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

Optimal. Leaf size=153 \[ \frac{\sqrt{\pi } f^a e^{-\frac{(e-b \log (f))^2}{4 c \log (f)}-d} \text{Erfi}\left (\frac{-b \log (f)-2 c x \log (f)+e}{2 \sqrt{c} \sqrt{\log (f)}}\right )}{4 \sqrt{c} \sqrt{\log (f)}}+\frac{\sqrt{\pi } f^a e^{d-\frac{(b \log (f)+e)^2}{4 c \log (f)}} \text{Erfi}\left (\frac{b \log (f)+2 c x \log (f)+e}{2 \sqrt{c} \sqrt{\log (f)}}\right )}{4 \sqrt{c} \sqrt{\log (f)}} \]

[Out]

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

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Rubi [A]  time = 0.300778, antiderivative size = 153, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {5512, 2287, 2234, 2204} \[ \frac{\sqrt{\pi } f^a e^{-\frac{(e-b \log (f))^2}{4 c \log (f)}-d} \text{Erfi}\left (\frac{-b \log (f)-2 c x \log (f)+e}{2 \sqrt{c} \sqrt{\log (f)}}\right )}{4 \sqrt{c} \sqrt{\log (f)}}+\frac{\sqrt{\pi } f^a e^{d-\frac{(b \log (f)+e)^2}{4 c \log (f)}} \text{Erfi}\left (\frac{b \log (f)+2 c x \log (f)+e}{2 \sqrt{c} \sqrt{\log (f)}}\right )}{4 \sqrt{c} \sqrt{\log (f)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.335413, size = 135, normalized size = 0.88 \[ \frac{\sqrt{\pi } f^{a-\frac{b^2}{4 c}} e^{-\frac{e (2 b \log (f)+e)}{4 c \log (f)}} \left ((\sinh (d)+\cosh (d)) \text{Erfi}\left (\frac{\log (f) (b+2 c x)+e}{2 \sqrt{c} \sqrt{\log (f)}}\right )-e^{\frac{b e}{c}} (\cosh (d)-\sinh (d)) \text{Erfi}\left (\frac{\log (f) (b+2 c x)-e}{2 \sqrt{c} \sqrt{\log (f)}}\right )\right )}{4 \sqrt{c} \sqrt{\log (f)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.116, size = 156, normalized size = 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\,d\ln \left ( f \right ) c+{e}^{2}}{4\,c\ln \left ( f \right ) }}}}{\it Erf} \left ( -\sqrt{-c\ln \left ( f \right ) }x+{\frac{e+b\ln \left ( f \right ) }{2}{\frac{1}{\sqrt{-c\ln \left ( f \right ) }}}} \right ){\frac{1}{\sqrt{-c\ln \left ( f \right ) }}}}+{\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\,d\ln \left ( f \right ) c+{e}^{2}}{4\,c\ln \left ( f \right ) }}}}{\it Erf} \left ( -\sqrt{-c\ln \left ( f \right ) }x+{\frac{b\ln \left ( f \right ) -e}{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)*sinh(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*ln(f)*c+e^2)/ln(f)/c)/(-c*ln(f))^(1/2)*erf(-(-c*ln(f))
^(1/2)*x+1/2*(e+b*ln(f))/(-c*ln(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*ln(f)*c+e^2)
/ln(f)/c)/(-c*ln(f))^(1/2)*erf(-(-c*ln(f))^(1/2)*x+1/2*(b*ln(f)-e)/(-c*ln(f))^(1/2))

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Maxima [A]  time = 1.05659, size = 174, normalized size = 1.14 \begin{align*} \frac{\sqrt{\pi } f^{a} \operatorname{erf}\left (\sqrt{-c \log \left (f\right )} x - \frac{b \log \left (f\right ) + e}{2 \, \sqrt{-c \log \left (f\right )}}\right ) e^{\left (d - \frac{{\left (b \log \left (f\right ) + e\right )}^{2}}{4 \, c \log \left (f\right )}\right )}}{4 \, \sqrt{-c \log \left (f\right )}} - \frac{\sqrt{\pi } f^{a} \operatorname{erf}\left (\sqrt{-c \log \left (f\right )} x - \frac{b \log \left (f\right ) - e}{2 \, \sqrt{-c \log \left (f\right )}}\right ) e^{\left (-d - \frac{{\left (b \log \left (f\right ) - e\right )}^{2}}{4 \, c \log \left (f\right )}\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+b*x+a)*sinh(e*x+d),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.95237, size = 717, normalized size = 4.69 \begin{align*} -\frac{\sqrt{-c \log \left (f\right )}{\left (\sqrt{\pi } \cosh \left (-\frac{{\left (b^{2} - 4 \, a c\right )} \log \left (f\right )^{2} + e^{2} - 2 \,{\left (2 \, c d - b e\right )} \log \left (f\right )}{4 \, c \log \left (f\right )}\right ) + \sqrt{\pi } \sinh \left (-\frac{{\left (b^{2} - 4 \, a c\right )} \log \left (f\right )^{2} + e^{2} - 2 \,{\left (2 \, c d - b e\right )} \log \left (f\right )}{4 \, c \log \left (f\right )}\right )\right )} \operatorname{erf}\left (\frac{{\left ({\left (2 \, c x + b\right )} \log \left (f\right ) + e\right )} \sqrt{-c \log \left (f\right )}}{2 \, c \log \left (f\right )}\right ) - \sqrt{-c \log \left (f\right )}{\left (\sqrt{\pi } \cosh \left (-\frac{{\left (b^{2} - 4 \, a c\right )} \log \left (f\right )^{2} + e^{2} + 2 \,{\left (2 \, c d - b e\right )} \log \left (f\right )}{4 \, c \log \left (f\right )}\right ) + \sqrt{\pi } \sinh \left (-\frac{{\left (b^{2} - 4 \, a c\right )} \log \left (f\right )^{2} + e^{2} + 2 \,{\left (2 \, c d - b e\right )} \log \left (f\right )}{4 \, c \log \left (f\right )}\right )\right )} \operatorname{erf}\left (\frac{{\left ({\left (2 \, c x + b\right )} \log \left (f\right ) - e\right )} \sqrt{-c \log \left (f\right )}}{2 \, c \log \left (f\right )}\right )}{4 \, c \log \left (f\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.23225, size = 228, normalized size = 1.49 \begin{align*} \frac{\sqrt{\pi } \operatorname{erf}\left (-\frac{1}{2} \, \sqrt{-c \log \left (f\right )}{\left (2 \, x + \frac{b \log \left (f\right ) - e}{c \log \left (f\right )}\right )}\right ) e^{\left (-\frac{b^{2} \log \left (f\right )^{2} - 4 \, a c \log \left (f\right )^{2} + 4 \, c d \log \left (f\right ) - 2 \, b e \log \left (f\right ) + e^{2}}{4 \, c \log \left (f\right )}\right )}}{4 \, \sqrt{-c \log \left (f\right )}} - \frac{\sqrt{\pi } \operatorname{erf}\left (-\frac{1}{2} \, \sqrt{-c \log \left (f\right )}{\left (2 \, x + \frac{b \log \left (f\right ) + e}{c \log \left (f\right )}\right )}\right ) e^{\left (-\frac{b^{2} \log \left (f\right )^{2} - 4 \, a c \log \left (f\right )^{2} - 4 \, c d \log \left (f\right ) + 2 \, b e \log \left (f\right ) + e^{2}}{4 \, c \log \left (f\right )}\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+b*x+a)*sinh(e*x+d),x, algorithm="giac")

[Out]

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