∫ fa+bx+cx2 sin3(d + ex + fx2)dx

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

Optimal. Leaf size=430 \[ \frac{3 i \sqrt{\pi } f^a \exp \left (-\frac{(e+i b \log (f))^2}{-4 c \log (f)+4 i f}-i d\right ) \text{Erf}\left (\frac{-b \log (f)+2 x (-c \log (f)+i f)+i e}{2 \sqrt{-c \log (f)+i f}}\right )}{16 \sqrt{-c \log (f)+i f}}-\frac{i \sqrt{\pi } f^a \exp \left (-\frac{(3 e+i b \log (f))^2}{4 (-c \log (f)+3 i f)}-3 i d\right ) \text{Erf}\left (\frac{-b \log (f)+2 x (-c \log (f)+3 i f)+3 i e}{2 \sqrt{-c \log (f)+3 i f}}\right )}{16 \sqrt{-c \log (f)+3 i f}}-\frac{3 i \sqrt{\pi } f^a \exp \left (\frac{(e-i b \log (f))^2}{4 c \log (f)+4 i f}+i d\right ) \text{Erfi}\left (\frac{b \log (f)+2 x (c \log (f)+i f)+i e}{2 \sqrt{c \log (f)+i f}}\right )}{16 \sqrt{c \log (f)+i f}}+\frac{i \sqrt{\pi } f^a \exp \left (3 i d-\frac{(b \log (f)+3 i e)^2}{4 (c \log (f)+3 i f)}\right ) \text{Erfi}\left (\frac{b \log (f)+2 x (c \log (f)+3 i f)+3 i e}{2 \sqrt{c \log (f)+3 i f}}\right )}{16 \sqrt{c \log (f)+3 i f}} \]

[Out]

(((3*I)/16)*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/16)*E^((-3*I)*d - (3*e + I*b*Log[f])^2/(4*

((3*I)*f - c*Log[f])))*f^a*Sqrt[Pi]*Erf[((3*I)*e - b*Log[f] + 2*x*((3*I)*f - c*Log[f]))/(2*Sqrt[(3*I)*f - c*Lo

g[f]])])/Sqrt[(3*I)*f - c*Log[f]] - (((3*I)/16)*E^(I*d + (e - I*b*Log[f])^2/((4*I)*f + 4*c*Log[f]))*f^a*Sqrt[P

i]*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]] + ((I/16)*E^((

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

+ c*Log[f]))/(2*Sqrt[(3*I)*f + c*Log[f]])])/Sqrt[(3*I)*f + c*Log[f]]

________________________________________________________________________________________

Rubi [A] time = 0.910781, antiderivative size = 430, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {4472, 2287, 2234, 2205, 2204} \[ \frac{3 i \sqrt{\pi } f^a \exp \left (-\frac{(e+i b \log (f))^2}{-4 c \log (f)+4 i f}-i d\right ) \text{Erf}\left (\frac{-b \log (f)+2 x (-c \log (f)+i f)+i e}{2 \sqrt{-c \log (f)+i f}}\right )}{16 \sqrt{-c \log (f)+i f}}-\frac{i \sqrt{\pi } f^a \exp \left (-\frac{(3 e+i b \log (f))^2}{4 (-c \log (f)+3 i f)}-3 i d\right ) \text{Erf}\left (\frac{-b \log (f)+2 x (-c \log (f)+3 i f)+3 i e}{2 \sqrt{-c \log (f)+3 i f}}\right )}{16 \sqrt{-c \log (f)+3 i f}}-\frac{3 i \sqrt{\pi } f^a \exp \left (\frac{(e-i b \log (f))^2}{4 c \log (f)+4 i f}+i d\right ) \text{Erfi}\left (\frac{b \log (f)+2 x (c \log (f)+i f)+i e}{2 \sqrt{c \log (f)+i f}}\right )}{16 \sqrt{c \log (f)+i f}}+\frac{i \sqrt{\pi } f^a \exp \left (3 i d-\frac{(b \log (f)+3 i e)^2}{4 (c \log (f)+3 i f)}\right ) \text{Erfi}\left (\frac{b \log (f)+2 x (c \log (f)+3 i f)+3 i e}{2 \sqrt{c \log (f)+3 i f}}\right )}{16 \sqrt{c \log (f)+3 i f}} \]

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(((3*I)/16)*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/16)*E^((-3*I)*d - (3*e + I*b*Log[f])^2/(4*

((3*I)*f - c*Log[f])))*f^a*Sqrt[Pi]*Erf[((3*I)*e - b*Log[f] + 2*x*((3*I)*f - c*Log[f]))/(2*Sqrt[(3*I)*f - c*Lo

g[f]])])/Sqrt[(3*I)*f - c*Log[f]] - (((3*I)/16)*E^(I*d + (e - I*b*Log[f])^2/((4*I)*f + 4*c*Log[f]))*f^a*Sqrt[P

i]*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]] + ((I/16)*E^((

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

+ c*Log[f]))/(2*Sqrt[(3*I)*f + c*Log[f]])])/Sqrt[(3*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 ^3\left (d+e x+f x^2\right ) \, dx &=\int \left (-\frac{1}{8} i e^{-3 i \left (d+e x+f x^2\right )} f^{a+b x+c x^2}+\frac{3}{8} i \exp \left (2 i d+2 i e x+2 i f x^2-3 i \left (d+e x+f x^2\right )\right ) f^{a+b x+c x^2}-\frac{3}{8} i \exp \left (4 i d+4 i e x+4 i f x^2-3 i \left (d+e x+f x^2\right )\right ) f^{a+b x+c x^2}+\frac{1}{8} i \exp \left (6 i d+6 i e x+6 i f x^2-3 i \left (d+e x+f x^2\right )\right ) f^{a+b x+c x^2}\right ) \, dx\\ &=-\left (\frac{1}{8} i \int e^{-3 i \left (d+e x+f x^2\right )} f^{a+b x+c x^2} \, dx\right )+\frac{1}{8} i \int \exp \left (6 i d+6 i e x+6 i f x^2-3 i \left (d+e x+f x^2\right )\right ) f^{a+b x+c x^2} \, dx+\frac{3}{8} i \int \exp \left (2 i d+2 i e x+2 i f x^2-3 i \left (d+e x+f x^2\right )\right ) f^{a+b x+c x^2} \, dx-\frac{3}{8} i \int \exp \left (4 i d+4 i e x+4 i f x^2-3 i \left (d+e x+f x^2\right )\right ) f^{a+b x+c x^2} \, dx\\ &=-\left (\frac{1}{8} i \int \exp \left (-3 i d+a \log (f)-x (3 i e-b \log (f))-x^2 (3 i f-c \log (f))\right ) \, dx\right )+\frac{1}{8} i \int \exp \left (3 i d+a \log (f)+x (3 i e+b \log (f))+x^2 (3 i f+c \log (f))\right ) \, dx+\frac{3}{8} 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{3}{8} 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}{8} \left (3 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}{8} \left (i \exp \left (-3 i d-\frac{(3 e+i b \log (f))^2}{4 (3 i f-c \log (f))}\right ) f^a\right ) \int \exp \left (\frac{(-3 i e+b \log (f)+2 x (-3 i f+c \log (f)))^2}{4 (-3 i f+c \log (f))}\right ) \, dx+\frac{1}{8} \left (i \exp \left (3 i d-\frac{(3 i e+b \log (f))^2}{4 (3 i f+c \log (f))}\right ) f^a\right ) \int \exp \left (\frac{(3 i e+b \log (f)+2 x (3 i f+c \log (f)))^2}{4 (3 i f+c \log (f))}\right ) \, dx-\frac{1}{8} \left (3 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{3 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 )}{16 \sqrt{i f-c \log (f)}}-\frac{i \exp \left (-3 i d-\frac{(3 e+i b \log (f))^2}{4 (3 i f-c \log (f))}\right ) f^a \sqrt{\pi } \text{erf}\left (\frac{3 i e-b \log (f)+2 x (3 i f-c \log (f))}{2 \sqrt{3 i f-c \log (f)}}\right )}{16 \sqrt{3 i f-c \log (f)}}-\frac{3 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 )}{16 \sqrt{i f+c \log (f)}}+\frac{i \exp \left (3 i d-\frac{(3 i e+b \log (f))^2}{4 (3 i f+c \log (f))}\right ) f^a \sqrt{\pi } \text{erfi}\left (\frac{3 i e+b \log (f)+2 x (3 i f+c \log (f))}{2 \sqrt{3 i f+c \log (f)}}\right )}{16 \sqrt{3 i f+c \log (f)}}\\ \end{align*}

Mathematica [B] time = 7.24826, size = 3835, normalized size = 8.92 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(f^a*Sqrt[Pi]*(-27*(-1)^(3/4)*E^(((I/4)*(-e^2 + (2*I)*b*e*Log[f] + b^2*Log[f]^2))/(f - I*c*Log[f]))*f^3*Cos[d]

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

+ 27*(-1)^(1/4)*c*E^(((I/4)*(-e^2 + (2*I)*b*e*Log[f] + b^2*Log[f]^2))/(f - I*c*Log[f]))*f^2*Cos[d]*Erfi[((-1)^

(1/4)*(e + 2*f*x - I*b*Log[f] - (2*I)*c*x*Log[f]))/(2*Sqrt[f - I*c*Log[f]])]*Log[f]*Sqrt[f - I*c*Log[f]] - 3*(

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

(e + 2*f*x - I*b*Log[f] - (2*I)*c*x*Log[f]))/(2*Sqrt[f - I*c*Log[f]])]*Log[f]^2*Sqrt[f - I*c*Log[f]] + 3*(-1)^

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

*f*x - I*b*Log[f] - (2*I)*c*x*Log[f]))/(2*Sqrt[f - I*c*Log[f]])]*Log[f]^3*Sqrt[f - I*c*Log[f]] + 3*(-1)^(3/4)*

E^(((I/4)*(-9*e^2 + (6*I)*b*e*Log[f] + b^2*Log[f]^2))/(3*f - I*c*Log[f]))*f^3*Cos[3*d]*Erfi[((-1)^(1/4)*(3*e +

6*f*x - I*b*Log[f] - (2*I)*c*x*Log[f]))/(2*Sqrt[3*f - I*c*Log[f]])]*Sqrt[3*f - I*c*Log[f]] - (-1)^(1/4)*c*E^(

((I/4)*(-9*e^2 + (6*I)*b*e*Log[f] + b^2*Log[f]^2))/(3*f - I*c*Log[f]))*f^2*Cos[3*d]*Erfi[((-1)^(1/4)*(3*e + 6*

f*x - I*b*Log[f] - (2*I)*c*x*Log[f]))/(2*Sqrt[3*f - I*c*Log[f]])]*Log[f]*Sqrt[3*f - I*c*Log[f]] + 3*(-1)^(3/4)

*c^2*E^(((I/4)*(-9*e^2 + (6*I)*b*e*Log[f] + b^2*Log[f]^2))/(3*f - I*c*Log[f]))*f*Cos[3*d]*Erfi[((-1)^(1/4)*(3*

e + 6*f*x - I*b*Log[f] - (2*I)*c*x*Log[f]))/(2*Sqrt[3*f - I*c*Log[f]])]*Log[f]^2*Sqrt[3*f - I*c*Log[f]] - (-1)

^(1/4)*c^3*E^(((I/4)*(-9*e^2 + (6*I)*b*e*Log[f] + b^2*Log[f]^2))/(3*f - I*c*Log[f]))*Cos[3*d]*Erfi[((-1)^(1/4)

*(3*e + 6*f*x - I*b*Log[f] - (2*I)*c*x*Log[f]))/(2*Sqrt[3*f - I*c*Log[f]])]*Log[f]^3*Sqrt[3*f - I*c*Log[f]] +

(27*(-1)^(1/4)*f^3*Cos[d]*Erfi[((-1)^(3/4)*(e + 2*f*x + I*b*Log[f] + (2*I)*c*x*Log[f]))/(2*Sqrt[f + I*c*Log[f]

])]*Sqrt[f + I*c*Log[f]])/E^(((I/4)*(-e^2 - (2*I)*b*e*Log[f] + b^2*Log[f]^2))/(f + I*c*Log[f])) - (27*(-1)^(3/

4)*c*f^2*Cos[d]*Erfi[((-1)^(3/4)*(e + 2*f*x + I*b*Log[f] + (2*I)*c*x*Log[f]))/(2*Sqrt[f + I*c*Log[f]])]*Log[f]

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

^2*f*Cos[d]*Erfi[((-1)^(3/4)*(e + 2*f*x + I*b*Log[f] + (2*I)*c*x*Log[f]))/(2*Sqrt[f + I*c*Log[f]])]*Log[f]^2*S

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

*Cos[d]*Erfi[((-1)^(3/4)*(e + 2*f*x + I*b*Log[f] + (2*I)*c*x*Log[f]))/(2*Sqrt[f + I*c*Log[f]])]*Log[f]^3*Sqrt[

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

[3*d]*Erfi[((-1)^(3/4)*(3*e + 6*f*x + I*b*Log[f] + (2*I)*c*x*Log[f]))/(2*Sqrt[3*f + I*c*Log[f]])]*Sqrt[3*f + I

*c*Log[f]])/E^(((I/4)*(-9*e^2 - (6*I)*b*e*Log[f] + b^2*Log[f]^2))/(3*f + I*c*Log[f])) + ((-1)^(3/4)*c*f^2*Cos[

3*d]*Erfi[((-1)^(3/4)*(3*e + 6*f*x + I*b*Log[f] + (2*I)*c*x*Log[f]))/(2*Sqrt[3*f + I*c*Log[f]])]*Log[f]*Sqrt[3

*f + I*c*Log[f]])/E^(((I/4)*(-9*e^2 - (6*I)*b*e*Log[f] + b^2*Log[f]^2))/(3*f + I*c*Log[f])) - (3*(-1)^(1/4)*c^

2*f*Cos[3*d]*Erfi[((-1)^(3/4)*(3*e + 6*f*x + I*b*Log[f] + (2*I)*c*x*Log[f]))/(2*Sqrt[3*f + I*c*Log[f]])]*Log[f

]^2*Sqrt[3*f + I*c*Log[f]])/E^(((I/4)*(-9*e^2 - (6*I)*b*e*Log[f] + b^2*Log[f]^2))/(3*f + I*c*Log[f])) + ((-1)^

(3/4)*c^3*Cos[3*d]*Erfi[((-1)^(3/4)*(3*e + 6*f*x + I*b*Log[f] + (2*I)*c*x*Log[f]))/(2*Sqrt[3*f + I*c*Log[f]])]

*Log[f]^3*Sqrt[3*f + I*c*Log[f]])/E^(((I/4)*(-9*e^2 - (6*I)*b*e*Log[f] + b^2*Log[f]^2))/(3*f + I*c*Log[f])) +

27*(-1)^(1/4)*E^(((I/4)*(-e^2 + (2*I)*b*e*Log[f] + b^2*Log[f]^2))/(f - I*c*Log[f]))*f^3*Erfi[((-1)^(1/4)*(e +

2*f*x - I*b*Log[f] - (2*I)*c*x*Log[f]))/(2*Sqrt[f - I*c*Log[f]])]*Sqrt[f - I*c*Log[f]]*Sin[d] + 27*(-1)^(3/4)*

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

Log[f] - (2*I)*c*x*Log[f]))/(2*Sqrt[f - I*c*Log[f]])]*Log[f]*Sqrt[f - I*c*Log[f]]*Sin[d] + 3*(-1)^(1/4)*c^2*E^

(((I/4)*(-e^2 + (2*I)*b*e*Log[f] + b^2*Log[f]^2))/(f - I*c*Log[f]))*f*Erfi[((-1)^(1/4)*(e + 2*f*x - I*b*Log[f]

- (2*I)*c*x*Log[f]))/(2*Sqrt[f - I*c*Log[f]])]*Log[f]^2*Sqrt[f - I*c*Log[f]]*Sin[d] + 3*(-1)^(3/4)*c^3*E^(((I

/4)*(-e^2 + (2*I)*b*e*Log[f] + b^2*Log[f]^2))/(f - I*c*Log[f]))*Erfi[((-1)^(1/4)*(e + 2*f*x - I*b*Log[f] - (2*

I)*c*x*Log[f]))/(2*Sqrt[f - I*c*Log[f]])]*Log[f]^3*Sqrt[f - I*c*Log[f]]*Sin[d] - (27*(-1)^(3/4)*f^3*Erfi[((-1)

^(3/4)*(e + 2*f*x + I*b*Log[f] + (2*I)*c*x*Log[f]))/(2*Sqrt[f + I*c*Log[f]])]*Sqrt[f + I*c*Log[f]]*Sin[d])/E^(

((I/4)*(-e^2 - (2*I)*b*e*Log[f] + b^2*Log[f]^2))/(f + I*c*Log[f])) - (27*(-1)^(1/4)*c*f^2*Erfi[((-1)^(3/4)*(e

+ 2*f*x + I*b*Log[f] + (2*I)*c*x*Log[f]))/(2*Sqrt[f + I*c*Log[f]])]*Log[f]*Sqrt[f + I*c*Log[f]]*Sin[d])/E^(((I

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

f*x + I*b*Log[f] + (2*I)*c*x*Log[f]))/(2*Sqrt[f + I*c*Log[f]])]*Log[f]^2*Sqrt[f + I*c*Log[f]]*Sin[d])/E^(((I/4

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

+ I*b*Log[f] + (2*I)*c*x*Log[f]))/(2*Sqrt[f + I*c*Log[f]])]*Log[f]^3*Sqrt[f + I*c*Log[f]]*Sin[d])/E^(((I/4)*(-

e^2 - (2*I)*b*e*Log[f] + b^2*Log[f]^2))/(f + I*c*Log[f])) - 3*(-1)^(1/4)*E^(((I/4)*(-9*e^2 + (6*I)*b*e*Log[f]

+ b^2*Log[f]^2))/(3*f - I*c*Log[f]))*f^3*Erfi[((-1)^(1/4)*(3*e + 6*f*x - I*b*Log[f] - (2*I)*c*x*Log[f]))/(2*Sq

rt[3*f - I*c*Log[f]])]*Sqrt[3*f - I*c*Log[f]]*Sin[3*d] - (-1)^(3/4)*c*E^(((I/4)*(-9*e^2 + (6*I)*b*e*Log[f] + b

^2*Log[f]^2))/(3*f - I*c*Log[f]))*f^2*Erfi[((-1)^(1/4)*(3*e + 6*f*x - I*b*Log[f] - (2*I)*c*x*Log[f]))/(2*Sqrt[

3*f - I*c*Log[f]])]*Log[f]*Sqrt[3*f - I*c*Log[f]]*Sin[3*d] - 3*(-1)^(1/4)*c^2*E^(((I/4)*(-9*e^2 + (6*I)*b*e*Lo

g[f] + b^2*Log[f]^2))/(3*f - I*c*Log[f]))*f*Erfi[((-1)^(1/4)*(3*e + 6*f*x - I*b*Log[f] - (2*I)*c*x*Log[f]))/(2

*Sqrt[3*f - I*c*Log[f]])]*Log[f]^2*Sqrt[3*f - I*c*Log[f]]*Sin[3*d] - (-1)^(3/4)*c^3*E^(((I/4)*(-9*e^2 + (6*I)*

b*e*Log[f] + b^2*Log[f]^2))/(3*f - I*c*Log[f]))*Erfi[((-1)^(1/4)*(3*e + 6*f*x - I*b*Log[f] - (2*I)*c*x*Log[f])

)/(2*Sqrt[3*f - I*c*Log[f]])]*Log[f]^3*Sqrt[3*f - I*c*Log[f]]*Sin[3*d] + (3*(-1)^(3/4)*f^3*Erfi[((-1)^(3/4)*(3

*e + 6*f*x + I*b*Log[f] + (2*I)*c*x*Log[f]))/(2*Sqrt[3*f + I*c*Log[f]])]*Sqrt[3*f + I*c*Log[f]]*Sin[3*d])/E^((

(I/4)*(-9*e^2 - (6*I)*b*e*Log[f] + b^2*Log[f]^2))/(3*f + I*c*Log[f])) + ((-1)^(1/4)*c*f^2*Erfi[((-1)^(3/4)*(3*

e + 6*f*x + I*b*Log[f] + (2*I)*c*x*Log[f]))/(2*Sqrt[3*f + I*c*Log[f]])]*Log[f]*Sqrt[3*f + I*c*Log[f]]*Sin[3*d]

)/E^(((I/4)*(-9*e^2 - (6*I)*b*e*Log[f] + b^2*Log[f]^2))/(3*f + I*c*Log[f])) + (3*(-1)^(3/4)*c^2*f*Erfi[((-1)^(

3/4)*(3*e + 6*f*x + I*b*Log[f] + (2*I)*c*x*Log[f]))/(2*Sqrt[3*f + I*c*Log[f]])]*Log[f]^2*Sqrt[3*f + I*c*Log[f]

]*Sin[3*d])/E^(((I/4)*(-9*e^2 - (6*I)*b*e*Log[f] + b^2*Log[f]^2))/(3*f + I*c*Log[f])) + ((-1)^(1/4)*c^3*Erfi[(

(-1)^(3/4)*(3*e + 6*f*x + I*b*Log[f] + (2*I)*c*x*Log[f]))/(2*Sqrt[3*f + I*c*Log[f]])]*Log[f]^3*Sqrt[3*f + I*c*

Log[f]]*Sin[3*d])/E^(((I/4)*(-9*e^2 - (6*I)*b*e*Log[f] + b^2*Log[f]^2))/(3*f + I*c*Log[f]))))/(16*(I*f - c*Log

[f])*(f - I*c*Log[f])*(3*f - I*c*Log[f])*(3*f + I*c*Log[f]))

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Maple [A] time = 0.611, size = 430, normalized size = 1. \begin{align*}{-{\frac{i}{16}}{f}^{a}\sqrt{\pi }{{\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+36\,df-9\,{e}^{2}}{12\,if+4\,c\ln \left ( f \right ) }}}}{\it Erf} \left ( -\sqrt{-c\ln \left ( f \right ) -3\,if}x+{\frac{3\,ie+b\ln \left ( f \right ) }{2}{\frac{1}{\sqrt{-c\ln \left ( f \right ) -3\,if}}}} \right ){\frac{1}{\sqrt{-c\ln \left ( f \right ) -3\,if}}}}+{{\frac{i}{16}}{f}^{a}\sqrt{\pi }{{\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+36\,df-9\,{e}^{2}}{4\,c\ln \left ( f \right ) -12\,if}}}}{\it Erf} \left ( -x\sqrt{3\,if-c\ln \left ( f \right ) }+{\frac{b\ln \left ( f \right ) -3\,ie}{2}{\frac{1}{\sqrt{3\,if-c\ln \left ( f \right ) }}}} \right ){\frac{1}{\sqrt{3\,if-c\ln \left ( f \right ) }}}}-{{\frac{3\,i}{16}}{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 ) }}}}+{{\frac{3\,i}{16}}{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}}}} \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)^3,x)

[Out]

-1/16*I*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+36*d*f-9*e^2)/(3*I*f+c*ln(f)))/(-c*ln(

f)-3*I*f)^(1/2)*erf(-(-c*ln(f)-3*I*f)^(1/2)*x+1/2*(3*I*e+b*ln(f))/(-c*ln(f)-3*I*f)^(1/2))+1/16*I*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+36*d*f-9*e^2)/(-3*I*f+c*ln(f)))/(3*I*f-c*ln(f))^(1/2)*erf(-

x*(3*I*f-c*ln(f))^(1/2)+1/2*(b*ln(f)-3*I*e)/(3*I*f-c*ln(f))^(1/2))-3/16*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))+3/16*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))

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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)*sin(f*x^2+e*x+d)^3,x, algorithm="maxima")

[Out]

Exception raised: IndexError

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Fricas [B] time = 0.698121, size = 2196, normalized size = 5.11 \begin{align*} \text{result too large to display} \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)^3,x, algorithm="fricas")

[Out]

1/16*(sqrt(pi)*(-I*c^3*log(f)^3 - 3*c^2*f*log(f)^2 - I*c*f^2*log(f) - 3*f^3)*sqrt(-c*log(f) - 3*I*f)*erf(1/2*(

18*f^2*x + (2*c^2*x + b*c)*log(f)^2 + 9*e*f + (3*I*c*e - 3*I*b*f)*log(f))*sqrt(-c*log(f) - 3*I*f)/(c^2*log(f)^

2 + 9*f^2))*e^(-1/4*((b^2*c - 4*a*c^2)*log(f)^3 + 27*I*e^2*f - 108*I*d*f^2 - (12*I*c^2*d - 6*I*b*c*e + 3*I*b^2

*f)*log(f)^2 - 9*(c*e^2 - 2*b*e*f + 4*a*f^2)*log(f))/(c^2*log(f)^2 + 9*f^2)) + sqrt(pi)*(3*I*c^3*log(f)^3 + 3*

c^2*f*log(f)^2 + 27*I*c*f^2*log(f) + 27*f^3)*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)*(-3*I*c^3*log(f)^3 + 3*c^2*f*log(f)^2 - 27*I*c*f^2*log(f) + 27*f^3)*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^3*log(f)^3 - 3*c^2

*f*log(f)^2 + I*c*f^2*log(f) - 3*f^3)*sqrt(-c*log(f) + 3*I*f)*erf(1/2*(18*f^2*x + (2*c^2*x + b*c)*log(f)^2 + 9

*e*f + (-3*I*c*e + 3*I*b*f)*log(f))*sqrt(-c*log(f) + 3*I*f)/(c^2*log(f)^2 + 9*f^2))*e^(-1/4*((b^2*c - 4*a*c^2)

*log(f)^3 - 27*I*e^2*f + 108*I*d*f^2 - (-12*I*c^2*d + 6*I*b*c*e - 3*I*b^2*f)*log(f)^2 - 9*(c*e^2 - 2*b*e*f + 4

*a*f^2)*log(f))/(c^2*log(f)^2 + 9*f^2)))/(c^4*log(f)^4 + 10*c^2*f^2*log(f)^2 + 9*f^4)

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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)*sin(f*x**2+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} \sin \left (f x^{2} + 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)*sin(f*x^2+e*x+d)^3,x, algorithm="giac")

[Out]

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

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