\(\int f^{a+b x+c x^2} \cos ^3(d+f x^2) \, dx\) [130]

Optimal result

Integrand size = 23, antiderivative size = 378 \[ \int f^{a+b x+c x^2} \cos ^3\left (d+f x^2\right ) \, dx=-\frac {3 e^{-i d+\frac {b^2 \log ^2(f)}{4 i f-4 c \log (f)}} f^a \sqrt {\pi } \text {erf}\left (\frac {b \log (f)-2 x (i f-c \log (f))}{2 \sqrt {i f-c \log (f)}}\right )}{16 \sqrt {i f-c \log (f)}}-\frac {e^{-3 i d+\frac {b^2 \log ^2(f)}{12 i f-4 c \log (f)}} f^a \sqrt {\pi } \text {erf}\left (\frac {b \log (f)-2 x (3 i f-c \log (f))}{2 \sqrt {3 i f-c \log (f)}}\right )}{16 \sqrt {3 i f-c \log (f)}}+\frac {3 e^{i d-\frac {b^2 \log ^2(f)}{4 i f+4 c \log (f)}} f^a \sqrt {\pi } \text {erfi}\left (\frac {b \log (f)+2 x (i f+c \log (f))}{2 \sqrt {i f+c \log (f)}}\right )}{16 \sqrt {i f+c \log (f)}}+\frac {e^{3 i d-\frac {b^2 \log ^2(f)}{4 (3 i f+c \log (f))}} f^a \sqrt {\pi } \text {erfi}\left (\frac {b \log (f)+2 x (3 i f+c \log (f))}{2 \sqrt {3 i f+c \log (f)}}\right )}{16 \sqrt {3 i f+c \log (f)}} \]

[Out]

-3/16*exp(-I*d+b^2*ln(f)^2/(4*I*f-4*c*ln(f)))*f^a*erf(1/2*(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+b^2*ln(f)^2/(12*I*f-4*c*ln(f)))*f^a*erf(1/2*(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)+3/16*exp(I*d-b^2*ln(f)^2/(4*I*f+4*c*ln(f)))*f^a*er
fi(1/2*(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*b^2*ln
(f)^2/(3*I*f+c*ln(f)))*f^a*erfi(1/2*(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 = 0.62 (sec) , antiderivative size = 378, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {4561, 2325, 2266, 2236, 2235} \[ \int f^{a+b x+c x^2} \cos ^3\left (d+f x^2\right ) \, dx=-\frac {3 \sqrt {\pi } f^a e^{\frac {b^2 \log ^2(f)}{-4 c \log (f)+4 i f}-i d} \text {erf}\left (\frac {b \log (f)-2 x (-c \log (f)+i f)}{2 \sqrt {-c \log (f)+i f}}\right )}{16 \sqrt {-c \log (f)+i f}}-\frac {\sqrt {\pi } f^a e^{\frac {b^2 \log ^2(f)}{-4 c \log (f)+12 i f}-3 i d} \text {erf}\left (\frac {b \log (f)-2 x (-c \log (f)+3 i f)}{2 \sqrt {-c \log (f)+3 i f}}\right )}{16 \sqrt {-c \log (f)+3 i f}}+\frac {\sqrt {\pi } f^a \exp \left (3 i d-\frac {b^2 \log ^2(f)}{4 (c \log (f)+3 i f)}\right ) \text {erfi}\left (\frac {b \log (f)+2 x (c \log (f)+3 i f)}{2 \sqrt {c \log (f)+3 i f}}\right )}{16 \sqrt {c \log (f)+3 i f}}+\frac {3 \sqrt {\pi } f^a e^{i d-\frac {b^2 \log ^2(f)}{4 c \log (f)+4 i f}} \text {erfi}\left (\frac {b \log (f)+2 x (c \log (f)+i f)}{2 \sqrt {c \log (f)+i f}}\right )}{16 \sqrt {c \log (f)+i f}} \]

[In]

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

[Out]

(-3*E^((-I)*d + (b^2*Log[f]^2)/((4*I)*f - 4*c*Log[f]))*f^a*Sqrt[Pi]*Erf[(b*Log[f] - 2*x*(I*f - c*Log[f]))/(2*S
qrt[I*f - c*Log[f]])])/(16*Sqrt[I*f - c*Log[f]]) - (E^((-3*I)*d + (b^2*Log[f]^2)/((12*I)*f - 4*c*Log[f]))*f^a*
Sqrt[Pi]*Erf[(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]]
) + (3*E^(I*d - (b^2*Log[f]^2)/((4*I)*f + 4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(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 - (b^2*Log[f]^2)/(4*((3*I)*f + c*Log[f])))*f^a
*Sqrt[Pi]*Erfi[(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 {3}{8} e^{-i d-i f x^2} f^{a+b x+c x^2}+\frac {3}{8} e^{i d+i f x^2} f^{a+b x+c x^2}+\frac {1}{8} e^{-3 i d-3 i f x^2} f^{a+b x+c x^2}+\frac {1}{8} e^{3 i d+3 i f x^2} f^{a+b x+c x^2}\right ) \, dx \\ & = \frac {1}{8} \int e^{-3 i d-3 i f x^2} f^{a+b x+c x^2} \, dx+\frac {1}{8} \int e^{3 i d+3 i f x^2} f^{a+b x+c x^2} \, dx+\frac {3}{8} \int e^{-i d-i f x^2} f^{a+b x+c x^2} \, dx+\frac {3}{8} \int e^{i d+i f x^2} f^{a+b x+c x^2} \, dx \\ & = \frac {1}{8} \int \exp \left (-3 i d+a \log (f)+b x \log (f)-x^2 (3 i f-c \log (f))\right ) \, dx+\frac {1}{8} \int \exp \left (3 i d+a \log (f)+b x \log (f)+x^2 (3 i f+c \log (f))\right ) \, dx+\frac {3}{8} \int \exp \left (-i d+a \log (f)+b x \log (f)-x^2 (i f-c \log (f))\right ) \, dx+\frac {3}{8} \int \exp \left (i d+a \log (f)+b x \log (f)+x^2 (i f+c \log (f))\right ) \, dx \\ & = \frac {1}{8} \left (3 e^{-i d+\frac {b^2 \log ^2(f)}{4 i f-4 c \log (f)}} f^a\right ) \int \exp \left (\frac {(b \log (f)+2 x (-i f+c \log (f)))^2}{4 (-i f+c \log (f))}\right ) \, dx+\frac {1}{8} \left (e^{-3 i d+\frac {b^2 \log ^2(f)}{12 i f-4 c \log (f)}} f^a\right ) \int \exp \left (\frac {(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 {b^2 \log ^2(f)}{4 (3 i f+c \log (f))}\right ) f^a\right ) \int \exp \left (\frac {(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 e^{i d-\frac {b^2 \log ^2(f)}{4 i f+4 c \log (f)}} f^a\right ) \int \exp \left (\frac {(b \log (f)+2 x (i f+c \log (f)))^2}{4 (i f+c \log (f))}\right ) \, dx \\ & = -\frac {3 e^{-i d+\frac {b^2 \log ^2(f)}{4 i f-4 c \log (f)}} f^a \sqrt {\pi } \text {erf}\left (\frac {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 {b^2 \log ^2(f)}{12 i f-4 c \log (f)}} f^a \sqrt {\pi } \text {erf}\left (\frac {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 {b^2 \log ^2(f)}{4 i f+4 c \log (f)}} f^a \sqrt {\pi } \text {erfi}\left (\frac {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 {b^2 \log ^2(f)}{4 (3 i f+c \log (f))}\right ) f^a \sqrt {\pi } \text {erfi}\left (\frac {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] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(3285\) vs. \(2(378)=756\).

Time = 7.58 (sec) , antiderivative size = 3285, normalized size of antiderivative = 8.69 \[ \int f^{a+b x+c x^2} \cos ^3\left (d+f x^2\right ) \, dx=\text {Result too large to show} \]

[In]

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

[Out]

(f^a*Sqrt[Pi]*(-27*(-1)^(3/4)*E^(((I/4)*b^2*Log[f]^2)/(f - I*c*Log[f]))*f^3*Cos[d]*Erfi[((-1)^(1/4)*(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)*b^2*
Log[f]^2)/(f - I*c*Log[f]))*f^2*Cos[d]*Erfi[((-1)^(1/4)*(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)*b^2*Log[f]^2)/(f - I*c*Log[f]))*f*Cos[d]
*Erfi[((-1)^(1/4)*(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)*b^2*Log[f]^2)/(f - I*c*Log[f]))*Cos[d]*Erfi[((-1)^(1/4)*(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)*b^2*Log[
f]^2)/(3*f - I*c*Log[f]))*f^3*Cos[3*d]*Erfi[((-1)^(1/4)*(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)*b^2*Log[f]^2)/(3*f - I*c*Log[f]))*f^2*Cos[3*d]*
Erfi[((-1)^(1/4)*(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*Lo
g[f]] - 3*(-1)^(3/4)*c^2*E^(((I/4)*b^2*Log[f]^2)/(3*f - I*c*Log[f]))*f*Cos[3*d]*Erfi[((-1)^(1/4)*(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)*b^2*Log[f]^2)/(3*f - I*c*Log[f]))*Cos[3*d]*Erfi[((-1)^(1/4)*(6*f*x - I*b*Log[f] - (2*I)*c*x*Log[f]))/(2*S
qrt[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)*(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)*b^2*Log[f]^2)/(f + I
*c*Log[f])) + (27*(-1)^(3/4)*c*f^2*Cos[d]*Erfi[((-1)^(3/4)*(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)*b^2*Log[f]^2)/(f + I*c*Log[f])) - (3*(-1)^(1/4)*c^2*f*C
os[d]*Erfi[((-1)^(3/4)*(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]])/E^(((I/4)*b^2*Log[f]^2)/(f + I*c*Log[f])) + (3*(-1)^(3/4)*c^3*Cos[d]*Erfi[((-1)^(3/4)*(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)*b^2*Log[f]^2)/(
f + I*c*Log[f])) - (3*(-1)^(1/4)*f^3*Cos[3*d]*Erfi[((-1)^(3/4)*(6*f*x + I*b*Log[f] + (2*I)*c*x*Log[f]))/(2*Sqr
t[3*f + I*c*Log[f]])]*Sqrt[3*f + I*c*Log[f]])/E^(((I/4)*b^2*Log[f]^2)/(3*f + I*c*Log[f])) + ((-1)^(3/4)*c*f^2*
Cos[3*d]*Erfi[((-1)^(3/4)*(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)*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)*(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)*
b^2*Log[f]^2)/(3*f + I*c*Log[f])) + ((-1)^(3/4)*c^3*Cos[3*d]*Erfi[((-1)^(3/4)*(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)*b^2*Log[f]^2)/(3*f + I*c*Log[f
])) + 27*(-1)^(1/4)*E^(((I/4)*b^2*Log[f]^2)/(f - I*c*Log[f]))*f^3*Erfi[((-1)^(1/4)*(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)*b^2*Log[f]^2)/
(f - I*c*Log[f]))*f^2*Erfi[((-1)^(1/4)*(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)*b^2*Log[f]^2)/(f - I*c*Log[f]))*f*Erfi[((-1)^(1/4)
*(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)*b^2*Log[f]^2)/(f - I*c*Log[f]))*Erfi[((-1)^(1/4)*(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)*(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)*b^2*Log[f
]^2)/(f + I*c*Log[f])) + (27*(-1)^(1/4)*c*f^2*Erfi[((-1)^(3/4)*(2*f*x + I*b*Log[f] + (2*I)*c*x*Log[f]))/(2*Sqr
t[f + I*c*Log[f]])]*Log[f]*Sqrt[f + I*c*Log[f]]*Sin[d])/E^(((I/4)*b^2*Log[f]^2)/(f + I*c*Log[f])) + (3*(-1)^(3
/4)*c^2*f*Erfi[((-1)^(3/4)*(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)*b^2*Log[f]^2)/(f + I*c*Log[f])) + (3*(-1)^(1/4)*c^3*Erfi[((-1)^(3/4)*(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)*b^2*
Log[f]^2)/(f + I*c*Log[f])) + 3*(-1)^(1/4)*E^(((I/4)*b^2*Log[f]^2)/(3*f - I*c*Log[f]))*f^3*Erfi[((-1)^(1/4)*(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] + (-1)^(3/4
)*c*E^(((I/4)*b^2*Log[f]^2)/(3*f - I*c*Log[f]))*f^2*Erfi[((-1)^(1/4)*(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)*b^2*Log[f]^2)/
(3*f - I*c*Log[f]))*f*Erfi[((-1)^(1/4)*(6*f*x - I*b*Log[f] - (2*I)*c*x*Log[f]))/(2*Sqrt[3*f - I*c*Log[f]])]*Lo
g[f]^2*Sqrt[3*f - I*c*Log[f]]*Sin[3*d] + (-1)^(3/4)*c^3*E^(((I/4)*b^2*Log[f]^2)/(3*f - I*c*Log[f]))*Erfi[((-1)
^(1/4)*(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]]*Si
n[3*d] + (3*(-1)^(3/4)*f^3*Erfi[((-1)^(3/4)*(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)*b^2*Log[f]^2)/(3*f + I*c*Log[f])) + ((-1)^(1/4)*c*f^2*Erfi[((-1)
^(3/4)*(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)*b^2*Log[f]^2)/(3*f + I*c*Log[f])) + (3*(-1)^(3/4)*c^2*f*Erfi[((-1)^(3/4)*(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)*b^2*Log[f]
^2)/(3*f + I*c*Log[f])) + ((-1)^(1/4)*c^3*Erfi[((-1)^(3/4)*(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)*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 = 1.70 (sec) , antiderivative size = 354, normalized size of antiderivative = 0.94

method	result	size
risch	\(-\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {\ln \left (f \right )^{2} b^{2}+12 i d \ln \left (f \right ) c +36 d f}{4 \left (c \ln \left (f \right )-3 i f \right )}} \operatorname {erf}\left (-x \sqrt {3 i f -c \ln \left (f \right )}+\frac {\ln \left (f \right ) b}{2 \sqrt {3 i f -c \ln \left (f \right )}}\right )}{16 \sqrt {3 i f -c \ln \left (f \right )}}-\frac {3 \sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {\ln \left (f \right )^{2} b^{2}+4 i d \ln \left (f \right ) c +4 d f}{4 \left (c \ln \left (f \right )-i f \right )}} \operatorname {erf}\left (-x \sqrt {i f -c \ln \left (f \right )}+\frac {\ln \left (f \right ) b}{2 \sqrt {i f -c \ln \left (f \right )}}\right )}{16 \sqrt {i f -c \ln \left (f \right )}}-\frac {3 \sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {\ln \left (f \right )^{2} b^{2}-4 i d \ln \left (f \right ) c +4 d f}{4 \left (i f +c \ln \left (f \right )\right )}} \operatorname {erf}\left (-\sqrt {-c \ln \left (f \right )-i f}\, x +\frac {\ln \left (f \right ) b}{2 \sqrt {-c \ln \left (f \right )-i f}}\right )}{16 \sqrt {-c \ln \left (f \right )-i f}}-\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {\ln \left (f \right )^{2} b^{2}-12 i d \ln \left (f \right ) c +36 d f}{4 \left (3 i f +c \ln \left (f \right )\right )}} \operatorname {erf}\left (-\sqrt {-c \ln \left (f \right )-3 i f}\, x +\frac {\ln \left (f \right ) b}{2 \sqrt {-c \ln \left (f \right )-3 i f}}\right )}{16 \sqrt {-c \ln \left (f \right )-3 i f}}\)	\(354\)

[In]

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

[Out]

-1/16*Pi^(1/2)*f^a*exp(-1/4*(ln(f)^2*b^2+12*I*d*ln(f)*c+36*d*f)/(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*ln(f)*b/(3*I*f-c*ln(f))^(1/2))-3/16*Pi^(1/2)*f^a*exp(-1/4*(ln(f)^2*b^2+4*I*d*ln(f)*c
+4*d*f)/(c*ln(f)-I*f))/(I*f-c*ln(f))^(1/2)*erf(-x*(I*f-c*ln(f))^(1/2)+1/2*ln(f)*b/(I*f-c*ln(f))^(1/2))-3/16*Pi
^(1/2)*f^a*exp(-1/4*(ln(f)^2*b^2-4*I*d*ln(f)*c+4*d*f)/(I*f+c*ln(f)))/(-c*ln(f)-I*f)^(1/2)*erf(-(-c*ln(f)-I*f)^
(1/2)*x+1/2*ln(f)*b/(-c*ln(f)-I*f)^(1/2))-1/16*Pi^(1/2)*f^a*exp(-1/4*(ln(f)^2*b^2-12*I*d*ln(f)*c+36*d*f)/(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*ln(f)*b/(-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. 725 vs. \(2 (289) = 578\).

Time = 0.29 (sec) , antiderivative size = 725, normalized size of antiderivative = 1.92 \[ \int f^{a+b x+c x^2} \cos ^3\left (d+f x^2\right ) \, dx=-\frac {\sqrt {\pi } {\left (c^{3} \log \left (f\right )^{3} - 3 i \, c^{2} f \log \left (f\right )^{2} + c f^{2} \log \left (f\right ) - 3 i \, f^{3}\right )} \sqrt {-c \log \left (f\right ) - 3 i \, f} \operatorname {erf}\left (\frac {{\left (18 \, f^{2} x - 3 i \, b f \log \left (f\right ) + {\left (2 \, c^{2} x + b c\right )} \log \left (f\right )^{2}\right )} \sqrt {-c \log \left (f\right ) - 3 i \, f}}{2 \, {\left (c^{2} \log \left (f\right )^{2} + 9 \, f^{2}\right )}}\right ) e^{\left (\frac {36 \, a f^{2} \log \left (f\right ) - {\left (b^{2} c - 4 \, a c^{2}\right )} \log \left (f\right )^{3} + 108 i \, d f^{2} - 3 \, {\left (-4 i \, c^{2} d - i \, b^{2} f\right )} \log \left (f\right )^{2}}{4 \, {\left (c^{2} \log \left (f\right )^{2} + 9 \, f^{2}\right )}}\right )} + \sqrt {\pi } {\left (c^{3} \log \left (f\right )^{3} + 3 i \, c^{2} f \log \left (f\right )^{2} + c f^{2} \log \left (f\right ) + 3 i \, f^{3}\right )} \sqrt {-c \log \left (f\right ) + 3 i \, f} \operatorname {erf}\left (\frac {{\left (18 \, f^{2} x + 3 i \, b f \log \left (f\right ) + {\left (2 \, c^{2} x + b c\right )} \log \left (f\right )^{2}\right )} \sqrt {-c \log \left (f\right ) + 3 i \, f}}{2 \, {\left (c^{2} \log \left (f\right )^{2} + 9 \, f^{2}\right )}}\right ) e^{\left (\frac {36 \, a f^{2} \log \left (f\right ) - {\left (b^{2} c - 4 \, a c^{2}\right )} \log \left (f\right )^{3} - 108 i \, d f^{2} - 3 \, {\left (4 i \, c^{2} d + i \, b^{2} f\right )} \log \left (f\right )^{2}}{4 \, {\left (c^{2} \log \left (f\right )^{2} + 9 \, f^{2}\right )}}\right )} + 3 \, \sqrt {\pi } {\left (c^{3} \log \left (f\right )^{3} - i \, c^{2} f \log \left (f\right )^{2} + 9 \, c f^{2} \log \left (f\right ) - 9 i \, f^{3}\right )} \sqrt {-c \log \left (f\right ) - i \, f} \operatorname {erf}\left (\frac {{\left (2 \, f^{2} x - i \, b f \log \left (f\right ) + {\left (2 \, c^{2} x + b c\right )} \log \left (f\right )^{2}\right )} \sqrt {-c \log \left (f\right ) - i \, f}}{2 \, {\left (c^{2} \log \left (f\right )^{2} + f^{2}\right )}}\right ) e^{\left (\frac {4 \, a f^{2} \log \left (f\right ) - {\left (b^{2} c - 4 \, a c^{2}\right )} \log \left (f\right )^{3} + 4 i \, d f^{2} + {\left (4 i \, c^{2} d + i \, b^{2} f\right )} \log \left (f\right )^{2}}{4 \, {\left (c^{2} \log \left (f\right )^{2} + f^{2}\right )}}\right )} + 3 \, \sqrt {\pi } {\left (c^{3} \log \left (f\right )^{3} + i \, c^{2} f \log \left (f\right )^{2} + 9 \, c f^{2} \log \left (f\right ) + 9 i \, f^{3}\right )} \sqrt {-c \log \left (f\right ) + i \, f} \operatorname {erf}\left (\frac {{\left (2 \, f^{2} x + i \, b f \log \left (f\right ) + {\left (2 \, c^{2} x + b c\right )} \log \left (f\right )^{2}\right )} \sqrt {-c \log \left (f\right ) + i \, f}}{2 \, {\left (c^{2} \log \left (f\right )^{2} + f^{2}\right )}}\right ) e^{\left (\frac {4 \, a f^{2} \log \left (f\right ) - {\left (b^{2} c - 4 \, a c^{2}\right )} \log \left (f\right )^{3} - 4 i \, d f^{2} + {\left (-4 i \, c^{2} d - i \, b^{2} f\right )} \log \left (f\right )^{2}}{4 \, {\left (c^{2} \log \left (f\right )^{2} + f^{2}\right )}}\right )}}{16 \, {\left (c^{4} \log \left (f\right )^{4} + 10 \, c^{2} f^{2} \log \left (f\right )^{2} + 9 \, f^{4}\right )}} \]

[In]

integrate(f^(c*x^2+b*x+a)*cos(f*x^2+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 - 3*I*b*f*log(f) + (2*c^2*x + b*c)*log(f)^2)*sqrt(-c*log(f) - 3*I*f)/(c^2*log(f)^2 + 9*f^2))*e^(1/4*(
36*a*f^2*log(f) - (b^2*c - 4*a*c^2)*log(f)^3 + 108*I*d*f^2 - 3*(-4*I*c^2*d - I*b^2*f)*log(f)^2)/(c^2*log(f)^2
+ 9*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 + 3*I*b*f*log(f) + (2*c^2*x + b*c)*log(f)^2)*sqrt(-c*log(f) + 3*I*f)/(c^2*log(f)^2 + 9*f^2))*e^(
1/4*(36*a*f^2*log(f) - (b^2*c - 4*a*c^2)*log(f)^3 - 108*I*d*f^2 - 3*(4*I*c^2*d + I*b^2*f)*log(f)^2)/(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 - I*b*f*log(f) + (2*c^2*x + b*c)*log(f)^2)*sqrt(-c*log(f) - I*f)/(c^2*log(f)^2 + f^2))*e^(1/4
*(4*a*f^2*log(f) - (b^2*c - 4*a*c^2)*log(f)^3 + 4*I*d*f^2 + (4*I*c^2*d + I*b^2*f)*log(f)^2)/(c^2*log(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 + I*b*f*log(f) + (2*c^2*x + b*c)*log(f)^2)*sqrt(-c*log(f) + I*f)/(c^2*log(f)^2 + f^2))*e^(1/4*(4*a*f^2*
log(f) - (b^2*c - 4*a*c^2)*log(f)^3 - 4*I*d*f^2 + (-4*I*c^2*d - I*b^2*f)*log(f)^2)/(c^2*log(f)^2 + f^2)))/(c^4
*log(f)^4 + 10*c^2*f^2*log(f)^2 + 9*f^4)

Sympy [F]

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

[In]

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

[Out]

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

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. 2456 vs. \(2 (289) = 578\).

Time = 0.28 (sec) , antiderivative size = 2456, normalized size of antiderivative = 6.50 \[ \int f^{a+b x+c x^2} \cos ^3\left (d+f x^2\right ) \, dx=\text {Too large to display} \]

[In]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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