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

Optimal. Leaf size=189 \[ \frac{\sqrt{\pi } f^a e^{i d-\frac{b^2 \log ^2(f)}{4 c \log (f)+4 i f}} \text{Erfi}\left (\frac{b \log (f)+2 x (c \log (f)+i f)}{2 \sqrt{c \log (f)+i f}}\right )}{4 \sqrt{c \log (f)+i f}}-\frac{\sqrt{\pi } f^a e^{\frac{b^2 \log ^2(f)}{-4 c \log (f)+4 i f}-i d} \text{Erf}\left (\frac{b \log (f)-2 x (-c \log (f)+i f)}{2 \sqrt{-c \log (f)+i f}}\right )}{4 \sqrt{-c \log (f)+i f}} \]

[Out]

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

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Rubi [A]  time = 0.265494, antiderivative size = 189, normalized size of antiderivative = 1., 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} \[ \frac{\sqrt{\pi } f^a e^{i d-\frac{b^2 \log ^2(f)}{4 c \log (f)+4 i f}} \text{Erfi}\left (\frac{b \log (f)+2 x (c \log (f)+i f)}{2 \sqrt{c \log (f)+i f}}\right )}{4 \sqrt{c \log (f)+i f}}-\frac{\sqrt{\pi } f^a e^{\frac{b^2 \log ^2(f)}{-4 c \log (f)+4 i f}-i d} \text{Erf}\left (\frac{b \log (f)-2 x (-c \log (f)+i f)}{2 \sqrt{-c \log (f)+i f}}\right )}{4 \sqrt{-c \log (f)+i f}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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+b x+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+b x+c x^2}+\frac{1}{2} e^{i d+i f x^2} f^{a+b x+c x^2}\right ) \, dx\\ &=\frac{1}{2} \int e^{-i d-i f x^2} f^{a+b x+c x^2} \, dx+\frac{1}{2} \int e^{i d+i f x^2} f^{a+b x+c x^2} \, dx\\ &=\frac{1}{2} \int \exp \left (-i d+a \log (f)+b x \log (f)-x^2 (i f-c \log (f))\right ) \, dx+\frac{1}{2} \int \exp \left (i d+a \log (f)+b x \log (f)+x^2 (i f+c \log (f))\right ) \, dx\\ &=\frac{1}{2} \left (e^{-i d+\frac{b^2 \log ^2(f)}{4 i f-4 c \log (f)}} f^a\right ) \int \exp \left (\frac{(b \log (f)+2 x (-i f+c \log (f)))^2}{4 (-i f+c \log (f))}\right ) \, dx+\frac{1}{2} \left (e^{i d-\frac{b^2 \log ^2(f)}{4 i f+4 c \log (f)}} f^a\right ) \int \exp \left (\frac{(b \log (f)+2 x (i f+c \log (f)))^2}{4 (i f+c \log (f))}\right ) \, dx\\ &=-\frac{e^{-i d+\frac{b^2 \log ^2(f)}{4 i f-4 c \log (f)}} f^a \sqrt{\pi } \text{erf}\left (\frac{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{b^2 \log ^2(f)}{4 i f+4 c \log (f)}} f^a \sqrt{\pi } \text{erfi}\left (\frac{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]  time = 0.982861, size = 231, normalized size = 1.22 \[ -\frac{(-1)^{3/4} \sqrt{\pi } f^a e^{\frac{b^2 \log ^2(f)}{-4 c \log (f)+4 i f}} \left (\sqrt{f-i c \log (f)} (f+i c \log (f)) (\cos (d)+i \sin (d)) e^{\frac{i b^2 f \log ^2(f)}{2 \left (c^2 \log ^2(f)+f^2\right )}} \text{Erfi}\left (\frac{\sqrt [4]{-1} (2 f x-i \log (f) (b+2 c x))}{2 \sqrt{f-i c \log (f)}}\right )+(f-i c \log (f)) \sqrt{f+i c \log (f)} (-\sin (d)-i \cos (d)) \text{Erfi}\left (\frac{(-1)^{3/4} (2 f x+i \log (f) (b+2 c x))}{2 \sqrt{f+i c \log (f)}}\right )\right )}{4 \left (c^2 \log ^2(f)+f^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.089, size = 178, normalized size = 0.9 \begin{align*} -{\frac{{f}^{a}\sqrt{\pi }}{4}{{\rm e}^{-{\frac{ \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}+4\,id\ln \left ( f \right ) c+4\,df}{4\,c\ln \left ( f \right ) -4\,if}}}}{\it Erf} \left ( -x\sqrt{if-c\ln \left ( f \right ) }+{\frac{b\ln \left ( f \right ) }{2}{\frac{1}{\sqrt{if-c\ln \left ( f \right ) }}}} \right ){\frac{1}{\sqrt{if-c\ln \left ( f \right ) }}}}-{\frac{{f}^{a}\sqrt{\pi }}{4}{{\rm e}^{-{\frac{ \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}-4\,id\ln \left ( f \right ) c+4\,df}{4\,if+4\,c\ln \left ( f \right ) }}}}{\it Erf} \left ( -\sqrt{-c\ln \left ( f \right ) -if}x+{\frac{b\ln \left ( f \right ) }{2}{\frac{1}{\sqrt{-c\ln \left ( f \right ) -if}}}} \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+b*x+a)*cos(f*x^2+d),x)

[Out]

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

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

[Out]

Exception raised: IndexError

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Fricas [B]  time = 0.547952, size = 778, normalized size = 4.12 \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 (\frac{{\left (2 \, f^{2} x - i \, b f \log \left (f\right ) +{\left (2 \, c^{2} x + b c\right )} \log \left (f\right )^{2}\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{4 \, a f^{2} \log \left (f\right ) -{\left (b^{2} c - 4 \, a c^{2}\right )} \log \left (f\right )^{3} + 4 i \, d f^{2} +{\left (4 i \, c^{2} d + i \, b^{2} f\right )} \log \left (f\right )^{2}}{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 + i \, b f \log \left (f\right ) +{\left (2 \, c^{2} x + b c\right )} \log \left (f\right )^{2}\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{4 \, a f^{2} \log \left (f\right ) -{\left (b^{2} c - 4 \, a c^{2}\right )} \log \left (f\right )^{3} - 4 i \, d f^{2} +{\left (-4 i \, c^{2} d - i \, b^{2} f\right )} \log \left (f\right )^{2}}{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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(c*x^2+b*x+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(1/2*(2*f^2*x - I*b*f*log(f) + (2*c^2*x + b*c)*log(f)
^2)*sqrt(-c*log(f) - I*f)/(c^2*log(f)^2 + f^2))*e^(1/4*(4*a*f^2*log(f) - (b^2*c - 4*a*c^2)*log(f)^3 + 4*I*d*f^
2 + (4*I*c^2*d + I*b^2*f)*log(f)^2)/(c^2*log(f)^2 + f^2)) + sqrt(pi)*(c*log(f) + I*f)*sqrt(-c*log(f) + I*f)*er
f(1/2*(2*f^2*x + I*b*f*log(f) + (2*c^2*x + b*c)*log(f)^2)*sqrt(-c*log(f) + I*f)/(c^2*log(f)^2 + f^2))*e^(1/4*(
4*a*f^2*log(f) - (b^2*c - 4*a*c^2)*log(f)^3 - 4*I*d*f^2 + (-4*I*c^2*d - I*b^2*f)*log(f)^2)/(c^2*log(f)^2 + f^2
)))/(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 + b x + 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+b*x+a)*cos(f*x**2+d),x)

[Out]

Integral(f**(a + b*x + 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} + b x + 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+b*x+a)*cos(f*x^2+d),x, algorithm="giac")

[Out]

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