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

Optimal. Leaf size=377 \[ -\frac{i \sqrt{\pi } f^a \exp \left (-\frac{9 e^2}{4 (-c \log (f)+3 i f)}-3 i d\right ) \text{Erf}\left (\frac{2 x (-c \log (f)+3 i f)+3 i e}{2 \sqrt{-c \log (f)+3 i f}}\right )}{16 \sqrt{-c \log (f)+3 i f}}+\frac{3 i \sqrt{\pi } f^a e^{-\frac{e^2}{-4 c \log (f)+4 i f}-i d} \text{Erf}\left (\frac{2 x (-c \log (f)+i f)+i e}{2 \sqrt{-c \log (f)+i f}}\right )}{16 \sqrt{-c \log (f)+i f}}-\frac{3 i \sqrt{\pi } f^a e^{\frac{e^2}{4 c \log (f)+4 i f}+i d} \text{Erfi}\left (\frac{2 x (c \log (f)+i f)+i e}{2 \sqrt{c \log (f)+i f}}\right )}{16 \sqrt{c \log (f)+i f}}+\frac{i \sqrt{\pi } f^a e^{\frac{9 e^2}{4 (c \log (f)+3 i f)}+3 i d} \text{Erfi}\left (\frac{2 x (c \log (f)+3 i f)+3 i e}{2 \sqrt{c \log (f)+3 i f}}\right )}{16 \sqrt{c \log (f)+3 i f}} \]

[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]]

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Rubi [A]  time = 0.655414, antiderivative size = 377, 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} \[ -\frac{i \sqrt{\pi } f^a \exp \left (-\frac{9 e^2}{4 (-c \log (f)+3 i f)}-3 i d\right ) \text{Erf}\left (\frac{2 x (-c \log (f)+3 i f)+3 i e}{2 \sqrt{-c \log (f)+3 i f}}\right )}{16 \sqrt{-c \log (f)+3 i f}}+\frac{3 i \sqrt{\pi } f^a e^{-\frac{e^2}{-4 c \log (f)+4 i f}-i d} \text{Erf}\left (\frac{2 x (-c \log (f)+i f)+i e}{2 \sqrt{-c \log (f)+i f}}\right )}{16 \sqrt{-c \log (f)+i f}}-\frac{3 i \sqrt{\pi } f^a e^{\frac{e^2}{4 c \log (f)+4 i f}+i d} \text{Erfi}\left (\frac{2 x (c \log (f)+i f)+i e}{2 \sqrt{c \log (f)+i f}}\right )}{16 \sqrt{c \log (f)+i f}}+\frac{i \sqrt{\pi } f^a e^{\frac{9 e^2}{4 (c \log (f)+3 i f)}+3 i d} \text{Erfi}\left (\frac{2 x (c \log (f)+3 i f)+3 i e}{2 \sqrt{c \log (f)+3 i f}}\right )}{16 \sqrt{c \log (f)+3 i f}} \]

Antiderivative was successfully verified.

[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 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+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+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]  time = 6.6193, size = 490, normalized size = 1.3 \[ \frac{\sqrt [4]{-1} \sqrt{\pi } f^a \left ((f-i c \log (f)) \left (\sqrt{3 f-i c \log (f)} \left (-c^2 \log ^2(f)+4 i c f \log (f)+3 f^2\right ) (\cos (3 d)+i \sin (3 d)) e^{\frac{9 e^2}{4 (c \log (f)+3 i f)}} \text{Erfi}\left (\frac{\sqrt [4]{-1} (-2 i c x \log (f)+3 e+6 f x)}{2 \sqrt{3 f-i c \log (f)}}\right )+(3 f-i c \log (f)) \left (3 \sqrt{f+i c \log (f)} (c \log (f)-3 i f) (\cos (d)-i \sin (d)) e^{\frac{e^2}{4 c \log (f)-4 i f}} \text{Erfi}\left (\frac{(-1)^{3/4} (2 i c x \log (f)+e+2 f x)}{2 \sqrt{f+i c \log (f)}}\right )+(f+i c \log (f)) \sqrt{3 f+i c \log (f)} (\sin (3 d)+i \cos (3 d)) e^{\frac{9 e^2}{4 (c \log (f)-3 i f)}} \text{Erfi}\left (\frac{(-1)^{3/4} (2 i c x \log (f)+3 e+6 f x)}{2 \sqrt{3 f+i c \log (f)}}\right )\right )\right )-3 \sqrt{f-i c \log (f)} \left (c^2 f \log ^2(f)+i c^3 \log ^3(f)+9 i c f^2 \log (f)+9 f^3\right ) (\cos (d)+i \sin (d)) e^{\frac{e^2}{4 c \log (f)+4 i f}} \text{Erfi}\left (\frac{\sqrt [4]{-1} (-2 i c x \log (f)+e+2 f x)}{2 \sqrt{f-i c \log (f)}}\right )\right )}{16 \left (10 c^2 f^2 \log ^2(f)+c^4 \log ^4(f)+9 f^4\right )} \]

Warning: Unable to verify antiderivative.

[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))

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Maple [A]  time = 0.613, size = 338, normalized size = 0.9 \begin{align*}{-{\frac{i}{16}}{f}^{a}\sqrt{\pi }{{\rm e}^{{\frac{12\,id\ln \left ( f \right ) c-36\,df+9\,{e}^{2}}{4\,c\ln \left ( f \right ) +12\,if}}}}{\it Erf} \left ( -\sqrt{-c\ln \left ( f \right ) -3\,if}x+{{\frac{3\,i}{2}}e{\frac{1}{\sqrt{-c\ln \left ( f \right ) -3\,if}}}} \right ){\frac{1}{\sqrt{-c\ln \left ( f \right ) -3\,if}}}}-{{\frac{i}{16}}\sqrt{\pi }{f}^{a}{{\rm e}^{-{\frac{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{3\,i}{2}}e{\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{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{i}{2}}e{\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{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{i}{2}}e{\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+a)*sin(f*x^2+e*x+d)^3,x)

[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)/(-
3*I*f+c*ln(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)/(-I*f+c*ln(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))

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

[Out]

Exception raised: IndexError

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Fricas [B]  time = 0.668306, size = 1840, normalized size = 4.88 \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+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)) + 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*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)) + 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*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)

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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+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} + 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+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)