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

Optimal. Leaf size=245 \[ -\frac {\sqrt {\pi } f^a e^{\frac {b^2 \log ^2(f)}{-4 c \log (f)+8 i f}-2 i d} \text {erf}\left (\frac {b \log (f)-2 x (-c \log (f)+2 i f)}{2 \sqrt {-c \log (f)+2 i f}}\right )}{8 \sqrt {-c \log (f)+2 i f}}+\frac {\sqrt {\pi } f^a e^{2 i d-\frac {b^2 \log ^2(f)}{4 c \log (f)+8 i f}} \text {erfi}\left (\frac {b \log (f)+2 x (c \log (f)+2 i f)}{2 \sqrt {c \log (f)+2 i f}}\right )}{8 \sqrt {c \log (f)+2 i f}}+\frac {\sqrt {\pi } f^{a-\frac {b^2}{4 c}} \text {erfi}\left (\frac {\sqrt {\log (f)} (b+2 c x)}{2 \sqrt {c}}\right )}{4 \sqrt {c} \sqrt {\log (f)}} \]

[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+b^2*ln
(f)^2/(8*I*f-4*c*ln(f)))*f^a*erf(1/2*(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-b^2*ln(f)^2/(8*I*f+4*c*ln(f)))*f^a*erfi(1/2*(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 = 0.41, antiderivative size = 245, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {4473, 2234, 2204, 2287, 2205} \[ -\frac {\sqrt {\pi } f^a e^{\frac {b^2 \log ^2(f)}{-4 c \log (f)+8 i f}-2 i d} \text {Erf}\left (\frac {b \log (f)-2 x (-c \log (f)+2 i f)}{2 \sqrt {-c \log (f)+2 i f}}\right )}{8 \sqrt {-c \log (f)+2 i f}}+\frac {\sqrt {\pi } f^a e^{2 i d-\frac {b^2 \log ^2(f)}{4 c \log (f)+8 i f}} \text {Erfi}\left (\frac {b \log (f)+2 x (c \log (f)+2 i f)}{2 \sqrt {c \log (f)+2 i f}}\right )}{8 \sqrt {c \log (f)+2 i f}}+\frac {\sqrt {\pi } f^{a-\frac {b^2}{4 c}} \text {Erfi}\left (\frac {\sqrt {\log (f)} (b+2 c x)}{2 \sqrt {c}}\right )}{4 \sqrt {c} \sqrt {\log (f)}} \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b*x + c*x^2)*Cos[d + 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 + (b^2*Log[f]^2)/((8*I)*f - 4*c*Log[f]))*f^a*Sqrt[Pi]*Erf[(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 - (b^2*Log[f]^2)/((8*I)*f + 4*c*Log[f]))*f^a*S
qrt[Pi]*Erfi[(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+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 f x^2} f^{a+b x+c x^2}+\frac {1}{4} e^{2 i d+2 i f x^2} f^{a+b x+c x^2}\right ) \, dx\\ &=\frac {1}{4} \int e^{-2 i d-2 i f x^2} f^{a+b x+c x^2} \, dx+\frac {1}{4} \int e^{2 i d+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)+b x \log (f)-x^2 (2 i f-c \log (f))\right ) \, dx+\frac {1}{4} \int \exp \left (2 i d+a \log (f)+b x \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 (e^{-2 i d+\frac {b^2 \log ^2(f)}{8 i f-4 c \log (f)}} f^a\right ) \int \exp \left (\frac {(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 (e^{2 i d-\frac {b^2 \log ^2(f)}{8 i f+4 c \log (f)}} f^a\right ) \int \exp \left (\frac {(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 {e^{-2 i d+\frac {b^2 \log ^2(f)}{8 i f-4 c \log (f)}} f^a \sqrt {\pi } \text {erf}\left (\frac {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 {e^{2 i d-\frac {b^2 \log ^2(f)}{8 i f+4 c \log (f)}} f^a \sqrt {\pi } \text {erfi}\left (\frac {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 [A]  time = 3.10, size = 301, normalized size = 1.23 \[ \frac {1}{8} \sqrt {\pi } f^a \left (\frac {2 f^{-\frac {b^2}{4 c}} \text {erfi}\left (\frac {\sqrt {\log (f)} (b+2 c x)}{2 \sqrt {c}}\right )}{\sqrt {c} \sqrt {\log (f)}}+\frac {\sqrt [4]{-1} e^{\frac {b^2 \log ^2(f)}{-4 c \log (f)+8 i f}} \left (\sqrt {2 f-i c \log (f)} (2 f+i c \log (f)) (\sin (2 d)-i \cos (2 d)) e^{\frac {i b^2 f \log ^2(f)}{c^2 \log ^2(f)+4 f^2}} \text {erfi}\left (\frac {\sqrt [4]{-1} (4 f x-i \log (f) (b+2 c x))}{2 \sqrt {2 f-i c \log (f)}}\right )-(2 f-i c \log (f)) \sqrt {2 f+i c \log (f)} (\cos (2 d)-i \sin (2 d)) \text {erfi}\left (\frac {(-1)^{3/4} (4 f x+i \log (f) (b+2 c x))}{2 \sqrt {2 f+i c \log (f)}}\right )\right )}{c^2 \log ^2(f)+4 f^2}\right ) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(f^a*Sqrt[Pi]*((2*Erfi[((b + 2*c*x)*Sqrt[Log[f]])/(2*Sqrt[c])])/(Sqrt[c]*f^(b^2/(4*c))*Sqrt[Log[f]]) + ((-1)^(
1/4)*E^((b^2*Log[f]^2)/((8*I)*f - 4*c*Log[f]))*(-(Erfi[((-1)^(3/4)*(4*f*x + I*(b + 2*c*x)*Log[f]))/(2*Sqrt[2*f
+ I*c*Log[f]])]*(2*f - I*c*Log[f])*Sqrt[2*f + I*c*Log[f]]*(Cos[2*d] - I*Sin[2*d])) + E^((I*b^2*f*Log[f]^2)/(4
*f^2 + c^2*Log[f]^2))*Erfi[((-1)^(1/4)*(4*f*x - I*(b + 2*c*x)*Log[f]))/(2*Sqrt[2*f - I*c*Log[f]])]*Sqrt[2*f -
I*c*Log[f]]*(2*f + I*c*Log[f])*((-I)*Cos[2*d] + Sin[2*d])))/(4*f^2 + c^2*Log[f]^2)))/8

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fricas [B]  time = 0.72, size = 400, normalized size = 1.63 \[ -\frac {\sqrt {\pi } {\left (c^{2} \log \relax (f)^{2} - 2 i \, c f \log \relax (f)\right )} \sqrt {-c \log \relax (f) - 2 i \, f} \operatorname {erf}\left (\frac {{\left (8 \, f^{2} x - 2 i \, b f \log \relax (f) + {\left (2 \, c^{2} x + b c\right )} \log \relax (f)^{2}\right )} \sqrt {-c \log \relax (f) - 2 i \, f}}{2 \, {\left (c^{2} \log \relax (f)^{2} + 4 \, f^{2}\right )}}\right ) e^{\left (\frac {16 \, a f^{2} \log \relax (f) - {\left (b^{2} c - 4 \, a c^{2}\right )} \log \relax (f)^{3} + 32 i \, d f^{2} + {\left (8 i \, c^{2} d + 2 i \, b^{2} f\right )} \log \relax (f)^{2}}{4 \, {\left (c^{2} \log \relax (f)^{2} + 4 \, f^{2}\right )}}\right )} + \sqrt {\pi } {\left (c^{2} \log \relax (f)^{2} + 2 i \, c f \log \relax (f)\right )} \sqrt {-c \log \relax (f) + 2 i \, f} \operatorname {erf}\left (\frac {{\left (8 \, f^{2} x + 2 i \, b f \log \relax (f) + {\left (2 \, c^{2} x + b c\right )} \log \relax (f)^{2}\right )} \sqrt {-c \log \relax (f) + 2 i \, f}}{2 \, {\left (c^{2} \log \relax (f)^{2} + 4 \, f^{2}\right )}}\right ) e^{\left (\frac {16 \, a f^{2} \log \relax (f) - {\left (b^{2} c - 4 \, a c^{2}\right )} \log \relax (f)^{3} - 32 i \, d f^{2} + {\left (-8 i \, c^{2} d - 2 i \, b^{2} f\right )} \log \relax (f)^{2}}{4 \, {\left (c^{2} \log \relax (f)^{2} + 4 \, f^{2}\right )}}\right )} + \frac {2 \, \sqrt {\pi } {\left (c^{2} \log \relax (f)^{2} + 4 \, f^{2}\right )} \sqrt {-c \log \relax (f)} \operatorname {erf}\left (\frac {{\left (2 \, c x + b\right )} \sqrt {-c \log \relax (f)}}{2 \, c}\right )}{f^{\frac {b^{2} - 4 \, a c}{4 \, c}}}}{8 \, {\left (c^{3} \log \relax (f)^{3} + 4 \, c f^{2} \log \relax (f)\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(c*x^2+b*x+a)*cos(f*x^2+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*I*b*f*log(f) + (2*
c^2*x + b*c)*log(f)^2)*sqrt(-c*log(f) - 2*I*f)/(c^2*log(f)^2 + 4*f^2))*e^(1/4*(16*a*f^2*log(f) - (b^2*c - 4*a*
c^2)*log(f)^3 + 32*I*d*f^2 + (8*I*c^2*d + 2*I*b^2*f)*log(f)^2)/(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*I*b*f*log(f) + (2*c^2*x + b*c)*log(f)^2)*sqrt
(-c*log(f) + 2*I*f)/(c^2*log(f)^2 + 4*f^2))*e^(1/4*(16*a*f^2*log(f) - (b^2*c - 4*a*c^2)*log(f)^3 - 32*I*d*f^2
+ (-8*I*c^2*d - 2*I*b^2*f)*log(f)^2)/(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 \[ \int f^{c x^{2} + b x + a} \cos \left (f x^{2} + d\right )^{2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.31, size = 227, normalized size = 0.93 \[ -\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {\ln \relax (f )^{2} b^{2}+8 i d \ln \relax (f ) c +16 d f}{4 \left (-2 i f +c \ln \relax (f )\right )}} \erf \left (-x \sqrt {2 i f -c \ln \relax (f )}+\frac {\ln \relax (f ) b}{2 \sqrt {2 i f -c \ln \relax (f )}}\right )}{8 \sqrt {2 i f -c \ln \relax (f )}}-\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {\ln \relax (f )^{2} b^{2}-8 i d \ln \relax (f ) c +16 d f}{4 \left (2 i f +c \ln \relax (f )\right )}} \erf \left (-\sqrt {-2 i f -c \ln \relax (f )}\, x +\frac {\ln \relax (f ) b}{2 \sqrt {-2 i f -c \ln \relax (f )}}\right )}{8 \sqrt {-2 i f -c \ln \relax (f )}}-\frac {\sqrt {\pi }\, f^{a} f^{-\frac {b^{2}}{4 c}} \erf \left (-\sqrt {-c \ln \relax (f )}\, x +\frac {b \ln \relax (f )}{2 \sqrt {-c \ln \relax (f )}}\right )}{4 \sqrt {-c \ln \relax (f )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*Pi^(1/2)*f^a*exp(-1/4*(ln(f)^2*b^2+8*I*d*ln(f)*c+16*d*f)/(-2*I*f+c*ln(f)))/(2*I*f-c*ln(f))^(1/2)*erf(-x*(
2*I*f-c*ln(f))^(1/2)+1/2*ln(f)*b/(2*I*f-c*ln(f))^(1/2))-1/8*Pi^(1/2)*f^a*exp(-1/4*(ln(f)^2*b^2-8*I*d*ln(f)*c+1
6*d*f)/(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*ln(f)*b/(-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)*erf(-(-c*ln(f))^(1/2)*x+1/2/(-c*ln(f))^(1/2)*b*ln(f))

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maxima [C]  time = 0.38, size = 997, normalized size = 4.07 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*(sqrt(pi)*sqrt(2*c^2*log(f)^2 + 8*f^2)*((I*f^a*f^(1/4*b^2/c)*cos(1/2*(16*d*f^2 + (4*c^2*d + b^2*f)*log(f)
^2)/(c^2*log(f)^2 + 4*f^2)) + f^a*f^(1/4*b^2/c)*sin(1/2*(16*d*f^2 + (4*c^2*d + 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))/sqrt(-c*log(f) + 2*I*f)) + (-I*f^a*f^(1/4*b^2/c)*cos(1/
2*(16*d*f^2 + (4*c^2*d + b^2*f)*log(f)^2)/(c^2*log(f)^2 + 4*f^2)) + f^a*f^(1/4*b^2/c)*sin(1/2*(16*d*f^2 + (4*c
^2*d + 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))/sqrt(-c*log(f) -
2*I*f)))*sqrt(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*f^(1/4*b^2/c)*cos(1/2*(16*d*f^2 + (4*c^2*d + b^2*f)*log(f)^2)/(c^2*log(f)^2 + 4*f^2)) - I*f^a*f^(1/4*b^2/
c)*sin(1/2*(16*d*f^2 + (4*c^2*d + 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))/sqrt(-c*log(f) + 2*I*f)) + (f^a*f^(1/4*b^2/c)*cos(1/2*(16*d*f^2 + (4*c^2*d + b^2*f)*log(f)^2)/(c^2*l
og(f)^2 + 4*f^2)) + I*f^a*f^(1/4*b^2/c)*sin(1/2*(16*d*f^2 + (4*c^2*d + 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))/sqrt(-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))*log(f)^2 + 4*f^(a
+ 2)*e^(1/4*b^2*c*log(f)^3/(c^2*log(f)^2 + 4*f^2)))*erf(-1/2*b*conjugate(1/sqrt(-c*log(f)))*log(f) + x*conjug
ate(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))*log(f)^2 + 4*f^(a + 2)*e^(1/4*b
^2*c*log(f)^3/(c^2*log(f)^2 + 4*f^2)))*erf(1/2*(2*c*x*log(f) + b*log(f))/sqrt(-c*log(f)))))/((c^2*e^(1/4*b^2*c
*log(f)^3/(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) + 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 \[ \int f^{c\,x^2+b\,x+a}\,{\cos \left (f\,x^2+d\right )}^2 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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