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

Optimal result

Integrand size = 24, antiderivative size = 208 \[ \int f^{a+b x+c x^2} \cos \left (d+e x+f x^2\right ) \, dx=\frac {e^{-i d-\frac {(e+i b \log (f))^2}{4 i f-4 c \log (f)}} f^a \sqrt {\pi } \text {erf}\left (\frac {i e-b \log (f)+2 x (i f-c \log (f))}{2 \sqrt {i f-c \log (f)}}\right )}{4 \sqrt {i f-c \log (f)}}+\frac {e^{i d+\frac {(e-i b \log (f))^2}{4 i f+4 c \log (f)}} f^a \sqrt {\pi } \text {erfi}\left (\frac {i e+b \log (f)+2 x (i f+c \log (f))}{2 \sqrt {i f+c \log (f)}}\right )}{4 \sqrt {i f+c \log (f)}} \]

[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)

Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {4561, 2325, 2266, 2236, 2235} \[ \int f^{a+b x+c x^2} \cos \left (d+e x+f x^2\right ) \, dx=\frac {\sqrt {\pi } f^a \exp \left (-\frac {(e+i b \log (f))^2}{-4 c \log (f)+4 i f}-i d\right ) \text {erf}\left (\frac {-b \log (f)+2 x (-c \log (f)+i f)+i e}{2 \sqrt {-c \log (f)+i f}}\right )}{4 \sqrt {-c \log (f)+i f}}+\frac {\sqrt {\pi } f^a \exp \left (\frac {(e-i b \log (f))^2}{4 c \log (f)+4 i f}+i d\right ) \text {erfi}\left (\frac {b \log (f)+2 x (c \log (f)+i f)+i e}{2 \sqrt {c \log (f)+i f}}\right )}{4 \sqrt {c \log (f)+i f}} \]

[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*} \text {integral}& = \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*}

Mathematica [A] (warning: unable to verify)

Time = 1.75 (sec) , antiderivative size = 348, normalized size of antiderivative = 1.67 \[ \int f^{a+b x+c x^2} \cos \left (d+e x+f x^2\right ) \, dx=\frac {\sqrt [4]{-1} e^{-\frac {1}{4} i \left (\frac {e^2}{f-i c \log (f)}+\frac {b^2 \log ^2(f)}{f+i c \log (f)}\right )} f^{\frac {f (-b e+a f)+a c^2 \log ^2(f)}{f^2+c^2 \log ^2(f)}} \sqrt {\pi } \left (-e^{\frac {i e^2 f}{2 \left (f^2+c^2 \log ^2(f)\right )}} f^{\frac {b e}{2 f-2 i c \log (f)}} \text {erfi}\left (\frac {(-1)^{3/4} (e+2 f x+i (b+2 c x) \log (f))}{2 \sqrt {f+i c \log (f)}}\right ) (f-i c \log (f)) \sqrt {f+i c \log (f)} (\cos (d)-i \sin (d))+e^{\frac {i b^2 f \log ^2(f)}{2 \left (f^2+c^2 \log ^2(f)\right )}} f^{\frac {b e}{2 f+2 i c \log (f)}} \text {erfi}\left (\frac {\sqrt [4]{-1} (e+2 f x-i (b+2 c x) \log (f))}{2 \sqrt {f-i c \log (f)}}\right ) \sqrt {f-i c \log (f)} (f+i c \log (f)) (-i \cos (d)+\sin (d))\right )}{4 \left (f^2+c^2 \log ^2(f)\right )} \]

[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))

Maple [A] (verified)

Time = 0.49 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.03

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

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

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. 377 vs. \(2 (155) = 310\).

Time = 0.26 (sec) , antiderivative size = 377, normalized size of antiderivative = 1.81 \[ \int f^{a+b x+c x^2} \cos \left (d+e x+f x^2\right ) \, dx=-\frac {\sqrt {\pi } {\left (c \log \left (f\right ) - i \, f\right )} \sqrt {-c \log \left (f\right ) - i \, f} \operatorname {erf}\left (\frac {{\left (2 \, f^{2} x + {\left (2 \, c^{2} x + b c\right )} \log \left (f\right )^{2} + e f + {\left (i \, c e - i \, b f\right )} \log \left (f\right )\right )} \sqrt {-c \log \left (f\right ) - i \, f}}{2 \, {\left (c^{2} \log \left (f\right )^{2} + f^{2}\right )}}\right ) e^{\left (-\frac {{\left (b^{2} c - 4 \, a c^{2}\right )} \log \left (f\right )^{3} + i \, e^{2} f - 4 i \, d f^{2} - {\left (4 i \, c^{2} d - 2 i \, b c e + i \, b^{2} f\right )} \log \left (f\right )^{2} - {\left (c e^{2} - 2 \, b e f + 4 \, a f^{2}\right )} \log \left (f\right )}{4 \, {\left (c^{2} \log \left (f\right )^{2} + f^{2}\right )}}\right )} + \sqrt {\pi } {\left (c \log \left (f\right ) + i \, f\right )} \sqrt {-c \log \left (f\right ) + i \, f} \operatorname {erf}\left (\frac {{\left (2 \, f^{2} x + {\left (2 \, c^{2} x + b c\right )} \log \left (f\right )^{2} + e f + {\left (-i \, c e + i \, b f\right )} \log \left (f\right )\right )} \sqrt {-c \log \left (f\right ) + i \, f}}{2 \, {\left (c^{2} \log \left (f\right )^{2} + f^{2}\right )}}\right ) e^{\left (-\frac {{\left (b^{2} c - 4 \, a c^{2}\right )} \log \left (f\right )^{3} - i \, e^{2} f + 4 i \, d f^{2} - {\left (-4 i \, c^{2} d + 2 i \, b c e - i \, b^{2} f\right )} \log \left (f\right )^{2} - {\left (c e^{2} - 2 \, b e f + 4 \, a f^{2}\right )} \log \left (f\right )}{4 \, {\left (c^{2} \log \left (f\right )^{2} + f^{2}\right )}}\right )}}{4 \, {\left (c^{2} \log \left (f\right )^{2} + f^{2}\right )}} \]

[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 + e*f + (I*c
*e - I*b*f)*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 + I*e^2*f
- 4*I*d*f^2 - (4*I*c^2*d - 2*I*b*c*e + I*b^2*f)*log(f)^2 - (c*e^2 - 2*b*e*f + 4*a*f^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 + e*f + (
-I*c*e + I*b*f)*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 - I*e^
2*f + 4*I*d*f^2 - (-4*I*c^2*d + 2*I*b*c*e - I*b^2*f)*log(f)^2 - (c*e^2 - 2*b*e*f + 4*a*f^2)*log(f))/(c^2*log(f
)^2 + f^2)))/(c^2*log(f)^2 + f^2)

Sympy [F]

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

[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)

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. 1008 vs. \(2 (155) = 310\).

Time = 0.25 (sec) , antiderivative size = 1008, normalized size of antiderivative = 4.85 \[ \int f^{a+b x+c x^2} \cos \left (d+e x+f x^2\right ) \, dx=\text {Too large to display} \]

[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^(1/4*c*e^2/(c^2*log(f)^2 + f^2))*f^a*cos(-1/4*(e^2*f - 4*d*f
^2 - (4*c^2*d - 2*b*c*e + b^2*f)*log(f)^2)/(c^2*log(f)^2 + f^2)) + f^(1/4*c*e^2/(c^2*log(f)^2 + f^2))*f^a*sin(
-1/4*(e^2*f - 4*d*f^2 - (4*c^2*d - 2*b*c*e + 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) - I*e)*sqrt(-c*log(f) + I*f)/(c*log(f) - I*f)) + (-I*f^(1/4*c*e^2/(c^2*log(f)^2 + f^2))*f^a*co
s(-1/4*(e^2*f - 4*d*f^2 - (4*c^2*d - 2*b*c*e + b^2*f)*log(f)^2)/(c^2*log(f)^2 + f^2)) + f^(1/4*c*e^2/(c^2*log(
f)^2 + f^2))*f^a*sin(-1/4*(e^2*f - 4*d*f^2 - (4*c^2*d - 2*b*c*e + 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) + 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^(1/4*c*e^2/(c^2*log(f)^2 + f^2))*f^a*cos(-1/4*(e^
2*f - 4*d*f^2 - (4*c^2*d - 2*b*c*e + b^2*f)*log(f)^2)/(c^2*log(f)^2 + f^2)) - I*f^(1/4*c*e^2/(c^2*log(f)^2 + f
^2))*f^a*sin(-1/4*(e^2*f - 4*d*f^2 - (4*c^2*d - 2*b*c*e + 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) - I*e)*sqrt(-c*log(f) + I*f)/(c*log(f) - I*f)) + (f^(1/4*c*e^2/(c^2*log(f)^2 + f^
2))*f^a*cos(-1/4*(e^2*f - 4*d*f^2 - (4*c^2*d - 2*b*c*e + b^2*f)*log(f)^2)/(c^2*log(f)^2 + f^2)) + I*f^(1/4*c*e
^2/(c^2*log(f)^2 + f^2))*f^a*sin(-1/4*(e^2*f - 4*d*f^2 - (4*c^2*d - 2*b*c*e + 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) + 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*e*f*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*e*f*log(f)/(c^2*log(f)^2 + f^2)))

Giac [F]

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

[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)

Mupad [F(-1)]

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

[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)