3.100 \(\int f^{a+b x+c x^2} \sin (d+e x+f x^2) \, dx\)

Optimal. Leaf size=212 \[ \frac{i \sqrt{\pi } f^a \exp \left (-\frac{(e+i b \log (f))^2}{-4 c \log (f)+4 i f}-i d\right ) \text{Erf}\left (\frac{-b \log (f)+2 x (-c \log (f)+i f)+i e}{2 \sqrt{-c \log (f)+i f}}\right )}{4 \sqrt{-c \log (f)+i f}}-\frac{i \sqrt{\pi } f^a \exp \left (\frac{(e-i b \log (f))^2}{4 c \log (f)+4 i f}+i d\right ) \text{Erfi}\left (\frac{b \log (f)+2 x (c \log (f)+i f)+i e}{2 \sqrt{c \log (f)+i f}}\right )}{4 \sqrt{c \log (f)+i f}} \]

[Out]

((I/4)*E^((-I)*d - (e + I*b*Log[f])^2/((4*I)*f - 4*c*Log[f]))*f^a*Sqrt[Pi]*Erf[(I*e - b*Log[f] + 2*x*(I*f - c*
Log[f]))/(2*Sqrt[I*f - c*Log[f]])])/Sqrt[I*f - c*Log[f]] - ((I/4)*E^(I*d + (e - I*b*Log[f])^2/((4*I)*f + 4*c*L
og[f]))*f^a*Sqrt[Pi]*Erfi[(I*e + b*Log[f] + 2*x*(I*f + c*Log[f]))/(2*Sqrt[I*f + c*Log[f]])])/Sqrt[I*f + c*Log[
f]]

________________________________________________________________________________________

Rubi [A]  time = 0.566377, antiderivative size = 212, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {4472, 2287, 2234, 2205, 2204} \[ \frac{i \sqrt{\pi } f^a \exp \left (-\frac{(e+i b \log (f))^2}{-4 c \log (f)+4 i f}-i d\right ) \text{Erf}\left (\frac{-b \log (f)+2 x (-c \log (f)+i f)+i e}{2 \sqrt{-c \log (f)+i f}}\right )}{4 \sqrt{-c \log (f)+i f}}-\frac{i \sqrt{\pi } f^a \exp \left (\frac{(e-i b \log (f))^2}{4 c \log (f)+4 i f}+i d\right ) \text{Erfi}\left (\frac{b \log (f)+2 x (c \log (f)+i f)+i e}{2 \sqrt{c \log (f)+i f}}\right )}{4 \sqrt{c \log (f)+i f}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((I/4)*E^((-I)*d - (e + I*b*Log[f])^2/((4*I)*f - 4*c*Log[f]))*f^a*Sqrt[Pi]*Erf[(I*e - b*Log[f] + 2*x*(I*f - c*
Log[f]))/(2*Sqrt[I*f - c*Log[f]])])/Sqrt[I*f - c*Log[f]] - ((I/4)*E^(I*d + (e - I*b*Log[f])^2/((4*I)*f + 4*c*L
og[f]))*f^a*Sqrt[Pi]*Erfi[(I*e + b*Log[f] + 2*x*(I*f + c*Log[f]))/(2*Sqrt[I*f + c*Log[f]])])/Sqrt[I*f + c*Log[
f]]

Rule 4472

Int[(F_)^(u_)*Sin[v_]^(n_.), x_Symbol] :> Int[ExpandTrigToExp[F^u, Sin[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 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 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} \sin \left (d+e x+f x^2\right ) \, dx &=\int \left (\frac{1}{2} i e^{-i d-i e x-i f x^2} f^{a+b x+c x^2}-\frac{1}{2} i e^{i d+i e x+i f x^2} f^{a+b x+c x^2}\right ) \, dx\\ &=\frac{1}{2} i \int e^{-i d-i e x-i f x^2} f^{a+b x+c x^2} \, dx-\frac{1}{2} i \int e^{i d+i e x+i f x^2} f^{a+b x+c x^2} \, dx\\ &=\frac{1}{2} i \int \exp \left (-i d+a \log (f)-x (i e-b \log (f))-x^2 (i f-c \log (f))\right ) \, dx-\frac{1}{2} i \int \exp \left (i d+a \log (f)+x (i e+b \log (f))+x^2 (i f+c \log (f))\right ) \, dx\\ &=\frac{1}{2} \left (i \exp \left (-i d-\frac{(e+i b \log (f))^2}{4 i f-4 c \log (f)}\right ) f^a\right ) \int \exp \left (\frac{(-i e+b \log (f)+2 x (-i f+c \log (f)))^2}{4 (-i f+c \log (f))}\right ) \, dx-\frac{1}{2} \left (i \exp \left (i d+\frac{(e-i b \log (f))^2}{4 i f+4 c \log (f)}\right ) f^a\right ) \int \exp \left (\frac{(i e+b \log (f)+2 x (i f+c \log (f)))^2}{4 (i f+c \log (f))}\right ) \, dx\\ &=\frac{i \exp \left (-i d-\frac{(e+i b \log (f))^2}{4 i f-4 c \log (f)}\right ) f^a \sqrt{\pi } \text{erf}\left (\frac{i e-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{i \exp \left (i d+\frac{(e-i b \log (f))^2}{4 i f+4 c \log (f)}\right ) f^a \sqrt{\pi } \text{erfi}\left (\frac{i e+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*}

Mathematica [A]  time = 2.18063, size = 347, normalized size = 1.64 \[ -\frac{\sqrt [4]{-1} \sqrt{\pi } f^{\frac{f (a f-b e)+a c^2 \log ^2(f)}{c^2 \log ^2(f)+f^2}} \exp \left (-\frac{1}{4} i \left (\frac{b^2 \log ^2(f)}{f+i c \log (f)}+\frac{e^2}{f-i c \log (f)}\right )\right ) \left (\sqrt{f-i c \log (f)} (f+i c \log (f)) (\cos (d)+i \sin (d)) e^{\frac{i b^2 f \log ^2(f)}{2 \left (c^2 \log ^2(f)+f^2\right )}} f^{\frac{b e}{2 f+2 i c \log (f)}} \text{Erfi}\left (\frac{\sqrt [4]{-1} (-i \log (f) (b+2 c x)+e+2 f x)}{2 \sqrt{f-i c \log (f)}}\right )+(f-i c \log (f)) \sqrt{f+i c \log (f)} (\sin (d)+i \cos (d)) e^{\frac{i e^2 f}{2 \left (c^2 \log ^2(f)+f^2\right )}} f^{\frac{b e}{2 f-2 i c \log (f)}} \text{Erfi}\left (\frac{(-1)^{3/4} (i \log (f) (b+2 c x)+e+2 f x)}{2 \sqrt{f+i c \log (f)}}\right )\right )}{4 \left (c^2 \log ^2(f)+f^2\right )} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-((-1)^(1/4)*f^((f*(-(b*e) + a*f) + a*c^2*Log[f]^2)/(f^2 + c^2*Log[f]^2))*Sqrt[Pi]*(E^(((I/2)*b^2*f*Log[f]^2)/
(f^2 + c^2*Log[f]^2))*f^((b*e)/(2*f + (2*I)*c*Log[f]))*Erfi[((-1)^(1/4)*(e + 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]) + E^(((I/2)*e^2*f)/(f^2 + c^
2*Log[f]^2))*f^((b*e)/(2*f - (2*I)*c*Log[f]))*Erfi[((-1)^(3/4)*(e + 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])))/(4*E^((I/4)*(e^2/(f - I*c*Log[f]) +
(b^2*Log[f]^2)/(f + I*c*Log[f])))*(f^2 + c^2*Log[f]^2))

________________________________________________________________________________________

Maple [A]  time = 0.289, size = 216, normalized size = 1. \begin{align*}{{\frac{i}{4}}{f}^{a}\sqrt{\pi }{{\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+4\,df-{e}^{2}}{4\,if+4\,c\ln \left ( f \right ) }}}}{\it Erf} \left ( -\sqrt{-c\ln \left ( f \right ) -if}x+{\frac{ie+b\ln \left ( f \right ) }{2}{\frac{1}{\sqrt{-c\ln \left ( f \right ) -if}}}} \right ){\frac{1}{\sqrt{-c\ln \left ( f \right ) -if}}}}-{{\frac{i}{4}}{f}^{a}\sqrt{\pi }{{\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+4\,df-{e}^{2}}{4\,c\ln \left ( f \right ) -4\,if}}}}{\it Erf} \left ( -x\sqrt{if-c\ln \left ( f \right ) }+{\frac{b\ln \left ( f \right ) -ie}{2}{\frac{1}{\sqrt{if-c\ln \left ( f \right ) }}}} \right ){\frac{1}{\sqrt{if-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)*sin(f*x^2+e*x+d),x)

[Out]

1/4*I*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+4*d*f-e^2)/(I*f+c*ln(f)))/(-c*ln(f)-I*f)^
(1/2)*erf(-(-c*ln(f)-I*f)^(1/2)*x+1/2*(I*e+b*ln(f))/(-c*ln(f)-I*f)^(1/2))-1/4*I*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+4*d*f-e^2)/(-I*f+c*ln(f)))/(I*f-c*ln(f))^(1/2)*erf(-x*(I*f-c*ln(f))^(1/2)+1/2
*(b*ln(f)-I*e)/(I*f-c*ln(f))^(1/2))

________________________________________________________________________________________

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)*sin(f*x^2+e*x+d),x, algorithm="maxima")

[Out]

Exception raised: IndexError

________________________________________________________________________________________

Fricas [B]  time = 0.565179, size = 938, normalized size = 4.42 \begin{align*} \frac{\sqrt{\pi }{\left (i \, c \log \left (f\right ) + f\right )} \sqrt{-c \log \left (f\right ) - i \, f} \operatorname{erf}\left (\frac{{\left (2 \, f^{2} x +{\left (2 \, c^{2} x + b c\right )} \log \left (f\right )^{2} + e f +{\left (i \, c e - i \, b f\right )} \log \left (f\right )\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{{\left (b^{2} c - 4 \, a c^{2}\right )} \log \left (f\right )^{3} + i \, e^{2} f - 4 i \, d f^{2} -{\left (4 i \, c^{2} d - 2 i \, b c e + i \, b^{2} f\right )} \log \left (f\right )^{2} -{\left (c e^{2} - 2 \, b e f + 4 \, a f^{2}\right )} \log \left (f\right )}{4 \,{\left (c^{2} \log \left (f\right )^{2} + f^{2}\right )}}\right )} + \sqrt{\pi }{\left (-i \, c \log \left (f\right ) + f\right )} \sqrt{-c \log \left (f\right ) + i \, f} \operatorname{erf}\left (\frac{{\left (2 \, f^{2} x +{\left (2 \, c^{2} x + b c\right )} \log \left (f\right )^{2} + e f +{\left (-i \, c e + i \, b f\right )} \log \left (f\right )\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{{\left (b^{2} c - 4 \, a c^{2}\right )} \log \left (f\right )^{3} - i \, e^{2} f + 4 i \, d f^{2} -{\left (-4 i \, c^{2} d + 2 i \, b c e - i \, b^{2} f\right )} \log \left (f\right )^{2} -{\left (c e^{2} - 2 \, b e f + 4 \, a f^{2}\right )} \log \left (f\right )}{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{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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