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

Optimal. Leaf size=245 \[ \frac{\sqrt{\pi } f^a e^{\frac{b^2 \log ^2(f)}{-4 c \log (f)+8 i f}-2 i d} \text{Erf}\left (\frac{b \log (f)-2 x (-c \log (f)+2 i f)}{2 \sqrt{-c \log (f)+2 i f}}\right )}{8 \sqrt{-c \log (f)+2 i f}}-\frac{\sqrt{\pi } f^a e^{2 i d-\frac{b^2 \log ^2(f)}{4 c \log (f)+8 i f}} \text{Erfi}\left (\frac{b \log (f)+2 x (c \log (f)+2 i f)}{2 \sqrt{c \log (f)+2 i f}}\right )}{8 \sqrt{c \log (f)+2 i f}}+\frac{\sqrt{\pi } f^{a-\frac{b^2}{4 c}} \text{Erfi}\left (\frac{\sqrt{\log (f)} (b+2 c x)}{2 \sqrt{c}}\right )}{4 \sqrt{c} \sqrt{\log (f)}} \]

[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 + (b^2*Log[f]^2)/((8*I)*f - 4*c*Log[f]))*f^a*Sqrt[Pi]*Erf[(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 - (b^2*Log[f]^2)/((8*I)*f + 4*c*Log[f]))*f^a*S
qrt[Pi]*Erfi[(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]])

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Rubi [A]  time = 0.462039, antiderivative size = 245, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {4472, 2234, 2204, 2287, 2205} \[ \frac{\sqrt{\pi } f^a e^{\frac{b^2 \log ^2(f)}{-4 c \log (f)+8 i f}-2 i d} \text{Erf}\left (\frac{b \log (f)-2 x (-c \log (f)+2 i f)}{2 \sqrt{-c \log (f)+2 i f}}\right )}{8 \sqrt{-c \log (f)+2 i f}}-\frac{\sqrt{\pi } f^a e^{2 i d-\frac{b^2 \log ^2(f)}{4 c \log (f)+8 i f}} \text{Erfi}\left (\frac{b \log (f)+2 x (c \log (f)+2 i f)}{2 \sqrt{c \log (f)+2 i f}}\right )}{8 \sqrt{c \log (f)+2 i f}}+\frac{\sqrt{\pi } f^{a-\frac{b^2}{4 c}} \text{Erfi}\left (\frac{\sqrt{\log (f)} (b+2 c x)}{2 \sqrt{c}}\right )}{4 \sqrt{c} \sqrt{\log (f)}} \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b*x + c*x^2)*Sin[d + 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 + (b^2*Log[f]^2)/((8*I)*f - 4*c*Log[f]))*f^a*Sqrt[Pi]*Erf[(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 - (b^2*Log[f]^2)/((8*I)*f + 4*c*Log[f]))*f^a*S
qrt[Pi]*Erfi[(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 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 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 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]

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 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]

Rubi steps

\begin{align*} \int f^{a+b x+c x^2} \sin ^2\left (d+f x^2\right ) \, dx &=\int \left (\frac{1}{2} f^{a+b x+c x^2}-\frac{1}{4} e^{-2 i d-2 i f x^2} f^{a+b x+c x^2}-\frac{1}{4} e^{2 i d+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 f x^2} f^{a+b x+c x^2} \, dx\right )-\frac{1}{4} \int e^{2 i d+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)+b x \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)+b x \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 (e^{-2 i d+\frac{b^2 \log ^2(f)}{8 i f-4 c \log (f)}} f^a\right ) \int \exp \left (\frac{(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 (e^{2 i d-\frac{b^2 \log ^2(f)}{8 i f+4 c \log (f)}} f^a\right ) \int \exp \left (\frac{(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{e^{-2 i d+\frac{b^2 \log ^2(f)}{8 i f-4 c \log (f)}} f^a \sqrt{\pi } \text{erf}\left (\frac{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{b^2 \log ^2(f)}{8 i f+4 c \log (f)}} f^a \sqrt{\pi } \text{erfi}\left (\frac{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 [A]  time = 3.08969, size = 299, normalized size = 1.22 \[ \frac{1}{8} \sqrt{\pi } f^a \left (\frac{2 f^{-\frac{b^2}{4 c}} \text{Erfi}\left (\frac{\sqrt{\log (f)} (b+2 c x)}{2 \sqrt{c}}\right )}{\sqrt{c} \sqrt{\log (f)}}+\frac{\sqrt [4]{-1} e^{\frac{b^2 \log ^2(f)}{-4 c \log (f)+8 i f}} \left (\sqrt{2 f-i c \log (f)} (2 f+i c \log (f)) (\cos (2 d)+i \sin (2 d)) e^{\frac{i b^2 f \log ^2(f)}{c^2 \log ^2(f)+4 f^2}} \text{Erf}\left (\frac{(-1)^{3/4} (4 f x-i \log (f) (b+2 c x))}{2 \sqrt{2 f-i c \log (f)}}\right )+\sqrt{2 f+i c \log (f)} (c \log (f)+2 i f) (\cos (2 d)-i \sin (2 d)) \text{Erf}\left (\frac{\sqrt [4]{-1} (4 f x+i \log (f) (b+2 c x))}{2 \sqrt{2 f+i c \log (f)}}\right )\right )}{c^2 \log ^2(f)+4 f^2}\right ) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [A]  time = 0.424, size = 227, normalized size = 0.9 \begin{align*}{\frac{{f}^{a}\sqrt{\pi }}{8}{{\rm e}^{-{\frac{ \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}+8\,id\ln \left ( f \right ) c+16\,df}{4\,c\ln \left ( f \right ) -8\,if}}}}{\it Erf} \left ( -x\sqrt{2\,if-c\ln \left ( f \right ) }+{\frac{b\ln \left ( f \right ) }{2}{\frac{1}{\sqrt{2\,if-c\ln \left ( f \right ) }}}} \right ){\frac{1}{\sqrt{2\,if-c\ln \left ( f \right ) }}}}+{\frac{{f}^{a}\sqrt{\pi }}{8}{{\rm e}^{-{\frac{ \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}-8\,id\ln \left ( f \right ) c+16\,df}{8\,if+4\,c\ln \left ( f \right ) }}}}{\it Erf} \left ( -\sqrt{-c\ln \left ( f \right ) -2\,if}x+{\frac{b\ln \left ( f \right ) }{2}{\frac{1}{\sqrt{-c\ln \left ( f \right ) -2\,if}}}} \right ){\frac{1}{\sqrt{-c\ln \left ( f \right ) -2\,if}}}}-{\frac{{f}^{a}\sqrt{\pi }}{4}{f}^{-{\frac{{b}^{2}}{4\,c}}}{\it Erf} \left ( -\sqrt{-c\ln \left ( f \right ) }x+{\frac{b\ln \left ( f \right ) }{2}{\frac{1}{\sqrt{-c\ln \left ( f \right ) }}}} \right ){\frac{1}{\sqrt{-c\ln \left ( f \right ) }}}} \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)^2,x)

[Out]

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

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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)^2,x, algorithm="maxima")

[Out]

Exception raised: IndexError

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Fricas [B]  time = 0.559602, size = 1025, normalized size = 4.18 \begin{align*} \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 - 2 i \, b f \log \left (f\right ) +{\left (2 \, c^{2} x + b c\right )} \log \left (f\right )^{2}\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{16 \, a f^{2} \log \left (f\right ) -{\left (b^{2} c - 4 \, a c^{2}\right )} \log \left (f\right )^{3} + 32 i \, d f^{2} +{\left (8 i \, c^{2} d + 2 i \, b^{2} f\right )} \log \left (f\right )^{2}}{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 + 2 i \, b f \log \left (f\right ) +{\left (2 \, c^{2} x + b c\right )} \log \left (f\right )^{2}\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{16 \, a f^{2} \log \left (f\right ) -{\left (b^{2} c - 4 \, a c^{2}\right )} \log \left (f\right )^{3} - 32 i \, d f^{2} +{\left (-8 i \, c^{2} d - 2 i \, b^{2} f\right )} \log \left (f\right )^{2}}{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 )}} \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)^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*I*b*f*log(f) + (2*c
^2*x + b*c)*log(f)^2)*sqrt(-c*log(f) - 2*I*f)/(c^2*log(f)^2 + 4*f^2))*e^(1/4*(16*a*f^2*log(f) - (b^2*c - 4*a*c
^2)*log(f)^3 + 32*I*d*f^2 + (8*I*c^2*d + 2*I*b^2*f)*log(f)^2)/(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)*erf(1/2*(8*f^2*x + 2*I*b*f*log(f) + (2*c^2*x + b*c)*log(f)^2)*sqrt(
-c*log(f) + 2*I*f)/(c^2*log(f)^2 + 4*f^2))*e^(1/4*(16*a*f^2*log(f) - (b^2*c - 4*a*c^2)*log(f)^3 - 32*I*d*f^2 +
(-8*I*c^2*d - 2*I*b^2*f)*log(f)^2)/(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*log(f))

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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)**2,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 )^{2}\,{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)^2,x, algorithm="giac")

[Out]

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