Integrand size = 12, antiderivative size = 88 \[ \int \frac {\cosh (f x)}{(d x)^{3/2}} \, dx=-\frac {2 \cosh (f x)}{d \sqrt {d x}}-\frac {\sqrt {f} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {f} \sqrt {d x}}{\sqrt {d}}\right )}{d^{3/2}}+\frac {\sqrt {f} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {f} \sqrt {d x}}{\sqrt {d}}\right )}{d^{3/2}} \] Output:
-2*cosh(f*x)/d/(d*x)^(1/2)-f^(1/2)*Pi^(1/2)*erf(f^(1/2)*(d*x)^(1/2)/d^(1/2 ))/d^(3/2)+f^(1/2)*Pi^(1/2)*erfi(f^(1/2)*(d*x)^(1/2)/d^(1/2))/d^(3/2)
Time = 0.03 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.76 \[ \int \frac {\cosh (f x)}{(d x)^{3/2}} \, dx=\frac {e^{-f x} x \left (-1-e^{2 f x}+e^{f x} \sqrt {-f x} \Gamma \left (\frac {1}{2},-f x\right )+e^{f x} \sqrt {f x} \Gamma \left (\frac {1}{2},f x\right )\right )}{(d x)^{3/2}} \] Input:
Integrate[Cosh[f*x]/(d*x)^(3/2),x]
Output:
(x*(-1 - E^(2*f*x) + E^(f*x)*Sqrt[-(f*x)]*Gamma[1/2, -(f*x)] + E^(f*x)*Sqr t[f*x]*Gamma[1/2, f*x]))/(E^(f*x)*(d*x)^(3/2))
Result contains complex when optimal does not.
Time = 0.43 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.20, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {3042, 3778, 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.
\(\displaystyle \int \frac {\cosh (f x)}{(d x)^{3/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sin \left (\frac {\pi }{2}+i f x\right )}{(d x)^{3/2}}dx\) |
\(\Big \downarrow \) 3778 |
\(\displaystyle -\frac {2 \cosh (f x)}{d \sqrt {d x}}+\frac {2 i f \int -\frac {i \sinh (f x)}{\sqrt {d x}}dx}{d}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \frac {2 f \int \frac {\sinh (f x)}{\sqrt {d x}}dx}{d}-\frac {2 \cosh (f x)}{d \sqrt {d x}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {2 \cosh (f x)}{d \sqrt {d x}}+\frac {2 f \int -\frac {i \sin (i f x)}{\sqrt {d x}}dx}{d}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -\frac {2 \cosh (f x)}{d \sqrt {d x}}-\frac {2 i f \int \frac {\sin (i f x)}{\sqrt {d x}}dx}{d}\) |
\(\Big \downarrow \) 3789 |
\(\displaystyle -\frac {2 \cosh (f x)}{d \sqrt {d x}}-\frac {2 i f \left (\frac {1}{2} i \int \frac {e^{f x}}{\sqrt {d x}}dx-\frac {1}{2} i \int \frac {e^{-f x}}{\sqrt {d x}}dx\right )}{d}\) |
\(\Big \downarrow \) 2611 |
\(\displaystyle -\frac {2 \cosh (f x)}{d \sqrt {d x}}-\frac {2 i f \left (\frac {i \int e^{f x}d\sqrt {d x}}{d}-\frac {i \int e^{-f x}d\sqrt {d x}}{d}\right )}{d}\) |
\(\Big \downarrow \) 2633 |
\(\displaystyle -\frac {2 \cosh (f x)}{d \sqrt {d x}}-\frac {2 i f \left (\frac {i \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {f} \sqrt {d x}}{\sqrt {d}}\right )}{2 \sqrt {d} \sqrt {f}}-\frac {i \int e^{-f x}d\sqrt {d x}}{d}\right )}{d}\) |
\(\Big \downarrow \) 2634 |
\(\displaystyle -\frac {2 \cosh (f x)}{d \sqrt {d x}}-\frac {2 i f \left (\frac {i \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {f} \sqrt {d x}}{\sqrt {d}}\right )}{2 \sqrt {d} \sqrt {f}}-\frac {i \sqrt {\pi } \text {erf}\left (\frac {\sqrt {f} \sqrt {d x}}{\sqrt {d}}\right )}{2 \sqrt {d} \sqrt {f}}\right )}{d}\) |
Input:
Int[Cosh[f*x]/(d*x)^(3/2),x]
Output:
(-2*Cosh[f*x])/(d*Sqrt[d*x]) - ((2*I)*f*(((-1/2*I)*Sqrt[Pi]*Erf[(Sqrt[f]*S qrt[d*x])/Sqrt[d]])/(Sqrt[d]*Sqrt[f]) + ((I/2)*Sqrt[Pi]*Erfi[(Sqrt[f]*Sqrt [d*x])/Sqrt[d]])/(Sqrt[d]*Sqrt[f])))/d
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 + 1)*(Sin[e + f*x]/(d*(m + 1))), x] - Simp[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]
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 = 0.28 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.31
method | result | size |
meijerg | \(-\frac {i \sqrt {\pi }\, x^{\frac {3}{2}} \sqrt {2}\, \left (i f \right )^{\frac {3}{2}} \left (-\frac {2 \sqrt {2}\, {\mathrm e}^{f x}}{\sqrt {\pi }\, \sqrt {x}\, \sqrt {i f}}-\frac {2 \sqrt {2}\, {\mathrm e}^{-f x}}{\sqrt {\pi }\, \sqrt {x}\, \sqrt {i f}}-\frac {2 \sqrt {2}\, \sqrt {f}\, \operatorname {erf}\left (\sqrt {x}\, \sqrt {f}\right )}{\sqrt {i f}}+\frac {2 \sqrt {2}\, \sqrt {f}\, \operatorname {erfi}\left (\sqrt {x}\, \sqrt {f}\right )}{\sqrt {i f}}\right )}{4 \left (d x \right )^{\frac {3}{2}} f}\) | \(115\) |
Input:
int(cosh(f*x)/(d*x)^(3/2),x,method=_RETURNVERBOSE)
Output:
-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)*e xp(-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)))
Leaf count of result is larger than twice the leaf count of optimal. 136 vs. \(2 (62) = 124\).
Time = 0.11 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.55 \[ \int \frac {\cosh (f x)}{(d x)^{3/2}} \, dx=-\frac {\sqrt {\pi } {\left (d x \cosh \left (f x\right ) + d x \sinh \left (f x\right )\right )} \sqrt {\frac {f}{d}} \operatorname {erf}\left (\sqrt {d x} \sqrt {\frac {f}{d}}\right ) + \sqrt {\pi } {\left (d x \cosh \left (f x\right ) + d x \sinh \left (f x\right )\right )} \sqrt {-\frac {f}{d}} \operatorname {erf}\left (\sqrt {d x} \sqrt {-\frac {f}{d}}\right ) + \sqrt {d x} {\left (\cosh \left (f x\right )^{2} + 2 \, \cosh \left (f x\right ) \sinh \left (f x\right ) + \sinh \left (f x\right )^{2} + 1\right )}}{d^{2} x \cosh \left (f x\right ) + d^{2} x \sinh \left (f x\right )} \] Input:
integrate(cosh(f*x)/(d*x)^(3/2),x, algorithm="fricas")
Output:
-(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)*sq rt(-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))
Result contains complex when optimal does not.
Time = 1.44 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.12 \[ \int \frac {\cosh (f x)}{(d x)^{3/2}} \, dx=- \frac {\sqrt {2} \sqrt {\pi } \sqrt {f} e^{- \frac {3 i \pi }{4}} S\left (\frac {\sqrt {2} \sqrt {f} \sqrt {x} e^{\frac {i \pi }{4}}}{\sqrt {\pi }}\right ) \Gamma \left (- \frac {1}{4}\right )}{2 d^{\frac {3}{2}} \Gamma \left (\frac {3}{4}\right )} + \frac {\cosh {\left (f x \right )} \Gamma \left (- \frac {1}{4}\right )}{2 d^{\frac {3}{2}} \sqrt {x} \Gamma \left (\frac {3}{4}\right )} \] Input:
integrate(cosh(f*x)/(d*x)**(3/2),x)
Output:
-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)*gamm a(-1/4)/(2*d**(3/2)*sqrt(x)*gamma(3/4))
Time = 0.04 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.86 \[ \int \frac {\cosh (f x)}{(d x)^{3/2}} \, dx=-\frac {\frac {f {\left (\frac {\sqrt {\pi } \operatorname {erf}\left (\sqrt {d x} \sqrt {\frac {f}{d}}\right )}{\sqrt {\frac {f}{d}}} - \frac {\sqrt {\pi } \operatorname {erf}\left (\sqrt {d x} \sqrt {-\frac {f}{d}}\right )}{\sqrt {-\frac {f}{d}}}\right )}}{d} + \frac {2 \, \cosh \left (f x\right )}{\sqrt {d x}}}{d} \] Input:
integrate(cosh(f*x)/(d*x)^(3/2),x, algorithm="maxima")
Output:
-(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
\[ \int \frac {\cosh (f x)}{(d x)^{3/2}} \, dx=\int { \frac {\cosh \left (f x\right )}{\left (d x\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate(cosh(f*x)/(d*x)^(3/2),x, algorithm="giac")
Output:
integrate(cosh(f*x)/(d*x)^(3/2), x)
Timed out. \[ \int \frac {\cosh (f x)}{(d x)^{3/2}} \, dx=\int \frac {\mathrm {cosh}\left (f\,x\right )}{{\left (d\,x\right )}^{3/2}} \,d x \] Input:
int(cosh(f*x)/(d*x)^(3/2),x)
Output:
int(cosh(f*x)/(d*x)^(3/2), x)
\[ \int \frac {\cosh (f x)}{(d x)^{3/2}} \, dx=\frac {\int \frac {\cosh \left (f x \right )}{\sqrt {x}\, x}d x}{\sqrt {d}\, d} \] Input:
int(cosh(f*x)/(d*x)^(3/2),x)
Output:
int(cosh(f*x)/(sqrt(x)*x),x)/(sqrt(d)*d)