Integrand size = 12, antiderivative size = 87 \[ \int \frac {\sinh (f x)}{(d x)^{3/2}} \, dx=\frac {\sqrt {f} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {f} \sqrt {d x}}{\sqrt {d}}\right )}{d^{3/2}}+\frac {\sqrt {f} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {f} \sqrt {d x}}{\sqrt {d}}\right )}{d^{3/2}}-\frac {2 \sinh (f x)}{d \sqrt {d x}} \] Output:
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)-2*sinh(f*x)/d/(d*x)^(1/2)
Time = 0.01 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.56 \[ \int \frac {\sinh (f x)}{(d x)^{3/2}} \, dx=\frac {x \left (\sqrt {-f x} \Gamma \left (\frac {1}{2},-f x\right )-\sqrt {f x} \Gamma \left (\frac {1}{2},f x\right )-2 \sinh (f x)\right )}{(d x)^{3/2}} \] Input:
Integrate[Sinh[f*x]/(d*x)^(3/2),x]
Output:
(x*(Sqrt[-(f*x)]*Gamma[1/2, -(f*x)] - Sqrt[f*x]*Gamma[1/2, f*x] - 2*Sinh[f *x]))/(d*x)^(3/2)
Result contains complex when optimal does not.
Time = 0.46 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.24, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {3042, 26, 3778, 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.
\(\displaystyle \int \frac {\sinh (f x)}{(d x)^{3/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\frac {i \sin (i f x)}{(d x)^{3/2}}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -i \int \frac {\sin (i f x)}{(d x)^{3/2}}dx\) |
\(\Big \downarrow \) 3778 |
\(\displaystyle -i \left (\frac {2 i f \int \frac {\cosh (f x)}{\sqrt {d x}}dx}{d}-\frac {2 i \sinh (f x)}{d \sqrt {d x}}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -i \left (\frac {2 i f \int \frac {\sin \left (i f x+\frac {\pi }{2}\right )}{\sqrt {d x}}dx}{d}-\frac {2 i \sinh (f x)}{d \sqrt {d x}}\right )\) |
\(\Big \downarrow \) 3788 |
\(\displaystyle -i \left (\frac {2 i f \left (\frac {1}{2} i \int -\frac {i e^{f x}}{\sqrt {d x}}dx-\frac {1}{2} i \int \frac {i e^{-f x}}{\sqrt {d x}}dx\right )}{d}-\frac {2 i \sinh (f x)}{d \sqrt {d x}}\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -i \left (\frac {2 i f \left (\frac {1}{2} \int \frac {e^{-f x}}{\sqrt {d x}}dx+\frac {1}{2} \int \frac {e^{f x}}{\sqrt {d x}}dx\right )}{d}-\frac {2 i \sinh (f x)}{d \sqrt {d x}}\right )\) |
\(\Big \downarrow \) 2611 |
\(\displaystyle -i \left (\frac {2 i f \left (\frac {\int e^{-f x}d\sqrt {d x}}{d}+\frac {\int e^{f x}d\sqrt {d x}}{d}\right )}{d}-\frac {2 i \sinh (f x)}{d \sqrt {d x}}\right )\) |
\(\Big \downarrow \) 2633 |
\(\displaystyle -i \left (\frac {2 i f \left (\frac {\int e^{-f x}d\sqrt {d x}}{d}+\frac {\sqrt {\pi } \text {erfi}\left (\frac {\sqrt {f} \sqrt {d x}}{\sqrt {d}}\right )}{2 \sqrt {d} \sqrt {f}}\right )}{d}-\frac {2 i \sinh (f x)}{d \sqrt {d x}}\right )\) |
\(\Big \downarrow \) 2634 |
\(\displaystyle -i \left (\frac {2 i f \left (\frac {\sqrt {\pi } \text {erf}\left (\frac {\sqrt {f} \sqrt {d x}}{\sqrt {d}}\right )}{2 \sqrt {d} \sqrt {f}}+\frac {\sqrt {\pi } \text {erfi}\left (\frac {\sqrt {f} \sqrt {d x}}{\sqrt {d}}\right )}{2 \sqrt {d} \sqrt {f}}\right )}{d}-\frac {2 i \sinh (f x)}{d \sqrt {d x}}\right )\) |
Input:
Int[Sinh[f*x]/(d*x)^(3/2),x]
Output:
(-I)*(((2*I)*f*((Sqrt[Pi]*Erf[(Sqrt[f]*Sqrt[d*x])/Sqrt[d]])/(2*Sqrt[d]*Sqr t[f]) + (Sqrt[Pi]*Erfi[(Sqrt[f]*Sqrt[d*x])/Sqrt[d]])/(2*Sqrt[d]*Sqrt[f]))) /d - ((2*I)*Sinh[f*x])/(d*Sqrt[d*x]))
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_.) + 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 = 0.09 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.38
method | result | size |
meijerg | \(-\frac {\sqrt {\pi }\, x^{\frac {3}{2}} \sqrt {2}\, \left (i f \right )^{\frac {3}{2}} \left (\frac {2 \sqrt {2}\, \sqrt {i f}\, {\mathrm e}^{-f x}}{\sqrt {\pi }\, \sqrt {x}\, f}-\frac {2 \sqrt {2}\, \sqrt {i f}\, {\mathrm e}^{f x}}{\sqrt {\pi }\, \sqrt {x}\, f}+\frac {2 \sqrt {i f}\, \sqrt {2}\, \operatorname {erf}\left (\sqrt {x}\, \sqrt {f}\right )}{\sqrt {f}}+\frac {2 \sqrt {i f}\, \sqrt {2}\, \operatorname {erfi}\left (\sqrt {x}\, \sqrt {f}\right )}{\sqrt {f}}\right )}{4 \left (d x \right )^{\frac {3}{2}} f}\) | \(120\) |
Input:
int(sinh(f*x)/(d*x)^(3/2),x,method=_RETURNVERBOSE)
Output:
-1/4*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)/f*exp(-f*x)-2/Pi^(1/2)/x^(1/2)*2^(1/2)*(I*f)^(1/2)/f *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)))
Leaf count of result is larger than twice the leaf count of optimal. 137 vs. \(2 (61) = 122\).
Time = 0.09 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.57 \[ \int \frac {\sinh (f x)}{(d x)^{3/2}} \, dx=\frac {\sqrt {\pi } {\left (d x \cosh \left (f x\right ) + d x \sinh \left (f x\right )\right )} \sqrt {\frac {f}{d}} \operatorname {erf}\left (\sqrt {d x} \sqrt {\frac {f}{d}}\right ) - \sqrt {\pi } {\left (d x \cosh \left (f x\right ) + d x \sinh \left (f x\right )\right )} \sqrt {-\frac {f}{d}} \operatorname {erf}\left (\sqrt {d x} \sqrt {-\frac {f}{d}}\right ) - \sqrt {d x} {\left (\cosh \left (f x\right )^{2} + 2 \, \cosh \left (f x\right ) \sinh \left (f x\right ) + \sinh \left (f x\right )^{2} - 1\right )}}{d^{2} x \cosh \left (f x\right ) + d^{2} x \sinh \left (f x\right )} \] Input:
integrate(sinh(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)*sqr t(-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 = 94, normalized size of antiderivative = 1.08 \[ \int \frac {\sinh (f x)}{(d x)^{3/2}} \, dx=\frac {\sqrt {2} \sqrt {\pi } \sqrt {f} e^{- \frac {i \pi }{4}} C\left (\frac {\sqrt {2} \sqrt {f} \sqrt {x} e^{\frac {i \pi }{4}}}{\sqrt {\pi }}\right ) \Gamma \left (\frac {1}{4}\right )}{2 d^{\frac {3}{2}} \Gamma \left (\frac {5}{4}\right )} - \frac {\sinh {\left (f x \right )} \Gamma \left (\frac {1}{4}\right )}{2 d^{\frac {3}{2}} \sqrt {x} \Gamma \left (\frac {5}{4}\right )} \] Input:
integrate(sinh(f*x)/(d*x)**(3/2),x)
Output:
sqrt(2)*sqrt(pi)*sqrt(f)*exp(-I*pi/4)*fresnelc(sqrt(2)*sqrt(f)*sqrt(x)*exp (I*pi/4)/sqrt(pi))*gamma(1/4)/(2*d**(3/2)*gamma(5/4)) - sinh(f*x)*gamma(1/ 4)/(2*d**(3/2)*sqrt(x)*gamma(5/4))
Time = 0.04 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.85 \[ \int \frac {\sinh (f x)}{(d x)^{3/2}} \, dx=\frac {\frac {f {\left (\frac {\sqrt {\pi } \operatorname {erf}\left (\sqrt {d x} \sqrt {\frac {f}{d}}\right )}{\sqrt {\frac {f}{d}}} + \frac {\sqrt {\pi } \operatorname {erf}\left (\sqrt {d x} \sqrt {-\frac {f}{d}}\right )}{\sqrt {-\frac {f}{d}}}\right )}}{d} - \frac {2 \, \sinh \left (f x\right )}{\sqrt {d x}}}{d} \] Input:
integrate(sinh(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)*s qrt(-f/d))/sqrt(-f/d))/d - 2*sinh(f*x)/sqrt(d*x))/d
\[ \int \frac {\sinh (f x)}{(d x)^{3/2}} \, dx=\int { \frac {\sinh \left (f x\right )}{\left (d x\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate(sinh(f*x)/(d*x)^(3/2),x, algorithm="giac")
Output:
integrate(sinh(f*x)/(d*x)^(3/2), x)
Timed out. \[ \int \frac {\sinh (f x)}{(d x)^{3/2}} \, dx=\int \frac {\mathrm {sinh}\left (f\,x\right )}{{\left (d\,x\right )}^{3/2}} \,d x \] Input:
int(sinh(f*x)/(d*x)^(3/2),x)
Output:
int(sinh(f*x)/(d*x)^(3/2), x)
\[ \int \frac {\sinh (f x)}{(d x)^{3/2}} \, dx=\frac {\int \frac {\sinh \left (f x \right )}{\sqrt {x}\, x}d x}{\sqrt {d}\, d} \] Input:
int(sinh(f*x)/(d*x)^(3/2),x)
Output:
int(sinh(f*x)/(sqrt(x)*x),x)/(sqrt(d)*d)