\(\int \sqrt {d x} \cosh (f x) \, dx\) [64]

Optimal result

Integrand size = 12, antiderivative size = 92 \[ \int \sqrt {d x} \cosh (f x) \, dx=\frac {\sqrt {d} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {f} \sqrt {d x}}{\sqrt {d}}\right )}{4 f^{3/2}}-\frac {\sqrt {d} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {f} \sqrt {d x}}{\sqrt {d}}\right )}{4 f^{3/2}}+\frac {\sqrt {d x} \sinh (f x)}{f} \]

[Out]

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

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 92, 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 = {3377, 3389, 2211, 2235, 2236} \[ \int \sqrt {d x} \cosh (f x) \, dx=\frac {\sqrt {\pi } \sqrt {d} \text {erf}\left (\frac {\sqrt {f} \sqrt {d x}}{\sqrt {d}}\right )}{4 f^{3/2}}-\frac {\sqrt {\pi } \sqrt {d} \text {erfi}\left (\frac {\sqrt {f} \sqrt {d x}}{\sqrt {d}}\right )}{4 f^{3/2}}+\frac {\sqrt {d x} \sinh (f x)}{f} \]

[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 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 3377

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 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 {\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 {\text {Subst}\left (\int e^{-\frac {f x^2}{d}} \, dx,x,\sqrt {d x}\right )}{2 f}-\frac {\text {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] (verified)

Time = 0.01 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.52 \[ \int \sqrt {d x} \cosh (f x) \, dx=-\frac {d \left (\sqrt {-f x} \Gamma \left (\frac {3}{2},-f x\right )+\sqrt {f x} \Gamma \left (\frac {3}{2},f x\right )\right )}{2 f^2 \sqrt {d x}} \]

[In]

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

[Out]

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

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.04 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.32

method	result	size
meijerg	\(-\frac {i \sqrt {\pi }\, \sqrt {d x}\, \sqrt {2}\, \left (\frac {\sqrt {x}\, \sqrt {2}\, \left (i f \right )^{\frac {3}{2}} {\mathrm e}^{f x}}{4 \sqrt {\pi }\, f}-\frac {\sqrt {x}\, \sqrt {2}\, \left (i f \right )^{\frac {3}{2}} {\mathrm e}^{-f x}}{4 \sqrt {\pi }\, f}+\frac {\left (i f \right )^{\frac {3}{2}} \sqrt {2}\, \operatorname {erf}\left (\sqrt {x}\, \sqrt {f}\right )}{8 f^{\frac {3}{2}}}-\frac {\left (i f \right )^{\frac {3}{2}} \sqrt {2}\, \operatorname {erfi}\left (\sqrt {x}\, \sqrt {f}\right )}{8 f^{\frac {3}{2}}}\right )}{\sqrt {x}\, \sqrt {i f}\, f}\)	\(121\)

[In]

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

[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)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 138 vs. \(2 (62) = 124\).

Time = 0.28 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.50 \[ \int \sqrt {d x} \cosh (f x) \, dx=\frac {\sqrt {\pi } {\left (d \cosh \left (f x\right ) + d \sinh \left (f x\right )\right )} \sqrt {\frac {f}{d}} \operatorname {erf}\left (\sqrt {d x} \sqrt {\frac {f}{d}}\right ) + \sqrt {\pi } {\left (d \cosh \left (f x\right ) + d \sinh \left (f x\right )\right )} \sqrt {-\frac {f}{d}} \operatorname {erf}\left (\sqrt {d x} \sqrt {-\frac {f}{d}}\right ) + 2 \, {\left (f \cosh \left (f x\right )^{2} + 2 \, f \cosh \left (f x\right ) \sinh \left (f x\right ) + f \sinh \left (f x\right )^{2} - f\right )} \sqrt {d x}}{4 \, {\left (f^{2} \cosh \left (f x\right ) + f^{2} \sinh \left (f x\right )\right )}} \]

[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))

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.83 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.09 \[ \int \sqrt {d x} \cosh (f x) \, dx=\frac {3 \sqrt {d} \sqrt {x} \sinh {\left (f x \right )} \Gamma \left (\frac {3}{4}\right )}{4 f \Gamma \left (\frac {7}{4}\right )} - \frac {3 \sqrt {2} \sqrt {\pi } \sqrt {d} e^{- \frac {3 i \pi }{4}} S\left (\frac {\sqrt {2} \sqrt {f} \sqrt {x} e^{\frac {i \pi }{4}}}{\sqrt {\pi }}\right ) \Gamma \left (\frac {3}{4}\right )}{8 f^{\frac {3}{2}} \Gamma \left (\frac {7}{4}\right )} \]

[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))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 148 vs. \(2 (62) = 124\).

Time = 0.19 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.61 \[ \int \sqrt {d x} \cosh (f x) \, dx=\frac {8 \, \left (d x\right )^{\frac {3}{2}} \cosh \left (f x\right ) + \frac {f {\left (\frac {3 \, \sqrt {\pi } d^{2} \operatorname {erf}\left (\sqrt {d x} \sqrt {\frac {f}{d}}\right )}{f^{2} \sqrt {\frac {f}{d}}} - \frac {3 \, \sqrt {\pi } d^{2} \operatorname {erf}\left (\sqrt {d x} \sqrt {-\frac {f}{d}}\right )}{f^{2} \sqrt {-\frac {f}{d}}} - \frac {2 \, {\left (2 \, \left (d x\right )^{\frac {3}{2}} d f - 3 \, \sqrt {d x} d^{2}\right )} e^{\left (f x\right )}}{f^{2}} - \frac {2 \, {\left (2 \, \left (d x\right )^{\frac {3}{2}} d f + 3 \, \sqrt {d x} d^{2}\right )} e^{\left (-f x\right )}}{f^{2}}\right )}}{d}}{12 \, d} \]

[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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.17 \[ \int \sqrt {d x} \cosh (f x) \, dx=-\frac {\frac {\sqrt {\pi } d^{2} \operatorname {erf}\left (-\frac {\sqrt {d f} \sqrt {d x}}{d}\right )}{\sqrt {d f} f} + \frac {2 \, \sqrt {d x} d e^{\left (-f x\right )}}{f}}{4 \, d} + \frac {\frac {\sqrt {\pi } d^{2} \operatorname {erf}\left (-\frac {\sqrt {-d f} \sqrt {d x}}{d}\right )}{\sqrt {-d f} f} + \frac {2 \, \sqrt {d x} d e^{\left (f x\right )}}{f}}{4 \, d} \]

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

Mupad [F(-1)]

Timed out. \[ \int \sqrt {d x} \cosh (f x) \, dx=\int \mathrm {cosh}\left (f\,x\right )\,\sqrt {d\,x} \,d x \]

[In]

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

[Out]

int(cosh(f*x)*(d*x)^(1/2), x)