Integrand size = 12, antiderivative size = 92 \[ \int \sqrt {d x} \sinh (f x) \, dx=\frac {\sqrt {d x} \cosh (f x)}{f}-\frac {\sqrt {d} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {f} \sqrt {d x}}{\sqrt {d}}\right )}{4 f^{3/2}}-\frac {\sqrt {d} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {f} \sqrt {d x}}{\sqrt {d}}\right )}{4 f^{3/2}} \] Output:
(d*x)^(1/2)*cosh(f*x)/f-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)
Time = 0.01 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.53 \[ \int \sqrt {d x} \sinh (f x) \, dx=\frac {d \left (-\sqrt {-f x} \Gamma \left (\frac {3}{2},-f x\right )+\sqrt {f x} \Gamma \left (\frac {3}{2},f x\right )\right )}{2 f^2 \sqrt {d x}} \] Input:
Integrate[Sqrt[d*x]*Sinh[f*x],x]
Output:
(d*(-(Sqrt[-(f*x)]*Gamma[3/2, -(f*x)]) + Sqrt[f*x]*Gamma[3/2, f*x]))/(2*f^ 2*Sqrt[d*x])
Result contains complex when optimal does not.
Time = 0.43 (sec) , antiderivative size = 110, 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, 26, 3777, 3042, 3788, 26, 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.
\(\displaystyle \int \sqrt {d x} \sinh (f x) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -i \sqrt {d x} \sin (i f x)dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -i \int \sqrt {d x} \sin (i f x)dx\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle -i \left (\frac {i \sqrt {d x} \cosh (f x)}{f}-\frac {i d \int \frac {\cosh (f x)}{\sqrt {d x}}dx}{2 f}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -i \left (\frac {i \sqrt {d x} \cosh (f x)}{f}-\frac {i d \int \frac {\sin \left (i f x+\frac {\pi }{2}\right )}{\sqrt {d x}}dx}{2 f}\right )\) |
\(\Big \downarrow \) 3788 |
\(\displaystyle -i \left (\frac {i \sqrt {d x} \cosh (f x)}{f}-\frac {i d \left (\frac {1}{2} i \int -\frac {i e^{f x}}{\sqrt {d x}}dx-\frac {1}{2} i \int \frac {i e^{-f x}}{\sqrt {d x}}dx\right )}{2 f}\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -i \left (\frac {i \sqrt {d x} \cosh (f x)}{f}-\frac {i d \left (\frac {1}{2} \int \frac {e^{-f x}}{\sqrt {d x}}dx+\frac {1}{2} \int \frac {e^{f x}}{\sqrt {d x}}dx\right )}{2 f}\right )\) |
\(\Big \downarrow \) 2611 |
\(\displaystyle -i \left (\frac {i \sqrt {d x} \cosh (f x)}{f}-\frac {i d \left (\frac {\int e^{-f x}d\sqrt {d x}}{d}+\frac {\int e^{f x}d\sqrt {d x}}{d}\right )}{2 f}\right )\) |
\(\Big \downarrow \) 2633 |
\(\displaystyle -i \left (\frac {i \sqrt {d x} \cosh (f x)}{f}-\frac {i d \left (\frac {\int e^{-f x}d\sqrt {d x}}{d}+\frac {\sqrt {\pi } \text {erfi}\left (\frac {\sqrt {f} \sqrt {d x}}{\sqrt {d}}\right )}{2 \sqrt {d} \sqrt {f}}\right )}{2 f}\right )\) |
\(\Big \downarrow \) 2634 |
\(\displaystyle -i \left (\frac {i \sqrt {d x} \cosh (f x)}{f}-\frac {i d \left (\frac {\sqrt {\pi } \text {erf}\left (\frac {\sqrt {f} \sqrt {d x}}{\sqrt {d}}\right )}{2 \sqrt {d} \sqrt {f}}+\frac {\sqrt {\pi } \text {erfi}\left (\frac {\sqrt {f} \sqrt {d x}}{\sqrt {d}}\right )}{2 \sqrt {d} \sqrt {f}}\right )}{2 f}\right )\) |
Input:
Int[Sqrt[d*x]*Sinh[f*x],x]
Output:
(-I)*((I*Sqrt[d*x]*Cosh[f*x])/f - ((I/2)*d*((Sqrt[Pi]*Erf[(Sqrt[f]*Sqrt[d* x])/Sqrt[d]])/(2*Sqrt[d]*Sqrt[f]) + (Sqrt[Pi]*Erfi[(Sqrt[f]*Sqrt[d*x])/Sqr t[d]])/(2*Sqrt[d]*Sqrt[f])))/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_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol ] :> Simp[I/2 Int[(c + d*x)^m/(E^(I*k*Pi)*E^(I*(e + f*x))), x], x] - Simp [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]
Result contains complex when optimal does not.
Time = 0.08 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.30
method | result | size |
meijerg | \(-\frac {\sqrt {d x}\, \sqrt {2}\, \sqrt {\pi }\, \left (\frac {\sqrt {x}\, \sqrt {2}\, \left (i f \right )^{\frac {5}{2}} {\mathrm e}^{-f x}}{4 \sqrt {\pi }\, f^{2}}+\frac {\sqrt {x}\, \sqrt {2}\, \left (i f \right )^{\frac {5}{2}} {\mathrm e}^{f x}}{4 \sqrt {\pi }\, f^{2}}-\frac {\left (i f \right )^{\frac {5}{2}} \sqrt {2}\, \operatorname {erf}\left (\sqrt {x}\, \sqrt {f}\right )}{8 f^{\frac {5}{2}}}-\frac {\left (i f \right )^{\frac {5}{2}} \sqrt {2}\, \operatorname {erfi}\left (\sqrt {x}\, \sqrt {f}\right )}{8 f^{\frac {5}{2}}}\right )}{\sqrt {x}\, \sqrt {i f}\, f}\) | \(120\) |
Input:
int((d*x)^(1/2)*sinh(f*x),x,method=_RETURNVERBOSE)
Output:
-(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)^(5/2)/f^2*exp(-f*x)+1/4/Pi^(1/2)*x^(1/2)*2^(1/2)*(I*f)^(5/2) /f^2*exp(f*x)-1/8*(I*f)^(5/2)*2^(1/2)/f^(5/2)*erf(x^(1/2)*f^(1/2))-1/8*(I* f)^(5/2)*2^(1/2)/f^(5/2)*erfi(x^(1/2)*f^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 137 vs. \(2 (62) = 124\).
Time = 0.09 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.49 \[ \int \sqrt {d x} \sinh (f x) \, dx=-\frac {\sqrt {\pi } {\left (d \cosh \left (f x\right ) + d \sinh \left (f x\right )\right )} \sqrt {\frac {f}{d}} \operatorname {erf}\left (\sqrt {d x} \sqrt {\frac {f}{d}}\right ) - \sqrt {\pi } {\left (d \cosh \left (f x\right ) + d \sinh \left (f x\right )\right )} \sqrt {-\frac {f}{d}} \operatorname {erf}\left (\sqrt {d x} \sqrt {-\frac {f}{d}}\right ) - 2 \, {\left (f \cosh \left (f x\right )^{2} + 2 \, f \cosh \left (f x\right ) \sinh \left (f x\right ) + f \sinh \left (f x\right )^{2} + f\right )} \sqrt {d x}}{4 \, {\left (f^{2} \cosh \left (f x\right ) + f^{2} \sinh \left (f x\right )\right )}} \] Input:
integrate((d*x)^(1/2)*sinh(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)*sq rt(d*x))/(f^2*cosh(f*x) + f^2*sinh(f*x))
Result contains complex when optimal does not.
Time = 0.94 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.08 \[ \int \sqrt {d x} \sinh (f x) \, dx=\frac {5 \sqrt {d} \sqrt {x} \cosh {\left (f x \right )} \Gamma \left (\frac {5}{4}\right )}{4 f \Gamma \left (\frac {9}{4}\right )} - \frac {5 \sqrt {2} \sqrt {\pi } \sqrt {d} e^{- \frac {i \pi }{4}} C\left (\frac {\sqrt {2} \sqrt {f} \sqrt {x} e^{\frac {i \pi }{4}}}{\sqrt {\pi }}\right ) \Gamma \left (\frac {5}{4}\right )}{8 f^{\frac {3}{2}} \Gamma \left (\frac {9}{4}\right )} \] Input:
integrate((d*x)**(1/2)*sinh(f*x),x)
Output:
5*sqrt(d)*sqrt(x)*cosh(f*x)*gamma(5/4)/(4*f*gamma(9/4)) - 5*sqrt(2)*sqrt(p i)*sqrt(d)*exp(-I*pi/4)*fresnelc(sqrt(2)*sqrt(f)*sqrt(x)*exp(I*pi/4)/sqrt( pi))*gamma(5/4)/(8*f**(3/2)*gamma(9/4))
Leaf count of result is larger than twice the leaf count of optimal. 149 vs. \(2 (62) = 124\).
Time = 0.04 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.62 \[ \int \sqrt {d x} \sinh (f x) \, dx=\frac {8 \, \left (d x\right )^{\frac {3}{2}} \sinh \left (f x\right ) - \frac {f {\left (\frac {3 \, \sqrt {\pi } d^{2} \operatorname {erf}\left (\sqrt {d x} \sqrt {\frac {f}{d}}\right )}{f^{2} \sqrt {\frac {f}{d}}} + \frac {3 \, \sqrt {\pi } d^{2} \operatorname {erf}\left (\sqrt {d x} \sqrt {-\frac {f}{d}}\right )}{f^{2} \sqrt {-\frac {f}{d}}} + \frac {2 \, {\left (2 \, \left (d x\right )^{\frac {3}{2}} d f - 3 \, \sqrt {d x} d^{2}\right )} e^{\left (f x\right )}}{f^{2}} - \frac {2 \, {\left (2 \, \left (d x\right )^{\frac {3}{2}} d f + 3 \, \sqrt {d x} d^{2}\right )} e^{\left (-f x\right )}}{f^{2}}\right )}}{d}}{12 \, d} \] Input:
integrate((d*x)^(1/2)*sinh(f*x),x, algorithm="maxima")
Output:
1/12*(8*(d*x)^(3/2)*sinh(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 = 0.12 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.17 \[ \int \sqrt {d x} \sinh (f x) \, dx=\frac {\frac {\sqrt {\pi } d^{2} \operatorname {erf}\left (-\frac {\sqrt {d f} \sqrt {d x}}{d}\right )}{\sqrt {d f} f} + \frac {2 \, \sqrt {d x} d e^{\left (-f x\right )}}{f}}{4 \, d} + \frac {\frac {\sqrt {\pi } d^{2} \operatorname {erf}\left (-\frac {\sqrt {-d f} \sqrt {d x}}{d}\right )}{\sqrt {-d f} f} + \frac {2 \, \sqrt {d x} d e^{\left (f x\right )}}{f}}{4 \, d} \] Input:
integrate((d*x)^(1/2)*sinh(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. \[ \int \sqrt {d x} \sinh (f x) \, dx=\int \mathrm {sinh}\left (f\,x\right )\,\sqrt {d\,x} \,d x \] Input:
int(sinh(f*x)*(d*x)^(1/2),x)
Output:
int(sinh(f*x)*(d*x)^(1/2), x)
\[ \int \sqrt {d x} \sinh (f x) \, dx=\frac {\sqrt {d}\, \left (\sqrt {\pi }\, e^{f x} \mathrm {erf}\left (\sqrt {x}\, \sqrt {f}\, i \right ) i +2 \sqrt {x}\, \sqrt {f}\, e^{2 f x}-\sqrt {f}\, e^{f x} \left (\int \frac {\sqrt {x}}{e^{f x} x}d x \right )+2 \sqrt {x}\, \sqrt {f}\right )}{4 \sqrt {f}\, e^{f x} f} \] Input:
int((d*x)^(1/2)*sinh(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)