Integrand size = 12, antiderivative size = 57 \[ \int \frac {\sinh \left (a+\frac {b}{x^2}\right )}{x^2} \, dx=\frac {e^{-a} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {b}}{x}\right )}{4 \sqrt {b}}-\frac {e^a \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b}}{x}\right )}{4 \sqrt {b}} \] Output:
1/4*Pi^(1/2)*erf(b^(1/2)/x)/b^(1/2)/exp(a)-1/4*exp(a)*Pi^(1/2)*erfi(b^(1/2 )/x)/b^(1/2)
Time = 0.02 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.88 \[ \int \frac {\sinh \left (a+\frac {b}{x^2}\right )}{x^2} \, dx=\frac {\sqrt {\pi } \left (\text {erf}\left (\frac {\sqrt {b}}{x}\right ) (\cosh (a)-\sinh (a))-\text {erfi}\left (\frac {\sqrt {b}}{x}\right ) (\cosh (a)+\sinh (a))\right )}{4 \sqrt {b}} \] Input:
Integrate[Sinh[a + b/x^2]/x^2,x]
Output:
(Sqrt[Pi]*(Erf[Sqrt[b]/x]*(Cosh[a] - Sinh[a]) - Erfi[Sqrt[b]/x]*(Cosh[a] + Sinh[a])))/(4*Sqrt[b])
Time = 0.29 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5869, 5821, 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.
\(\displaystyle \int \frac {\sinh \left (a+\frac {b}{x^2}\right )}{x^2} \, dx\) |
\(\Big \downarrow \) 5869 |
\(\displaystyle -\int \sinh \left (a+\frac {b}{x^2}\right )d\frac {1}{x}\) |
\(\Big \downarrow \) 5821 |
\(\displaystyle \frac {1}{2} \int e^{-a-\frac {b}{x^2}}d\frac {1}{x}-\frac {1}{2} \int e^{a+\frac {b}{x^2}}d\frac {1}{x}\) |
\(\Big \downarrow \) 2633 |
\(\displaystyle \frac {1}{2} \int e^{-a-\frac {b}{x^2}}d\frac {1}{x}-\frac {\sqrt {\pi } e^a \text {erfi}\left (\frac {\sqrt {b}}{x}\right )}{4 \sqrt {b}}\) |
\(\Big \downarrow \) 2634 |
\(\displaystyle \frac {\sqrt {\pi } e^{-a} \text {erf}\left (\frac {\sqrt {b}}{x}\right )}{4 \sqrt {b}}-\frac {\sqrt {\pi } e^a \text {erfi}\left (\frac {\sqrt {b}}{x}\right )}{4 \sqrt {b}}\) |
Input:
Int[Sinh[a + b/x^2]/x^2,x]
Output:
(Sqrt[Pi]*Erf[Sqrt[b]/x])/(4*Sqrt[b]*E^a) - (E^a*Sqrt[Pi]*Erfi[Sqrt[b]/x]) /(4*Sqrt[b])
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[Sinh[(c_.) + (d_.)*(x_)^(n_)], x_Symbol] :> Simp[1/2 Int[E^(c + d*x^n ), x], x] - Simp[1/2 Int[E^(-c - d*x^n), x], x] /; FreeQ[{c, d}, x] && IG tQ[n, 1]
Int[(x_)^(m_.)*((a_.) + (b_.)*Sinh[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbo l] :> -Subst[Int[(a + b*Sinh[c + d/x^n])^p/x^(m + 2), x], x, 1/x] /; FreeQ[ {a, b, c, d}, x] && IntegerQ[p] && ILtQ[n, 0] && IntegerQ[m]
Time = 0.20 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.77
method | result | size |
risch | \(\frac {\sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {b}}{x}\right ) {\mathrm e}^{-a}}{4 \sqrt {b}}-\frac {{\mathrm e}^{a} \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {-b}}{x}\right )}{4 \sqrt {-b}}\) | \(44\) |
meijerg | \(\frac {\sqrt {\pi }\, \cosh \left (a \right ) \sqrt {2}\, \sqrt {i b}\, \left (-\frac {\left (i b \right )^{\frac {3}{2}} \sqrt {2}\, \operatorname {erf}\left (\frac {\sqrt {b}}{x}\right )}{2 b^{\frac {3}{2}}}+\frac {\left (i b \right )^{\frac {3}{2}} \sqrt {2}\, \operatorname {erfi}\left (\frac {\sqrt {b}}{x}\right )}{2 b^{\frac {3}{2}}}\right )}{4 b}+\frac {i \sqrt {\pi }\, \sinh \left (a \right ) \sqrt {2}\, \sqrt {i b}\, \left (\frac {\sqrt {i b}\, \sqrt {2}\, \operatorname {erf}\left (\frac {\sqrt {b}}{x}\right )}{2 \sqrt {b}}+\frac {\sqrt {i b}\, \sqrt {2}\, \operatorname {erfi}\left (\frac {\sqrt {b}}{x}\right )}{2 \sqrt {b}}\right )}{4 b}\) | \(131\) |
Input:
int(sinh(a+b/x^2)/x^2,x,method=_RETURNVERBOSE)
Output:
1/4*Pi^(1/2)*erf(b^(1/2)/x)/b^(1/2)/exp(a)-1/4*exp(a)*Pi^(1/2)/(-b)^(1/2)* erf((-b)^(1/2)/x)
Time = 0.08 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.91 \[ \int \frac {\sinh \left (a+\frac {b}{x^2}\right )}{x^2} \, dx=\frac {\sqrt {\pi } \sqrt {-b} {\left (\cosh \left (a\right ) + \sinh \left (a\right )\right )} \operatorname {erf}\left (\frac {\sqrt {-b}}{x}\right ) + \sqrt {\pi } \sqrt {b} {\left (\cosh \left (a\right ) - \sinh \left (a\right )\right )} \operatorname {erf}\left (\frac {\sqrt {b}}{x}\right )}{4 \, b} \] Input:
integrate(sinh(a+b/x^2)/x^2,x, algorithm="fricas")
Output:
1/4*(sqrt(pi)*sqrt(-b)*(cosh(a) + sinh(a))*erf(sqrt(-b)/x) + sqrt(pi)*sqrt (b)*(cosh(a) - sinh(a))*erf(sqrt(b)/x))/b
\[ \int \frac {\sinh \left (a+\frac {b}{x^2}\right )}{x^2} \, dx=\int \frac {\sinh {\left (a + \frac {b}{x^{2}} \right )}}{x^{2}}\, dx \] Input:
integrate(sinh(a+b/x**2)/x**2,x)
Output:
Integral(sinh(a + b/x**2)/x**2, x)
Time = 0.09 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.09 \[ \int \frac {\sinh \left (a+\frac {b}{x^2}\right )}{x^2} \, dx=-\frac {1}{2} \, b {\left (\frac {e^{\left (-a\right )} \Gamma \left (\frac {3}{2}, \frac {b}{x^{2}}\right )}{x^{3} \left (\frac {b}{x^{2}}\right )^{\frac {3}{2}}} + \frac {e^{a} \Gamma \left (\frac {3}{2}, -\frac {b}{x^{2}}\right )}{x^{3} \left (-\frac {b}{x^{2}}\right )^{\frac {3}{2}}}\right )} - \frac {\sinh \left (a + \frac {b}{x^{2}}\right )}{x} \] Input:
integrate(sinh(a+b/x^2)/x^2,x, algorithm="maxima")
Output:
-1/2*b*(e^(-a)*gamma(3/2, b/x^2)/(x^3*(b/x^2)^(3/2)) + e^a*gamma(3/2, -b/x ^2)/(x^3*(-b/x^2)^(3/2))) - sinh(a + b/x^2)/x
\[ \int \frac {\sinh \left (a+\frac {b}{x^2}\right )}{x^2} \, dx=\int { \frac {\sinh \left (a + \frac {b}{x^{2}}\right )}{x^{2}} \,d x } \] Input:
integrate(sinh(a+b/x^2)/x^2,x, algorithm="giac")
Output:
integrate(sinh(a + b/x^2)/x^2, x)
Timed out. \[ \int \frac {\sinh \left (a+\frac {b}{x^2}\right )}{x^2} \, dx=\int \frac {\mathrm {sinh}\left (a+\frac {b}{x^2}\right )}{x^2} \,d x \] Input:
int(sinh(a + b/x^2)/x^2,x)
Output:
int(sinh(a + b/x^2)/x^2, x)
\[ \int \frac {\sinh \left (a+\frac {b}{x^2}\right )}{x^2} \, dx=\frac {-e^{\frac {2 a \,x^{2}+2 b}{x^{2}}}-2 e^{\frac {a \,x^{2}+b}{x^{2}}} \left (\int \frac {e^{\frac {a \,x^{2}+b}{x^{2}}}}{x^{4}}d x \right ) b x -2 e^{\frac {a \,x^{2}+b}{x^{2}}} \left (\int \frac {1}{e^{\frac {a \,x^{2}+b}{x^{2}}} x^{4}}d x \right ) b x +1}{2 e^{\frac {a \,x^{2}+b}{x^{2}}} x} \] Input:
int(sinh(a+b/x^2)/x^2,x)
Output:
( - e**((2*a*x**2 + 2*b)/x**2) - 2*e**((a*x**2 + b)/x**2)*int(e**((a*x**2 + b)/x**2)/x**4,x)*b*x - 2*e**((a*x**2 + b)/x**2)*int(1/(e**((a*x**2 + b)/ x**2)*x**4),x)*b*x + 1)/(2*e**((a*x**2 + b)/x**2)*x)