3.67 \(\int \frac{\cosh (f x)}{(d x)^{5/2}} \, dx\)

Optimal. Leaf size=114 \[ \frac{2 \sqrt{\pi } f^{3/2} \text{Erf}\left (\frac{\sqrt{f} \sqrt{d x}}{\sqrt{d}}\right )}{3 d^{5/2}}+\frac{2 \sqrt{\pi } f^{3/2} \text{Erfi}\left (\frac{\sqrt{f} \sqrt{d x}}{\sqrt{d}}\right )}{3 d^{5/2}}-\frac{4 f \sinh (f x)}{3 d^2 \sqrt{d x}}-\frac{2 \cosh (f x)}{3 d (d x)^{3/2}} \]

[Out]

(-2*Cosh[f*x])/(3*d*(d*x)^(3/2)) + (2*f^(3/2)*Sqrt[Pi]*Erf[(Sqrt[f]*Sqrt[d*x])/Sqrt[d]])/(3*d^(5/2)) + (2*f^(3
/2)*Sqrt[Pi]*Erfi[(Sqrt[f]*Sqrt[d*x])/Sqrt[d]])/(3*d^(5/2)) - (4*f*Sinh[f*x])/(3*d^2*Sqrt[d*x])

________________________________________________________________________________________

Rubi [A]  time = 0.14846, antiderivative size = 114, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {3297, 3307, 2180, 2204, 2205} \[ \frac{2 \sqrt{\pi } f^{3/2} \text{Erf}\left (\frac{\sqrt{f} \sqrt{d x}}{\sqrt{d}}\right )}{3 d^{5/2}}+\frac{2 \sqrt{\pi } f^{3/2} \text{Erfi}\left (\frac{\sqrt{f} \sqrt{d x}}{\sqrt{d}}\right )}{3 d^{5/2}}-\frac{4 f \sinh (f x)}{3 d^2 \sqrt{d x}}-\frac{2 \cosh (f x)}{3 d (d x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[f*x]/(d*x)^(5/2),x]

[Out]

(-2*Cosh[f*x])/(3*d*(d*x)^(3/2)) + (2*f^(3/2)*Sqrt[Pi]*Erf[(Sqrt[f]*Sqrt[d*x])/Sqrt[d]])/(3*d^(5/2)) + (2*f^(3
/2)*Sqrt[Pi]*Erfi[(Sqrt[f]*Sqrt[d*x])/Sqrt[d]])/(3*d^(5/2)) - (4*f*Sinh[f*x])/(3*d^2*Sqrt[d*x])

Rule 3297

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

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)}{(d x)^{5/2}} \, dx &=-\frac{2 \cosh (f x)}{3 d (d x)^{3/2}}+\frac{(2 f) \int \frac{\sinh (f x)}{(d x)^{3/2}} \, dx}{3 d}\\ &=-\frac{2 \cosh (f x)}{3 d (d x)^{3/2}}-\frac{4 f \sinh (f x)}{3 d^2 \sqrt{d x}}+\frac{\left (4 f^2\right ) \int \frac{\cosh (f x)}{\sqrt{d x}} \, dx}{3 d^2}\\ &=-\frac{2 \cosh (f x)}{3 d (d x)^{3/2}}-\frac{4 f \sinh (f x)}{3 d^2 \sqrt{d x}}+\frac{\left (2 f^2\right ) \int \frac{e^{-f x}}{\sqrt{d x}} \, dx}{3 d^2}+\frac{\left (2 f^2\right ) \int \frac{e^{f x}}{\sqrt{d x}} \, dx}{3 d^2}\\ &=-\frac{2 \cosh (f x)}{3 d (d x)^{3/2}}-\frac{4 f \sinh (f x)}{3 d^2 \sqrt{d x}}+\frac{\left (4 f^2\right ) \operatorname{Subst}\left (\int e^{-\frac{f x^2}{d}} \, dx,x,\sqrt{d x}\right )}{3 d^3}+\frac{\left (4 f^2\right ) \operatorname{Subst}\left (\int e^{\frac{f x^2}{d}} \, dx,x,\sqrt{d x}\right )}{3 d^3}\\ &=-\frac{2 \cosh (f x)}{3 d (d x)^{3/2}}+\frac{2 f^{3/2} \sqrt{\pi } \text{erf}\left (\frac{\sqrt{f} \sqrt{d x}}{\sqrt{d}}\right )}{3 d^{5/2}}+\frac{2 f^{3/2} \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{f} \sqrt{d x}}{\sqrt{d}}\right )}{3 d^{5/2}}-\frac{4 f \sinh (f x)}{3 d^2 \sqrt{d x}}\\ \end{align*}

Mathematica [A]  time = 0.0930811, size = 78, normalized size = 0.68 \[ \frac{x \left (-4 (-f x)^{3/2} \text{Gamma}\left (\frac{1}{2},-f x\right )+e^{-f x} \left (-4 e^{f x} (f x)^{3/2} \text{Gamma}\left (\frac{1}{2},f x\right )+4 f x-2\right )-2 e^{f x} (2 f x+1)\right )}{6 (d x)^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[f*x]/(d*x)^(5/2),x]

[Out]

(x*(-2*E^(f*x)*(1 + 2*f*x) - 4*(-(f*x))^(3/2)*Gamma[1/2, -(f*x)] + (-2 + 4*f*x - 4*E^(f*x)*(f*x)^(3/2)*Gamma[1
/2, f*x])/E^(f*x)))/(6*(d*x)^(5/2))

________________________________________________________________________________________

Maple [C]  time = 0.027, size = 126, normalized size = 1.1 \begin{align*}{\frac{-{\frac{i}{8}}\sqrt{\pi }\sqrt{2}}{f}{x}^{{\frac{5}{2}}} \left ( if \right ) ^{{\frac{5}{2}}} \left ( -{\frac{8\,\sqrt{2}{{\rm e}^{-fx}}}{3\,\sqrt{\pi }} \left ( -fx+{\frac{1}{2}} \right ){x}^{-{\frac{3}{2}}} \left ( if \right ) ^{-{\frac{3}{2}}}}-{\frac{8\,\sqrt{2}{{\rm e}^{fx}}}{3\,\sqrt{\pi }} \left ( fx+{\frac{1}{2}} \right ){x}^{-{\frac{3}{2}}} \left ( if \right ) ^{-{\frac{3}{2}}}}+{\frac{8\,\sqrt{2}}{3}{f}^{{\frac{3}{2}}}{\it Erf} \left ( \sqrt{x}\sqrt{f} \right ) \left ( if \right ) ^{-{\frac{3}{2}}}}+{\frac{8\,\sqrt{2}}{3}{f}^{{\frac{3}{2}}}{\it erfi} \left ( \sqrt{x}\sqrt{f} \right ) \left ( if \right ) ^{-{\frac{3}{2}}}} \right ) \left ( dx \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(f*x)/(d*x)^(5/2),x)

[Out]

-1/8*I*Pi^(1/2)/(d*x)^(5/2)*x^(5/2)*2^(1/2)*(I*f)^(5/2)/f*(-8/3/Pi^(1/2)/x^(3/2)*2^(1/2)/(I*f)^(3/2)*(-f*x+1/2
)*exp(-f*x)-8/3/Pi^(1/2)/x^(3/2)*2^(1/2)/(I*f)^(3/2)*(f*x+1/2)*exp(f*x)+8/3/(I*f)^(3/2)*2^(1/2)*f^(3/2)*erf(x^
(1/2)*f^(1/2))+8/3/(I*f)^(3/2)*2^(1/2)*f^(3/2)*erfi(x^(1/2)*f^(1/2)))

________________________________________________________________________________________

Maxima [A]  time = 1.26112, size = 78, normalized size = 0.68 \begin{align*} \frac{\frac{f{\left (\frac{\sqrt{f x} \Gamma \left (-\frac{1}{2}, f x\right )}{\sqrt{d x}} - \frac{\sqrt{-f x} \Gamma \left (-\frac{1}{2}, -f x\right )}{\sqrt{d x}}\right )}}{d} - \frac{2 \, \cosh \left (f x\right )}{\left (d x\right )^{\frac{3}{2}}}}{3 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(f*x)/(d*x)^(5/2),x, algorithm="maxima")

[Out]

1/3*(f*(sqrt(f*x)*gamma(-1/2, f*x)/sqrt(d*x) - sqrt(-f*x)*gamma(-1/2, -f*x)/sqrt(d*x))/d - 2*cosh(f*x)/(d*x)^(
3/2))/d

________________________________________________________________________________________

Fricas [B]  time = 1.85452, size = 452, normalized size = 3.96 \begin{align*} \frac{2 \, \sqrt{\pi }{\left (d f x^{2} \cosh \left (f x\right ) + d f x^{2} \sinh \left (f x\right )\right )} \sqrt{\frac{f}{d}} \operatorname{erf}\left (\sqrt{d x} \sqrt{\frac{f}{d}}\right ) - 2 \, \sqrt{\pi }{\left (d f x^{2} \cosh \left (f x\right ) + d f x^{2} \sinh \left (f x\right )\right )} \sqrt{-\frac{f}{d}} \operatorname{erf}\left (\sqrt{d x} \sqrt{-\frac{f}{d}}\right ) -{\left ({\left (2 \, f x + 1\right )} \cosh \left (f x\right )^{2} + 2 \,{\left (2 \, f x + 1\right )} \cosh \left (f x\right ) \sinh \left (f x\right ) +{\left (2 \, f x + 1\right )} \sinh \left (f x\right )^{2} - 2 \, f x + 1\right )} \sqrt{d x}}{3 \,{\left (d^{3} x^{2} \cosh \left (f x\right ) + d^{3} x^{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)^(5/2),x, algorithm="fricas")

[Out]

1/3*(2*sqrt(pi)*(d*f*x^2*cosh(f*x) + d*f*x^2*sinh(f*x))*sqrt(f/d)*erf(sqrt(d*x)*sqrt(f/d)) - 2*sqrt(pi)*(d*f*x
^2*cosh(f*x) + d*f*x^2*sinh(f*x))*sqrt(-f/d)*erf(sqrt(d*x)*sqrt(-f/d)) - ((2*f*x + 1)*cosh(f*x)^2 + 2*(2*f*x +
1)*cosh(f*x)*sinh(f*x) + (2*f*x + 1)*sinh(f*x)^2 - 2*f*x + 1)*sqrt(d*x))/(d^3*x^2*cosh(f*x) + d^3*x^2*sinh(f*
x))

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(f*x)/(d*x)**(5/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cosh \left (f x\right )}{\left (d x\right )^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(f*x)/(d*x)^(5/2),x, algorithm="giac")

[Out]

integrate(cosh(f*x)/(d*x)^(5/2), x)