3.2.3 \(\int f^{a+b x+c x^2} \sin (a+b x+e x^2) \, dx\) [103]

Optimal. Leaf size=213 \[ \frac {i e^{-\left ((i-\log (f)) \left (a-\frac {b^2 (i-\log (f))}{4 i e-4 c \log (f)}\right )\right )} \sqrt {\pi } \text {Erf}\left (\frac {b (i-\log (f))+2 x (i e-c \log (f))}{2 \sqrt {i e-c \log (f)}}\right )}{4 \sqrt {i e-c \log (f)}}-\frac {i e^{(i+\log (f)) \left (a-\frac {b^2 (i+\log (f))}{4 i e+4 c \log (f)}\right )} \sqrt {\pi } \text {Erfi}\left (\frac {b (i+\log (f))+2 x (i e+c \log (f))}{2 \sqrt {i e+c \log (f)}}\right )}{4 \sqrt {i e+c \log (f)}} \]

[Out]

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

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Rubi [A]
time = 0.56, antiderivative size = 213, normalized size of antiderivative = 1.00, 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 = {4560, 2325, 2266, 2236, 2235} \begin {gather*} \frac {i \sqrt {\pi } \exp \left (-\left ((-\log (f)+i) \left (a-\frac {b^2 (-\log (f)+i)}{-4 c \log (f)+4 i e}\right )\right )\right ) \text {Erf}\left (\frac {b (-\log (f)+i)+2 x (-c \log (f)+i e)}{2 \sqrt {-c \log (f)+i e}}\right )}{4 \sqrt {-c \log (f)+i e}}-\frac {i \sqrt {\pi } \exp \left ((\log (f)+i) \left (a-\frac {b^2 (\log (f)+i)}{4 c \log (f)+4 i e}\right )\right ) \text {Erfi}\left (\frac {b (\log (f)+i)+2 x (c \log (f)+i e)}{2 \sqrt {c \log (f)+i e}}\right )}{4 \sqrt {c \log (f)+i e}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((I/4)*Sqrt[Pi]*Erf[(b*(I - Log[f]) + 2*x*(I*e - c*Log[f]))/(2*Sqrt[I*e - c*Log[f]])])/(E^((I - Log[f])*(a - (
b^2*(I - Log[f]))/((4*I)*e - 4*c*Log[f])))*Sqrt[I*e - c*Log[f]]) - ((I/4)*E^((I + Log[f])*(a - (b^2*(I + Log[f
]))/((4*I)*e + 4*c*Log[f])))*Sqrt[Pi]*Erfi[(b*(I + Log[f]) + 2*x*(I*e + c*Log[f]))/(2*Sqrt[I*e + c*Log[f]])])/
Sqrt[I*e + 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 4560

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]

Rubi steps

\begin {align*} \int f^{a+b x+c x^2} \sin \left (a+b x+e x^2\right ) \, dx &=\int \left (\frac {1}{2} i e^{-i a-i b x-i e x^2} f^{a+b x+c x^2}-\frac {1}{2} i e^{i a+i b x+i e x^2} f^{a+b x+c x^2}\right ) \, dx\\ &=\frac {1}{2} i \int e^{-i a-i b x-i e x^2} f^{a+b x+c x^2} \, dx-\frac {1}{2} i \int e^{i a+i b x+i e x^2} f^{a+b x+c x^2} \, dx\\ &=\frac {1}{2} i \int \exp \left (-a (i-\log (f))-b x (i-\log (f))-x^2 (i e-c \log (f))\right ) \, dx-\frac {1}{2} i \int \exp \left (a (i+\log (f))+b x (i+\log (f))+x^2 (i e+c \log (f))\right ) \, dx\\ &=\frac {1}{2} \left (i \exp \left (-(i-\log (f)) \left (a-\frac {b^2 (i-\log (f))}{4 i e-4 c \log (f)}\right )\right )\right ) \int \exp \left (\frac {(-b (i-\log (f))+2 x (-i e+c \log (f)))^2}{4 (-i e+c \log (f))}\right ) \, dx-\frac {1}{2} \left (i \exp \left ((i+\log (f)) \left (a-\frac {b^2 (i+\log (f))}{4 i e+4 c \log (f)}\right )\right )\right ) \int \exp \left (\frac {(b (i+\log (f))+2 x (i e+c \log (f)))^2}{4 (i e+c \log (f))}\right ) \, dx\\ &=\frac {i \exp \left (-(i-\log (f)) \left (a-\frac {b^2 (i-\log (f))}{4 i e-4 c \log (f)}\right )\right ) \sqrt {\pi } \text {erf}\left (\frac {b (i-\log (f))+2 x (i e-c \log (f))}{2 \sqrt {i e-c \log (f)}}\right )}{4 \sqrt {i e-c \log (f)}}-\frac {i \exp \left ((i+\log (f)) \left (a-\frac {b^2 (i+\log (f))}{4 i e+4 c \log (f)}\right )\right ) \sqrt {\pi } \text {erfi}\left (\frac {b (i+\log (f))+2 x (i e+c \log (f))}{2 \sqrt {i e+c \log (f)}}\right )}{4 \sqrt {i e+c \log (f)}}\\ \end {align*}

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Mathematica [A]
time = 1.95, size = 324, normalized size = 1.52 \begin {gather*} \frac {e^{-\frac {b^2 c \log ^3(f)}{2 \left (e^2+c^2 \log ^2(f)\right )}} f^{a-\frac {b^2}{2 (e-i c \log (f))}} \sqrt {\pi } \left (-e^{\frac {1}{4} b^2 \left (\frac {1}{-i e+c \log (f)}+\frac {\log ^2(f)}{i e+c \log (f)}\right )} f^{\frac {i b^2 c \log (f)}{e^2+c^2 \log ^2(f)}} \text {Erfi}\left (\frac {-i (b+2 e x)+(b+2 c x) \log (f)}{2 \sqrt {-i e+c \log (f)}}\right ) (e-i c \log (f)) \sqrt {-i e+c \log (f)} (\cos (a)-i \sin (a))+e^{\frac {1}{4} b^2 \left (\frac {\log ^2(f)}{-i e+c \log (f)}+\frac {1}{i e+c \log (f)}\right )} \text {Erfi}\left (\frac {-i (b+2 e x)-(b+2 c x) \log (f)}{2 \sqrt {i e+c \log (f)}}\right ) (e+i c \log (f)) \sqrt {i e+c \log (f)} (\cos (a)+i \sin (a))\right )}{4 \left (e^2+c^2 \log ^2(f)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.91, size = 217, normalized size = 1.02

method	result	size
risch	\(\frac {i \sqrt {\pi }\, f^{a} {\mathrm e}^{\frac {-\ln \left (f \right )^{2} b^{2}+4 i \ln \left (f \right ) a c -2 i \ln \left (f \right ) b^{2}-4 a e +b^{2}}{4 i e +4 c \ln \left (f \right )}} \erf \left (-\sqrt {-c \ln \left (f \right )-i e}\, x +\frac {b \ln \left (f \right )+i b}{2 \sqrt {-c \ln \left (f \right )-i e}}\right )}{4 \sqrt {-c \ln \left (f \right )-i e}}-\frac {i \sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {\ln \left (f \right )^{2} b^{2}+4 i \ln \left (f \right ) a c -2 i \ln \left (f \right ) b^{2}+4 a e -b^{2}}{4 \left (c \ln \left (f \right )-i e \right )}} \erf \left (-\sqrt {i e -c \ln \left (f \right )}\, x +\frac {b \ln \left (f \right )-i b}{2 \sqrt {i e -c \ln \left (f \right )}}\right )}{4 \sqrt {i e -c \ln \left (f \right )}}\)	\(217\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(c*x^2+b*x+a)*sin(e*x^2+b*x+a),x,method=_RETURNVERBOSE)

[Out]

1/4*I*Pi^(1/2)*f^a*exp(1/4*(-ln(f)^2*b^2+4*I*ln(f)*a*c-2*I*ln(f)*b^2-4*a*e+b^2)/(I*e+c*ln(f)))/(-c*ln(f)-I*e)^
(1/2)*erf(-(-c*ln(f)-I*e)^(1/2)*x+1/2*(b*ln(f)+I*b)/(-c*ln(f)-I*e)^(1/2))-1/4*I*Pi^(1/2)*f^a*exp(-1/4*(ln(f)^2
*b^2+4*I*ln(f)*a*c-2*I*ln(f)*b^2+4*a*e-b^2)/(c*ln(f)-I*e))/(I*e-c*ln(f))^(1/2)*erf(-(I*e-c*ln(f))^(1/2)*x+1/2*
(b*ln(f)-I*b)/(I*e-c*ln(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. 1012 vs. \(2 (161) = 322\).
time = 0.30, size = 1012, normalized size = 4.75 \begin {gather*} \frac {\sqrt {\pi } \sqrt {2 \, c^{2} \log \left (f\right )^{2} + 2 \, e^{2}} {\left ({\left (f^{\frac {b^{2} c}{4 \, {\left (c^{2} \log \left (f\right )^{2} + e^{2}\right )}}} f^{a} \cos \left (-\frac {b^{2} e + {\left (2 \, b^{2} c - 4 \, a c^{2} - b^{2} e\right )} \log \left (f\right )^{2} - 4 \, a e^{2}}{4 \, {\left (c^{2} \log \left (f\right )^{2} + e^{2}\right )}}\right ) - i \, f^{\frac {b^{2} c}{4 \, {\left (c^{2} \log \left (f\right )^{2} + e^{2}\right )}}} f^{a} \sin \left (-\frac {b^{2} e + {\left (2 \, b^{2} c - 4 \, a c^{2} - b^{2} e\right )} \log \left (f\right )^{2} - 4 \, a e^{2}}{4 \, {\left (c^{2} \log \left (f\right )^{2} + e^{2}\right )}}\right )\right )} \operatorname {erf}\left (\frac {{\left (2 \, {\left (c \log \left (f\right ) - i \, e\right )} x + b \log \left (f\right ) - i \, b\right )} \sqrt {-c \log \left (f\right ) + i \, e}}{2 \, {\left (c \log \left (f\right ) - i \, e\right )}}\right ) + {\left (f^{\frac {b^{2} c}{4 \, {\left (c^{2} \log \left (f\right )^{2} + e^{2}\right )}}} f^{a} \cos \left (-\frac {b^{2} e + {\left (2 \, b^{2} c - 4 \, a c^{2} - b^{2} e\right )} \log \left (f\right )^{2} - 4 \, a e^{2}}{4 \, {\left (c^{2} \log \left (f\right )^{2} + e^{2}\right )}}\right ) + i \, f^{\frac {b^{2} c}{4 \, {\left (c^{2} \log \left (f\right )^{2} + e^{2}\right )}}} f^{a} \sin \left (-\frac {b^{2} e + {\left (2 \, b^{2} c - 4 \, a c^{2} - b^{2} e\right )} \log \left (f\right )^{2} - 4 \, a e^{2}}{4 \, {\left (c^{2} \log \left (f\right )^{2} + e^{2}\right )}}\right )\right )} \operatorname {erf}\left (\frac {{\left (2 \, {\left (c \log \left (f\right ) + i \, e\right )} x + b \log \left (f\right ) + i \, b\right )} \sqrt {-c \log \left (f\right ) - i \, e}}{2 \, {\left (c \log \left (f\right ) + i \, e\right )}}\right )\right )} \sqrt {c \log \left (f\right ) + \sqrt {c^{2} \log \left (f\right )^{2} + e^{2}}} + \sqrt {\pi } \sqrt {2 \, c^{2} \log \left (f\right )^{2} + 2 \, e^{2}} {\left ({\left (i \, f^{\frac {b^{2} c}{4 \, {\left (c^{2} \log \left (f\right )^{2} + e^{2}\right )}}} f^{a} \cos \left (-\frac {b^{2} e + {\left (2 \, b^{2} c - 4 \, a c^{2} - b^{2} e\right )} \log \left (f\right )^{2} - 4 \, a e^{2}}{4 \, {\left (c^{2} \log \left (f\right )^{2} + e^{2}\right )}}\right ) + f^{\frac {b^{2} c}{4 \, {\left (c^{2} \log \left (f\right )^{2} + e^{2}\right )}}} f^{a} \sin \left (-\frac {b^{2} e + {\left (2 \, b^{2} c - 4 \, a c^{2} - b^{2} e\right )} \log \left (f\right )^{2} - 4 \, a e^{2}}{4 \, {\left (c^{2} \log \left (f\right )^{2} + e^{2}\right )}}\right )\right )} \operatorname {erf}\left (\frac {{\left (2 \, {\left (c \log \left (f\right ) - i \, e\right )} x + b \log \left (f\right ) - i \, b\right )} \sqrt {-c \log \left (f\right ) + i \, e}}{2 \, {\left (c \log \left (f\right ) - i \, e\right )}}\right ) + {\left (-i \, f^{\frac {b^{2} c}{4 \, {\left (c^{2} \log \left (f\right )^{2} + e^{2}\right )}}} f^{a} \cos \left (-\frac {b^{2} e + {\left (2 \, b^{2} c - 4 \, a c^{2} - b^{2} e\right )} \log \left (f\right )^{2} - 4 \, a e^{2}}{4 \, {\left (c^{2} \log \left (f\right )^{2} + e^{2}\right )}}\right ) + f^{\frac {b^{2} c}{4 \, {\left (c^{2} \log \left (f\right )^{2} + e^{2}\right )}}} f^{a} \sin \left (-\frac {b^{2} e + {\left (2 \, b^{2} c - 4 \, a c^{2} - b^{2} e\right )} \log \left (f\right )^{2} - 4 \, a e^{2}}{4 \, {\left (c^{2} \log \left (f\right )^{2} + e^{2}\right )}}\right )\right )} \operatorname {erf}\left (\frac {{\left (2 \, {\left (c \log \left (f\right ) + i \, e\right )} x + b \log \left (f\right ) + i \, b\right )} \sqrt {-c \log \left (f\right ) - i \, e}}{2 \, {\left (c \log \left (f\right ) + i \, e\right )}}\right )\right )} \sqrt {-c \log \left (f\right ) + \sqrt {c^{2} \log \left (f\right )^{2} + e^{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} + e^{2}\right )}} + \frac {b^{2} e \log \left (f\right )}{2 \, {\left (c^{2} \log \left (f\right )^{2} + e^{2}\right )}}\right )} \log \left (f\right )^{2} + e^{\left (\frac {b^{2} c \log \left (f\right )^{3}}{4 \, {\left (c^{2} \log \left (f\right )^{2} + e^{2}\right )}} + \frac {b^{2} e \log \left (f\right )}{2 \, {\left (c^{2} \log \left (f\right )^{2} + e^{2}\right )}} + 2\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(sqrt(pi)*sqrt(2*c^2*log(f)^2 + 2*e^2)*((f^(1/4*b^2*c/(c^2*log(f)^2 + e^2))*f^a*cos(-1/4*(b^2*e + (2*b^2*c
- 4*a*c^2 - b^2*e)*log(f)^2 - 4*a*e^2)/(c^2*log(f)^2 + e^2)) - I*f^(1/4*b^2*c/(c^2*log(f)^2 + e^2))*f^a*sin(-
1/4*(b^2*e + (2*b^2*c - 4*a*c^2 - b^2*e)*log(f)^2 - 4*a*e^2)/(c^2*log(f)^2 + e^2)))*erf(1/2*(2*(c*log(f) - I*e
)*x + b*log(f) - I*b)*sqrt(-c*log(f) + I*e)/(c*log(f) - I*e)) + (f^(1/4*b^2*c/(c^2*log(f)^2 + e^2))*f^a*cos(-1
/4*(b^2*e + (2*b^2*c - 4*a*c^2 - b^2*e)*log(f)^2 - 4*a*e^2)/(c^2*log(f)^2 + e^2)) + I*f^(1/4*b^2*c/(c^2*log(f)
^2 + e^2))*f^a*sin(-1/4*(b^2*e + (2*b^2*c - 4*a*c^2 - b^2*e)*log(f)^2 - 4*a*e^2)/(c^2*log(f)^2 + e^2)))*erf(1/
2*(2*(c*log(f) + I*e)*x + b*log(f) + I*b)*sqrt(-c*log(f) - I*e)/(c*log(f) + I*e)))*sqrt(c*log(f) + sqrt(c^2*lo
g(f)^2 + e^2)) + sqrt(pi)*sqrt(2*c^2*log(f)^2 + 2*e^2)*((I*f^(1/4*b^2*c/(c^2*log(f)^2 + e^2))*f^a*cos(-1/4*(b^
2*e + (2*b^2*c - 4*a*c^2 - b^2*e)*log(f)^2 - 4*a*e^2)/(c^2*log(f)^2 + e^2)) + f^(1/4*b^2*c/(c^2*log(f)^2 + e^2
))*f^a*sin(-1/4*(b^2*e + (2*b^2*c - 4*a*c^2 - b^2*e)*log(f)^2 - 4*a*e^2)/(c^2*log(f)^2 + e^2)))*erf(1/2*(2*(c*
log(f) - I*e)*x + b*log(f) - I*b)*sqrt(-c*log(f) + I*e)/(c*log(f) - I*e)) + (-I*f^(1/4*b^2*c/(c^2*log(f)^2 + e
^2))*f^a*cos(-1/4*(b^2*e + (2*b^2*c - 4*a*c^2 - b^2*e)*log(f)^2 - 4*a*e^2)/(c^2*log(f)^2 + e^2)) + f^(1/4*b^2*
c/(c^2*log(f)^2 + e^2))*f^a*sin(-1/4*(b^2*e + (2*b^2*c - 4*a*c^2 - b^2*e)*log(f)^2 - 4*a*e^2)/(c^2*log(f)^2 +
e^2)))*erf(1/2*(2*(c*log(f) + I*e)*x + b*log(f) + I*b)*sqrt(-c*log(f) - I*e)/(c*log(f) + I*e)))*sqrt(-c*log(f)
+ sqrt(c^2*log(f)^2 + e^2)))/(c^2*e^(1/4*b^2*c*log(f)^3/(c^2*log(f)^2 + e^2) + 1/2*b^2*e*log(f)/(c^2*log(f)^2
+ e^2))*log(f)^2 + e^(1/4*b^2*c*log(f)^3/(c^2*log(f)^2 + e^2) + 1/2*b^2*e*log(f)/(c^2*log(f)^2 + e^2) + 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. 384 vs. \(2 (161) = 322\).
time = 3.31, size = 384, normalized size = 1.80 \begin {gather*} \frac {\sqrt {\pi } {\left (i \, c \log \left (f\right ) + e\right )} \sqrt {-c \log \left (f\right ) - i \, e} \operatorname {erf}\left (\frac {{\left ({\left (2 \, c^{2} x + b c\right )} \log \left (f\right )^{2} + 2 \, x e^{2} + b e + {\left (i \, b c - i \, b e\right )} \log \left (f\right )\right )} \sqrt {-c \log \left (f\right ) - i \, e}}{2 \, {\left (c^{2} \log \left (f\right )^{2} + e^{2}\right )}}\right ) e^{\left (-\frac {{\left (b^{2} c - 4 \, a c^{2}\right )} \log \left (f\right )^{3} + i \, b^{2} e - {\left (-2 i \, b^{2} c + 4 i \, a c^{2} + i \, b^{2} e\right )} \log \left (f\right )^{2} - 4 i \, a e^{2} - {\left (b^{2} c - 2 \, b^{2} e + 4 \, a e^{2}\right )} \log \left (f\right )}{4 \, {\left (c^{2} \log \left (f\right )^{2} + e^{2}\right )}}\right )} + \sqrt {\pi } {\left (-i \, c \log \left (f\right ) + e\right )} \sqrt {-c \log \left (f\right ) + i \, e} \operatorname {erf}\left (\frac {{\left ({\left (2 \, c^{2} x + b c\right )} \log \left (f\right )^{2} + 2 \, x e^{2} + b e + {\left (-i \, b c + i \, b e\right )} \log \left (f\right )\right )} \sqrt {-c \log \left (f\right ) + i \, e}}{2 \, {\left (c^{2} \log \left (f\right )^{2} + e^{2}\right )}}\right ) e^{\left (-\frac {{\left (b^{2} c - 4 \, a c^{2}\right )} \log \left (f\right )^{3} - i \, b^{2} e - {\left (2 i \, b^{2} c - 4 i \, a c^{2} - i \, b^{2} e\right )} \log \left (f\right )^{2} + 4 i \, a e^{2} - {\left (b^{2} c - 2 \, b^{2} e + 4 \, a e^{2}\right )} \log \left (f\right )}{4 \, {\left (c^{2} \log \left (f\right )^{2} + e^{2}\right )}}\right )}}{4 \, {\left (c^{2} \log \left (f\right )^{2} + e^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(sqrt(pi)*(I*c*log(f) + e)*sqrt(-c*log(f) - I*e)*erf(1/2*((2*c^2*x + b*c)*log(f)^2 + 2*x*e^2 + b*e + (I*b*
c - I*b*e)*log(f))*sqrt(-c*log(f) - I*e)/(c^2*log(f)^2 + e^2))*e^(-1/4*((b^2*c - 4*a*c^2)*log(f)^3 + I*b^2*e -
(-2*I*b^2*c + 4*I*a*c^2 + I*b^2*e)*log(f)^2 - 4*I*a*e^2 - (b^2*c - 2*b^2*e + 4*a*e^2)*log(f))/(c^2*log(f)^2 +
e^2)) + sqrt(pi)*(-I*c*log(f) + e)*sqrt(-c*log(f) + I*e)*erf(1/2*((2*c^2*x + b*c)*log(f)^2 + 2*x*e^2 + b*e +
(-I*b*c + I*b*e)*log(f))*sqrt(-c*log(f) + I*e)/(c^2*log(f)^2 + e^2))*e^(-1/4*((b^2*c - 4*a*c^2)*log(f)^3 - I*b
^2*e - (2*I*b^2*c - 4*I*a*c^2 - I*b^2*e)*log(f)^2 + 4*I*a*e^2 - (b^2*c - 2*b^2*e + 4*a*e^2)*log(f))/(c^2*log(f
)^2 + e^2)))/(c^2*log(f)^2 + e^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}} \sin {\left (a + b x + e 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)*sin(e*x**2+b*x+a),x)

[Out]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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