3.124 \(\int f^{a+c x^2} \cos ^3(d+e x+f x^2) \, dx\)
Optimal. Leaf size=369 \[ \frac{\sqrt{\pi } f^a \exp \left (-\frac{9 e^2}{4 (-c \log (f)+3 i f)}-3 i d\right ) \text{Erf}\left (\frac{2 x (-c \log (f)+3 i f)+3 i e}{2 \sqrt{-c \log (f)+3 i f}}\right )}{16 \sqrt{-c \log (f)+3 i f}}+\frac{3 \sqrt{\pi } f^a e^{-\frac{e^2}{-4 c \log (f)+4 i f}-i d} \text{Erf}\left (\frac{2 x (-c \log (f)+i f)+i e}{2 \sqrt{-c \log (f)+i f}}\right )}{16 \sqrt{-c \log (f)+i f}}+\frac{3 \sqrt{\pi } f^a e^{\frac{e^2}{4 c \log (f)+4 i f}+i d} \text{Erfi}\left (\frac{2 x (c \log (f)+i f)+i e}{2 \sqrt{c \log (f)+i f}}\right )}{16 \sqrt{c \log (f)+i f}}+\frac{\sqrt{\pi } f^a e^{\frac{9 e^2}{4 (c \log (f)+3 i f)}+3 i d} \text{Erfi}\left (\frac{2 x (c \log (f)+3 i f)+3 i e}{2 \sqrt{c \log (f)+3 i f}}\right )}{16 \sqrt{c \log (f)+3 i f}} \]
[Out]
(3*E^((-I)*d - e^2/((4*I)*f - 4*c*Log[f]))*f^a*Sqrt[Pi]*Erf[(I*e + 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 - (9*e^2)/(4*((3*I)*f - c*Log[f])))*f^a*Sqrt[Pi]*Erf[((3*I)*e +
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 + e^2/((4*
I)*f + 4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(I*e + 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 + (9*e^2)/(4*((3*I)*f + c*Log[f])))*f^a*Sqrt[Pi]*Erfi[((3*I)*e + 2*x*((3*I)*f + c*Log[
f]))/(2*Sqrt[(3*I)*f + c*Log[f]])])/(16*Sqrt[(3*I)*f + c*Log[f]])
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Rubi [A] time = 0.582931, antiderivative size = 369, 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} \[ \frac{\sqrt{\pi } f^a \exp \left (-\frac{9 e^2}{4 (-c \log (f)+3 i f)}-3 i d\right ) \text{Erf}\left (\frac{2 x (-c \log (f)+3 i f)+3 i e}{2 \sqrt{-c \log (f)+3 i f}}\right )}{16 \sqrt{-c \log (f)+3 i f}}+\frac{3 \sqrt{\pi } f^a e^{-\frac{e^2}{-4 c \log (f)+4 i f}-i d} \text{Erf}\left (\frac{2 x (-c \log (f)+i f)+i e}{2 \sqrt{-c \log (f)+i f}}\right )}{16 \sqrt{-c \log (f)+i f}}+\frac{3 \sqrt{\pi } f^a e^{\frac{e^2}{4 c \log (f)+4 i f}+i d} \text{Erfi}\left (\frac{2 x (c \log (f)+i f)+i e}{2 \sqrt{c \log (f)+i f}}\right )}{16 \sqrt{c \log (f)+i f}}+\frac{\sqrt{\pi } f^a e^{\frac{9 e^2}{4 (c \log (f)+3 i f)}+3 i d} \text{Erfi}\left (\frac{2 x (c \log (f)+3 i f)+3 i e}{2 \sqrt{c \log (f)+3 i f}}\right )}{16 \sqrt{c \log (f)+3 i f}} \]
Antiderivative was successfully verified.
[In]
Int[f^(a + c*x^2)*Cos[d + e*x + f*x^2]^3,x]
[Out]
(3*E^((-I)*d - e^2/((4*I)*f - 4*c*Log[f]))*f^a*Sqrt[Pi]*Erf[(I*e + 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 - (9*e^2)/(4*((3*I)*f - c*Log[f])))*f^a*Sqrt[Pi]*Erf[((3*I)*e +
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 + e^2/((4*
I)*f + 4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(I*e + 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 + (9*e^2)/(4*((3*I)*f + c*Log[f])))*f^a*Sqrt[Pi]*Erfi[((3*I)*e + 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+c x^2} \cos ^3\left (d+e x+f x^2\right ) \, dx &=\int \left (\frac{1}{8} e^{-3 i \left (d+e x+f x^2\right )} f^{a+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+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+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+c x^2}\right ) \, dx\\ &=\frac{1}{8} \int e^{-3 i \left (d+e x+f x^2\right )} f^{a+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+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+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+c x^2} \, dx\\ &=\frac{1}{8} \int \exp \left (-3 i d-3 i e x+a \log (f)-x^2 (3 i f-c \log (f))\right ) \, dx+\frac{1}{8} \int \exp \left (3 i d+3 i e x+a \log (f)+x^2 (3 i f+c \log (f))\right ) \, dx+\frac{3}{8} \int \exp \left (-i d-i e x+a \log (f)-x^2 (i f-c \log (f))\right ) \, dx+\frac{3}{8} \int \exp \left (i d+i e x+a \log (f)+x^2 (i f+c \log (f))\right ) \, dx\\ &=\frac{1}{8} \left (3 e^{-i d-\frac{e^2}{4 i f-4 c \log (f)}} f^a\right ) \int \exp \left (\frac{(-i e+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{9 e^2}{4 (3 i f-c \log (f))}} f^a\right ) \int \exp \left (\frac{(-3 i e+2 x (-3 i f+c \log (f)))^2}{4 (-3 i f+c \log (f))}\right ) \, dx+\frac{1}{8} \left (e^{3 i d+\frac{9 e^2}{4 (3 i f+c \log (f))}} f^a\right ) \int \exp \left (\frac{(3 i e+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{e^2}{4 i f+4 c \log (f)}} f^a\right ) \int \exp \left (\frac{(i e+2 x (i f+c \log (f)))^2}{4 (i f+c \log (f))}\right ) \, dx\\ &=\frac{3 e^{-i d-\frac{e^2}{4 i f-4 c \log (f)}} f^a \sqrt{\pi } \text{erf}\left (\frac{i e+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{9 e^2}{4 (3 i f-c \log (f))}} f^a \sqrt{\pi } \text{erf}\left (\frac{3 i e+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^2}{4 i f+4 c \log (f)}} f^a \sqrt{\pi } \text{erfi}\left (\frac{i e+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{9 e^2}{4 (3 i f+c \log (f))}} f^a \sqrt{\pi } \text{erfi}\left (\frac{3 i e+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 = 7.01844, size = 2997, normalized size = 8.12 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[f^(a + c*x^2)*Cos[d + e*x + f*x^2]^3,x]
[Out]
(f^a*Sqrt[Pi]*((-27*(-1)^(3/4)*f^3*Cos[d]*Erfi[((-1)^(1/4)*(e + 2*f*x - (2*I)*c*x*Log[f]))/(2*Sqrt[f - I*c*Log
[f]])]*Sqrt[f - I*c*Log[f]])/E^(((I/4)*e^2)/(f - I*c*Log[f])) + (27*(-1)^(1/4)*c*f^2*Cos[d]*Erfi[((-1)^(1/4)*(
e + 2*f*x - (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)/(f - I*c*
Log[f])) - (3*(-1)^(3/4)*c^2*f*Cos[d]*Erfi[((-1)^(1/4)*(e + 2*f*x - (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)*e^2)/(f - I*c*Log[f])) + (3*(-1)^(1/4)*c^3*Cos[d]*Erfi[((-1)^(1/4)
*(e + 2*f*x - (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)/(f -
I*c*Log[f])) - (3*(-1)^(3/4)*f^3*Cos[3*d]*Erfi[((-1)^(1/4)*(3*e + 6*f*x - (2*I)*c*x*Log[f]))/(2*Sqrt[3*f - I*c
*Log[f]])]*Sqrt[3*f - I*c*Log[f]])/E^((((9*I)/4)*e^2)/(3*f - I*c*Log[f])) + ((-1)^(1/4)*c*f^2*Cos[3*d]*Erfi[((
-1)^(1/4)*(3*e + 6*f*x - (2*I)*c*x*Log[f]))/(2*Sqrt[3*f - I*c*Log[f]])]*Log[f]*Sqrt[3*f - I*c*Log[f]])/E^((((9
*I)/4)*e^2)/(3*f - I*c*Log[f])) - (3*(-1)^(3/4)*c^2*f*Cos[3*d]*Erfi[((-1)^(1/4)*(3*e + 6*f*x - (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^((((9*I)/4)*e^2)/(3*f - I*c*Log[f])) + ((-1
)^(1/4)*c^3*Cos[3*d]*Erfi[((-1)^(1/4)*(3*e + 6*f*x - (2*I)*c*x*Log[f]))/(2*Sqrt[3*f - I*c*Log[f]])]*Log[f]^3*S
qrt[3*f - I*c*Log[f]])/E^((((9*I)/4)*e^2)/(3*f - I*c*Log[f])) - 27*(-1)^(1/4)*E^(((I/4)*e^2)/(f + I*c*Log[f]))
*f^3*Cos[d]*Erfi[((-1)^(3/4)*(e + 2*f*x + (2*I)*c*x*Log[f]))/(2*Sqrt[f + I*c*Log[f]])]*Sqrt[f + I*c*Log[f]] +
27*(-1)^(3/4)*c*E^(((I/4)*e^2)/(f + I*c*Log[f]))*f^2*Cos[d]*Erfi[((-1)^(3/4)*(e + 2*f*x + (2*I)*c*x*Log[f]))/(
2*Sqrt[f + I*c*Log[f]])]*Log[f]*Sqrt[f + I*c*Log[f]] - 3*(-1)^(1/4)*c^2*E^(((I/4)*e^2)/(f + I*c*Log[f]))*f*Cos
[d]*Erfi[((-1)^(3/4)*(e + 2*f*x + (2*I)*c*x*Log[f]))/(2*Sqrt[f + I*c*Log[f]])]*Log[f]^2*Sqrt[f + I*c*Log[f]] +
3*(-1)^(3/4)*c^3*E^(((I/4)*e^2)/(f + I*c*Log[f]))*Cos[d]*Erfi[((-1)^(3/4)*(e + 2*f*x + (2*I)*c*x*Log[f]))/(2*
Sqrt[f + I*c*Log[f]])]*Log[f]^3*Sqrt[f + I*c*Log[f]] - 3*(-1)^(1/4)*E^((((9*I)/4)*e^2)/(3*f + I*c*Log[f]))*f^3
*Cos[3*d]*Erfi[((-1)^(3/4)*(3*e + 6*f*x + (2*I)*c*x*Log[f]))/(2*Sqrt[3*f + I*c*Log[f]])]*Sqrt[3*f + I*c*Log[f]
] + (-1)^(3/4)*c*E^((((9*I)/4)*e^2)/(3*f + I*c*Log[f]))*f^2*Cos[3*d]*Erfi[((-1)^(3/4)*(3*e + 6*f*x + (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)^(1/4)*c^2*E^((((9*I)/4)*e^2)/(3*f
+ I*c*Log[f]))*f*Cos[3*d]*Erfi[((-1)^(3/4)*(3*e + 6*f*x + (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)^(3/4)*c^3*E^((((9*I)/4)*e^2)/(3*f + I*c*Log[f]))*Cos[3*d]*Erfi[((-1)^(3/4)*
(3*e + 6*f*x + (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*Erfi[((-1)^(1/4)*(e + 2*f*x - (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)/(f - I*c*Log[f])) + (27*(-1)^(3/4)*c*f^2*Erfi[((-1)^(1/4)*(e + 2*f*x - (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)*e^2)/(f - I*c*Log[f])) + (3*(-1)^(1/4)*c^2*f
*Erfi[((-1)^(1/4)*(e + 2*f*x - (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)/(f - I*c*Log[f])) + (3*(-1)^(3/4)*c^3*Erfi[((-1)^(1/4)*(e + 2*f*x - (2*I)*c*x*Log[f]))/(2*S
qrt[f - I*c*Log[f]])]*Log[f]^3*Sqrt[f - I*c*Log[f]]*Sin[d])/E^(((I/4)*e^2)/(f - I*c*Log[f])) + 27*(-1)^(3/4)*E
^(((I/4)*e^2)/(f + I*c*Log[f]))*f^3*Erfi[((-1)^(3/4)*(e + 2*f*x + (2*I)*c*x*Log[f]))/(2*Sqrt[f + I*c*Log[f]])]
*Sqrt[f + I*c*Log[f]]*Sin[d] + 27*(-1)^(1/4)*c*E^(((I/4)*e^2)/(f + I*c*Log[f]))*f^2*Erfi[((-1)^(3/4)*(e + 2*f*
x + (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)^(3/4)*c^2*E^(((I/
4)*e^2)/(f + I*c*Log[f]))*f*Erfi[((-1)^(3/4)*(e + 2*f*x + (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)^(1/4)*c^3*E^(((I/4)*e^2)/(f + I*c*Log[f]))*Erfi[((-1)^(3/4)*(e + 2*f*x
+ (2*I)*c*x*Log[f]))/(2*Sqrt[f + I*c*Log[f]])]*Log[f]^3*Sqrt[f + I*c*Log[f]]*Sin[d] + (3*(-1)^(1/4)*f^3*Erfi[(
(-1)^(1/4)*(3*e + 6*f*x - (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^((
((9*I)/4)*e^2)/(3*f - I*c*Log[f])) + ((-1)^(3/4)*c*f^2*Erfi[((-1)^(1/4)*(3*e + 6*f*x - (2*I)*c*x*Log[f]))/(2*S
qrt[3*f - I*c*Log[f]])]*Log[f]*Sqrt[3*f - I*c*Log[f]]*Sin[3*d])/E^((((9*I)/4)*e^2)/(3*f - I*c*Log[f])) + (3*(-
1)^(1/4)*c^2*f*Erfi[((-1)^(1/4)*(3*e + 6*f*x - (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^((((9*I)/4)*e^2)/(3*f - I*c*Log[f])) + ((-1)^(3/4)*c^3*Erfi[((-1)^(1/4)*(3*e + 6*f
*x - (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^((((9*I)/4)*e^
2)/(3*f - I*c*Log[f])) + 3*(-1)^(3/4)*E^((((9*I)/4)*e^2)/(3*f + I*c*Log[f]))*f^3*Erfi[((-1)^(3/4)*(3*e + 6*f*x
+ (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)^(1/4)*c*E^((((9*I)/4)
*e^2)/(3*f + I*c*Log[f]))*f^2*Erfi[((-1)^(3/4)*(3*e + 6*f*x + (2*I)*c*x*Log[f]))/(2*Sqrt[3*f + I*c*Log[f]])]*L
og[f]*Sqrt[3*f + I*c*Log[f]]*Sin[3*d] + 3*(-1)^(3/4)*c^2*E^((((9*I)/4)*e^2)/(3*f + I*c*Log[f]))*f*Erfi[((-1)^(
3/4)*(3*e + 6*f*x + (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)^(1/4)*c^3*E^((((9*I)/4)*e^2)/(3*f + I*c*Log[f]))*Erfi[((-1)^(3/4)*(3*e + 6*f*x + (2*I)*c*x*Log[f]))/(2*Sq
rt[3*f + I*c*Log[f]])]*Log[f]^3*Sqrt[3*f + I*c*Log[f]]*Sin[3*d]))/(16*(f - I*c*Log[f])*(3*f - I*c*Log[f])*(f +
I*c*Log[f])*(3*f + I*c*Log[f]))
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Maple [A] time = 0.31, size = 334, normalized size = 0.9 \begin{align*}{\frac{{f}^{a}\sqrt{\pi }}{16}{{\rm e}^{-{\frac{12\,id\ln \left ( f \right ) c+36\,df-9\,{e}^{2}}{4\,c\ln \left ( f \right ) -12\,if}}}}{\it Erf} \left ( x\sqrt{3\,if-c\ln \left ( f \right ) }+{{\frac{3\,i}{2}}e{\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{4\,id\ln \left ( f \right ) c+4\,df-{e}^{2}}{4\,c\ln \left ( f \right ) -4\,if}}}}{\it Erf} \left ( x\sqrt{if-c\ln \left ( f \right ) }+{{\frac{i}{2}}e{\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{4\,id\ln \left ( f \right ) c-4\,df+{e}^{2}}{4\,if+4\,c\ln \left ( f \right ) }}}}{\it Erf} \left ( -\sqrt{-c\ln \left ( f \right ) -if}x+{{\frac{i}{2}}e{\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{12\,id\ln \left ( f \right ) c-36\,df+9\,{e}^{2}}{12\,if+4\,c\ln \left ( f \right ) }}}}{\it Erf} \left ( -\sqrt{-c\ln \left ( f \right ) -3\,if}x+{{\frac{3\,i}{2}}e{\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+a)*cos(f*x^2+e*x+d)^3,x)
[Out]
1/16*Pi^(1/2)*f^a*exp(-3/4*(4*I*d*ln(f)*c+12*d*f-3*e^2)/(-3*I*f+c*ln(f)))/(3*I*f-c*ln(f))^(1/2)*erf(x*(3*I*f-c
*ln(f))^(1/2)+3/2*I*e/(3*I*f-c*ln(f))^(1/2))+3/16*Pi^(1/2)*f^a*exp(-1/4*(4*I*d*ln(f)*c+4*d*f-e^2)/(-I*f+c*ln(f
)))/(I*f-c*ln(f))^(1/2)*erf(x*(I*f-c*ln(f))^(1/2)+1/2*I*e/(I*f-c*ln(f))^(1/2))-3/16*Pi^(1/2)*f^a*exp(1/4*(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/(-c*ln(f)-I*f)^(1
/2))-1/16*Pi^(1/2)*f^a*exp(3/4*(4*I*d*ln(f)*c-12*d*f+3*e^2)/(3*I*f+c*ln(f)))/(-c*ln(f)-3*I*f)^(1/2)*erf(-(-c*l
n(f)-3*I*f)^(1/2)*x+3/2*I*e/(-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+a)*cos(f*x^2+e*x+d)^3,x, algorithm="maxima")
[Out]
Exception raised: IndexError
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Fricas [B] time = 0.662307, size = 1839, normalized size = 4.98 \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+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*(
2*c^2*x*log(f)^2 + 18*f^2*x + 3*I*c*e*log(f) + 9*e*f)*sqrt(-c*log(f) - 3*I*f)/(c^2*log(f)^2 + 9*f^2))*e^(1/4*(
4*a*c^2*log(f)^3 + 12*I*c^2*d*log(f)^2 - 27*I*e^2*f + 108*I*d*f^2 + 9*(c*e^2 + 4*a*f^2)*log(f))/(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*(2*c^2*x*log(f)^2 + 18*f^2*x - 3*I*c*e*log(f) + 9*e*f)*sqrt(-c*log(f) + 3*I*f)/(c^2*log(f)^2 + 9*f^2))*e^(
1/4*(4*a*c^2*log(f)^3 - 12*I*c^2*d*log(f)^2 + 27*I*e^2*f - 108*I*d*f^2 + 9*(c*e^2 + 4*a*f^2)*log(f))/(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*c^2*x*log(f)^2 + 2*f^2*x + I*c*e*log(f) + e*f)*sqrt(-c*log(f) - I*f)/(c^2*log(f)^2 + f^2))*e^(
1/4*(4*a*c^2*log(f)^3 + 4*I*c^2*d*log(f)^2 - I*e^2*f + 4*I*d*f^2 + (c*e^2 + 4*a*f^2)*log(f))/(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*c^2*x*log(f)^2 + 2*f^2*x - I*c*e*log(f) + e*f)*sqrt(-c*log(f) + I*f)/(c^2*log(f)^2 + f^2))*e^(1/4*(4*a*c
^2*log(f)^3 - 4*I*c^2*d*log(f)^2 + I*e^2*f - 4*I*d*f^2 + (c*e^2 + 4*a*f^2)*log(f))/(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+a)*cos(f*x**2+e*x+d)**3,x)
[Out]
Timed out
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int f^{c x^{2} + a} \cos \left (f x^{2} + e x + d\right )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(f^(c*x^2+a)*cos(f*x^2+e*x+d)^3,x, algorithm="giac")
[Out]
integrate(f^(c*x^2 + a)*cos(f*x^2 + e*x + d)^3, x)