\(\int x^2 \sinh (a+\frac {b}{x^2}) \, dx\) [43]

Optimal result

Integrand size = 12, antiderivative size = 86 \[ \int x^2 \sinh \left (a+\frac {b}{x^2}\right ) \, dx=\frac {2}{3} b x \cosh \left (a+\frac {b}{x^2}\right )+\frac {1}{3} b^{3/2} e^{-a} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {b}}{x}\right )-\frac {1}{3} b^{3/2} e^a \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b}}{x}\right )+\frac {1}{3} x^3 \sinh \left (a+\frac {b}{x^2}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5454, 5434, 5435, 5406, 2235, 2236} \[ \int x^2 \sinh \left (a+\frac {b}{x^2}\right ) \, dx=\frac {1}{3} \sqrt {\pi } e^{-a} b^{3/2} \text {erf}\left (\frac {\sqrt {b}}{x}\right )-\frac {1}{3} \sqrt {\pi } e^a b^{3/2} \text {erfi}\left (\frac {\sqrt {b}}{x}\right )+\frac {2}{3} b x \cosh \left (a+\frac {b}{x^2}\right )+\frac {1}{3} x^3 \sinh \left (a+\frac {b}{x^2}\right ) \]

[In]

Int[x^2*Sinh[a + b/x^2],x]

[Out]

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

Rule 2235

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 2236

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] /; FreeQ[{F, a, b, c, d}, x] && NegQ[b]

Rule 5406

Int[Sinh[(c_.) + (d_.)*(x_)^(n_)], x_Symbol] :> Dist[1/2, Int[E^(c + d*x^n), x], x] - Dist[1/2, Int[E^(-c - d*
x^n), x], x] /; FreeQ[{c, d}, x] && IGtQ[n, 1]

Rule 5434

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] - Dist[d*(n/(e^n*(m + 1))), Int[(e*x)^(m + n)*Cosh[c + d*x^n], x], x] /; FreeQ[{c, d, e}, x] && IGtQ[
n, 0] && LtQ[m, -1]

Rule 5435

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

Rule 5454

Int[(x_)^(m_.)*((a_.) + (b_.)*Sinh[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol] :> -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]

Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {\sinh \left (a+b x^2\right )}{x^4} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {1}{3} x^3 \sinh \left (a+\frac {b}{x^2}\right )-\frac {1}{3} (2 b) \text {Subst}\left (\int \frac {\cosh \left (a+b x^2\right )}{x^2} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {2}{3} b x \cosh \left (a+\frac {b}{x^2}\right )+\frac {1}{3} x^3 \sinh \left (a+\frac {b}{x^2}\right )-\frac {1}{3} \left (4 b^2\right ) \text {Subst}\left (\int \sinh \left (a+b x^2\right ) \, dx,x,\frac {1}{x}\right ) \\ & = \frac {2}{3} b x \cosh \left (a+\frac {b}{x^2}\right )+\frac {1}{3} x^3 \sinh \left (a+\frac {b}{x^2}\right )+\frac {1}{3} \left (2 b^2\right ) \text {Subst}\left (\int e^{-a-b x^2} \, dx,x,\frac {1}{x}\right )-\frac {1}{3} \left (2 b^2\right ) \text {Subst}\left (\int e^{a+b x^2} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {2}{3} b x \cosh \left (a+\frac {b}{x^2}\right )+\frac {1}{3} b^{3/2} e^{-a} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {b}}{x}\right )-\frac {1}{3} b^{3/2} e^a \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b}}{x}\right )+\frac {1}{3} x^3 \sinh \left (a+\frac {b}{x^2}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.98 \[ \int x^2 \sinh \left (a+\frac {b}{x^2}\right ) \, dx=\frac {1}{3} \left (2 b x \cosh \left (a+\frac {b}{x^2}\right )+b^{3/2} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {b}}{x}\right ) (\cosh (a)-\sinh (a))-b^{3/2} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b}}{x}\right ) (\cosh (a)+\sinh (a))+x^3 \sinh \left (a+\frac {b}{x^2}\right )\right ) \]

[In]

Integrate[x^2*Sinh[a + b/x^2],x]

[Out]

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

Maple [A] (verified)

Time = 0.57 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.20

method	result	size
risch	\(-\frac {{\mathrm e}^{-a} x^{3} {\mathrm e}^{-\frac {b}{x^{2}}}}{6}+\frac {b^{\frac {3}{2}} \operatorname {erf}\left (\frac {\sqrt {b}}{x}\right ) \sqrt {\pi }\, {\mathrm e}^{-a}}{3}+\frac {{\mathrm e}^{-a} {\mathrm e}^{-\frac {b}{x^{2}}} b x}{3}+\frac {{\mathrm e}^{a} x^{3} {\mathrm e}^{\frac {b}{x^{2}}}}{6}+\frac {{\mathrm e}^{a} b x \,{\mathrm e}^{\frac {b}{x^{2}}}}{3}-\frac {{\mathrm e}^{a} b^{2} \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {-b}}{x}\right )}{3 \sqrt {-b}}\)	\(103\)
meijerg	\(-\frac {b \sqrt {\pi }\, \cosh \left (a \right ) \sqrt {2}\, \sqrt {i b}\, \left (-\frac {4 x^{3} \sqrt {2}\, \left (\frac {2 b}{x^{2}}+1\right ) {\mathrm e}^{\frac {b}{x^{2}}}}{3 \sqrt {\pi }\, \sqrt {i b}\, b}+\frac {4 x^{3} \sqrt {2}\, \left (-\frac {2 b}{x^{2}}+1\right ) {\mathrm e}^{-\frac {b}{x^{2}}}}{3 \sqrt {\pi }\, \sqrt {i b}\, b}-\frac {8 \sqrt {2}\, \sqrt {b}\, \operatorname {erf}\left (\frac {\sqrt {b}}{x}\right )}{3 \sqrt {i b}}+\frac {8 \sqrt {2}\, \sqrt {b}\, \operatorname {erfi}\left (\frac {\sqrt {b}}{x}\right )}{3 \sqrt {i b}}\right )}{16}-\frac {i b \sqrt {\pi }\, \sinh \left (a \right ) \sqrt {2}\, \sqrt {i b}\, \left (-\frac {8 x^{3} \sqrt {2}\, \left (-\frac {b}{x^{2}}+\frac {1}{2}\right ) {\mathrm e}^{-\frac {b}{x^{2}}}}{3 \sqrt {\pi }\, \left (i b \right )^{\frac {3}{2}}}-\frac {8 x^{3} \sqrt {2}\, \left (\frac {b}{x^{2}}+\frac {1}{2}\right ) {\mathrm e}^{\frac {b}{x^{2}}}}{3 \sqrt {\pi }\, \left (i b \right )^{\frac {3}{2}}}+\frac {8 \sqrt {2}\, b^{\frac {3}{2}} \operatorname {erf}\left (\frac {\sqrt {b}}{x}\right )}{3 \left (i b \right )^{\frac {3}{2}}}+\frac {8 \sqrt {2}\, b^{\frac {3}{2}} \operatorname {erfi}\left (\frac {\sqrt {b}}{x}\right )}{3 \left (i b \right )^{\frac {3}{2}}}\right )}{16}\)	\(258\)

[In]

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

[Out]

-1/6/exp(a)*x^3*exp(-b/x^2)+1/3*b^(3/2)*erf(b^(1/2)/x)*Pi^(1/2)/exp(a)+1/3/exp(a)*exp(-b/x^2)*b*x+1/6*exp(a)*x
^3*exp(b/x^2)+1/3*exp(a)*b*x*exp(b/x^2)-1/3*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. 267 vs. \(2 (64) = 128\).

Time = 0.25 (sec) , antiderivative size = 267, normalized size of antiderivative = 3.10 \[ \int x^2 \sinh \left (a+\frac {b}{x^2}\right ) \, dx=-\frac {x^{3} - {\left (x^{3} + 2 \, b x\right )} \cosh \left (\frac {a x^{2} + b}{x^{2}}\right )^{2} - 2 \, \sqrt {\pi } {\left (b \cosh \left (a\right ) \cosh \left (\frac {a x^{2} + b}{x^{2}}\right ) + b \cosh \left (\frac {a x^{2} + b}{x^{2}}\right ) \sinh \left (a\right ) + {\left (b \cosh \left (a\right ) + b \sinh \left (a\right )\right )} \sinh \left (\frac {a x^{2} + b}{x^{2}}\right )\right )} \sqrt {-b} \operatorname {erf}\left (\frac {\sqrt {-b}}{x}\right ) - 2 \, \sqrt {\pi } {\left (b \cosh \left (a\right ) \cosh \left (\frac {a x^{2} + b}{x^{2}}\right ) - b \cosh \left (\frac {a x^{2} + b}{x^{2}}\right ) \sinh \left (a\right ) + {\left (b \cosh \left (a\right ) - b \sinh \left (a\right )\right )} \sinh \left (\frac {a x^{2} + b}{x^{2}}\right )\right )} \sqrt {b} \operatorname {erf}\left (\frac {\sqrt {b}}{x}\right ) - 2 \, {\left (x^{3} + 2 \, b x\right )} \cosh \left (\frac {a x^{2} + b}{x^{2}}\right ) \sinh \left (\frac {a x^{2} + b}{x^{2}}\right ) - {\left (x^{3} + 2 \, b x\right )} \sinh \left (\frac {a x^{2} + b}{x^{2}}\right )^{2} - 2 \, b x}{6 \, {\left (\cosh \left (\frac {a x^{2} + b}{x^{2}}\right ) + \sinh \left (\frac {a x^{2} + b}{x^{2}}\right )\right )}} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

Integral(x**2*sinh(a + b/x**2), x)

Maxima [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.67 \[ \int x^2 \sinh \left (a+\frac {b}{x^2}\right ) \, dx=\frac {1}{3} \, x^{3} \sinh \left (a + \frac {b}{x^{2}}\right ) + \frac {1}{6} \, {\left (x \sqrt {\frac {b}{x^{2}}} e^{\left (-a\right )} \Gamma \left (-\frac {1}{2}, \frac {b}{x^{2}}\right ) + x \sqrt {-\frac {b}{x^{2}}} e^{a} \Gamma \left (-\frac {1}{2}, -\frac {b}{x^{2}}\right )\right )} b \]

[In]

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

[Out]

1/3*x^3*sinh(a + b/x^2) + 1/6*(x*sqrt(b/x^2)*e^(-a)*gamma(-1/2, b/x^2) + x*sqrt(-b/x^2)*e^a*gamma(-1/2, -b/x^2
))*b

Giac [F]

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

[In]

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

[Out]

integrate(x^2*sinh(a + b/x^2), x)

Mupad [F(-1)]

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

[In]

int(x^2*sinh(a + b/x^2),x)

[Out]

int(x^2*sinh(a + b/x^2), x)