3.65 \(\int \frac{\cosh (f x)}{\sqrt{d x}} \, dx\)

Optimal. Leaf size=77 \[ \frac{\sqrt{\pi } \text{Erf}\left (\frac{\sqrt{f} \sqrt{d x}}{\sqrt{d}}\right )}{2 \sqrt{d} \sqrt{f}}+\frac{\sqrt{\pi } \text{Erfi}\left (\frac{\sqrt{f} \sqrt{d x}}{\sqrt{d}}\right )}{2 \sqrt{d} \sqrt{f}} \]

[Out]

(Sqrt[Pi]*Erf[(Sqrt[f]*Sqrt[d*x])/Sqrt[d]])/(2*Sqrt[d]*Sqrt[f]) + (Sqrt[Pi]*Erfi[(Sqrt[f]*Sqrt[d*x])/Sqrt[d]])
/(2*Sqrt[d]*Sqrt[f])

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Rubi [A]  time = 0.0774998, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {3307, 2180, 2204, 2205} \[ \frac{\sqrt{\pi } \text{Erf}\left (\frac{\sqrt{f} \sqrt{d x}}{\sqrt{d}}\right )}{2 \sqrt{d} \sqrt{f}}+\frac{\sqrt{\pi } \text{Erfi}\left (\frac{\sqrt{f} \sqrt{d x}}{\sqrt{d}}\right )}{2 \sqrt{d} \sqrt{f}} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[f*x]/Sqrt[d*x],x]

[Out]

(Sqrt[Pi]*Erf[(Sqrt[f]*Sqrt[d*x])/Sqrt[d]])/(2*Sqrt[d]*Sqrt[f]) + (Sqrt[Pi]*Erfi[(Sqrt[f]*Sqrt[d*x])/Sqrt[d]])
/(2*Sqrt[d]*Sqrt[f])

Rule 3307

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol] :> Dist[I/2, Int[(c + d*x)^m/(E^(
I*k*Pi)*E^(I*(e + f*x))), x], x] - Dist[I/2, Int[(c + d*x)^m*E^(I*k*Pi)*E^(I*(e + f*x)), x], x] /; FreeQ[{c, d
, e, f, m}, x] && IntegerQ[2*k]

Rule 2180

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[F^(g*(e - (c*
f)/d) + (f*g*x^2)/d), x], x, Sqrt[c + d*x]], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !$UseGamma === True

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 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 \frac{\cosh (f x)}{\sqrt{d x}} \, dx &=\frac{1}{2} \int \frac{e^{-f x}}{\sqrt{d x}} \, dx+\frac{1}{2} \int \frac{e^{f x}}{\sqrt{d x}} \, dx\\ &=\frac{\operatorname{Subst}\left (\int e^{-\frac{f x^2}{d}} \, dx,x,\sqrt{d x}\right )}{d}+\frac{\operatorname{Subst}\left (\int e^{\frac{f x^2}{d}} \, dx,x,\sqrt{d x}\right )}{d}\\ &=\frac{\sqrt{\pi } \text{erf}\left (\frac{\sqrt{f} \sqrt{d x}}{\sqrt{d}}\right )}{2 \sqrt{d} \sqrt{f}}+\frac{\sqrt{\pi } \text{erfi}\left (\frac{\sqrt{f} \sqrt{d x}}{\sqrt{d}}\right )}{2 \sqrt{d} \sqrt{f}}\\ \end{align*}

Mathematica [A]  time = 0.0077995, size = 48, normalized size = 0.62 \[ \frac{\sqrt{-f x} \text{Gamma}\left (\frac{1}{2},-f x\right )-\sqrt{f x} \text{Gamma}\left (\frac{1}{2},f x\right )}{2 f \sqrt{d x}} \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[f*x]/Sqrt[d*x],x]

[Out]

(Sqrt[-(f*x)]*Gamma[1/2, -(f*x)] - Sqrt[f*x]*Gamma[1/2, f*x])/(2*f*Sqrt[d*x])

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Maple [C]  time = 0.022, size = 72, normalized size = 0.9 \begin{align*}{\frac{-{\frac{i}{2}}\sqrt{\pi }\sqrt{2}}{f}\sqrt{x}\sqrt{if} \left ({\frac{\sqrt{2}}{2}\sqrt{if}{\it Erf} \left ( \sqrt{x}\sqrt{f} \right ){\frac{1}{\sqrt{f}}}}+{\frac{\sqrt{2}}{2}\sqrt{if}{\it erfi} \left ( \sqrt{x}\sqrt{f} \right ){\frac{1}{\sqrt{f}}}} \right ){\frac{1}{\sqrt{dx}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(f*x)/(d*x)^(1/2),x)

[Out]

-1/2*I*Pi^(1/2)/(d*x)^(1/2)*x^(1/2)*2^(1/2)*(I*f)^(1/2)/f*(1/2*(I*f)^(1/2)*2^(1/2)/f^(1/2)*erf(x^(1/2)*f^(1/2)
)+1/2*(I*f)^(1/2)*2^(1/2)/f^(1/2)*erfi(x^(1/2)*f^(1/2)))

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Maxima [B]  time = 1.05695, size = 158, normalized size = 2.05 \begin{align*} \frac{4 \, \sqrt{d x} \cosh \left (f x\right ) - \frac{{\left (\frac{2 \, \sqrt{d x} d e^{\left (f x\right )}}{f} + \frac{2 \, \sqrt{d x} d e^{\left (-f x\right )}}{f} - \frac{\sqrt{\pi } d \operatorname{erf}\left (\sqrt{d x} \sqrt{\frac{f}{d}}\right )}{f \sqrt{\frac{f}{d}}} - \frac{\sqrt{\pi } d \operatorname{erf}\left (\sqrt{d x} \sqrt{-\frac{f}{d}}\right )}{f \sqrt{-\frac{f}{d}}}\right )} f}{d}}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(f*x)/(d*x)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.83252, size = 136, normalized size = 1.77 \begin{align*} \frac{\sqrt{\pi } \sqrt{\frac{f}{d}} \operatorname{erf}\left (\sqrt{d x} \sqrt{\frac{f}{d}}\right ) - \sqrt{\pi } \sqrt{-\frac{f}{d}} \operatorname{erf}\left (\sqrt{d x} \sqrt{-\frac{f}{d}}\right )}{2 \, f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(f*x)/(d*x)^(1/2),x, algorithm="fricas")

[Out]

1/2*(sqrt(pi)*sqrt(f/d)*erf(sqrt(d*x)*sqrt(f/d)) - sqrt(pi)*sqrt(-f/d)*erf(sqrt(d*x)*sqrt(-f/d)))/f

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Sympy [C]  time = 1.64281, size = 66, normalized size = 0.86 \begin{align*} \frac{\sqrt{2} \sqrt{\pi } e^{- \frac{i \pi }{4}} C\left (\frac{\sqrt{2} \sqrt{f} \sqrt{x} e^{\frac{i \pi }{4}}}{\sqrt{\pi }}\right ) \Gamma \left (\frac{1}{4}\right )}{4 \sqrt{d} \sqrt{f} \Gamma \left (\frac{5}{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(f*x)/(d*x)**(1/2),x)

[Out]

sqrt(2)*sqrt(pi)*exp(-I*pi/4)*fresnelc(sqrt(2)*sqrt(f)*sqrt(x)*exp(I*pi/4)/sqrt(pi))*gamma(1/4)/(4*sqrt(d)*sqr
t(f)*gamma(5/4))

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Giac [A]  time = 1.29224, size = 81, normalized size = 1.05 \begin{align*} -\frac{\frac{\sqrt{\pi } d \operatorname{erf}\left (-\frac{\sqrt{d f} \sqrt{d x}}{d}\right )}{\sqrt{d f}} + \frac{\sqrt{\pi } d \operatorname{erf}\left (-\frac{\sqrt{-d f} \sqrt{d x}}{d}\right )}{\sqrt{-d f}}}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(f*x)/(d*x)^(1/2),x, algorithm="giac")

[Out]

-1/2*(sqrt(pi)*d*erf(-sqrt(d*f)*sqrt(d*x)/d)/sqrt(d*f) + sqrt(pi)*d*erf(-sqrt(-d*f)*sqrt(d*x)/d)/sqrt(-d*f))/d