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

Optimal. Leaf size=92 \[ \frac{\sqrt{\pi } \sqrt{d} \text{Erf}\left (\frac{\sqrt{f} \sqrt{d x}}{\sqrt{d}}\right )}{4 f^{3/2}}-\frac{\sqrt{\pi } \sqrt{d} \text{Erfi}\left (\frac{\sqrt{f} \sqrt{d x}}{\sqrt{d}}\right )}{4 f^{3/2}}+\frac{\sqrt{d x} \sinh (f x)}{f} \]

[Out]

(Sqrt[d]*Sqrt[Pi]*Erf[(Sqrt[f]*Sqrt[d*x])/Sqrt[d]])/(4*f^(3/2)) - (Sqrt[d]*Sqrt[Pi]*Erfi[(Sqrt[f]*Sqrt[d*x])/S
qrt[d]])/(4*f^(3/2)) + (Sqrt[d*x]*Sinh[f*x])/f

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Rubi [A]  time = 0.106476, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {3296, 3308, 2180, 2204, 2205} \[ \frac{\sqrt{\pi } \sqrt{d} \text{Erf}\left (\frac{\sqrt{f} \sqrt{d x}}{\sqrt{d}}\right )}{4 f^{3/2}}-\frac{\sqrt{\pi } \sqrt{d} \text{Erfi}\left (\frac{\sqrt{f} \sqrt{d x}}{\sqrt{d}}\right )}{4 f^{3/2}}+\frac{\sqrt{d x} \sinh (f x)}{f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[d]*Sqrt[Pi]*Erf[(Sqrt[f]*Sqrt[d*x])/Sqrt[d]])/(4*f^(3/2)) - (Sqrt[d]*Sqrt[Pi]*Erfi[(Sqrt[f]*Sqrt[d*x])/S
qrt[d]])/(4*f^(3/2)) + (Sqrt[d*x]*Sinh[f*x])/f

Rule 3296

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> -Simp[((c + d*x)^m*Cos[e + f*x])/f, x] +
Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

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]

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 \sqrt{d x} \cosh (f x) \, dx &=\frac{\sqrt{d x} \sinh (f x)}{f}-\frac{d \int \frac{\sinh (f x)}{\sqrt{d x}} \, dx}{2 f}\\ &=\frac{\sqrt{d x} \sinh (f x)}{f}+\frac{d \int \frac{e^{-f x}}{\sqrt{d x}} \, dx}{4 f}-\frac{d \int \frac{e^{f x}}{\sqrt{d x}} \, dx}{4 f}\\ &=\frac{\sqrt{d x} \sinh (f x)}{f}+\frac{\operatorname{Subst}\left (\int e^{-\frac{f x^2}{d}} \, dx,x,\sqrt{d x}\right )}{2 f}-\frac{\operatorname{Subst}\left (\int e^{\frac{f x^2}{d}} \, dx,x,\sqrt{d x}\right )}{2 f}\\ &=\frac{\sqrt{d} \sqrt{\pi } \text{erf}\left (\frac{\sqrt{f} \sqrt{d x}}{\sqrt{d}}\right )}{4 f^{3/2}}-\frac{\sqrt{d} \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{f} \sqrt{d x}}{\sqrt{d}}\right )}{4 f^{3/2}}+\frac{\sqrt{d x} \sinh (f x)}{f}\\ \end{align*}

Mathematica [A]  time = 0.0111476, size = 48, normalized size = 0.52 \[ -\frac{d \left (\sqrt{-f x} \text{Gamma}\left (\frac{3}{2},-f x\right )+\sqrt{f x} \text{Gamma}\left (\frac{3}{2},f x\right )\right )}{2 f^2 \sqrt{d x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(d*(Sqrt[-(f*x)]*Gamma[3/2, -(f*x)] + Sqrt[f*x]*Gamma[3/2, f*x]))/(2*f^2*Sqrt[d*x])

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Maple [C]  time = 0.019, size = 121, normalized size = 1.3 \begin{align*}{\frac{-i\sqrt{\pi }\sqrt{2}}{f}\sqrt{dx} \left ({\frac{\sqrt{2}{{\rm e}^{fx}}}{4\,\sqrt{\pi }f}\sqrt{x} \left ( if \right ) ^{{\frac{3}{2}}}}-{\frac{\sqrt{2}{{\rm e}^{-fx}}}{4\,\sqrt{\pi }f}\sqrt{x} \left ( if \right ) ^{{\frac{3}{2}}}}+{\frac{\sqrt{2}}{8} \left ( if \right ) ^{{\frac{3}{2}}}{\it Erf} \left ( \sqrt{x}\sqrt{f} \right ){f}^{-{\frac{3}{2}}}}-{\frac{\sqrt{2}}{8} \left ( if \right ) ^{{\frac{3}{2}}}{\it erfi} \left ( \sqrt{x}\sqrt{f} \right ){f}^{-{\frac{3}{2}}}} \right ){\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{if}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-I*Pi^(1/2)*(d*x)^(1/2)/x^(1/2)*2^(1/2)/(I*f)^(1/2)/f*(1/4/Pi^(1/2)*x^(1/2)*2^(1/2)*(I*f)^(3/2)/f*exp(f*x)-1/4
/Pi^(1/2)*x^(1/2)*2^(1/2)*(I*f)^(3/2)/f*exp(-f*x)+1/8*(I*f)^(3/2)*2^(1/2)/f^(3/2)*erf(x^(1/2)*f^(1/2))-1/8*(I*
f)^(3/2)*2^(1/2)/f^(3/2)*erfi(x^(1/2)*f^(1/2)))

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Maxima [B]  time = 1.1235, size = 200, normalized size = 2.17 \begin{align*} \frac{8 \, \left (d x\right )^{\frac{3}{2}} \cosh \left (f x\right ) + \frac{f{\left (\frac{3 \, \sqrt{\pi } d^{2} \operatorname{erf}\left (\sqrt{d x} \sqrt{\frac{f}{d}}\right )}{f^{2} \sqrt{\frac{f}{d}}} - \frac{3 \, \sqrt{\pi } d^{2} \operatorname{erf}\left (\sqrt{d x} \sqrt{-\frac{f}{d}}\right )}{f^{2} \sqrt{-\frac{f}{d}}} - \frac{2 \,{\left (2 \, \left (d x\right )^{\frac{3}{2}} d f - 3 \, \sqrt{d x} d^{2}\right )} e^{\left (f x\right )}}{f^{2}} - \frac{2 \,{\left (2 \, \left (d x\right )^{\frac{3}{2}} d f + 3 \, \sqrt{d x} d^{2}\right )} e^{\left (-f x\right )}}{f^{2}}\right )}}{d}}{12 \, 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]

1/12*(8*(d*x)^(3/2)*cosh(f*x) + f*(3*sqrt(pi)*d^2*erf(sqrt(d*x)*sqrt(f/d))/(f^2*sqrt(f/d)) - 3*sqrt(pi)*d^2*er
f(sqrt(d*x)*sqrt(-f/d))/(f^2*sqrt(-f/d)) - 2*(2*(d*x)^(3/2)*d*f - 3*sqrt(d*x)*d^2)*e^(f*x)/f^2 - 2*(2*(d*x)^(3
/2)*d*f + 3*sqrt(d*x)*d^2)*e^(-f*x)/f^2)/d)/d

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Fricas [B]  time = 1.84988, size = 355, normalized size = 3.86 \begin{align*} \frac{\sqrt{\pi }{\left (d \cosh \left (f x\right ) + d \sinh \left (f x\right )\right )} \sqrt{\frac{f}{d}} \operatorname{erf}\left (\sqrt{d x} \sqrt{\frac{f}{d}}\right ) + \sqrt{\pi }{\left (d \cosh \left (f x\right ) + d \sinh \left (f x\right )\right )} \sqrt{-\frac{f}{d}} \operatorname{erf}\left (\sqrt{d x} \sqrt{-\frac{f}{d}}\right ) + 2 \,{\left (f \cosh \left (f x\right )^{2} + 2 \, f \cosh \left (f x\right ) \sinh \left (f x\right ) + f \sinh \left (f x\right )^{2} - f\right )} \sqrt{d x}}{4 \,{\left (f^{2} \cosh \left (f x\right ) + f^{2} \sinh \left (f x\right )\right )}} \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]

1/4*(sqrt(pi)*(d*cosh(f*x) + d*sinh(f*x))*sqrt(f/d)*erf(sqrt(d*x)*sqrt(f/d)) + sqrt(pi)*(d*cosh(f*x) + d*sinh(
f*x))*sqrt(-f/d)*erf(sqrt(d*x)*sqrt(-f/d)) + 2*(f*cosh(f*x)^2 + 2*f*cosh(f*x)*sinh(f*x) + f*sinh(f*x)^2 - f)*s
qrt(d*x))/(f^2*cosh(f*x) + f^2*sinh(f*x))

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Sympy [C]  time = 3.5449, size = 100, normalized size = 1.09 \begin{align*} \frac{3 \sqrt{d} \sqrt{x} \sinh{\left (f x \right )} \Gamma \left (\frac{3}{4}\right )}{4 f \Gamma \left (\frac{7}{4}\right )} - \frac{3 \sqrt{2} \sqrt{\pi } \sqrt{d} 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 )}{8 f^{\frac{3}{2}} \Gamma \left (\frac{7}{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]

3*sqrt(d)*sqrt(x)*sinh(f*x)*gamma(3/4)/(4*f*gamma(7/4)) - 3*sqrt(2)*sqrt(pi)*sqrt(d)*exp(-3*I*pi/4)*fresnels(s
qrt(2)*sqrt(f)*sqrt(x)*exp(I*pi/4)/sqrt(pi))*gamma(3/4)/(8*f**(3/2)*gamma(7/4))

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Giac [A]  time = 1.21625, size = 138, normalized size = 1.5 \begin{align*} -\frac{\frac{\sqrt{\pi } d^{2} \operatorname{erf}\left (-\frac{\sqrt{d f} \sqrt{d x}}{d}\right )}{\sqrt{d f} f} - \frac{\sqrt{\pi } d^{2} \operatorname{erf}\left (-\frac{\sqrt{-d f} \sqrt{d x}}{d}\right )}{\sqrt{-d f} f} - \frac{2 \, \sqrt{d x} d e^{\left (f x\right )}}{f} + \frac{2 \, \sqrt{d x} d e^{\left (-f x\right )}}{f}}{4 \, 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]

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