\(\int \frac {\cosh (f x)}{(d x)^{3/2}} \, dx\) [66]

Optimal result

Integrand size = 12, antiderivative size = 88 \[ \int \frac {\cosh (f x)}{(d x)^{3/2}} \, dx=-\frac {2 \cosh (f x)}{d \sqrt {d x}}-\frac {\sqrt {f} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {f} \sqrt {d x}}{\sqrt {d}}\right )}{d^{3/2}}+\frac {\sqrt {f} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {f} \sqrt {d x}}{\sqrt {d}}\right )}{d^{3/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {3378, 3389, 2211, 2235, 2236} \[ \int \frac {\cosh (f x)}{(d x)^{3/2}} \, dx=-\frac {\sqrt {\pi } \sqrt {f} \text {erf}\left (\frac {\sqrt {f} \sqrt {d x}}{\sqrt {d}}\right )}{d^{3/2}}+\frac {\sqrt {\pi } \sqrt {f} \text {erfi}\left (\frac {\sqrt {f} \sqrt {d x}}{\sqrt {d}}\right )}{d^{3/2}}-\frac {2 \cosh (f x)}{d \sqrt {d x}} \]

[In]

Int[Cosh[f*x]/(d*x)^(3/2),x]

[Out]

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

Rule 2211

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] &&  !TrueQ[$UseGamma]

Rule 2235

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 2236

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 3378

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 3389

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.76 \[ \int \frac {\cosh (f x)}{(d x)^{3/2}} \, dx=\frac {e^{-f x} x \left (-1-e^{2 f x}+e^{f x} \sqrt {-f x} \Gamma \left (\frac {1}{2},-f x\right )+e^{f x} \sqrt {f x} \Gamma \left (\frac {1}{2},f x\right )\right )}{(d x)^{3/2}} \]

[In]

Integrate[Cosh[f*x]/(d*x)^(3/2),x]

[Out]

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

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.03 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.31

method	result	size
meijerg	\(-\frac {i \sqrt {\pi }\, x^{\frac {3}{2}} \sqrt {2}\, \left (i f \right )^{\frac {3}{2}} \left (-\frac {2 \sqrt {2}\, {\mathrm e}^{f x}}{\sqrt {\pi }\, \sqrt {x}\, \sqrt {i f}}-\frac {2 \sqrt {2}\, {\mathrm e}^{-f x}}{\sqrt {\pi }\, \sqrt {x}\, \sqrt {i f}}-\frac {2 \sqrt {2}\, \sqrt {f}\, \operatorname {erf}\left (\sqrt {x}\, \sqrt {f}\right )}{\sqrt {i f}}+\frac {2 \sqrt {2}\, \sqrt {f}\, \operatorname {erfi}\left (\sqrt {x}\, \sqrt {f}\right )}{\sqrt {i f}}\right )}{4 \left (d x \right )^{\frac {3}{2}} f}\)	\(115\)

[In]

int(cosh(f*x)/(d*x)^(3/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 136 vs. \(2 (62) = 124\).

Time = 0.27 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.55 \[ \int \frac {\cosh (f x)}{(d x)^{3/2}} \, dx=-\frac {\sqrt {\pi } {\left (d x \cosh \left (f x\right ) + d x \sinh \left (f x\right )\right )} \sqrt {\frac {f}{d}} \operatorname {erf}\left (\sqrt {d x} \sqrt {\frac {f}{d}}\right ) + \sqrt {\pi } {\left (d x \cosh \left (f x\right ) + d x \sinh \left (f x\right )\right )} \sqrt {-\frac {f}{d}} \operatorname {erf}\left (\sqrt {d x} \sqrt {-\frac {f}{d}}\right ) + \sqrt {d x} {\left (\cosh \left (f x\right )^{2} + 2 \, \cosh \left (f x\right ) \sinh \left (f x\right ) + \sinh \left (f x\right )^{2} + 1\right )}}{d^{2} x \cosh \left (f x\right ) + d^{2} x \sinh \left (f x\right )} \]

[In]

integrate(cosh(f*x)/(d*x)^(3/2),x, algorithm="fricas")

[Out]

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.42 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.12 \[ \int \frac {\cosh (f x)}{(d x)^{3/2}} \, dx=- \frac {\sqrt {2} \sqrt {\pi } \sqrt {f} e^{- \frac {3 i \pi }{4}} S\left (\frac {\sqrt {2} \sqrt {f} \sqrt {x} e^{\frac {i \pi }{4}}}{\sqrt {\pi }}\right ) \Gamma \left (- \frac {1}{4}\right )}{2 d^{\frac {3}{2}} \Gamma \left (\frac {3}{4}\right )} + \frac {\cosh {\left (f x \right )} \Gamma \left (- \frac {1}{4}\right )}{2 d^{\frac {3}{2}} \sqrt {x} \Gamma \left (\frac {3}{4}\right )} \]

[In]

integrate(cosh(f*x)/(d*x)**(3/2),x)

[Out]

-sqrt(2)*sqrt(pi)*sqrt(f)*exp(-3*I*pi/4)*fresnels(sqrt(2)*sqrt(f)*sqrt(x)*exp(I*pi/4)/sqrt(pi))*gamma(-1/4)/(2
*d**(3/2)*gamma(3/4)) + cosh(f*x)*gamma(-1/4)/(2*d**(3/2)*sqrt(x)*gamma(3/4))

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.86 \[ \int \frac {\cosh (f x)}{(d x)^{3/2}} \, dx=-\frac {\frac {f {\left (\frac {\sqrt {\pi } \operatorname {erf}\left (\sqrt {d x} \sqrt {\frac {f}{d}}\right )}{\sqrt {\frac {f}{d}}} - \frac {\sqrt {\pi } \operatorname {erf}\left (\sqrt {d x} \sqrt {-\frac {f}{d}}\right )}{\sqrt {-\frac {f}{d}}}\right )}}{d} + \frac {2 \, \cosh \left (f x\right )}{\sqrt {d x}}}{d} \]

[In]

integrate(cosh(f*x)/(d*x)^(3/2),x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int \frac {\cosh (f x)}{(d x)^{3/2}} \, dx=\int { \frac {\cosh \left (f x\right )}{\left (d x\right )^{\frac {3}{2}}} \,d x } \]

[In]

integrate(cosh(f*x)/(d*x)^(3/2),x, algorithm="giac")

[Out]

integrate(cosh(f*x)/(d*x)^(3/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\cosh (f x)}{(d x)^{3/2}} \, dx=\int \frac {\mathrm {cosh}\left (f\,x\right )}{{\left (d\,x\right )}^{3/2}} \,d x \]

[In]

int(cosh(f*x)/(d*x)^(3/2),x)

[Out]

int(cosh(f*x)/(d*x)^(3/2), x)