\(\int f^{a+b x+c x^2} \sin ^2(d+e x+f x^2) \, dx\) [101]

Optimal result

Integrand size = 26, antiderivative size = 268 \[ \int f^{a+b x+c x^2} \sin ^2\left (d+e x+f x^2\right ) \, dx=\frac {f^{a-\frac {b^2}{4 c}} \sqrt {\pi } \text {erfi}\left (\frac {(b+2 c x) \sqrt {\log (f)}}{2 \sqrt {c}}\right )}{4 \sqrt {c} \sqrt {\log (f)}}-\frac {e^{-2 i d-\frac {(2 e+i b \log (f))^2}{8 i f-4 c \log (f)}} f^a \sqrt {\pi } \text {erf}\left (\frac {2 i e-b \log (f)+2 x (2 i f-c \log (f))}{2 \sqrt {2 i f-c \log (f)}}\right )}{8 \sqrt {2 i f-c \log (f)}}-\frac {e^{2 i d+\frac {(2 e-i b \log (f))^2}{8 i f+4 c \log (f)}} f^a \sqrt {\pi } \text {erfi}\left (\frac {2 i e+b \log (f)+2 x (2 i f+c \log (f))}{2 \sqrt {2 i f+c \log (f)}}\right )}{8 \sqrt {2 i f+c \log (f)}} \]

[Out]

1/4*f^(a-1/4*b^2/c)*erfi(1/2*(2*c*x+b)*ln(f)^(1/2)/c^(1/2))*Pi^(1/2)/c^(1/2)/ln(f)^(1/2)-1/8*exp(-2*I*d-(2*e+I
*b*ln(f))^2/(8*I*f-4*c*ln(f)))*f^a*erf(1/2*(2*I*e-b*ln(f)+2*x*(2*I*f-c*ln(f)))/(2*I*f-c*ln(f))^(1/2))*Pi^(1/2)
/(2*I*f-c*ln(f))^(1/2)-1/8*exp(2*I*d+(2*e-I*b*ln(f))^2/(8*I*f+4*c*ln(f)))*f^a*erfi(1/2*(2*I*e+b*ln(f)+2*x*(2*I
*f+c*ln(f)))/(2*I*f+c*ln(f))^(1/2))*Pi^(1/2)/(2*I*f+c*ln(f))^(1/2)

Rubi [A] (verified)

Time = 0.72 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {4560, 2266, 2235, 2325, 2236} \[ \int f^{a+b x+c x^2} \sin ^2\left (d+e x+f x^2\right ) \, dx=\frac {\sqrt {\pi } f^{a-\frac {b^2}{4 c}} \text {erfi}\left (\frac {\sqrt {\log (f)} (b+2 c x)}{2 \sqrt {c}}\right )}{4 \sqrt {c} \sqrt {\log (f)}}-\frac {\sqrt {\pi } f^a \exp \left (-\frac {(2 e+i b \log (f))^2}{-4 c \log (f)+8 i f}-2 i d\right ) \text {erf}\left (\frac {-b \log (f)+2 x (-c \log (f)+2 i f)+2 i e}{2 \sqrt {-c \log (f)+2 i f}}\right )}{8 \sqrt {-c \log (f)+2 i f}}-\frac {\sqrt {\pi } f^a \exp \left (\frac {(2 e-i b \log (f))^2}{4 c \log (f)+8 i f}+2 i d\right ) \text {erfi}\left (\frac {b \log (f)+2 x (c \log (f)+2 i f)+2 i e}{2 \sqrt {c \log (f)+2 i f}}\right )}{8 \sqrt {c \log (f)+2 i f}} \]

[In]

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

[Out]

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

Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1120\) vs. \(2(268)=536\).

Time = 6.53 (sec) , antiderivative size = 1120, normalized size of antiderivative = 4.18 \[ \int f^{a+b x+c x^2} \sin ^2\left (d+e x+f x^2\right ) \, dx=\frac {f^a \sqrt {\pi } \left (8 \sqrt {c} f^{2-\frac {b^2}{4 c}} \text {erfi}\left (\frac {(b+2 c x) \sqrt {\log (f)}}{2 \sqrt {c}}\right ) \sqrt {\log (f)}+2 c^{5/2} f^{-\frac {b^2}{4 c}} \text {erfi}\left (\frac {(b+2 c x) \sqrt {\log (f)}}{2 \sqrt {c}}\right ) \log ^{\frac {5}{2}}(f)+2 \sqrt [4]{-1} c e^{\frac {i \left (-4 e^2+4 i b e \log (f)+b^2 \log ^2(f)\right )}{4 (2 f-i c \log (f))}} f \cos (2 d) \text {erf}\left (\frac {(-1)^{3/4} (2 e+4 f x-i b \log (f)-2 i c x \log (f))}{2 \sqrt {2 f-i c \log (f)}}\right ) \log (f) \sqrt {2 f-i c \log (f)}+(-1)^{3/4} c^2 e^{\frac {i \left (-4 e^2+4 i b e \log (f)+b^2 \log ^2(f)\right )}{4 (2 f-i c \log (f))}} \cos (2 d) \text {erf}\left (\frac {(-1)^{3/4} (2 e+4 f x-i b \log (f)-2 i c x \log (f))}{2 \sqrt {2 f-i c \log (f)}}\right ) \log ^2(f) \sqrt {2 f-i c \log (f)}+2 (-1)^{3/4} c e^{-\frac {i \left (-4 e^2-4 i b e \log (f)+b^2 \log ^2(f)\right )}{4 (2 f+i c \log (f))}} f \cos (2 d) \text {erf}\left (\frac {\sqrt [4]{-1} (2 e+4 f x+i b \log (f)+2 i c x \log (f))}{2 \sqrt {2 f+i c \log (f)}}\right ) \log (f) \sqrt {2 f+i c \log (f)}+\sqrt [4]{-1} c^2 e^{-\frac {i \left (-4 e^2-4 i b e \log (f)+b^2 \log ^2(f)\right )}{4 (2 f+i c \log (f))}} \cos (2 d) \text {erf}\left (\frac {\sqrt [4]{-1} (2 e+4 f x+i b \log (f)+2 i c x \log (f))}{2 \sqrt {2 f+i c \log (f)}}\right ) \log ^2(f) \sqrt {2 f+i c \log (f)}+2 (-1)^{3/4} c e^{\frac {i \left (-4 e^2+4 i b e \log (f)+b^2 \log ^2(f)\right )}{4 (2 f-i c \log (f))}} f \text {erf}\left (\frac {(-1)^{3/4} (2 e+4 f x-i b \log (f)-2 i c x \log (f))}{2 \sqrt {2 f-i c \log (f)}}\right ) \log (f) \sqrt {2 f-i c \log (f)} \sin (2 d)-\sqrt [4]{-1} c^2 e^{\frac {i \left (-4 e^2+4 i b e \log (f)+b^2 \log ^2(f)\right )}{4 (2 f-i c \log (f))}} \text {erf}\left (\frac {(-1)^{3/4} (2 e+4 f x-i b \log (f)-2 i c x \log (f))}{2 \sqrt {2 f-i c \log (f)}}\right ) \log ^2(f) \sqrt {2 f-i c \log (f)} \sin (2 d)+2 \sqrt [4]{-1} c e^{-\frac {i \left (-4 e^2-4 i b e \log (f)+b^2 \log ^2(f)\right )}{4 (2 f+i c \log (f))}} f \text {erf}\left (\frac {\sqrt [4]{-1} (2 e+4 f x+i b \log (f)+2 i c x \log (f))}{2 \sqrt {2 f+i c \log (f)}}\right ) \log (f) \sqrt {2 f+i c \log (f)} \sin (2 d)-(-1)^{3/4} c^2 e^{-\frac {i \left (-4 e^2-4 i b e \log (f)+b^2 \log ^2(f)\right )}{4 (2 f+i c \log (f))}} \text {erf}\left (\frac {\sqrt [4]{-1} (2 e+4 f x+i b \log (f)+2 i c x \log (f))}{2 \sqrt {2 f+i c \log (f)}}\right ) \log ^2(f) \sqrt {2 f+i c \log (f)} \sin (2 d)\right )}{8 c \log (f) (2 f-i c \log (f)) (2 f+i c \log (f))} \]

[In]

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

[Out]

(f^a*Sqrt[Pi]*(8*Sqrt[c]*f^(2 - b^2/(4*c))*Erfi[((b + 2*c*x)*Sqrt[Log[f]])/(2*Sqrt[c])]*Sqrt[Log[f]] + (2*c^(5
/2)*Erfi[((b + 2*c*x)*Sqrt[Log[f]])/(2*Sqrt[c])]*Log[f]^(5/2))/f^(b^2/(4*c)) + 2*(-1)^(1/4)*c*E^(((I/4)*(-4*e^
2 + (4*I)*b*e*Log[f] + b^2*Log[f]^2))/(2*f - I*c*Log[f]))*f*Cos[2*d]*Erf[((-1)^(3/4)*(2*e + 4*f*x - I*b*Log[f]
- (2*I)*c*x*Log[f]))/(2*Sqrt[2*f - I*c*Log[f]])]*Log[f]*Sqrt[2*f - I*c*Log[f]] + (-1)^(3/4)*c^2*E^(((I/4)*(-4
*e^2 + (4*I)*b*e*Log[f] + b^2*Log[f]^2))/(2*f - I*c*Log[f]))*Cos[2*d]*Erf[((-1)^(3/4)*(2*e + 4*f*x - I*b*Log[f
] - (2*I)*c*x*Log[f]))/(2*Sqrt[2*f - I*c*Log[f]])]*Log[f]^2*Sqrt[2*f - I*c*Log[f]] + (2*(-1)^(3/4)*c*f*Cos[2*d
]*Erf[((-1)^(1/4)*(2*e + 4*f*x + I*b*Log[f] + (2*I)*c*x*Log[f]))/(2*Sqrt[2*f + I*c*Log[f]])]*Log[f]*Sqrt[2*f +
I*c*Log[f]])/E^(((I/4)*(-4*e^2 - (4*I)*b*e*Log[f] + b^2*Log[f]^2))/(2*f + I*c*Log[f])) + ((-1)^(1/4)*c^2*Cos[
2*d]*Erf[((-1)^(1/4)*(2*e + 4*f*x + I*b*Log[f] + (2*I)*c*x*Log[f]))/(2*Sqrt[2*f + I*c*Log[f]])]*Log[f]^2*Sqrt[
2*f + I*c*Log[f]])/E^(((I/4)*(-4*e^2 - (4*I)*b*e*Log[f] + b^2*Log[f]^2))/(2*f + I*c*Log[f])) + 2*(-1)^(3/4)*c*
E^(((I/4)*(-4*e^2 + (4*I)*b*e*Log[f] + b^2*Log[f]^2))/(2*f - I*c*Log[f]))*f*Erf[((-1)^(3/4)*(2*e + 4*f*x - I*b
*Log[f] - (2*I)*c*x*Log[f]))/(2*Sqrt[2*f - I*c*Log[f]])]*Log[f]*Sqrt[2*f - I*c*Log[f]]*Sin[2*d] - (-1)^(1/4)*c
^2*E^(((I/4)*(-4*e^2 + (4*I)*b*e*Log[f] + b^2*Log[f]^2))/(2*f - I*c*Log[f]))*Erf[((-1)^(3/4)*(2*e + 4*f*x - I*
b*Log[f] - (2*I)*c*x*Log[f]))/(2*Sqrt[2*f - I*c*Log[f]])]*Log[f]^2*Sqrt[2*f - I*c*Log[f]]*Sin[2*d] + (2*(-1)^(
1/4)*c*f*Erf[((-1)^(1/4)*(2*e + 4*f*x + I*b*Log[f] + (2*I)*c*x*Log[f]))/(2*Sqrt[2*f + I*c*Log[f]])]*Log[f]*Sqr
t[2*f + I*c*Log[f]]*Sin[2*d])/E^(((I/4)*(-4*e^2 - (4*I)*b*e*Log[f] + b^2*Log[f]^2))/(2*f + I*c*Log[f])) - ((-1
)^(3/4)*c^2*Erf[((-1)^(1/4)*(2*e + 4*f*x + I*b*Log[f] + (2*I)*c*x*Log[f]))/(2*Sqrt[2*f + I*c*Log[f]])]*Log[f]^
2*Sqrt[2*f + I*c*Log[f]]*Sin[2*d])/E^(((I/4)*(-4*e^2 - (4*I)*b*e*Log[f] + b^2*Log[f]^2))/(2*f + I*c*Log[f]))))
/(8*c*Log[f]*(2*f - I*c*Log[f])*(2*f + I*c*Log[f]))

Maple [A] (verified)

Time = 0.77 (sec) , antiderivative size = 263, normalized size of antiderivative = 0.98

method	result	size
risch	\(\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {\ln \left (f \right )^{2} b^{2}-4 i \ln \left (f \right ) b e +8 i d \ln \left (f \right ) c +16 d f -4 e^{2}}{4 \left (c \ln \left (f \right )-2 i f \right )}} \operatorname {erf}\left (-x \sqrt {2 i f -c \ln \left (f \right )}+\frac {b \ln \left (f \right )-2 i e}{2 \sqrt {2 i f -c \ln \left (f \right )}}\right )}{8 \sqrt {2 i f -c \ln \left (f \right )}}+\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {\ln \left (f \right )^{2} b^{2}+4 i \ln \left (f \right ) b e -8 i d \ln \left (f \right ) c +16 d f -4 e^{2}}{4 \left (2 i f +c \ln \left (f \right )\right )}} \operatorname {erf}\left (-\sqrt {-c \ln \left (f \right )-2 i f}\, x +\frac {2 i e +b \ln \left (f \right )}{2 \sqrt {-c \ln \left (f \right )-2 i f}}\right )}{8 \sqrt {-c \ln \left (f \right )-2 i f}}-\frac {\sqrt {\pi }\, f^{-\frac {b^{2}}{4 c}} f^{a} \operatorname {erf}\left (-\sqrt {-c \ln \left (f \right )}\, x +\frac {\ln \left (f \right ) b}{2 \sqrt {-c \ln \left (f \right )}}\right )}{4 \sqrt {-c \ln \left (f \right )}}\)	\(263\)

[In]

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

[Out]

1/8*Pi^(1/2)*f^a*exp(-1/4*(ln(f)^2*b^2-4*I*ln(f)*b*e+8*I*d*ln(f)*c+16*d*f-4*e^2)/(c*ln(f)-2*I*f))/(2*I*f-c*ln(
f))^(1/2)*erf(-x*(2*I*f-c*ln(f))^(1/2)+1/2*(b*ln(f)-2*I*e)/(2*I*f-c*ln(f))^(1/2))+1/8*Pi^(1/2)*f^a*exp(-1/4*(l
n(f)^2*b^2+4*I*ln(f)*b*e-8*I*d*ln(f)*c+16*d*f-4*e^2)/(2*I*f+c*ln(f)))/(-c*ln(f)-2*I*f)^(1/2)*erf(-(-c*ln(f)-2*
I*f)^(1/2)*x+1/2*(2*I*e+b*ln(f))/(-c*ln(f)-2*I*f)^(1/2))-1/4*Pi^(1/2)*f^(-1/4*b^2/c)*f^a/(-c*ln(f))^(1/2)*erf(
-(-c*ln(f))^(1/2)*x+1/2*ln(f)*b/(-c*ln(f))^(1/2))

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 470 vs. \(2 (199) = 398\).

Time = 0.26 (sec) , antiderivative size = 470, normalized size of antiderivative = 1.75 \[ \int f^{a+b x+c x^2} \sin ^2\left (d+e x+f x^2\right ) \, dx=\frac {\sqrt {\pi } {\left (c^{2} \log \left (f\right )^{2} - 2 i \, c f \log \left (f\right )\right )} \sqrt {-c \log \left (f\right ) - 2 i \, f} \operatorname {erf}\left (\frac {{\left (8 \, f^{2} x + {\left (2 \, c^{2} x + b c\right )} \log \left (f\right )^{2} + 4 \, e f - 2 \, {\left (-i \, c e + i \, b f\right )} \log \left (f\right )\right )} \sqrt {-c \log \left (f\right ) - 2 i \, f}}{2 \, {\left (c^{2} \log \left (f\right )^{2} + 4 \, f^{2}\right )}}\right ) e^{\left (-\frac {{\left (b^{2} c - 4 \, a c^{2}\right )} \log \left (f\right )^{3} + 8 i \, e^{2} f - 32 i \, d f^{2} + 2 \, {\left (-4 i \, c^{2} d + 2 i \, b c e - i \, b^{2} f\right )} \log \left (f\right )^{2} - 4 \, {\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} + 4 \, f^{2}\right )}}\right )} + \sqrt {\pi } {\left (c^{2} \log \left (f\right )^{2} + 2 i \, c f \log \left (f\right )\right )} \sqrt {-c \log \left (f\right ) + 2 i \, f} \operatorname {erf}\left (\frac {{\left (8 \, f^{2} x + {\left (2 \, c^{2} x + b c\right )} \log \left (f\right )^{2} + 4 \, e f - 2 \, {\left (i \, c e - i \, b f\right )} \log \left (f\right )\right )} \sqrt {-c \log \left (f\right ) + 2 i \, f}}{2 \, {\left (c^{2} \log \left (f\right )^{2} + 4 \, f^{2}\right )}}\right ) e^{\left (-\frac {{\left (b^{2} c - 4 \, a c^{2}\right )} \log \left (f\right )^{3} - 8 i \, e^{2} f + 32 i \, d f^{2} + 2 \, {\left (4 i \, c^{2} d - 2 i \, b c e + i \, b^{2} f\right )} \log \left (f\right )^{2} - 4 \, {\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} + 4 \, f^{2}\right )}}\right )} - \frac {2 \, \sqrt {\pi } {\left (c^{2} \log \left (f\right )^{2} + 4 \, f^{2}\right )} \sqrt {-c \log \left (f\right )} \operatorname {erf}\left (\frac {{\left (2 \, c x + b\right )} \sqrt {-c \log \left (f\right )}}{2 \, c}\right )}{f^{\frac {b^{2} - 4 \, a c}{4 \, c}}}}{8 \, {\left (c^{3} \log \left (f\right )^{3} + 4 \, c f^{2} \log \left (f\right )\right )}} \]

[In]

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

[Out]

1/8*(sqrt(pi)*(c^2*log(f)^2 - 2*I*c*f*log(f))*sqrt(-c*log(f) - 2*I*f)*erf(1/2*(8*f^2*x + (2*c^2*x + b*c)*log(f
)^2 + 4*e*f - 2*(-I*c*e + I*b*f)*log(f))*sqrt(-c*log(f) - 2*I*f)/(c^2*log(f)^2 + 4*f^2))*e^(-1/4*((b^2*c - 4*a
*c^2)*log(f)^3 + 8*I*e^2*f - 32*I*d*f^2 + 2*(-4*I*c^2*d + 2*I*b*c*e - I*b^2*f)*log(f)^2 - 4*(c*e^2 - 2*b*e*f +
4*a*f^2)*log(f))/(c^2*log(f)^2 + 4*f^2)) + sqrt(pi)*(c^2*log(f)^2 + 2*I*c*f*log(f))*sqrt(-c*log(f) + 2*I*f)*e
rf(1/2*(8*f^2*x + (2*c^2*x + b*c)*log(f)^2 + 4*e*f - 2*(I*c*e - I*b*f)*log(f))*sqrt(-c*log(f) + 2*I*f)/(c^2*lo
g(f)^2 + 4*f^2))*e^(-1/4*((b^2*c - 4*a*c^2)*log(f)^3 - 8*I*e^2*f + 32*I*d*f^2 + 2*(4*I*c^2*d - 2*I*b*c*e + I*b
^2*f)*log(f)^2 - 4*(c*e^2 - 2*b*e*f + 4*a*f^2)*log(f))/(c^2*log(f)^2 + 4*f^2)) - 2*sqrt(pi)*(c^2*log(f)^2 + 4*
f^2)*sqrt(-c*log(f))*erf(1/2*(2*c*x + b)*sqrt(-c*log(f))/c)/f^(1/4*(b^2 - 4*a*c)/c))/(c^3*log(f)^3 + 4*c*f^2*l
og(f))

Sympy [F]

\[ \int f^{a+b x+c x^2} \sin ^2\left (d+e x+f x^2\right ) \, dx=\int f^{a + b x + c x^{2}} \sin ^{2}{\left (d + e x + f x^{2} \right )}\, dx \]

[In]

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

[Out]

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

Maxima [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.27 (sec) , antiderivative size = 1487, normalized size of antiderivative = 5.55 \[ \int f^{a+b x+c x^2} \sin ^2\left (d+e x+f x^2\right ) \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/16*(sqrt(pi)*sqrt(2*c^2*log(f)^2 + 8*f^2)*((I*f^a*cos(-1/2*(4*e^2*f - 16*d*f^2 - (4*c^2*d - 2*b*c*e + b^2*f)
*log(f)^2)/(c^2*log(f)^2 + 4*f^2))*e^(c*e^2*log(f)/(c^2*log(f)^2 + 4*f^2) + 1/4*b^2*log(f)/c) + f^a*e^(c*e^2*l
og(f)/(c^2*log(f)^2 + 4*f^2) + 1/4*b^2*log(f)/c)*sin(-1/2*(4*e^2*f - 16*d*f^2 - (4*c^2*d - 2*b*c*e + b^2*f)*lo
g(f)^2)/(c^2*log(f)^2 + 4*f^2)))*erf(1/2*(2*(c*log(f) - 2*I*f)*x + b*log(f) - 2*I*e)*sqrt(-c*log(f) + 2*I*f)/(
c*log(f) - 2*I*f)) + (-I*f^a*cos(-1/2*(4*e^2*f - 16*d*f^2 - (4*c^2*d - 2*b*c*e + b^2*f)*log(f)^2)/(c^2*log(f)^
2 + 4*f^2))*e^(c*e^2*log(f)/(c^2*log(f)^2 + 4*f^2) + 1/4*b^2*log(f)/c) + f^a*e^(c*e^2*log(f)/(c^2*log(f)^2 + 4
*f^2) + 1/4*b^2*log(f)/c)*sin(-1/2*(4*e^2*f - 16*d*f^2 - (4*c^2*d - 2*b*c*e + b^2*f)*log(f)^2)/(c^2*log(f)^2 +
4*f^2)))*erf(1/2*(2*(c*log(f) + 2*I*f)*x + b*log(f) + 2*I*e)*sqrt(-c*log(f) - 2*I*f)/(c*log(f) + 2*I*f)))*sqr
t(c*log(f) + sqrt(c^2*log(f)^2 + 4*f^2))*sqrt(-c*log(f)) - sqrt(pi)*sqrt(2*c^2*log(f)^2 + 8*f^2)*((f^a*cos(-1/
2*(4*e^2*f - 16*d*f^2 - (4*c^2*d - 2*b*c*e + b^2*f)*log(f)^2)/(c^2*log(f)^2 + 4*f^2))*e^(c*e^2*log(f)/(c^2*log
(f)^2 + 4*f^2) + 1/4*b^2*log(f)/c) - I*f^a*e^(c*e^2*log(f)/(c^2*log(f)^2 + 4*f^2) + 1/4*b^2*log(f)/c)*sin(-1/2
*(4*e^2*f - 16*d*f^2 - (4*c^2*d - 2*b*c*e + b^2*f)*log(f)^2)/(c^2*log(f)^2 + 4*f^2)))*erf(1/2*(2*(c*log(f) - 2
*I*f)*x + b*log(f) - 2*I*e)*sqrt(-c*log(f) + 2*I*f)/(c*log(f) - 2*I*f)) + (f^a*cos(-1/2*(4*e^2*f - 16*d*f^2 -
(4*c^2*d - 2*b*c*e + b^2*f)*log(f)^2)/(c^2*log(f)^2 + 4*f^2))*e^(c*e^2*log(f)/(c^2*log(f)^2 + 4*f^2) + 1/4*b^2
*log(f)/c) + I*f^a*e^(c*e^2*log(f)/(c^2*log(f)^2 + 4*f^2) + 1/4*b^2*log(f)/c)*sin(-1/2*(4*e^2*f - 16*d*f^2 - (
4*c^2*d - 2*b*c*e + b^2*f)*log(f)^2)/(c^2*log(f)^2 + 4*f^2)))*erf(1/2*(2*(c*log(f) + 2*I*f)*x + b*log(f) + 2*I
*e)*sqrt(-c*log(f) - 2*I*f)/(c*log(f) + 2*I*f)))*sqrt(-c*log(f) + sqrt(c^2*log(f)^2 + 4*f^2))*sqrt(-c*log(f))
+ 2*sqrt(pi)*((c^2*f^a*e^(1/4*b^2*c*log(f)^3/(c^2*log(f)^2 + 4*f^2) + 2*b*e*f*log(f)/(c^2*log(f)^2 + 4*f^2))*l
og(f)^2 + 4*f^(a + 2)*e^(1/4*b^2*c*log(f)^3/(c^2*log(f)^2 + 4*f^2) + 2*b*e*f*log(f)/(c^2*log(f)^2 + 4*f^2)))*e
rf(-1/2*b*conjugate(1/sqrt(-c*log(f)))*log(f) + x*conjugate(sqrt(-c*log(f)))) - (c^2*f^a*e^(1/4*b^2*c*log(f)^3
/(c^2*log(f)^2 + 4*f^2) + 2*b*e*f*log(f)/(c^2*log(f)^2 + 4*f^2))*log(f)^2 + 4*f^(a + 2)*e^(1/4*b^2*c*log(f)^3/
(c^2*log(f)^2 + 4*f^2) + 2*b*e*f*log(f)/(c^2*log(f)^2 + 4*f^2)))*erf(1/2*(2*c*x*log(f) + b*log(f))/sqrt(-c*log
(f)))))/((c^2*e^(1/4*b^2*c*log(f)^3/(c^2*log(f)^2 + 4*f^2) + 2*b*e*f*log(f)/(c^2*log(f)^2 + 4*f^2) + 1/4*b^2*l
og(f)/c)*log(f)^2 + 4*f^2*e^(1/4*b^2*c*log(f)^3/(c^2*log(f)^2 + 4*f^2) + 2*b*e*f*log(f)/(c^2*log(f)^2 + 4*f^2)
+ 1/4*b^2*log(f)/c))*sqrt(-c*log(f)))

Giac [F]

\[ \int f^{a+b x+c x^2} \sin ^2\left (d+e x+f x^2\right ) \, dx=\int { f^{c x^{2} + b x + a} \sin \left (f x^{2} + e x + d\right )^{2} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int f^{a+b x+c x^2} \sin ^2\left (d+e x+f x^2\right ) \, dx=\int f^{c\,x^2+b\,x+a}\,{\sin \left (f\,x^2+e\,x+d\right )}^2 \,d x \]

[In]

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

[Out]

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