3.2.19 \(\int f^{a+c x^2} \cos (d+f x^2) \, dx\) [119]

Optimal. Leaf size=103 \[ \frac {e^{-i d} f^a \sqrt {\pi } \text {Erf}\left (x \sqrt {i f-c \log (f)}\right )}{4 \sqrt {i f-c \log (f)}}+\frac {e^{i d} f^a \sqrt {\pi } \text {Erfi}\left (x \sqrt {i f+c \log (f)}\right )}{4 \sqrt {i f+c \log (f)}} \]

[Out]

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

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Rubi [A]
time = 0.11, antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {4561, 2325, 2236, 2235} \begin {gather*} \frac {\sqrt {\pi } e^{-i d} f^a \text {Erf}\left (x \sqrt {-c \log (f)+i f}\right )}{4 \sqrt {-c \log (f)+i f}}+\frac {\sqrt {\pi } e^{i d} f^a \text {Erfi}\left (x \sqrt {c \log (f)+i f}\right )}{4 \sqrt {c \log (f)+i f}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[f^(a + c*x^2)*Cos[d + f*x^2],x]

[Out]

(f^a*Sqrt[Pi]*Erf[x*Sqrt[I*f - c*Log[f]]])/(4*E^(I*d)*Sqrt[I*f - c*Log[f]]) + (E^(I*d)*f^a*Sqrt[Pi]*Erfi[x*Sqr
t[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 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+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+c x^2}+\frac {1}{2} e^{i d+i f x^2} f^{a+c x^2}\right ) \, dx\\ &=\frac {1}{2} \int e^{-i d-i f x^2} f^{a+c x^2} \, dx+\frac {1}{2} \int e^{i d+i f x^2} f^{a+c x^2} \, dx\\ &=\frac {1}{2} \int e^{-i d+a \log (f)-x^2 (i f-c \log (f))} \, dx+\frac {1}{2} \int e^{i d+a \log (f)+x^2 (i f+c \log (f))} \, dx\\ &=\frac {e^{-i d} f^a \sqrt {\pi } \text {erf}\left (x \sqrt {i f-c \log (f)}\right )}{4 \sqrt {i f-c \log (f)}}+\frac {e^{i d} f^a \sqrt {\pi } \text {erfi}\left (x \sqrt {i f+c \log (f)}\right )}{4 \sqrt {i f+c \log (f)}}\\ \end {align*}

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Mathematica [A]
time = 0.51, size = 170, normalized size = 1.65 \begin {gather*} -\frac {(-1)^{3/4} f^a \sqrt {\pi } \left (\text {Erfi}\left (\sqrt [4]{-1} x \sqrt {f-i c \log (f)}\right ) \sqrt {f-i c \log (f)} (f+i c \log (f)) (\cos (d)+i \sin (d))+\sqrt {f+i c \log (f)} \left (f \cos (d) \text {Erf}\left (\frac {(1+i) x \sqrt {f+i c \log (f)}}{\sqrt {2}}\right )-\text {Erfi}\left ((-1)^{3/4} x \sqrt {f+i c \log (f)}\right ) (c \cos (d) \log (f)+(f-i c \log (f)) \sin (d))\right )\right )}{4 \left (f^2+c^2 \log ^2(f)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.16, size = 82, normalized size = 0.80

method	result	size
risch	\(\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{-i d} \erf \left (x \sqrt {i f -c \ln \left (f \right )}\right )}{4 \sqrt {i f -c \ln \left (f \right )}}+\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{i d} \erf \left (\sqrt {-c \ln \left (f \right )-i f}\, x \right )}{4 \sqrt {-c \ln \left (f \right )-i f}}\)	\(82\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(c*x^2+a)*cos(f*x^2+d),x,method=_RETURNVERBOSE)

[Out]

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

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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. 205 vs. \(2 (73) = 146\).
time = 0.28, size = 205, normalized size = 1.99 \begin {gather*} -\frac {\sqrt {\pi } \sqrt {2 \, c^{2} \log \left (f\right )^{2} + 2 \, f^{2}} {\left (f^{a} {\left (i \, \cos \left (d\right ) + \sin \left (d\right )\right )} \operatorname {erf}\left (\sqrt {-c \log \left (f\right ) + i \, f} x\right ) + f^{a} {\left (-i \, \cos \left (d\right ) + \sin \left (d\right )\right )} \operatorname {erf}\left (\sqrt {-c \log \left (f\right ) - i \, f} x\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 (f^{a} {\left (\cos \left (d\right ) - i \, \sin \left (d\right )\right )} \operatorname {erf}\left (\sqrt {-c \log \left (f\right ) + i \, f} x\right ) + f^{a} {\left (\cos \left (d\right ) + i \, \sin \left (d\right )\right )} \operatorname {erf}\left (\sqrt {-c \log \left (f\right ) - i \, f} x\right )\right )} \sqrt {-c \log \left (f\right ) + \sqrt {c^{2} \log \left (f\right )^{2} + f^{2}}}}{8 \, {\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+a)*cos(f*x^2+d),x, algorithm="maxima")

[Out]

-1/8*(sqrt(pi)*sqrt(2*c^2*log(f)^2 + 2*f^2)*(f^a*(I*cos(d) + sin(d))*erf(sqrt(-c*log(f) + I*f)*x) + f^a*(-I*co
s(d) + sin(d))*erf(sqrt(-c*log(f) - I*f)*x))*sqrt(c*log(f) + sqrt(c^2*log(f)^2 + f^2)) - sqrt(pi)*sqrt(2*c^2*l
og(f)^2 + 2*f^2)*(f^a*(cos(d) - I*sin(d))*erf(sqrt(-c*log(f) + I*f)*x) + f^a*(cos(d) + I*sin(d))*erf(sqrt(-c*l
og(f) - I*f)*x))*sqrt(-c*log(f) + sqrt(c^2*log(f)^2 + f^2)))/(c^2*log(f)^2 + f^2)

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Fricas [A]
time = 2.24, size = 109, normalized size = 1.06 \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 (\sqrt {-c \log \left (f\right ) - i \, f} x\right ) e^{\left (a \log \left (f\right ) + i \, d\right )} + \sqrt {\pi } {\left (c \log \left (f\right ) + i \, f\right )} \sqrt {-c \log \left (f\right ) + i \, f} \operatorname {erf}\left (\sqrt {-c \log \left (f\right ) + i \, f} x\right ) e^{\left (a \log \left (f\right ) - i \, d\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+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(sqrt(-c*log(f) - I*f)*x)*e^(a*log(f) + I*d) + sqrt(p
i)*(c*log(f) + I*f)*sqrt(-c*log(f) + I*f)*erf(sqrt(-c*log(f) + I*f)*x)*e^(a*log(f) - I*d))/(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 + 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+a)*cos(f*x**2+d),x)

[Out]

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

[Out]

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

[Out]

int(f^(a + c*x^2)*cos(d + f*x^2), x)

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