\(\int f^{a+b x+c x^2} \cos ^3(d+e x+f x^2) \, dx\) [133]

Optimal result

Integrand size = 26, antiderivative size = 422 \[ \int f^{a+b x+c x^2} \cos ^3\left (d+e x+f x^2\right ) \, dx=\frac {3 e^{-i d-\frac {(e+i b \log (f))^2}{4 i f-4 c \log (f)}} f^a \sqrt {\pi } \text {erf}\left (\frac {i e-b \log (f)+2 x (i f-c \log (f))}{2 \sqrt {i f-c \log (f)}}\right )}{16 \sqrt {i f-c \log (f)}}+\frac {e^{-3 i d-\frac {(3 e+i b \log (f))^2}{4 (3 i f-c \log (f))}} f^a \sqrt {\pi } \text {erf}\left (\frac {3 i e-b \log (f)+2 x (3 i f-c \log (f))}{2 \sqrt {3 i f-c \log (f)}}\right )}{16 \sqrt {3 i f-c \log (f)}}+\frac {3 e^{i d+\frac {(e-i b \log (f))^2}{4 i f+4 c \log (f)}} f^a \sqrt {\pi } \text {erfi}\left (\frac {i e+b \log (f)+2 x (i f+c \log (f))}{2 \sqrt {i f+c \log (f)}}\right )}{16 \sqrt {i f+c \log (f)}}+\frac {e^{3 i d-\frac {(3 i e+b \log (f))^2}{4 (3 i f+c \log (f))}} f^a \sqrt {\pi } \text {erfi}\left (\frac {3 i e+b \log (f)+2 x (3 i f+c \log (f))}{2 \sqrt {3 i f+c \log (f)}}\right )}{16 \sqrt {3 i f+c \log (f)}} \]

[Out]

3/16*exp(-I*d-(e+I*b*ln(f))^2/(4*I*f-4*c*ln(f)))*f^a*erf(1/2*(I*e-b*ln(f)+2*x*(I*f-c*ln(f)))/(I*f-c*ln(f))^(1/
2))*Pi^(1/2)/(I*f-c*ln(f))^(1/2)+1/16*exp(-3*I*d-1/4*(3*e+I*b*ln(f))^2/(3*I*f-c*ln(f)))*f^a*erf(1/2*(3*I*e-b*l
n(f)+2*x*(3*I*f-c*ln(f)))/(3*I*f-c*ln(f))^(1/2))*Pi^(1/2)/(3*I*f-c*ln(f))^(1/2)+3/16*exp(I*d+(e-I*b*ln(f))^2/(
4*I*f+4*c*ln(f)))*f^a*erfi(1/2*(I*e+b*ln(f)+2*x*(I*f+c*ln(f)))/(I*f+c*ln(f))^(1/2))*Pi^(1/2)/(I*f+c*ln(f))^(1/
2)+1/16*exp(3*I*d-1/4*(3*I*e+b*ln(f))^2/(3*I*f+c*ln(f)))*f^a*erfi(1/2*(3*I*e+b*ln(f)+2*x*(3*I*f+c*ln(f)))/(3*I
*f+c*ln(f))^(1/2))*Pi^(1/2)/(3*I*f+c*ln(f))^(1/2)

Rubi [A] (verified)

Time = 1.04 (sec) , antiderivative size = 422, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {4561, 2325, 2266, 2236, 2235} \[ \int f^{a+b x+c x^2} \cos ^3\left (d+e x+f x^2\right ) \, dx=\frac {3 \sqrt {\pi } f^a \exp \left (-\frac {(e+i b \log (f))^2}{-4 c \log (f)+4 i f}-i d\right ) \text {erf}\left (\frac {-b \log (f)+2 x (-c \log (f)+i f)+i e}{2 \sqrt {-c \log (f)+i f}}\right )}{16 \sqrt {-c \log (f)+i f}}+\frac {\sqrt {\pi } f^a \exp \left (-\frac {(3 e+i b \log (f))^2}{4 (-c \log (f)+3 i f)}-3 i d\right ) \text {erf}\left (\frac {-b \log (f)+2 x (-c \log (f)+3 i f)+3 i e}{2 \sqrt {-c \log (f)+3 i f}}\right )}{16 \sqrt {-c \log (f)+3 i f}}+\frac {3 \sqrt {\pi } f^a \exp \left (\frac {(e-i b \log (f))^2}{4 c \log (f)+4 i f}+i d\right ) \text {erfi}\left (\frac {b \log (f)+2 x (c \log (f)+i f)+i e}{2 \sqrt {c \log (f)+i f}}\right )}{16 \sqrt {c \log (f)+i f}}+\frac {\sqrt {\pi } f^a \exp \left (3 i d-\frac {(b \log (f)+3 i e)^2}{4 (c \log (f)+3 i f)}\right ) \text {erfi}\left (\frac {b \log (f)+2 x (c \log (f)+3 i f)+3 i e}{2 \sqrt {c \log (f)+3 i f}}\right )}{16 \sqrt {c \log (f)+3 i f}} \]

[In]

Int[f^(a + b*x + c*x^2)*Cos[d + e*x + f*x^2]^3,x]

[Out]

(3*E^((-I)*d - (e + I*b*Log[f])^2/((4*I)*f - 4*c*Log[f]))*f^a*Sqrt[Pi]*Erf[(I*e - b*Log[f] + 2*x*(I*f - c*Log[
f]))/(2*Sqrt[I*f - c*Log[f]])])/(16*Sqrt[I*f - c*Log[f]]) + (E^((-3*I)*d - (3*e + I*b*Log[f])^2/(4*((3*I)*f -
c*Log[f])))*f^a*Sqrt[Pi]*Erf[((3*I)*e - b*Log[f] + 2*x*((3*I)*f - c*Log[f]))/(2*Sqrt[(3*I)*f - c*Log[f]])])/(1
6*Sqrt[(3*I)*f - c*Log[f]]) + (3*E^(I*d + (e - I*b*Log[f])^2/((4*I)*f + 4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(I*e +
b*Log[f] + 2*x*(I*f + c*Log[f]))/(2*Sqrt[I*f + c*Log[f]])])/(16*Sqrt[I*f + c*Log[f]]) + (E^((3*I)*d - ((3*I)*e
+ b*Log[f])^2/(4*((3*I)*f + c*Log[f])))*f^a*Sqrt[Pi]*Erfi[((3*I)*e + b*Log[f] + 2*x*((3*I)*f + c*Log[f]))/(2*
Sqrt[(3*I)*f + c*Log[f]])])/(16*Sqrt[(3*I)*f + c*Log[f]])

Rule 2235

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt[Pi]*(Erfi[(c + d*x)*Rt[b*Log[F], 2
]]/(2*d*Rt[b*Log[F], 2])), x] /; FreeQ[{F, a, b, c, d}, x] && PosQ[b]

Rule 2236

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt[Pi]*(Erf[(c + d*x)*Rt[(-b)*Log[F],
2]]/(2*d*Rt[(-b)*Log[F], 2])), x] /; FreeQ[{F, a, b, c, d}, x] && NegQ[b]

Rule 2266

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

Rule 2325

Int[(u_.)*(F_)^(v_)*(G_)^(w_), x_Symbol] :> With[{z = v*Log[F] + w*Log[G]}, Int[u*NormalizeIntegrand[E^z, x],
x] /; BinomialQ[z, x] || (PolynomialQ[z, x] && LeQ[Exponent[z, x], 2])] /; FreeQ[{F, G}, x]

Rule 4561

Int[Cos[v_]^(n_.)*(F_)^(u_), x_Symbol] :> Int[ExpandTrigToExp[F^u, Cos[v]^n, x], x] /; FreeQ[F, x] && (LinearQ
[u, x] || PolyQ[u, x, 2]) && (LinearQ[v, x] || PolyQ[v, x, 2]) && IGtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{8} e^{-3 i \left (d+e x+f x^2\right )} f^{a+b x+c x^2}+\frac {3}{8} \exp \left (2 i d+2 i e x+2 i f x^2-3 i \left (d+e x+f x^2\right )\right ) f^{a+b x+c x^2}+\frac {3}{8} \exp \left (4 i d+4 i e x+4 i f x^2-3 i \left (d+e x+f x^2\right )\right ) f^{a+b x+c x^2}+\frac {1}{8} \exp \left (6 i d+6 i e x+6 i f x^2-3 i \left (d+e x+f x^2\right )\right ) f^{a+b x+c x^2}\right ) \, dx \\ & = \frac {1}{8} \int e^{-3 i \left (d+e x+f x^2\right )} f^{a+b x+c x^2} \, dx+\frac {1}{8} \int \exp \left (6 i d+6 i e x+6 i f x^2-3 i \left (d+e x+f x^2\right )\right ) f^{a+b x+c x^2} \, dx+\frac {3}{8} \int \exp \left (2 i d+2 i e x+2 i f x^2-3 i \left (d+e x+f x^2\right )\right ) f^{a+b x+c x^2} \, dx+\frac {3}{8} \int \exp \left (4 i d+4 i e x+4 i f x^2-3 i \left (d+e x+f x^2\right )\right ) f^{a+b x+c x^2} \, dx \\ & = \frac {1}{8} \int \exp \left (-3 i d+a \log (f)-x (3 i e-b \log (f))-x^2 (3 i f-c \log (f))\right ) \, dx+\frac {1}{8} \int \exp \left (3 i d+a \log (f)+x (3 i e+b \log (f))+x^2 (3 i f+c \log (f))\right ) \, dx+\frac {3}{8} \int \exp \left (-i d+a \log (f)-x (i e-b \log (f))-x^2 (i f-c \log (f))\right ) \, dx+\frac {3}{8} \int \exp \left (i d+a \log (f)+x (i e+b \log (f))+x^2 (i f+c \log (f))\right ) \, dx \\ & = \frac {1}{8} \left (3 \exp \left (-i d-\frac {(e+i b \log (f))^2}{4 i f-4 c \log (f)}\right ) f^a\right ) \int \exp \left (\frac {(-i e+b \log (f)+2 x (-i f+c \log (f)))^2}{4 (-i f+c \log (f))}\right ) \, dx+\frac {1}{8} \left (\exp \left (-3 i d-\frac {(3 e+i b \log (f))^2}{4 (3 i f-c \log (f))}\right ) f^a\right ) \int \exp \left (\frac {(-3 i e+b \log (f)+2 x (-3 i f+c \log (f)))^2}{4 (-3 i f+c \log (f))}\right ) \, dx+\frac {1}{8} \left (\exp \left (3 i d-\frac {(3 i e+b \log (f))^2}{4 (3 i f+c \log (f))}\right ) f^a\right ) \int \exp \left (\frac {(3 i e+b \log (f)+2 x (3 i f+c \log (f)))^2}{4 (3 i f+c \log (f))}\right ) \, dx+\frac {1}{8} \left (3 \exp \left (i d+\frac {(e-i b \log (f))^2}{4 i f+4 c \log (f)}\right ) f^a\right ) \int \exp \left (\frac {(i e+b \log (f)+2 x (i f+c \log (f)))^2}{4 (i f+c \log (f))}\right ) \, dx \\ & = \frac {3 \exp \left (-i d-\frac {(e+i b \log (f))^2}{4 i f-4 c \log (f)}\right ) f^a \sqrt {\pi } \text {erf}\left (\frac {i e-b \log (f)+2 x (i f-c \log (f))}{2 \sqrt {i f-c \log (f)}}\right )}{16 \sqrt {i f-c \log (f)}}+\frac {\exp \left (-3 i d-\frac {(3 e+i b \log (f))^2}{4 (3 i f-c \log (f))}\right ) f^a \sqrt {\pi } \text {erf}\left (\frac {3 i e-b \log (f)+2 x (3 i f-c \log (f))}{2 \sqrt {3 i f-c \log (f)}}\right )}{16 \sqrt {3 i f-c \log (f)}}+\frac {3 \exp \left (i d+\frac {(e-i b \log (f))^2}{4 i f+4 c \log (f)}\right ) f^a \sqrt {\pi } \text {erfi}\left (\frac {i e+b \log (f)+2 x (i f+c \log (f))}{2 \sqrt {i f+c \log (f)}}\right )}{16 \sqrt {i f+c \log (f)}}+\frac {\exp \left (3 i d-\frac {(3 i e+b \log (f))^2}{4 (3 i f+c \log (f))}\right ) f^a \sqrt {\pi } \text {erfi}\left (\frac {3 i e+b \log (f)+2 x (3 i f+c \log (f))}{2 \sqrt {3 i f+c \log (f)}}\right )}{16 \sqrt {3 i f+c \log (f)}} \\ \end{align*}

Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(3829\) vs. \(2(422)=844\).

Time = 6.98 (sec) , antiderivative size = 3829, normalized size of antiderivative = 9.07 \[ \int f^{a+b x+c x^2} \cos ^3\left (d+e x+f x^2\right ) \, dx=\text {Result too large to show} \]

[In]

Integrate[f^(a + b*x + c*x^2)*Cos[d + e*x + f*x^2]^3,x]

[Out]

(f^a*Sqrt[Pi]*(-27*(-1)^(3/4)*E^(((I/4)*(-e^2 + (2*I)*b*e*Log[f] + b^2*Log[f]^2))/(f - I*c*Log[f]))*f^3*Cos[d]
*Erfi[((-1)^(1/4)*(e + 2*f*x - I*b*Log[f] - (2*I)*c*x*Log[f]))/(2*Sqrt[f - I*c*Log[f]])]*Sqrt[f - I*c*Log[f]]
+ 27*(-1)^(1/4)*c*E^(((I/4)*(-e^2 + (2*I)*b*e*Log[f] + b^2*Log[f]^2))/(f - I*c*Log[f]))*f^2*Cos[d]*Erfi[((-1)^
(1/4)*(e + 2*f*x - I*b*Log[f] - (2*I)*c*x*Log[f]))/(2*Sqrt[f - I*c*Log[f]])]*Log[f]*Sqrt[f - I*c*Log[f]] - 3*(
-1)^(3/4)*c^2*E^(((I/4)*(-e^2 + (2*I)*b*e*Log[f] + b^2*Log[f]^2))/(f - I*c*Log[f]))*f*Cos[d]*Erfi[((-1)^(1/4)*
(e + 2*f*x - I*b*Log[f] - (2*I)*c*x*Log[f]))/(2*Sqrt[f - I*c*Log[f]])]*Log[f]^2*Sqrt[f - I*c*Log[f]] + 3*(-1)^
(1/4)*c^3*E^(((I/4)*(-e^2 + (2*I)*b*e*Log[f] + b^2*Log[f]^2))/(f - I*c*Log[f]))*Cos[d]*Erfi[((-1)^(1/4)*(e + 2
*f*x - I*b*Log[f] - (2*I)*c*x*Log[f]))/(2*Sqrt[f - I*c*Log[f]])]*Log[f]^3*Sqrt[f - I*c*Log[f]] - 3*(-1)^(3/4)*
E^(((I/4)*(-9*e^2 + (6*I)*b*e*Log[f] + b^2*Log[f]^2))/(3*f - I*c*Log[f]))*f^3*Cos[3*d]*Erfi[((-1)^(1/4)*(3*e +
6*f*x - I*b*Log[f] - (2*I)*c*x*Log[f]))/(2*Sqrt[3*f - I*c*Log[f]])]*Sqrt[3*f - I*c*Log[f]] + (-1)^(1/4)*c*E^(
((I/4)*(-9*e^2 + (6*I)*b*e*Log[f] + b^2*Log[f]^2))/(3*f - I*c*Log[f]))*f^2*Cos[3*d]*Erfi[((-1)^(1/4)*(3*e + 6*
f*x - I*b*Log[f] - (2*I)*c*x*Log[f]))/(2*Sqrt[3*f - I*c*Log[f]])]*Log[f]*Sqrt[3*f - I*c*Log[f]] - 3*(-1)^(3/4)
*c^2*E^(((I/4)*(-9*e^2 + (6*I)*b*e*Log[f] + b^2*Log[f]^2))/(3*f - I*c*Log[f]))*f*Cos[3*d]*Erfi[((-1)^(1/4)*(3*
e + 6*f*x - I*b*Log[f] - (2*I)*c*x*Log[f]))/(2*Sqrt[3*f - I*c*Log[f]])]*Log[f]^2*Sqrt[3*f - I*c*Log[f]] + (-1)
^(1/4)*c^3*E^(((I/4)*(-9*e^2 + (6*I)*b*e*Log[f] + b^2*Log[f]^2))/(3*f - I*c*Log[f]))*Cos[3*d]*Erfi[((-1)^(1/4)
*(3*e + 6*f*x - I*b*Log[f] - (2*I)*c*x*Log[f]))/(2*Sqrt[3*f - I*c*Log[f]])]*Log[f]^3*Sqrt[3*f - I*c*Log[f]] -
(27*(-1)^(1/4)*f^3*Cos[d]*Erfi[((-1)^(3/4)*(e + 2*f*x + I*b*Log[f] + (2*I)*c*x*Log[f]))/(2*Sqrt[f + I*c*Log[f]
])]*Sqrt[f + I*c*Log[f]])/E^(((I/4)*(-e^2 - (2*I)*b*e*Log[f] + b^2*Log[f]^2))/(f + I*c*Log[f])) + (27*(-1)^(3/
4)*c*f^2*Cos[d]*Erfi[((-1)^(3/4)*(e + 2*f*x + I*b*Log[f] + (2*I)*c*x*Log[f]))/(2*Sqrt[f + I*c*Log[f]])]*Log[f]
*Sqrt[f + I*c*Log[f]])/E^(((I/4)*(-e^2 - (2*I)*b*e*Log[f] + b^2*Log[f]^2))/(f + I*c*Log[f])) - (3*(-1)^(1/4)*c
^2*f*Cos[d]*Erfi[((-1)^(3/4)*(e + 2*f*x + I*b*Log[f] + (2*I)*c*x*Log[f]))/(2*Sqrt[f + I*c*Log[f]])]*Log[f]^2*S
qrt[f + I*c*Log[f]])/E^(((I/4)*(-e^2 - (2*I)*b*e*Log[f] + b^2*Log[f]^2))/(f + I*c*Log[f])) + (3*(-1)^(3/4)*c^3
*Cos[d]*Erfi[((-1)^(3/4)*(e + 2*f*x + I*b*Log[f] + (2*I)*c*x*Log[f]))/(2*Sqrt[f + I*c*Log[f]])]*Log[f]^3*Sqrt[
f + I*c*Log[f]])/E^(((I/4)*(-e^2 - (2*I)*b*e*Log[f] + b^2*Log[f]^2))/(f + I*c*Log[f])) - (3*(-1)^(1/4)*f^3*Cos
[3*d]*Erfi[((-1)^(3/4)*(3*e + 6*f*x + I*b*Log[f] + (2*I)*c*x*Log[f]))/(2*Sqrt[3*f + I*c*Log[f]])]*Sqrt[3*f + I
*c*Log[f]])/E^(((I/4)*(-9*e^2 - (6*I)*b*e*Log[f] + b^2*Log[f]^2))/(3*f + I*c*Log[f])) + ((-1)^(3/4)*c*f^2*Cos[
3*d]*Erfi[((-1)^(3/4)*(3*e + 6*f*x + I*b*Log[f] + (2*I)*c*x*Log[f]))/(2*Sqrt[3*f + I*c*Log[f]])]*Log[f]*Sqrt[3
*f + I*c*Log[f]])/E^(((I/4)*(-9*e^2 - (6*I)*b*e*Log[f] + b^2*Log[f]^2))/(3*f + I*c*Log[f])) - (3*(-1)^(1/4)*c^
2*f*Cos[3*d]*Erfi[((-1)^(3/4)*(3*e + 6*f*x + I*b*Log[f] + (2*I)*c*x*Log[f]))/(2*Sqrt[3*f + I*c*Log[f]])]*Log[f
]^2*Sqrt[3*f + I*c*Log[f]])/E^(((I/4)*(-9*e^2 - (6*I)*b*e*Log[f] + b^2*Log[f]^2))/(3*f + I*c*Log[f])) + ((-1)^
(3/4)*c^3*Cos[3*d]*Erfi[((-1)^(3/4)*(3*e + 6*f*x + I*b*Log[f] + (2*I)*c*x*Log[f]))/(2*Sqrt[3*f + I*c*Log[f]])]
*Log[f]^3*Sqrt[3*f + I*c*Log[f]])/E^(((I/4)*(-9*e^2 - (6*I)*b*e*Log[f] + b^2*Log[f]^2))/(3*f + I*c*Log[f])) +
27*(-1)^(1/4)*E^(((I/4)*(-e^2 + (2*I)*b*e*Log[f] + b^2*Log[f]^2))/(f - I*c*Log[f]))*f^3*Erfi[((-1)^(1/4)*(e +
2*f*x - I*b*Log[f] - (2*I)*c*x*Log[f]))/(2*Sqrt[f - I*c*Log[f]])]*Sqrt[f - I*c*Log[f]]*Sin[d] + 27*(-1)^(3/4)*
c*E^(((I/4)*(-e^2 + (2*I)*b*e*Log[f] + b^2*Log[f]^2))/(f - I*c*Log[f]))*f^2*Erfi[((-1)^(1/4)*(e + 2*f*x - I*b*
Log[f] - (2*I)*c*x*Log[f]))/(2*Sqrt[f - I*c*Log[f]])]*Log[f]*Sqrt[f - I*c*Log[f]]*Sin[d] + 3*(-1)^(1/4)*c^2*E^
(((I/4)*(-e^2 + (2*I)*b*e*Log[f] + b^2*Log[f]^2))/(f - I*c*Log[f]))*f*Erfi[((-1)^(1/4)*(e + 2*f*x - I*b*Log[f]
- (2*I)*c*x*Log[f]))/(2*Sqrt[f - I*c*Log[f]])]*Log[f]^2*Sqrt[f - I*c*Log[f]]*Sin[d] + 3*(-1)^(3/4)*c^3*E^(((I
/4)*(-e^2 + (2*I)*b*e*Log[f] + b^2*Log[f]^2))/(f - I*c*Log[f]))*Erfi[((-1)^(1/4)*(e + 2*f*x - I*b*Log[f] - (2*
I)*c*x*Log[f]))/(2*Sqrt[f - I*c*Log[f]])]*Log[f]^3*Sqrt[f - I*c*Log[f]]*Sin[d] + (27*(-1)^(3/4)*f^3*Erfi[((-1)
^(3/4)*(e + 2*f*x + I*b*Log[f] + (2*I)*c*x*Log[f]))/(2*Sqrt[f + I*c*Log[f]])]*Sqrt[f + I*c*Log[f]]*Sin[d])/E^(
((I/4)*(-e^2 - (2*I)*b*e*Log[f] + b^2*Log[f]^2))/(f + I*c*Log[f])) + (27*(-1)^(1/4)*c*f^2*Erfi[((-1)^(3/4)*(e
+ 2*f*x + I*b*Log[f] + (2*I)*c*x*Log[f]))/(2*Sqrt[f + I*c*Log[f]])]*Log[f]*Sqrt[f + I*c*Log[f]]*Sin[d])/E^(((I
/4)*(-e^2 - (2*I)*b*e*Log[f] + b^2*Log[f]^2))/(f + I*c*Log[f])) + (3*(-1)^(3/4)*c^2*f*Erfi[((-1)^(3/4)*(e + 2*
f*x + I*b*Log[f] + (2*I)*c*x*Log[f]))/(2*Sqrt[f + I*c*Log[f]])]*Log[f]^2*Sqrt[f + I*c*Log[f]]*Sin[d])/E^(((I/4
)*(-e^2 - (2*I)*b*e*Log[f] + b^2*Log[f]^2))/(f + I*c*Log[f])) + (3*(-1)^(1/4)*c^3*Erfi[((-1)^(3/4)*(e + 2*f*x
+ I*b*Log[f] + (2*I)*c*x*Log[f]))/(2*Sqrt[f + I*c*Log[f]])]*Log[f]^3*Sqrt[f + I*c*Log[f]]*Sin[d])/E^(((I/4)*(-
e^2 - (2*I)*b*e*Log[f] + b^2*Log[f]^2))/(f + I*c*Log[f])) + 3*(-1)^(1/4)*E^(((I/4)*(-9*e^2 + (6*I)*b*e*Log[f]
+ b^2*Log[f]^2))/(3*f - I*c*Log[f]))*f^3*Erfi[((-1)^(1/4)*(3*e + 6*f*x - I*b*Log[f] - (2*I)*c*x*Log[f]))/(2*Sq
rt[3*f - I*c*Log[f]])]*Sqrt[3*f - I*c*Log[f]]*Sin[3*d] + (-1)^(3/4)*c*E^(((I/4)*(-9*e^2 + (6*I)*b*e*Log[f] + b
^2*Log[f]^2))/(3*f - I*c*Log[f]))*f^2*Erfi[((-1)^(1/4)*(3*e + 6*f*x - I*b*Log[f] - (2*I)*c*x*Log[f]))/(2*Sqrt[
3*f - I*c*Log[f]])]*Log[f]*Sqrt[3*f - I*c*Log[f]]*Sin[3*d] + 3*(-1)^(1/4)*c^2*E^(((I/4)*(-9*e^2 + (6*I)*b*e*Lo
g[f] + b^2*Log[f]^2))/(3*f - I*c*Log[f]))*f*Erfi[((-1)^(1/4)*(3*e + 6*f*x - I*b*Log[f] - (2*I)*c*x*Log[f]))/(2
*Sqrt[3*f - I*c*Log[f]])]*Log[f]^2*Sqrt[3*f - I*c*Log[f]]*Sin[3*d] + (-1)^(3/4)*c^3*E^(((I/4)*(-9*e^2 + (6*I)*
b*e*Log[f] + b^2*Log[f]^2))/(3*f - I*c*Log[f]))*Erfi[((-1)^(1/4)*(3*e + 6*f*x - I*b*Log[f] - (2*I)*c*x*Log[f])
)/(2*Sqrt[3*f - I*c*Log[f]])]*Log[f]^3*Sqrt[3*f - I*c*Log[f]]*Sin[3*d] + (3*(-1)^(3/4)*f^3*Erfi[((-1)^(3/4)*(3
*e + 6*f*x + I*b*Log[f] + (2*I)*c*x*Log[f]))/(2*Sqrt[3*f + I*c*Log[f]])]*Sqrt[3*f + I*c*Log[f]]*Sin[3*d])/E^((
(I/4)*(-9*e^2 - (6*I)*b*e*Log[f] + b^2*Log[f]^2))/(3*f + I*c*Log[f])) + ((-1)^(1/4)*c*f^2*Erfi[((-1)^(3/4)*(3*
e + 6*f*x + I*b*Log[f] + (2*I)*c*x*Log[f]))/(2*Sqrt[3*f + I*c*Log[f]])]*Log[f]*Sqrt[3*f + I*c*Log[f]]*Sin[3*d]
)/E^(((I/4)*(-9*e^2 - (6*I)*b*e*Log[f] + b^2*Log[f]^2))/(3*f + I*c*Log[f])) + (3*(-1)^(3/4)*c^2*f*Erfi[((-1)^(
3/4)*(3*e + 6*f*x + I*b*Log[f] + (2*I)*c*x*Log[f]))/(2*Sqrt[3*f + I*c*Log[f]])]*Log[f]^2*Sqrt[3*f + I*c*Log[f]
]*Sin[3*d])/E^(((I/4)*(-9*e^2 - (6*I)*b*e*Log[f] + b^2*Log[f]^2))/(3*f + I*c*Log[f])) + ((-1)^(1/4)*c^3*Erfi[(
(-1)^(3/4)*(3*e + 6*f*x + I*b*Log[f] + (2*I)*c*x*Log[f]))/(2*Sqrt[3*f + I*c*Log[f]])]*Log[f]^3*Sqrt[3*f + I*c*
Log[f]]*Sin[3*d])/E^(((I/4)*(-9*e^2 - (6*I)*b*e*Log[f] + b^2*Log[f]^2))/(3*f + I*c*Log[f]))))/(16*(f - I*c*Log
[f])*(3*f - I*c*Log[f])*(f + I*c*Log[f])*(3*f + I*c*Log[f]))

Maple [A] (verified)

Time = 2.16 (sec) , antiderivative size = 426, normalized size of antiderivative = 1.01

method	result	size
risch	\(-\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {\ln \left (f \right )^{2} b^{2}-6 i \ln \left (f \right ) b e +12 i d \ln \left (f \right ) c +36 d f -9 e^{2}}{4 \left (c \ln \left (f \right )-3 i f \right )}} \operatorname {erf}\left (-x \sqrt {3 i f -c \ln \left (f \right )}+\frac {b \ln \left (f \right )-3 i e}{2 \sqrt {3 i f -c \ln \left (f \right )}}\right )}{16 \sqrt {3 i f -c \ln \left (f \right )}}-\frac {3 \sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {\ln \left (f \right )^{2} b^{2}-2 i \ln \left (f \right ) b e +4 i d \ln \left (f \right ) c +4 d f -e^{2}}{4 \left (c \ln \left (f \right )-i f \right )}} \operatorname {erf}\left (-x \sqrt {i f -c \ln \left (f \right )}+\frac {b \ln \left (f \right )-i e}{2 \sqrt {i f -c \ln \left (f \right )}}\right )}{16 \sqrt {i f -c \ln \left (f \right )}}-\frac {3 \sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {\ln \left (f \right )^{2} b^{2}+2 i \ln \left (f \right ) b e -4 i d \ln \left (f \right ) c +4 d f -e^{2}}{4 \left (i f +c \ln \left (f \right )\right )}} \operatorname {erf}\left (-\sqrt {-c \ln \left (f \right )-i f}\, x +\frac {i e +b \ln \left (f \right )}{2 \sqrt {-c \ln \left (f \right )-i f}}\right )}{16 \sqrt {-c \ln \left (f \right )-i f}}-\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {\ln \left (f \right )^{2} b^{2}+6 i \ln \left (f \right ) b e -12 i d \ln \left (f \right ) c +36 d f -9 e^{2}}{4 \left (3 i f +c \ln \left (f \right )\right )}} \operatorname {erf}\left (-\sqrt {-c \ln \left (f \right )-3 i f}\, x +\frac {3 i e +b \ln \left (f \right )}{2 \sqrt {-c \ln \left (f \right )-3 i f}}\right )}{16 \sqrt {-c \ln \left (f \right )-3 i f}}\)	\(426\)

[In]

int(f^(c*x^2+b*x+a)*cos(f*x^2+e*x+d)^3,x,method=_RETURNVERBOSE)

[Out]

-1/16*Pi^(1/2)*f^a*exp(-1/4*(ln(f)^2*b^2-6*I*ln(f)*b*e+12*I*d*ln(f)*c+36*d*f-9*e^2)/(c*ln(f)-3*I*f))/(3*I*f-c*
ln(f))^(1/2)*erf(-x*(3*I*f-c*ln(f))^(1/2)+1/2*(b*ln(f)-3*I*e)/(3*I*f-c*ln(f))^(1/2))-3/16*Pi^(1/2)*f^a*exp(-1/
4*(ln(f)^2*b^2-2*I*ln(f)*b*e+4*I*d*ln(f)*c+4*d*f-e^2)/(c*ln(f)-I*f))/(I*f-c*ln(f))^(1/2)*erf(-x*(I*f-c*ln(f))^
(1/2)+1/2*(b*ln(f)-I*e)/(I*f-c*ln(f))^(1/2))-3/16*Pi^(1/2)*f^a*exp(-1/4*(ln(f)^2*b^2+2*I*ln(f)*b*e-4*I*d*ln(f)
*c+4*d*f-e^2)/(I*f+c*ln(f)))/(-c*ln(f)-I*f)^(1/2)*erf(-(-c*ln(f)-I*f)^(1/2)*x+1/2*(I*e+b*ln(f))/(-c*ln(f)-I*f)
^(1/2))-1/16*Pi^(1/2)*f^a*exp(-1/4*(ln(f)^2*b^2+6*I*ln(f)*b*e-12*I*d*ln(f)*c+36*d*f-9*e^2)/(3*I*f+c*ln(f)))/(-
c*ln(f)-3*I*f)^(1/2)*erf(-(-c*ln(f)-3*I*f)^(1/2)*x+1/2*(3*I*e+b*ln(f))/(-c*ln(f)-3*I*f)^(1/2))

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 859 vs. \(2 (312) = 624\).

Time = 0.30 (sec) , antiderivative size = 859, normalized size of antiderivative = 2.04 \[ \int f^{a+b x+c x^2} \cos ^3\left (d+e x+f x^2\right ) \, dx=\text {Too large to display} \]

[In]

integrate(f^(c*x^2+b*x+a)*cos(f*x^2+e*x+d)^3,x, algorithm="fricas")

[Out]

-1/16*(sqrt(pi)*(c^3*log(f)^3 - 3*I*c^2*f*log(f)^2 + c*f^2*log(f) - 3*I*f^3)*sqrt(-c*log(f) - 3*I*f)*erf(1/2*(
18*f^2*x + (2*c^2*x + b*c)*log(f)^2 + 9*e*f - 3*(-I*c*e + I*b*f)*log(f))*sqrt(-c*log(f) - 3*I*f)/(c^2*log(f)^2
+ 9*f^2))*e^(-1/4*((b^2*c - 4*a*c^2)*log(f)^3 + 27*I*e^2*f - 108*I*d*f^2 + 3*(-4*I*c^2*d + 2*I*b*c*e - I*b^2*
f)*log(f)^2 - 9*(c*e^2 - 2*b*e*f + 4*a*f^2)*log(f))/(c^2*log(f)^2 + 9*f^2)) + 3*sqrt(pi)*(c^3*log(f)^3 - I*c^2
*f*log(f)^2 + 9*c*f^2*log(f) - 9*I*f^3)*sqrt(-c*log(f) - I*f)*erf(1/2*(2*f^2*x + (2*c^2*x + b*c)*log(f)^2 + e*
f + (I*c*e - I*b*f)*log(f))*sqrt(-c*log(f) - I*f)/(c^2*log(f)^2 + f^2))*e^(-1/4*((b^2*c - 4*a*c^2)*log(f)^3 +
I*e^2*f - 4*I*d*f^2 - (4*I*c^2*d - 2*I*b*c*e + I*b^2*f)*log(f)^2 - (c*e^2 - 2*b*e*f + 4*a*f^2)*log(f))/(c^2*lo
g(f)^2 + f^2)) + 3*sqrt(pi)*(c^3*log(f)^3 + I*c^2*f*log(f)^2 + 9*c*f^2*log(f) + 9*I*f^3)*sqrt(-c*log(f) + I*f)
*erf(1/2*(2*f^2*x + (2*c^2*x + b*c)*log(f)^2 + e*f + (-I*c*e + I*b*f)*log(f))*sqrt(-c*log(f) + I*f)/(c^2*log(f
)^2 + f^2))*e^(-1/4*((b^2*c - 4*a*c^2)*log(f)^3 - I*e^2*f + 4*I*d*f^2 - (-4*I*c^2*d + 2*I*b*c*e - I*b^2*f)*log
(f)^2 - (c*e^2 - 2*b*e*f + 4*a*f^2)*log(f))/(c^2*log(f)^2 + f^2)) + sqrt(pi)*(c^3*log(f)^3 + 3*I*c^2*f*log(f)^
2 + c*f^2*log(f) + 3*I*f^3)*sqrt(-c*log(f) + 3*I*f)*erf(1/2*(18*f^2*x + (2*c^2*x + b*c)*log(f)^2 + 9*e*f - 3*(
I*c*e - I*b*f)*log(f))*sqrt(-c*log(f) + 3*I*f)/(c^2*log(f)^2 + 9*f^2))*e^(-1/4*((b^2*c - 4*a*c^2)*log(f)^3 - 2
7*I*e^2*f + 108*I*d*f^2 + 3*(4*I*c^2*d - 2*I*b*c*e + I*b^2*f)*log(f)^2 - 9*(c*e^2 - 2*b*e*f + 4*a*f^2)*log(f))
/(c^2*log(f)^2 + 9*f^2)))/(c^4*log(f)^4 + 10*c^2*f^2*log(f)^2 + 9*f^4)

Sympy [F(-1)]

Timed out. \[ \int f^{a+b x+c x^2} \cos ^3\left (d+e x+f x^2\right ) \, dx=\text {Timed out} \]

[In]

integrate(f**(c*x**2+b*x+a)*cos(f*x**2+e*x+d)**3,x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 4348 vs. \(2 (312) = 624\).

Time = 0.31 (sec) , antiderivative size = 4348, normalized size of antiderivative = 10.30 \[ \int f^{a+b x+c x^2} \cos ^3\left (d+e x+f x^2\right ) \, dx=\text {Too large to display} \]

[In]

integrate(f^(c*x^2+b*x+a)*cos(f*x^2+e*x+d)^3,x, algorithm="maxima")

[Out]

-1/32*(sqrt(pi)*sqrt(2*c^2*log(f)^2 + 18*f^2)*(((I*c^2*f^a*e^(1/4*b^2*c*log(f)^3/(c^2*log(f)^2 + f^2) + 9/4*c*
e^2*log(f)/(c^2*log(f)^2 + 9*f^2) + 1/2*b*e*f*log(f)/(c^2*log(f)^2 + f^2))*log(f)^2 + I*f^(a + 2)*e^(1/4*b^2*c
*log(f)^3/(c^2*log(f)^2 + f^2) + 9/4*c*e^2*log(f)/(c^2*log(f)^2 + 9*f^2) + 1/2*b*e*f*log(f)/(c^2*log(f)^2 + f^
2)))*cos(-3/4*(9*e^2*f - 36*d*f^2 - (4*c^2*d - 2*b*c*e + b^2*f)*log(f)^2)/(c^2*log(f)^2 + 9*f^2)) + (c^2*f^a*e
^(1/4*b^2*c*log(f)^3/(c^2*log(f)^2 + f^2) + 9/4*c*e^2*log(f)/(c^2*log(f)^2 + 9*f^2) + 1/2*b*e*f*log(f)/(c^2*lo
g(f)^2 + f^2))*log(f)^2 + f^(a + 2)*e^(1/4*b^2*c*log(f)^3/(c^2*log(f)^2 + f^2) + 9/4*c*e^2*log(f)/(c^2*log(f)^
2 + 9*f^2) + 1/2*b*e*f*log(f)/(c^2*log(f)^2 + f^2)))*sin(-3/4*(9*e^2*f - 36*d*f^2 - (4*c^2*d - 2*b*c*e + b^2*f
)*log(f)^2)/(c^2*log(f)^2 + 9*f^2)))*erf(1/2*(2*(c*log(f) - 3*I*f)*x + b*log(f) - 3*I*e)*sqrt(-c*log(f) + 3*I*
f)/(c*log(f) - 3*I*f)) + ((-I*c^2*f^a*e^(1/4*b^2*c*log(f)^3/(c^2*log(f)^2 + f^2) + 9/4*c*e^2*log(f)/(c^2*log(f
)^2 + 9*f^2) + 1/2*b*e*f*log(f)/(c^2*log(f)^2 + f^2))*log(f)^2 - I*f^(a + 2)*e^(1/4*b^2*c*log(f)^3/(c^2*log(f)
^2 + f^2) + 9/4*c*e^2*log(f)/(c^2*log(f)^2 + 9*f^2) + 1/2*b*e*f*log(f)/(c^2*log(f)^2 + f^2)))*cos(-3/4*(9*e^2*
f - 36*d*f^2 - (4*c^2*d - 2*b*c*e + b^2*f)*log(f)^2)/(c^2*log(f)^2 + 9*f^2)) + (c^2*f^a*e^(1/4*b^2*c*log(f)^3/
(c^2*log(f)^2 + f^2) + 9/4*c*e^2*log(f)/(c^2*log(f)^2 + 9*f^2) + 1/2*b*e*f*log(f)/(c^2*log(f)^2 + f^2))*log(f)
^2 + f^(a + 2)*e^(1/4*b^2*c*log(f)^3/(c^2*log(f)^2 + f^2) + 9/4*c*e^2*log(f)/(c^2*log(f)^2 + 9*f^2) + 1/2*b*e*
f*log(f)/(c^2*log(f)^2 + f^2)))*sin(-3/4*(9*e^2*f - 36*d*f^2 - (4*c^2*d - 2*b*c*e + b^2*f)*log(f)^2)/(c^2*log(
f)^2 + 9*f^2)))*erf(1/2*(2*(c*log(f) + 3*I*f)*x + b*log(f) + 3*I*e)*sqrt(-c*log(f) - 3*I*f)/(c*log(f) + 3*I*f)
))*sqrt(c*log(f) + sqrt(c^2*log(f)^2 + 9*f^2)) - 3*sqrt(pi)*sqrt(2*c^2*log(f)^2 + 2*f^2)*(((-I*c^2*f^a*e^(1/4*
b^2*c*log(f)^3/(c^2*log(f)^2 + 9*f^2) + 1/4*c*e^2*log(f)/(c^2*log(f)^2 + f^2) + 9/2*b*e*f*log(f)/(c^2*log(f)^2
+ 9*f^2))*log(f)^2 - 9*I*f^(a + 2)*e^(1/4*b^2*c*log(f)^3/(c^2*log(f)^2 + 9*f^2) + 1/4*c*e^2*log(f)/(c^2*log(f
)^2 + f^2) + 9/2*b*e*f*log(f)/(c^2*log(f)^2 + 9*f^2)))*cos(-1/4*(e^2*f - 4*d*f^2 - (4*c^2*d - 2*b*c*e + b^2*f)
*log(f)^2)/(c^2*log(f)^2 + f^2)) - (c^2*f^a*e^(1/4*b^2*c*log(f)^3/(c^2*log(f)^2 + 9*f^2) + 1/4*c*e^2*log(f)/(c
^2*log(f)^2 + f^2) + 9/2*b*e*f*log(f)/(c^2*log(f)^2 + 9*f^2))*log(f)^2 + 9*f^(a + 2)*e^(1/4*b^2*c*log(f)^3/(c^
2*log(f)^2 + 9*f^2) + 1/4*c*e^2*log(f)/(c^2*log(f)^2 + f^2) + 9/2*b*e*f*log(f)/(c^2*log(f)^2 + 9*f^2)))*sin(-1
/4*(e^2*f - 4*d*f^2 - (4*c^2*d - 2*b*c*e + b^2*f)*log(f)^2)/(c^2*log(f)^2 + f^2)))*erf(1/2*(2*(c*log(f) - I*f)
*x + b*log(f) - I*e)*sqrt(-c*log(f) + I*f)/(c*log(f) - I*f)) + ((I*c^2*f^a*e^(1/4*b^2*c*log(f)^3/(c^2*log(f)^2
+ 9*f^2) + 1/4*c*e^2*log(f)/(c^2*log(f)^2 + f^2) + 9/2*b*e*f*log(f)/(c^2*log(f)^2 + 9*f^2))*log(f)^2 + 9*I*f^
(a + 2)*e^(1/4*b^2*c*log(f)^3/(c^2*log(f)^2 + 9*f^2) + 1/4*c*e^2*log(f)/(c^2*log(f)^2 + f^2) + 9/2*b*e*f*log(f
)/(c^2*log(f)^2 + 9*f^2)))*cos(-1/4*(e^2*f - 4*d*f^2 - (4*c^2*d - 2*b*c*e + b^2*f)*log(f)^2)/(c^2*log(f)^2 + f
^2)) - (c^2*f^a*e^(1/4*b^2*c*log(f)^3/(c^2*log(f)^2 + 9*f^2) + 1/4*c*e^2*log(f)/(c^2*log(f)^2 + f^2) + 9/2*b*e
*f*log(f)/(c^2*log(f)^2 + 9*f^2))*log(f)^2 + 9*f^(a + 2)*e^(1/4*b^2*c*log(f)^3/(c^2*log(f)^2 + 9*f^2) + 1/4*c*
e^2*log(f)/(c^2*log(f)^2 + f^2) + 9/2*b*e*f*log(f)/(c^2*log(f)^2 + 9*f^2)))*sin(-1/4*(e^2*f - 4*d*f^2 - (4*c^2
*d - 2*b*c*e + b^2*f)*log(f)^2)/(c^2*log(f)^2 + f^2)))*erf(1/2*(2*(c*log(f) + I*f)*x + b*log(f) + I*e)*sqrt(-c
*log(f) - I*f)/(c*log(f) + I*f)))*sqrt(c*log(f) + sqrt(c^2*log(f)^2 + f^2)) - sqrt(pi)*sqrt(2*c^2*log(f)^2 + 1
8*f^2)*(((c^2*f^a*e^(1/4*b^2*c*log(f)^3/(c^2*log(f)^2 + f^2) + 9/4*c*e^2*log(f)/(c^2*log(f)^2 + 9*f^2) + 1/2*b
*e*f*log(f)/(c^2*log(f)^2 + f^2))*log(f)^2 + f^(a + 2)*e^(1/4*b^2*c*log(f)^3/(c^2*log(f)^2 + f^2) + 9/4*c*e^2*
log(f)/(c^2*log(f)^2 + 9*f^2) + 1/2*b*e*f*log(f)/(c^2*log(f)^2 + f^2)))*cos(-3/4*(9*e^2*f - 36*d*f^2 - (4*c^2*
d - 2*b*c*e + b^2*f)*log(f)^2)/(c^2*log(f)^2 + 9*f^2)) - (I*c^2*f^a*e^(1/4*b^2*c*log(f)^3/(c^2*log(f)^2 + f^2)
+ 9/4*c*e^2*log(f)/(c^2*log(f)^2 + 9*f^2) + 1/2*b*e*f*log(f)/(c^2*log(f)^2 + f^2))*log(f)^2 + I*f^(a + 2)*e^(
1/4*b^2*c*log(f)^3/(c^2*log(f)^2 + f^2) + 9/4*c*e^2*log(f)/(c^2*log(f)^2 + 9*f^2) + 1/2*b*e*f*log(f)/(c^2*log(
f)^2 + f^2)))*sin(-3/4*(9*e^2*f - 36*d*f^2 - (4*c^2*d - 2*b*c*e + b^2*f)*log(f)^2)/(c^2*log(f)^2 + 9*f^2)))*er
f(1/2*(2*(c*log(f) - 3*I*f)*x + b*log(f) - 3*I*e)*sqrt(-c*log(f) + 3*I*f)/(c*log(f) - 3*I*f)) + ((c^2*f^a*e^(1
/4*b^2*c*log(f)^3/(c^2*log(f)^2 + f^2) + 9/4*c*e^2*log(f)/(c^2*log(f)^2 + 9*f^2) + 1/2*b*e*f*log(f)/(c^2*log(f
)^2 + f^2))*log(f)^2 + f^(a + 2)*e^(1/4*b^2*c*log(f)^3/(c^2*log(f)^2 + f^2) + 9/4*c*e^2*log(f)/(c^2*log(f)^2 +
9*f^2) + 1/2*b*e*f*log(f)/(c^2*log(f)^2 + f^2)))*cos(-3/4*(9*e^2*f - 36*d*f^2 - (4*c^2*d - 2*b*c*e + b^2*f)*l
og(f)^2)/(c^2*log(f)^2 + 9*f^2)) - (-I*c^2*f^a*e^(1/4*b^2*c*log(f)^3/(c^2*log(f)^2 + f^2) + 9/4*c*e^2*log(f)/(
c^2*log(f)^2 + 9*f^2) + 1/2*b*e*f*log(f)/(c^2*log(f)^2 + f^2))*log(f)^2 - I*f^(a + 2)*e^(1/4*b^2*c*log(f)^3/(c
^2*log(f)^2 + f^2) + 9/4*c*e^2*log(f)/(c^2*log(f)^2 + 9*f^2) + 1/2*b*e*f*log(f)/(c^2*log(f)^2 + f^2)))*sin(-3/
4*(9*e^2*f - 36*d*f^2 - (4*c^2*d - 2*b*c*e + b^2*f)*log(f)^2)/(c^2*log(f)^2 + 9*f^2)))*erf(1/2*(2*(c*log(f) +
3*I*f)*x + b*log(f) + 3*I*e)*sqrt(-c*log(f) - 3*I*f)/(c*log(f) + 3*I*f)))*sqrt(-c*log(f) + sqrt(c^2*log(f)^2 +
9*f^2)) - 3*sqrt(pi)*sqrt(2*c^2*log(f)^2 + 2*f^2)*(((c^2*f^a*e^(1/4*b^2*c*log(f)^3/(c^2*log(f)^2 + 9*f^2) + 1
/4*c*e^2*log(f)/(c^2*log(f)^2 + f^2) + 9/2*b*e*f*log(f)/(c^2*log(f)^2 + 9*f^2))*log(f)^2 + 9*f^(a + 2)*e^(1/4*
b^2*c*log(f)^3/(c^2*log(f)^2 + 9*f^2) + 1/4*c*e^2*log(f)/(c^2*log(f)^2 + f^2) + 9/2*b*e*f*log(f)/(c^2*log(f)^2
+ 9*f^2)))*cos(-1/4*(e^2*f - 4*d*f^2 - (4*c^2*d - 2*b*c*e + b^2*f)*log(f)^2)/(c^2*log(f)^2 + f^2)) + (-I*c^2*
f^a*e^(1/4*b^2*c*log(f)^3/(c^2*log(f)^2 + 9*f^2) + 1/4*c*e^2*log(f)/(c^2*log(f)^2 + f^2) + 9/2*b*e*f*log(f)/(c
^2*log(f)^2 + 9*f^2))*log(f)^2 - 9*I*f^(a + 2)*e^(1/4*b^2*c*log(f)^3/(c^2*log(f)^2 + 9*f^2) + 1/4*c*e^2*log(f)
/(c^2*log(f)^2 + f^2) + 9/2*b*e*f*log(f)/(c^2*log(f)^2 + 9*f^2)))*sin(-1/4*(e^2*f - 4*d*f^2 - (4*c^2*d - 2*b*c
*e + b^2*f)*log(f)^2)/(c^2*log(f)^2 + f^2)))*erf(1/2*(2*(c*log(f) - I*f)*x + b*log(f) - I*e)*sqrt(-c*log(f) +
I*f)/(c*log(f) - I*f)) + ((c^2*f^a*e^(1/4*b^2*c*log(f)^3/(c^2*log(f)^2 + 9*f^2) + 1/4*c*e^2*log(f)/(c^2*log(f)
^2 + f^2) + 9/2*b*e*f*log(f)/(c^2*log(f)^2 + 9*f^2))*log(f)^2 + 9*f^(a + 2)*e^(1/4*b^2*c*log(f)^3/(c^2*log(f)^
2 + 9*f^2) + 1/4*c*e^2*log(f)/(c^2*log(f)^2 + f^2) + 9/2*b*e*f*log(f)/(c^2*log(f)^2 + 9*f^2)))*cos(-1/4*(e^2*f
- 4*d*f^2 - (4*c^2*d - 2*b*c*e + b^2*f)*log(f)^2)/(c^2*log(f)^2 + f^2)) + (I*c^2*f^a*e^(1/4*b^2*c*log(f)^3/(c
^2*log(f)^2 + 9*f^2) + 1/4*c*e^2*log(f)/(c^2*log(f)^2 + f^2) + 9/2*b*e*f*log(f)/(c^2*log(f)^2 + 9*f^2))*log(f)
^2 + 9*I*f^(a + 2)*e^(1/4*b^2*c*log(f)^3/(c^2*log(f)^2 + 9*f^2) + 1/4*c*e^2*log(f)/(c^2*log(f)^2 + f^2) + 9/2*
b*e*f*log(f)/(c^2*log(f)^2 + 9*f^2)))*sin(-1/4*(e^2*f - 4*d*f^2 - (4*c^2*d - 2*b*c*e + b^2*f)*log(f)^2)/(c^2*l
og(f)^2 + f^2)))*erf(1/2*(2*(c*log(f) + I*f)*x + b*log(f) + I*e)*sqrt(-c*log(f) - I*f)/(c*log(f) + I*f)))*sqrt
(-c*log(f) + sqrt(c^2*log(f)^2 + f^2)))/(c^4*e^(1/4*b^2*c*log(f)^3/(c^2*log(f)^2 + 9*f^2) + 1/4*b^2*c*log(f)^3
/(c^2*log(f)^2 + f^2) + 9/2*b*e*f*log(f)/(c^2*log(f)^2 + 9*f^2) + 1/2*b*e*f*log(f)/(c^2*log(f)^2 + f^2))*log(f
)^4 + 10*c^2*f^2*e^(1/4*b^2*c*log(f)^3/(c^2*log(f)^2 + 9*f^2) + 1/4*b^2*c*log(f)^3/(c^2*log(f)^2 + f^2) + 9/2*
b*e*f*log(f)/(c^2*log(f)^2 + 9*f^2) + 1/2*b*e*f*log(f)/(c^2*log(f)^2 + f^2))*log(f)^2 + 9*f^4*e^(1/4*b^2*c*log
(f)^3/(c^2*log(f)^2 + 9*f^2) + 1/4*b^2*c*log(f)^3/(c^2*log(f)^2 + f^2) + 9/2*b*e*f*log(f)/(c^2*log(f)^2 + 9*f^
2) + 1/2*b*e*f*log(f)/(c^2*log(f)^2 + f^2)))

Giac [F]

\[ \int f^{a+b x+c x^2} \cos ^3\left (d+e x+f x^2\right ) \, dx=\int { f^{c x^{2} + b x + a} \cos \left (f x^{2} + e x + d\right )^{3} \,d x } \]

[In]

integrate(f^(c*x^2+b*x+a)*cos(f*x^2+e*x+d)^3,x, algorithm="giac")

[Out]

integrate(f^(c*x^2 + b*x + a)*cos(f*x^2 + e*x + d)^3, x)

Mupad [F(-1)]

Timed out. \[ \int f^{a+b x+c x^2} \cos ^3\left (d+e x+f x^2\right ) \, dx=\int f^{c\,x^2+b\,x+a}\,{\cos \left (f\,x^2+e\,x+d\right )}^3 \,d x \]

[In]

int(f^(a + b*x + c*x^2)*cos(d + e*x + f*x^2)^3,x)

[Out]

int(f^(a + b*x + c*x^2)*cos(d + e*x + f*x^2)^3, x)