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

Optimal. Leaf size=268 \[ \frac{\sqrt{\pi } f^{a-\frac{b^2}{4 c}} \text{Erfi}\left (\frac{\sqrt{\log (f)} (b+2 c x)}{2 \sqrt{c}}\right )}{4 \sqrt{c} \sqrt{\log (f)}}+\frac{\sqrt{\pi } f^a \exp \left (-\frac{(2 e+i b \log (f))^2}{-4 c \log (f)+8 i f}-2 i d\right ) \text{Erf}\left (\frac{-b \log (f)+2 x (-c \log (f)+2 i f)+2 i e}{2 \sqrt{-c \log (f)+2 i f}}\right )}{8 \sqrt{-c \log (f)+2 i f}}+\frac{\sqrt{\pi } f^a \exp \left (\frac{(2 e-i b \log (f))^2}{4 c \log (f)+8 i f}+2 i d\right ) \text{Erfi}\left (\frac{b \log (f)+2 x (c \log (f)+2 i f)+2 i e}{2 \sqrt{c \log (f)+2 i f}}\right )}{8 \sqrt{c \log (f)+2 i f}} \]

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

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Rubi [A]  time = 0.460815, antiderivative size = 268, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {4473, 2234, 2204, 2287, 2205} \[ \frac{\sqrt{\pi } f^{a-\frac{b^2}{4 c}} \text{Erfi}\left (\frac{\sqrt{\log (f)} (b+2 c x)}{2 \sqrt{c}}\right )}{4 \sqrt{c} \sqrt{\log (f)}}+\frac{\sqrt{\pi } f^a \exp \left (-\frac{(2 e+i b \log (f))^2}{-4 c \log (f)+8 i f}-2 i d\right ) \text{Erf}\left (\frac{-b \log (f)+2 x (-c \log (f)+2 i f)+2 i e}{2 \sqrt{-c \log (f)+2 i f}}\right )}{8 \sqrt{-c \log (f)+2 i f}}+\frac{\sqrt{\pi } f^a \exp \left (\frac{(2 e-i b \log (f))^2}{4 c \log (f)+8 i f}+2 i d\right ) \text{Erfi}\left (\frac{b \log (f)+2 x (c \log (f)+2 i f)+2 i e}{2 \sqrt{c \log (f)+2 i f}}\right )}{8 \sqrt{c \log (f)+2 i f}} \]

Antiderivative was successfully verified.

[In]

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

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

Mathematica [B]  time = 6.73518, size = 1118, normalized size = 4.17 \[ \frac{f^a \sqrt{\pi } \left (8 \sqrt{c} \text{Erfi}\left (\frac{(b+2 c x) \sqrt{\log (f)}}{2 \sqrt{c}}\right ) \sqrt{\log (f)} f^{2-\frac{b^2}{4 c}}+2 c^{5/2} \text{Erfi}\left (\frac{(b+2 c x) \sqrt{\log (f)}}{2 \sqrt{c}}\right ) \log ^{\frac{5}{2}}(f) f^{-\frac{b^2}{4 c}}+2 \sqrt [4]{-1} c e^{\frac{i \left (-4 e^2+4 i b \log (f) e+b^2 \log ^2(f)\right )}{4 (2 f-i c \log (f))}} \text{Erfi}\left (\frac{\sqrt [4]{-1} (2 e+4 f x-i b \log (f)-2 i c x \log (f))}{2 \sqrt{2 f-i c \log (f)}}\right ) \log (f) \sqrt{2 f-i c \log (f)} \sin (2 d) f+2 (-1)^{3/4} c e^{-\frac{i \left (-4 e^2-4 i b \log (f) e+b^2 \log ^2(f)\right )}{4 (2 f+i c \log (f))}} \text{Erfi}\left (\frac{(-1)^{3/4} (2 e+4 f x+i b \log (f)+2 i c x \log (f))}{2 \sqrt{2 f+i c \log (f)}}\right ) \log (f) \sqrt{2 f+i c \log (f)} \sin (2 d) f-2 (-1)^{3/4} c e^{\frac{i \left (-4 e^2+4 i b \log (f) e+b^2 \log ^2(f)\right )}{4 (2 f-i c \log (f))}} \cos (2 d) \text{Erfi}\left (\frac{\sqrt [4]{-1} (2 e+4 f x-i b \log (f)-2 i c x \log (f))}{2 \sqrt{2 f-i c \log (f)}}\right ) \log (f) \sqrt{2 f-i c \log (f)} f-2 \sqrt [4]{-1} c e^{-\frac{i \left (-4 e^2-4 i b \log (f) e+b^2 \log ^2(f)\right )}{4 (2 f+i c \log (f))}} \cos (2 d) \text{Erfi}\left (\frac{(-1)^{3/4} (2 e+4 f x+i b \log (f)+2 i c x \log (f))}{2 \sqrt{2 f+i c \log (f)}}\right ) \log (f) \sqrt{2 f+i c \log (f)} f+(-1)^{3/4} c^2 e^{\frac{i \left (-4 e^2+4 i b \log (f) e+b^2 \log ^2(f)\right )}{4 (2 f-i c \log (f))}} \text{Erfi}\left (\frac{\sqrt [4]{-1} (2 e+4 f x-i b \log (f)-2 i c x \log (f))}{2 \sqrt{2 f-i c \log (f)}}\right ) \log ^2(f) \sqrt{2 f-i c \log (f)} \sin (2 d)+\sqrt [4]{-1} c^2 e^{-\frac{i \left (-4 e^2-4 i b \log (f) e+b^2 \log ^2(f)\right )}{4 (2 f+i c \log (f))}} \text{Erfi}\left (\frac{(-1)^{3/4} (2 e+4 f x+i b \log (f)+2 i c x \log (f))}{2 \sqrt{2 f+i c \log (f)}}\right ) \log ^2(f) \sqrt{2 f+i c \log (f)} \sin (2 d)+\sqrt [4]{-1} c^2 e^{\frac{i \left (-4 e^2+4 i b \log (f) e+b^2 \log ^2(f)\right )}{4 (2 f-i c \log (f))}} \cos (2 d) \text{Erfi}\left (\frac{\sqrt [4]{-1} (2 e+4 f x-i b \log (f)-2 i c x \log (f))}{2 \sqrt{2 f-i c \log (f)}}\right ) \log ^2(f) \sqrt{2 f-i c \log (f)}+(-1)^{3/4} c^2 e^{-\frac{i \left (-4 e^2-4 i b \log (f) e+b^2 \log ^2(f)\right )}{4 (2 f+i c \log (f))}} \cos (2 d) \text{Erfi}\left (\frac{(-1)^{3/4} (2 e+4 f x+i b \log (f)+2 i c x \log (f))}{2 \sqrt{2 f+i c \log (f)}}\right ) \log ^2(f) \sqrt{2 f+i c \log (f)}\right )}{8 c \log (f) (2 f-i c \log (f)) (2 f+i c \log (f))} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(f^a*Sqrt[Pi]*(8*Sqrt[c]*f^(2 - b^2/(4*c))*Erfi[((b + 2*c*x)*Sqrt[Log[f]])/(2*Sqrt[c])]*Sqrt[Log[f]] + (2*c^(5
/2)*Erfi[((b + 2*c*x)*Sqrt[Log[f]])/(2*Sqrt[c])]*Log[f]^(5/2))/f^(b^2/(4*c)) - 2*(-1)^(3/4)*c*E^(((I/4)*(-4*e^
2 + (4*I)*b*e*Log[f] + b^2*Log[f]^2))/(2*f - I*c*Log[f]))*f*Cos[2*d]*Erfi[((-1)^(1/4)*(2*e + 4*f*x - I*b*Log[f
] - (2*I)*c*x*Log[f]))/(2*Sqrt[2*f - I*c*Log[f]])]*Log[f]*Sqrt[2*f - I*c*Log[f]] + (-1)^(1/4)*c^2*E^(((I/4)*(-
4*e^2 + (4*I)*b*e*Log[f] + b^2*Log[f]^2))/(2*f - I*c*Log[f]))*Cos[2*d]*Erfi[((-1)^(1/4)*(2*e + 4*f*x - I*b*Log
[f] - (2*I)*c*x*Log[f]))/(2*Sqrt[2*f - I*c*Log[f]])]*Log[f]^2*Sqrt[2*f - I*c*Log[f]] - (2*(-1)^(1/4)*c*f*Cos[2
*d]*Erfi[((-1)^(3/4)*(2*e + 4*f*x + I*b*Log[f] + (2*I)*c*x*Log[f]))/(2*Sqrt[2*f + I*c*Log[f]])]*Log[f]*Sqrt[2*
f + I*c*Log[f]])/E^(((I/4)*(-4*e^2 - (4*I)*b*e*Log[f] + b^2*Log[f]^2))/(2*f + I*c*Log[f])) + ((-1)^(3/4)*c^2*C
os[2*d]*Erfi[((-1)^(3/4)*(2*e + 4*f*x + I*b*Log[f] + (2*I)*c*x*Log[f]))/(2*Sqrt[2*f + I*c*Log[f]])]*Log[f]^2*S
qrt[2*f + I*c*Log[f]])/E^(((I/4)*(-4*e^2 - (4*I)*b*e*Log[f] + b^2*Log[f]^2))/(2*f + I*c*Log[f])) + 2*(-1)^(1/4
)*c*E^(((I/4)*(-4*e^2 + (4*I)*b*e*Log[f] + b^2*Log[f]^2))/(2*f - I*c*Log[f]))*f*Erfi[((-1)^(1/4)*(2*e + 4*f*x
- I*b*Log[f] - (2*I)*c*x*Log[f]))/(2*Sqrt[2*f - I*c*Log[f]])]*Log[f]*Sqrt[2*f - I*c*Log[f]]*Sin[2*d] + (-1)^(3
/4)*c^2*E^(((I/4)*(-4*e^2 + (4*I)*b*e*Log[f] + b^2*Log[f]^2))/(2*f - I*c*Log[f]))*Erfi[((-1)^(1/4)*(2*e + 4*f*
x - I*b*Log[f] - (2*I)*c*x*Log[f]))/(2*Sqrt[2*f - I*c*Log[f]])]*Log[f]^2*Sqrt[2*f - I*c*Log[f]]*Sin[2*d] + (2*
(-1)^(3/4)*c*f*Erfi[((-1)^(3/4)*(2*e + 4*f*x + I*b*Log[f] + (2*I)*c*x*Log[f]))/(2*Sqrt[2*f + I*c*Log[f]])]*Log
[f]*Sqrt[2*f + I*c*Log[f]]*Sin[2*d])/E^(((I/4)*(-4*e^2 - (4*I)*b*e*Log[f] + b^2*Log[f]^2))/(2*f + I*c*Log[f]))
+ ((-1)^(1/4)*c^2*Erfi[((-1)^(3/4)*(2*e + 4*f*x + I*b*Log[f] + (2*I)*c*x*Log[f]))/(2*Sqrt[2*f + I*c*Log[f]])]
*Log[f]^2*Sqrt[2*f + I*c*Log[f]]*Sin[2*d])/E^(((I/4)*(-4*e^2 - (4*I)*b*e*Log[f] + b^2*Log[f]^2))/(2*f + I*c*Lo
g[f]))))/(8*c*Log[f]*(2*f - I*c*Log[f])*(2*f + I*c*Log[f]))

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Maple [A]  time = 0.176, size = 263, normalized size = 1. \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+16\,df-4\,{e}^{2}}{4\,c\ln \left ( f \right ) -8\,if}}}}{\it Erf} \left ( -x\sqrt{2\,if-c\ln \left ( f \right ) }+{\frac{b\ln \left ( f \right ) -2\,ie}{2}{\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{ \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}+4\,i\ln \left ( f \right ) be-8\,id\ln \left ( f \right ) c+16\,df-4\,{e}^{2}}{8\,if+4\,c\ln \left ( f \right ) }}}}{\it Erf} \left ( -\sqrt{-c\ln \left ( f \right ) -2\,if}x+{\frac{2\,ie+b\ln \left ( f \right ) }{2}{\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}{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(f*x^2+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+16*d*f-4*e^2)/(-2*I*f+c*ln(f)))/(2*I*f-c*l
n(f))^(1/2)*erf(-x*(2*I*f-c*ln(f))^(1/2)+1/2*(b*ln(f)-2*I*e)/(2*I*f-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+16*d*f-4*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+1/2*(2*I*e+b*ln(f))/(-c*ln(f)-2*I*f)^(1/2))-1/4*Pi^(1/2)*f^a*f^(-1/4*b^2/c)/(-c*ln(f))^(1/2)*er
f(-(-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(f*x^2+e*x+d)^2,x, algorithm="maxima")

[Out]

Exception raised: IndexError

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Fricas [B]  time = 0.575358, size = 1206, normalized size = 4.5 \begin{align*} -\frac{\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 (8 \, f^{2} x +{\left (2 \, c^{2} x + b c\right )} \log \left (f\right )^{2} + 4 \, e f +{\left (2 i \, c e - 2 i \, b f\right )} \log \left (f\right )\right )} \sqrt{-c \log \left (f\right ) - 2 i \, f}}{2 \,{\left (c^{2} \log \left (f\right )^{2} + 4 \, f^{2}\right )}}\right ) e^{\left (-\frac{{\left (b^{2} c - 4 \, a c^{2}\right )} \log \left (f\right )^{3} + 8 i \, e^{2} f - 32 i \, d f^{2} -{\left (8 i \, c^{2} d - 4 i \, b c e + 2 i \, b^{2} f\right )} \log \left (f\right )^{2} - 4 \,{\left (c e^{2} - 2 \, b e f + 4 \, a f^{2}\right )} \log \left (f\right )}{4 \,{\left (c^{2} \log \left (f\right )^{2} + 4 \, f^{2}\right )}}\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 (8 \, f^{2} x +{\left (2 \, c^{2} x + b c\right )} \log \left (f\right )^{2} + 4 \, e f +{\left (-2 i \, c e + 2 i \, b f\right )} \log \left (f\right )\right )} \sqrt{-c \log \left (f\right ) + 2 i \, f}}{2 \,{\left (c^{2} \log \left (f\right )^{2} + 4 \, f^{2}\right )}}\right ) e^{\left (-\frac{{\left (b^{2} c - 4 \, a c^{2}\right )} \log \left (f\right )^{3} - 8 i \, e^{2} f + 32 i \, d f^{2} -{\left (-8 i \, c^{2} d + 4 i \, b c e - 2 i \, b^{2} f\right )} \log \left (f\right )^{2} - 4 \,{\left (c e^{2} - 2 \, b e f + 4 \, a f^{2}\right )} \log \left (f\right )}{4 \,{\left (c^{2} \log \left (f\right )^{2} + 4 \, f^{2}\right )}}\right )} + \frac{2 \, \sqrt{\pi }{\left (c^{2} \log \left (f\right )^{2} + 4 \, f^{2}\right )} \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 \,{\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+b*x+a)*cos(f*x^2+e*x+d)^2,x, algorithm="fricas")

[Out]

-1/8*(sqrt(pi)*(c^2*log(f)^2 - 2*I*c*f*log(f))*sqrt(-c*log(f) - 2*I*f)*erf(1/2*(8*f^2*x + (2*c^2*x + b*c)*log(
f)^2 + 4*e*f + (2*I*c*e - 2*I*b*f)*log(f))*sqrt(-c*log(f) - 2*I*f)/(c^2*log(f)^2 + 4*f^2))*e^(-1/4*((b^2*c - 4
*a*c^2)*log(f)^3 + 8*I*e^2*f - 32*I*d*f^2 - (8*I*c^2*d - 4*I*b*c*e + 2*I*b^2*f)*log(f)^2 - 4*(c*e^2 - 2*b*e*f
+ 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(1/2*(8*f^2*x + (2*c^2*x + b*c)*log(f)^2 + 4*e*f + (-2*I*c*e + 2*I*b*f)*log(f))*sqrt(-c*log(f) + 2*I*f)/(c^
2*log(f)^2 + 4*f^2))*e^(-1/4*((b^2*c - 4*a*c^2)*log(f)^3 - 8*I*e^2*f + 32*I*d*f^2 - (-8*I*c^2*d + 4*I*b*c*e -
2*I*b^2*f)*log(f)^2 - 4*(c*e^2 - 2*b*e*f + 4*a*f^2)*log(f))/(c^2*log(f)^2 + 4*f^2)) + 2*sqrt(pi)*(c^2*log(f)^2
+ 4*f^2)*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^3*log(f)^3 + 4*c*
f^2*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(f*x**2+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 (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+b*x+a)*cos(f*x^2+e*x+d)^2,x, algorithm="giac")

[Out]

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