3.2.28 \(\int f^{a+b x+c x^2} \cos (d+f x^2) \, dx\) [128]

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

[Out]

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

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Rubi [A]
time = 0.24, antiderivative size = 189, normalized size of antiderivative = 1.00, 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 = {4561, 2325, 2266, 2236, 2235} \begin {gather*} \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}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*(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
*Sqrt[I*f - c*Log[f]])])/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 2235

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 2236

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 2266

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 2325

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 4561

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 \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*}

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Mathematica [A]
time = 1.04, size = 231, normalized size = 1.22 \begin {gather*} -\frac {(-1)^{3/4} e^{\frac {b^2 \log ^2(f)}{4 i f-4 c \log (f)}} f^a \sqrt {\pi } \left (\text {Erfi}\left (\frac {(-1)^{3/4} (2 f x+i (b+2 c x) \log (f))}{2 \sqrt {f+i c \log (f)}}\right ) (f-i c \log (f)) \sqrt {f+i c \log (f)} (-i \cos (d)-\sin (d))+e^{\frac {i b^2 f \log ^2(f)}{2 \left (f^2+c^2 \log ^2(f)\right )}} \text {Erfi}\left (\frac {\sqrt [4]{-1} (2 f x-i (b+2 c x) \log (f))}{2 \sqrt {f-i c \log (f)}}\right ) \sqrt {f-i c \log (f)} (f+i c \log (f)) (\cos (d)+i \sin (d))\right )}{4 \left (f^2+c^2 \log ^2(f)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*((-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)*Log[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])))/(f^2 + c^2*Log[f]^2)

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Maple [A]
time = 0.15, size = 178, normalized size = 0.94

method	result	size
risch	\(-\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {\ln \left (f \right )^{2} b^{2}+4 i d \ln \left (f \right ) c +4 d f}{4 \left (-i f +c \ln \left (f \right )\right )}} \erf \left (-x \sqrt {i f -c \ln \left (f \right )}+\frac {\ln \left (f \right ) b}{2 \sqrt {i f -c \ln \left (f \right )}}\right )}{4 \sqrt {i f -c \ln \left (f \right )}}-\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {\ln \left (f \right )^{2} b^{2}-4 i d \ln \left (f \right ) c +4 d f}{4 \left (i f +c \ln \left (f \right )\right )}} \erf \left (-\sqrt {-c \ln \left (f \right )-i f}\, x +\frac {\ln \left (f \right ) b}{2 \sqrt {-c \ln \left (f \right )-i f}}\right )}{4 \sqrt {-c \ln \left (f \right )-i f}}\)	\(178\)

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,method=_RETURNVERBOSE)

[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 [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 648 vs. \(2 (145) = 290\).
time = 0.29, size = 648, normalized size = 3.43 \begin {gather*} \frac {\sqrt {\pi } \sqrt {2 \, c^{2} \log \left (f\right )^{2} + 2 \, f^{2}} {\left ({\left (i \, f^{a} \cos \left (\frac {4 \, d f^{2} + {\left (4 \, c^{2} d + b^{2} f\right )} \log \left (f\right )^{2}}{4 \, {\left (c^{2} \log \left (f\right )^{2} + f^{2}\right )}}\right ) + f^{a} \sin \left (\frac {4 \, d f^{2} + {\left (4 \, c^{2} d + b^{2} f\right )} \log \left (f\right )^{2}}{4 \, {\left (c^{2} \log \left (f\right )^{2} + f^{2}\right )}}\right )\right )} \operatorname {erf}\left (\frac {2 \, {\left (c \log \left (f\right ) - i \, f\right )} x + b \log \left (f\right )}{2 \, \sqrt {-c \log \left (f\right ) + i \, f}}\right ) + {\left (-i \, f^{a} \cos \left (\frac {4 \, d f^{2} + {\left (4 \, c^{2} d + b^{2} f\right )} \log \left (f\right )^{2}}{4 \, {\left (c^{2} \log \left (f\right )^{2} + f^{2}\right )}}\right ) + f^{a} \sin \left (\frac {4 \, d f^{2} + {\left (4 \, c^{2} d + b^{2} f\right )} \log \left (f\right )^{2}}{4 \, {\left (c^{2} \log \left (f\right )^{2} + f^{2}\right )}}\right )\right )} \operatorname {erf}\left (\frac {2 \, {\left (c \log \left (f\right ) + i \, f\right )} x + b \log \left (f\right )}{2 \, \sqrt {-c \log \left (f\right ) - i \, f}}\right )\right )} \sqrt {c \log \left (f\right ) + \sqrt {c^{2} \log \left (f\right )^{2} + f^{2}}} - \sqrt {\pi } \sqrt {2 \, c^{2} \log \left (f\right )^{2} + 2 \, f^{2}} {\left ({\left (f^{a} \cos \left (\frac {4 \, d f^{2} + {\left (4 \, c^{2} d + b^{2} f\right )} \log \left (f\right )^{2}}{4 \, {\left (c^{2} \log \left (f\right )^{2} + f^{2}\right )}}\right ) - i \, f^{a} \sin \left (\frac {4 \, d f^{2} + {\left (4 \, c^{2} d + b^{2} f\right )} \log \left (f\right )^{2}}{4 \, {\left (c^{2} \log \left (f\right )^{2} + f^{2}\right )}}\right )\right )} \operatorname {erf}\left (\frac {2 \, {\left (c \log \left (f\right ) - i \, f\right )} x + b \log \left (f\right )}{2 \, \sqrt {-c \log \left (f\right ) + i \, f}}\right ) + {\left (f^{a} \cos \left (\frac {4 \, d f^{2} + {\left (4 \, c^{2} d + b^{2} f\right )} \log \left (f\right )^{2}}{4 \, {\left (c^{2} \log \left (f\right )^{2} + f^{2}\right )}}\right ) + i \, f^{a} \sin \left (\frac {4 \, d f^{2} + {\left (4 \, c^{2} d + b^{2} f\right )} \log \left (f\right )^{2}}{4 \, {\left (c^{2} \log \left (f\right )^{2} + f^{2}\right )}}\right )\right )} \operatorname {erf}\left (\frac {2 \, {\left (c \log \left (f\right ) + i \, f\right )} x + b \log \left (f\right )}{2 \, \sqrt {-c \log \left (f\right ) - i \, f}}\right )\right )} \sqrt {-c \log \left (f\right ) + \sqrt {c^{2} \log \left (f\right )^{2} + f^{2}}}}{8 \, {\left (c^{2} e^{\left (\frac {b^{2} c \log \left (f\right )^{3}}{4 \, {\left (c^{2} \log \left (f\right )^{2} + f^{2}\right )}}\right )} \log \left (f\right )^{2} + f^{2} e^{\left (\frac {b^{2} c \log \left (f\right )^{3}}{4 \, {\left (c^{2} \log \left (f\right )^{2} + f^{2}\right )}}\right )}\right )}} \end {gather*}

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]

1/8*(sqrt(pi)*sqrt(2*c^2*log(f)^2 + 2*f^2)*((I*f^a*cos(1/4*(4*d*f^2 + (4*c^2*d + b^2*f)*log(f)^2)/(c^2*log(f)^
2 + f^2)) + f^a*sin(1/4*(4*d*f^2 + (4*c^2*d + b^2*f)*log(f)^2)/(c^2*log(f)^2 + f^2)))*erf(1/2*(2*(c*log(f) - I
*f)*x + b*log(f))/sqrt(-c*log(f) + I*f)) + (-I*f^a*cos(1/4*(4*d*f^2 + (4*c^2*d + b^2*f)*log(f)^2)/(c^2*log(f)^
2 + f^2)) + f^a*sin(1/4*(4*d*f^2 + (4*c^2*d + b^2*f)*log(f)^2)/(c^2*log(f)^2 + f^2)))*erf(1/2*(2*(c*log(f) + I
*f)*x + b*log(f))/sqrt(-c*log(f) - I*f)))*sqrt(c*log(f) + sqrt(c^2*log(f)^2 + f^2)) - sqrt(pi)*sqrt(2*c^2*log(
f)^2 + 2*f^2)*((f^a*cos(1/4*(4*d*f^2 + (4*c^2*d + b^2*f)*log(f)^2)/(c^2*log(f)^2 + f^2)) - I*f^a*sin(1/4*(4*d*
f^2 + (4*c^2*d + b^2*f)*log(f)^2)/(c^2*log(f)^2 + f^2)))*erf(1/2*(2*(c*log(f) - I*f)*x + b*log(f))/sqrt(-c*log
(f) + I*f)) + (f^a*cos(1/4*(4*d*f^2 + (4*c^2*d + b^2*f)*log(f)^2)/(c^2*log(f)^2 + f^2)) + I*f^a*sin(1/4*(4*d*f
^2 + (4*c^2*d + b^2*f)*log(f)^2)/(c^2*log(f)^2 + f^2)))*erf(1/2*(2*(c*log(f) + I*f)*x + b*log(f))/sqrt(-c*log(
f) - I*f)))*sqrt(-c*log(f) + sqrt(c^2*log(f)^2 + f^2)))/(c^2*e^(1/4*b^2*c*log(f)^3/(c^2*log(f)^2 + f^2))*log(f
)^2 + f^2*e^(1/4*b^2*c*log(f)^3/(c^2*log(f)^2 + f^2)))

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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 311 vs. \(2 (145) = 290\).
time = 2.70, size = 311, normalized size = 1.65 \begin {gather*} -\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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int f^{a + b x + c x^{2}} \cos {\left (d + f x^{2} \right )}\, dx \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int f^{c\,x^2+b\,x+a}\,\cos \left (f\,x^2+d\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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