Integrand size = 12, antiderivative size = 86 \[ \int x^2 \sinh \left (a+\frac {b}{x^2}\right ) \, dx=\frac {2}{3} b x \cosh \left (a+\frac {b}{x^2}\right )+\frac {1}{3} b^{3/2} e^{-a} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {b}}{x}\right )-\frac {1}{3} b^{3/2} e^a \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b}}{x}\right )+\frac {1}{3} x^3 \sinh \left (a+\frac {b}{x^2}\right ) \]
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)
Time = 0.06 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.98 \[ \int x^2 \sinh \left (a+\frac {b}{x^2}\right ) \, dx=\frac {1}{3} \left (2 b x \cosh \left (a+\frac {b}{x^2}\right )+b^{3/2} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {b}}{x}\right ) (\cosh (a)-\sinh (a))-b^{3/2} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b}}{x}\right ) (\cosh (a)+\sinh (a))+x^3 \sinh \left (a+\frac {b}{x^2}\right )\right ) \]
(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
Time = 0.45 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.08, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5869, 5849, 5850, 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.
\(\displaystyle \int x^2 \sinh \left (a+\frac {b}{x^2}\right ) \, dx\) |
\(\Big \downarrow \) 5869 |
\(\displaystyle -\int x^4 \sinh \left (a+\frac {b}{x^2}\right )d\frac {1}{x}\) |
\(\Big \downarrow \) 5849 |
\(\displaystyle \frac {1}{3} x^3 \sinh \left (a+\frac {b}{x^2}\right )-\frac {2}{3} b \int x^2 \cosh \left (a+\frac {b}{x^2}\right )d\frac {1}{x}\) |
\(\Big \downarrow \) 5850 |
\(\displaystyle \frac {1}{3} x^3 \sinh \left (a+\frac {b}{x^2}\right )-\frac {2}{3} b \left (2 b \int \sinh \left (a+\frac {b}{x^2}\right )d\frac {1}{x}-x \cosh \left (a+\frac {b}{x^2}\right )\right )\) |
\(\Big \downarrow \) 5821 |
\(\displaystyle \frac {1}{3} x^3 \sinh \left (a+\frac {b}{x^2}\right )-\frac {2}{3} b \left (2 b \left (\frac {1}{2} \int e^{a+\frac {b}{x^2}}d\frac {1}{x}-\frac {1}{2} \int e^{-a-\frac {b}{x^2}}d\frac {1}{x}\right )-x \cosh \left (a+\frac {b}{x^2}\right )\right )\) |
\(\Big \downarrow \) 2633 |
\(\displaystyle \frac {1}{3} x^3 \sinh \left (a+\frac {b}{x^2}\right )-\frac {2}{3} b \left (2 b \left (\frac {\sqrt {\pi } e^a \text {erfi}\left (\frac {\sqrt {b}}{x}\right )}{4 \sqrt {b}}-\frac {1}{2} \int e^{-a-\frac {b}{x^2}}d\frac {1}{x}\right )-x \cosh \left (a+\frac {b}{x^2}\right )\right )\) |
\(\Big \downarrow \) 2634 |
\(\displaystyle \frac {1}{3} x^3 \sinh \left (a+\frac {b}{x^2}\right )-\frac {2}{3} b \left (2 b \left (\frac {\sqrt {\pi } e^a \text {erfi}\left (\frac {\sqrt {b}}{x}\right )}{4 \sqrt {b}}-\frac {\sqrt {\pi } e^{-a} \text {erf}\left (\frac {\sqrt {b}}{x}\right )}{4 \sqrt {b}}\right )-x \cosh \left (a+\frac {b}{x^2}\right )\right )\) |
(-2*b*(-(x*Cosh[a + b/x^2]) + 2*b*(-1/4*(Sqrt[Pi]*Erf[Sqrt[b]/x])/(Sqrt[b] *E^a) + (E^a*Sqrt[Pi]*Erfi[Sqrt[b]/x])/(4*Sqrt[b]))))/3 + (x^3*Sinh[a + b/ x^2])/3
3.1.43.3.1 Defintions of rubi rules used
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[((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]
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] - Simp[d*(n/(e^n*(m + 1))) In t[(e*x)^(m + n)*Sinh[c + d*x^n], x], x] /; FreeQ[{c, d, e}, x] && IGtQ[n, 0 ] && LtQ[m, -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 = 0.57 (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 | \(-\frac {b \sqrt {\pi }\, \cosh \left (a \right ) \sqrt {2}\, \sqrt {i b}\, \left (-\frac {4 x^{3} \sqrt {2}\, \left (\frac {2 b}{x^{2}}+1\right ) {\mathrm e}^{\frac {b}{x^{2}}}}{3 \sqrt {\pi }\, \sqrt {i b}\, b}+\frac {4 x^{3} \sqrt {2}\, \left (-\frac {2 b}{x^{2}}+1\right ) {\mathrm e}^{-\frac {b}{x^{2}}}}{3 \sqrt {\pi }\, \sqrt {i b}\, b}-\frac {8 \sqrt {2}\, \sqrt {b}\, \operatorname {erf}\left (\frac {\sqrt {b}}{x}\right )}{3 \sqrt {i b}}+\frac {8 \sqrt {2}\, \sqrt {b}\, \operatorname {erfi}\left (\frac {\sqrt {b}}{x}\right )}{3 \sqrt {i b}}\right )}{16}-\frac {i b \sqrt {\pi }\, \sinh \left (a \right ) \sqrt {2}\, \sqrt {i b}\, \left (-\frac {8 x^{3} \sqrt {2}\, \left (-\frac {b}{x^{2}}+\frac {1}{2}\right ) {\mathrm e}^{-\frac {b}{x^{2}}}}{3 \sqrt {\pi }\, \left (i b \right )^{\frac {3}{2}}}-\frac {8 x^{3} \sqrt {2}\, \left (\frac {b}{x^{2}}+\frac {1}{2}\right ) {\mathrm e}^{\frac {b}{x^{2}}}}{3 \sqrt {\pi }\, \left (i b \right )^{\frac {3}{2}}}+\frac {8 \sqrt {2}\, b^{\frac {3}{2}} \operatorname {erf}\left (\frac {\sqrt {b}}{x}\right )}{3 \left (i b \right )^{\frac {3}{2}}}+\frac {8 \sqrt {2}\, b^{\frac {3}{2}} \operatorname {erfi}\left (\frac {\sqrt {b}}{x}\right )}{3 \left (i b \right )^{\frac {3}{2}}}\right )}{16}\) | \(258\) |
-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)
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 \[ \int x^2 \sinh \left (a+\frac {b}{x^2}\right ) \, dx=-\frac {x^{3} - {\left (x^{3} + 2 \, b x\right )} \cosh \left (\frac {a x^{2} + b}{x^{2}}\right )^{2} - 2 \, \sqrt {\pi } {\left (b \cosh \left (a\right ) \cosh \left (\frac {a x^{2} + b}{x^{2}}\right ) + b \cosh \left (\frac {a x^{2} + b}{x^{2}}\right ) \sinh \left (a\right ) + {\left (b \cosh \left (a\right ) + b \sinh \left (a\right )\right )} \sinh \left (\frac {a x^{2} + b}{x^{2}}\right )\right )} \sqrt {-b} \operatorname {erf}\left (\frac {\sqrt {-b}}{x}\right ) - 2 \, \sqrt {\pi } {\left (b \cosh \left (a\right ) \cosh \left (\frac {a x^{2} + b}{x^{2}}\right ) - b \cosh \left (\frac {a x^{2} + b}{x^{2}}\right ) \sinh \left (a\right ) + {\left (b \cosh \left (a\right ) - b \sinh \left (a\right )\right )} \sinh \left (\frac {a x^{2} + b}{x^{2}}\right )\right )} \sqrt {b} \operatorname {erf}\left (\frac {\sqrt {b}}{x}\right ) - 2 \, {\left (x^{3} + 2 \, b x\right )} \cosh \left (\frac {a x^{2} + b}{x^{2}}\right ) \sinh \left (\frac {a x^{2} + b}{x^{2}}\right ) - {\left (x^{3} + 2 \, b x\right )} \sinh \left (\frac {a x^{2} + b}{x^{2}}\right )^{2} - 2 \, b x}{6 \, {\left (\cosh \left (\frac {a x^{2} + b}{x^{2}}\right ) + \sinh \left (\frac {a x^{2} + b}{x^{2}}\right )\right )}} \]
-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*s inh(a))*sinh((a*x^2 + b)/x^2))*sqrt(-b)*erf(sqrt(-b)/x) - 2*sqrt(pi)*(b*co sh(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))
\[ \int x^2 \sinh \left (a+\frac {b}{x^2}\right ) \, dx=\int x^{2} \sinh {\left (a + \frac {b}{x^{2}} \right )}\, dx \]
Time = 0.25 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.67 \[ \int x^2 \sinh \left (a+\frac {b}{x^2}\right ) \, dx=\frac {1}{3} \, x^{3} \sinh \left (a + \frac {b}{x^{2}}\right ) + \frac {1}{6} \, {\left (x \sqrt {\frac {b}{x^{2}}} e^{\left (-a\right )} \Gamma \left (-\frac {1}{2}, \frac {b}{x^{2}}\right ) + x \sqrt {-\frac {b}{x^{2}}} e^{a} \Gamma \left (-\frac {1}{2}, -\frac {b}{x^{2}}\right )\right )} b \]
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
\[ \int x^2 \sinh \left (a+\frac {b}{x^2}\right ) \, dx=\int { x^{2} \sinh \left (a + \frac {b}{x^{2}}\right ) \,d x } \]
Timed out. \[ \int x^2 \sinh \left (a+\frac {b}{x^2}\right ) \, dx=\int x^2\,\mathrm {sinh}\left (a+\frac {b}{x^2}\right ) \,d x \]