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

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

[Out]

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

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Rubi [A]  time = 0.320949, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {5512, 2287, 2234, 2205, 2204} \[ \frac{\sqrt{\pi } f^a e^{d-\frac{e^2}{4 (c \log (f)+f)}} \text{Erfi}\left (\frac{2 x (c \log (f)+f)+e}{2 \sqrt{c \log (f)+f}}\right )}{4 \sqrt{c \log (f)+f}}-\frac{\sqrt{\pi } f^a e^{\frac{e^2}{4 f-4 c \log (f)}-d} \text{Erf}\left (\frac{2 x (f-c \log (f))+e}{2 \sqrt{f-c \log (f)}}\right )}{4 \sqrt{f-c \log (f)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.729534, size = 166, normalized size = 1.19 \[ \frac{\sqrt{\pi } f^a e^{-\frac{e^2}{4 (c \log (f)+f)}} \left (\sqrt{f-c \log (f)} (\sinh (d)+\cosh (d)) \text{Erfi}\left (\frac{2 c x \log (f)+e+2 f x}{2 \sqrt{c \log (f)+f}}\right )-\sqrt{c \log (f)+f} (\cosh (d)-\sinh (d)) e^{\frac{e^2 f}{2 f^2-2 c^2 \log ^2(f)}} \text{Erf}\left (\frac{-2 c x \log (f)+e+2 f x}{2 \sqrt{f-c \log (f)}}\right )\right )}{4 \sqrt{f-c \log (f)} \sqrt{c \log (f)+f}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.175, size = 147, normalized size = 1.1 \begin{align*} -{\frac{\sqrt{\pi }{f}^{a}}{4}{{\rm e}^{{\frac{4\,d\ln \left ( f \right ) c+4\,df-{e}^{2}}{4\,c\ln \left ( f \right ) +4\,f}}}}{\it Erf} \left ( -\sqrt{-c\ln \left ( f \right ) -f}x+{\frac{e}{2}{\frac{1}{\sqrt{-c\ln \left ( f \right ) -f}}}} \right ){\frac{1}{\sqrt{-c\ln \left ( f \right ) -f}}}}-{\frac{\sqrt{\pi }{f}^{a}}{4}{{\rm e}^{-{\frac{4\,d\ln \left ( f \right ) c-4\,df+{e}^{2}}{4\,c\ln \left ( f \right ) -4\,f}}}}{\it Erf} \left ( x\sqrt{f-c\ln \left ( f \right ) }+{\frac{e}{2}{\frac{1}{\sqrt{f-c\ln \left ( f \right ) }}}} \right ){\frac{1}{\sqrt{f-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),x)

[Out]

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

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Maxima [A]  time = 1.04879, size = 171, normalized size = 1.22 \begin{align*} \frac{\sqrt{\pi } f^{a} \operatorname{erf}\left (\sqrt{-c \log \left (f\right ) - f} x - \frac{e}{2 \, \sqrt{-c \log \left (f\right ) - f}}\right ) e^{\left (d - \frac{e^{2}}{4 \,{\left (c \log \left (f\right ) + f\right )}}\right )}}{4 \, \sqrt{-c \log \left (f\right ) - f}} - \frac{\sqrt{\pi } f^{a} \operatorname{erf}\left (\sqrt{-c \log \left (f\right ) + f} x + \frac{e}{2 \, \sqrt{-c \log \left (f\right ) + f}}\right ) e^{\left (-d - \frac{e^{2}}{4 \,{\left (c \log \left (f\right ) - f\right )}}\right )}}{4 \, \sqrt{-c \log \left (f\right ) + 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),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 2.03711, size = 853, normalized size = 6.09 \begin{align*} \frac{{\left (\sqrt{\pi }{\left (c \log \left (f\right ) + f\right )} \cosh \left (\frac{4 \, a c \log \left (f\right )^{2} - e^{2} + 4 \, d f - 4 \,{\left (c d + a f\right )} \log \left (f\right )}{4 \,{\left (c \log \left (f\right ) - f\right )}}\right ) + \sqrt{\pi }{\left (c \log \left (f\right ) + f\right )} \sinh \left (\frac{4 \, a c \log \left (f\right )^{2} - e^{2} + 4 \, d f - 4 \,{\left (c d + a f\right )} \log \left (f\right )}{4 \,{\left (c \log \left (f\right ) - f\right )}}\right )\right )} \sqrt{-c \log \left (f\right ) + f} \operatorname{erf}\left (\frac{{\left (2 \, c x \log \left (f\right ) - 2 \, f x - e\right )} \sqrt{-c \log \left (f\right ) + f}}{2 \,{\left (c \log \left (f\right ) - f\right )}}\right ) -{\left (\sqrt{\pi }{\left (c \log \left (f\right ) - f\right )} \cosh \left (\frac{4 \, a c \log \left (f\right )^{2} - e^{2} + 4 \, d f + 4 \,{\left (c d + a f\right )} \log \left (f\right )}{4 \,{\left (c \log \left (f\right ) + f\right )}}\right ) + \sqrt{\pi }{\left (c \log \left (f\right ) - f\right )} \sinh \left (\frac{4 \, a c \log \left (f\right )^{2} - e^{2} + 4 \, d f + 4 \,{\left (c d + a f\right )} \log \left (f\right )}{4 \,{\left (c \log \left (f\right ) + f\right )}}\right )\right )} \sqrt{-c \log \left (f\right ) - f} \operatorname{erf}\left (\frac{{\left (2 \, c x \log \left (f\right ) + 2 \, f x + e\right )} \sqrt{-c \log \left (f\right ) - f}}{2 \,{\left (c \log \left (f\right ) + f\right )}}\right )}{4 \,{\left (c^{2} \log \left (f\right )^{2} - f^{2}\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),x, algorithm="fricas")

[Out]

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

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

[Out]

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

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Giac [A]  time = 1.3036, size = 232, normalized size = 1.66 \begin{align*} -\frac{\sqrt{\pi } \operatorname{erf}\left (-\frac{1}{2} \, \sqrt{-c \log \left (f\right ) - f}{\left (2 \, x + \frac{e}{c \log \left (f\right ) + f}\right )}\right ) e^{\left (\frac{4 \, a c \log \left (f\right )^{2} + 4 \, c d \log \left (f\right ) + 4 \, a f \log \left (f\right ) + 4 \, d f - e^{2}}{4 \,{\left (c \log \left (f\right ) + f\right )}}\right )}}{4 \, \sqrt{-c \log \left (f\right ) - f}} + \frac{\sqrt{\pi } \operatorname{erf}\left (-\frac{1}{2} \, \sqrt{-c \log \left (f\right ) + f}{\left (2 \, x - \frac{e}{c \log \left (f\right ) - f}\right )}\right ) e^{\left (\frac{4 \, a c \log \left (f\right )^{2} - 4 \, c d \log \left (f\right ) - 4 \, a f \log \left (f\right ) + 4 \, d f - e^{2}}{4 \,{\left (c \log \left (f\right ) - f\right )}}\right )}}{4 \, \sqrt{-c \log \left (f\right ) + 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),x, algorithm="giac")

[Out]

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