(2*c)])/(16*c^(5/2)) + (e*(2*c*d - b*e)*Sin[2*a + 2
*b*x + 2*c*x^2])/(16*c^2) + (e*(d + e*x)*Sin[2*a + 2*b*x + 2*c*x^2])/(8*c)
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]
Rule 3464
Int[Cos[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]*((d_.) + (e_.)*(x_))^(m_), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1)*S
in[a + b*x + c*x^2])/(2*c), x] + (-Dist[(e^2*(m - 1))/(2*c), Int[(d + e*x)^(m - 2)*Sin[a + b*x + c*x^2], x], x
] - Dist[(b*e - 2*c*d)/(2*c), Int[(d + e*x)^(m - 1)*Cos[a + b*x + c*x^2], x], x]) /; FreeQ[{a, b, c, d, e}, x]
&& NeQ[b*e - 2*c*d, 0] && GtQ[m, 1]
Rule 3447
Int[Sin[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[Cos[(b^2 - 4*a*c)/(4*c)], Int[Sin[(b + 2*c*x)^2/
(4*c)], x], x] - Dist[Sin[(b^2 - 4*a*c)/(4*c)], Int[Cos[(b + 2*c*x)^2/(4*c)], x], x] /; FreeQ[{a, b, c}, x] &&
NeQ[b^2 - 4*a*c, 0]
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 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 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]
Rubi steps
\begin{align*} \int (d+e x)^2 \cos ^2\left (a+b x+c x^2\right ) \, dx &=\int \left (\frac{1}{2} (d+e x)^2+\frac{1}{2} (d+e x)^2 \cos \left (2 a+2 b x+2 c x^2\right )\right ) \, dx\\ &=\frac{(d+e x)^3}{6 e}+\frac{1}{2} \int (d+e x)^2 \cos \left (2 a+2 b x+2 c x^2\right ) \, dx\\ &=\frac{(d+e x)^3}{6 e}+\frac{e (d+e x) \sin \left (2 a+2 b x+2 c x^2\right )}{8 c}-\frac{e^2 \int \sin \left (2 a+2 b x+2 c x^2\right ) \, dx}{8 c}+\frac{(2 c d-b e) \int (d+e x) \cos \left (2 a+2 b x+2 c x^2\right ) \, dx}{4 c}\\ &=\frac{(d+e x)^3}{6 e}+\frac{e (2 c d-b e) \sin \left (2 a+2 b x+2 c x^2\right )}{16 c^2}+\frac{e (d+e x) \sin \left (2 a+2 b x+2 c x^2\right )}{8 c}+\frac{(2 c d-b e)^2 \int \cos \left (2 a+2 b x+2 c x^2\right ) \, dx}{8 c^2}-\frac{\left (e^2 \cos \left (2 a-\frac{b^2}{2 c}\right )\right ) \int \sin \left (\frac{(2 b+4 c x)^2}{8 c}\right ) \, dx}{8 c}-\frac{\left (e^2 \sin \left (2 a-\frac{b^2}{2 c}\right )\right ) \int \cos \left (\frac{(2 b+4 c x)^2}{8 c}\right ) \, dx}{8 c}\\ &=\frac{(d+e x)^3}{6 e}-\frac{e^2 \sqrt{\pi } \cos \left (2 a-\frac{b^2}{2 c}\right ) S\left (\frac{b+2 c x}{\sqrt{c} \sqrt{\pi }}\right )}{16 c^{3/2}}-\frac{e^2 \sqrt{\pi } C\left (\frac{b+2 c x}{\sqrt{c} \sqrt{\pi }}\right ) \sin \left (2 a-\frac{b^2}{2 c}\right )}{16 c^{3/2}}+\frac{e (2 c d-b e) \sin \left (2 a+2 b x+2 c x^2\right )}{16 c^2}+\frac{e (d+e x) \sin \left (2 a+2 b x+2 c x^2\right )}{8 c}+\frac{\left ((2 c d-b e)^2 \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}{8 c^2}-\frac{\left ((2 c d-b e)^2 \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}{8 c^2}\\ &=\frac{(d+e x)^3}{6 e}+\frac{(2 c d-b e)^2 \sqrt{\pi } \cos \left (2 a-\frac{b^2}{2 c}\right ) C\left (\frac{b+2 c x}{\sqrt{c} \sqrt{\pi }}\right )}{16 c^{5/2}}-\frac{e^2 \sqrt{\pi } \cos \left (2 a-\frac{b^2}{2 c}\right ) S\left (\frac{b+2 c x}{\sqrt{c} \sqrt{\pi }}\right )}{16 c^{3/2}}-\frac{e^2 \sqrt{\pi } C\left (\frac{b+2 c x}{\sqrt{c} \sqrt{\pi }}\right ) \sin \left (2 a-\frac{b^2}{2 c}\right )}{16 c^{3/2}}-\frac{(2 c d-b e)^2 \sqrt{\pi } S\left (\frac{b+2 c x}{\sqrt{c} \sqrt{\pi }}\right ) \sin \left (2 a-\frac{b^2}{2 c}\right )}{16 c^{5/2}}+\frac{e (2 c d-b e) \sin \left (2 a+2 b x+2 c x^2\right )}{16 c^2}+\frac{e (d+e x) \sin \left (2 a+2 b x+2 c x^2\right )}{8 c}\\ \end{align*}
Mathematica [A] time = 1.17733, size = 215, normalized size = 0.74 \[ \frac{3 \sqrt{\pi } \text{FresnelC}\left (\frac{b+2 c x}{\sqrt{\pi } \sqrt{c}}\right ) \left (\cos \left (2 a-\frac{b^2}{2 c}\right ) (b e-2 c d)^2-c e^2 \sin \left (2 a-\frac{b^2}{2 c}\right )\right )-3 \sqrt{\pi } S\left (\frac{b+2 c x}{\sqrt{c} \sqrt{\pi }}\right ) \left (\sin \left (2 a-\frac{b^2}{2 c}\right ) (b e-2 c d)^2+c e^2 \cos \left (2 a-\frac{b^2}{2 c}\right )\right )+\sqrt{c} \left (3 e \sin (2 (a+x (b+c x))) (-b e+4 c d+2 c e x)+8 c^2 x \left (3 d^2+3 d e x+e^2 x^2\right )\right )}{48 c^{5/2}} \]
Antiderivative was successfully verified.
[In]
Integrate[(d + e*x)^2*Cos[a + b*x + c*x^2]^2,x]
[Out]
(3*Sqrt[Pi]*FresnelC[(b + 2*c*x)/(Sqrt[c]*Sqrt[Pi])]*((-2*c*d + b*e)^2*Cos[2*a - b^2/(2*c)] - c*e^2*Sin[2*a -
b^2/(2*c)]) - 3*Sqrt[Pi]*FresnelS[(b + 2*c*x)/(Sqrt[c]*Sqrt[Pi])]*(c*e^2*Cos[2*a - b^2/(2*c)] + (-2*c*d + b*e)
^2*Sin[2*a - b^2/(2*c)]) + Sqrt[c]*(8*c^2*x*(3*d^2 + 3*d*e*x + e^2*x^2) + 3*e*(4*c*d - b*e + 2*c*e*x)*Sin[2*(a
+ x*(b + c*x))]))/(48*c^(5/2))
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Maple [A] time = 0.035, size = 378, normalized size = 1.3 \begin{align*}{\frac{{e}^{2}x\sin \left ( 2\,c{x}^{2}+2\,bx+2\,a \right ) }{8\,c}}-{\frac{{e}^{2}b}{4\,c} \left ({\frac{\sin \left ( 2\,c{x}^{2}+2\,bx+2\,a \right ) }{4\,c}}-{\frac{b\sqrt{\pi }}{4} \left ( \cos \left ({\frac{-4\,ca+{b}^{2}}{2\,c}} \right ){\it FresnelC} \left ({\frac{2\,cx+b}{\sqrt{\pi }}{\frac{1}{\sqrt{c}}}} \right ) +\sin \left ({\frac{-4\,ca+{b}^{2}}{2\,c}} \right ){\it FresnelS} \left ({\frac{2\,cx+b}{\sqrt{\pi }}{\frac{1}{\sqrt{c}}}} \right ) \right ){c}^{-{\frac{3}{2}}}} \right ) }-{\frac{{e}^{2}\sqrt{\pi }}{16} \left ( \cos \left ({\frac{-4\,ca+{b}^{2}}{2\,c}} \right ){\it FresnelS} \left ({\frac{2\,cx+b}{\sqrt{\pi }}{\frac{1}{\sqrt{c}}}} \right ) -\sin \left ({\frac{-4\,ca+{b}^{2}}{2\,c}} \right ){\it FresnelC} \left ({\frac{2\,cx+b}{\sqrt{\pi }}{\frac{1}{\sqrt{c}}}} \right ) \right ){c}^{-{\frac{3}{2}}}}+{\frac{de\sin \left ( 2\,c{x}^{2}+2\,bx+2\,a \right ) }{4\,c}}-{\frac{deb\sqrt{\pi }}{4} \left ( \cos \left ({\frac{-4\,ca+{b}^{2}}{2\,c}} \right ){\it FresnelC} \left ({\frac{2\,cx+b}{\sqrt{\pi }}{\frac{1}{\sqrt{c}}}} \right ) +\sin \left ({\frac{-4\,ca+{b}^{2}}{2\,c}} \right ){\it FresnelS} \left ({\frac{2\,cx+b}{\sqrt{\pi }}{\frac{1}{\sqrt{c}}}} \right ) \right ){c}^{-{\frac{3}{2}}}}+{\frac{\sqrt{\pi }{d}^{2}}{4} \left ( \cos \left ({\frac{-4\,ca+{b}^{2}}{2\,c}} \right ){\it FresnelC} \left ({\frac{2\,cx+b}{\sqrt{\pi }}{\frac{1}{\sqrt{c}}}} \right ) +\sin \left ({\frac{-4\,ca+{b}^{2}}{2\,c}} \right ){\it FresnelS} \left ({\frac{2\,cx+b}{\sqrt{\pi }}{\frac{1}{\sqrt{c}}}} \right ) \right ){\frac{1}{\sqrt{c}}}}+{\frac{{x}^{2}de}{2}}+{\frac{{d}^{2}x}{2}}+{\frac{{x}^{3}{e}^{2}}{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((e*x+d)^2*cos(c*x^2+b*x+a)^2,x)
[Out]
1/8*e^2/c*x*sin(2*c*x^2+2*b*x+2*a)-1/4*e^2*b/c*(1/4*sin(2*c*x^2+2*b*x+2*a)/c-1/4*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
))))-1/16*e^2/c^(3/2)*Pi^(1/2)*(cos(1/2*(-4*a*c+b^2)/c)*FresnelS((2*c*x+b)/c^(1/2)/Pi^(1/2))-sin(1/2*(-4*a*c+b
^2)/c)*FresnelC((2*c*x+b)/c^(1/2)/Pi^(1/2)))+1/4*d*e/c*sin(2*c*x^2+2*b*x+2*a)-1/4*d*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^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)))+1/2*x^2*d*e+1/2*d^2*x+1/6*x^3*e^2
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Maxima [C] time = 4.07912, size = 6029, normalized size = 20.72 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)^2*cos(c*x^2+b*x+a)^2,x, algorithm="maxima")
[Out]
1/32*(sqrt(2)*sqrt(pi)*(((cos(1/4*pi + 1/2*arctan2(0, c)) + cos(-1/4*pi + 1/2*arctan2(0, c)) - I*sin(1/4*pi +
1/2*arctan2(0, c)) + I*sin(-1/4*pi + 1/2*arctan2(0, c)))*cos(-1/2*(b^2 - 4*a*c)/c) - (I*cos(1/4*pi + 1/2*arcta
n2(0, c)) + I*cos(-1/4*pi + 1/2*arctan2(0, c)) + sin(1/4*pi + 1/2*arctan2(0, c)) - sin(-1/4*pi + 1/2*arctan2(0
, c)))*sin(-1/2*(b^2 - 4*a*c)/c))*erf((2*I*c*x + I*b)/sqrt(2*I*c)) - ((cos(1/4*pi + 1/2*arctan2(0, c)) + cos(-
1/4*pi + 1/2*arctan2(0, c)) + I*sin(1/4*pi + 1/2*arctan2(0, c)) - I*sin(-1/4*pi + 1/2*arctan2(0, c)))*cos(-1/2
*(b^2 - 4*a*c)/c) + (I*cos(1/4*pi + 1/2*arctan2(0, c)) + I*cos(-1/4*pi + 1/2*arctan2(0, c)) - sin(1/4*pi + 1/2
*arctan2(0, c)) + sin(-1/4*pi + 1/2*arctan2(0, c)))*sin(-1/2*(b^2 - 4*a*c)/c))*erf((2*I*c*x + I*b)/sqrt(-2*I*c
)))*sqrt(abs(c)) + 16*x*abs(c))*d^2/abs(c) + 1/16*sqrt(2)*(sqrt(2)*(4*c^2*x^2 - c*(I*e^(1/2*(4*I*c^2*x^2 + 4*I
*b*c*x + I*b^2)/c) - 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) + 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))*s
qrt((4*c^2*x^2 + 4*b*c*x + b^2)/abs(c)) - ((sqrt(pi)*(erf(sqrt(1/2)*sqrt((4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c))
- 1) + 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*sqrt(pi)*(erf(sqrt(1/2)*sqrt((4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c)) - 1) + I*sqrt(pi)*(erf(sqrt(1/2)*sq
rt(-(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*(sqrt(pi)*(erf(sqrt(1/2)*sq
rt((4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c)) - 1) + 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*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*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)*cos(1/2*arctan2(2*(4*c^2*x^2 + 4*b*c*x + b^2)/c, 0)) - ((-I*sqrt(pi)*(erf(sqrt(1/2)*sqrt((4*I*
c^2*x^2 + 4*I*b*c*x + I*b^2)/c)) - 1) + I*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) - (sqrt(pi)*(erf(sqrt(1/2)*sqrt((4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c)) - 1)
+ 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*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*sqrt(pi)*(erf(sqrt(1/2)*sq
rt(-(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*(sqrt(pi)*(erf(sqrt(1/2)*sqr
t((4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c)) - 1) + 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)*sin(1/2*arctan2(2*(4*c^2*x^2 + 4*b*c*x + b^2)/c, 0)))*d*e/(c^2*sq
rt((4*c^2*x^2 + 4*b*c*x + b^2)/abs(c))) + 1/192*sqrt(2)*(2*sqrt(2)*(8*c^3*x^3*abs(c) + b*c*(3*I*e^(1/2*(4*I*c^
2*x^2 + 4*I*b*c*x + I*b^2)/c) - 3*I*e^(-1/2*(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c))*abs(c)*cos(-1/2*(b^2 - 4*a*c
)/c) - 3*b*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))*abs(
c)*sin(-1/2*(b^2 - 4*a*c)/c))*((4*c^2*x^2 + 4*b*c*x + b^2)/abs(c))^(3/2) - (6*b^3*(gamma(3/2, 1/2*(4*I*c^2*x^2
+ 4*I*b*c*x + I*b^2)/c) + gamma(3/2, -1/2*(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c))*abs(c)*cos(-1/2*(b^2 - 4*a*c)
/c) - b^3*(6*I*gamma(3/2, 1/2*(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c) - 6*I*gamma(3/2, -1/2*(4*I*c^2*x^2 + 4*I*b*
c*x + I*b^2)/c))*abs(c)*sin(-1/2*(b^2 - 4*a*c)/c) + (48*c^3*(gamma(3/2, 1/2*(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/
c) + gamma(3/2, -1/2*(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c))*abs(c)*cos(-1/2*(b^2 - 4*a*c)/c) - c^3*(48*I*gamma(
3/2, 1/2*(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c) - 48*I*gamma(3/2, -1/2*(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c))*abs
(c)*sin(-1/2*(b^2 - 4*a*c)/c))*x^3 + (72*b*c^2*(gamma(3/2, 1/2*(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c) + gamma(3/
2, -1/2*(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c))*abs(c)*cos(-1/2*(b^2 - 4*a*c)/c) - b*c^2*(72*I*gamma(3/2, 1/2*(4
*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c) - 72*I*gamma(3/2, -1/2*(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c))*abs(c)*sin(-1/
2*(b^2 - 4*a*c)/c))*x^2 + (36*b^2*c*(gamma(3/2, 1/2*(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c) + gamma(3/2, -1/2*(4*
I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c))*abs(c)*cos(-1/2*(b^2 - 4*a*c)/c) - b^2*c*(36*I*gamma(3/2, 1/2*(4*I*c^2*x^2
+ 4*I*b*c*x + I*b^2)/c) - 36*I*gamma(3/2, -1/2*(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c))*abs(c)*sin(-1/2*(b^2 - 4*
a*c)/c))*x)*cos(3/2*arctan2(2*(4*c^2*x^2 + 4*b*c*x + b^2)/c, 0)) + (3*(sqrt(pi)*(erf(sqrt(1/2)*sqrt((4*I*c^2*x
^2 + 4*I*b*c*x + I*b^2)/c)) - 1) + sqrt(pi)*(erf(sqrt(1/2)*sqrt(-(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c)) - 1))*b
^5*cos(-1/2*(b^2 - 4*a*c)/c) + (-3*I*sqrt(pi)*(erf(sqrt(1/2)*sqrt((4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c)) - 1) +
3*I*sqrt(pi)*(erf(sqrt(1/2)*sqrt(-(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c)) - 1))*b^5*sin(-1/2*(b^2 - 4*a*c)/c) +
(24*(sqrt(pi)*(erf(sqrt(1/2)*sqrt((4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c)) - 1) + 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*c^3*cos(-1/2*(b^2 - 4*a*c)/c) + (-24*I*sqrt(pi)*(erf(sqrt(1/2)
*sqrt((4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c)) - 1) + 24*I*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*c^3*sin(-1/2*(b^2 - 4*a*c)/c))*x^3 + (36*(sqrt(pi)*(erf(sqrt(1/2)*sqrt((4*I*c^2*x^2 +
4*I*b*c*x + I*b^2)/c)) - 1) + sqrt(pi)*(erf(sqrt(1/2)*sqrt(-(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c)) - 1))*b^3*c^
2*cos(-1/2*(b^2 - 4*a*c)/c) + (-36*I*sqrt(pi)*(erf(sqrt(1/2)*sqrt((4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c)) - 1) +
36*I*sqrt(pi)*(erf(sqrt(1/2)*sqrt(-(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c)) - 1))*b^3*c^2*sin(-1/2*(b^2 - 4*a*c)
/c))*x^2 + (18*(sqrt(pi)*(erf(sqrt(1/2)*sqrt((4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c)) - 1) + sqrt(pi)*(erf(sqrt(1
/2)*sqrt(-(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c)) - 1))*b^4*c*cos(-1/2*(b^2 - 4*a*c)/c) + (-18*I*sqrt(pi)*(erf(s
qrt(1/2)*sqrt((4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c)) - 1) + 18*I*sqrt(pi)*(erf(sqrt(1/2)*sqrt(-(4*I*c^2*x^2 + 4
*I*b*c*x + I*b^2)/c)) - 1))*b^4*c*sin(-1/2*(b^2 - 4*a*c)/c))*x)*cos(1/2*arctan2(2*(4*c^2*x^2 + 4*b*c*x + b^2)/
c, 0)) + (b^3*(6*I*gamma(3/2, 1/2*(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c) - 6*I*gamma(3/2, -1/2*(4*I*c^2*x^2 + 4*
I*b*c*x + I*b^2)/c))*abs(c)*cos(-1/2*(b^2 - 4*a*c)/c) + 6*b^3*(gamma(3/2, 1/2*(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2
)/c) + gamma(3/2, -1/2*(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c))*abs(c)*sin(-1/2*(b^2 - 4*a*c)/c) + (c^3*(48*I*gam
ma(3/2, 1/2*(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c) - 48*I*gamma(3/2, -1/2*(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c))*
abs(c)*cos(-1/2*(b^2 - 4*a*c)/c) + 48*c^3*(gamma(3/2, 1/2*(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c) + gamma(3/2, -1
/2*(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c))*abs(c)*sin(-1/2*(b^2 - 4*a*c)/c))*x^3 + (b*c^2*(72*I*gamma(3/2, 1/2*(
4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c) - 72*I*gamma(3/2, -1/2*(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c))*abs(c)*cos(-1
/2*(b^2 - 4*a*c)/c) + 72*b*c^2*(gamma(3/2, 1/2*(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c) + gamma(3/2, -1/2*(4*I*c^2
*x^2 + 4*I*b*c*x + I*b^2)/c))*abs(c)*sin(-1/2*(b^2 - 4*a*c)/c))*x^2 + (b^2*c*(36*I*gamma(3/2, 1/2*(4*I*c^2*x^2
+ 4*I*b*c*x + I*b^2)/c) - 36*I*gamma(3/2, -1/2*(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c))*abs(c)*cos(-1/2*(b^2 - 4
*a*c)/c) + 36*b^2*c*(gamma(3/2, 1/2*(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c) + gamma(3/2, -1/2*(4*I*c^2*x^2 + 4*I*
b*c*x + I*b^2)/c))*abs(c)*sin(-1/2*(b^2 - 4*a*c)/c))*x)*sin(3/2*arctan2(2*(4*c^2*x^2 + 4*b*c*x + b^2)/c, 0)) +
((-3*I*sqrt(pi)*(erf(sqrt(1/2)*sqrt((4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c)) - 1) + 3*I*sqrt(pi)*(erf(sqrt(1/2)*
sqrt(-(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c)) - 1))*b^5*cos(-1/2*(b^2 - 4*a*c)/c) - 3*(sqrt(pi)*(erf(sqrt(1/2)*s
qrt((4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c)) - 1) + sqrt(pi)*(erf(sqrt(1/2)*sqrt(-(4*I*c^2*x^2 + 4*I*b*c*x + I*b^
2)/c)) - 1))*b^5*sin(-1/2*(b^2 - 4*a*c)/c) + ((-24*I*sqrt(pi)*(erf(sqrt(1/2)*sqrt((4*I*c^2*x^2 + 4*I*b*c*x + I
*b^2)/c)) - 1) + 24*I*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*c^3*cos(-1
/2*(b^2 - 4*a*c)/c) - 24*(sqrt(pi)*(erf(sqrt(1/2)*sqrt((4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c)) - 1) + 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*c^3*sin(-1/2*(b^2 - 4*a*c)/c))*x^3 + ((-36
*I*sqrt(pi)*(erf(sqrt(1/2)*sqrt((4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c)) - 1) + 36*I*sqrt(pi)*(erf(sqrt(1/2)*sqrt
(-(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c)) - 1))*b^3*c^2*cos(-1/2*(b^2 - 4*a*c)/c) - 36*(sqrt(pi)*(erf(sqrt(1/2)*
sqrt((4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c)) - 1) + sqrt(pi)*(erf(sqrt(1/2)*sqrt(-(4*I*c^2*x^2 + 4*I*b*c*x + I*b
^2)/c)) - 1))*b^3*c^2*sin(-1/2*(b^2 - 4*a*c)/c))*x^2 + ((-18*I*sqrt(pi)*(erf(sqrt(1/2)*sqrt((4*I*c^2*x^2 + 4*I
*b*c*x + I*b^2)/c)) - 1) + 18*I*sqrt(pi)*(erf(sqrt(1/2)*sqrt(-(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c)) - 1))*b^4*
c*cos(-1/2*(b^2 - 4*a*c)/c) - 18*(sqrt(pi)*(erf(sqrt(1/2)*sqrt((4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c)) - 1) + sq
rt(pi)*(erf(sqrt(1/2)*sqrt(-(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c)) - 1))*b^4*c*sin(-1/2*(b^2 - 4*a*c)/c))*x)*si
n(1/2*arctan2(2*(4*c^2*x^2 + 4*b*c*x + b^2)/c, 0)))*e^2/(c^3*((4*c^2*x^2 + 4*b*c*x + b^2)/abs(c))^(3/2)*abs(c)
)
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Fricas [A] time = 1.60503, size = 608, normalized size = 2.09 \begin{align*} \frac{8 \, c^{3} e^{2} x^{3} + 24 \, c^{3} d e x^{2} + 24 \, c^{3} d^{2} x + 6 \,{\left (2 \, c^{2} e^{2} x + 4 \, c^{2} d e - b c e^{2}\right )} \cos \left (c x^{2} + b x + a\right ) \sin \left (c x^{2} + b x + a\right ) - 3 \,{\left (\pi c e^{2} \sin \left (-\frac{b^{2} - 4 \, a c}{2 \, c}\right ) - \pi{\left (4 \, c^{2} d^{2} - 4 \, b c d e + b^{2} e^{2}\right )} \cos \left (-\frac{b^{2} - 4 \, a c}{2 \, c}\right )\right )} \sqrt{\frac{c}{\pi }} \operatorname{C}\left (\frac{{\left (2 \, c x + b\right )} \sqrt{\frac{c}{\pi }}}{c}\right ) - 3 \,{\left (\pi c e^{2} \cos \left (-\frac{b^{2} - 4 \, a c}{2 \, c}\right ) + \pi{\left (4 \, c^{2} d^{2} - 4 \, b c d e + b^{2} e^{2}\right )} \sin \left (-\frac{b^{2} - 4 \, a c}{2 \, c}\right )\right )} \sqrt{\frac{c}{\pi }} \operatorname{S}\left (\frac{{\left (2 \, c x + b\right )} \sqrt{\frac{c}{\pi }}}{c}\right )}{48 \, c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)^2*cos(c*x^2+b*x+a)^2,x, algorithm="fricas")
[Out]
1/48*(8*c^3*e^2*x^3 + 24*c^3*d*e*x^2 + 24*c^3*d^2*x + 6*(2*c^2*e^2*x + 4*c^2*d*e - b*c*e^2)*cos(c*x^2 + b*x +
a)*sin(c*x^2 + b*x + a) - 3*(pi*c*e^2*sin(-1/2*(b^2 - 4*a*c)/c) - pi*(4*c^2*d^2 - 4*b*c*d*e + b^2*e^2)*cos(-1/
2*(b^2 - 4*a*c)/c))*sqrt(c/pi)*fresnel_cos((2*c*x + b)*sqrt(c/pi)/c) - 3*(pi*c*e^2*cos(-1/2*(b^2 - 4*a*c)/c) +
pi*(4*c^2*d^2 - 4*b*c*d*e + b^2*e^2)*sin(-1/2*(b^2 - 4*a*c)/c))*sqrt(c/pi)*fresnel_sin((2*c*x + b)*sqrt(c/pi)
/c))/c^3
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d + e x\right )^{2} \cos ^{2}{\left (a + b x + c x^{2} \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)**2*cos(c*x**2+b*x+a)**2,x)
[Out]
Integral((d + e*x)**2*cos(a + b*x + c*x**2)**2, x)
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Giac [C] time = 1.35039, size = 724, normalized size = 2.49 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)^2*cos(c*x^2+b*x+a)^2,x, algorithm="giac")
[Out]
1/6*x^3*e^2 + 1/2*d*x^2*e + 1/2*d^2*x - 1/8*sqrt(pi)*d^2*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^2*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)*b*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 - 2*c)/c)/(sqrt(c)*(-I*c/abs(c) + 1)) - I*d*e^(2*I*c*x^2 + 2*I
*b*x + 2*I*a + 1))/c + 1/8*(sqrt(pi)*b*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 - 2*c)/c)/(sqrt(c)*(I*c/abs(c) + 1)) + I*d*e^(-2*I*c*x^2 - 2*I*b*x - 2*I*a + 1))/c - 1/32*((c*(2*I*x + I*b
/c) - 2*I*b)*e^(2*I*c*x^2 + 2*I*b*x + 2*I*a + 2) + sqrt(pi)*(b^2 + I*c)*erf(-1/2*sqrt(c)*(2*x + b/c)*(-I*c/abs
(c) + 1))*e^(-1/2*(I*b^2 - 4*I*a*c - 4*c)/c)/(sqrt(c)*(-I*c/abs(c) + 1)))/c^2 - 1/32*((c*(-2*I*x - I*b/c) + 2*
I*b)*e^(-2*I*c*x^2 - 2*I*b*x - 2*I*a + 2) + sqrt(pi)*(b^2 - I*c)*erf(-1/2*sqrt(c)*(2*x + b/c)*(I*c/abs(c) + 1)
)*e^(-1/2*(-I*b^2 + 4*I*a*c - 4*c)/c)/(sqrt(c)*(I*c/abs(c) + 1)))/c^2