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

Optimal. Leaf size=161 \[ -\frac{\sqrt{\pi } f^a e^{-\frac{e^2}{c \log (f)}-2 d} \text{Erfi}\left (\frac{e-c x \log (f)}{\sqrt{c} \sqrt{\log (f)}}\right )}{8 \sqrt{c} \sqrt{\log (f)}}+\frac{\sqrt{\pi } f^a e^{2 d-\frac{e^2}{c \log (f)}} \text{Erfi}\left (\frac{c x \log (f)+e}{\sqrt{c} \sqrt{\log (f)}}\right )}{8 \sqrt{c} \sqrt{\log (f)}}-\frac{\sqrt{\pi } f^a \text{Erfi}\left (\sqrt{c} x \sqrt{\log (f)}\right )}{4 \sqrt{c} \sqrt{\log (f)}} \]

[Out]

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

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Rubi [A]  time = 0.226126, antiderivative size = 161, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {5512, 2204, 2287, 2234} \[ -\frac{\sqrt{\pi } f^a e^{-\frac{e^2}{c \log (f)}-2 d} \text{Erfi}\left (\frac{e-c x \log (f)}{\sqrt{c} \sqrt{\log (f)}}\right )}{8 \sqrt{c} \sqrt{\log (f)}}+\frac{\sqrt{\pi } f^a e^{2 d-\frac{e^2}{c \log (f)}} \text{Erfi}\left (\frac{c x \log (f)+e}{\sqrt{c} \sqrt{\log (f)}}\right )}{8 \sqrt{c} \sqrt{\log (f)}}-\frac{\sqrt{\pi } f^a \text{Erfi}\left (\sqrt{c} x \sqrt{\log (f)}\right )}{4 \sqrt{c} \sqrt{\log (f)}} \]

Antiderivative was successfully verified.

[In]

Int[f^(a + c*x^2)*Sinh[d + e*x]^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/(c*Log[f]))*f^a*Sqrt[Pi
]*Erfi[(e - c*x*Log[f])/(Sqrt[c]*Sqrt[Log[f]])])/(8*Sqrt[c]*Sqrt[Log[f]]) + (E^(2*d - e^2/(c*Log[f]))*f^a*Sqrt
[Pi]*Erfi[(e + c*x*Log[f])/(Sqrt[c]*Sqrt[Log[f]])])/(8*Sqrt[c]*Sqrt[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]

Rubi steps

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

Mathematica [A]  time = 0.26411, size = 131, normalized size = 0.81 \[ \frac{\sqrt{\pi } f^a e^{-\frac{e^2}{c \log (f)}} \left ((\cosh (2 d)-\sinh (2 d)) \text{Erfi}\left (\frac{c x \log (f)-e}{\sqrt{c} \sqrt{\log (f)}}\right )+(\sinh (2 d)+\cosh (2 d)) \text{Erfi}\left (\frac{c x \log (f)+e}{\sqrt{c} \sqrt{\log (f)}}\right )-2 e^{\frac{e^2}{c \log (f)}} \text{Erfi}\left (\sqrt{c} x \sqrt{\log (f)}\right )\right )}{8 \sqrt{c} \sqrt{\log (f)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.122, size = 139, normalized size = 0.9 \begin{align*}{\frac{\sqrt{\pi }{f}^{a}}{8}{{\rm e}^{-{\frac{2\,d\ln \left ( f \right ) c+{e}^{2}}{c\ln \left ( f \right ) }}}}{\it Erf} \left ( \sqrt{-c\ln \left ( f \right ) }x+{e{\frac{1}{\sqrt{-c\ln \left ( f \right ) }}}} \right ){\frac{1}{\sqrt{-c\ln \left ( f \right ) }}}}-{\frac{\sqrt{\pi }{f}^{a}}{8}{{\rm e}^{{\frac{2\,d\ln \left ( f \right ) c-{e}^{2}}{c\ln \left ( f \right ) }}}}{\it Erf} \left ( -\sqrt{-c\ln \left ( f \right ) }x+{e{\frac{1}{\sqrt{-c\ln \left ( f \right ) }}}} \right ){\frac{1}{\sqrt{-c\ln \left ( f \right ) }}}}-{\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(e*x+d)^2,x)

[Out]

1/8*Pi^(1/2)*f^a*exp(-(2*d*ln(f)*c+e^2)/ln(f)/c)/(-c*ln(f))^(1/2)*erf((-c*ln(f))^(1/2)*x+e/(-c*ln(f))^(1/2))-1
/8*Pi^(1/2)*f^a*exp((2*d*ln(f)*c-e^2)/ln(f)/c)/(-c*ln(f))^(1/2)*erf(-(-c*ln(f))^(1/2)*x+e/(-c*ln(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 = 1.07114, size = 177, normalized size = 1.1 \begin{align*} \frac{\sqrt{\pi } f^{a} \operatorname{erf}\left (\sqrt{-c \log \left (f\right )} x - \frac{e}{\sqrt{-c \log \left (f\right )}}\right ) e^{\left (2 \, d - \frac{e^{2}}{c \log \left (f\right )}\right )}}{8 \, \sqrt{-c \log \left (f\right )}} + \frac{\sqrt{\pi } f^{a} \operatorname{erf}\left (\sqrt{-c \log \left (f\right )} x + \frac{e}{\sqrt{-c \log \left (f\right )}}\right ) e^{\left (-2 \, d - \frac{e^{2}}{c \log \left (f\right )}\right )}}{8 \, \sqrt{-c \log \left (f\right )}} - \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(e*x+d)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.80786, size = 676, normalized size = 4.2 \begin{align*} \frac{2 \, \sqrt{-c \log \left (f\right )}{\left (\sqrt{\pi } \cosh \left (a \log \left (f\right )\right ) + \sqrt{\pi } \sinh \left (a \log \left (f\right )\right )\right )} \operatorname{erf}\left (\sqrt{-c \log \left (f\right )} x\right ) - \sqrt{-c \log \left (f\right )}{\left (\sqrt{\pi } \cosh \left (\frac{a c \log \left (f\right )^{2} + 2 \, c d \log \left (f\right ) - e^{2}}{c \log \left (f\right )}\right ) + \sqrt{\pi } \sinh \left (\frac{a c \log \left (f\right )^{2} + 2 \, c d \log \left (f\right ) - e^{2}}{c \log \left (f\right )}\right )\right )} \operatorname{erf}\left (\frac{{\left (c x \log \left (f\right ) + e\right )} \sqrt{-c \log \left (f\right )}}{c \log \left (f\right )}\right ) - \sqrt{-c \log \left (f\right )}{\left (\sqrt{\pi } \cosh \left (\frac{a c \log \left (f\right )^{2} - 2 \, c d \log \left (f\right ) - e^{2}}{c \log \left (f\right )}\right ) + \sqrt{\pi } \sinh \left (\frac{a c \log \left (f\right )^{2} - 2 \, c d \log \left (f\right ) - e^{2}}{c \log \left (f\right )}\right )\right )} \operatorname{erf}\left (\frac{{\left (c x \log \left (f\right ) - e\right )} \sqrt{-c \log \left (f\right )}}{c \log \left (f\right )}\right )}{8 \, 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(e*x+d)^2,x, algorithm="fricas")

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.29808, size = 203, normalized size = 1.26 \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 )}{\left (x + \frac{e}{c \log \left (f\right )}\right )}\right ) e^{\left (\frac{a c \log \left (f\right )^{2} + 2 \, c d \log \left (f\right ) - e^{2}}{c \log \left (f\right )}\right )}}{8 \, \sqrt{-c \log \left (f\right )}} - \frac{\sqrt{\pi } \operatorname{erf}\left (-\sqrt{-c \log \left (f\right )}{\left (x - \frac{e}{c \log \left (f\right )}\right )}\right ) e^{\left (\frac{a c \log \left (f\right )^{2} - 2 \, c d \log \left (f\right ) - e^{2}}{c \log \left (f\right )}\right )}}{8 \, \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(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))*(x + e/(c*log(f))
))*e^((a*c*log(f)^2 + 2*c*d*log(f) - e^2)/(c*log(f)))/sqrt(-c*log(f)) - 1/8*sqrt(pi)*erf(-sqrt(-c*log(f))*(x -
e/(c*log(f))))*e^((a*c*log(f)^2 - 2*c*d*log(f) - e^2)/(c*log(f)))/sqrt(-c*log(f))