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

Optimal. Leaf size=346 \[ -\frac{3 \sqrt{\pi } f^a e^{\frac{(e+i b \log (f))^2}{4 c \log (f)}-i d} \text{Erfi}\left (\frac{-b \log (f)-2 c x \log (f)+i e}{2 \sqrt{c} \sqrt{\log (f)}}\right )}{16 \sqrt{c} \sqrt{\log (f)}}-\frac{\sqrt{\pi } f^a e^{\frac{(3 e+i b \log (f))^2}{4 c \log (f)}-3 i d} \text{Erfi}\left (\frac{-b \log (f)-2 c x \log (f)+3 i e}{2 \sqrt{c} \sqrt{\log (f)}}\right )}{16 \sqrt{c} \sqrt{\log (f)}}+\frac{3 \sqrt{\pi } f^a e^{\frac{(e-i b \log (f))^2}{4 c \log (f)}+i d} \text{Erfi}\left (\frac{b \log (f)+2 c x \log (f)+i e}{2 \sqrt{c} \sqrt{\log (f)}}\right )}{16 \sqrt{c} \sqrt{\log (f)}}+\frac{\sqrt{\pi } f^a e^{3 i d-\frac{(b \log (f)+3 i e)^2}{4 c \log (f)}} \text{Erfi}\left (\frac{b \log (f)+2 c x \log (f)+3 i e}{2 \sqrt{c} \sqrt{\log (f)}}\right )}{16 \sqrt{c} \sqrt{\log (f)}} \]

[Out]

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

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Rubi [A]  time = 0.415833, antiderivative size = 346, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {4473, 2287, 2234, 2204} \[ -\frac{3 \sqrt{\pi } f^a e^{\frac{(e+i b \log (f))^2}{4 c \log (f)}-i d} \text{Erfi}\left (\frac{-b \log (f)-2 c x \log (f)+i e}{2 \sqrt{c} \sqrt{\log (f)}}\right )}{16 \sqrt{c} \sqrt{\log (f)}}-\frac{\sqrt{\pi } f^a e^{\frac{(3 e+i b \log (f))^2}{4 c \log (f)}-3 i d} \text{Erfi}\left (\frac{-b \log (f)-2 c x \log (f)+3 i e}{2 \sqrt{c} \sqrt{\log (f)}}\right )}{16 \sqrt{c} \sqrt{\log (f)}}+\frac{3 \sqrt{\pi } f^a e^{\frac{(e-i b \log (f))^2}{4 c \log (f)}+i d} \text{Erfi}\left (\frac{b \log (f)+2 c x \log (f)+i e}{2 \sqrt{c} \sqrt{\log (f)}}\right )}{16 \sqrt{c} \sqrt{\log (f)}}+\frac{\sqrt{\pi } f^a e^{3 i d-\frac{(b \log (f)+3 i e)^2}{4 c \log (f)}} \text{Erfi}\left (\frac{b \log (f)+2 c x \log (f)+3 i e}{2 \sqrt{c} \sqrt{\log (f)}}\right )}{16 \sqrt{c} \sqrt{\log (f)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*E^((-I)*d + (e + I*b*Log[f])^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(I*e - b*Log[f] - 2*c*x*Log[f])/(2*Sqrt[c]*
Sqrt[Log[f]])])/(16*Sqrt[c]*Sqrt[Log[f]]) - (E^((-3*I)*d + (3*e + I*b*Log[f])^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erf
i[((3*I)*e - b*Log[f] - 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])])/(16*Sqrt[c]*Sqrt[Log[f]]) + (3*E^(I*d + (e -
I*b*Log[f])^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(I*e + b*Log[f] + 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])])/(16*S
qrt[c]*Sqrt[Log[f]]) + (E^((3*I)*d - ((3*I)*e + b*Log[f])^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[((3*I)*e + b*Log[f
] + 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+b x+c x^2} \cos ^3(d+e x) \, dx &=\int \left (\frac{3}{8} e^{-i d-i e x} f^{a+b x+c x^2}+\frac{3}{8} e^{i d+i e x} f^{a+b x+c x^2}+\frac{1}{8} e^{-3 i d-3 i e x} f^{a+b x+c x^2}+\frac{1}{8} e^{3 i d+3 i e x} f^{a+b x+c x^2}\right ) \, dx\\ &=\frac{1}{8} \int e^{-3 i d-3 i e x} f^{a+b x+c x^2} \, dx+\frac{1}{8} \int e^{3 i d+3 i e x} f^{a+b x+c x^2} \, dx+\frac{3}{8} \int e^{-i d-i e x} f^{a+b x+c x^2} \, dx+\frac{3}{8} \int e^{i d+i e x} f^{a+b x+c x^2} \, dx\\ &=\frac{1}{8} \int \exp \left (-3 i d+a \log (f)+c x^2 \log (f)-x (3 i e-b \log (f))\right ) \, dx+\frac{1}{8} \int \exp \left (3 i d+a \log (f)+c x^2 \log (f)+x (3 i e+b \log (f))\right ) \, dx+\frac{3}{8} \int \exp \left (-i d+a \log (f)+c x^2 \log (f)-x (i e-b \log (f))\right ) \, dx+\frac{3}{8} \int \exp \left (i d+a \log (f)+c x^2 \log (f)+x (i e+b \log (f))\right ) \, dx\\ &=\frac{1}{8} \left (3 e^{i d+\frac{(e-i b \log (f))^2}{4 c \log (f)}} f^a\right ) \int \exp \left (\frac{(i e+b \log (f)+2 c x \log (f))^2}{4 c \log (f)}\right ) \, dx+\frac{1}{8} \left (3 e^{-i d+\frac{(e+i b \log (f))^2}{4 c \log (f)}} f^a\right ) \int \exp \left (\frac{(-i e+b \log (f)+2 c x \log (f))^2}{4 c \log (f)}\right ) \, dx+\frac{1}{8} \left (\exp \left (-3 i d+\frac{(3 e+i b \log (f))^2}{4 c \log (f)}\right ) f^a\right ) \int \exp \left (\frac{(-3 i e+b \log (f)+2 c x \log (f))^2}{4 c \log (f)}\right ) \, dx+\frac{1}{8} \left (e^{3 i d-\frac{(3 i e+b \log (f))^2}{4 c \log (f)}} f^a\right ) \int \exp \left (\frac{(3 i e+b \log (f)+2 c x \log (f))^2}{4 c \log (f)}\right ) \, dx\\ &=-\frac{3 e^{-i d+\frac{(e+i b \log (f))^2}{4 c \log (f)}} f^a \sqrt{\pi } \text{erfi}\left (\frac{i e-b \log (f)-2 c x \log (f)}{2 \sqrt{c} \sqrt{\log (f)}}\right )}{16 \sqrt{c} \sqrt{\log (f)}}-\frac{\exp \left (-3 i d+\frac{(3 e+i b \log (f))^2}{4 c \log (f)}\right ) f^a \sqrt{\pi } \text{erfi}\left (\frac{3 i e-b \log (f)-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-i b \log (f))^2}{4 c \log (f)}} f^a \sqrt{\pi } \text{erfi}\left (\frac{i e+b \log (f)+2 c x \log (f)}{2 \sqrt{c} \sqrt{\log (f)}}\right )}{16 \sqrt{c} \sqrt{\log (f)}}+\frac{e^{3 i d-\frac{(3 i e+b \log (f))^2}{4 c \log (f)}} f^a \sqrt{\pi } \text{erfi}\left (\frac{3 i e+b \log (f)+2 c x \log (f)}{2 \sqrt{c} \sqrt{\log (f)}}\right )}{16 \sqrt{c} \sqrt{\log (f)}}\\ \end{align*}

Mathematica [A]  time = 0.966964, size = 386, normalized size = 1.12 \[ \frac{\sqrt{\pi } f^{a-\frac{b^2}{4 c}} e^{\frac{e (e-6 i b \log (f))}{4 c \log (f)}} \left (i \sin (3 d) e^{\frac{2 e^2}{c \log (f)}} \text{Erfi}\left (\frac{\log (f) (b+2 c x)+3 i e}{2 \sqrt{c} \sqrt{\log (f)}}\right )+\cos (3 d) e^{\frac{2 e^2}{c \log (f)}} \text{Erfi}\left (\frac{\log (f) (b+2 c x)+3 i e}{2 \sqrt{c} \sqrt{\log (f)}}\right )-i \sin (3 d) e^{\frac{e (2 e+3 i b \log (f))}{c \log (f)}} \text{Erfi}\left (\frac{\log (f) (b+2 c x)-3 i e}{2 \sqrt{c} \sqrt{\log (f)}}\right )+\cos (3 d) e^{\frac{e (2 e+3 i b \log (f))}{c \log (f)}} \text{Erfi}\left (\frac{\log (f) (b+2 c x)-3 i e}{2 \sqrt{c} \sqrt{\log (f)}}\right )+3 e^{\frac{2 i b e}{c}} (\cos (d)-i \sin (d)) \text{Erfi}\left (\frac{\log (f) (b+2 c x)-i e}{2 \sqrt{c} \sqrt{\log (f)}}\right )+3 e^{\frac{i b e}{c}} (\cos (d)+i \sin (d)) \text{Erfi}\left (\frac{\log (f) (b+2 c x)+i e}{2 \sqrt{c} \sqrt{\log (f)}}\right )\right )}{16 \sqrt{c} \sqrt{\log (f)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(E^((e*(e - (6*I)*b*Log[f]))/(4*c*Log[f]))*f^(a - b^2/(4*c))*Sqrt[Pi]*(E^((e*(2*e + (3*I)*b*Log[f]))/(c*Log[f]
))*Cos[3*d]*Erfi[((-3*I)*e + (b + 2*c*x)*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])] + E^((2*e^2)/(c*Log[f]))*Cos[3*d]*E
rfi[((3*I)*e + (b + 2*c*x)*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])] + 3*E^(((2*I)*b*e)/c)*Erfi[((-I)*e + (b + 2*c*x)*
Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])]*(Cos[d] - I*Sin[d]) + 3*E^((I*b*e)/c)*Erfi[(I*e + (b + 2*c*x)*Log[f])/(2*Sqr
t[c]*Sqrt[Log[f]])]*(Cos[d] + I*Sin[d]) - I*E^((e*(2*e + (3*I)*b*Log[f]))/(c*Log[f]))*Erfi[((-3*I)*e + (b + 2*
c*x)*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])]*Sin[3*d] + I*E^((2*e^2)/(c*Log[f]))*Erfi[((3*I)*e + (b + 2*c*x)*Log[f])
/(2*Sqrt[c]*Sqrt[Log[f]])]*Sin[3*d]))/(16*Sqrt[c]*Sqrt[Log[f]])

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Maple [A]  time = 0.237, size = 334, normalized size = 1. \begin{align*} -{\frac{{f}^{a}\sqrt{\pi }}{16}{{\rm e}^{-{\frac{ \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}-6\,i\ln \left ( f \right ) be+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{b\ln \left ( f \right ) -3\,ie}{2}{\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{ \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}-2\,i\ln \left ( f \right ) be+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{b\ln \left ( f \right ) -ie}{2}{\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{ \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}+2\,i\ln \left ( f \right ) be-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{ie+b\ln \left ( f \right ) }{2}{\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{ \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}+6\,i\ln \left ( f \right ) be-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\,ie+b\ln \left ( f \right ) }{2}{\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+b*x+a)*cos(e*x+d)^3,x)

[Out]

-1/16*Pi^(1/2)*f^a*exp(-1/4*(ln(f)^2*b^2-6*I*ln(f)*b*e+12*I*d*ln(f)*c-9*e^2)/ln(f)/c)/(-c*ln(f))^(1/2)*erf(-(-
c*ln(f))^(1/2)*x+1/2*(b*ln(f)-3*I*e)/(-c*ln(f))^(1/2))-3/16*Pi^(1/2)*f^a*exp(-1/4*(ln(f)^2*b^2-2*I*ln(f)*b*e+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*(b*ln(f)-I*e)/(-c*ln(f))^(1/2))-3/16*P
i^(1/2)*f^a*exp(-1/4*(ln(f)^2*b^2+2*I*ln(f)*b*e-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+b*ln(f))/(-c*ln(f))^(1/2))-1/16*Pi^(1/2)*f^a*exp(-1/4*(ln(f)^2*b^2+6*I*ln(f)*b*e-12*I*d*ln(f)*
c-9*e^2)/ln(f)/c)/(-c*ln(f))^(1/2)*erf(-(-c*ln(f))^(1/2)*x+1/2*(3*I*e+b*ln(f))/(-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+b*x+a)*cos(e*x+d)^3,x, algorithm="maxima")

[Out]

Exception raised: IndexError

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Fricas [A]  time = 0.521183, size = 964, normalized size = 2.79 \begin{align*} -\frac{3 \, \sqrt{\pi } \sqrt{-c \log \left (f\right )} \operatorname{erf}\left (\frac{{\left ({\left (2 \, c x + b\right )} \log \left (f\right ) + i \, e\right )} \sqrt{-c \log \left (f\right )}}{2 \, c \log \left (f\right )}\right ) e^{\left (-\frac{{\left (b^{2} - 4 \, a c\right )} \log \left (f\right )^{2} - e^{2} -{\left (4 i \, c d - 2 i \, b e\right )} \log \left (f\right )}{4 \, c \log \left (f\right )}\right )} + 3 \, \sqrt{\pi } \sqrt{-c \log \left (f\right )} \operatorname{erf}\left (\frac{{\left ({\left (2 \, c x + b\right )} \log \left (f\right ) - i \, e\right )} \sqrt{-c \log \left (f\right )}}{2 \, c \log \left (f\right )}\right ) e^{\left (-\frac{{\left (b^{2} - 4 \, a c\right )} \log \left (f\right )^{2} - e^{2} -{\left (-4 i \, c d + 2 i \, b e\right )} \log \left (f\right )}{4 \, c \log \left (f\right )}\right )} + \sqrt{\pi } \sqrt{-c \log \left (f\right )} \operatorname{erf}\left (\frac{{\left ({\left (2 \, c x + b\right )} \log \left (f\right ) + 3 i \, e\right )} \sqrt{-c \log \left (f\right )}}{2 \, c \log \left (f\right )}\right ) e^{\left (-\frac{{\left (b^{2} - 4 \, a c\right )} \log \left (f\right )^{2} - 9 \, e^{2} -{\left (12 i \, c d - 6 i \, b e\right )} \log \left (f\right )}{4 \, c \log \left (f\right )}\right )} + \sqrt{\pi } \sqrt{-c \log \left (f\right )} \operatorname{erf}\left (\frac{{\left ({\left (2 \, c x + b\right )} \log \left (f\right ) - 3 i \, e\right )} \sqrt{-c \log \left (f\right )}}{2 \, c \log \left (f\right )}\right ) e^{\left (-\frac{{\left (b^{2} - 4 \, a c\right )} \log \left (f\right )^{2} - 9 \, e^{2} -{\left (-12 i \, c d + 6 i \, b e\right )} \log \left (f\right )}{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+b*x+a)*cos(e*x+d)^3,x, algorithm="fricas")

[Out]

-1/16*(3*sqrt(pi)*sqrt(-c*log(f))*erf(1/2*((2*c*x + b)*log(f) + I*e)*sqrt(-c*log(f))/(c*log(f)))*e^(-1/4*((b^2
- 4*a*c)*log(f)^2 - e^2 - (4*I*c*d - 2*I*b*e)*log(f))/(c*log(f))) + 3*sqrt(pi)*sqrt(-c*log(f))*erf(1/2*((2*c*
x + b)*log(f) - I*e)*sqrt(-c*log(f))/(c*log(f)))*e^(-1/4*((b^2 - 4*a*c)*log(f)^2 - e^2 - (-4*I*c*d + 2*I*b*e)*
log(f))/(c*log(f))) + sqrt(pi)*sqrt(-c*log(f))*erf(1/2*((2*c*x + b)*log(f) + 3*I*e)*sqrt(-c*log(f))/(c*log(f))
)*e^(-1/4*((b^2 - 4*a*c)*log(f)^2 - 9*e^2 - (12*I*c*d - 6*I*b*e)*log(f))/(c*log(f))) + sqrt(pi)*sqrt(-c*log(f)
)*erf(1/2*((2*c*x + b)*log(f) - 3*I*e)*sqrt(-c*log(f))/(c*log(f)))*e^(-1/4*((b^2 - 4*a*c)*log(f)^2 - 9*e^2 - (
-12*I*c*d + 6*I*b*e)*log(f))/(c*log(f))))/(c*log(f))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

[Out]

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