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

Optimal. Leaf size=103 \[ \frac{\sqrt{\pi } e^{-i d} f^a \text{Erf}\left (x \sqrt{-c \log (f)+i f}\right )}{4 \sqrt{-c \log (f)+i f}}+\frac{\sqrt{\pi } e^{i d} f^a \text{Erfi}\left (x \sqrt{c \log (f)+i f}\right )}{4 \sqrt{c \log (f)+i f}} \]

[Out]

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

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Rubi [A]  time = 0.154279, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {4473, 2287, 2205, 2204} \[ \frac{\sqrt{\pi } e^{-i d} f^a \text{Erf}\left (x \sqrt{-c \log (f)+i f}\right )}{4 \sqrt{-c \log (f)+i f}}+\frac{\sqrt{\pi } e^{i d} f^a \text{Erfi}\left (x \sqrt{c \log (f)+i f}\right )}{4 \sqrt{c \log (f)+i f}} \]

Antiderivative was successfully verified.

[In]

Int[f^(a + c*x^2)*Cos[d + f*x^2],x]

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.476822, size = 170, normalized size = 1.65 \[ -\frac{(-1)^{3/4} \sqrt{\pi } f^a \left (\sqrt{f+i c \log (f)} \left (f \cos (d) \text{Erf}\left (\frac{(1+i) x \sqrt{f+i c \log (f)}}{\sqrt{2}}\right )-\text{Erfi}\left ((-1)^{3/4} x \sqrt{f+i c \log (f)}\right ) (c \cos (d) \log (f)+\sin (d) (f-i c \log (f)))\right )+\sqrt{f-i c \log (f)} (f+i c \log (f)) (\cos (d)+i \sin (d)) \text{Erfi}\left (\sqrt [4]{-1} x \sqrt{f-i c \log (f)}\right )\right )}{4 \left (c^2 \log ^2(f)+f^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.063, size = 82, normalized size = 0.8 \begin{align*}{\frac{{f}^{a}\sqrt{\pi }{{\rm e}^{-id}}}{4}{\it Erf} \left ( x\sqrt{if-c\ln \left ( f \right ) } \right ){\frac{1}{\sqrt{if-c\ln \left ( f \right ) }}}}+{\frac{{f}^{a}\sqrt{\pi }{{\rm e}^{id}}}{4}{\it Erf} \left ( \sqrt{-c\ln \left ( f \right ) -if}x \right ){\frac{1}{\sqrt{-c\ln \left ( f \right ) -if}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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+d),x, algorithm="maxima")

[Out]

Exception raised: IndexError

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Fricas [A]  time = 0.504205, size = 301, normalized size = 2.92 \begin{align*} -\frac{\sqrt{\pi }{\left (c \log \left (f\right ) - i \, f\right )} \sqrt{-c \log \left (f\right ) - i \, f} \operatorname{erf}\left (\sqrt{-c \log \left (f\right ) - i \, f} x\right ) e^{\left (a \log \left (f\right ) + i \, d\right )} + \sqrt{\pi }{\left (c \log \left (f\right ) + i \, f\right )} \sqrt{-c \log \left (f\right ) + i \, f} \operatorname{erf}\left (\sqrt{-c \log \left (f\right ) + i \, f} x\right ) e^{\left (a \log \left (f\right ) - i \, d\right )}}{4 \,{\left (c^{2} \log \left (f\right )^{2} + f^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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

[Out]

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