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

Optimal. Leaf size=183 \[ \frac{\sqrt{\pi } f^a e^{\frac{e^2}{2 f-c \log (f)}-2 d} \text{Erf}\left (\frac{x (2 f-c \log (f))+e}{\sqrt{2 f-c \log (f)}}\right )}{8 \sqrt{2 f-c \log (f)}}+\frac{\sqrt{\pi } f^a e^{2 d-\frac{e^2}{c \log (f)+2 f}} \text{Erfi}\left (\frac{x (c \log (f)+2 f)+e}{\sqrt{c \log (f)+2 f}}\right )}{8 \sqrt{c \log (f)+2 f}}-\frac{\sqrt{\pi } f^a \text{Erfi}\left (\sqrt{c} x \sqrt{\log (f)}\right )}{4 \sqrt{c} \sqrt{\log (f)}} \]

[Out]

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

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Rubi [A]  time = 0.333797, antiderivative size = 183, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {5512, 2204, 2287, 2234, 2205} \[ \frac{\sqrt{\pi } f^a e^{\frac{e^2}{2 f-c \log (f)}-2 d} \text{Erf}\left (\frac{x (2 f-c \log (f))+e}{\sqrt{2 f-c \log (f)}}\right )}{8 \sqrt{2 f-c \log (f)}}+\frac{\sqrt{\pi } f^a e^{2 d-\frac{e^2}{c \log (f)+2 f}} \text{Erfi}\left (\frac{x (c \log (f)+2 f)+e}{\sqrt{c \log (f)+2 f}}\right )}{8 \sqrt{c \log (f)+2 f}}-\frac{\sqrt{\pi } f^a \text{Erfi}\left (\sqrt{c} x \sqrt{\log (f)}\right )}{4 \sqrt{c} \sqrt{\log (f)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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 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]

Rubi steps

\begin{align*} \int f^{a+c x^2} \sinh ^2\left (d+e x+f x^2\right ) \, dx &=\int \left (-\frac{1}{2} f^{a+c x^2}+\frac{1}{4} e^{-2 d-2 e x-2 f x^2} f^{a+c x^2}+\frac{1}{4} e^{2 d+2 e x+2 f x^2} f^{a+c x^2}\right ) \, dx\\ &=\frac{1}{4} \int e^{-2 d-2 e x-2 f x^2} f^{a+c x^2} \, dx+\frac{1}{4} \int e^{2 d+2 e x+2 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 d-2 e x+a \log (f)-x^2 (2 f-c \log (f))\right ) \, dx+\frac{1}{4} \int \exp \left (2 d+2 e x+a \log (f)+x^2 (2 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{1}{4} \left (e^{-2 d+\frac{e^2}{2 f-c \log (f)}} f^a\right ) \int \exp \left (\frac{(-2 e+2 x (-2 f+c \log (f)))^2}{4 (-2 f+c \log (f))}\right ) \, dx+\frac{1}{4} \left (e^{2 d-\frac{e^2}{2 f+c \log (f)}} f^a\right ) \int \exp \left (\frac{(2 e+2 x (2 f+c \log (f)))^2}{4 (2 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 d+\frac{e^2}{2 f-c \log (f)}} f^a \sqrt{\pi } \text{erf}\left (\frac{e+x (2 f-c \log (f))}{\sqrt{2 f-c \log (f)}}\right )}{8 \sqrt{2 f-c \log (f)}}+\frac{e^{2 d-\frac{e^2}{2 f+c \log (f)}} f^a \sqrt{\pi } \text{erfi}\left (\frac{e+x (2 f+c \log (f))}{\sqrt{2 f+c \log (f)}}\right )}{8 \sqrt{2 f+c \log (f)}}\\ \end{align*}

Mathematica [A]  time = 1.42745, size = 258, normalized size = 1.41 \[ \frac{\sqrt{\pi } f^a e^{\frac{e^2}{2 f-c \log (f)}} \left (2 \left (4 f^2-c^2 \log ^2(f)\right ) e^{\frac{e^2}{c \log (f)-2 f}} \text{Erfi}\left (\sqrt{c} x \sqrt{\log (f)}\right )-\sqrt{c} \sqrt{\log (f)} \left ((2 f-c \log (f)) \sqrt{c \log (f)+2 f} (\sinh (2 d)+\cosh (2 d)) e^{\frac{4 e^2 f}{c^2 \log ^2(f)-4 f^2}} \text{Erfi}\left (\frac{c x \log (f)+e+2 f x}{\sqrt{c \log (f)+2 f}}\right )+\sqrt{2 f-c \log (f)} (c \log (f)+2 f) (\cosh (2 d)-\sinh (2 d)) \text{Erf}\left (\frac{-c x \log (f)+e+2 f x}{\sqrt{2 f-c \log (f)}}\right )\right )\right )}{8 \sqrt{c} \sqrt{\log (f)} \left (c^2 \log ^2(f)-4 f^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(E^(e^2/(2*f - c*Log[f]))*f^a*Sqrt[Pi]*(2*E^(e^2/(-2*f + c*Log[f]))*Erfi[Sqrt[c]*x*Sqrt[Log[f]]]*(4*f^2 - c^2*
Log[f]^2) - Sqrt[c]*Sqrt[Log[f]]*(Erf[(e + 2*f*x - c*x*Log[f])/Sqrt[2*f - c*Log[f]]]*Sqrt[2*f - c*Log[f]]*(2*f
+ c*Log[f])*(Cosh[2*d] - Sinh[2*d]) + E^((4*e^2*f)/(-4*f^2 + c^2*Log[f]^2))*Erfi[(e + 2*f*x + c*x*Log[f])/Sqr
t[2*f + c*Log[f]]]*(2*f - c*Log[f])*Sqrt[2*f + c*Log[f]]*(Cosh[2*d] + Sinh[2*d]))))/(8*Sqrt[c]*Sqrt[Log[f]]*(-
4*f^2 + c^2*Log[f]^2))

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Maple [A]  time = 0.185, size = 177, normalized size = 1. \begin{align*}{\frac{\sqrt{\pi }{f}^{a}}{8}{{\rm e}^{-{\frac{2\,d\ln \left ( f \right ) c-4\,df+{e}^{2}}{-2\,f+c\ln \left ( f \right ) }}}}{\it Erf} \left ( x\sqrt{2\,f-c\ln \left ( f \right ) }+{e{\frac{1}{\sqrt{2\,f-c\ln \left ( f \right ) }}}} \right ){\frac{1}{\sqrt{2\,f-c\ln \left ( f \right ) }}}}-{\frac{\sqrt{\pi }{f}^{a}}{8}{{\rm e}^{{\frac{2\,d\ln \left ( f \right ) c+4\,df-{e}^{2}}{2\,f+c\ln \left ( f \right ) }}}}{\it Erf} \left ( -\sqrt{-c\ln \left ( f \right ) -2\,f}x+{e{\frac{1}{\sqrt{-c\ln \left ( f \right ) -2\,f}}}} \right ){\frac{1}{\sqrt{-c\ln \left ( f \right ) -2\,f}}}}-{\frac{\sqrt{\pi }{f}^{a}}{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)*sinh(f*x^2+e*x+d)^2,x)

[Out]

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

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Maxima [A]  time = 1.07094, size = 217, normalized size = 1.19 \begin{align*} \frac{\sqrt{\pi } f^{a} \operatorname{erf}\left (\sqrt{-c \log \left (f\right ) - 2 \, f} x - \frac{e}{\sqrt{-c \log \left (f\right ) - 2 \, f}}\right ) e^{\left (2 \, d - \frac{e^{2}}{c \log \left (f\right ) + 2 \, f}\right )}}{8 \, \sqrt{-c \log \left (f\right ) - 2 \, f}} + \frac{\sqrt{\pi } f^{a} \operatorname{erf}\left (\sqrt{-c \log \left (f\right ) + 2 \, f} x + \frac{e}{\sqrt{-c \log \left (f\right ) + 2 \, f}}\right ) e^{\left (-2 \, d - \frac{e^{2}}{c \log \left (f\right ) - 2 \, f}\right )}}{8 \, \sqrt{-c \log \left (f\right ) + 2 \, f}} - \frac{\sqrt{\pi } f^{a} \operatorname{erf}\left (\sqrt{-c \log \left (f\right )} x\right )}{4 \, \sqrt{-c \log \left (f\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.93488, size = 1118, normalized size = 6.11 \begin{align*} \frac{2 \,{\left (\sqrt{\pi }{\left (c^{2} \log \left (f\right )^{2} - 4 \, f^{2}\right )} \cosh \left (a \log \left (f\right )\right ) + \sqrt{\pi }{\left (c^{2} \log \left (f\right )^{2} - 4 \, f^{2}\right )} \sinh \left (a \log \left (f\right )\right )\right )} \sqrt{-c \log \left (f\right )} \operatorname{erf}\left (\sqrt{-c \log \left (f\right )} x\right ) -{\left (\sqrt{\pi }{\left (c^{2} \log \left (f\right )^{2} + 2 \, c f \log \left (f\right )\right )} \cosh \left (\frac{a c \log \left (f\right )^{2} - e^{2} + 4 \, d f - 2 \,{\left (c d + a f\right )} \log \left (f\right )}{c \log \left (f\right ) - 2 \, f}\right ) + \sqrt{\pi }{\left (c^{2} \log \left (f\right )^{2} + 2 \, c f \log \left (f\right )\right )} \sinh \left (\frac{a c \log \left (f\right )^{2} - e^{2} + 4 \, d f - 2 \,{\left (c d + a f\right )} \log \left (f\right )}{c \log \left (f\right ) - 2 \, f}\right )\right )} \sqrt{-c \log \left (f\right ) + 2 \, f} \operatorname{erf}\left (\frac{{\left (c x \log \left (f\right ) - 2 \, f x - e\right )} \sqrt{-c \log \left (f\right ) + 2 \, f}}{c \log \left (f\right ) - 2 \, f}\right ) -{\left (\sqrt{\pi }{\left (c^{2} \log \left (f\right )^{2} - 2 \, c f \log \left (f\right )\right )} \cosh \left (\frac{a c \log \left (f\right )^{2} - e^{2} + 4 \, d f + 2 \,{\left (c d + a f\right )} \log \left (f\right )}{c \log \left (f\right ) + 2 \, f}\right ) + \sqrt{\pi }{\left (c^{2} \log \left (f\right )^{2} - 2 \, c f \log \left (f\right )\right )} \sinh \left (\frac{a c \log \left (f\right )^{2} - e^{2} + 4 \, d f + 2 \,{\left (c d + a f\right )} \log \left (f\right )}{c \log \left (f\right ) + 2 \, f}\right )\right )} \sqrt{-c \log \left (f\right ) - 2 \, f} \operatorname{erf}\left (\frac{{\left (c x \log \left (f\right ) + 2 \, f x + e\right )} \sqrt{-c \log \left (f\right ) - 2 \, f}}{c \log \left (f\right ) + 2 \, f}\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)*sinh(f*x^2+e*x+d)^2,x, algorithm="fricas")

[Out]

1/8*(2*(sqrt(pi)*(c^2*log(f)^2 - 4*f^2)*cosh(a*log(f)) + sqrt(pi)*(c^2*log(f)^2 - 4*f^2)*sinh(a*log(f)))*sqrt(
-c*log(f))*erf(sqrt(-c*log(f))*x) - (sqrt(pi)*(c^2*log(f)^2 + 2*c*f*log(f))*cosh((a*c*log(f)^2 - e^2 + 4*d*f -
2*(c*d + a*f)*log(f))/(c*log(f) - 2*f)) + sqrt(pi)*(c^2*log(f)^2 + 2*c*f*log(f))*sinh((a*c*log(f)^2 - e^2 + 4
*d*f - 2*(c*d + a*f)*log(f))/(c*log(f) - 2*f)))*sqrt(-c*log(f) + 2*f)*erf((c*x*log(f) - 2*f*x - e)*sqrt(-c*log
(f) + 2*f)/(c*log(f) - 2*f)) - (sqrt(pi)*(c^2*log(f)^2 - 2*c*f*log(f))*cosh((a*c*log(f)^2 - e^2 + 4*d*f + 2*(c
*d + a*f)*log(f))/(c*log(f) + 2*f)) + sqrt(pi)*(c^2*log(f)^2 - 2*c*f*log(f))*sinh((a*c*log(f)^2 - e^2 + 4*d*f
+ 2*(c*d + a*f)*log(f))/(c*log(f) + 2*f)))*sqrt(-c*log(f) - 2*f)*erf((c*x*log(f) + 2*f*x + e)*sqrt(-c*log(f) -
2*f)/(c*log(f) + 2*f)))/(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}} \sinh ^{2}{\left (d + e x + f x^{2} \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.2743, size = 267, normalized size = 1.46 \begin{align*} \frac{\sqrt{\pi } f^{a} \operatorname{erf}\left (-\sqrt{-c \log \left (f\right )} x\right )}{4 \, \sqrt{-c \log \left (f\right )}} - \frac{\sqrt{\pi } \operatorname{erf}\left (-\sqrt{-c \log \left (f\right ) - 2 \, f}{\left (x + \frac{e}{c \log \left (f\right ) + 2 \, f}\right )}\right ) e^{\left (\frac{a c \log \left (f\right )^{2} + 2 \, c d \log \left (f\right ) + 2 \, a f \log \left (f\right ) + 4 \, d f - e^{2}}{c \log \left (f\right ) + 2 \, f}\right )}}{8 \, \sqrt{-c \log \left (f\right ) - 2 \, f}} - \frac{\sqrt{\pi } \operatorname{erf}\left (-\sqrt{-c \log \left (f\right ) + 2 \, f}{\left (x - \frac{e}{c \log \left (f\right ) - 2 \, f}\right )}\right ) e^{\left (\frac{a c \log \left (f\right )^{2} - 2 \, c d \log \left (f\right ) - 2 \, a f \log \left (f\right ) + 4 \, d f - e^{2}}{c \log \left (f\right ) - 2 \, f}\right )}}{8 \, \sqrt{-c \log \left (f\right ) + 2 \, f}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*sqrt(pi)*f^a*erf(-sqrt(-c*log(f))*x)/sqrt(-c*log(f)) - 1/8*sqrt(pi)*erf(-sqrt(-c*log(f) - 2*f)*(x + e/(c*l
og(f) + 2*f)))*e^((a*c*log(f)^2 + 2*c*d*log(f) + 2*a*f*log(f) + 4*d*f - e^2)/(c*log(f) + 2*f))/sqrt(-c*log(f)
- 2*f) - 1/8*sqrt(pi)*erf(-sqrt(-c*log(f) + 2*f)*(x - e/(c*log(f) - 2*f)))*e^((a*c*log(f)^2 - 2*c*d*log(f) - 2
*a*f*log(f) + 4*d*f - e^2)/(c*log(f) - 2*f))/sqrt(-c*log(f) + 2*f)