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

Optimal. Leaf size=378 \[ -\frac{3 \sqrt{\pi } f^a e^{\frac{b^2 \log ^2(f)}{-4 c \log (f)+4 i f}-i d} \text{Erf}\left (\frac{b \log (f)-2 x (-c \log (f)+i f)}{2 \sqrt{-c \log (f)+i f}}\right )}{16 \sqrt{-c \log (f)+i f}}-\frac{\sqrt{\pi } f^a e^{\frac{b^2 \log ^2(f)}{-4 c \log (f)+12 i f}-3 i d} \text{Erf}\left (\frac{b \log (f)-2 x (-c \log (f)+3 i f)}{2 \sqrt{-c \log (f)+3 i f}}\right )}{16 \sqrt{-c \log (f)+3 i f}}+\frac{\sqrt{\pi } f^a \exp \left (3 i d-\frac{b^2 \log ^2(f)}{4 (c \log (f)+3 i f)}\right ) \text{Erfi}\left (\frac{b \log (f)+2 x (c \log (f)+3 i f)}{2 \sqrt{c \log (f)+3 i f}}\right )}{16 \sqrt{c \log (f)+3 i f}}+\frac{3 \sqrt{\pi } f^a e^{i d-\frac{b^2 \log ^2(f)}{4 c \log (f)+4 i f}} \text{Erfi}\left (\frac{b \log (f)+2 x (c \log (f)+i f)}{2 \sqrt{c \log (f)+i f}}\right )}{16 \sqrt{c \log (f)+i f}} \]

[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
]])

________________________________________________________________________________________

Rubi [A]  time = 0.521957, antiderivative size = 378, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {4473, 2287, 2234, 2205, 2204} \[ -\frac{3 \sqrt{\pi } f^a e^{\frac{b^2 \log ^2(f)}{-4 c \log (f)+4 i f}-i d} \text{Erf}\left (\frac{b \log (f)-2 x (-c \log (f)+i f)}{2 \sqrt{-c \log (f)+i f}}\right )}{16 \sqrt{-c \log (f)+i f}}-\frac{\sqrt{\pi } f^a e^{\frac{b^2 \log ^2(f)}{-4 c \log (f)+12 i f}-3 i d} \text{Erf}\left (\frac{b \log (f)-2 x (-c \log (f)+3 i f)}{2 \sqrt{-c \log (f)+3 i f}}\right )}{16 \sqrt{-c \log (f)+3 i f}}+\frac{\sqrt{\pi } f^a \exp \left (3 i d-\frac{b^2 \log ^2(f)}{4 (c \log (f)+3 i f)}\right ) \text{Erfi}\left (\frac{b \log (f)+2 x (c \log (f)+3 i f)}{2 \sqrt{c \log (f)+3 i f}}\right )}{16 \sqrt{c \log (f)+3 i f}}+\frac{3 \sqrt{\pi } f^a e^{i d-\frac{b^2 \log ^2(f)}{4 c \log (f)+4 i f}} \text{Erfi}\left (\frac{b \log (f)+2 x (c \log (f)+i f)}{2 \sqrt{c \log (f)+i f}}\right )}{16 \sqrt{c \log (f)+i f}} \]

Antiderivative was successfully verified.

[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 4473

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]

Rule 2287

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 2234

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 2205

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 2204

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]

Rubi steps

\begin{align*} \int f^{a+b x+c x^2} \cos ^3\left (d+f x^2\right ) \, dx &=\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]  time = 6.97349, size = 3285, normalized size = 8.69 \[ \text{Result too large to show} \]

Antiderivative was successfully verified.

[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]  time = 0.316, size = 354, normalized size = 0.9 \begin{align*} -{\frac{{f}^{a}\sqrt{\pi }}{16}{{\rm e}^{-{\frac{ \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}+12\,id\ln \left ( f \right ) c+36\,df}{4\,c\ln \left ( f \right ) -12\,if}}}}{\it Erf} \left ( -x\sqrt{3\,if-c\ln \left ( f \right ) }+{\frac{b\ln \left ( f \right ) }{2}{\frac{1}{\sqrt{3\,if-c\ln \left ( f \right ) }}}} \right ){\frac{1}{\sqrt{3\,if-c\ln \left ( f \right ) }}}}-{\frac{3\,{f}^{a}\sqrt{\pi }}{16}{{\rm e}^{-{\frac{ \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}+4\,id\ln \left ( f \right ) c+4\,df}{4\,c\ln \left ( f \right ) -4\,if}}}}{\it Erf} \left ( -x\sqrt{if-c\ln \left ( f \right ) }+{\frac{b\ln \left ( f \right ) }{2}{\frac{1}{\sqrt{if-c\ln \left ( f \right ) }}}} \right ){\frac{1}{\sqrt{if-c\ln \left ( f \right ) }}}}-{\frac{3\,{f}^{a}\sqrt{\pi }}{16}{{\rm e}^{-{\frac{ \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}-4\,id\ln \left ( f \right ) c+4\,df}{4\,if+4\,c\ln \left ( f \right ) }}}}{\it Erf} \left ( -\sqrt{-c\ln \left ( f \right ) -if}x+{\frac{b\ln \left ( f \right ) }{2}{\frac{1}{\sqrt{-c\ln \left ( f \right ) -if}}}} \right ){\frac{1}{\sqrt{-c\ln \left ( f \right ) -if}}}}-{\frac{{f}^{a}\sqrt{\pi }}{16}{{\rm e}^{-{\frac{ \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}-12\,id\ln \left ( f \right ) c+36\,df}{12\,if+4\,c\ln \left ( f \right ) }}}}{\it Erf} \left ( -\sqrt{-c\ln \left ( f \right ) -3\,if}x+{\frac{b\ln \left ( f \right ) }{2}{\frac{1}{\sqrt{-c\ln \left ( f \right ) -3\,if}}}} \right ){\frac{1}{\sqrt{-c\ln \left ( f \right ) -3\,if}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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)/(-3*I*f+c*ln(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)/(-I*f+c*ln(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))

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: IndexError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: IndexError

________________________________________________________________________________________

Fricas [B]  time = 0.660477, size = 1852, normalized size = 4.9 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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 + (12*I*c^2*d + 3*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 + (-12*I*c^2*d - 3*I*b^2*f)*log(f)^2)/(c^2*log
(f)^2 + 9*f^2)) + sqrt(pi)*(3*c^3*log(f)^3 - 3*I*c^2*f*log(f)^2 + 27*c*f^2*log(f) - 27*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)) + sqrt(pi)*(3*c^3*log(f)^3 + 3*I*c^2*f*log(f)^2 + 27*c*f^2*log(f) + 27*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(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int f^{c x^{2} + b x + a} \cos \left (f x^{2} + d\right )^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)