Integrand size = 12, antiderivative size = 92 \[ \int \sqrt {d x} \cosh (f x) \, dx=\frac {\sqrt {d} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {f} \sqrt {d x}}{\sqrt {d}}\right )}{4 f^{3/2}}-\frac {\sqrt {d} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {f} \sqrt {d x}}{\sqrt {d}}\right )}{4 f^{3/2}}+\frac {\sqrt {d x} \sinh (f x)}{f} \] Output:
1/4*d^(1/2)*Pi^(1/2)*erf(f^(1/2)*(d*x)^(1/2)/d^(1/2))/f^(3/2)-1/4*d^(1/2)* Pi^(1/2)*erfi(f^(1/2)*(d*x)^(1/2)/d^(1/2))/f^(3/2)+(d*x)^(1/2)*sinh(f*x)/f
Time = 0.01 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.52 \[ \int \sqrt {d x} \cosh (f x) \, dx=-\frac {d \left (\sqrt {-f x} \Gamma \left (\frac {3}{2},-f x\right )+\sqrt {f x} \Gamma \left (\frac {3}{2},f x\right )\right )}{2 f^2 \sqrt {d x}} \] Input:
Integrate[Sqrt[d*x]*Cosh[f*x],x]
Output:
-1/2*(d*(Sqrt[-(f*x)]*Gamma[3/2, -(f*x)] + Sqrt[f*x]*Gamma[3/2, f*x]))/(f^ 2*Sqrt[d*x])
Result contains complex when optimal does not.
Time = 0.45 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.16, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {3042, 3777, 26, 3042, 26, 3789, 2611, 2633, 2634}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \sqrt {d x} \cosh (f x) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sqrt {d x} \sin \left (\frac {\pi }{2}+i f x\right )dx\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle \frac {\sqrt {d x} \sinh (f x)}{f}-\frac {i d \int -\frac {i \sinh (f x)}{\sqrt {d x}}dx}{2 f}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \frac {\sqrt {d x} \sinh (f x)}{f}-\frac {d \int \frac {\sinh (f x)}{\sqrt {d x}}dx}{2 f}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sqrt {d x} \sinh (f x)}{f}-\frac {d \int -\frac {i \sin (i f x)}{\sqrt {d x}}dx}{2 f}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \frac {\sqrt {d x} \sinh (f x)}{f}+\frac {i d \int \frac {\sin (i f x)}{\sqrt {d x}}dx}{2 f}\) |
\(\Big \downarrow \) 3789 |
\(\displaystyle \frac {\sqrt {d x} \sinh (f x)}{f}+\frac {i d \left (\frac {1}{2} i \int \frac {e^{f x}}{\sqrt {d x}}dx-\frac {1}{2} i \int \frac {e^{-f x}}{\sqrt {d x}}dx\right )}{2 f}\) |
\(\Big \downarrow \) 2611 |
\(\displaystyle \frac {\sqrt {d x} \sinh (f x)}{f}+\frac {i d \left (\frac {i \int e^{f x}d\sqrt {d x}}{d}-\frac {i \int e^{-f x}d\sqrt {d x}}{d}\right )}{2 f}\) |
\(\Big \downarrow \) 2633 |
\(\displaystyle \frac {\sqrt {d x} \sinh (f x)}{f}+\frac {i d \left (\frac {i \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {f} \sqrt {d x}}{\sqrt {d}}\right )}{2 \sqrt {d} \sqrt {f}}-\frac {i \int e^{-f x}d\sqrt {d x}}{d}\right )}{2 f}\) |
\(\Big \downarrow \) 2634 |
\(\displaystyle \frac {\sqrt {d x} \sinh (f x)}{f}+\frac {i d \left (\frac {i \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {f} \sqrt {d x}}{\sqrt {d}}\right )}{2 \sqrt {d} \sqrt {f}}-\frac {i \sqrt {\pi } \text {erf}\left (\frac {\sqrt {f} \sqrt {d x}}{\sqrt {d}}\right )}{2 \sqrt {d} \sqrt {f}}\right )}{2 f}\) |
Input:
Int[Sqrt[d*x]*Cosh[f*x],x]
Output:
((I/2)*d*(((-1/2*I)*Sqrt[Pi]*Erf[(Sqrt[f]*Sqrt[d*x])/Sqrt[d]])/(Sqrt[d]*Sq rt[f]) + ((I/2)*Sqrt[Pi]*Erfi[(Sqrt[f]*Sqrt[d*x])/Sqrt[d]])/(Sqrt[d]*Sqrt[ f])))/f + (Sqrt[d*x]*Sinh[f*x])/f
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] : > Simp[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]
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]
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] /; Fr eeQ[{F, a, b, c, d}, x] && NegQ[b]
Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[( -(c + d*x)^m)*(Cos[e + f*x]/f), x] + Simp[d*(m/f) Int[(c + d*x)^(m - 1)*C os[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[I /2 Int[(c + d*x)^m/E^(I*(e + f*x)), x], x] - Simp[I/2 Int[(c + d*x)^m*E ^(I*(e + f*x)), x], x] /; FreeQ[{c, d, e, f, m}, x]
Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.32
method | result | size |
meijerg | \(-\frac {i \sqrt {d x}\, \sqrt {2}\, \sqrt {\pi }\, \left (\frac {\sqrt {x}\, \sqrt {2}\, \left (i f \right )^{\frac {3}{2}} {\mathrm e}^{f x}}{4 \sqrt {\pi }\, f}-\frac {\sqrt {x}\, \sqrt {2}\, \left (i f \right )^{\frac {3}{2}} {\mathrm e}^{-f x}}{4 \sqrt {\pi }\, f}+\frac {\left (i f \right )^{\frac {3}{2}} \sqrt {2}\, \operatorname {erf}\left (\sqrt {x}\, \sqrt {f}\right )}{8 f^{\frac {3}{2}}}-\frac {\left (i f \right )^{\frac {3}{2}} \sqrt {2}\, \operatorname {erfi}\left (\sqrt {x}\, \sqrt {f}\right )}{8 f^{\frac {3}{2}}}\right )}{\sqrt {x}\, \sqrt {i f}\, f}\) | \(121\) |
Input:
int((d*x)^(1/2)*cosh(f*x),x,method=_RETURNVERBOSE)
Output:
-I*(d*x)^(1/2)/x^(1/2)*2^(1/2)/(I*f)^(1/2)*Pi^(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)))
Leaf count of result is larger than twice the leaf count of optimal. 138 vs. \(2 (62) = 124\).
Time = 0.08 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.50 \[ \int \sqrt {d x} \cosh (f x) \, dx=\frac {\sqrt {\pi } {\left (d \cosh \left (f x\right ) + d \sinh \left (f x\right )\right )} \sqrt {\frac {f}{d}} \operatorname {erf}\left (\sqrt {d x} \sqrt {\frac {f}{d}}\right ) + \sqrt {\pi } {\left (d \cosh \left (f x\right ) + d \sinh \left (f x\right )\right )} \sqrt {-\frac {f}{d}} \operatorname {erf}\left (\sqrt {d x} \sqrt {-\frac {f}{d}}\right ) + 2 \, {\left (f \cosh \left (f x\right )^{2} + 2 \, f \cosh \left (f x\right ) \sinh \left (f x\right ) + f \sinh \left (f x\right )^{2} - f\right )} \sqrt {d x}}{4 \, {\left (f^{2} \cosh \left (f x\right ) + f^{2} \sinh \left (f x\right )\right )}} \] Input:
integrate((d*x)^(1/2)*cosh(f*x),x, algorithm="fricas")
Output:
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)*sqr t(d*x))/(f^2*cosh(f*x) + f^2*sinh(f*x))
Result contains complex when optimal does not.
Time = 0.76 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.09 \[ \int \sqrt {d x} \cosh (f x) \, dx=\frac {3 \sqrt {d} \sqrt {x} \sinh {\left (f x \right )} \Gamma \left (\frac {3}{4}\right )}{4 f \Gamma \left (\frac {7}{4}\right )} - \frac {3 \sqrt {2} \sqrt {\pi } \sqrt {d} e^{- \frac {3 i \pi }{4}} S\left (\frac {\sqrt {2} \sqrt {f} \sqrt {x} e^{\frac {i \pi }{4}}}{\sqrt {\pi }}\right ) \Gamma \left (\frac {3}{4}\right )}{8 f^{\frac {3}{2}} \Gamma \left (\frac {7}{4}\right )} \] Input:
integrate((d*x)**(1/2)*cosh(f*x),x)
Output:
3*sqrt(d)*sqrt(x)*sinh(f*x)*gamma(3/4)/(4*f*gamma(7/4)) - 3*sqrt(2)*sqrt(p i)*sqrt(d)*exp(-3*I*pi/4)*fresnels(sqrt(2)*sqrt(f)*sqrt(x)*exp(I*pi/4)/sqr t(pi))*gamma(3/4)/(8*f**(3/2)*gamma(7/4))
Leaf count of result is larger than twice the leaf count of optimal. 148 vs. \(2 (62) = 124\).
Time = 0.04 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.61 \[ \int \sqrt {d x} \cosh (f x) \, dx=\frac {8 \, \left (d x\right )^{\frac {3}{2}} \cosh \left (f x\right ) + \frac {f {\left (\frac {3 \, \sqrt {\pi } d^{2} \operatorname {erf}\left (\sqrt {d x} \sqrt {\frac {f}{d}}\right )}{f^{2} \sqrt {\frac {f}{d}}} - \frac {3 \, \sqrt {\pi } d^{2} \operatorname {erf}\left (\sqrt {d x} \sqrt {-\frac {f}{d}}\right )}{f^{2} \sqrt {-\frac {f}{d}}} - \frac {2 \, {\left (2 \, \left (d x\right )^{\frac {3}{2}} d f - 3 \, \sqrt {d x} d^{2}\right )} e^{\left (f x\right )}}{f^{2}} - \frac {2 \, {\left (2 \, \left (d x\right )^{\frac {3}{2}} d f + 3 \, \sqrt {d x} d^{2}\right )} e^{\left (-f x\right )}}{f^{2}}\right )}}{d}}{12 \, d} \] Input:
integrate((d*x)^(1/2)*cosh(f*x),x, algorithm="maxima")
Output:
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*erf(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
Time = 0.13 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.17 \[ \int \sqrt {d x} \cosh (f x) \, dx=-\frac {\frac {\sqrt {\pi } d^{2} \operatorname {erf}\left (-\frac {\sqrt {d f} \sqrt {d x}}{d}\right )}{\sqrt {d f} f} + \frac {2 \, \sqrt {d x} d e^{\left (-f x\right )}}{f}}{4 \, d} + \frac {\frac {\sqrt {\pi } d^{2} \operatorname {erf}\left (-\frac {\sqrt {-d f} \sqrt {d x}}{d}\right )}{\sqrt {-d f} f} + \frac {2 \, \sqrt {d x} d e^{\left (f x\right )}}{f}}{4 \, d} \] Input:
integrate((d*x)^(1/2)*cosh(f*x),x, algorithm="giac")
Output:
-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
Timed out. \[ \int \sqrt {d x} \cosh (f x) \, dx=\int \mathrm {cosh}\left (f\,x\right )\,\sqrt {d\,x} \,d x \] Input:
int(cosh(f*x)*(d*x)^(1/2),x)
Output:
int(cosh(f*x)*(d*x)^(1/2), x)
\[ \int \sqrt {d x} \cosh (f x) \, dx=\frac {\sqrt {d}\, \left (\sqrt {\pi }\, e^{f x} \mathrm {erf}\left (\sqrt {x}\, \sqrt {f}\, i \right ) i +2 \sqrt {x}\, \sqrt {f}\, e^{2 f x}+\sqrt {f}\, e^{f x} \left (\int \frac {\sqrt {x}}{e^{f x} x}d x \right )-2 \sqrt {x}\, \sqrt {f}\right )}{4 \sqrt {f}\, e^{f x} f} \] Input:
int((d*x)^(1/2)*cosh(f*x),x)
Output:
(sqrt(d)*(sqrt(pi)*e**(f*x)*erf(sqrt(x)*sqrt(f)*i)*i + 2*sqrt(x)*sqrt(f)*e **(2*f*x) + sqrt(f)*e**(f*x)*int(sqrt(x)/(e**(f*x)*x),x) - 2*sqrt(x)*sqrt( f)))/(4*sqrt(f)*e**(f*x)*f)