3.29 \(\int (d+e x)^2 \sin (a+b x+c x^2) \, dx\)
Optimal. Leaf size=285 \[ \frac{\sqrt{\frac{\pi }{2}} \sin \left (a-\frac{b^2}{4 c}\right ) (2 c d-b e)^2 \text{FresnelC}\left (\frac{b+2 c x}{\sqrt{2 \pi } \sqrt{c}}\right )}{4 c^{5/2}}+\frac{\sqrt{\frac{\pi }{2}} \cos \left (a-\frac{b^2}{4 c}\right ) (2 c d-b e)^2 S\left (\frac{b+2 c x}{\sqrt{c} \sqrt{2 \pi }}\right )}{4 c^{5/2}}+\frac{\sqrt{\frac{\pi }{2}} e^2 \cos \left (a-\frac{b^2}{4 c}\right ) \text{FresnelC}\left (\frac{b+2 c x}{\sqrt{2 \pi } \sqrt{c}}\right )}{2 c^{3/2}}-\frac{\sqrt{\frac{\pi }{2}} e^2 \sin \left (a-\frac{b^2}{4 c}\right ) S\left (\frac{b+2 c x}{\sqrt{c} \sqrt{2 \pi }}\right )}{2 c^{3/2}}-\frac{e (2 c d-b e) \cos \left (a+b x+c x^2\right )}{4 c^2}-\frac{e (d+e x) \cos \left (a+b x+c x^2\right )}{2 c} \]
[Out]
-(e*(2*c*d - b*e)*Cos[a + b*x + c*x^2])/(4*c^2) - (e*(d + e*x)*Cos[a + b*x + c*x^2])/(2*c) + (e^2*Sqrt[Pi/2]*C
os[a - b^2/(4*c)]*FresnelC[(b + 2*c*x)/(Sqrt[c]*Sqrt[2*Pi])])/(2*c^(3/2)) + ((2*c*d - b*e)^2*Sqrt[Pi/2]*Cos[a
- b^2/(4*c)]*FresnelS[(b + 2*c*x)/(Sqrt[c]*Sqrt[2*Pi])])/(4*c^(5/2)) + ((2*c*d - b*e)^2*Sqrt[Pi/2]*FresnelC[(b
+ 2*c*x)/(Sqrt[c]*Sqrt[2*Pi])]*Sin[a - b^2/(4*c)])/(4*c^(5/2)) - (e^2*Sqrt[Pi/2]*FresnelS[(b + 2*c*x)/(Sqrt[c
]*Sqrt[2*Pi])]*Sin[a - b^2/(4*c)])/(2*c^(3/2))
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Rubi [A] time = 0.270835, antiderivative size = 285, normalized size of antiderivative = 1.,
number of steps used = 8, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316, Rules used =
{3463, 3448, 3352, 3351, 3461, 3447} \[ \frac{\sqrt{\frac{\pi }{2}} \sin \left (a-\frac{b^2}{4 c}\right ) (2 c d-b e)^2 \text{FresnelC}\left (\frac{b+2 c x}{\sqrt{2 \pi } \sqrt{c}}\right )}{4 c^{5/2}}+\frac{\sqrt{\frac{\pi }{2}} \cos \left (a-\frac{b^2}{4 c}\right ) (2 c d-b e)^2 S\left (\frac{b+2 c x}{\sqrt{c} \sqrt{2 \pi }}\right )}{4 c^{5/2}}+\frac{\sqrt{\frac{\pi }{2}} e^2 \cos \left (a-\frac{b^2}{4 c}\right ) \text{FresnelC}\left (\frac{b+2 c x}{\sqrt{2 \pi } \sqrt{c}}\right )}{2 c^{3/2}}-\frac{\sqrt{\frac{\pi }{2}} e^2 \sin \left (a-\frac{b^2}{4 c}\right ) S\left (\frac{b+2 c x}{\sqrt{c} \sqrt{2 \pi }}\right )}{2 c^{3/2}}-\frac{e (2 c d-b e) \cos \left (a+b x+c x^2\right )}{4 c^2}-\frac{e (d+e x) \cos \left (a+b x+c x^2\right )}{2 c} \]
Antiderivative was successfully verified.
[In]
Int[(d + e*x)^2*Sin[a + b*x + c*x^2],x]
[Out]
-(e*(2*c*d - b*e)*Cos[a + b*x + c*x^2])/(4*c^2) - (e*(d + e*x)*Cos[a + b*x + c*x^2])/(2*c) + (e^2*Sqrt[Pi/2]*C
os[a - b^2/(4*c)]*FresnelC[(b + 2*c*x)/(Sqrt[c]*Sqrt[2*Pi])])/(2*c^(3/2)) + ((2*c*d - b*e)^2*Sqrt[Pi/2]*Cos[a
- b^2/(4*c)]*FresnelS[(b + 2*c*x)/(Sqrt[c]*Sqrt[2*Pi])])/(4*c^(5/2)) + ((2*c*d - b*e)^2*Sqrt[Pi/2]*FresnelC[(b
+ 2*c*x)/(Sqrt[c]*Sqrt[2*Pi])]*Sin[a - b^2/(4*c)])/(4*c^(5/2)) - (e^2*Sqrt[Pi/2]*FresnelS[(b + 2*c*x)/(Sqrt[c
]*Sqrt[2*Pi])]*Sin[a - b^2/(4*c)])/(2*c^(3/2))
Rule 3463
Int[((d_.) + (e_.)*(x_))^(m_)*Sin[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> -Simp[(e*(d + e*x)^(m - 1)*
Cos[a + b*x + c*x^2])/(2*c), x] + (Dist[(e^2*(m - 1))/(2*c), Int[(d + e*x)^(m - 2)*Cos[a + b*x + c*x^2], x], x
] - Dist[(b*e - 2*c*d)/(2*c), Int[(d + e*x)^(m - 1)*Sin[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 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 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 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 3461
Int[((d_.) + (e_.)*(x_))*Sin[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> -Simp[(e*Cos[a + b*x + c*x^2])/(
2*c), x] + Dist[(2*c*d - b*e)/(2*c), Int[Sin[a + b*x + c*x^2], x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*
d - b*e, 0]
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]
Rubi steps
\begin{align*} \int (d+e x)^2 \sin \left (a+b x+c x^2\right ) \, dx &=-\frac{e (d+e x) \cos \left (a+b x+c x^2\right )}{2 c}+\frac{e^2 \int \cos \left (a+b x+c x^2\right ) \, dx}{2 c}-\frac{(-2 c d+b e) \int (d+e x) \sin \left (a+b x+c x^2\right ) \, dx}{2 c}\\ &=-\frac{e (2 c d-b e) \cos \left (a+b x+c x^2\right )}{4 c^2}-\frac{e (d+e x) \cos \left (a+b x+c x^2\right )}{2 c}+\frac{(2 c d-b e)^2 \int \sin \left (a+b x+c x^2\right ) \, dx}{4 c^2}+\frac{\left (e^2 \cos \left (a-\frac{b^2}{4 c}\right )\right ) \int \cos \left (\frac{(b+2 c x)^2}{4 c}\right ) \, dx}{2 c}-\frac{\left (e^2 \sin \left (a-\frac{b^2}{4 c}\right )\right ) \int \sin \left (\frac{(b+2 c x)^2}{4 c}\right ) \, dx}{2 c}\\ &=-\frac{e (2 c d-b e) \cos \left (a+b x+c x^2\right )}{4 c^2}-\frac{e (d+e x) \cos \left (a+b x+c x^2\right )}{2 c}+\frac{e^2 \sqrt{\frac{\pi }{2}} \cos \left (a-\frac{b^2}{4 c}\right ) C\left (\frac{b+2 c x}{\sqrt{c} \sqrt{2 \pi }}\right )}{2 c^{3/2}}-\frac{e^2 \sqrt{\frac{\pi }{2}} S\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{\left ((2 c d-b e)^2 \cos \left (a-\frac{b^2}{4 c}\right )\right ) \int \sin \left (\frac{(b+2 c x)^2}{4 c}\right ) \, dx}{4 c^2}+\frac{\left ((2 c d-b e)^2 \sin \left (a-\frac{b^2}{4 c}\right )\right ) \int \cos \left (\frac{(b+2 c x)^2}{4 c}\right ) \, dx}{4 c^2}\\ &=-\frac{e (2 c d-b e) \cos \left (a+b x+c x^2\right )}{4 c^2}-\frac{e (d+e x) \cos \left (a+b x+c x^2\right )}{2 c}+\frac{e^2 \sqrt{\frac{\pi }{2}} \cos \left (a-\frac{b^2}{4 c}\right ) C\left (\frac{b+2 c x}{\sqrt{c} \sqrt{2 \pi }}\right )}{2 c^{3/2}}+\frac{(2 c d-b e)^2 \sqrt{\frac{\pi }{2}} \cos \left (a-\frac{b^2}{4 c}\right ) S\left (\frac{b+2 c x}{\sqrt{c} \sqrt{2 \pi }}\right )}{4 c^{5/2}}+\frac{(2 c d-b e)^2 \sqrt{\frac{\pi }{2}} C\left (\frac{b+2 c x}{\sqrt{c} \sqrt{2 \pi }}\right ) \sin \left (a-\frac{b^2}{4 c}\right )}{4 c^{5/2}}-\frac{e^2 \sqrt{\frac{\pi }{2}} S\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}}\\ \end{align*}
Mathematica [A] time = 1.30037, size = 186, normalized size = 0.65 \[ \frac{\sqrt{2 \pi } \text{FresnelC}\left (\frac{b+2 c x}{\sqrt{2 \pi } \sqrt{c}}\right ) \left (\sin \left (a-\frac{b^2}{4 c}\right ) (b e-2 c d)^2+2 c e^2 \cos \left (a-\frac{b^2}{4 c}\right )\right )+\sqrt{2 \pi } S\left (\frac{b+2 c x}{\sqrt{c} \sqrt{2 \pi }}\right ) \left (\cos \left (a-\frac{b^2}{4 c}\right ) (b e-2 c d)^2-2 c e^2 \sin \left (a-\frac{b^2}{4 c}\right )\right )+2 \sqrt{c} e \cos (a+x (b+c x)) (b e-2 c (2 d+e x))}{8 c^{5/2}} \]
Antiderivative was successfully verified.
[In]
Integrate[(d + e*x)^2*Sin[a + b*x + c*x^2],x]
[Out]
(2*Sqrt[c]*e*(b*e - 2*c*(2*d + e*x))*Cos[a + x*(b + c*x)] + Sqrt[2*Pi]*FresnelS[(b + 2*c*x)/(Sqrt[c]*Sqrt[2*Pi
])]*((-2*c*d + b*e)^2*Cos[a - b^2/(4*c)] - 2*c*e^2*Sin[a - b^2/(4*c)]) + Sqrt[2*Pi]*FresnelC[(b + 2*c*x)/(Sqrt
[c]*Sqrt[2*Pi])]*(2*c*e^2*Cos[a - b^2/(4*c)] + (-2*c*d + b*e)^2*Sin[a - b^2/(4*c)]))/(8*c^(5/2))
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Maple [A] time = 0.01, size = 399, normalized size = 1.4 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((e*x+d)^2*sin(c*x^2+b*x+a),x)
[Out]
-1/2*e^2/c*x*cos(c*x^2+b*x+a)-1/2*e^2*b/c*(-1/2*cos(c*x^2+b*x+a)/c-1/4*b/c^(3/2)*2^(1/2)*Pi^(1/2)*(cos((1/4*b^
2-c*a)/c)*FresnelS(2^(1/2)/Pi^(1/2)/c^(1/2)*(c*x+1/2*b))-sin((1/4*b^2-c*a)/c)*FresnelC(2^(1/2)/Pi^(1/2)/c^(1/2
)*(c*x+1/2*b))))+1/4*e^2/c^(3/2)*2^(1/2)*Pi^(1/2)*(cos((1/4*b^2-c*a)/c)*FresnelC(2^(1/2)/Pi^(1/2)/c^(1/2)*(c*x
+1/2*b))+sin((1/4*b^2-c*a)/c)*FresnelS(2^(1/2)/Pi^(1/2)/c^(1/2)*(c*x+1/2*b)))-d*e/c*cos(c*x^2+b*x+a)-1/2*d*e*b
/c^(3/2)*2^(1/2)*Pi^(1/2)*(cos((1/4*b^2-c*a)/c)*FresnelS(2^(1/2)/Pi^(1/2)/c^(1/2)*(c*x+1/2*b))-sin((1/4*b^2-c*
a)/c)*FresnelC(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^2*(cos((1/4*b^2-c*a)/c)*F
resnelS(2^(1/2)/Pi^(1/2)/c^(1/2)*(c*x+1/2*b))-sin((1/4*b^2-c*a)/c)*FresnelC(2^(1/2)/Pi^(1/2)/c^(1/2)*(c*x+1/2*
b)))
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Maxima [C] time = 3.95157, size = 5825, normalized size = 20.44 \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*sin(c*x^2+b*x+a),x, algorithm="maxima")
[Out]
-1/8*sqrt(pi)*(((-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*ar
ctan2(0, c)) + sin(-1/4*pi + 1/2*arctan2(0, c)))*cos(-1/4*(b^2 - 4*a*c)/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)))*s
in(-1/4*(b^2 - 4*a*c)/c))*erf(1/2*(2*I*c*x + I*b)/sqrt(I*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)))*cos(-1/4*(b^2
- 4*a*c)/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*arctan
2(0, c)) - I*sin(-1/4*pi + 1/2*arctan2(0, c)))*sin(-1/4*(b^2 - 4*a*c)/c))*erf(1/2*(2*I*c*x + I*b)/sqrt(-I*c)))
*d^2/sqrt(abs(c)) - 1/4*(((I*sqrt(pi)*(erf(1/2*sqrt((4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c)) - 1) - I*sqrt(pi)*(e
rf(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) + (sqrt(pi)*(erf(1/2*sq
rt((4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c)) - 1) + 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*sqrt(pi)*(erf(1/2*sqrt((4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c)) - 1)
- 2*I*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) + 2*(s
qrt(pi)*(erf(1/2*sqrt((4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c)) - 1) + 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)*cos(1/2*arctan2((4*c^2*x^2 + 4*b*c*x + b^2)/c, 0))
+ ((sqrt(pi)*(erf(1/2*sqrt((4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c)) - 1) + 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*sqrt(pi)*(erf(1/2*sqrt((4*I*c^2*x^2 + 4*I*b*
c*x + I*b^2)/c)) - 1) + I*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*(sqrt(pi)*(erf(1/2*sqrt((4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c)) - 1) + 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) + (-2*I*sqrt(pi)*(erf(1/2*sqrt((4*
I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c)) - 1) + 2*I*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)*sin(1/2*arctan2((4*c^2*x^2 + 4*b*c*x + b^2)/c, 0)) + (2*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))*cos(-1/4*(b^2 - 4*a*c)/c) + c*(
2*I*e^(1/4*(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c) - 2*I*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)/abs(c)))*d*e/(c^2*sqrt((4*c^2*x^2 + 4*b*c*x + b^2)/abs(c)))
+ 1/16*((4*b*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))*ab
s(c)*cos(-1/4*(b^2 - 4*a*c)/c) + b*c*(4*I*e^(1/4*(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c) - 4*I*e^(-1/4*(4*I*c^2*x
^2 + 4*I*b*c*x + I*b^2)/c))*abs(c)*sin(-1/4*(b^2 - 4*a*c)/c))*((4*c^2*x^2 + 4*b*c*x + b^2)/abs(c))^(3/2) + (b^
3*(-4*I*gamma(3/2, 1/4*(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c) + 4*I*gamma(3/2, -1/4*(4*I*c^2*x^2 + 4*I*b*c*x + I
*b^2)/c))*abs(c)*cos(-1/4*(b^2 - 4*a*c)/c) - 4*b^3*(gamma(3/2, 1/4*(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c) + gamm
a(3/2, -1/4*(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c))*abs(c)*sin(-1/4*(b^2 - 4*a*c)/c) + (c^3*(-32*I*gamma(3/2, 1/
4*(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c) + 32*I*gamma(3/2, -1/4*(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c))*abs(c)*cos
(-1/4*(b^2 - 4*a*c)/c) - 32*c^3*(gamma(3/2, 1/4*(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c) + gamma(3/2, -1/4*(4*I*c^
2*x^2 + 4*I*b*c*x + I*b^2)/c))*abs(c)*sin(-1/4*(b^2 - 4*a*c)/c))*x^3 + (b*c^2*(-48*I*gamma(3/2, 1/4*(4*I*c^2*x
^2 + 4*I*b*c*x + I*b^2)/c) + 48*I*gamma(3/2, -1/4*(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c))*abs(c)*cos(-1/4*(b^2 -
4*a*c)/c) - 48*b*c^2*(gamma(3/2, 1/4*(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c) + gamma(3/2, -1/4*(4*I*c^2*x^2 + 4*
I*b*c*x + I*b^2)/c))*abs(c)*sin(-1/4*(b^2 - 4*a*c)/c))*x^2 + (b^2*c*(-24*I*gamma(3/2, 1/4*(4*I*c^2*x^2 + 4*I*b
*c*x + I*b^2)/c) + 24*I*gamma(3/2, -1/4*(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c))*abs(c)*cos(-1/4*(b^2 - 4*a*c)/c)
- 24*b^2*c*(gamma(3/2, 1/4*(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c) + gamma(3/2, -1/4*(4*I*c^2*x^2 + 4*I*b*c*x +
I*b^2)/c))*abs(c)*sin(-1/4*(b^2 - 4*a*c)/c))*x)*cos(3/2*arctan2((4*c^2*x^2 + 4*b*c*x + b^2)/c, 0)) + ((I*sqrt(
pi)*(erf(1/2*sqrt((4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c)) - 1) - I*sqrt(pi)*(erf(1/2*sqrt(-(4*I*c^2*x^2 + 4*I*b*
c*x + I*b^2)/c)) - 1))*b^5*cos(-1/4*(b^2 - 4*a*c)/c) + (sqrt(pi)*(erf(1/2*sqrt((4*I*c^2*x^2 + 4*I*b*c*x + I*b^
2)/c)) - 1) + sqrt(pi)*(erf(1/2*sqrt(-(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c)) - 1))*b^5*sin(-1/4*(b^2 - 4*a*c)/c
) + ((8*I*sqrt(pi)*(erf(1/2*sqrt((4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c)) - 1) - 8*I*sqrt(pi)*(erf(1/2*sqrt(-(4*I
*c^2*x^2 + 4*I*b*c*x + I*b^2)/c)) - 1))*b^2*c^3*cos(-1/4*(b^2 - 4*a*c)/c) + 8*(sqrt(pi)*(erf(1/2*sqrt((4*I*c^2
*x^2 + 4*I*b*c*x + I*b^2)/c)) - 1) + sqrt(pi)*(erf(1/2*sqrt(-(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c)) - 1))*b^2*c
^3*sin(-1/4*(b^2 - 4*a*c)/c))*x^3 + ((12*I*sqrt(pi)*(erf(1/2*sqrt((4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c)) - 1) -
12*I*sqrt(pi)*(erf(1/2*sqrt(-(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c)) - 1))*b^3*c^2*cos(-1/4*(b^2 - 4*a*c)/c) +
12*(sqrt(pi)*(erf(1/2*sqrt((4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c)) - 1) + sqrt(pi)*(erf(1/2*sqrt(-(4*I*c^2*x^2 +
4*I*b*c*x + I*b^2)/c)) - 1))*b^3*c^2*sin(-1/4*(b^2 - 4*a*c)/c))*x^2 + ((6*I*sqrt(pi)*(erf(1/2*sqrt((4*I*c^2*x
^2 + 4*I*b*c*x + I*b^2)/c)) - 1) - 6*I*sqrt(pi)*(erf(1/2*sqrt(-(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c)) - 1))*b^4
*c*cos(-1/4*(b^2 - 4*a*c)/c) + 6*(sqrt(pi)*(erf(1/2*sqrt((4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c)) - 1) + sqrt(pi)
*(erf(1/2*sqrt(-(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c)) - 1))*b^4*c*sin(-1/4*(b^2 - 4*a*c)/c))*x)*cos(1/2*arctan
2((4*c^2*x^2 + 4*b*c*x + b^2)/c, 0)) - (4*b^3*(gamma(3/2, 1/4*(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c) + gamma(3/2
, -1/4*(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c))*abs(c)*cos(-1/4*(b^2 - 4*a*c)/c) - b^3*(4*I*gamma(3/2, 1/4*(4*I*c
^2*x^2 + 4*I*b*c*x + I*b^2)/c) - 4*I*gamma(3/2, -1/4*(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c))*abs(c)*sin(-1/4*(b^
2 - 4*a*c)/c) + (32*c^3*(gamma(3/2, 1/4*(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c) + gamma(3/2, -1/4*(4*I*c^2*x^2 +
4*I*b*c*x + I*b^2)/c))*abs(c)*cos(-1/4*(b^2 - 4*a*c)/c) - c^3*(32*I*gamma(3/2, 1/4*(4*I*c^2*x^2 + 4*I*b*c*x +
I*b^2)/c) - 32*I*gamma(3/2, -1/4*(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c))*abs(c)*sin(-1/4*(b^2 - 4*a*c)/c))*x^3 +
(48*b*c^2*(gamma(3/2, 1/4*(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c) + gamma(3/2, -1/4*(4*I*c^2*x^2 + 4*I*b*c*x + I
*b^2)/c))*abs(c)*cos(-1/4*(b^2 - 4*a*c)/c) - b*c^2*(48*I*gamma(3/2, 1/4*(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c) -
48*I*gamma(3/2, -1/4*(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c))*abs(c)*sin(-1/4*(b^2 - 4*a*c)/c))*x^2 + (24*b^2*c*
(gamma(3/2, 1/4*(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c) + gamma(3/2, -1/4*(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c))*a
bs(c)*cos(-1/4*(b^2 - 4*a*c)/c) - b^2*c*(24*I*gamma(3/2, 1/4*(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c) - 24*I*gamma
(3/2, -1/4*(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c))*abs(c)*sin(-1/4*(b^2 - 4*a*c)/c))*x)*sin(3/2*arctan2((4*c^2*x
^2 + 4*b*c*x + b^2)/c, 0)) + ((sqrt(pi)*(erf(1/2*sqrt((4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c)) - 1) + sqrt(pi)*(e
rf(1/2*sqrt(-(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c)) - 1))*b^5*cos(-1/4*(b^2 - 4*a*c)/c) + (-I*sqrt(pi)*(erf(1/2
*sqrt((4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c)) - 1) + I*sqrt(pi)*(erf(1/2*sqrt(-(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)
/c)) - 1))*b^5*sin(-1/4*(b^2 - 4*a*c)/c) + (8*(sqrt(pi)*(erf(1/2*sqrt((4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c)) -
1) + sqrt(pi)*(erf(1/2*sqrt(-(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c)) - 1))*b^2*c^3*cos(-1/4*(b^2 - 4*a*c)/c) + (
-8*I*sqrt(pi)*(erf(1/2*sqrt((4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c)) - 1) + 8*I*sqrt(pi)*(erf(1/2*sqrt(-(4*I*c^2*
x^2 + 4*I*b*c*x + I*b^2)/c)) - 1))*b^2*c^3*sin(-1/4*(b^2 - 4*a*c)/c))*x^3 + (12*(sqrt(pi)*(erf(1/2*sqrt((4*I*c
^2*x^2 + 4*I*b*c*x + I*b^2)/c)) - 1) + sqrt(pi)*(erf(1/2*sqrt(-(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c)) - 1))*b^3
*c^2*cos(-1/4*(b^2 - 4*a*c)/c) + (-12*I*sqrt(pi)*(erf(1/2*sqrt((4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c)) - 1) + 12
*I*sqrt(pi)*(erf(1/2*sqrt(-(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c)) - 1))*b^3*c^2*sin(-1/4*(b^2 - 4*a*c)/c))*x^2
+ (6*(sqrt(pi)*(erf(1/2*sqrt((4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c)) - 1) + sqrt(pi)*(erf(1/2*sqrt(-(4*I*c^2*x^2
+ 4*I*b*c*x + I*b^2)/c)) - 1))*b^4*c*cos(-1/4*(b^2 - 4*a*c)/c) + (-6*I*sqrt(pi)*(erf(1/2*sqrt((4*I*c^2*x^2 +
4*I*b*c*x + I*b^2)/c)) - 1) + 6*I*sqrt(pi)*(erf(1/2*sqrt(-(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c)) - 1))*b^4*c*si
n(-1/4*(b^2 - 4*a*c)/c))*x)*sin(1/2*arctan2((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 = 1.79653, size = 567, normalized size = 1.99 \begin{align*} \frac{\sqrt{2}{\left (2 \, \pi c e^{2} \cos \left (-\frac{b^{2} - 4 \, a c}{4 \, 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}{4 \, c}\right )\right )} \sqrt{\frac{c}{\pi }} \operatorname{C}\left (\frac{\sqrt{2}{\left (2 \, c x + b\right )} \sqrt{\frac{c}{\pi }}}{2 \, c}\right ) - \sqrt{2}{\left (2 \, \pi c e^{2} \sin \left (-\frac{b^{2} - 4 \, a c}{4 \, 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}{4 \, c}\right )\right )} \sqrt{\frac{c}{\pi }} \operatorname{S}\left (\frac{\sqrt{2}{\left (2 \, c x + b\right )} \sqrt{\frac{c}{\pi }}}{2 \, c}\right ) - 2 \,{\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 )}{8 \, c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)^2*sin(c*x^2+b*x+a),x, algorithm="fricas")
[Out]
1/8*(sqrt(2)*(2*pi*c*e^2*cos(-1/4*(b^2 - 4*a*c)/c) + pi*(4*c^2*d^2 - 4*b*c*d*e + b^2*e^2)*sin(-1/4*(b^2 - 4*a*
c)/c))*sqrt(c/pi)*fresnel_cos(1/2*sqrt(2)*(2*c*x + b)*sqrt(c/pi)/c) - sqrt(2)*(2*pi*c*e^2*sin(-1/4*(b^2 - 4*a*
c)/c) - pi*(4*c^2*d^2 - 4*b*c*d*e + b^2*e^2)*cos(-1/4*(b^2 - 4*a*c)/c))*sqrt(c/pi)*fresnel_sin(1/2*sqrt(2)*(2*
c*x + b)*sqrt(c/pi)/c) - 2*(2*c^2*e^2*x + 4*c^2*d*e - b*c*e^2)*cos(c*x^2 + b*x + a))/c^3
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d + e x\right )^{2} \sin{\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*sin(c*x**2+b*x+a),x)
[Out]
Integral((d + e*x)**2*sin(a + b*x + c*x**2), x)
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Giac [C] time = 1.2609, size = 768, normalized size = 2.69 \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*sin(c*x^2+b*x+a),x, algorithm="giac")
[Out]
1/4*I*sqrt(2)*sqrt(pi)*d^2*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))) - 1/4*I*sqrt(2)*sqrt(pi)*d^2*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))) - 1/4*(I*sqrt(2)*sqrt(pi)*b*d
*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 - 4*c)/c)/((-I*c/abs(c)
+ 1)*sqrt(abs(c))) + 2*d*e^(I*c*x^2 + I*b*x + I*a + 1))/c - 1/4*(-I*sqrt(2)*sqrt(pi)*b*d*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 - 4*c)/c)/((I*c/abs(c) + 1)*sqrt(abs(c))) +
2*d*e^(-I*c*x^2 - I*b*x - I*a + 1))/c - 1/16*(-I*sqrt(2)*sqrt(pi)*(b^2 + 2*I*c)*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 - 8*c)/c)/((-I*c/abs(c) + 1)*sqrt(abs(c))) - 2*I*(c*(2
*I*x + I*b/c) - 2*I*b)*e^(I*c*x^2 + I*b*x + I*a + 2))/c^2 - 1/16*(I*sqrt(2)*sqrt(pi)*(b^2 - 2*I*c)*erf(-1/4*sq
rt(2)*(2*x + b/c)*(I*c/abs(c) + 1)*sqrt(abs(c)))*e^(-1/4*(-I*b^2 + 4*I*a*c - 8*c)/c)/((I*c/abs(c) + 1)*sqrt(ab
s(c))) - 2*I*(c*(2*I*x + I*b/c) - 2*I*b)*e^(-I*c*x^2 - I*b*x - I*a + 2))/c^2