\(\int \sinh (a+\frac {b}{x^2}) \, dx\) [45]

Optimal result

Integrand size = 8, antiderivative size = 67 \[ \int \sinh \left (a+\frac {b}{x^2}\right ) \, dx=-\frac {1}{2} \sqrt {b} e^{-a} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {b}}{x}\right )-\frac {1}{2} \sqrt {b} e^a \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b}}{x}\right )+x \sinh \left (a+\frac {b}{x^2}\right ) \] Output:

-1/2*b^(1/2)*Pi^(1/2)*erf(b^(1/2)/x)/exp(a)-1/2*b^(1/2)*exp(a)*Pi^(1/2)*er 
fi(b^(1/2)/x)+x*sinh(a+b/x^2)
 

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.04 \[ \int \sinh \left (a+\frac {b}{x^2}\right ) \, dx=x \cosh \left (\frac {b}{x^2}\right ) \sinh (a)-\frac {1}{2} \sqrt {b} \sqrt {\pi } \left (\text {erf}\left (\frac {\sqrt {b}}{x}\right ) (\cosh (a)-\sinh (a))+\text {erfi}\left (\frac {\sqrt {b}}{x}\right ) (\cosh (a)+\sinh (a))\right )+x \cosh (a) \sinh \left (\frac {b}{x^2}\right ) \] Input:

Integrate[Sinh[a + b/x^2],x]
 

Output:

x*Cosh[b/x^2]*Sinh[a] - (Sqrt[b]*Sqrt[Pi]*(Erf[Sqrt[b]/x]*(Cosh[a] - Sinh[ 
a]) + Erfi[Sqrt[b]/x]*(Cosh[a] + Sinh[a])))/2 + x*Cosh[a]*Sinh[b/x^2]
 

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.06, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {5825, 5849, 5822, 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.

\(\displaystyle \int \sinh \left (a+\frac {b}{x^2}\right ) \, dx\)

\(\Big \downarrow \) 5825

\(\displaystyle -\int x^2 \sinh \left (a+\frac {b}{x^2}\right )d\frac {1}{x}\)

\(\Big \downarrow \) 5849

\(\displaystyle x \sinh \left (a+\frac {b}{x^2}\right )-2 b \int \cosh \left (a+\frac {b}{x^2}\right )d\frac {1}{x}\)

\(\Big \downarrow \) 5822

\(\displaystyle x \sinh \left (a+\frac {b}{x^2}\right )-2 b \left (\frac {1}{2} \int e^{-a-\frac {b}{x^2}}d\frac {1}{x}+\frac {1}{2} \int e^{a+\frac {b}{x^2}}d\frac {1}{x}\right )\)

\(\Big \downarrow \) 2633

\(\displaystyle x \sinh \left (a+\frac {b}{x^2}\right )-2 b \left (\frac {1}{2} \int e^{-a-\frac {b}{x^2}}d\frac {1}{x}+\frac {\sqrt {\pi } e^a \text {erfi}\left (\frac {\sqrt {b}}{x}\right )}{4 \sqrt {b}}\right )\)

\(\Big \downarrow \) 2634

\(\displaystyle x \sinh \left (a+\frac {b}{x^2}\right )-2 b \left (\frac {\sqrt {\pi } e^{-a} \text {erf}\left (\frac {\sqrt {b}}{x}\right )}{4 \sqrt {b}}+\frac {\sqrt {\pi } e^a \text {erfi}\left (\frac {\sqrt {b}}{x}\right )}{4 \sqrt {b}}\right )\)

Input:

Int[Sinh[a + b/x^2],x]
 

Output:

-2*b*((Sqrt[Pi]*Erf[Sqrt[b]/x])/(4*Sqrt[b]*E^a) + (E^a*Sqrt[Pi]*Erfi[Sqrt[ 
b]/x])/(4*Sqrt[b])) + x*Sinh[a + b/x^2]
 

Defintions of rubi rules used

rule 2633

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]
 

rule 2634

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]
 

rule 5822

Int[Cosh[(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]
 

rule 5825

Int[((a_.) + (b_.)*Sinh[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol] :> -Subs 
t[Int[(a + b*Sinh[c + d/x^n])^p/x^2, x], x, 1/x] /; FreeQ[{a, b, c, d}, x] 
&& ILtQ[n, 0] && IntegerQ[p]
 

rule 5849

Int[((e_.)*(x_))^(m_)*Sinh[(c_.) + (d_.)*(x_)^(n_)], x_Symbol] :> Simp[(e*x 
)^(m + 1)*(Sinh[c + d*x^n]/(e*(m + 1))), x] - Simp[d*(n/(e^n*(m + 1)))   In 
t[(e*x)^(m + n)*Cosh[c + d*x^n], x], x] /; FreeQ[{c, d, e}, x] && IGtQ[n, 0 
] && LtQ[m, -1]
 

Maple [A] (verified)

Time = 0.14 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.04

method	result	size
risch	\(-\frac {\sqrt {b}\, \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {b}}{x}\right ) {\mathrm e}^{-a}}{2}-\frac {{\mathrm e}^{-a} x \,{\mathrm e}^{-\frac {b}{x^{2}}}}{2}+\frac {{\mathrm e}^{a} {\mathrm e}^{\frac {b}{x^{2}}} x}{2}-\frac {{\mathrm e}^{a} b \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {-b}}{x}\right )}{2 \sqrt {-b}}\)	\(70\)
meijerg	\(\frac {i \sqrt {\pi }\, \cosh \left (a \right ) \sqrt {2}\, \sqrt {i b}\, \left (\frac {2 x \sqrt {2}\, \sqrt {i b}\, {\mathrm e}^{-\frac {b}{x^{2}}}}{\sqrt {\pi }\, b}-\frac {2 x \sqrt {2}\, \sqrt {i b}\, {\mathrm e}^{\frac {b}{x^{2}}}}{\sqrt {\pi }\, b}+\frac {2 \sqrt {i b}\, \sqrt {2}\, \operatorname {erf}\left (\frac {\sqrt {b}}{x}\right )}{\sqrt {b}}+\frac {2 \sqrt {i b}\, \sqrt {2}\, \operatorname {erfi}\left (\frac {\sqrt {b}}{x}\right )}{\sqrt {b}}\right )}{8}-\frac {\sqrt {\pi }\, \sinh \left (a \right ) \sqrt {2}\, \sqrt {i b}\, \left (-\frac {2 x \sqrt {2}\, {\mathrm e}^{\frac {b}{x^{2}}}}{\sqrt {\pi }\, \sqrt {i b}}-\frac {2 x \sqrt {2}\, {\mathrm e}^{-\frac {b}{x^{2}}}}{\sqrt {\pi }\, \sqrt {i b}}-\frac {2 \sqrt {2}\, \sqrt {b}\, \operatorname {erf}\left (\frac {\sqrt {b}}{x}\right )}{\sqrt {i b}}+\frac {2 \sqrt {2}\, \sqrt {b}\, \operatorname {erfi}\left (\frac {\sqrt {b}}{x}\right )}{\sqrt {i b}}\right )}{8}\)	\(217\)

Input:

int(sinh(a+b/x^2),x,method=_RETURNVERBOSE)
 

Output:

-1/2*b^(1/2)*Pi^(1/2)*erf(b^(1/2)/x)/exp(a)-1/2/exp(a)*x*exp(-b/x^2)+1/2*e 
xp(a)*exp(b/x^2)*x-1/2*exp(a)*b*Pi^(1/2)/(-b)^(1/2)*erf((-b)^(1/2)/x)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 228 vs. \(2 (49) = 98\).

Time = 0.10 (sec) , antiderivative size = 228, normalized size of antiderivative = 3.40 \[ \int \sinh \left (a+\frac {b}{x^2}\right ) \, dx=\frac {x \cosh \left (\frac {a x^{2} + b}{x^{2}}\right )^{2} + \sqrt {\pi } {\left (\cosh \left (a\right ) \cosh \left (\frac {a x^{2} + b}{x^{2}}\right ) + \cosh \left (\frac {a x^{2} + b}{x^{2}}\right ) \sinh \left (a\right ) + {\left (\cosh \left (a\right ) + \sinh \left (a\right )\right )} \sinh \left (\frac {a x^{2} + b}{x^{2}}\right )\right )} \sqrt {-b} \operatorname {erf}\left (\frac {\sqrt {-b}}{x}\right ) - \sqrt {\pi } {\left (\cosh \left (a\right ) \cosh \left (\frac {a x^{2} + b}{x^{2}}\right ) - \cosh \left (\frac {a x^{2} + b}{x^{2}}\right ) \sinh \left (a\right ) + {\left (\cosh \left (a\right ) - \sinh \left (a\right )\right )} \sinh \left (\frac {a x^{2} + b}{x^{2}}\right )\right )} \sqrt {b} \operatorname {erf}\left (\frac {\sqrt {b}}{x}\right ) + 2 \, x \cosh \left (\frac {a x^{2} + b}{x^{2}}\right ) \sinh \left (\frac {a x^{2} + b}{x^{2}}\right ) + x \sinh \left (\frac {a x^{2} + b}{x^{2}}\right )^{2} - x}{2 \, {\left (\cosh \left (\frac {a x^{2} + b}{x^{2}}\right ) + \sinh \left (\frac {a x^{2} + b}{x^{2}}\right )\right )}} \] Input:

integrate(sinh(a+b/x^2),x, algorithm="fricas")
 

Output:

1/2*(x*cosh((a*x^2 + b)/x^2)^2 + sqrt(pi)*(cosh(a)*cosh((a*x^2 + b)/x^2) + 
 cosh((a*x^2 + b)/x^2)*sinh(a) + (cosh(a) + sinh(a))*sinh((a*x^2 + b)/x^2) 
)*sqrt(-b)*erf(sqrt(-b)/x) - sqrt(pi)*(cosh(a)*cosh((a*x^2 + b)/x^2) - cos 
h((a*x^2 + b)/x^2)*sinh(a) + (cosh(a) - sinh(a))*sinh((a*x^2 + b)/x^2))*sq 
rt(b)*erf(sqrt(b)/x) + 2*x*cosh((a*x^2 + b)/x^2)*sinh((a*x^2 + b)/x^2) + x 
*sinh((a*x^2 + b)/x^2)^2 - x)/(cosh((a*x^2 + b)/x^2) + sinh((a*x^2 + b)/x^ 
2))
 

Sympy [F]

\[ \int \sinh \left (a+\frac {b}{x^2}\right ) \, dx=\int \sinh {\left (a + \frac {b}{x^{2}} \right )}\, dx \] Input:

integrate(sinh(a+b/x**2),x)
 

Output:

Integral(sinh(a + b/x**2), x)
 

Maxima [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.06 \[ \int \sinh \left (a+\frac {b}{x^2}\right ) \, dx=-\frac {1}{2} \, b {\left (\frac {\sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {\frac {b}{x^{2}}}\right ) - 1\right )} e^{\left (-a\right )}}{x \sqrt {\frac {b}{x^{2}}}} + \frac {\sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {-\frac {b}{x^{2}}}\right ) - 1\right )} e^{a}}{x \sqrt {-\frac {b}{x^{2}}}}\right )} + x \sinh \left (a + \frac {b}{x^{2}}\right ) \] Input:

integrate(sinh(a+b/x^2),x, algorithm="maxima")
 

Output:

-1/2*b*(sqrt(pi)*(erf(sqrt(b/x^2)) - 1)*e^(-a)/(x*sqrt(b/x^2)) + sqrt(pi)* 
(erf(sqrt(-b/x^2)) - 1)*e^a/(x*sqrt(-b/x^2))) + x*sinh(a + b/x^2)
 

Giac [F]

\[ \int \sinh \left (a+\frac {b}{x^2}\right ) \, dx=\int { \sinh \left (a + \frac {b}{x^{2}}\right ) \,d x } \] Input:

integrate(sinh(a+b/x^2),x, algorithm="giac")
 

Output:

integrate(sinh(a + b/x^2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \sinh \left (a+\frac {b}{x^2}\right ) \, dx=\int \mathrm {sinh}\left (a+\frac {b}{x^2}\right ) \,d x \] Input:

int(sinh(a + b/x^2),x)
 

Output:

int(sinh(a + b/x^2), x)
 

Reduce [F]

\[ \int \sinh \left (a+\frac {b}{x^2}\right ) \, dx=\frac {-2 \cosh \left (\frac {a \,x^{2}+b}{x^{2}}\right ) b -4 \left (\int \frac {\sinh \left (\frac {a \,x^{2}+b}{x^{2}}\right )}{x^{4}}d x \right ) b^{2} x +\sinh \left (\frac {a \,x^{2}+b}{x^{2}}\right ) x^{2}}{x} \] Input:

int(sinh(a+b/x^2),x)
 

Output:

( - 2*cosh((a*x**2 + b)/x**2)*b - 4*int(sinh((a*x**2 + b)/x**2)/x**4,x)*b* 
*2*x + sinh((a*x**2 + b)/x**2)*x**2)/x