3.2.16 \(\int f^{a+c x^2} \cos (d+e x) \, dx\) [116]

Optimal. Leaf size=147 \[ -\frac {e^{-i d+\frac {e^2}{4 c \log (f)}} f^a \sqrt {\pi } \text {Erfi}\left (\frac {i e-2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{4 \sqrt {c} \sqrt {\log (f)}}+\frac {e^{i d+\frac {e^2}{4 c \log (f)}} f^a \sqrt {\pi } \text {Erfi}\left (\frac {i e+2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{4 \sqrt {c} \sqrt {\log (f)}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.14, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4561, 2325, 2266, 2235} \begin {gather*} \frac {\sqrt {\pi } f^a e^{\frac {e^2}{4 c \log (f)}+i d} \text {Erfi}\left (\frac {2 c x \log (f)+i e}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{4 \sqrt {c} \sqrt {\log (f)}}-\frac {\sqrt {\pi } f^a e^{\frac {e^2}{4 c \log (f)}-i d} \text {Erfi}\left (\frac {-2 c x \log (f)+i e}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{4 \sqrt {c} \sqrt {\log (f)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[f^(a + c*x^2)*Cos[d + e*x],x]

[Out]

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

________________________________________________________________________________________

Mathematica [A]
time = 0.16, size = 116, normalized size = 0.79 \begin {gather*} \frac {e^{\frac {e^2}{4 c \log (f)}} f^a \sqrt {\pi } \left (\text {Erfi}\left (\frac {-i e+2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right ) (\cos (d)-i \sin (d))+\text {Erfi}\left (\frac {i e+2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right ) (\cos (d)+i \sin (d))\right )}{4 \sqrt {c} \sqrt {\log (f)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 0.16, size = 121, normalized size = 0.82

method	result	size
risch	\(\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {4 i d \ln \left (f \right ) c -e^{2}}{4 \ln \left (f \right ) c}} \erf \left (\sqrt {-c \ln \left (f \right )}\, x +\frac {i e}{2 \sqrt {-c \ln \left (f \right )}}\right )}{4 \sqrt {-c \ln \left (f \right )}}-\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{\frac {4 i d \ln \left (f \right ) c +e^{2}}{4 \ln \left (f \right ) c}} \erf \left (-\sqrt {-c \ln \left (f \right )}\, x +\frac {i e}{2 \sqrt {-c \ln \left (f \right )}}\right )}{4 \sqrt {-c \ln \left (f \right )}}\)	\(121\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.29, size = 204, normalized size = 1.39 \begin {gather*} -\frac {\sqrt {\pi } {\left (f^{a} {\left (\cos \left (d\right ) - i \, \sin \left (d\right )\right )} \operatorname {erf}\left (x \overline {\sqrt {-c \log \left (f\right )}} + \frac {1}{2} i \, \overline {\frac {1}{\sqrt {-c \log \left (f\right )}}} e\right ) e^{\left (\frac {e^{2}}{4 \, c \log \left (f\right )}\right )} + f^{a} {\left (\cos \left (d\right ) + i \, \sin \left (d\right )\right )} \operatorname {erf}\left (x \overline {\sqrt {-c \log \left (f\right )}} - \frac {1}{2} i \, \overline {\frac {1}{\sqrt {-c \log \left (f\right )}}} e\right ) e^{\left (\frac {e^{2}}{4 \, c \log \left (f\right )}\right )} - f^{a} {\left (\cos \left (d\right ) + i \, \sin \left (d\right )\right )} \operatorname {erf}\left (\frac {2 \, c x \log \left (f\right ) + i \, e}{2 \, \sqrt {-c \log \left (f\right )}}\right ) e^{\left (\frac {e^{2}}{4 \, c \log \left (f\right )}\right )} - f^{a} {\left (\cos \left (d\right ) - i \, \sin \left (d\right )\right )} \operatorname {erf}\left (\frac {2 \, c x \log \left (f\right ) - i \, e}{2 \, \sqrt {-c \log \left (f\right )}}\right ) e^{\left (\frac {e^{2}}{4 \, c \log \left (f\right )}\right )}\right )} \sqrt {-c \log \left (f\right )}}{8 \, c \log \left (f\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*sqrt(pi)*(f^a*(cos(d) - I*sin(d))*erf(x*conjugate(sqrt(-c*log(f))) + 1/2*I*conjugate(1/sqrt(-c*log(f)))*e
)*e^(1/4*e^2/(c*log(f))) + f^a*(cos(d) + I*sin(d))*erf(x*conjugate(sqrt(-c*log(f))) - 1/2*I*conjugate(1/sqrt(-
c*log(f)))*e)*e^(1/4*e^2/(c*log(f))) - f^a*(cos(d) + I*sin(d))*erf(1/2*(2*c*x*log(f) + I*e)/sqrt(-c*log(f)))*e
^(1/4*e^2/(c*log(f))) - f^a*(cos(d) - I*sin(d))*erf(1/2*(2*c*x*log(f) - I*e)/sqrt(-c*log(f)))*e^(1/4*e^2/(c*lo
g(f))))*sqrt(-c*log(f))/(c*log(f))

________________________________________________________________________________________

Fricas [A]
time = 1.85, size = 142, normalized size = 0.97 \begin {gather*} -\frac {\sqrt {\pi } \sqrt {-c \log \left (f\right )} \operatorname {erf}\left (\frac {{\left (2 \, c x \log \left (f\right ) + i \, e\right )} \sqrt {-c \log \left (f\right )}}{2 \, c \log \left (f\right )}\right ) e^{\left (\frac {4 \, a c \log \left (f\right )^{2} + 4 i \, c d \log \left (f\right ) + e^{2}}{4 \, c \log \left (f\right )}\right )} + \sqrt {\pi } \sqrt {-c \log \left (f\right )} \operatorname {erf}\left (\frac {{\left (2 \, c x \log \left (f\right ) - i \, e\right )} \sqrt {-c \log \left (f\right )}}{2 \, c \log \left (f\right )}\right ) e^{\left (\frac {4 \, a c \log \left (f\right )^{2} - 4 i \, c d \log \left (f\right ) + e^{2}}{4 \, c \log \left (f\right )}\right )}}{4 \, c \log \left (f\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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(e*x+d),x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

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

[Out]

int(f^(a + c*x^2)*cos(d + e*x), x)

________________________________________________________________________________________