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]

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

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Rubi [A]  time = 0.46, antiderivative size = 268, normalized size of antiderivative = 1.00, 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 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 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 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 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 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]

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*}

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Mathematica [B]  time = 6.74, 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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fricas [B]  time = 0.68, size = 468, normalized size = 1.75 \[ -\frac {\sqrt {\pi } {\left (c^{2} \log \relax (f)^{2} - 2 i \, c f \log \relax (f)\right )} \sqrt {-c \log \relax (f) - 2 i \, f} \operatorname {erf}\left (\frac {{\left (8 \, f^{2} x + {\left (2 \, c^{2} x + b c\right )} \log \relax (f)^{2} + 4 \, e f + {\left (2 i \, c e - 2 i \, b f\right )} \log \relax (f)\right )} \sqrt {-c \log \relax (f) - 2 i \, f}}{2 \, {\left (c^{2} \log \relax (f)^{2} + 4 \, f^{2}\right )}}\right ) e^{\left (-\frac {{\left (b^{2} c - 4 \, a c^{2}\right )} \log \relax (f)^{3} + 8 i \, e^{2} f - 32 i \, d f^{2} - {\left (8 i \, c^{2} d - 4 i \, b c e + 2 i \, b^{2} f\right )} \log \relax (f)^{2} - 4 \, {\left (c e^{2} - 2 \, b e f + 4 \, a f^{2}\right )} \log \relax (f)}{4 \, {\left (c^{2} \log \relax (f)^{2} + 4 \, f^{2}\right )}}\right )} + \sqrt {\pi } {\left (c^{2} \log \relax (f)^{2} + 2 i \, c f \log \relax (f)\right )} \sqrt {-c \log \relax (f) + 2 i \, f} \operatorname {erf}\left (\frac {{\left (8 \, f^{2} x + {\left (2 \, c^{2} x + b c\right )} \log \relax (f)^{2} + 4 \, e f + {\left (-2 i \, c e + 2 i \, b f\right )} \log \relax (f)\right )} \sqrt {-c \log \relax (f) + 2 i \, f}}{2 \, {\left (c^{2} \log \relax (f)^{2} + 4 \, f^{2}\right )}}\right ) e^{\left (-\frac {{\left (b^{2} c - 4 \, a c^{2}\right )} \log \relax (f)^{3} - 8 i \, e^{2} f + 32 i \, d f^{2} - {\left (-8 i \, c^{2} d + 4 i \, b c e - 2 i \, b^{2} f\right )} \log \relax (f)^{2} - 4 \, {\left (c e^{2} - 2 \, b e f + 4 \, a f^{2}\right )} \log \relax (f)}{4 \, {\left (c^{2} \log \relax (f)^{2} + 4 \, f^{2}\right )}}\right )} + \frac {2 \, \sqrt {\pi } {\left (c^{2} \log \relax (f)^{2} + 4 \, f^{2}\right )} \sqrt {-c \log \relax (f)} \operatorname {erf}\left (\frac {{\left (2 \, c x + b\right )} \sqrt {-c \log \relax (f)}}{2 \, c}\right )}{f^{\frac {b^{2} - 4 \, a c}{4 \, c}}}}{8 \, {\left (c^{3} \log \relax (f)^{3} + 4 \, c f^{2} \log \relax (f)\right )}} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int f^{c x^{2} + b x + a} \cos \left (f x^{2} + e x + d\right )^{2}\,{d x} \]

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)

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maple [A]  time = 0.33, size = 263, normalized size = 0.98 \[ -\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {\ln \relax (f )^{2} b^{2}-4 i \ln \relax (f ) b e +8 i d \ln \relax (f ) c +16 d f -4 e^{2}}{4 \left (-2 i f +c \ln \relax (f )\right )}} \erf \left (-x \sqrt {2 i f -c \ln \relax (f )}+\frac {b \ln \relax (f )-2 i e}{2 \sqrt {2 i f -c \ln \relax (f )}}\right )}{8 \sqrt {2 i f -c \ln \relax (f )}}-\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {\ln \relax (f )^{2} b^{2}+4 i \ln \relax (f ) b e -8 i d \ln \relax (f ) c +16 d f -4 e^{2}}{4 \left (2 i f +c \ln \relax (f )\right )}} \erf \left (-\sqrt {-2 i f -c \ln \relax (f )}\, x +\frac {2 i e +b \ln \relax (f )}{2 \sqrt {-2 i f -c \ln \relax (f )}}\right )}{8 \sqrt {-2 i f -c \ln \relax (f )}}-\frac {\sqrt {\pi }\, f^{a} f^{-\frac {b^{2}}{4 c}} \erf \left (-\sqrt {-c \ln \relax (f )}\, x +\frac {b \ln \relax (f )}{2 \sqrt {-c \ln \relax (f )}}\right )}{4 \sqrt {-c \ln \relax (f )}} \]

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)))/(-2*I*f-c*ln(f))^(1/2)*erf(-(-2*I*f-c*
ln(f))^(1/2)*x+1/2*(2*I*e+b*ln(f))/(-2*I*f-c*ln(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 [C]  time = 0.42, size = 1487, normalized size = 5.55 \[ \text {result too large to display} \]

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]

-1/16*(sqrt(pi)*sqrt(2*c^2*log(f)^2 + 8*f^2)*((I*f^a*cos(-1/2*(4*e^2*f - 16*d*f^2 - (4*c^2*d - 2*b*c*e + b^2*f
)*log(f)^2)/(c^2*log(f)^2 + 4*f^2))*e^(c*e^2*log(f)/(c^2*log(f)^2 + 4*f^2) + 1/4*b^2*log(f)/c) + f^a*e^(c*e^2*
log(f)/(c^2*log(f)^2 + 4*f^2) + 1/4*b^2*log(f)/c)*sin(-1/2*(4*e^2*f - 16*d*f^2 - (4*c^2*d - 2*b*c*e + b^2*f)*l
og(f)^2)/(c^2*log(f)^2 + 4*f^2)))*erf(1/2*(2*(c*log(f) - 2*I*f)*x + b*log(f) - 2*I*e)*sqrt(-c*log(f) + 2*I*f)/
(c*log(f) - 2*I*f)) + (-I*f^a*cos(-1/2*(4*e^2*f - 16*d*f^2 - (4*c^2*d - 2*b*c*e + b^2*f)*log(f)^2)/(c^2*log(f)
^2 + 4*f^2))*e^(c*e^2*log(f)/(c^2*log(f)^2 + 4*f^2) + 1/4*b^2*log(f)/c) + f^a*e^(c*e^2*log(f)/(c^2*log(f)^2 +
4*f^2) + 1/4*b^2*log(f)/c)*sin(-1/2*(4*e^2*f - 16*d*f^2 - (4*c^2*d - 2*b*c*e + b^2*f)*log(f)^2)/(c^2*log(f)^2
+ 4*f^2)))*erf(1/2*(2*(c*log(f) + 2*I*f)*x + b*log(f) + 2*I*e)*sqrt(-c*log(f) - 2*I*f)/(c*log(f) + 2*I*f)))*sq
rt(c*log(f) + sqrt(c^2*log(f)^2 + 4*f^2))*sqrt(-c*log(f)) - sqrt(pi)*sqrt(2*c^2*log(f)^2 + 8*f^2)*((f^a*cos(-1
/2*(4*e^2*f - 16*d*f^2 - (4*c^2*d - 2*b*c*e + b^2*f)*log(f)^2)/(c^2*log(f)^2 + 4*f^2))*e^(c*e^2*log(f)/(c^2*lo
g(f)^2 + 4*f^2) + 1/4*b^2*log(f)/c) - I*f^a*e^(c*e^2*log(f)/(c^2*log(f)^2 + 4*f^2) + 1/4*b^2*log(f)/c)*sin(-1/
2*(4*e^2*f - 16*d*f^2 - (4*c^2*d - 2*b*c*e + b^2*f)*log(f)^2)/(c^2*log(f)^2 + 4*f^2)))*erf(1/2*(2*(c*log(f) -
2*I*f)*x + b*log(f) - 2*I*e)*sqrt(-c*log(f) + 2*I*f)/(c*log(f) - 2*I*f)) + (f^a*cos(-1/2*(4*e^2*f - 16*d*f^2 -
(4*c^2*d - 2*b*c*e + b^2*f)*log(f)^2)/(c^2*log(f)^2 + 4*f^2))*e^(c*e^2*log(f)/(c^2*log(f)^2 + 4*f^2) + 1/4*b^
2*log(f)/c) + I*f^a*e^(c*e^2*log(f)/(c^2*log(f)^2 + 4*f^2) + 1/4*b^2*log(f)/c)*sin(-1/2*(4*e^2*f - 16*d*f^2 -
(4*c^2*d - 2*b*c*e + b^2*f)*log(f)^2)/(c^2*log(f)^2 + 4*f^2)))*erf(1/2*(2*(c*log(f) + 2*I*f)*x + b*log(f) + 2*
I*e)*sqrt(-c*log(f) - 2*I*f)/(c*log(f) + 2*I*f)))*sqrt(-c*log(f) + sqrt(c^2*log(f)^2 + 4*f^2))*sqrt(-c*log(f))
- 2*sqrt(pi)*((c^2*f^a*e^(1/4*b^2*c*log(f)^3/(c^2*log(f)^2 + 4*f^2) + 2*b*e*f*log(f)/(c^2*log(f)^2 + 4*f^2))*
log(f)^2 + 4*f^(a + 2)*e^(1/4*b^2*c*log(f)^3/(c^2*log(f)^2 + 4*f^2) + 2*b*e*f*log(f)/(c^2*log(f)^2 + 4*f^2)))*
erf(-1/2*b*conjugate(1/sqrt(-c*log(f)))*log(f) + x*conjugate(sqrt(-c*log(f)))) - (c^2*f^a*e^(1/4*b^2*c*log(f)^
3/(c^2*log(f)^2 + 4*f^2) + 2*b*e*f*log(f)/(c^2*log(f)^2 + 4*f^2))*log(f)^2 + 4*f^(a + 2)*e^(1/4*b^2*c*log(f)^3
/(c^2*log(f)^2 + 4*f^2) + 2*b*e*f*log(f)/(c^2*log(f)^2 + 4*f^2)))*erf(1/2*(2*c*x*log(f) + b*log(f))/sqrt(-c*lo
g(f)))))/((c^2*e^(1/4*b^2*c*log(f)^3/(c^2*log(f)^2 + 4*f^2) + 2*b*e*f*log(f)/(c^2*log(f)^2 + 4*f^2) + 1/4*b^2*
log(f)/c)*log(f)^2 + 4*f^2*e^(1/4*b^2*c*log(f)^3/(c^2*log(f)^2 + 4*f^2) + 2*b*e*f*log(f)/(c^2*log(f)^2 + 4*f^2
) + 1/4*b^2*log(f)/c))*sqrt(-c*log(f)))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int f^{c\,x^2+b\,x+a}\,{\cos \left (f\,x^2+e\,x+d\right )}^2 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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