3.122 \(\int f^{a+c x^2} \cos (d+e x+f x^2) \, dx\)

Optimal. Leaf size=183 \[ \frac {\sqrt {\pi } f^a e^{-\frac {e^2}{-4 c \log (f)+4 i f}-i d} \text {erf}\left (\frac {2 x (-c \log (f)+i f)+i e}{2 \sqrt {-c \log (f)+i f}}\right )}{4 \sqrt {-c \log (f)+i f}}+\frac {\sqrt {\pi } f^a e^{\frac {e^2}{4 c \log (f)+4 i f}+i d} \text {erfi}\left (\frac {2 x (c \log (f)+i f)+i e}{2 \sqrt {c \log (f)+i f}}\right )}{4 \sqrt {c \log (f)+i f}} \]

[Out]

1/4*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*l
n(f))^(1/2)+1/4*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)

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Rubi [A]  time = 0.30, antiderivative size = 183, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {4473, 2287, 2234, 2205, 2204} \[ \frac {\sqrt {\pi } f^a e^{-\frac {e^2}{-4 c \log (f)+4 i f}-i d} \text {Erf}\left (\frac {2 x (-c \log (f)+i f)+i e}{2 \sqrt {-c \log (f)+i f}}\right )}{4 \sqrt {-c \log (f)+i f}}+\frac {\sqrt {\pi } f^a e^{\frac {e^2}{4 c \log (f)+4 i f}+i d} \text {Erfi}\left (\frac {2 x (c \log (f)+i f)+i e}{2 \sqrt {c \log (f)+i f}}\right )}{4 \sqrt {c \log (f)+i f}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(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]]
)])/(4*Sqrt[I*f - c*Log[f]]) + (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]])])/(4*Sqrt[I*f + c*Log[f]])

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]

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 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 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 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]

Rubi steps

\begin {align*} \int f^{a+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+c x^2}+\frac {1}{2} e^{i d+i e x+i f x^2} f^{a+c x^2}\right ) \, dx\\ &=\frac {1}{2} \int e^{-i d-i e x-i f x^2} f^{a+c x^2} \, dx+\frac {1}{2} \int e^{i d+i e x+i f x^2} f^{a+c x^2} \, dx\\ &=\frac {1}{2} \int \exp \left (-i d-i e x+a \log (f)-x^2 (i f-c \log (f))\right ) \, dx+\frac {1}{2} \int \exp \left (i d+i e x+a \log (f)+x^2 (i f+c \log (f))\right ) \, dx\\ &=\frac {1}{2} \left (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}{2} \left (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 {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 )}{4 \sqrt {i f-c \log (f)}}+\frac {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 )}{4 \sqrt {i f+c \log (f)}}\\ \end {align*}

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Mathematica [A]  time = 0.95, size = 217, normalized size = 1.19 \[ \frac {\sqrt [4]{-1} \sqrt {\pi } f^a e^{\frac {e^2}{4 c \log (f)+4 i f}} \left (\sqrt {f-i c \log (f)} (c \log (f)-i f) (\cos (d)+i \sin (d)) \text {erfi}\left (\frac {\sqrt [4]{-1} (-2 i c x \log (f)+e+2 f x)}{2 \sqrt {f-i c \log (f)}}\right )-(f-i c \log (f)) \sqrt {f+i c \log (f)} (\cos (d)-i \sin (d)) e^{\frac {i e^2 f}{2 \left (c^2 \log ^2(f)+f^2\right )}} \text {erfi}\left (\frac {(-1)^{3/4} (2 i c x \log (f)+e+2 f x)}{2 \sqrt {f+i c \log (f)}}\right )\right )}{4 \left (c^2 \log ^2(f)+f^2\right )} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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fricas [B]  time = 0.82, size = 301, normalized size = 1.64 \[ -\frac {\sqrt {\pi } {\left (c \log \relax (f) - i \, f\right )} \sqrt {-c \log \relax (f) - i \, f} \operatorname {erf}\left (\frac {{\left (2 \, c^{2} x \log \relax (f)^{2} + 2 \, f^{2} x + i \, c e \log \relax (f) + e f\right )} \sqrt {-c \log \relax (f) - i \, f}}{2 \, {\left (c^{2} \log \relax (f)^{2} + f^{2}\right )}}\right ) e^{\left (\frac {4 \, a c^{2} \log \relax (f)^{3} + 4 i \, c^{2} d \log \relax (f)^{2} - i \, e^{2} f + 4 i \, d f^{2} + {\left (c e^{2} + 4 \, a f^{2}\right )} \log \relax (f)}{4 \, {\left (c^{2} \log \relax (f)^{2} + f^{2}\right )}}\right )} + \sqrt {\pi } {\left (c \log \relax (f) + i \, f\right )} \sqrt {-c \log \relax (f) + i \, f} \operatorname {erf}\left (\frac {{\left (2 \, c^{2} x \log \relax (f)^{2} + 2 \, f^{2} x - i \, c e \log \relax (f) + e f\right )} \sqrt {-c \log \relax (f) + i \, f}}{2 \, {\left (c^{2} \log \relax (f)^{2} + f^{2}\right )}}\right ) e^{\left (\frac {4 \, a c^{2} \log \relax (f)^{3} - 4 i \, c^{2} d \log \relax (f)^{2} + i \, e^{2} f - 4 i \, d f^{2} + {\left (c e^{2} + 4 \, a f^{2}\right )} \log \relax (f)}{4 \, {\left (c^{2} \log \relax (f)^{2} + f^{2}\right )}}\right )}}{4 \, {\left (c^{2} \log \relax (f)^{2} + f^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(c*x^2+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*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)*(c*log(f) + I*f)*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^2*
log(f)^2 + f^2)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int f^{c x^{2} + a} \cos \left (f x^{2} + e x + d\right )\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.17, size = 167, normalized size = 0.91 \[ \frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {4 d f +4 i d \ln \relax (f ) c -e^{2}}{4 \left (-i f +c \ln \relax (f )\right )}} \erf \left (x \sqrt {i f -c \ln \relax (f )}+\frac {i e}{2 \sqrt {i f -c \ln \relax (f )}}\right )}{4 \sqrt {i f -c \ln \relax (f )}}-\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{\frac {-4 d f +4 i d \ln \relax (f ) c +e^{2}}{4 i f +4 c \ln \relax (f )}} \erf \left (-\sqrt {-i f -c \ln \relax (f )}\, x +\frac {i e}{2 \sqrt {-i f -c \ln \relax (f )}}\right )}{4 \sqrt {-i f -c \ln \relax (f )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*Pi^(1/2)*f^a*exp(-1/4*(4*d*f+4*I*d*ln(f)*c-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))-1/4*Pi^(1/2)*f^a*exp(1/4*(-4*d*f+4*I*d*ln(f)*c+e^2)/(I*f+c*ln(f)))/(-I*f-c*ln
(f))^(1/2)*erf(-(-I*f-c*ln(f))^(1/2)*x+1/2*I*e/(-I*f-c*ln(f))^(1/2))

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maxima [B]  time = 0.37, size = 761, normalized size = 4.16 \[ \frac {\sqrt {\pi } \sqrt {2 \, c^{2} \log \relax (f)^{2} + 2 \, f^{2}} {\left ({\left (i \, f^{\frac {c e^{2}}{4 \, {\left (c^{2} \log \relax (f)^{2} + f^{2}\right )}}} f^{a} \cos \left (\frac {4 \, c^{2} d \log \relax (f)^{2} - e^{2} f + 4 \, d f^{2}}{4 \, {\left (c^{2} \log \relax (f)^{2} + f^{2}\right )}}\right ) + f^{\frac {c e^{2}}{4 \, {\left (c^{2} \log \relax (f)^{2} + f^{2}\right )}}} f^{a} \sin \left (\frac {4 \, c^{2} d \log \relax (f)^{2} - e^{2} f + 4 \, d f^{2}}{4 \, {\left (c^{2} \log \relax (f)^{2} + f^{2}\right )}}\right )\right )} \operatorname {erf}\left (\frac {2 \, {\left (c \log \relax (f) - i \, f\right )} x - i \, e}{2 \, \sqrt {-c \log \relax (f) + i \, f}}\right ) + {\left (-i \, f^{\frac {c e^{2}}{4 \, {\left (c^{2} \log \relax (f)^{2} + f^{2}\right )}}} f^{a} \cos \left (\frac {4 \, c^{2} d \log \relax (f)^{2} - e^{2} f + 4 \, d f^{2}}{4 \, {\left (c^{2} \log \relax (f)^{2} + f^{2}\right )}}\right ) + f^{\frac {c e^{2}}{4 \, {\left (c^{2} \log \relax (f)^{2} + f^{2}\right )}}} f^{a} \sin \left (\frac {4 \, c^{2} d \log \relax (f)^{2} - e^{2} f + 4 \, d f^{2}}{4 \, {\left (c^{2} \log \relax (f)^{2} + f^{2}\right )}}\right )\right )} \operatorname {erf}\left (\frac {2 \, {\left (c \log \relax (f) + i \, f\right )} x + i \, e}{2 \, \sqrt {-c \log \relax (f) - i \, f}}\right )\right )} \sqrt {c \log \relax (f) + \sqrt {c^{2} \log \relax (f)^{2} + f^{2}}} - \sqrt {\pi } \sqrt {2 \, c^{2} \log \relax (f)^{2} + 2 \, f^{2}} {\left ({\left (f^{\frac {c e^{2}}{4 \, {\left (c^{2} \log \relax (f)^{2} + f^{2}\right )}}} f^{a} \cos \left (\frac {4 \, c^{2} d \log \relax (f)^{2} - e^{2} f + 4 \, d f^{2}}{4 \, {\left (c^{2} \log \relax (f)^{2} + f^{2}\right )}}\right ) - i \, f^{\frac {c e^{2}}{4 \, {\left (c^{2} \log \relax (f)^{2} + f^{2}\right )}}} f^{a} \sin \left (\frac {4 \, c^{2} d \log \relax (f)^{2} - e^{2} f + 4 \, d f^{2}}{4 \, {\left (c^{2} \log \relax (f)^{2} + f^{2}\right )}}\right )\right )} \operatorname {erf}\left (\frac {2 \, {\left (c \log \relax (f) - i \, f\right )} x - i \, e}{2 \, \sqrt {-c \log \relax (f) + i \, f}}\right ) + {\left (f^{\frac {c e^{2}}{4 \, {\left (c^{2} \log \relax (f)^{2} + f^{2}\right )}}} f^{a} \cos \left (\frac {4 \, c^{2} d \log \relax (f)^{2} - e^{2} f + 4 \, d f^{2}}{4 \, {\left (c^{2} \log \relax (f)^{2} + f^{2}\right )}}\right ) + i \, f^{\frac {c e^{2}}{4 \, {\left (c^{2} \log \relax (f)^{2} + f^{2}\right )}}} f^{a} \sin \left (\frac {4 \, c^{2} d \log \relax (f)^{2} - e^{2} f + 4 \, d f^{2}}{4 \, {\left (c^{2} \log \relax (f)^{2} + f^{2}\right )}}\right )\right )} \operatorname {erf}\left (\frac {2 \, {\left (c \log \relax (f) + i \, f\right )} x + i \, e}{2 \, \sqrt {-c \log \relax (f) - i \, f}}\right )\right )} \sqrt {-c \log \relax (f) + \sqrt {c^{2} \log \relax (f)^{2} + f^{2}}}}{8 \, {\left (c^{2} \log \relax (f)^{2} + f^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(c*x^2+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*(4*c^2*d*log(f)^
2 - e^2*f + 4*d*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*(4*c^2*d*log(f)^2
- e^2*f + 4*d*f^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*f^
(1/4*c*e^2/(c^2*log(f)^2 + f^2))*f^a*cos(1/4*(4*c^2*d*log(f)^2 - e^2*f + 4*d*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*(4*c^2*d*log(f)^2 - e^2*f + 4*d*f^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)*sqr
t(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*(4*c^2*d*log(f)^2 - e^2*f + 4*d*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*(4*c^2*d*log(f)^2 - e^2*f + 4*d*f^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)) + (f^(1/4*c*e^2/(c^2*log(
f)^2 + f^2))*f^a*cos(1/4*(4*c^2*d*log(f)^2 - e^2*f + 4*d*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*(4*c^2*d*log(f)^2 - e^2*f + 4*d*f^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^2*log(f)^2 + f^2)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int f^{c\,x^2+a}\,\cos \left (f\,x^2+e\,x+d\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int f^{a + c x^{2}} \cos {\left (d + e x + f x^{2} \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(f**(a + c*x**2)*cos(d + e*x + f*x**2), x)

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