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

Optimal. Leaf size=211 \[ \frac{\sqrt{\pi } f^a e^{-\frac{e^2}{-c \log (f)+2 i f}-2 i d} \text{Erf}\left (\frac{x (-c \log (f)+2 i f)+i e}{\sqrt{-c \log (f)+2 i f}}\right )}{8 \sqrt{-c \log (f)+2 i f}}+\frac{\sqrt{\pi } f^a e^{\frac{e^2}{c \log (f)+2 i f}+2 i d} \text{Erfi}\left (\frac{x (c \log (f)+2 i f)+i e}{\sqrt{c \log (f)+2 i f}}\right )}{8 \sqrt{c \log (f)+2 i f}}+\frac{\sqrt{\pi } f^a \text{Erfi}\left (\sqrt{c} x \sqrt{\log (f)}\right )}{4 \sqrt{c} \sqrt{\log (f)}} \]

[Out]

(f^a*Sqrt[Pi]*Erfi[Sqrt[c]*x*Sqrt[Log[f]]])/(4*Sqrt[c]*Sqrt[Log[f]]) + (E^((-2*I)*d - e^2/((2*I)*f - c*Log[f])
)*f^a*Sqrt[Pi]*Erf[(I*e + x*((2*I)*f - c*Log[f]))/Sqrt[(2*I)*f - c*Log[f]]])/(8*Sqrt[(2*I)*f - c*Log[f]]) + (E
^((2*I)*d + e^2/((2*I)*f + c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(I*e + x*((2*I)*f + c*Log[f]))/Sqrt[(2*I)*f + c*Log[f]
]])/(8*Sqrt[(2*I)*f + c*Log[f]])

________________________________________________________________________________________

Rubi [A]  time = 0.3583, antiderivative size = 211, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {4473, 2204, 2287, 2234, 2205} \[ \frac{\sqrt{\pi } f^a e^{-\frac{e^2}{-c \log (f)+2 i f}-2 i d} \text{Erf}\left (\frac{x (-c \log (f)+2 i f)+i e}{\sqrt{-c \log (f)+2 i f}}\right )}{8 \sqrt{-c \log (f)+2 i f}}+\frac{\sqrt{\pi } f^a e^{\frac{e^2}{c \log (f)+2 i f}+2 i d} \text{Erfi}\left (\frac{x (c \log (f)+2 i f)+i e}{\sqrt{c \log (f)+2 i f}}\right )}{8 \sqrt{c \log (f)+2 i f}}+\frac{\sqrt{\pi } f^a \text{Erfi}\left (\sqrt{c} x \sqrt{\log (f)}\right )}{4 \sqrt{c} \sqrt{\log (f)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(f^a*Sqrt[Pi]*Erfi[Sqrt[c]*x*Sqrt[Log[f]]])/(4*Sqrt[c]*Sqrt[Log[f]]) + (E^((-2*I)*d - e^2/((2*I)*f - c*Log[f])
)*f^a*Sqrt[Pi]*Erf[(I*e + x*((2*I)*f - c*Log[f]))/Sqrt[(2*I)*f - c*Log[f]]])/(8*Sqrt[(2*I)*f - c*Log[f]]) + (E
^((2*I)*d + e^2/((2*I)*f + c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(I*e + x*((2*I)*f + c*Log[f]))/Sqrt[(2*I)*f + c*Log[f]
]])/(8*Sqrt[(2*I)*f + c*Log[f]])

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]

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 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 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 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]

Rubi steps

\begin{align*} \int f^{a+c x^2} \cos ^2\left (d+e x+f x^2\right ) \, dx &=\int \left (\frac{1}{2} f^{a+c x^2}+\frac{1}{4} e^{-2 i d-2 i e x-2 i f x^2} f^{a+c x^2}+\frac{1}{4} e^{2 i d+2 i e x+2 i f x^2} f^{a+c x^2}\right ) \, dx\\ &=\frac{1}{4} \int e^{-2 i d-2 i e x-2 i f x^2} f^{a+c x^2} \, dx+\frac{1}{4} \int e^{2 i d+2 i e x+2 i f x^2} f^{a+c x^2} \, dx+\frac{1}{2} \int f^{a+c x^2} \, dx\\ &=\frac{f^a \sqrt{\pi } \text{erfi}\left (\sqrt{c} x \sqrt{\log (f)}\right )}{4 \sqrt{c} \sqrt{\log (f)}}+\frac{1}{4} \int \exp \left (-2 i d-2 i e x+a \log (f)-x^2 (2 i f-c \log (f))\right ) \, dx+\frac{1}{4} \int \exp \left (2 i d+2 i e x+a \log (f)+x^2 (2 i f+c \log (f))\right ) \, dx\\ &=\frac{f^a \sqrt{\pi } \text{erfi}\left (\sqrt{c} x \sqrt{\log (f)}\right )}{4 \sqrt{c} \sqrt{\log (f)}}+\frac{1}{4} \left (e^{-2 i d-\frac{e^2}{2 i f-c \log (f)}} f^a\right ) \int \exp \left (\frac{(-2 i e+2 x (-2 i f+c \log (f)))^2}{4 (-2 i f+c \log (f))}\right ) \, dx+\frac{1}{4} \left (e^{2 i d+\frac{e^2}{2 i f+c \log (f)}} f^a\right ) \int \exp \left (\frac{(2 i e+2 x (2 i f+c \log (f)))^2}{4 (2 i f+c \log (f))}\right ) \, dx\\ &=\frac{f^a \sqrt{\pi } \text{erfi}\left (\sqrt{c} x \sqrt{\log (f)}\right )}{4 \sqrt{c} \sqrt{\log (f)}}+\frac{e^{-2 i d-\frac{e^2}{2 i f-c \log (f)}} f^a \sqrt{\pi } \text{erf}\left (\frac{i e+x (2 i f-c \log (f))}{\sqrt{2 i f-c \log (f)}}\right )}{8 \sqrt{2 i f-c \log (f)}}+\frac{e^{2 i d+\frac{e^2}{2 i f+c \log (f)}} f^a \sqrt{\pi } \text{erfi}\left (\frac{i e+x (2 i f+c \log (f))}{\sqrt{2 i f+c \log (f)}}\right )}{8 \sqrt{2 i f+c \log (f)}}\\ \end{align*}

Mathematica [A]  time = 2.29754, size = 252, normalized size = 1.19 \[ \frac{1}{8} \sqrt{\pi } f^a \left (\frac{2 \text{Erfi}\left (\sqrt{c} x \sqrt{\log (f)}\right )}{\sqrt{c} \sqrt{\log (f)}}+\frac{\sqrt [4]{-1} \left (\sqrt{2 f-i c \log (f)} (2 f+i c \log (f)) (\sin (2 d)-i \cos (2 d)) e^{\frac{e^2}{c \log (f)+2 i f}} \text{Erfi}\left (\frac{\sqrt [4]{-1} (-i c x \log (f)+e+2 f x)}{\sqrt{2 f-i c \log (f)}}\right )-(2 f-i c \log (f)) \sqrt{2 f+i c \log (f)} (\cos (2 d)-i \sin (2 d)) e^{\frac{e^2}{c \log (f)-2 i f}} \text{Erfi}\left (\frac{(-1)^{3/4} (i c x \log (f)+e+2 f x)}{\sqrt{2 f+i c \log (f)}}\right )\right )}{c^2 \log ^2(f)+4 f^2}\right ) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(f^a*Sqrt[Pi]*((2*Erfi[Sqrt[c]*x*Sqrt[Log[f]]])/(Sqrt[c]*Sqrt[Log[f]]) + ((-1)^(1/4)*(-(E^(e^2/((-2*I)*f + c*L
og[f]))*Erfi[((-1)^(3/4)*(e + 2*f*x + I*c*x*Log[f]))/Sqrt[2*f + I*c*Log[f]]]*(2*f - I*c*Log[f])*Sqrt[2*f + I*c
*Log[f]]*(Cos[2*d] - I*Sin[2*d])) + E^(e^2/((2*I)*f + c*Log[f]))*Erfi[((-1)^(1/4)*(e + 2*f*x - I*c*x*Log[f]))/
Sqrt[2*f - I*c*Log[f]]]*Sqrt[2*f - I*c*Log[f]]*(2*f + I*c*Log[f])*((-I)*Cos[2*d] + Sin[2*d])))/(4*f^2 + c^2*Lo
g[f]^2)))/8

________________________________________________________________________________________

Maple [A]  time = 0.138, size = 191, normalized size = 0.9 \begin{align*}{\frac{{f}^{a}\sqrt{\pi }}{8}{{\rm e}^{-{\frac{2\,id\ln \left ( f \right ) c+4\,df-{e}^{2}}{-2\,if+c\ln \left ( f \right ) }}}}{\it Erf} \left ( x\sqrt{2\,if-c\ln \left ( f \right ) }+{ie{\frac{1}{\sqrt{2\,if-c\ln \left ( f \right ) }}}} \right ){\frac{1}{\sqrt{2\,if-c\ln \left ( f \right ) }}}}-{\frac{{f}^{a}\sqrt{\pi }}{8}{{\rm e}^{{\frac{2\,id\ln \left ( f \right ) c-4\,df+{e}^{2}}{2\,if+c\ln \left ( f \right ) }}}}{\it Erf} \left ( -\sqrt{-c\ln \left ( f \right ) -2\,if}x+{ie{\frac{1}{\sqrt{-c\ln \left ( f \right ) -2\,if}}}} \right ){\frac{1}{\sqrt{-c\ln \left ( f \right ) -2\,if}}}}+{\frac{{f}^{a}\sqrt{\pi }}{4}{\it Erf} \left ( \sqrt{-c\ln \left ( f \right ) }x \right ){\frac{1}{\sqrt{-c\ln \left ( f \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*Pi^(1/2)*f^a*exp(-(2*I*d*ln(f)*c+4*d*f-e^2)/(-2*I*f+c*ln(f)))/(2*I*f-c*ln(f))^(1/2)*erf(x*(2*I*f-c*ln(f))^
(1/2)+I*e/(2*I*f-c*ln(f))^(1/2))-1/8*Pi^(1/2)*f^a*exp((2*I*d*ln(f)*c-4*d*f+e^2)/(2*I*f+c*ln(f)))/(-c*ln(f)-2*I
*f)^(1/2)*erf(-(-c*ln(f)-2*I*f)^(1/2)*x+I*e/(-c*ln(f)-2*I*f)^(1/2))+1/4*f^a*Pi^(1/2)/(-c*ln(f))^(1/2)*erf((-c*
ln(f))^(1/2)*x)

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: IndexError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: IndexError

________________________________________________________________________________________

Fricas [B]  time = 0.560042, size = 927, normalized size = 4.39 \begin{align*} -\frac{2 \, \sqrt{\pi }{\left (c^{2} \log \left (f\right )^{2} + 4 \, f^{2}\right )} \sqrt{-c \log \left (f\right )} f^{a} \operatorname{erf}\left (\sqrt{-c \log \left (f\right )} x\right ) + \sqrt{\pi }{\left (c^{2} \log \left (f\right )^{2} - 2 i \, c f \log \left (f\right )\right )} \sqrt{-c \log \left (f\right ) - 2 i \, f} \operatorname{erf}\left (\frac{{\left (c^{2} x \log \left (f\right )^{2} + 4 \, f^{2} x + i \, c e \log \left (f\right ) + 2 \, e f\right )} \sqrt{-c \log \left (f\right ) - 2 i \, f}}{c^{2} \log \left (f\right )^{2} + 4 \, f^{2}}\right ) e^{\left (\frac{a c^{2} \log \left (f\right )^{3} + 2 i \, c^{2} d \log \left (f\right )^{2} - 2 i \, e^{2} f + 8 i \, d f^{2} +{\left (c e^{2} + 4 \, a f^{2}\right )} \log \left (f\right )}{c^{2} \log \left (f\right )^{2} + 4 \, f^{2}}\right )} + \sqrt{\pi }{\left (c^{2} \log \left (f\right )^{2} + 2 i \, c f \log \left (f\right )\right )} \sqrt{-c \log \left (f\right ) + 2 i \, f} \operatorname{erf}\left (\frac{{\left (c^{2} x \log \left (f\right )^{2} + 4 \, f^{2} x - i \, c e \log \left (f\right ) + 2 \, e f\right )} \sqrt{-c \log \left (f\right ) + 2 i \, f}}{c^{2} \log \left (f\right )^{2} + 4 \, f^{2}}\right ) e^{\left (\frac{a c^{2} \log \left (f\right )^{3} - 2 i \, c^{2} d \log \left (f\right )^{2} + 2 i \, e^{2} f - 8 i \, d f^{2} +{\left (c e^{2} + 4 \, a f^{2}\right )} \log \left (f\right )}{c^{2} \log \left (f\right )^{2} + 4 \, f^{2}}\right )}}{8 \,{\left (c^{3} \log \left (f\right )^{3} + 4 \, c f^{2} \log \left (f\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(2*sqrt(pi)*(c^2*log(f)^2 + 4*f^2)*sqrt(-c*log(f))*f^a*erf(sqrt(-c*log(f))*x) + sqrt(pi)*(c^2*log(f)^2 -
2*I*c*f*log(f))*sqrt(-c*log(f) - 2*I*f)*erf((c^2*x*log(f)^2 + 4*f^2*x + I*c*e*log(f) + 2*e*f)*sqrt(-c*log(f) -
2*I*f)/(c^2*log(f)^2 + 4*f^2))*e^((a*c^2*log(f)^3 + 2*I*c^2*d*log(f)^2 - 2*I*e^2*f + 8*I*d*f^2 + (c*e^2 + 4*a
*f^2)*log(f))/(c^2*log(f)^2 + 4*f^2)) + sqrt(pi)*(c^2*log(f)^2 + 2*I*c*f*log(f))*sqrt(-c*log(f) + 2*I*f)*erf((
c^2*x*log(f)^2 + 4*f^2*x - I*c*e*log(f) + 2*e*f)*sqrt(-c*log(f) + 2*I*f)/(c^2*log(f)^2 + 4*f^2))*e^((a*c^2*log
(f)^3 - 2*I*c^2*d*log(f)^2 + 2*I*e^2*f - 8*I*d*f^2 + (c*e^2 + 4*a*f^2)*log(f))/(c^2*log(f)^2 + 4*f^2)))/(c^3*l
og(f)^3 + 4*c*f^2*log(f))

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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