3.2.31 \(\int f^{a+b x+c x^2} \cos (d+e x+f x^2) \, dx\) [131]

Optimal. Leaf size=208 \[ \frac {e^{-i d-\frac {(e+i b \log (f))^2}{4 i f-4 c \log (f)}} f^a \sqrt {\pi } \text {Erf}\left (\frac {i e-b \log (f)+2 x (i f-c \log (f))}{2 \sqrt {i f-c \log (f)}}\right )}{4 \sqrt {i f-c \log (f)}}+\frac {e^{i d+\frac {(e-i b \log (f))^2}{4 i f+4 c \log (f)}} f^a \sqrt {\pi } \text {Erfi}\left (\frac {i e+b \log (f)+2 x (i f+c \log (f))}{2 \sqrt {i f+c \log (f)}}\right )}{4 \sqrt {i f+c \log (f)}} \]

[Out]

1/4*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)+1/4*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)

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Rubi [A]
time = 0.33, antiderivative size = 208, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {4561, 2325, 2266, 2236, 2235} \begin {gather*} \frac {\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 )}{4 \sqrt {-c \log (f)+i f}}+\frac {\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 )}{4 \sqrt {c \log (f)+i f}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(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]])])/(4*Sqrt[I*f - c*Log[f]]) + (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]])])/(4*Sqrt[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+b x+c x^2} \cos \left (d+e x+f x^2\right ) \, dx &=\int \left (\frac {1}{2} e^{-i d-i e x-i f x^2} f^{a+b x+c x^2}+\frac {1}{2} e^{i d+i e x+i f x^2} f^{a+b x+c x^2}\right ) \, dx\\ &=\frac {1}{2} \int e^{-i d-i e x-i f x^2} f^{a+b x+c x^2} \, dx+\frac {1}{2} \int e^{i d+i e x+i f x^2} f^{a+b x+c x^2} \, dx\\ &=\frac {1}{2} \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}{2} \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}{2} \left (\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}{2} \left (\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 {\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 )}{4 \sqrt {i f-c \log (f)}}+\frac {\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 )}{4 \sqrt {i f+c \log (f)}}\\ \end {align*}

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Mathematica [A]
time = 2.21, size = 348, normalized size = 1.67 \begin {gather*} \frac {\sqrt [4]{-1} e^{-\frac {1}{4} i \left (\frac {e^2}{f-i c \log (f)}+\frac {b^2 \log ^2(f)}{f+i c \log (f)}\right )} f^{\frac {f (-b e+a f)+a c^2 \log ^2(f)}{f^2+c^2 \log ^2(f)}} \sqrt {\pi } \left (-e^{\frac {i e^2 f}{2 \left (f^2+c^2 \log ^2(f)\right )}} f^{\frac {b e}{2 f-2 i c \log (f)}} \text {Erfi}\left (\frac {(-1)^{3/4} (e+2 f x+i (b+2 c x) \log (f))}{2 \sqrt {f+i c \log (f)}}\right ) (f-i c \log (f)) \sqrt {f+i c \log (f)} (\cos (d)-i \sin (d))+e^{\frac {i b^2 f \log ^2(f)}{2 \left (f^2+c^2 \log ^2(f)\right )}} f^{\frac {b e}{2 f+2 i c \log (f)}} \text {Erfi}\left (\frac {\sqrt [4]{-1} (e+2 f x-i (b+2 c x) \log (f))}{2 \sqrt {f-i c \log (f)}}\right ) \sqrt {f-i c \log (f)} (f+i c \log (f)) (-i \cos (d)+\sin (d))\right )}{4 \left (f^2+c^2 \log ^2(f)\right )} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [A]
time = 0.23, size = 214, normalized size = 1.03

method	result	size
risch	\(-\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {\ln \left (f \right )^{2} b^{2}-2 i \ln \left (f \right ) b e +4 i d \ln \left (f \right ) c +4 d f -e^{2}}{4 \left (-i f +c \ln \left (f \right )\right )}} \erf \left (-x \sqrt {i f -c \ln \left (f \right )}+\frac {b \ln \left (f \right )-i e}{2 \sqrt {i f -c \ln \left (f \right )}}\right )}{4 \sqrt {i f -c \ln \left (f \right )}}-\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {\ln \left (f \right )^{2} b^{2}+2 i \ln \left (f \right ) b e -4 i d \ln \left (f \right ) c +4 d f -e^{2}}{4 \left (i f +c \ln \left (f \right )\right )}} \erf \left (-\sqrt {-c \ln \left (f \right )-i f}\, x +\frac {i e +b \ln \left (f \right )}{2 \sqrt {-c \ln \left (f \right )-i f}}\right )}{4 \sqrt {-c \ln \left (f \right )-i f}}\)	\(214\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*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)))/(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))-1/4*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))

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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. 1006 vs. \(2 (159) = 318\).
time = 0.30, size = 1006, normalized size = 4.84 \begin {gather*} -\frac {\sqrt {\pi } \sqrt {2 \, c^{2} \log \left (f\right )^{2} + 2 \, f^{2}} {\left ({\left (i \, f^{a} f^{\frac {c e^{2}}{4 \, {\left (c^{2} \log \left (f\right )^{2} + f^{2}\right )}}} \cos \left (\frac {4 \, d f^{2} + {\left (4 \, c^{2} d + b^{2} f - 2 \, b c e\right )} \log \left (f\right )^{2} - f e^{2}}{4 \, {\left (c^{2} \log \left (f\right )^{2} + f^{2}\right )}}\right ) + f^{a} f^{\frac {c e^{2}}{4 \, {\left (c^{2} \log \left (f\right )^{2} + f^{2}\right )}}} \sin \left (\frac {4 \, d f^{2} + {\left (4 \, c^{2} d + b^{2} f - 2 \, b c e\right )} \log \left (f\right )^{2} - f e^{2}}{4 \, {\left (c^{2} \log \left (f\right )^{2} + f^{2}\right )}}\right )\right )} \operatorname {erf}\left (\frac {{\left (2 \, {\left (c \log \left (f\right ) - i \, f\right )} x + b \log \left (f\right ) - i \, e\right )} \sqrt {-c \log \left (f\right ) + i \, f}}{2 \, {\left (c \log \left (f\right ) - i \, f\right )}}\right ) + {\left (-i \, f^{a} f^{\frac {c e^{2}}{4 \, {\left (c^{2} \log \left (f\right )^{2} + f^{2}\right )}}} \cos \left (\frac {4 \, d f^{2} + {\left (4 \, c^{2} d + b^{2} f - 2 \, b c e\right )} \log \left (f\right )^{2} - f e^{2}}{4 \, {\left (c^{2} \log \left (f\right )^{2} + f^{2}\right )}}\right ) + f^{a} f^{\frac {c e^{2}}{4 \, {\left (c^{2} \log \left (f\right )^{2} + f^{2}\right )}}} \sin \left (\frac {4 \, d f^{2} + {\left (4 \, c^{2} d + b^{2} f - 2 \, b c e\right )} \log \left (f\right )^{2} - f e^{2}}{4 \, {\left (c^{2} \log \left (f\right )^{2} + f^{2}\right )}}\right )\right )} \operatorname {erf}\left (\frac {{\left (2 \, {\left (c \log \left (f\right ) + i \, f\right )} x + b \log \left (f\right ) + i \, e\right )} \sqrt {-c \log \left (f\right ) - i \, f}}{2 \, {\left (c \log \left (f\right ) + i \, f\right )}}\right )\right )} \sqrt {c \log \left (f\right ) + \sqrt {c^{2} \log \left (f\right )^{2} + f^{2}}} - \sqrt {\pi } \sqrt {2 \, c^{2} \log \left (f\right )^{2} + 2 \, f^{2}} {\left ({\left (f^{a} f^{\frac {c e^{2}}{4 \, {\left (c^{2} \log \left (f\right )^{2} + f^{2}\right )}}} \cos \left (\frac {4 \, d f^{2} + {\left (4 \, c^{2} d + b^{2} f - 2 \, b c e\right )} \log \left (f\right )^{2} - f e^{2}}{4 \, {\left (c^{2} \log \left (f\right )^{2} + f^{2}\right )}}\right ) - i \, f^{a} f^{\frac {c e^{2}}{4 \, {\left (c^{2} \log \left (f\right )^{2} + f^{2}\right )}}} \sin \left (\frac {4 \, d f^{2} + {\left (4 \, c^{2} d + b^{2} f - 2 \, b c e\right )} \log \left (f\right )^{2} - f e^{2}}{4 \, {\left (c^{2} \log \left (f\right )^{2} + f^{2}\right )}}\right )\right )} \operatorname {erf}\left (\frac {{\left (2 \, {\left (c \log \left (f\right ) - i \, f\right )} x + b \log \left (f\right ) - i \, e\right )} \sqrt {-c \log \left (f\right ) + i \, f}}{2 \, {\left (c \log \left (f\right ) - i \, f\right )}}\right ) + {\left (f^{a} f^{\frac {c e^{2}}{4 \, {\left (c^{2} \log \left (f\right )^{2} + f^{2}\right )}}} \cos \left (\frac {4 \, d f^{2} + {\left (4 \, c^{2} d + b^{2} f - 2 \, b c e\right )} \log \left (f\right )^{2} - f e^{2}}{4 \, {\left (c^{2} \log \left (f\right )^{2} + f^{2}\right )}}\right ) + i \, f^{a} f^{\frac {c e^{2}}{4 \, {\left (c^{2} \log \left (f\right )^{2} + f^{2}\right )}}} \sin \left (\frac {4 \, d f^{2} + {\left (4 \, c^{2} d + b^{2} f - 2 \, b c e\right )} \log \left (f\right )^{2} - f e^{2}}{4 \, {\left (c^{2} \log \left (f\right )^{2} + f^{2}\right )}}\right )\right )} \operatorname {erf}\left (\frac {{\left (2 \, {\left (c \log \left (f\right ) + i \, f\right )} x + b \log \left (f\right ) + i \, e\right )} \sqrt {-c \log \left (f\right ) - i \, f}}{2 \, {\left (c \log \left (f\right ) + i \, f\right )}}\right )\right )} \sqrt {-c \log \left (f\right ) + \sqrt {c^{2} \log \left (f\right )^{2} + f^{2}}}}{8 \, {\left (c^{2} e^{\left (\frac {b^{2} c \log \left (f\right )^{3}}{4 \, {\left (c^{2} \log \left (f\right )^{2} + f^{2}\right )}} + \frac {b f e \log \left (f\right )}{2 \, {\left (c^{2} \log \left (f\right )^{2} + f^{2}\right )}}\right )} \log \left (f\right )^{2} + f^{2} e^{\left (\frac {b^{2} c \log \left (f\right )^{3}}{4 \, {\left (c^{2} \log \left (f\right )^{2} + f^{2}\right )}} + \frac {b f e \log \left (f\right )}{2 \, {\left (c^{2} \log \left (f\right )^{2} + f^{2}\right )}}\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(sqrt(pi)*sqrt(2*c^2*log(f)^2 + 2*f^2)*((I*f^a*f^(1/4*c*e^2/(c^2*log(f)^2 + f^2))*cos(1/4*(4*d*f^2 + (4*c
^2*d + b^2*f - 2*b*c*e)*log(f)^2 - f*e^2)/(c^2*log(f)^2 + f^2)) + f^a*f^(1/4*c*e^2/(c^2*log(f)^2 + f^2))*sin(1
/4*(4*d*f^2 + (4*c^2*d + b^2*f - 2*b*c*e)*log(f)^2 - f*e^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*f^a*f^(1/4*c*e^2/(c^2*log(f)^2 + f^2))*cos(
1/4*(4*d*f^2 + (4*c^2*d + b^2*f - 2*b*c*e)*log(f)^2 - f*e^2)/(c^2*log(f)^2 + f^2)) + f^a*f^(1/4*c*e^2/(c^2*log
(f)^2 + f^2))*sin(1/4*(4*d*f^2 + (4*c^2*d + b^2*f - 2*b*c*e)*log(f)^2 - f*e^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 + 2*f^2)*((f^a*f^(1/4*c*e^2/(c^2*log(f)^2 + f^2))*cos(1/4*(4*d*f^2
+ (4*c^2*d + b^2*f - 2*b*c*e)*log(f)^2 - f*e^2)/(c^2*log(f)^2 + f^2)) - I*f^a*f^(1/4*c*e^2/(c^2*log(f)^2 + f^
2))*sin(1/4*(4*d*f^2 + (4*c^2*d + b^2*f - 2*b*c*e)*log(f)^2 - f*e^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)) + (f^a*f^(1/4*c*e^2/(c^2*log(f)^2 + f^2)
)*cos(1/4*(4*d*f^2 + (4*c^2*d + b^2*f - 2*b*c*e)*log(f)^2 - f*e^2)/(c^2*log(f)^2 + f^2)) + I*f^a*f^(1/4*c*e^2/
(c^2*log(f)^2 + f^2))*sin(1/4*(4*d*f^2 + (4*c^2*d + b^2*f - 2*b*c*e)*log(f)^2 - f*e^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)))/(c^2*e^(1/4*b^2*c*log(f)^3/(c^2*log(f)^2 + f^2) + 1/2*b*f*e*log(f)/(c^2*log(f)^2 + f^2)
)*log(f)^2 + f^2*e^(1/4*b^2*c*log(f)^3/(c^2*log(f)^2 + f^2) + 1/2*b*f*e*log(f)/(c^2*log(f)^2 + f^2)))

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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. 381 vs. \(2 (159) = 318\).
time = 3.13, size = 381, normalized size = 1.83 \begin {gather*} -\frac {\sqrt {\pi } {\left (c \log \left (f\right ) + i \, f\right )} \sqrt {-c \log \left (f\right ) + i \, f} \operatorname {erf}\left (\frac {{\left (2 \, f^{2} x + {\left (2 \, c^{2} x + b c\right )} \log \left (f\right )^{2} + f e + {\left (i \, b f - i \, c e\right )} \log \left (f\right )\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 {{\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 + 2 i \, b c e\right )} \log \left (f\right )^{2} - i \, f e^{2} - {\left (4 \, a f^{2} - 2 \, b f e + c e^{2}\right )} \log \left (f\right )}{4 \, {\left (c^{2} \log \left (f\right )^{2} + f^{2}\right )}}\right )} + \sqrt {\pi } {\left (c \log \left (f\right ) - i \, f\right )} \sqrt {-c \log \left (f\right ) - i \, f} \operatorname {erf}\left (\frac {{\left (2 \, f^{2} x + {\left (2 \, c^{2} x + b c\right )} \log \left (f\right )^{2} + f e + {\left (-i \, b f + i \, c e\right )} \log \left (f\right )\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 {{\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 - 2 i \, b c e\right )} \log \left (f\right )^{2} + i \, f e^{2} - {\left (4 \, a f^{2} - 2 \, b f e + c e^{2}\right )} \log \left (f\right )}{4 \, {\left (c^{2} \log \left (f\right )^{2} + f^{2}\right )}}\right )}}{4 \, {\left (c^{2} \log \left (f\right )^{2} + f^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(sqrt(pi)*(c*log(f) + I*f)*sqrt(-c*log(f) + I*f)*erf(1/2*(2*f^2*x + (2*c^2*x + b*c)*log(f)^2 + f*e + (I*b
*f - I*c*e)*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 + 4*I*d*f^
2 - (-4*I*c^2*d - I*b^2*f + 2*I*b*c*e)*log(f)^2 - I*f*e^2 - (4*a*f^2 - 2*b*f*e + c*e^2)*log(f))/(c^2*log(f)^2
+ f^2)) + sqrt(pi)*(c*log(f) - I*f)*sqrt(-c*log(f) - I*f)*erf(1/2*(2*f^2*x + (2*c^2*x + b*c)*log(f)^2 + f*e +
(-I*b*f + I*c*e)*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 - 4*I
*d*f^2 - (4*I*c^2*d + I*b^2*f - 2*I*b*c*e)*log(f)^2 + I*f*e^2 - (4*a*f^2 - 2*b*f*e + c*e^2)*log(f))/(c^2*log(f
)^2 + f^2)))/(c^2*log(f)^2 + f^2)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int f^{a + b x + c x^{2}} \cos {\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+b*x+a)*cos(f*x**2+e*x+d),x)

[Out]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int f^{c\,x^2+b\,x+a}\,\cos \left (f\,x^2+e\,x+d\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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