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

Optimal. Leaf size=231 \[ \frac{\sqrt{\pi } f^{a-\frac{b^2}{4 c}} \text{Erfi}\left (\frac{\sqrt{\log (f)} (b+2 c x)}{2 \sqrt{c}}\right )}{4 \sqrt{c} \sqrt{\log (f)}}-\frac{\sqrt{\pi } f^a e^{\frac{(2 e+i b \log (f))^2}{4 c \log (f)}-2 i d} \text{Erfi}\left (\frac{-b \log (f)-2 c x \log (f)+2 i e}{2 \sqrt{c} \sqrt{\log (f)}}\right )}{8 \sqrt{c} \sqrt{\log (f)}}+\frac{\sqrt{\pi } f^a e^{2 i d-\frac{(b \log (f)+2 i e)^2}{4 c \log (f)}} \text{Erfi}\left (\frac{b \log (f)+2 c x \log (f)+2 i e}{2 \sqrt{c} \sqrt{\log (f)}}\right )}{8 \sqrt{c} \sqrt{\log (f)}} \]

[Out]

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

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Rubi [A]  time = 0.281563, antiderivative size = 231, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {4473, 2234, 2204, 2287} \[ \frac{\sqrt{\pi } f^{a-\frac{b^2}{4 c}} \text{Erfi}\left (\frac{\sqrt{\log (f)} (b+2 c x)}{2 \sqrt{c}}\right )}{4 \sqrt{c} \sqrt{\log (f)}}-\frac{\sqrt{\pi } f^a e^{\frac{(2 e+i b \log (f))^2}{4 c \log (f)}-2 i d} \text{Erfi}\left (\frac{-b \log (f)-2 c x \log (f)+2 i e}{2 \sqrt{c} \sqrt{\log (f)}}\right )}{8 \sqrt{c} \sqrt{\log (f)}}+\frac{\sqrt{\pi } f^a e^{2 i d-\frac{(b \log (f)+2 i e)^2}{4 c \log (f)}} \text{Erfi}\left (\frac{b \log (f)+2 c x \log (f)+2 i e}{2 \sqrt{c} \sqrt{\log (f)}}\right )}{8 \sqrt{c} \sqrt{\log (f)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.638655, size = 204, normalized size = 0.88 \[ \frac{\sqrt{\pi } e^{-\frac{i b e}{c}} f^{a-\frac{b^2}{4 c}} \left ((\cos (2 d)+i \sin (2 d)) e^{\frac{e^2}{c \log (f)}} \text{Erfi}\left (\frac{\log (f) (b+2 c x)+2 i e}{2 \sqrt{c} \sqrt{\log (f)}}\right )+(\cos (2 d)-i \sin (2 d)) e^{\frac{e (e+2 i b \log (f))}{c \log (f)}} \text{Erfi}\left (\frac{\log (f) (b+2 c x)-2 i e}{2 \sqrt{c} \sqrt{\log (f)}}\right )+2 e^{\frac{i b e}{c}} \text{Erfi}\left (\frac{\sqrt{\log (f)} (b+2 c x)}{2 \sqrt{c}}\right )\right )}{8 \sqrt{c} \sqrt{\log (f)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.106, size = 217, normalized size = 0.9 \begin{align*} -{\frac{{f}^{a}\sqrt{\pi }}{8}{{\rm e}^{-{\frac{ \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}-4\,i\ln \left ( f \right ) be+8\,id\ln \left ( f \right ) c-4\,{e}^{2}}{4\,c\ln \left ( f \right ) }}}}{\it Erf} \left ( -\sqrt{-c\ln \left ( f \right ) }x+{\frac{b\ln \left ( f \right ) -2\,ie}{2}{\frac{1}{\sqrt{-c\ln \left ( f \right ) }}}} \right ){\frac{1}{\sqrt{-c\ln \left ( f \right ) }}}}-{\frac{{f}^{a}\sqrt{\pi }}{8}{{\rm e}^{-{\frac{ \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}+4\,i\ln \left ( f \right ) be-8\,id\ln \left ( f \right ) c-4\,{e}^{2}}{4\,c\ln \left ( f \right ) }}}}{\it Erf} \left ( -\sqrt{-c\ln \left ( f \right ) }x+{\frac{2\,ie+b\ln \left ( f \right ) }{2}{\frac{1}{\sqrt{-c\ln \left ( f \right ) }}}} \right ){\frac{1}{\sqrt{-c\ln \left ( f \right ) }}}}-{\frac{{f}^{a}\sqrt{\pi }}{4}{f}^{-{\frac{{b}^{2}}{4\,c}}}{\it Erf} \left ( -\sqrt{-c\ln \left ( f \right ) }x+{\frac{b\ln \left ( f \right ) }{2}{\frac{1}{\sqrt{-c\ln \left ( f \right ) }}}} \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+b*x+a)*cos(e*x+d)^2,x)

[Out]

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

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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(e*x+d)^2,x, algorithm="maxima")

[Out]

Exception raised: IndexError

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Fricas [A]  time = 0.507513, size = 618, normalized size = 2.68 \begin{align*} -\frac{\sqrt{\pi } \sqrt{-c \log \left (f\right )} \operatorname{erf}\left (\frac{{\left ({\left (2 \, c x + b\right )} \log \left (f\right ) + 2 i \, e\right )} \sqrt{-c \log \left (f\right )}}{2 \, c \log \left (f\right )}\right ) e^{\left (-\frac{{\left (b^{2} - 4 \, a c\right )} \log \left (f\right )^{2} - 4 \, e^{2} -{\left (8 i \, c d - 4 i \, b e\right )} \log \left (f\right )}{4 \, c \log \left (f\right )}\right )} + \sqrt{\pi } \sqrt{-c \log \left (f\right )} \operatorname{erf}\left (\frac{{\left ({\left (2 \, c x + b\right )} \log \left (f\right ) - 2 i \, e\right )} \sqrt{-c \log \left (f\right )}}{2 \, c \log \left (f\right )}\right ) e^{\left (-\frac{{\left (b^{2} - 4 \, a c\right )} \log \left (f\right )^{2} - 4 \, e^{2} -{\left (-8 i \, c d + 4 i \, b e\right )} \log \left (f\right )}{4 \, c \log \left (f\right )}\right )} + \frac{2 \, \sqrt{\pi } \sqrt{-c \log \left (f\right )} \operatorname{erf}\left (\frac{{\left (2 \, c x + b\right )} \sqrt{-c \log \left (f\right )}}{2 \, c}\right )}{f^{\frac{b^{2} - 4 \, a c}{4 \, c}}}}{8 \, c \log \left (f\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

[Out]

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