\(\int f^{a+c x^2} \sin ^3(d+e x+f x^2) \, dx\) [93]

Optimal result

Integrand size = 23, antiderivative size = 377 \[ \int f^{a+c x^2} \sin ^3\left (d+e x+f x^2\right ) \, dx=\frac {3 i e^{-i d-\frac {e^2}{4 i f-4 c \log (f)}} f^a \sqrt {\pi } \text {erf}\left (\frac {i e+2 x (i f-c \log (f))}{2 \sqrt {i f-c \log (f)}}\right )}{16 \sqrt {i f-c \log (f)}}-\frac {i e^{-3 i d-\frac {9 e^2}{4 (3 i f-c \log (f))}} f^a \sqrt {\pi } \text {erf}\left (\frac {3 i e+2 x (3 i f-c \log (f))}{2 \sqrt {3 i f-c \log (f)}}\right )}{16 \sqrt {3 i f-c \log (f)}}-\frac {3 i e^{i d+\frac {e^2}{4 i f+4 c \log (f)}} f^a \sqrt {\pi } \text {erfi}\left (\frac {i e+2 x (i f+c \log (f))}{2 \sqrt {i f+c \log (f)}}\right )}{16 \sqrt {i f+c \log (f)}}+\frac {i e^{3 i d+\frac {9 e^2}{4 (3 i f+c \log (f))}} f^a \sqrt {\pi } \text {erfi}\left (\frac {3 i e+2 x (3 i f+c \log (f))}{2 \sqrt {3 i f+c \log (f)}}\right )}{16 \sqrt {3 i f+c \log (f)}} \]

[Out]

3/16*I*exp(-I*d-e^2/(4*I*f-4*c*ln(f)))*f^a*erf(1/2*(I*e+2*x*(I*f-c*ln(f)))/(I*f-c*ln(f))^(1/2))*Pi^(1/2)/(I*f-
c*ln(f))^(1/2)-1/16*I*exp(-3*I*d-9/4*e^2/(3*I*f-c*ln(f)))*f^a*erf(1/2*(3*I*e+2*x*(3*I*f-c*ln(f)))/(3*I*f-c*ln(
f))^(1/2))*Pi^(1/2)/(3*I*f-c*ln(f))^(1/2)-3/16*I*exp(I*d+e^2/(4*I*f+4*c*ln(f)))*f^a*erfi(1/2*(I*e+2*x*(I*f+c*l
n(f)))/(I*f+c*ln(f))^(1/2))*Pi^(1/2)/(I*f+c*ln(f))^(1/2)+1/16*I*exp(3*I*d+9/4*e^2/(3*I*f+c*ln(f)))*f^a*erfi(1/
2*(3*I*e+2*x*(3*I*f+c*ln(f)))/(3*I*f+c*ln(f))^(1/2))*Pi^(1/2)/(3*I*f+c*ln(f))^(1/2)

Rubi [A] (verified)

Time = 0.81 (sec) , antiderivative size = 377, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {4560, 2325, 2266, 2236, 2235} \[ \int f^{a+c x^2} \sin ^3\left (d+e x+f x^2\right ) \, dx=-\frac {i \sqrt {\pi } f^a \exp \left (-\frac {9 e^2}{4 (-c \log (f)+3 i f)}-3 i d\right ) \text {erf}\left (\frac {2 x (-c \log (f)+3 i f)+3 i e}{2 \sqrt {-c \log (f)+3 i f}}\right )}{16 \sqrt {-c \log (f)+3 i f}}+\frac {3 i \sqrt {\pi } f^a e^{-\frac {e^2}{-4 c \log (f)+4 i f}-i d} \text {erf}\left (\frac {2 x (-c \log (f)+i f)+i e}{2 \sqrt {-c \log (f)+i f}}\right )}{16 \sqrt {-c \log (f)+i f}}-\frac {3 i \sqrt {\pi } f^a e^{\frac {e^2}{4 c \log (f)+4 i f}+i d} \text {erfi}\left (\frac {2 x (c \log (f)+i f)+i e}{2 \sqrt {c \log (f)+i f}}\right )}{16 \sqrt {c \log (f)+i f}}+\frac {i \sqrt {\pi } f^a e^{\frac {9 e^2}{4 (c \log (f)+3 i f)}+3 i d} \text {erfi}\left (\frac {2 x (c \log (f)+3 i f)+3 i e}{2 \sqrt {c \log (f)+3 i f}}\right )}{16 \sqrt {c \log (f)+3 i f}} \]

[In]

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

[Out]

(((3*I)/16)*E^((-I)*d - e^2/((4*I)*f - 4*c*Log[f]))*f^a*Sqrt[Pi]*Erf[(I*e + 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 - (9*e^2)/(4*((3*I)*f - c*Log[f])))*f^a*Sqrt[Pi]*Erf
[((3*I)*e + 2*x*((3*I)*f - c*Log[f]))/(2*Sqrt[(3*I)*f - c*Log[f]])])/Sqrt[(3*I)*f - c*Log[f]] - (((3*I)/16)*E^
(I*d + e^2/((4*I)*f + 4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(I*e + 2*x*(I*f + c*Log[f]))/(2*Sqrt[I*f + c*Log[f]])])/S
qrt[I*f + c*Log[f]] + ((I/16)*E^((3*I)*d + (9*e^2)/(4*((3*I)*f + c*Log[f])))*f^a*Sqrt[Pi]*Erfi[((3*I)*e + 2*x*
((3*I)*f + c*Log[f]))/(2*Sqrt[(3*I)*f + c*Log[f]])])/Sqrt[(3*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}{8} i e^{-3 i \left (d+e x+f x^2\right )} f^{a+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+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+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+c x^2}\right ) \, dx \\ & = -\left (\frac {1}{8} i \int e^{-3 i \left (d+e x+f x^2\right )} f^{a+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+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+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+c x^2} \, dx \\ & = -\left (\frac {1}{8} i \int \exp \left (-3 i d-3 i e x+a \log (f)-x^2 (3 i f-c \log (f))\right ) \, dx\right )+\frac {1}{8} i \int \exp \left (3 i d+3 i e x+a \log (f)+x^2 (3 i f+c \log (f))\right ) \, dx+\frac {3}{8} i \int \exp \left (-i d-i e x+a \log (f)-x^2 (i f-c \log (f))\right ) \, dx-\frac {3}{8} i \int \exp \left (i d+i e x+a \log (f)+x^2 (i f+c \log (f))\right ) \, dx \\ & = \frac {1}{8} \left (3 i e^{-i d-\frac {e^2}{4 i f-4 c \log (f)}} f^a\right ) \int \exp \left (\frac {(-i e+2 x (-i f+c \log (f)))^2}{4 (-i f+c \log (f))}\right ) \, dx-\frac {1}{8} \left (i e^{-3 i d-\frac {9 e^2}{4 (3 i f-c \log (f))}} f^a\right ) \int \exp \left (\frac {(-3 i e+2 x (-3 i f+c \log (f)))^2}{4 (-3 i f+c \log (f))}\right ) \, dx+\frac {1}{8} \left (i e^{3 i d+\frac {9 e^2}{4 (3 i f+c \log (f))}} f^a\right ) \int \exp \left (\frac {(3 i e+2 x (3 i f+c \log (f)))^2}{4 (3 i f+c \log (f))}\right ) \, dx-\frac {1}{8} \left (3 i e^{i d+\frac {e^2}{4 i f+4 c \log (f)}} f^a\right ) \int \exp \left (\frac {(i e+2 x (i f+c \log (f)))^2}{4 (i f+c \log (f))}\right ) \, dx \\ & = \frac {3 i e^{-i d-\frac {e^2}{4 i f-4 c \log (f)}} f^a \sqrt {\pi } \text {erf}\left (\frac {i e+2 x (i f-c \log (f))}{2 \sqrt {i f-c \log (f)}}\right )}{16 \sqrt {i f-c \log (f)}}-\frac {i e^{-3 i d-\frac {9 e^2}{4 (3 i f-c \log (f))}} f^a \sqrt {\pi } \text {erf}\left (\frac {3 i e+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 e^{i d+\frac {e^2}{4 i f+4 c \log (f)}} f^a \sqrt {\pi } \text {erfi}\left (\frac {i e+2 x (i f+c \log (f))}{2 \sqrt {i f+c \log (f)}}\right )}{16 \sqrt {i f+c \log (f)}}+\frac {i e^{3 i d+\frac {9 e^2}{4 (3 i f+c \log (f))}} f^a \sqrt {\pi } \text {erfi}\left (\frac {3 i e+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 [A] (verified)

Time = 5.14 (sec) , antiderivative size = 490, normalized size of antiderivative = 1.30 \[ \int f^{a+c x^2} \sin ^3\left (d+e x+f x^2\right ) \, dx=\frac {\sqrt [4]{-1} f^a \sqrt {\pi } \left (-3 e^{\frac {e^2}{4 i f+4 c \log (f)}} \text {erfi}\left (\frac {\sqrt [4]{-1} (e+2 f x-2 i c x \log (f))}{2 \sqrt {f-i c \log (f)}}\right ) \sqrt {f-i c \log (f)} \left (9 f^3+9 i c f^2 \log (f)+c^2 f \log ^2(f)+i c^3 \log ^3(f)\right ) (\cos (d)+i \sin (d))+(f-i c \log (f)) \left (e^{\frac {9 e^2}{4 (3 i f+c \log (f))}} \text {erfi}\left (\frac {\sqrt [4]{-1} (3 e+6 f x-2 i c x \log (f))}{2 \sqrt {3 f-i c \log (f)}}\right ) \sqrt {3 f-i c \log (f)} \left (3 f^2+4 i c f \log (f)-c^2 \log ^2(f)\right ) (\cos (3 d)+i \sin (3 d))+(3 f-i c \log (f)) \left (3 e^{\frac {e^2}{-4 i f+4 c \log (f)}} \text {erfi}\left (\frac {(-1)^{3/4} (e+2 f x+2 i c x \log (f))}{2 \sqrt {f+i c \log (f)}}\right ) \sqrt {f+i c \log (f)} (-3 i f+c \log (f)) (\cos (d)-i \sin (d))+e^{\frac {9 e^2}{4 (-3 i f+c \log (f))}} \text {erfi}\left (\frac {(-1)^{3/4} (3 e+6 f x+2 i c x \log (f))}{2 \sqrt {3 f+i c \log (f)}}\right ) (f+i c \log (f)) \sqrt {3 f+i c \log (f)} (i \cos (3 d)+\sin (3 d))\right )\right )\right )}{16 \left (9 f^4+10 c^2 f^2 \log ^2(f)+c^4 \log ^4(f)\right )} \]

[In]

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

[Out]

((-1)^(1/4)*f^a*Sqrt[Pi]*(-3*E^(e^2/((4*I)*f + 4*c*Log[f]))*Erfi[((-1)^(1/4)*(e + 2*f*x - (2*I)*c*x*Log[f]))/(
2*Sqrt[f - I*c*Log[f]])]*Sqrt[f - I*c*Log[f]]*(9*f^3 + (9*I)*c*f^2*Log[f] + c^2*f*Log[f]^2 + I*c^3*Log[f]^3)*(
Cos[d] + I*Sin[d]) + (f - I*c*Log[f])*(E^((9*e^2)/(4*((3*I)*f + c*Log[f])))*Erfi[((-1)^(1/4)*(3*e + 6*f*x - (2
*I)*c*x*Log[f]))/(2*Sqrt[3*f - I*c*Log[f]])]*Sqrt[3*f - I*c*Log[f]]*(3*f^2 + (4*I)*c*f*Log[f] - c^2*Log[f]^2)*
(Cos[3*d] + I*Sin[3*d]) + (3*f - I*c*Log[f])*(3*E^(e^2/((-4*I)*f + 4*c*Log[f]))*Erfi[((-1)^(3/4)*(e + 2*f*x +
(2*I)*c*x*Log[f]))/(2*Sqrt[f + I*c*Log[f]])]*Sqrt[f + I*c*Log[f]]*((-3*I)*f + c*Log[f])*(Cos[d] - I*Sin[d]) +
E^((9*e^2)/(4*((-3*I)*f + c*Log[f])))*Erfi[((-1)^(3/4)*(3*e + 6*f*x + (2*I)*c*x*Log[f]))/(2*Sqrt[3*f + I*c*Log
[f]])]*(f + I*c*Log[f])*Sqrt[3*f + I*c*Log[f]]*(I*Cos[3*d] + Sin[3*d])))))/(16*(9*f^4 + 10*c^2*f^2*Log[f]^2 +
c^4*Log[f]^4))

Maple [A] (verified)

Time = 1.59 (sec) , antiderivative size = 338, normalized size of antiderivative = 0.90

method	result	size
risch	\(-\frac {i \sqrt {\pi }\, f^{a} {\mathrm e}^{\frac {3 i d \ln \left (f \right ) c -9 d f +\frac {9 e^{2}}{4}}{3 i f +c \ln \left (f \right )}} \operatorname {erf}\left (-\sqrt {-c \ln \left (f \right )-3 i f}\, x +\frac {3 i e}{2 \sqrt {-c \ln \left (f \right )-3 i f}}\right )}{16 \sqrt {-c \ln \left (f \right )-3 i f}}-\frac {i \sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {3 \left (4 i d \ln \left (f \right ) c +12 d f -3 e^{2}\right )}{4 \left (c \ln \left (f \right )-3 i f \right )}} \operatorname {erf}\left (x \sqrt {3 i f -c \ln \left (f \right )}+\frac {3 i e}{2 \sqrt {3 i f -c \ln \left (f \right )}}\right )}{16 \sqrt {3 i f -c \ln \left (f \right )}}+\frac {3 i \sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {4 i d \ln \left (f \right ) c +4 d f -e^{2}}{4 \left (c \ln \left (f \right )-i f \right )}} \operatorname {erf}\left (x \sqrt {i f -c \ln \left (f \right )}+\frac {i e}{2 \sqrt {i f -c \ln \left (f \right )}}\right )}{16 \sqrt {i f -c \ln \left (f \right )}}+\frac {3 i \sqrt {\pi }\, f^{a} {\mathrm e}^{\frac {4 i d \ln \left (f \right ) c -4 d f +e^{2}}{4 i f +4 c \ln \left (f \right )}} \operatorname {erf}\left (-\sqrt {-c \ln \left (f \right )-i f}\, x +\frac {i e}{2 \sqrt {-c \ln \left (f \right )-i f}}\right )}{16 \sqrt {-c \ln \left (f \right )-i f}}\)	\(338\)

[In]

int(f^(c*x^2+a)*sin(f*x^2+e*x+d)^3,x,method=_RETURNVERBOSE)

[Out]

-1/16*I*Pi^(1/2)*f^a*exp(3/4*(4*I*d*ln(f)*c-12*d*f+3*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+3/2*I*e/(-c*ln(f)-3*I*f)^(1/2))-1/16*I*Pi^(1/2)*f^a*exp(-3/4*(4*I*d*ln(f)*c+12*d*f-3*e^2)/(c
*ln(f)-3*I*f))/(3*I*f-c*ln(f))^(1/2)*erf(x*(3*I*f-c*ln(f))^(1/2)+3/2*I*e/(3*I*f-c*ln(f))^(1/2))+3/16*I*Pi^(1/2
)*f^a*exp(-1/4*(4*I*d*ln(f)*c+4*d*f-e^2)/(c*ln(f)-I*f))/(I*f-c*ln(f))^(1/2)*erf(x*(I*f-c*ln(f))^(1/2)+1/2*I*e/
(I*f-c*ln(f))^(1/2))+3/16*I*Pi^(1/2)*f^a*exp(1/4*(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/(-c*ln(f)-I*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. 713 vs. \(2 (269) = 538\).

Time = 0.29 (sec) , antiderivative size = 713, normalized size of antiderivative = 1.89 \[ \int f^{a+c x^2} \sin ^3\left (d+e x+f x^2\right ) \, dx=\frac {\sqrt {\pi } {\left (-i \, c^{3} \log \left (f\right )^{3} - 3 \, c^{2} f \log \left (f\right )^{2} - i \, c f^{2} \log \left (f\right ) - 3 \, f^{3}\right )} \sqrt {-c \log \left (f\right ) - 3 i \, f} \operatorname {erf}\left (\frac {{\left (2 \, c^{2} x \log \left (f\right )^{2} + 18 \, f^{2} x + 3 i \, c e \log \left (f\right ) + 9 \, e f\right )} \sqrt {-c \log \left (f\right ) - 3 i \, f}}{2 \, {\left (c^{2} \log \left (f\right )^{2} + 9 \, f^{2}\right )}}\right ) e^{\left (\frac {4 \, a c^{2} \log \left (f\right )^{3} + 12 i \, c^{2} d \log \left (f\right )^{2} - 27 i \, e^{2} f + 108 i \, d f^{2} + 9 \, {\left (c e^{2} + 4 \, a f^{2}\right )} \log \left (f\right )}{4 \, {\left (c^{2} \log \left (f\right )^{2} + 9 \, f^{2}\right )}}\right )} + \sqrt {\pi } {\left (i \, c^{3} \log \left (f\right )^{3} - 3 \, c^{2} f \log \left (f\right )^{2} + i \, c f^{2} \log \left (f\right ) - 3 \, f^{3}\right )} \sqrt {-c \log \left (f\right ) + 3 i \, f} \operatorname {erf}\left (\frac {{\left (2 \, c^{2} x \log \left (f\right )^{2} + 18 \, f^{2} x - 3 i \, c e \log \left (f\right ) + 9 \, e f\right )} \sqrt {-c \log \left (f\right ) + 3 i \, f}}{2 \, {\left (c^{2} \log \left (f\right )^{2} + 9 \, f^{2}\right )}}\right ) e^{\left (\frac {4 \, a c^{2} \log \left (f\right )^{3} - 12 i \, c^{2} d \log \left (f\right )^{2} + 27 i \, e^{2} f - 108 i \, d f^{2} + 9 \, {\left (c e^{2} + 4 \, a f^{2}\right )} \log \left (f\right )}{4 \, {\left (c^{2} \log \left (f\right )^{2} + 9 \, f^{2}\right )}}\right )} - 3 \, \sqrt {\pi } {\left (-i \, c^{3} \log \left (f\right )^{3} - c^{2} f \log \left (f\right )^{2} - 9 i \, c f^{2} \log \left (f\right ) - 9 \, f^{3}\right )} \sqrt {-c \log \left (f\right ) - i \, f} \operatorname {erf}\left (\frac {{\left (2 \, c^{2} x \log \left (f\right )^{2} + 2 \, f^{2} x + i \, c e \log \left (f\right ) + e f\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 {4 \, a c^{2} \log \left (f\right )^{3} + 4 i \, c^{2} d \log \left (f\right )^{2} - i \, e^{2} f + 4 i \, d f^{2} + {\left (c e^{2} + 4 \, a f^{2}\right )} \log \left (f\right )}{4 \, {\left (c^{2} \log \left (f\right )^{2} + f^{2}\right )}}\right )} - 3 \, \sqrt {\pi } {\left (i \, c^{3} \log \left (f\right )^{3} - c^{2} f \log \left (f\right )^{2} + 9 i \, c f^{2} \log \left (f\right ) - 9 \, f^{3}\right )} \sqrt {-c \log \left (f\right ) + i \, f} \operatorname {erf}\left (\frac {{\left (2 \, c^{2} x \log \left (f\right )^{2} + 2 \, f^{2} x - i \, c e \log \left (f\right ) + e f\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 {4 \, a c^{2} \log \left (f\right )^{3} - 4 i \, c^{2} d \log \left (f\right )^{2} + i \, e^{2} f - 4 i \, d f^{2} + {\left (c e^{2} + 4 \, a f^{2}\right )} \log \left (f\right )}{4 \, {\left (c^{2} \log \left (f\right )^{2} + f^{2}\right )}}\right )}}{16 \, {\left (c^{4} \log \left (f\right )^{4} + 10 \, c^{2} f^{2} \log \left (f\right )^{2} + 9 \, f^{4}\right )}} \]

[In]

integrate(f^(c*x^2+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*(
2*c^2*x*log(f)^2 + 18*f^2*x + 3*I*c*e*log(f) + 9*e*f)*sqrt(-c*log(f) - 3*I*f)/(c^2*log(f)^2 + 9*f^2))*e^(1/4*(
4*a*c^2*log(f)^3 + 12*I*c^2*d*log(f)^2 - 27*I*e^2*f + 108*I*d*f^2 + 9*(c*e^2 + 4*a*f^2)*log(f))/(c^2*log(f)^2
+ 9*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*(2*c^2*x*log(f)^2 + 18*f^2*x - 3*I*c*e*log(f) + 9*e*f)*sqrt(-c*log(f) + 3*I*f)/(c^2*log(f)^2 + 9*f^2))*e^(
1/4*(4*a*c^2*log(f)^3 - 12*I*c^2*d*log(f)^2 + 27*I*e^2*f - 108*I*d*f^2 + 9*(c*e^2 + 4*a*f^2)*log(f))/(c^2*log(
f)^2 + 9*f^2)) - 3*sqrt(pi)*(-I*c^3*log(f)^3 - c^2*f*log(f)^2 - 9*I*c*f^2*log(f) - 9*f^3)*sqrt(-c*log(f) - I*f
)*erf(1/2*(2*c^2*x*log(f)^2 + 2*f^2*x + I*c*e*log(f) + e*f)*sqrt(-c*log(f) - I*f)/(c^2*log(f)^2 + f^2))*e^(1/4
*(4*a*c^2*log(f)^3 + 4*I*c^2*d*log(f)^2 - I*e^2*f + 4*I*d*f^2 + (c*e^2 + 4*a*f^2)*log(f))/(c^2*log(f)^2 + f^2)
) - 3*sqrt(pi)*(I*c^3*log(f)^3 - c^2*f*log(f)^2 + 9*I*c*f^2*log(f) - 9*f^3)*sqrt(-c*log(f) + I*f)*erf(1/2*(2*c
^2*x*log(f)^2 + 2*f^2*x - I*c*e*log(f) + e*f)*sqrt(-c*log(f) + I*f)/(c^2*log(f)^2 + f^2))*e^(1/4*(4*a*c^2*log(
f)^3 - 4*I*c^2*d*log(f)^2 + I*e^2*f - 4*I*d*f^2 + (c*e^2 + 4*a*f^2)*log(f))/(c^2*log(f)^2 + f^2)))/(c^4*log(f)
^4 + 10*c^2*f^2*log(f)^2 + 9*f^4)

Sympy [F]

\[ \int f^{a+c x^2} \sin ^3\left (d+e x+f x^2\right ) \, dx=\int f^{a + c x^{2}} \sin ^{3}{\left (d + e x + f x^{2} \right )}\, dx \]

[In]

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

[Out]

Integral(f**(a + c*x**2)*sin(d + e*x + f*x**2)**3, x)

Maxima [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2175 vs. \(2 (269) = 538\).

Time = 0.28 (sec) , antiderivative size = 2175, normalized size of antiderivative = 5.77 \[ \int f^{a+c x^2} \sin ^3\left (d+e x+f x^2\right ) \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/32*(sqrt(pi)*sqrt(2*c^2*log(f)^2 + 18*f^2)*(((c^2*f^(9/4*c*e^2/(c^2*log(f)^2 + 9*f^2))*f^a*log(f)^2 + f^(9/4
*c*e^2/(c^2*log(f)^2 + 9*f^2))*f^(a + 2))*cos(3/4*(4*c^2*d*log(f)^2 - 9*e^2*f + 36*d*f^2)/(c^2*log(f)^2 + 9*f^
2)) + (-I*c^2*f^(9/4*c*e^2/(c^2*log(f)^2 + 9*f^2))*f^a*log(f)^2 - I*f^(9/4*c*e^2/(c^2*log(f)^2 + 9*f^2))*f^(a
+ 2))*sin(3/4*(4*c^2*d*log(f)^2 - 9*e^2*f + 36*d*f^2)/(c^2*log(f)^2 + 9*f^2)))*erf(1/2*(2*(c*log(f) - 3*I*f)*x
- 3*I*e)/sqrt(-c*log(f) + 3*I*f)) + ((c^2*f^(9/4*c*e^2/(c^2*log(f)^2 + 9*f^2))*f^a*log(f)^2 + f^(9/4*c*e^2/(c
^2*log(f)^2 + 9*f^2))*f^(a + 2))*cos(3/4*(4*c^2*d*log(f)^2 - 9*e^2*f + 36*d*f^2)/(c^2*log(f)^2 + 9*f^2)) + (I*
c^2*f^(9/4*c*e^2/(c^2*log(f)^2 + 9*f^2))*f^a*log(f)^2 + I*f^(9/4*c*e^2/(c^2*log(f)^2 + 9*f^2))*f^(a + 2))*sin(
3/4*(4*c^2*d*log(f)^2 - 9*e^2*f + 36*d*f^2)/(c^2*log(f)^2 + 9*f^2)))*erf(1/2*(2*(c*log(f) + 3*I*f)*x + 3*I*e)/
sqrt(-c*log(f) - 3*I*f)))*sqrt(c*log(f) + sqrt(c^2*log(f)^2 + 9*f^2)) - 3*sqrt(pi)*sqrt(2*c^2*log(f)^2 + 2*f^2
)*(((c^2*f^(1/4*c*e^2/(c^2*log(f)^2 + f^2))*f^a*log(f)^2 + 9*f^(1/4*c*e^2/(c^2*log(f)^2 + f^2))*f^(a + 2))*cos
(1/4*(4*c^2*d*log(f)^2 - e^2*f + 4*d*f^2)/(c^2*log(f)^2 + f^2)) + (-I*c^2*f^(1/4*c*e^2/(c^2*log(f)^2 + f^2))*f
^a*log(f)^2 - 9*I*f^(1/4*c*e^2/(c^2*log(f)^2 + f^2))*f^(a + 2))*sin(1/4*(4*c^2*d*log(f)^2 - e^2*f + 4*d*f^2)/(
c^2*log(f)^2 + f^2)))*erf(1/2*(2*(c*log(f) - I*f)*x - I*e)/sqrt(-c*log(f) + I*f)) + ((c^2*f^(1/4*c*e^2/(c^2*lo
g(f)^2 + f^2))*f^a*log(f)^2 + 9*f^(1/4*c*e^2/(c^2*log(f)^2 + f^2))*f^(a + 2))*cos(1/4*(4*c^2*d*log(f)^2 - e^2*
f + 4*d*f^2)/(c^2*log(f)^2 + f^2)) + (I*c^2*f^(1/4*c*e^2/(c^2*log(f)^2 + f^2))*f^a*log(f)^2 + 9*I*f^(1/4*c*e^2
/(c^2*log(f)^2 + f^2))*f^(a + 2))*sin(1/4*(4*c^2*d*log(f)^2 - e^2*f + 4*d*f^2)/(c^2*log(f)^2 + f^2)))*erf(1/2*
(2*(c*log(f) + I*f)*x + I*e)/sqrt(-c*log(f) - I*f)))*sqrt(c*log(f) + sqrt(c^2*log(f)^2 + f^2)) + sqrt(pi)*sqrt
(2*c^2*log(f)^2 + 18*f^2)*(((I*c^2*f^(9/4*c*e^2/(c^2*log(f)^2 + 9*f^2))*f^a*log(f)^2 + I*f^(9/4*c*e^2/(c^2*log
(f)^2 + 9*f^2))*f^(a + 2))*cos(3/4*(4*c^2*d*log(f)^2 - 9*e^2*f + 36*d*f^2)/(c^2*log(f)^2 + 9*f^2)) + (c^2*f^(9
/4*c*e^2/(c^2*log(f)^2 + 9*f^2))*f^a*log(f)^2 + f^(9/4*c*e^2/(c^2*log(f)^2 + 9*f^2))*f^(a + 2))*sin(3/4*(4*c^2
*d*log(f)^2 - 9*e^2*f + 36*d*f^2)/(c^2*log(f)^2 + 9*f^2)))*erf(1/2*(2*(c*log(f) - 3*I*f)*x - 3*I*e)/sqrt(-c*lo
g(f) + 3*I*f)) + ((-I*c^2*f^(9/4*c*e^2/(c^2*log(f)^2 + 9*f^2))*f^a*log(f)^2 - I*f^(9/4*c*e^2/(c^2*log(f)^2 + 9
*f^2))*f^(a + 2))*cos(3/4*(4*c^2*d*log(f)^2 - 9*e^2*f + 36*d*f^2)/(c^2*log(f)^2 + 9*f^2)) + (c^2*f^(9/4*c*e^2/
(c^2*log(f)^2 + 9*f^2))*f^a*log(f)^2 + f^(9/4*c*e^2/(c^2*log(f)^2 + 9*f^2))*f^(a + 2))*sin(3/4*(4*c^2*d*log(f)
^2 - 9*e^2*f + 36*d*f^2)/(c^2*log(f)^2 + 9*f^2)))*erf(1/2*(2*(c*log(f) + 3*I*f)*x + 3*I*e)/sqrt(-c*log(f) - 3*
I*f)))*sqrt(-c*log(f) + sqrt(c^2*log(f)^2 + 9*f^2)) - 3*sqrt(pi)*sqrt(2*c^2*log(f)^2 + 2*f^2)*(((I*c^2*f^(1/4*
c*e^2/(c^2*log(f)^2 + f^2))*f^a*log(f)^2 + 9*I*f^(1/4*c*e^2/(c^2*log(f)^2 + f^2))*f^(a + 2))*cos(1/4*(4*c^2*d*
log(f)^2 - e^2*f + 4*d*f^2)/(c^2*log(f)^2 + f^2)) + (c^2*f^(1/4*c*e^2/(c^2*log(f)^2 + f^2))*f^a*log(f)^2 + 9*f
^(1/4*c*e^2/(c^2*log(f)^2 + f^2))*f^(a + 2))*sin(1/4*(4*c^2*d*log(f)^2 - e^2*f + 4*d*f^2)/(c^2*log(f)^2 + f^2)
))*erf(1/2*(2*(c*log(f) - I*f)*x - I*e)/sqrt(-c*log(f) + I*f)) + ((-I*c^2*f^(1/4*c*e^2/(c^2*log(f)^2 + f^2))*f
^a*log(f)^2 - 9*I*f^(1/4*c*e^2/(c^2*log(f)^2 + f^2))*f^(a + 2))*cos(1/4*(4*c^2*d*log(f)^2 - e^2*f + 4*d*f^2)/(
c^2*log(f)^2 + f^2)) + (c^2*f^(1/4*c*e^2/(c^2*log(f)^2 + f^2))*f^a*log(f)^2 + 9*f^(1/4*c*e^2/(c^2*log(f)^2 + f
^2))*f^(a + 2))*sin(1/4*(4*c^2*d*log(f)^2 - e^2*f + 4*d*f^2)/(c^2*log(f)^2 + f^2)))*erf(1/2*(2*(c*log(f) + I*f
)*x + I*e)/sqrt(-c*log(f) - I*f)))*sqrt(-c*log(f) + sqrt(c^2*log(f)^2 + f^2)))/(c^4*log(f)^4 + 10*c^2*f^2*log(
f)^2 + 9*f^4)

Giac [F]

\[ \int f^{a+c x^2} \sin ^3\left (d+e x+f x^2\right ) \, dx=\int { f^{c x^{2} + a} \sin \left (f x^{2} + e x + d\right )^{3} \,d x } \]

[In]

integrate(f^(c*x^2+a)*sin(f*x^2+e*x+d)^3,x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int f^{a+c x^2} \sin ^3\left (d+e x+f x^2\right ) \, dx=\int f^{c\,x^2+a}\,{\sin \left (f\,x^2+e\,x+d\right )}^3 \,d x \]

[In]

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

[Out]

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