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

Optimal. Leaf size=140 \[ -\frac{\sqrt{\pi } e^{-2 i d} f^a \text{Erf}\left (x \sqrt{-c \log (f)+2 i f}\right )}{8 \sqrt{-c \log (f)+2 i f}}-\frac{\sqrt{\pi } e^{2 i d} f^a \text{Erfi}\left (x \sqrt{c \log (f)+2 i f}\right )}{8 \sqrt{c \log (f)+2 i f}}+\frac{\sqrt{\pi } f^a \text{Erfi}\left (\sqrt{c} x \sqrt{\log (f)}\right )}{4 \sqrt{c} \sqrt{\log (f)}} \]

[Out]

(f^a*Sqrt[Pi]*Erfi[Sqrt[c]*x*Sqrt[Log[f]]])/(4*Sqrt[c]*Sqrt[Log[f]]) - (f^a*Sqrt[Pi]*Erf[x*Sqrt[(2*I)*f - c*Lo
g[f]]])/(8*E^((2*I)*d)*Sqrt[(2*I)*f - c*Log[f]]) - (E^((2*I)*d)*f^a*Sqrt[Pi]*Erfi[x*Sqrt[(2*I)*f + c*Log[f]]])
/(8*Sqrt[(2*I)*f + c*Log[f]])

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Rubi [A]  time = 0.227606, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {4472, 2204, 2287, 2205} \[ -\frac{\sqrt{\pi } e^{-2 i d} f^a \text{Erf}\left (x \sqrt{-c \log (f)+2 i f}\right )}{8 \sqrt{-c \log (f)+2 i f}}-\frac{\sqrt{\pi } e^{2 i d} f^a \text{Erfi}\left (x \sqrt{c \log (f)+2 i f}\right )}{8 \sqrt{c \log (f)+2 i f}}+\frac{\sqrt{\pi } f^a \text{Erfi}\left (\sqrt{c} x \sqrt{\log (f)}\right )}{4 \sqrt{c} \sqrt{\log (f)}} \]

Antiderivative was successfully verified.

[In]

Int[f^(a + c*x^2)*Sin[d + f*x^2]^2,x]

[Out]

(f^a*Sqrt[Pi]*Erfi[Sqrt[c]*x*Sqrt[Log[f]]])/(4*Sqrt[c]*Sqrt[Log[f]]) - (f^a*Sqrt[Pi]*Erf[x*Sqrt[(2*I)*f - c*Lo
g[f]]])/(8*E^((2*I)*d)*Sqrt[(2*I)*f - c*Log[f]]) - (E^((2*I)*d)*f^a*Sqrt[Pi]*Erfi[x*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 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+c x^2} \sin ^2\left (d+f x^2\right ) \, dx &=\int \left (\frac{1}{2} f^{a+c x^2}-\frac{1}{4} e^{-2 i d-2 i f x^2} f^{a+c x^2}-\frac{1}{4} e^{2 i d+2 i f x^2} f^{a+c x^2}\right ) \, dx\\ &=-\left (\frac{1}{4} \int e^{-2 i d-2 i f x^2} f^{a+c x^2} \, dx\right )-\frac{1}{4} \int e^{2 i d+2 i f x^2} f^{a+c x^2} \, dx+\frac{1}{2} \int f^{a+c x^2} \, dx\\ &=\frac{f^a \sqrt{\pi } \text{erfi}\left (\sqrt{c} x \sqrt{\log (f)}\right )}{4 \sqrt{c} \sqrt{\log (f)}}-\frac{1}{4} \int \exp \left (-2 i d+a \log (f)-x^2 (2 i f-c \log (f))\right ) \, dx-\frac{1}{4} \int \exp \left (2 i d+a \log (f)+x^2 (2 i f+c \log (f))\right ) \, dx\\ &=\frac{f^a \sqrt{\pi } \text{erfi}\left (\sqrt{c} x \sqrt{\log (f)}\right )}{4 \sqrt{c} \sqrt{\log (f)}}-\frac{e^{-2 i d} f^a \sqrt{\pi } \text{erf}\left (x \sqrt{2 i f-c \log (f)}\right )}{8 \sqrt{2 i f-c \log (f)}}-\frac{e^{2 i d} f^a \sqrt{\pi } \text{erfi}\left (x \sqrt{2 i f+c \log (f)}\right )}{8 \sqrt{2 i f+c \log (f)}}\\ \end{align*}

Mathematica [A]  time = 0.7892, size = 188, normalized size = 1.34 \[ \frac{1}{8} \sqrt{\pi } f^a \left (\frac{2 \text{Erfi}\left (\sqrt{c} x \sqrt{\log (f)}\right )}{\sqrt{c} \sqrt{\log (f)}}+\frac{\sqrt [4]{-1} \left (\sqrt{2 f+i c \log (f)} (c \log (f)+2 i f) (\cos (2 d)-i \sin (2 d)) \text{Erf}\left (\sqrt [4]{-1} x \sqrt{2 f+i c \log (f)}\right )+\sqrt{2 f-i c \log (f)} (2 f+i c \log (f)) (\cos (2 d)+i \sin (2 d)) \text{Erf}\left ((-1)^{3/4} x \sqrt{2 f-i c \log (f)}\right )\right )}{c^2 \log ^2(f)+4 f^2}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(f^a*Sqrt[Pi]*((2*Erfi[Sqrt[c]*x*Sqrt[Log[f]]])/(Sqrt[c]*Sqrt[Log[f]]) + ((-1)^(1/4)*(Erf[(-1)^(1/4)*x*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]) + Erf[(-1)^(3/4)*x*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 = 0.15, size = 107, normalized size = 0.8 \begin{align*} -{\frac{{f}^{a}\sqrt{\pi }{{\rm e}^{-2\,id}}}{8}{\it Erf} \left ( x\sqrt{2\,if-c\ln \left ( f \right ) } \right ){\frac{1}{\sqrt{2\,if-c\ln \left ( f \right ) }}}}-{\frac{{f}^{a}\sqrt{\pi }{{\rm e}^{2\,id}}}{8}{\it Erf} \left ( \sqrt{-c\ln \left ( f \right ) -2\,if}x \right ){\frac{1}{\sqrt{-c\ln \left ( f \right ) -2\,if}}}}+{\frac{{f}^{a}\sqrt{\pi }}{4}{\it Erf} \left ( \sqrt{-c\ln \left ( f \right ) }x \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+a)*sin(f*x^2+d)^2,x)

[Out]

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

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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+d)^2,x, algorithm="maxima")

[Out]

Exception raised: IndexError

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Fricas [A]  time = 0.505152, size = 479, normalized size = 3.42 \begin{align*} -\frac{2 \, \sqrt{\pi }{\left (c^{2} \log \left (f\right )^{2} + 4 \, f^{2}\right )} \sqrt{-c \log \left (f\right )} f^{a} \operatorname{erf}\left (\sqrt{-c \log \left (f\right )} x\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 (\sqrt{-c \log \left (f\right ) - 2 i \, f} x\right ) e^{\left (a \log \left (f\right ) + 2 i \, d\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 (\sqrt{-c \log \left (f\right ) + 2 i \, f} x\right ) e^{\left (a \log \left (f\right ) - 2 i \, d\right )}}{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+a)*sin(f*x^2+d)^2,x, algorithm="fricas")

[Out]

-1/8*(2*sqrt(pi)*(c^2*log(f)^2 + 4*f^2)*sqrt(-c*log(f))*f^a*erf(sqrt(-c*log(f))*x) - sqrt(pi)*(c^2*log(f)^2 -
2*I*c*f*log(f))*sqrt(-c*log(f) - 2*I*f)*erf(sqrt(-c*log(f) - 2*I*f)*x)*e^(a*log(f) + 2*I*d) - sqrt(pi)*(c^2*lo
g(f)^2 + 2*I*c*f*log(f))*sqrt(-c*log(f) + 2*I*f)*erf(sqrt(-c*log(f) + 2*I*f)*x)*e^(a*log(f) - 2*I*d))/(c^3*log
(f)^3 + 4*c*f^2*log(f))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int f^{a + c x^{2}} \sin ^{2}{\left (d + f x^{2} \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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} + d\right )^{2}\,{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+d)^2,x, algorithm="giac")

[Out]

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