3.99 \(\int f^{a+b x+c x^2} \sin ^3(d+f x^2) \, dx\)
Optimal. Leaf size=386 \[ -\frac{3 i \sqrt{\pi } f^a e^{\frac{b^2 \log ^2(f)}{-4 c \log (f)+4 i f}-i d} \text{Erf}\left (\frac{b \log (f)-2 x (-c \log (f)+i f)}{2 \sqrt{-c \log (f)+i f}}\right )}{16 \sqrt{-c \log (f)+i f}}+\frac{i \sqrt{\pi } f^a e^{\frac{b^2 \log ^2(f)}{-4 c \log (f)+12 i f}-3 i d} \text{Erf}\left (\frac{b \log (f)-2 x (-c \log (f)+3 i f)}{2 \sqrt{-c \log (f)+3 i f}}\right )}{16 \sqrt{-c \log (f)+3 i f}}+\frac{i \sqrt{\pi } f^a \exp \left (3 i d-\frac{b^2 \log ^2(f)}{4 (c \log (f)+3 i f)}\right ) \text{Erfi}\left (\frac{b \log (f)+2 x (c \log (f)+3 i f)}{2 \sqrt{c \log (f)+3 i f}}\right )}{16 \sqrt{c \log (f)+3 i f}}-\frac{3 i \sqrt{\pi } f^a e^{i d-\frac{b^2 \log ^2(f)}{4 c \log (f)+4 i f}} \text{Erfi}\left (\frac{b \log (f)+2 x (c \log (f)+i f)}{2 \sqrt{c \log (f)+i f}}\right )}{16 \sqrt{c \log (f)+i f}} \]
[Out]
(((-3*I)/16)*E^((-I)*d + (b^2*Log[f]^2)/((4*I)*f - 4*c*Log[f]))*f^a*Sqrt[Pi]*Erf[(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 + (b^2*Log[f]^2)/((12*I)*f - 4*c*Lo
g[f]))*f^a*Sqrt[Pi]*Erf[(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]] - (((3*I)/16)*E^(I*d - (b^2*Log[f]^2)/((4*I)*f + 4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(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 - (b^2*Log[f]^2)/(4*((3*I)*f
+ c*Log[f])))*f^a*Sqrt[Pi]*Erfi[(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 = 0.572557, antiderivative size = 386, normalized size of antiderivative = 1.,
number of steps used = 14, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used
= {4472, 2287, 2234, 2205, 2204} \[ -\frac{3 i \sqrt{\pi } f^a e^{\frac{b^2 \log ^2(f)}{-4 c \log (f)+4 i f}-i d} \text{Erf}\left (\frac{b \log (f)-2 x (-c \log (f)+i f)}{2 \sqrt{-c \log (f)+i f}}\right )}{16 \sqrt{-c \log (f)+i f}}+\frac{i \sqrt{\pi } f^a e^{\frac{b^2 \log ^2(f)}{-4 c \log (f)+12 i f}-3 i d} \text{Erf}\left (\frac{b \log (f)-2 x (-c \log (f)+3 i f)}{2 \sqrt{-c \log (f)+3 i f}}\right )}{16 \sqrt{-c \log (f)+3 i f}}+\frac{i \sqrt{\pi } f^a \exp \left (3 i d-\frac{b^2 \log ^2(f)}{4 (c \log (f)+3 i f)}\right ) \text{Erfi}\left (\frac{b \log (f)+2 x (c \log (f)+3 i f)}{2 \sqrt{c \log (f)+3 i f}}\right )}{16 \sqrt{c \log (f)+3 i f}}-\frac{3 i \sqrt{\pi } f^a e^{i d-\frac{b^2 \log ^2(f)}{4 c \log (f)+4 i f}} \text{Erfi}\left (\frac{b \log (f)+2 x (c \log (f)+i f)}{2 \sqrt{c \log (f)+i f}}\right )}{16 \sqrt{c \log (f)+i f}} \]
Antiderivative was successfully verified.
[In]
Int[f^(a + b*x + c*x^2)*Sin[d + f*x^2]^3,x]
[Out]
(((-3*I)/16)*E^((-I)*d + (b^2*Log[f]^2)/((4*I)*f - 4*c*Log[f]))*f^a*Sqrt[Pi]*Erf[(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 + (b^2*Log[f]^2)/((12*I)*f - 4*c*Lo
g[f]))*f^a*Sqrt[Pi]*Erf[(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]] - (((3*I)/16)*E^(I*d - (b^2*Log[f]^2)/((4*I)*f + 4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(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 - (b^2*Log[f]^2)/(4*((3*I)*f
+ c*Log[f])))*f^a*Sqrt[Pi]*Erfi[(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+f x^2\right ) \, dx &=\int \left (\frac{3}{8} i e^{-i d-i f x^2} f^{a+b x+c x^2}-\frac{3}{8} i e^{i d+i f x^2} f^{a+b x+c x^2}-\frac{1}{8} i e^{-3 i d-3 i f x^2} f^{a+b x+c x^2}+\frac{1}{8} i e^{3 i d+3 i f x^2} f^{a+b x+c x^2}\right ) \, dx\\ &=-\left (\frac{1}{8} i \int e^{-3 i d-3 i f x^2} f^{a+b x+c x^2} \, dx\right )+\frac{1}{8} i \int e^{3 i d+3 i f x^2} f^{a+b x+c x^2} \, dx+\frac{3}{8} i \int e^{-i d-i f x^2} f^{a+b x+c x^2} \, dx-\frac{3}{8} i \int e^{i d+i f x^2} f^{a+b x+c x^2} \, dx\\ &=-\left (\frac{1}{8} i \int \exp \left (-3 i d+a \log (f)+b x \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)+b x \log (f)+x^2 (3 i f+c \log (f))\right ) \, dx+\frac{3}{8} i \int \exp \left (-i d+a \log (f)+b x \log (f)-x^2 (i f-c \log (f))\right ) \, dx-\frac{3}{8} i \int \exp \left (i d+a \log (f)+b x \log (f)+x^2 (i f+c \log (f))\right ) \, dx\\ &=\frac{1}{8} \left (3 i e^{-i d+\frac{b^2 \log ^2(f)}{4 i f-4 c \log (f)}} f^a\right ) \int \exp \left (\frac{(b \log (f)+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{b^2 \log ^2(f)}{12 i f-4 c \log (f)}} f^a\right ) \int \exp \left (\frac{(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{b^2 \log ^2(f)}{4 (3 i f+c \log (f))}\right ) f^a\right ) \int \exp \left (\frac{(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 e^{i d-\frac{b^2 \log ^2(f)}{4 i f+4 c \log (f)}} f^a\right ) \int \exp \left (\frac{(b \log (f)+2 x (i f+c \log (f)))^2}{4 (i f+c \log (f))}\right ) \, dx\\ &=-\frac{3 i e^{-i d+\frac{b^2 \log ^2(f)}{4 i f-4 c \log (f)}} f^a \sqrt{\pi } \text{erf}\left (\frac{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 e^{-3 i d+\frac{b^2 \log ^2(f)}{12 i f-4 c \log (f)}} f^a \sqrt{\pi } \text{erf}\left (\frac{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 e^{i d-\frac{b^2 \log ^2(f)}{4 i f+4 c \log (f)}} f^a \sqrt{\pi } \text{erfi}\left (\frac{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{b^2 \log ^2(f)}{4 (3 i f+c \log (f))}\right ) f^a \sqrt{\pi } \text{erfi}\left (\frac{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 = 7.01153, size = 3291, normalized size = 8.53 \[ \text{Result too large to show} \]
Antiderivative was successfully verified.
[In]
Integrate[f^(a + b*x + c*x^2)*Sin[d + f*x^2]^3,x]
[Out]
(f^a*Sqrt[Pi]*(-27*(-1)^(3/4)*E^(((I/4)*b^2*Log[f]^2)/(f - I*c*Log[f]))*f^3*Cos[d]*Erfi[((-1)^(1/4)*(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)*b^2*
Log[f]^2)/(f - I*c*Log[f]))*f^2*Cos[d]*Erfi[((-1)^(1/4)*(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)*b^2*Log[f]^2)/(f - I*c*Log[f]))*f*Cos[d]
*Erfi[((-1)^(1/4)*(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)*b^2*Log[f]^2)/(f - I*c*Log[f]))*Cos[d]*Erfi[((-1)^(1/4)*(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)*b^2*Log[
f]^2)/(3*f - I*c*Log[f]))*f^3*Cos[3*d]*Erfi[((-1)^(1/4)*(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)*b^2*Log[f]^2)/(3*f - I*c*Log[f]))*f^2*Cos[3*d]*
Erfi[((-1)^(1/4)*(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*Lo
g[f]] + 3*(-1)^(3/4)*c^2*E^(((I/4)*b^2*Log[f]^2)/(3*f - I*c*Log[f]))*f*Cos[3*d]*Erfi[((-1)^(1/4)*(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)*b^2*Log[f]^2)/(3*f - I*c*Log[f]))*Cos[3*d]*Erfi[((-1)^(1/4)*(6*f*x - I*b*Log[f] - (2*I)*c*x*Log[f]))/(2*S
qrt[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)*(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)*b^2*Log[f]^2)/(f + I
*c*Log[f])) - (27*(-1)^(3/4)*c*f^2*Cos[d]*Erfi[((-1)^(3/4)*(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)*b^2*Log[f]^2)/(f + I*c*Log[f])) + (3*(-1)^(1/4)*c^2*f*C
os[d]*Erfi[((-1)^(3/4)*(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]])/E^(((I/4)*b^2*Log[f]^2)/(f + I*c*Log[f])) - (3*(-1)^(3/4)*c^3*Cos[d]*Erfi[((-1)^(3/4)*(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)*b^2*Log[f]^2)/(
f + I*c*Log[f])) - (3*(-1)^(1/4)*f^3*Cos[3*d]*Erfi[((-1)^(3/4)*(6*f*x + I*b*Log[f] + (2*I)*c*x*Log[f]))/(2*Sqr
t[3*f + I*c*Log[f]])]*Sqrt[3*f + I*c*Log[f]])/E^(((I/4)*b^2*Log[f]^2)/(3*f + I*c*Log[f])) + ((-1)^(3/4)*c*f^2*
Cos[3*d]*Erfi[((-1)^(3/4)*(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)*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)*(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)*
b^2*Log[f]^2)/(3*f + I*c*Log[f])) + ((-1)^(3/4)*c^3*Cos[3*d]*Erfi[((-1)^(3/4)*(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)*b^2*Log[f]^2)/(3*f + I*c*Log[f
])) + 27*(-1)^(1/4)*E^(((I/4)*b^2*Log[f]^2)/(f - I*c*Log[f]))*f^3*Erfi[((-1)^(1/4)*(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)*b^2*Log[f]^2)/
(f - I*c*Log[f]))*f^2*Erfi[((-1)^(1/4)*(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)*b^2*Log[f]^2)/(f - I*c*Log[f]))*f*Erfi[((-1)^(1/4)
*(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)*b^2*Log[f]^2)/(f - I*c*Log[f]))*Erfi[((-1)^(1/4)*(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)*(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)*b^2*Log[f
]^2)/(f + I*c*Log[f])) - (27*(-1)^(1/4)*c*f^2*Erfi[((-1)^(3/4)*(2*f*x + I*b*Log[f] + (2*I)*c*x*Log[f]))/(2*Sqr
t[f + I*c*Log[f]])]*Log[f]*Sqrt[f + I*c*Log[f]]*Sin[d])/E^(((I/4)*b^2*Log[f]^2)/(f + I*c*Log[f])) - (3*(-1)^(3
/4)*c^2*f*Erfi[((-1)^(3/4)*(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)*b^2*Log[f]^2)/(f + I*c*Log[f])) - (3*(-1)^(1/4)*c^3*Erfi[((-1)^(3/4)*(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)*b^2*
Log[f]^2)/(f + I*c*Log[f])) - 3*(-1)^(1/4)*E^(((I/4)*b^2*Log[f]^2)/(3*f - I*c*Log[f]))*f^3*Erfi[((-1)^(1/4)*(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] - (-1)^(3/4
)*c*E^(((I/4)*b^2*Log[f]^2)/(3*f - I*c*Log[f]))*f^2*Erfi[((-1)^(1/4)*(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)*b^2*Log[f]^2)/
(3*f - I*c*Log[f]))*f*Erfi[((-1)^(1/4)*(6*f*x - I*b*Log[f] - (2*I)*c*x*Log[f]))/(2*Sqrt[3*f - I*c*Log[f]])]*Lo
g[f]^2*Sqrt[3*f - I*c*Log[f]]*Sin[3*d] - (-1)^(3/4)*c^3*E^(((I/4)*b^2*Log[f]^2)/(3*f - I*c*Log[f]))*Erfi[((-1)
^(1/4)*(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]]*Si
n[3*d] + (3*(-1)^(3/4)*f^3*Erfi[((-1)^(3/4)*(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)*b^2*Log[f]^2)/(3*f + I*c*Log[f])) + ((-1)^(1/4)*c*f^2*Erfi[((-1)
^(3/4)*(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)*b^2*Log[f]^2)/(3*f + I*c*Log[f])) + (3*(-1)^(3/4)*c^2*f*Erfi[((-1)^(3/4)*(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)*b^2*Log[f]
^2)/(3*f + I*c*Log[f])) + ((-1)^(1/4)*c^3*Erfi[((-1)^(3/4)*(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)*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 = 0.629, size = 358, normalized size = 0.9 \begin{align*}{-{\frac{i}{16}}{f}^{a}\sqrt{\pi }{{\rm e}^{-{\frac{ \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}-12\,id\ln \left ( f \right ) c+36\,df}{12\,if+4\,c\ln \left ( f \right ) }}}}{\it Erf} \left ( -\sqrt{-c\ln \left ( f \right ) -3\,if}x+{\frac{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}+12\,id\ln \left ( f \right ) c+36\,df}{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 ) }{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}+4\,id\ln \left ( f \right ) c+4\,df}{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 ) }{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}-4\,id\ln \left ( f \right ) c+4\,df}{4\,if+4\,c\ln \left ( f \right ) }}}}{\it Erf} \left ( -\sqrt{-c\ln \left ( f \right ) -if}x+{\frac{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+d)^3,x)
[Out]
-1/16*I*Pi^(1/2)*f^a*exp(-1/4*(ln(f)^2*b^2-12*I*d*ln(f)*c+36*d*f)/(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*ln(f)*b/(-c*ln(f)-3*I*f)^(1/2))+1/16*I*Pi^(1/2)*f^a*exp(-1/4*(ln(f)^2*b^2+12*I*d
*ln(f)*c+36*d*f)/(-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*ln(f)*b/(3*I*f-c*ln(
f))^(1/2))-3/16*I*Pi^(1/2)*f^a*exp(-1/4*(ln(f)^2*b^2+4*I*d*ln(f)*c+4*d*f)/(-I*f+c*ln(f)))/(I*f-c*ln(f))^(1/2)*
erf(-x*(I*f-c*ln(f))^(1/2)+1/2*ln(f)*b/(I*f-c*ln(f))^(1/2))+3/16*I*Pi^(1/2)*f^a*exp(-1/4*(ln(f)^2*b^2-4*I*d*ln
(f)*c+4*d*f)/(I*f+c*ln(f)))/(-c*ln(f)-I*f)^(1/2)*erf(-(-c*ln(f)-I*f)^(1/2)*x+1/2*ln(f)*b/(-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+d)^3,x, algorithm="maxima")
[Out]
Exception raised: IndexError
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Fricas [B] time = 0.672334, size = 1854, normalized size = 4.8 \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+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 - 3*I*b*f*log(f) + (2*c^2*x + b*c)*log(f)^2)*sqrt(-c*log(f) - 3*I*f)/(c^2*log(f)^2 + 9*f^2))*e^(1/4*(
36*a*f^2*log(f) - (b^2*c - 4*a*c^2)*log(f)^3 + 108*I*d*f^2 + (12*I*c^2*d + 3*I*b^2*f)*log(f)^2)/(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*(18*f^2*x + 3*I*b*f*log(f) + (2*c^2*x + b*c)*log(f)^2)*sqrt(-c*log(f) + 3*I*f)/(c^2*log(f)^2 + 9*f^2))*e^(
1/4*(36*a*f^2*log(f) - (b^2*c - 4*a*c^2)*log(f)^3 - 108*I*d*f^2 + (-12*I*c^2*d - 3*I*b^2*f)*log(f)^2)/(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 - I*b*f*log(f) + (2*c^2*x + b*c)*log(f)^2)*sqrt(-c*log(f) - I*f)/(c^2*log(f)^2 + f^2))*
e^(1/4*(4*a*f^2*log(f) - (b^2*c - 4*a*c^2)*log(f)^3 + 4*I*d*f^2 + (4*I*c^2*d + I*b^2*f)*log(f)^2)/(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 + I*b*f*log(f) + (2*c^2*x + b*c)*log(f)^2)*sqrt(-c*log(f) + I*f)/(c^2*log(f)^2 + f^2))*e^(1/
4*(4*a*f^2*log(f) - (b^2*c - 4*a*c^2)*log(f)^3 - 4*I*d*f^2 + (-4*I*c^2*d - I*b^2*f)*log(f)^2)/(c^2*log(f)^2 +
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+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} + 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+d)^3,x, algorithm="giac")
[Out]
integrate(f^(c*x^2 + b*x + a)*sin(f*x^2 + d)^3, x)