3.2.24 \(\int f^{a+c x^2} \cos ^3(d+e x+f x^2) \, dx\) [124]
Optimal. Leaf size=369 \[ \frac {3 e^{-i d-\frac {e^2}{4 i f-4 c \log (f)}} f^a \sqrt {\pi } \text {Erf}\left (\frac {i e+2 x (i f-c \log (f))}{2 \sqrt {i f-c \log (f)}}\right )}{16 \sqrt {i f-c \log (f)}}+\frac {e^{-3 i d-\frac {9 e^2}{4 (3 i f-c \log (f))}} f^a \sqrt {\pi } \text {Erf}\left (\frac {3 i e+2 x (3 i f-c \log (f))}{2 \sqrt {3 i f-c \log (f)}}\right )}{16 \sqrt {3 i f-c \log (f)}}+\frac {3 e^{i d+\frac {e^2}{4 i f+4 c \log (f)}} f^a \sqrt {\pi } \text {Erfi}\left (\frac {i e+2 x (i f+c \log (f))}{2 \sqrt {i f+c \log (f)}}\right )}{16 \sqrt {i f+c \log (f)}}+\frac {e^{3 i d+\frac {9 e^2}{4 (3 i f+c \log (f))}} f^a \sqrt {\pi } \text {Erfi}\left (\frac {3 i e+2 x (3 i f+c \log (f))}{2 \sqrt {3 i f+c \log (f)}}\right )}{16 \sqrt {3 i f+c \log (f)}} \]
[Out]
3/16*exp(-I*d-e^2/(4*I*f-4*c*ln(f)))*f^a*erf(1/2*(I*e+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-9/4*e^2/(3*I*f-c*ln(f)))*f^a*erf(1/2*(3*I*e+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^2/(4*I*f+4*c*ln(f)))*f^a*erfi(1/2*(I*e+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+9/4*e^2/(3*I*f+c*ln(f)))*f^a*erfi(1/2*(3*I*e
+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)
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Rubi [A]
time = 0.43, antiderivative size = 369, normalized size of antiderivative = 1.00, 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 = {4561, 2325,
2266, 2236, 2235} \begin {gather*} \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}} \end {gather*}
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 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*} \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*}
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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice
the leaf count of optimal. \(2997\) vs. \(2(369)=738\).
time = 7.01, size = 2997, normalized size = 8.12 \begin {gather*} \text {Result too large to show} \end {gather*}
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)*...
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Maple [A]
time = 0.81, size = 334, normalized size = 0.91
| | |
| method |
result |
size |
| | |
| risch |
\(\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {3 \left (4 i d \ln \left (f \right ) c +12 d f -3 e^{2}\right )}{4 \left (-3 i f +c \ln \left (f \right )\right )}} \erf \left (x \sqrt {3 i f -c \ln \left (f \right )}+\frac {3 i e}{2 \sqrt {3 i f -c \ln \left (f \right )}}\right )}{16 \sqrt {3 i f -c \ln \left (f \right )}}+\frac {3 \sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {4 i d \ln \left (f \right ) c +4 d f -e^{2}}{4 \left (-i f +c \ln \left (f \right )\right )}} \erf \left (x \sqrt {i f -c \ln \left (f \right )}+\frac {i e}{2 \sqrt {i f -c \ln \left (f \right )}}\right )}{16 \sqrt {i f -c \ln \left (f \right )}}-\frac {3 \sqrt {\pi }\, f^{a} {\mathrm e}^{\frac {4 i d \ln \left (f \right ) c -4 d f +e^{2}}{4 i f +4 c \ln \left (f \right )}} \erf \left (-\sqrt {-c \ln \left (f \right )-i f}\, x +\frac {i e}{2 \sqrt {-c \ln \left (f \right )-i f}}\right )}{16 \sqrt {-c \ln \left (f \right )-i f}}-\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{\frac {3 i d \ln \left (f \right ) c -9 d f +\frac {9 e^{2}}{4}}{3 i f +c \ln \left (f \right )}} \erf \left (-\sqrt {-c \ln \left (f \right )-3 i f}\, x +\frac {3 i e}{2 \sqrt {-c \ln \left (f \right )-3 i f}}\right )}{16 \sqrt {-c \ln \left (f \right )-3 i f}}\) |
\(334\) |
| | |
|
|
|
|
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,method=_RETURNVERBOSE)
[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))
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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than
twice the leaf count of optimal. 2140 vs. \(2 (269) = 538\).
time = 0.33, size = 2140, normalized size = 5.80 \begin {gather*} \text {Too large to display} \end {gather*}
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]
1/32*(sqrt(pi)*sqrt(2*c^2*log(f)^2 + 18*f^2)*(((I*c^2*f^a*f^(9/4*c*e^2/(c^2*log(f)^2 + 9*f^2))*log(f)^2 + I*f^
(a + 2)*f^(9/4*c*e^2/(c^2*log(f)^2 + 9*f^2)))*cos(3/4*(4*c^2*d*log(f)^2 + 36*d*f^2 - 9*f*e^2)/(c^2*log(f)^2 +
9*f^2)) + (c^2*f^a*f^(9/4*c*e^2/(c^2*log(f)^2 + 9*f^2))*log(f)^2 + f^(a + 2)*f^(9/4*c*e^2/(c^2*log(f)^2 + 9*f^
2)))*sin(3/4*(4*c^2*d*log(f)^2 + 36*d*f^2 - 9*f*e^2)/(c^2*log(f)^2 + 9*f^2)))*erf(1/2*(2*(c*log(f) - 3*I*f)*x
- 3*I*e)/sqrt(-c*log(f) + 3*I*f)) + ((-I*c^2*f^a*f^(9/4*c*e^2/(c^2*log(f)^2 + 9*f^2))*log(f)^2 - I*f^(a + 2)*f
^(9/4*c*e^2/(c^2*log(f)^2 + 9*f^2)))*cos(3/4*(4*c^2*d*log(f)^2 + 36*d*f^2 - 9*f*e^2)/(c^2*log(f)^2 + 9*f^2)) +
(c^2*f^a*f^(9/4*c*e^2/(c^2*log(f)^2 + 9*f^2))*log(f)^2 + f^(a + 2)*f^(9/4*c*e^2/(c^2*log(f)^2 + 9*f^2)))*sin(
3/4*(4*c^2*d*log(f)^2 + 36*d*f^2 - 9*f*e^2)/(c^2*log(f)^2 + 9*f^2)))*erf(1/2*(2*(c*log(f) + 3*I*f)*x + 3*I*e)/
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*f^(1/4*c*e^2/(c^2*log(f)^2 + f^2))*log(f)^2 - 9*I*f^(a + 2)*f^(1/4*c*e^2/(c^2*log(f)^2 + f^2))
)*cos(1/4*(4*c^2*d*log(f)^2 + 4*d*f^2 - f*e^2)/(c^2*log(f)^2 + f^2)) - (c^2*f^a*f^(1/4*c*e^2/(c^2*log(f)^2 + f
^2))*log(f)^2 + 9*f^(a + 2)*f^(1/4*c*e^2/(c^2*log(f)^2 + f^2)))*sin(1/4*(4*c^2*d*log(f)^2 + 4*d*f^2 - f*e^2)/(
c^2*log(f)^2 + f^2)))*erf(1/2*(2*(c*log(f) - I*f)*x - I*e)/sqrt(-c*log(f) + I*f)) + ((I*c^2*f^a*f^(1/4*c*e^2/(
c^2*log(f)^2 + f^2))*log(f)^2 + 9*I*f^(a + 2)*f^(1/4*c*e^2/(c^2*log(f)^2 + f^2)))*cos(1/4*(4*c^2*d*log(f)^2 +
4*d*f^2 - f*e^2)/(c^2*log(f)^2 + f^2)) - (c^2*f^a*f^(1/4*c*e^2/(c^2*log(f)^2 + f^2))*log(f)^2 + 9*f^(a + 2)*f^
(1/4*c*e^2/(c^2*log(f)^2 + f^2)))*sin(1/4*(4*c^2*d*log(f)^2 + 4*d*f^2 - f*e^2)/(c^2*log(f)^2 + f^2)))*erf(1/2*
(2*(c*log(f) + I*f)*x + I*e)/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*f^(9/4*c*e^2/(c^2*log(f)^2 + 9*f^2))*log(f)^2 + f^(a + 2)*f^(9/4*c*e^2/(c
^2*log(f)^2 + 9*f^2)))*cos(3/4*(4*c^2*d*log(f)^2 + 36*d*f^2 - 9*f*e^2)/(c^2*log(f)^2 + 9*f^2)) - (I*c^2*f^a*f^
(9/4*c*e^2/(c^2*log(f)^2 + 9*f^2))*log(f)^2 + I*f^(a + 2)*f^(9/4*c*e^2/(c^2*log(f)^2 + 9*f^2)))*sin(3/4*(4*c^2
*d*log(f)^2 + 36*d*f^2 - 9*f*e^2)/(c^2*log(f)^2 + 9*f^2)))*erf(1/2*(2*(c*log(f) - 3*I*f)*x - 3*I*e)/sqrt(-c*lo
g(f) + 3*I*f)) + ((c^2*f^a*f^(9/4*c*e^2/(c^2*log(f)^2 + 9*f^2))*log(f)^2 + f^(a + 2)*f^(9/4*c*e^2/(c^2*log(f)^
2 + 9*f^2)))*cos(3/4*(4*c^2*d*log(f)^2 + 36*d*f^2 - 9*f*e^2)/(c^2*log(f)^2 + 9*f^2)) - (-I*c^2*f^a*f^(9/4*c*e^
2/(c^2*log(f)^2 + 9*f^2))*log(f)^2 - I*f^(a + 2)*f^(9/4*c*e^2/(c^2*log(f)^2 + 9*f^2)))*sin(3/4*(4*c^2*d*log(f)
^2 + 36*d*f^2 - 9*f*e^2)/(c^2*log(f)^2 + 9*f^2)))*erf(1/2*(2*(c*log(f) + 3*I*f)*x + 3*I*e)/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)*(((c^2*f^a*f^(1/
4*c*e^2/(c^2*log(f)^2 + f^2))*log(f)^2 + 9*f^(a + 2)*f^(1/4*c*e^2/(c^2*log(f)^2 + f^2)))*cos(1/4*(4*c^2*d*log(
f)^2 + 4*d*f^2 - f*e^2)/(c^2*log(f)^2 + f^2)) + (-I*c^2*f^a*f^(1/4*c*e^2/(c^2*log(f)^2 + f^2))*log(f)^2 - 9*I*
f^(a + 2)*f^(1/4*c*e^2/(c^2*log(f)^2 + f^2)))*sin(1/4*(4*c^2*d*log(f)^2 + 4*d*f^2 - f*e^2)/(c^2*log(f)^2 + f^2
)))*erf(1/2*(2*(c*log(f) - I*f)*x - I*e)/sqrt(-c*log(f) + I*f)) + ((c^2*f^a*f^(1/4*c*e^2/(c^2*log(f)^2 + f^2))
*log(f)^2 + 9*f^(a + 2)*f^(1/4*c*e^2/(c^2*log(f)^2 + f^2)))*cos(1/4*(4*c^2*d*log(f)^2 + 4*d*f^2 - f*e^2)/(c^2*
log(f)^2 + f^2)) + (I*c^2*f^a*f^(1/4*c*e^2/(c^2*log(f)^2 + f^2))*log(f)^2 + 9*I*f^(a + 2)*f^(1/4*c*e^2/(c^2*lo
g(f)^2 + f^2)))*sin(1/4*(4*c^2*d*log(f)^2 + 4*d*f^2 - f*e^2)/(c^2*log(f)^2 + f^2)))*erf(1/2*(2*(c*log(f) + I*f
)*x + I*e)/sqrt(-c*log(f) - I*f)))*sqrt(-c*log(f) + sqrt(c^2*log(f)^2 + f^2)))/(c^4*log(f)^4 + 10*c^2*f^2*log(
f)^2 + 9*f^4)
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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 707 vs. \(2 (269) = 538\).
time = 2.59, size = 707, normalized size = 1.92 \begin {gather*} -\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 (2 \, c^{2} x \log \left (f\right )^{2} + 18 \, f^{2} x + 3 i \, c e \log \left (f\right ) + 9 \, f e\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 {4 \, a c^{2} \log \left (f\right )^{3} + 12 i \, c^{2} d \log \left (f\right )^{2} + 108 i \, d f^{2} - 27 i \, f e^{2} + 9 \, {\left (4 \, a f^{2} + c e^{2}\right )} \log \left (f\right )}{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 (2 \, c^{2} x \log \left (f\right )^{2} + 18 \, f^{2} x - 3 i \, c e \log \left (f\right ) + 9 \, f e\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 {4 \, a c^{2} \log \left (f\right )^{3} - 12 i \, c^{2} d \log \left (f\right )^{2} - 108 i \, d f^{2} + 27 i \, f e^{2} + 9 \, {\left (4 \, a f^{2} + c e^{2}\right )} \log \left (f\right )}{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 \, c^{2} x \log \left (f\right )^{2} + 2 \, f^{2} x + i \, c e \log \left (f\right ) + f e\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 c^{2} \log \left (f\right )^{3} + 4 i \, c^{2} d \log \left (f\right )^{2} + 4 i \, d f^{2} - i \, f e^{2} + {\left (4 \, a f^{2} + c e^{2}\right )} \log \left (f\right )}{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 \, c^{2} x \log \left (f\right )^{2} + 2 \, f^{2} x - i \, c e \log \left (f\right ) + f e\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 c^{2} \log \left (f\right )^{3} - 4 i \, c^{2} d \log \left (f\right )^{2} - 4 i \, d f^{2} + i \, f e^{2} + {\left (4 \, a f^{2} + c e^{2}\right )} \log \left (f\right )}{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 )}} \end {gather*}
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*f*e)*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 + 108*I*d*f^2 - 27*I*f*e^2 + 9*(4*a*f^2 + c*e^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*f*e)*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 - 108*I*d*f^2 + 27*I*f*e^2 + 9*(4*a*f^2 + c*e^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*c^2*x*log(f)^2 + 2*f^2*x + I*c*e*log(f) + f*e)*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 + 4*I*d*f^2 - I*f*e^2 + (4*a*f^2 + c*e^2)*log(f))/(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*c^
2*x*log(f)^2 + 2*f^2*x - I*c*e*log(f) + f*e)*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 - 4*I*d*f^2 + I*f*e^2 + (4*a*f^2 + c*e^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)
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int f^{a + c x^{2}} \cos ^{3}{\left (d + e x + f x^{2} \right )}\, dx \end {gather*}
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]
Integral(f**(a + c*x**2)*cos(d + e*x + f*x**2)**3, x)
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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)
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int f^{c\,x^2+a}\,{\cos \left (f\,x^2+e\,x+d\right )}^3 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(f^(a + c*x^2)*cos(d + e*x + f*x^2)^3,x)
[Out]
int(f^(a + c*x^2)*cos(d + e*x + f*x^2)^3, x)
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