\(\int \frac {\sinh (a+\frac {b}{x^2})}{x^6} \, dx\) [51]

Optimal result

Integrand size = 12, antiderivative size = 93 \[ \int \frac {\sinh \left (a+\frac {b}{x^2}\right )}{x^6} \, dx=-\frac {\cosh \left (a+\frac {b}{x^2}\right )}{2 b x^3}+\frac {3 e^{-a} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {b}}{x}\right )}{16 b^{5/2}}-\frac {3 e^a \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b}}{x}\right )}{16 b^{5/2}}+\frac {3 \sinh \left (a+\frac {b}{x^2}\right )}{4 b^2 x} \] Output:

-1/2*cosh(a+b/x^2)/b/x^3+3/16*Pi^(1/2)*erf(b^(1/2)/x)/b^(5/2)/exp(a)-3/16* 
exp(a)*Pi^(1/2)*erfi(b^(1/2)/x)/b^(5/2)+3/4*sinh(a+b/x^2)/b^2/x
 

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.04 \[ \int \frac {\sinh \left (a+\frac {b}{x^2}\right )}{x^6} \, dx=\frac {3 \sqrt {\pi } x^3 \text {erf}\left (\frac {\sqrt {b}}{x}\right ) (\cosh (a)-\sinh (a))-3 \sqrt {\pi } x^3 \text {erfi}\left (\frac {\sqrt {b}}{x}\right ) (\cosh (a)+\sinh (a))+4 \sqrt {b} \left (-2 b \cosh \left (a+\frac {b}{x^2}\right )+3 x^2 \sinh \left (a+\frac {b}{x^2}\right )\right )}{16 b^{5/2} x^3} \] Input:

Integrate[Sinh[a + b/x^2]/x^6,x]
 

Output:

(3*Sqrt[Pi]*x^3*Erf[Sqrt[b]/x]*(Cosh[a] - Sinh[a]) - 3*Sqrt[Pi]*x^3*Erfi[S 
qrt[b]/x]*(Cosh[a] + Sinh[a]) + 4*Sqrt[b]*(-2*b*Cosh[a + b/x^2] + 3*x^2*Si 
nh[a + b/x^2]))/(16*b^(5/2)*x^3)
 

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.17, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5869, 5847, 5848, 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.

\(\displaystyle \int \frac {\sinh \left (a+\frac {b}{x^2}\right )}{x^6} \, dx\)

\(\Big \downarrow \) 5869

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

\(\Big \downarrow \) 5847

\(\displaystyle \frac {3 \int \frac {\cosh \left (a+\frac {b}{x^2}\right )}{x^2}d\frac {1}{x}}{2 b}-\frac {\cosh \left (a+\frac {b}{x^2}\right )}{2 b x^3}\)

\(\Big \downarrow \) 5848

\(\displaystyle \frac {3 \left (\frac {\sinh \left (a+\frac {b}{x^2}\right )}{2 b x}-\frac {\int \sinh \left (a+\frac {b}{x^2}\right )d\frac {1}{x}}{2 b}\right )}{2 b}-\frac {\cosh \left (a+\frac {b}{x^2}\right )}{2 b x^3}\)

\(\Big \downarrow \) 5821

\(\displaystyle \frac {3 \left (\frac {\sinh \left (a+\frac {b}{x^2}\right )}{2 b x}-\frac {\frac {1}{2} \int e^{a+\frac {b}{x^2}}d\frac {1}{x}-\frac {1}{2} \int e^{-a-\frac {b}{x^2}}d\frac {1}{x}}{2 b}\right )}{2 b}-\frac {\cosh \left (a+\frac {b}{x^2}\right )}{2 b x^3}\)

\(\Big \downarrow \) 2633

\(\displaystyle \frac {3 \left (\frac {\sinh \left (a+\frac {b}{x^2}\right )}{2 b x}-\frac {\frac {\sqrt {\pi } e^a \text {erfi}\left (\frac {\sqrt {b}}{x}\right )}{4 \sqrt {b}}-\frac {1}{2} \int e^{-a-\frac {b}{x^2}}d\frac {1}{x}}{2 b}\right )}{2 b}-\frac {\cosh \left (a+\frac {b}{x^2}\right )}{2 b x^3}\)

\(\Big \downarrow \) 2634

\(\displaystyle \frac {3 \left (\frac {\sinh \left (a+\frac {b}{x^2}\right )}{2 b x}-\frac {\frac {\sqrt {\pi } e^a \text {erfi}\left (\frac {\sqrt {b}}{x}\right )}{4 \sqrt {b}}-\frac {\sqrt {\pi } e^{-a} \text {erf}\left (\frac {\sqrt {b}}{x}\right )}{4 \sqrt {b}}}{2 b}\right )}{2 b}-\frac {\cosh \left (a+\frac {b}{x^2}\right )}{2 b x^3}\)

Input:

Int[Sinh[a + b/x^2]/x^6,x]
 

Output:

-1/2*Cosh[a + b/x^2]/(b*x^3) + (3*(-1/2*(-1/4*(Sqrt[Pi]*Erf[Sqrt[b]/x])/(S 
qrt[b]*E^a) + (E^a*Sqrt[Pi]*Erfi[Sqrt[b]/x])/(4*Sqrt[b]))/b + Sinh[a + b/x 
^2]/(2*b*x)))/(2*b)
 

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 5821

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]
 

rule 5847

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

rule 5848

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

rule 5869

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]
 

Maple [A] (verified)

Time = 0.27 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.26

method	result	size
risch	\(-\frac {{\mathrm e}^{-a} {\mathrm e}^{-\frac {b}{x^{2}}}}{4 b \,x^{3}}-\frac {3 \,{\mathrm e}^{-a} {\mathrm e}^{-\frac {b}{x^{2}}}}{8 b^{2} x}+\frac {3 \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {b}}{x}\right ) {\mathrm e}^{-a}}{16 b^{\frac {5}{2}}}-\frac {{\mathrm e}^{a} {\mathrm e}^{\frac {b}{x^{2}}}}{4 x^{3} b}+\frac {3 \,{\mathrm e}^{a} {\mathrm e}^{\frac {b}{x^{2}}}}{8 b^{2} x}-\frac {3 \,{\mathrm e}^{a} \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {-b}}{x}\right )}{16 b^{2} \sqrt {-b}}\)	\(117\)
meijerg	\(-\frac {\sqrt {\pi }\, \cosh \left (a \right ) \sqrt {2}\, \sqrt {i b}\, \left (-\frac {\sqrt {2}\, \left (i b \right )^{\frac {7}{2}} \left (-\frac {14 b}{x^{2}}+21\right ) {\mathrm e}^{\frac {b}{x^{2}}}}{112 \sqrt {\pi }\, x \,b^{3}}+\frac {\sqrt {2}\, \left (i b \right )^{\frac {7}{2}} \left (\frac {14 b}{x^{2}}+21\right ) {\mathrm e}^{-\frac {b}{x^{2}}}}{112 \sqrt {\pi }\, x \,b^{3}}-\frac {3 \left (i b \right )^{\frac {7}{2}} \sqrt {2}\, \operatorname {erf}\left (\frac {\sqrt {b}}{x}\right )}{32 b^{\frac {7}{2}}}+\frac {3 \left (i b \right )^{\frac {7}{2}} \sqrt {2}\, \operatorname {erfi}\left (\frac {\sqrt {b}}{x}\right )}{32 b^{\frac {7}{2}}}\right )}{b^{3}}-\frac {i \sqrt {\pi }\, \sinh \left (a \right ) \sqrt {2}\, \sqrt {i b}\, \left (-\frac {\sqrt {2}\, \left (i b \right )^{\frac {5}{2}} \left (\frac {10 b}{x^{2}}+15\right ) {\mathrm e}^{-\frac {b}{x^{2}}}}{80 \sqrt {\pi }\, x \,b^{2}}-\frac {\sqrt {2}\, \left (i b \right )^{\frac {5}{2}} \left (-\frac {10 b}{x^{2}}+15\right ) {\mathrm e}^{\frac {b}{x^{2}}}}{80 \sqrt {\pi }\, x \,b^{2}}+\frac {3 \left (i b \right )^{\frac {5}{2}} \sqrt {2}\, \operatorname {erf}\left (\frac {\sqrt {b}}{x}\right )}{32 b^{\frac {5}{2}}}+\frac {3 \left (i b \right )^{\frac {5}{2}} \sqrt {2}\, \operatorname {erfi}\left (\frac {\sqrt {b}}{x}\right )}{32 b^{\frac {5}{2}}}\right )}{b^{3}}\)	\(269\)

Input:

int(sinh(a+b/x^2)/x^6,x,method=_RETURNVERBOSE)
 

Output:

-1/4/exp(a)/b/x^3*exp(-b/x^2)-3/8/exp(a)/b^2/x*exp(-b/x^2)+3/16*Pi^(1/2)*e 
rf(b^(1/2)/x)/b^(5/2)/exp(a)-1/4*exp(a)*exp(b/x^2)/x^3/b+3/8*exp(a)/b^2*ex 
p(b/x^2)/x-3/16*exp(a)/b^2*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. 313 vs. \(2 (71) = 142\).

Time = 0.11 (sec) , antiderivative size = 313, normalized size of antiderivative = 3.37 \[ \int \frac {\sinh \left (a+\frac {b}{x^2}\right )}{x^6} \, dx=-\frac {6 \, b x^{2} - 2 \, {\left (3 \, b x^{2} - 2 \, b^{2}\right )} \cosh \left (\frac {a x^{2} + b}{x^{2}}\right )^{2} - 3 \, \sqrt {\pi } {\left (x^{3} \cosh \left (a\right ) \cosh \left (\frac {a x^{2} + b}{x^{2}}\right ) + x^{3} \cosh \left (\frac {a x^{2} + b}{x^{2}}\right ) \sinh \left (a\right ) + {\left (x^{3} \cosh \left (a\right ) + x^{3} \sinh \left (a\right )\right )} \sinh \left (\frac {a x^{2} + b}{x^{2}}\right )\right )} \sqrt {-b} \operatorname {erf}\left (\frac {\sqrt {-b}}{x}\right ) - 3 \, \sqrt {\pi } {\left (x^{3} \cosh \left (a\right ) \cosh \left (\frac {a x^{2} + b}{x^{2}}\right ) - x^{3} \cosh \left (\frac {a x^{2} + b}{x^{2}}\right ) \sinh \left (a\right ) + {\left (x^{3} \cosh \left (a\right ) - x^{3} \sinh \left (a\right )\right )} \sinh \left (\frac {a x^{2} + b}{x^{2}}\right )\right )} \sqrt {b} \operatorname {erf}\left (\frac {\sqrt {b}}{x}\right ) - 4 \, {\left (3 \, b x^{2} - 2 \, b^{2}\right )} \cosh \left (\frac {a x^{2} + b}{x^{2}}\right ) \sinh \left (\frac {a x^{2} + b}{x^{2}}\right ) - 2 \, {\left (3 \, b x^{2} - 2 \, b^{2}\right )} \sinh \left (\frac {a x^{2} + b}{x^{2}}\right )^{2} + 4 \, b^{2}}{16 \, {\left (b^{3} x^{3} \cosh \left (\frac {a x^{2} + b}{x^{2}}\right ) + b^{3} x^{3} \sinh \left (\frac {a x^{2} + b}{x^{2}}\right )\right )}} \] Input:

integrate(sinh(a+b/x^2)/x^6,x, algorithm="fricas")
 

Output:

-1/16*(6*b*x^2 - 2*(3*b*x^2 - 2*b^2)*cosh((a*x^2 + b)/x^2)^2 - 3*sqrt(pi)* 
(x^3*cosh(a)*cosh((a*x^2 + b)/x^2) + x^3*cosh((a*x^2 + b)/x^2)*sinh(a) + ( 
x^3*cosh(a) + x^3*sinh(a))*sinh((a*x^2 + b)/x^2))*sqrt(-b)*erf(sqrt(-b)/x) 
 - 3*sqrt(pi)*(x^3*cosh(a)*cosh((a*x^2 + b)/x^2) - x^3*cosh((a*x^2 + b)/x^ 
2)*sinh(a) + (x^3*cosh(a) - x^3*sinh(a))*sinh((a*x^2 + b)/x^2))*sqrt(b)*er 
f(sqrt(b)/x) - 4*(3*b*x^2 - 2*b^2)*cosh((a*x^2 + b)/x^2)*sinh((a*x^2 + b)/ 
x^2) - 2*(3*b*x^2 - 2*b^2)*sinh((a*x^2 + b)/x^2)^2 + 4*b^2)/(b^3*x^3*cosh( 
(a*x^2 + b)/x^2) + b^3*x^3*sinh((a*x^2 + b)/x^2))
 

Sympy [F]

\[ \int \frac {\sinh \left (a+\frac {b}{x^2}\right )}{x^6} \, dx=\int \frac {\sinh {\left (a + \frac {b}{x^{2}} \right )}}{x^{6}}\, dx \] Input:

integrate(sinh(a+b/x**2)/x**6,x)
 

Output:

Integral(sinh(a + b/x**2)/x**6, x)
 

Maxima [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.67 \[ \int \frac {\sinh \left (a+\frac {b}{x^2}\right )}{x^6} \, dx=-\frac {1}{10} \, b {\left (\frac {e^{\left (-a\right )} \Gamma \left (\frac {7}{2}, \frac {b}{x^{2}}\right )}{x^{7} \left (\frac {b}{x^{2}}\right )^{\frac {7}{2}}} + \frac {e^{a} \Gamma \left (\frac {7}{2}, -\frac {b}{x^{2}}\right )}{x^{7} \left (-\frac {b}{x^{2}}\right )^{\frac {7}{2}}}\right )} - \frac {\sinh \left (a + \frac {b}{x^{2}}\right )}{5 \, x^{5}} \] Input:

integrate(sinh(a+b/x^2)/x^6,x, algorithm="maxima")
 

Output:

-1/10*b*(e^(-a)*gamma(7/2, b/x^2)/(x^7*(b/x^2)^(7/2)) + e^a*gamma(7/2, -b/ 
x^2)/(x^7*(-b/x^2)^(7/2))) - 1/5*sinh(a + b/x^2)/x^5
 

Giac [F]

\[ \int \frac {\sinh \left (a+\frac {b}{x^2}\right )}{x^6} \, dx=\int { \frac {\sinh \left (a + \frac {b}{x^{2}}\right )}{x^{6}} \,d x } \] Input:

integrate(sinh(a+b/x^2)/x^6,x, algorithm="giac")
 

Output:

integrate(sinh(a + b/x^2)/x^6, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sinh \left (a+\frac {b}{x^2}\right )}{x^6} \, dx=\int \frac {\mathrm {sinh}\left (a+\frac {b}{x^2}\right )}{x^6} \,d x \] Input:

int(sinh(a + b/x^2)/x^6,x)
 

Output:

int(sinh(a + b/x^2)/x^6, x)
                                                                                    
                                                                                    
 

Reduce [F]

\[ \int \frac {\sinh \left (a+\frac {b}{x^2}\right )}{x^6} \, dx=\int \frac {\sinh \left (\frac {a \,x^{2}+b}{x^{2}}\right )}{x^{6}}d x \] Input:

int(sinh(a+b/x^2)/x^6,x)
 

Output:

int(sinh((a*x**2 + b)/x**2)/x**6,x)