3.345 \(\int f^{a+b x} \sinh (d+e x+f x^2) \, dx\)

Optimal. Leaf size=115 \[ \frac{1}{4} \sqrt{\pi } f^{a-\frac{1}{2}} e^{d-\frac{(b \log (f)+e)^2}{4 f}} \text{Erfi}\left (\frac{b \log (f)+e+2 f x}{2 \sqrt{f}}\right )-\frac{1}{4} \sqrt{\pi } f^{a-\frac{1}{2}} e^{\frac{(e-b \log (f))^2}{4 f}-d} \text{Erf}\left (\frac{-b \log (f)+e+2 f x}{2 \sqrt{f}}\right ) \]

[Out]

-(E^(-d + (e - b*Log[f])^2/(4*f))*f^(-1/2 + a)*Sqrt[Pi]*Erf[(e + 2*f*x - b*Log[f])/(2*Sqrt[f])])/4 + (E^(d - (
e + b*Log[f])^2/(4*f))*f^(-1/2 + a)*Sqrt[Pi]*Erfi[(e + 2*f*x + b*Log[f])/(2*Sqrt[f])])/4

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Rubi [A]  time = 0.224596, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {5512, 2287, 2234, 2205, 2204} \[ \frac{1}{4} \sqrt{\pi } f^{a-\frac{1}{2}} e^{d-\frac{(b \log (f)+e)^2}{4 f}} \text{Erfi}\left (\frac{b \log (f)+e+2 f x}{2 \sqrt{f}}\right )-\frac{1}{4} \sqrt{\pi } f^{a-\frac{1}{2}} e^{\frac{(e-b \log (f))^2}{4 f}-d} \text{Erf}\left (\frac{-b \log (f)+e+2 f x}{2 \sqrt{f}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b*x)*Sinh[d + e*x + f*x^2],x]

[Out]

-(E^(-d + (e - b*Log[f])^2/(4*f))*f^(-1/2 + a)*Sqrt[Pi]*Erf[(e + 2*f*x - b*Log[f])/(2*Sqrt[f])])/4 + (E^(d - (
e + b*Log[f])^2/(4*f))*f^(-1/2 + a)*Sqrt[Pi]*Erfi[(e + 2*f*x + b*Log[f])/(2*Sqrt[f])])/4

Rule 5512

Int[(F_)^(u_)*Sinh[v_]^(n_.), x_Symbol] :> Int[ExpandTrigToExp[F^u, Sinh[v]^n, x], x] /; FreeQ[F, x] && (Linea
rQ[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} \sinh \left (d+e x+f x^2\right ) \, dx &=\int \left (-\frac{1}{2} e^{-d-e x-f x^2} f^{a+b x}+\frac{1}{2} e^{d+e x+f x^2} f^{a+b x}\right ) \, dx\\ &=-\left (\frac{1}{2} \int e^{-d-e x-f x^2} f^{a+b x} \, dx\right )+\frac{1}{2} \int e^{d+e x+f x^2} f^{a+b x} \, dx\\ &=-\left (\frac{1}{2} \int e^{-d-f x^2+a \log (f)-x (e-b \log (f))} \, dx\right )+\frac{1}{2} \int e^{d+f x^2+a \log (f)+x (e+b \log (f))} \, dx\\ &=-\left (\frac{1}{2} \left (e^{-d+\frac{(e-b \log (f))^2}{4 f}} f^a\right ) \int e^{-\frac{(-e-2 f x+b \log (f))^2}{4 f}} \, dx\right )+\frac{1}{2} \left (e^{d-\frac{(e+b \log (f))^2}{4 f}} f^a\right ) \int e^{\frac{(e+2 f x+b \log (f))^2}{4 f}} \, dx\\ &=-\frac{1}{4} e^{-d+\frac{(e-b \log (f))^2}{4 f}} f^{-\frac{1}{2}+a} \sqrt{\pi } \text{erf}\left (\frac{e+2 f x-b \log (f)}{2 \sqrt{f}}\right )+\frac{1}{4} e^{d-\frac{(e+b \log (f))^2}{4 f}} f^{-\frac{1}{2}+a} \sqrt{\pi } \text{erfi}\left (\frac{e+2 f x+b \log (f)}{2 \sqrt{f}}\right )\\ \end{align*}

Mathematica [A]  time = 0.310203, size = 124, normalized size = 1.08 \[ \frac{1}{4} \sqrt{\pi } f^{a-\frac{b e+f}{2 f}} e^{-\frac{b^2 \log ^2(f)+e^2}{4 f}} \left ((\sinh (d)+\cosh (d)) \text{Erfi}\left (\frac{b \log (f)+e+2 f x}{2 \sqrt{f}}\right )-(\cosh (d)-\sinh (d)) e^{\frac{b^2 \log ^2(f)+e^2}{2 f}} \text{Erf}\left (\frac{-b \log (f)+e+2 f x}{2 \sqrt{f}}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[f^(a + b*x)*Sinh[d + e*x + f*x^2],x]

[Out]

(f^(a - (b*e + f)/(2*f))*Sqrt[Pi]*(-(E^((e^2 + b^2*Log[f]^2)/(2*f))*Erf[(e + 2*f*x - b*Log[f])/(2*Sqrt[f])]*(C
osh[d] - Sinh[d])) + Erfi[(e + 2*f*x + b*Log[f])/(2*Sqrt[f])]*(Cosh[d] + Sinh[d])))/(4*E^((e^2 + b^2*Log[f]^2)
/(4*f)))

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Maple [A]  time = 0.109, size = 126, normalized size = 1.1 \begin{align*} -{\frac{\sqrt{\pi }{f}^{a}}{4}{{\rm e}^{-{\frac{ \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}+2\,\ln \left ( f \right ) be-4\,df+{e}^{2}}{4\,f}}}}{\it Erf} \left ( -\sqrt{-f}x+{\frac{e+b\ln \left ( f \right ) }{2}{\frac{1}{\sqrt{-f}}}} \right ){\frac{1}{\sqrt{-f}}}}+{\frac{\sqrt{\pi }{f}^{a}}{4}{{\rm e}^{{\frac{ \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}-2\,\ln \left ( f \right ) be-4\,df+{e}^{2}}{4\,f}}}}{\it Erf} \left ( -\sqrt{f}x+{\frac{b\ln \left ( f \right ) -e}{2}{\frac{1}{\sqrt{f}}}} \right ){\frac{1}{\sqrt{f}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(b*x+a)*sinh(f*x^2+e*x+d),x)

[Out]

-1/4*Pi^(1/2)*f^a*exp(-1/4*(ln(f)^2*b^2+2*ln(f)*b*e-4*d*f+e^2)/f)/(-f)^(1/2)*erf(-(-f)^(1/2)*x+1/2*(e+b*ln(f))
/(-f)^(1/2))+1/4*Pi^(1/2)*f^a*exp(1/4*(ln(f)^2*b^2-2*ln(f)*b*e-4*d*f+e^2)/f)/f^(1/2)*erf(-f^(1/2)*x+1/2*(b*ln(
f)-e)/f^(1/2))

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Maxima [A]  time = 1.07543, size = 138, normalized size = 1.2 \begin{align*} -\frac{1}{4} \, \sqrt{\pi } f^{a - \frac{1}{2}} \operatorname{erf}\left (\sqrt{f} x - \frac{b \log \left (f\right ) - e}{2 \, \sqrt{f}}\right ) e^{\left (-d + \frac{{\left (b \log \left (f\right ) - e\right )}^{2}}{4 \, f}\right )} + \frac{\sqrt{\pi } f^{a} \operatorname{erf}\left (\sqrt{-f} x - \frac{b \log \left (f\right ) + e}{2 \, \sqrt{-f}}\right ) e^{\left (d - \frac{{\left (b \log \left (f\right ) + e\right )}^{2}}{4 \, f}\right )}}{4 \, \sqrt{-f}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(b*x+a)*sinh(f*x^2+e*x+d),x, algorithm="maxima")

[Out]

-1/4*sqrt(pi)*f^(a - 1/2)*erf(sqrt(f)*x - 1/2*(b*log(f) - e)/sqrt(f))*e^(-d + 1/4*(b*log(f) - e)^2/f) + 1/4*sq
rt(pi)*f^a*erf(sqrt(-f)*x - 1/2*(b*log(f) + e)/sqrt(-f))*e^(d - 1/4*(b*log(f) + e)^2/f)/sqrt(-f)

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Fricas [B]  time = 1.84538, size = 699, normalized size = 6.08 \begin{align*} -\frac{\sqrt{\pi } \sqrt{-f} \cosh \left (\frac{b^{2} \log \left (f\right )^{2} + e^{2} - 4 \, d f + 2 \,{\left (b e - 2 \, a f\right )} \log \left (f\right )}{4 \, f}\right ) \operatorname{erf}\left (\frac{{\left (2 \, f x + b \log \left (f\right ) + e\right )} \sqrt{-f}}{2 \, f}\right ) - \sqrt{\pi } \sqrt{f} \cosh \left (\frac{b^{2} \log \left (f\right )^{2} + e^{2} - 4 \, d f - 2 \,{\left (b e - 2 \, a f\right )} \log \left (f\right )}{4 \, f}\right ) \operatorname{erf}\left (-\frac{2 \, f x - b \log \left (f\right ) + e}{2 \, \sqrt{f}}\right ) - \sqrt{\pi } \sqrt{-f} \operatorname{erf}\left (\frac{{\left (2 \, f x + b \log \left (f\right ) + e\right )} \sqrt{-f}}{2 \, f}\right ) \sinh \left (\frac{b^{2} \log \left (f\right )^{2} + e^{2} - 4 \, d f + 2 \,{\left (b e - 2 \, a f\right )} \log \left (f\right )}{4 \, f}\right ) - \sqrt{\pi } \sqrt{f} \operatorname{erf}\left (-\frac{2 \, f x - b \log \left (f\right ) + e}{2 \, \sqrt{f}}\right ) \sinh \left (\frac{b^{2} \log \left (f\right )^{2} + e^{2} - 4 \, d f - 2 \,{\left (b e - 2 \, a f\right )} \log \left (f\right )}{4 \, f}\right )}{4 \, f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(b*x+a)*sinh(f*x^2+e*x+d),x, algorithm="fricas")

[Out]

-1/4*(sqrt(pi)*sqrt(-f)*cosh(1/4*(b^2*log(f)^2 + e^2 - 4*d*f + 2*(b*e - 2*a*f)*log(f))/f)*erf(1/2*(2*f*x + b*l
og(f) + e)*sqrt(-f)/f) - sqrt(pi)*sqrt(f)*cosh(1/4*(b^2*log(f)^2 + e^2 - 4*d*f - 2*(b*e - 2*a*f)*log(f))/f)*er
f(-1/2*(2*f*x - b*log(f) + e)/sqrt(f)) - sqrt(pi)*sqrt(-f)*erf(1/2*(2*f*x + b*log(f) + e)*sqrt(-f)/f)*sinh(1/4
*(b^2*log(f)^2 + e^2 - 4*d*f + 2*(b*e - 2*a*f)*log(f))/f) - sqrt(pi)*sqrt(f)*erf(-1/2*(2*f*x - b*log(f) + e)/s
qrt(f))*sinh(1/4*(b^2*log(f)^2 + e^2 - 4*d*f - 2*(b*e - 2*a*f)*log(f))/f))/f

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f**(b*x+a)*sinh(f*x**2+e*x+d),x)

[Out]

Integral(f**(a + b*x)*sinh(d + e*x + f*x**2), x)

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Giac [A]  time = 1.28214, size = 181, normalized size = 1.57 \begin{align*} \frac{\sqrt{\pi } \operatorname{erf}\left (-\frac{1}{2} \, \sqrt{f}{\left (2 \, x - \frac{b \log \left (f\right ) - e}{f}\right )}\right ) e^{\left (\frac{b^{2} \log \left (f\right )^{2} + 4 \, a f \log \left (f\right ) - 2 \, b e \log \left (f\right ) - 4 \, d f + e^{2}}{4 \, f}\right )}}{4 \, \sqrt{f}} - \frac{\sqrt{\pi } \operatorname{erf}\left (-\frac{1}{2} \, \sqrt{-f}{\left (2 \, x + \frac{b \log \left (f\right ) + e}{f}\right )}\right ) e^{\left (-\frac{b^{2} \log \left (f\right )^{2} - 4 \, a f \log \left (f\right ) + 2 \, b e \log \left (f\right ) - 4 \, d f + e^{2}}{4 \, f}\right )}}{4 \, \sqrt{-f}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(b*x+a)*sinh(f*x^2+e*x+d),x, algorithm="giac")

[Out]

1/4*sqrt(pi)*erf(-1/2*sqrt(f)*(2*x - (b*log(f) - e)/f))*e^(1/4*(b^2*log(f)^2 + 4*a*f*log(f) - 2*b*e*log(f) - 4
*d*f + e^2)/f)/sqrt(f) - 1/4*sqrt(pi)*erf(-1/2*sqrt(-f)*(2*x + (b*log(f) + e)/f))*e^(-1/4*(b^2*log(f)^2 - 4*a*
f*log(f) + 2*b*e*log(f) - 4*d*f + e^2)/f)/sqrt(-f)