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]

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

________________________________________________________________________________________

Rubi [A]  time = 0.08, antiderivative size = 77, normalized size of antiderivative = 1.00, 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 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]

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]

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 = 0.01, size = 48, normalized size = 0.62 \[ \frac {\sqrt {-f x} \Gamma \left (\frac {1}{2},-f x\right )-\sqrt {f x} \Gamma \left (\frac {1}{2},f x\right )}{2 f \sqrt {d x}} \]

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])

________________________________________________________________________________________

fricas [A]  time = 0.46, size = 59, normalized size = 0.77 \[ \frac {\sqrt {\pi } \sqrt {\frac {f}{d}} \operatorname {erf}\left (\sqrt {d x} \sqrt {\frac {f}{d}}\right ) - \sqrt {\pi } \sqrt {-\frac {f}{d}} \operatorname {erf}\left (\sqrt {d x} \sqrt {-\frac {f}{d}}\right )}{2 \, f} \]

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

________________________________________________________________________________________

giac [A]  time = 0.12, size = 60, normalized size = 0.78 \[ -\frac {\frac {\sqrt {\pi } d \operatorname {erf}\left (-\frac {\sqrt {d f} \sqrt {d x}}{d}\right )}{\sqrt {d f}} + \frac {\sqrt {\pi } d \operatorname {erf}\left (-\frac {\sqrt {-d f} \sqrt {d x}}{d}\right )}{\sqrt {-d f}}}{2 \, d} \]

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

________________________________________________________________________________________

maple [C]  time = 0.09, size = 72, normalized size = 0.94 \[ -\frac {i \sqrt {\pi }\, \sqrt {x}\, \sqrt {2}\, \sqrt {i f}\, \left (\frac {\sqrt {i f}\, \sqrt {2}\, \erf \left (\sqrt {x}\, \sqrt {f}\right )}{2 \sqrt {f}}+\frac {\sqrt {i f}\, \sqrt {2}\, \erfi \left (\sqrt {x}\, \sqrt {f}\right )}{2 \sqrt {f}}\right )}{2 \sqrt {d x}\, f} \]

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)))

________________________________________________________________________________________

maxima [B]  time = 0.50, size = 117, normalized size = 1.52 \[ \frac {4 \, \sqrt {d x} \cosh \left (f x\right ) - \frac {{\left (\frac {2 \, \sqrt {d x} d e^{\left (f x\right )}}{f} + \frac {2 \, \sqrt {d x} d e^{\left (-f x\right )}}{f} - \frac {\sqrt {\pi } d \operatorname {erf}\left (\sqrt {d x} \sqrt {\frac {f}{d}}\right )}{f \sqrt {\frac {f}{d}}} - \frac {\sqrt {\pi } d \operatorname {erf}\left (\sqrt {d x} \sqrt {-\frac {f}{d}}\right )}{f \sqrt {-\frac {f}{d}}}\right )} f}{d}}{2 \, d} \]

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

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\mathrm {cosh}\left (f\,x\right )}{\sqrt {d\,x}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(cosh(f*x)/(d*x)^(1/2), x)

________________________________________________________________________________________

sympy [C]  time = 0.96, size = 66, normalized size = 0.86 \[ \frac {\sqrt {2} \sqrt {\pi } e^{- \frac {i \pi }{4}} C\left (\frac {\sqrt {2} \sqrt {f} \sqrt {x} e^{\frac {i \pi }{4}}}{\sqrt {\pi }}\right ) \Gamma \left (\frac {1}{4}\right )}{4 \sqrt {d} \sqrt {f} \Gamma \left (\frac {5}{4}\right )} \]

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))

________________________________________________________________________________________