3.62 \(\int \frac {\sinh (f x)}{\sqrt {d x}} \, dx\)

Optimal. Leaf size=77 \[ \frac {\sqrt {\pi } \text {erfi}\left (\frac {\sqrt {f} \sqrt {d x}}{\sqrt {d}}\right )}{2 \sqrt {d} \sqrt {f}}-\frac {\sqrt {\pi } \text {erf}\left (\frac {\sqrt {f} \sqrt {d x}}{\sqrt {d}}\right )}{2 \sqrt {d} \sqrt {f}} \]

[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)

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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 = {3308, 2180, 2204, 2205} \[ \frac {\sqrt {\pi } \text {Erfi}\left (\frac {\sqrt {f} \sqrt {d x}}{\sqrt {d}}\right )}{2 \sqrt {d} \sqrt {f}}-\frac {\sqrt {\pi } \text {Erf}\left (\frac {\sqrt {f} \sqrt {d x}}{\sqrt {d}}\right )}{2 \sqrt {d} \sqrt {f}} \]

Antiderivative was successfully verified.

[In]

Int[Sinh[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 3308

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

Rubi steps

\begin {align*} \int \frac {\sinh (f x)}{\sqrt {d x}} \, dx &=-\left (\frac {1}{2} \int \frac {e^{-f x}}{\sqrt {d x}} \, dx\right )+\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*}

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Mathematica [A]  time = 0.01, size = 47, normalized size = 0.61 \[ \frac {\sqrt {-f x} \Gamma \left (\frac {1}{2},-f x\right )+\sqrt {f x} \Gamma \left (\frac {1}{2},f x\right )}{2 f \sqrt {d x}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sinh[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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fricas [A]  time = 0.67, size = 58, normalized size = 0.75 \[ -\frac {\sqrt {\pi } \sqrt {\frac {f}{d}} \operatorname {erf}\left (\sqrt {d x} \sqrt {\frac {f}{d}}\right ) + \sqrt {\pi } \sqrt {-\frac {f}{d}} \operatorname {erf}\left (\sqrt {d x} \sqrt {-\frac {f}{d}}\right )}{2 \, f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(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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giac [A]  time = 0.18, size = 61, normalized size = 0.79 \[ \frac {\frac {\sqrt {\pi } d \operatorname {erf}\left (-\frac {\sqrt {d f} \sqrt {d x}}{d}\right )}{\sqrt {d f}} - \frac {\sqrt {\pi } d \operatorname {erf}\left (-\frac {\sqrt {-d f} \sqrt {d x}}{d}\right )}{\sqrt {-d f}}}{2 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(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

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maple [C]  time = 0.03, size = 71, normalized size = 0.92 \[ -\frac {\sqrt {\pi }\, \sqrt {x}\, \sqrt {2}\, \sqrt {i f}\, \left (-\frac {\left (i f \right )^{\frac {3}{2}} \sqrt {2}\, \erf \left (\sqrt {x}\, \sqrt {f}\right )}{2 f^{\frac {3}{2}}}+\frac {\left (i f \right )^{\frac {3}{2}} \sqrt {2}\, \erfi \left (\sqrt {x}\, \sqrt {f}\right )}{2 f^{\frac {3}{2}}}\right )}{2 \sqrt {d x}\, f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*Pi^(1/2)/(d*x)^(1/2)*x^(1/2)*2^(1/2)*(I*f)^(1/2)/f*(-1/2*(I*f)^(3/2)*2^(1/2)/f^(3/2)*erf(x^(1/2)*f^(1/2))
+1/2*(I*f)^(3/2)*2^(1/2)/f^(3/2)*erfi(x^(1/2)*f^(1/2)))

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maxima [B]  time = 0.33, size = 116, normalized size = 1.51 \[ \frac {4 \, \sqrt {d x} \sinh \left (f x\right ) - \frac {{\left (\frac {2 \, \sqrt {d x} d e^{\left (f x\right )}}{f} - \frac {2 \, \sqrt {d x} d e^{\left (-f x\right )}}{f} + \frac {\sqrt {\pi } d \operatorname {erf}\left (\sqrt {d x} \sqrt {\frac {f}{d}}\right )}{f \sqrt {\frac {f}{d}}} - \frac {\sqrt {\pi } d \operatorname {erf}\left (\sqrt {d x} \sqrt {-\frac {f}{d}}\right )}{f \sqrt {-\frac {f}{d}}}\right )} f}{d}}{2 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(4*sqrt(d*x)*sinh(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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\mathrm {sinh}\left (f\,x\right )}{\sqrt {d\,x}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(sinh(f*x)/(d*x)^(1/2), x)

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sympy [C]  time = 0.96, size = 70, normalized size = 0.91 \[ \frac {3 \sqrt {2} \sqrt {\pi } e^{- \frac {3 i \pi }{4}} S\left (\frac {\sqrt {2} \sqrt {f} \sqrt {x} e^{\frac {i \pi }{4}}}{\sqrt {\pi }}\right ) \Gamma \left (\frac {3}{4}\right )}{4 \sqrt {d} \sqrt {f} \Gamma \left (\frac {7}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*sqrt(2)*sqrt(pi)*exp(-3*I*pi/4)*fresnels(sqrt(2)*sqrt(f)*sqrt(x)*exp(I*pi/4)/sqrt(pi))*gamma(3/4)/(4*sqrt(d)
*sqrt(f)*gamma(7/4))

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