3.118 \(\int f^{a+c x^2} \cos ^3(d+e x) \, dx\)

Optimal. Leaf size=293 \[ -\frac{3 \sqrt{\pi } f^a e^{\frac{e^2}{4 c \log (f)}-i d} \text{Erfi}\left (\frac{-2 c x \log (f)+i e}{2 \sqrt{c} \sqrt{\log (f)}}\right )}{16 \sqrt{c} \sqrt{\log (f)}}-\frac{\sqrt{\pi } f^a e^{\frac{9 e^2}{4 c \log (f)}-3 i d} \text{Erfi}\left (\frac{-2 c x \log (f)+3 i e}{2 \sqrt{c} \sqrt{\log (f)}}\right )}{16 \sqrt{c} \sqrt{\log (f)}}+\frac{3 \sqrt{\pi } f^a e^{\frac{e^2}{4 c \log (f)}+i d} \text{Erfi}\left (\frac{2 c x \log (f)+i e}{2 \sqrt{c} \sqrt{\log (f)}}\right )}{16 \sqrt{c} \sqrt{\log (f)}}+\frac{\sqrt{\pi } f^a e^{\frac{9 e^2}{4 c \log (f)}+3 i d} \text{Erfi}\left (\frac{2 c x \log (f)+3 i e}{2 \sqrt{c} \sqrt{\log (f)}}\right )}{16 \sqrt{c} \sqrt{\log (f)}} \]

[Out]

(-3*E^((-I)*d + e^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(I*e - 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])])/(16*Sqrt[c
]*Sqrt[Log[f]]) - (E^((-3*I)*d + (9*e^2)/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[((3*I)*e - 2*c*x*Log[f])/(2*Sqrt[c]*S
qrt[Log[f]])])/(16*Sqrt[c]*Sqrt[Log[f]]) + (3*E^(I*d + e^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(I*e + 2*c*x*Log[f]
)/(2*Sqrt[c]*Sqrt[Log[f]])])/(16*Sqrt[c]*Sqrt[Log[f]]) + (E^((3*I)*d + (9*e^2)/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi
[((3*I)*e + 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])])/(16*Sqrt[c]*Sqrt[Log[f]])

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Rubi [A]  time = 0.330865, antiderivative size = 293, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {4473, 2287, 2234, 2204} \[ -\frac{3 \sqrt{\pi } f^a e^{\frac{e^2}{4 c \log (f)}-i d} \text{Erfi}\left (\frac{-2 c x \log (f)+i e}{2 \sqrt{c} \sqrt{\log (f)}}\right )}{16 \sqrt{c} \sqrt{\log (f)}}-\frac{\sqrt{\pi } f^a e^{\frac{9 e^2}{4 c \log (f)}-3 i d} \text{Erfi}\left (\frac{-2 c x \log (f)+3 i e}{2 \sqrt{c} \sqrt{\log (f)}}\right )}{16 \sqrt{c} \sqrt{\log (f)}}+\frac{3 \sqrt{\pi } f^a e^{\frac{e^2}{4 c \log (f)}+i d} \text{Erfi}\left (\frac{2 c x \log (f)+i e}{2 \sqrt{c} \sqrt{\log (f)}}\right )}{16 \sqrt{c} \sqrt{\log (f)}}+\frac{\sqrt{\pi } f^a e^{\frac{9 e^2}{4 c \log (f)}+3 i d} \text{Erfi}\left (\frac{2 c x \log (f)+3 i e}{2 \sqrt{c} \sqrt{\log (f)}}\right )}{16 \sqrt{c} \sqrt{\log (f)}} \]

Antiderivative was successfully verified.

[In]

Int[f^(a + c*x^2)*Cos[d + e*x]^3,x]

[Out]

(-3*E^((-I)*d + e^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(I*e - 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])])/(16*Sqrt[c
]*Sqrt[Log[f]]) - (E^((-3*I)*d + (9*e^2)/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[((3*I)*e - 2*c*x*Log[f])/(2*Sqrt[c]*S
qrt[Log[f]])])/(16*Sqrt[c]*Sqrt[Log[f]]) + (3*E^(I*d + e^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(I*e + 2*c*x*Log[f]
)/(2*Sqrt[c]*Sqrt[Log[f]])])/(16*Sqrt[c]*Sqrt[Log[f]]) + (E^((3*I)*d + (9*e^2)/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi
[((3*I)*e + 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])])/(16*Sqrt[c]*Sqrt[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 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 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]

Rubi steps

\begin{align*} \int f^{a+c x^2} \cos ^3(d+e x) \, dx &=\int \left (\frac{3}{8} e^{-i d-i e x} f^{a+c x^2}+\frac{3}{8} e^{i d+i e x} f^{a+c x^2}+\frac{1}{8} e^{-3 i d-3 i e x} f^{a+c x^2}+\frac{1}{8} e^{3 i d+3 i e x} f^{a+c x^2}\right ) \, dx\\ &=\frac{1}{8} \int e^{-3 i d-3 i e x} f^{a+c x^2} \, dx+\frac{1}{8} \int e^{3 i d+3 i e x} f^{a+c x^2} \, dx+\frac{3}{8} \int e^{-i d-i e x} f^{a+c x^2} \, dx+\frac{3}{8} \int e^{i d+i e x} f^{a+c x^2} \, dx\\ &=\frac{1}{8} \int e^{-3 i d-3 i e x+a \log (f)+c x^2 \log (f)} \, dx+\frac{1}{8} \int e^{3 i d+3 i e x+a \log (f)+c x^2 \log (f)} \, dx+\frac{3}{8} \int e^{-i d-i e x+a \log (f)+c x^2 \log (f)} \, dx+\frac{3}{8} \int e^{i d+i e x+a \log (f)+c x^2 \log (f)} \, dx\\ &=\frac{1}{8} \left (3 e^{-i d+\frac{e^2}{4 c \log (f)}} f^a\right ) \int e^{\frac{(-i e+2 c x \log (f))^2}{4 c \log (f)}} \, dx+\frac{1}{8} \left (3 e^{i d+\frac{e^2}{4 c \log (f)}} f^a\right ) \int e^{\frac{(i e+2 c x \log (f))^2}{4 c \log (f)}} \, dx+\frac{1}{8} \left (e^{-3 i d+\frac{9 e^2}{4 c \log (f)}} f^a\right ) \int e^{\frac{(-3 i e+2 c x \log (f))^2}{4 c \log (f)}} \, dx+\frac{1}{8} \left (e^{3 i d+\frac{9 e^2}{4 c \log (f)}} f^a\right ) \int e^{\frac{(3 i e+2 c x \log (f))^2}{4 c \log (f)}} \, dx\\ &=-\frac{3 e^{-i d+\frac{e^2}{4 c \log (f)}} f^a \sqrt{\pi } \text{erfi}\left (\frac{i e-2 c x \log (f)}{2 \sqrt{c} \sqrt{\log (f)}}\right )}{16 \sqrt{c} \sqrt{\log (f)}}-\frac{e^{-3 i d+\frac{9 e^2}{4 c \log (f)}} f^a \sqrt{\pi } \text{erfi}\left (\frac{3 i e-2 c x \log (f)}{2 \sqrt{c} \sqrt{\log (f)}}\right )}{16 \sqrt{c} \sqrt{\log (f)}}+\frac{3 e^{i d+\frac{e^2}{4 c \log (f)}} f^a \sqrt{\pi } \text{erfi}\left (\frac{i e+2 c x \log (f)}{2 \sqrt{c} \sqrt{\log (f)}}\right )}{16 \sqrt{c} \sqrt{\log (f)}}+\frac{e^{3 i d+\frac{9 e^2}{4 c \log (f)}} f^a \sqrt{\pi } \text{erfi}\left (\frac{3 i e+2 c x \log (f)}{2 \sqrt{c} \sqrt{\log (f)}}\right )}{16 \sqrt{c} \sqrt{\log (f)}}\\ \end{align*}

Mathematica [A]  time = 0.422011, size = 218, normalized size = 0.74 \[ \frac{\sqrt{\pi } f^a e^{\frac{e^2}{4 c \log (f)}} \left (e^{\frac{2 e^2}{c \log (f)}} \left ((\cos (3 d)-i \sin (3 d)) \text{Erfi}\left (\frac{2 c x \log (f)-3 i e}{2 \sqrt{c} \sqrt{\log (f)}}\right )+(\cos (3 d)+i \sin (3 d)) \text{Erfi}\left (\frac{2 c x \log (f)+3 i e}{2 \sqrt{c} \sqrt{\log (f)}}\right )\right )+3 (\cos (d)-i \sin (d)) \text{Erfi}\left (\frac{2 c x \log (f)-i e}{2 \sqrt{c} \sqrt{\log (f)}}\right )+3 (\cos (d)+i \sin (d)) \text{Erfi}\left (\frac{2 c x \log (f)+i e}{2 \sqrt{c} \sqrt{\log (f)}}\right )\right )}{16 \sqrt{c} \sqrt{\log (f)}} \]

Antiderivative was successfully verified.

[In]

Integrate[f^(a + c*x^2)*Cos[d + e*x]^3,x]

[Out]

(E^(e^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*(3*Erfi[((-I)*e + 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])]*(Cos[d] - I*Sin[d
]) + 3*Erfi[(I*e + 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])]*(Cos[d] + I*Sin[d]) + E^((2*e^2)/(c*Log[f]))*(Erfi[
((-3*I)*e + 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])]*(Cos[3*d] - I*Sin[3*d]) + Erfi[((3*I)*e + 2*c*x*Log[f])/(2
*Sqrt[c]*Sqrt[Log[f]])]*(Cos[3*d] + I*Sin[3*d]))))/(16*Sqrt[c]*Sqrt[Log[f]])

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Maple [A]  time = 0.202, size = 242, normalized size = 0.8 \begin{align*}{\frac{{f}^{a}\sqrt{\pi }}{16}{{\rm e}^{-{\frac{12\,id\ln \left ( f \right ) c-9\,{e}^{2}}{4\,c\ln \left ( f \right ) }}}}{\it Erf} \left ( \sqrt{-c\ln \left ( f \right ) }x+{{\frac{3\,i}{2}}e{\frac{1}{\sqrt{-c\ln \left ( f \right ) }}}} \right ){\frac{1}{\sqrt{-c\ln \left ( f \right ) }}}}+{\frac{3\,{f}^{a}\sqrt{\pi }}{16}{{\rm e}^{-{\frac{4\,id\ln \left ( f \right ) c-{e}^{2}}{4\,c\ln \left ( f \right ) }}}}{\it Erf} \left ( \sqrt{-c\ln \left ( f \right ) }x+{{\frac{i}{2}}e{\frac{1}{\sqrt{-c\ln \left ( f \right ) }}}} \right ){\frac{1}{\sqrt{-c\ln \left ( f \right ) }}}}-{\frac{3\,{f}^{a}\sqrt{\pi }}{16}{{\rm e}^{{\frac{4\,id\ln \left ( f \right ) c+{e}^{2}}{4\,c\ln \left ( f \right ) }}}}{\it Erf} \left ( -\sqrt{-c\ln \left ( f \right ) }x+{{\frac{i}{2}}e{\frac{1}{\sqrt{-c\ln \left ( f \right ) }}}} \right ){\frac{1}{\sqrt{-c\ln \left ( f \right ) }}}}-{\frac{{f}^{a}\sqrt{\pi }}{16}{{\rm e}^{{\frac{12\,id\ln \left ( f \right ) c+9\,{e}^{2}}{4\,c\ln \left ( f \right ) }}}}{\it Erf} \left ( -\sqrt{-c\ln \left ( f \right ) }x+{{\frac{3\,i}{2}}e{\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+a)*cos(e*x+d)^3,x)

[Out]

1/16*Pi^(1/2)*f^a*exp(-3/4*(4*I*d*ln(f)*c-3*e^2)/ln(f)/c)/(-c*ln(f))^(1/2)*erf((-c*ln(f))^(1/2)*x+3/2*I*e/(-c*
ln(f))^(1/2))+3/16*Pi^(1/2)*f^a*exp(-1/4*(4*I*d*ln(f)*c-e^2)/ln(f)/c)/(-c*ln(f))^(1/2)*erf((-c*ln(f))^(1/2)*x+
1/2*I*e/(-c*ln(f))^(1/2))-3/16*Pi^(1/2)*f^a*exp(1/4*(4*I*d*ln(f)*c+e^2)/ln(f)/c)/(-c*ln(f))^(1/2)*erf(-(-c*ln(
f))^(1/2)*x+1/2*I*e/(-c*ln(f))^(1/2))-1/16*Pi^(1/2)*f^a*exp(3/4*(4*I*d*ln(f)*c+3*e^2)/ln(f)/c)/(-c*ln(f))^(1/2
)*erf(-(-c*ln(f))^(1/2)*x+3/2*I*e/(-c*ln(f))^(1/2))

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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+a)*cos(e*x+d)^3,x, algorithm="maxima")

[Out]

Exception raised: IndexError

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Fricas [A]  time = 0.528473, size = 815, normalized size = 2.78 \begin{align*} -\frac{\sqrt{\pi } \sqrt{-c \log \left (f\right )} \operatorname{erf}\left (\frac{{\left (2 \, c x \log \left (f\right ) + 3 i \, e\right )} \sqrt{-c \log \left (f\right )}}{2 \, c \log \left (f\right )}\right ) e^{\left (\frac{4 \, a c \log \left (f\right )^{2} + 12 i \, c d \log \left (f\right ) + 9 \, e^{2}}{4 \, c \log \left (f\right )}\right )} + 3 \, \sqrt{\pi } \sqrt{-c \log \left (f\right )} \operatorname{erf}\left (\frac{{\left (2 \, c x \log \left (f\right ) + i \, e\right )} \sqrt{-c \log \left (f\right )}}{2 \, c \log \left (f\right )}\right ) e^{\left (\frac{4 \, a c \log \left (f\right )^{2} + 4 i \, c d \log \left (f\right ) + e^{2}}{4 \, c \log \left (f\right )}\right )} + 3 \, \sqrt{\pi } \sqrt{-c \log \left (f\right )} \operatorname{erf}\left (\frac{{\left (2 \, c x \log \left (f\right ) - i \, e\right )} \sqrt{-c \log \left (f\right )}}{2 \, c \log \left (f\right )}\right ) e^{\left (\frac{4 \, a c \log \left (f\right )^{2} - 4 i \, c d \log \left (f\right ) + e^{2}}{4 \, c \log \left (f\right )}\right )} + \sqrt{\pi } \sqrt{-c \log \left (f\right )} \operatorname{erf}\left (\frac{{\left (2 \, c x \log \left (f\right ) - 3 i \, e\right )} \sqrt{-c \log \left (f\right )}}{2 \, c \log \left (f\right )}\right ) e^{\left (\frac{4 \, a c \log \left (f\right )^{2} - 12 i \, c d \log \left (f\right ) + 9 \, e^{2}}{4 \, c \log \left (f\right )}\right )}}{16 \, c \log \left (f\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(c*x^2+a)*cos(e*x+d)^3,x, algorithm="fricas")

[Out]

-1/16*(sqrt(pi)*sqrt(-c*log(f))*erf(1/2*(2*c*x*log(f) + 3*I*e)*sqrt(-c*log(f))/(c*log(f)))*e^(1/4*(4*a*c*log(f
)^2 + 12*I*c*d*log(f) + 9*e^2)/(c*log(f))) + 3*sqrt(pi)*sqrt(-c*log(f))*erf(1/2*(2*c*x*log(f) + I*e)*sqrt(-c*l
og(f))/(c*log(f)))*e^(1/4*(4*a*c*log(f)^2 + 4*I*c*d*log(f) + e^2)/(c*log(f))) + 3*sqrt(pi)*sqrt(-c*log(f))*erf
(1/2*(2*c*x*log(f) - I*e)*sqrt(-c*log(f))/(c*log(f)))*e^(1/4*(4*a*c*log(f)^2 - 4*I*c*d*log(f) + e^2)/(c*log(f)
)) + sqrt(pi)*sqrt(-c*log(f))*erf(1/2*(2*c*x*log(f) - 3*I*e)*sqrt(-c*log(f))/(c*log(f)))*e^(1/4*(4*a*c*log(f)^
2 - 12*I*c*d*log(f) + 9*e^2)/(c*log(f))))/(c*log(f))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int f^{a + c x^{2}} \cos ^{3}{\left (d + e x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f**(c*x**2+a)*cos(e*x+d)**3,x)

[Out]

Integral(f**(a + c*x**2)*cos(d + e*x)**3, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int f^{c x^{2} + a} \cos \left (e x + d\right )^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(c*x^2+a)*cos(e*x+d)^3,x, algorithm="giac")

[Out]

integrate(f^(c*x^2 + a)*cos(e*x + d)^3, x)