Integrand size = 10, antiderivative size = 42 \[ \int x \sinh \left (a+\frac {b}{x^2}\right ) \, dx=-\frac {1}{2} b \cosh (a) \text {Chi}\left (\frac {b}{x^2}\right )+\frac {1}{2} x^2 \sinh \left (a+\frac {b}{x^2}\right )-\frac {1}{2} b \sinh (a) \text {Shi}\left (\frac {b}{x^2}\right ) \]
Time = 0.02 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.93 \[ \int x \sinh \left (a+\frac {b}{x^2}\right ) \, dx=\frac {1}{2} \left (-b \cosh (a) \text {Chi}\left (\frac {b}{x^2}\right )+x^2 \sinh \left (a+\frac {b}{x^2}\right )-b \sinh (a) \text {Shi}\left (\frac {b}{x^2}\right )\right ) \]
Result contains complex when optimal does not.
Time = 0.45 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.10, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.100, Rules used = {5843, 3042, 26, 3778, 3042, 3784, 26, 3042, 26, 3779, 3782}
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.
\(\displaystyle \int x \sinh \left (a+\frac {b}{x^2}\right ) \, dx\) |
\(\Big \downarrow \) 5843 |
\(\displaystyle -\frac {1}{2} \int x^4 \sinh \left (a+\frac {b}{x^2}\right )d\frac {1}{x^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {1}{2} \int -i x^4 \sin \left (i a+\frac {i b}{x^2}\right )d\frac {1}{x^2}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \frac {1}{2} i \int x^4 \sin \left (i a+\frac {i b}{x^2}\right )d\frac {1}{x^2}\) |
\(\Big \downarrow \) 3778 |
\(\displaystyle \frac {1}{2} i \left (i b \int x^2 \cosh \left (a+\frac {b}{x^2}\right )d\frac {1}{x^2}-i x^2 \sinh \left (a+\frac {b}{x^2}\right )\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{2} i \left (i b \int x^2 \sin \left (i a+\frac {i b}{x^2}+\frac {\pi }{2}\right )d\frac {1}{x^2}-i x^2 \sinh \left (a+\frac {b}{x^2}\right )\right )\) |
\(\Big \downarrow \) 3784 |
\(\displaystyle \frac {1}{2} i \left (i b \left (\cosh (a) \int x^2 \cosh \left (\frac {b}{x^2}\right )d\frac {1}{x^2}-i \sinh (a) \int i x^2 \sinh \left (\frac {b}{x^2}\right )d\frac {1}{x^2}\right )-i x^2 \sinh \left (a+\frac {b}{x^2}\right )\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \frac {1}{2} i \left (i b \left (\sinh (a) \int x^2 \sinh \left (\frac {b}{x^2}\right )d\frac {1}{x^2}+\cosh (a) \int x^2 \cosh \left (\frac {b}{x^2}\right )d\frac {1}{x^2}\right )-i x^2 \sinh \left (a+\frac {b}{x^2}\right )\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{2} i \left (i b \left (\sinh (a) \int -i x^2 \sin \left (\frac {i b}{x^2}\right )d\frac {1}{x^2}+\cosh (a) \int x^2 \sin \left (\frac {i b}{x^2}+\frac {\pi }{2}\right )d\frac {1}{x^2}\right )-i x^2 \sinh \left (a+\frac {b}{x^2}\right )\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \frac {1}{2} i \left (i b \left (\cosh (a) \int x^2 \sin \left (\frac {i b}{x^2}+\frac {\pi }{2}\right )d\frac {1}{x^2}-i \sinh (a) \int x^2 \sin \left (\frac {i b}{x^2}\right )d\frac {1}{x^2}\right )-i x^2 \sinh \left (a+\frac {b}{x^2}\right )\right )\) |
\(\Big \downarrow \) 3779 |
\(\displaystyle \frac {1}{2} i \left (i b \left (\sinh (a) \text {Shi}\left (\frac {b}{x^2}\right )+\cosh (a) \int x^2 \sin \left (\frac {i b}{x^2}+\frac {\pi }{2}\right )d\frac {1}{x^2}\right )-i x^2 \sinh \left (a+\frac {b}{x^2}\right )\right )\) |
\(\Big \downarrow \) 3782 |
\(\displaystyle \frac {1}{2} i \left (i b \left (\cosh (a) \text {Chi}\left (\frac {b}{x^2}\right )+\sinh (a) \text {Shi}\left (\frac {b}{x^2}\right )\right )-i x^2 \sinh \left (a+\frac {b}{x^2}\right )\right )\) |
(I/2)*((-I)*x^2*Sinh[a + b/x^2] + I*b*(Cosh[a]*CoshIntegral[b/x^2] + Sinh[ a]*SinhIntegral[b/x^2]))
3.1.44.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(c + d*x)^(m + 1)*(Sin[e + f*x]/(d*(m + 1))), x] - Simp[f/(d*(m + 1)) Int[( c + d*x)^(m + 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && LtQ[m, - 1]
Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbo l] :> Simp[I*(SinhIntegral[c*f*(fz/d) + f*fz*x]/d), x] /; FreeQ[{c, d, e, f , fz}, x] && EqQ[d*e - c*f*fz*I, 0]
Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbo l] :> Simp[CoshIntegral[c*f*(fz/d) + f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz }, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]
Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[Cos[(d* e - c*f)/d] Int[Sin[c*(f/d) + f*x]/(c + d*x), x], x] + Simp[Sin[(d*e - c* f)/d] Int[Cos[c*(f/d) + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f}, x] && NeQ[d*e - c*f, 0]
Int[(x_)^(m_.)*((a_.) + (b_.)*Sinh[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbo l] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*Sinh[c + d*x] )^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IntegerQ[Simplif y[(m + 1)/n]] && (EqQ[p, 1] || EqQ[m, n - 1] || (IntegerQ[p] && GtQ[Simplif y[(m + 1)/n], 0]))
Time = 0.59 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.74
method | result | size |
risch | \(\frac {{\mathrm e}^{\frac {2 a \,x^{2}+b}{x^{2}}} {\mathrm e}^{-a} x^{2}}{4}+\frac {{\mathrm e}^{2 a} {\mathrm e}^{-a} \operatorname {Ei}_{1}\left (-\frac {b}{x^{2}}\right ) b}{4}-\frac {{\mathrm e}^{-a} x^{2} {\mathrm e}^{-\frac {b}{x^{2}}}}{4}+\frac {{\mathrm e}^{-a} b \,\operatorname {Ei}_{1}\left (\frac {b}{x^{2}}\right )}{4}\) | \(73\) |
meijerg | \(-\frac {b \sqrt {\pi }\, \cosh \left (a \right ) \left (\frac {4}{\sqrt {\pi }}-\frac {4 x^{2} \sinh \left (\frac {b}{x^{2}}\right )}{\sqrt {\pi }\, b}+\frac {4 \,\operatorname {Chi}\left (\frac {b}{x^{2}}\right )-4 \ln \left (\frac {b}{x^{2}}\right )-4 \gamma }{\sqrt {\pi }}+\frac {4 \gamma -4-8 \ln \left (x \right )+4 \ln \left (i b \right )}{\sqrt {\pi }}\right )}{8}-\frac {i b \sqrt {\pi }\, \sinh \left (a \right ) \left (\frac {4 i x^{2} \cosh \left (\frac {b}{x^{2}}\right )}{b \sqrt {\pi }}-\frac {4 i \operatorname {Shi}\left (\frac {b}{x^{2}}\right )}{\sqrt {\pi }}\right )}{8}\) | \(117\) |
1/4*exp((2*a*x^2+b)/x^2)*exp(-a)*x^2+1/4*exp(2*a)*exp(-a)*Ei(1,-b/x^2)*b-1 /4*exp(-a)*x^2*exp(-b/x^2)+1/4*exp(-a)*b*Ei(1,b/x^2)
Time = 0.24 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.50 \[ \int x \sinh \left (a+\frac {b}{x^2}\right ) \, dx=\frac {1}{2} \, x^{2} \sinh \left (\frac {a x^{2} + b}{x^{2}}\right ) - \frac {1}{4} \, {\left (b {\rm Ei}\left (\frac {b}{x^{2}}\right ) + b {\rm Ei}\left (-\frac {b}{x^{2}}\right )\right )} \cosh \left (a\right ) - \frac {1}{4} \, {\left (b {\rm Ei}\left (\frac {b}{x^{2}}\right ) - b {\rm Ei}\left (-\frac {b}{x^{2}}\right )\right )} \sinh \left (a\right ) \]
1/2*x^2*sinh((a*x^2 + b)/x^2) - 1/4*(b*Ei(b/x^2) + b*Ei(-b/x^2))*cosh(a) - 1/4*(b*Ei(b/x^2) - b*Ei(-b/x^2))*sinh(a)
\[ \int x \sinh \left (a+\frac {b}{x^2}\right ) \, dx=\int x \sinh {\left (a + \frac {b}{x^{2}} \right )}\, dx \]
Time = 0.25 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.93 \[ \int x \sinh \left (a+\frac {b}{x^2}\right ) \, dx=\frac {1}{2} \, x^{2} \sinh \left (a + \frac {b}{x^{2}}\right ) - \frac {1}{4} \, {\left ({\rm Ei}\left (-\frac {b}{x^{2}}\right ) e^{\left (-a\right )} + {\rm Ei}\left (\frac {b}{x^{2}}\right ) e^{a}\right )} b \]
Leaf count of result is larger than twice the leaf count of optimal. 193 vs. \(2 (36) = 72\).
Time = 0.27 (sec) , antiderivative size = 193, normalized size of antiderivative = 4.60 \[ \int x \sinh \left (a+\frac {b}{x^2}\right ) \, dx=-\frac {a b^{2} {\rm Ei}\left (a - \frac {a x^{2} + b}{x^{2}}\right ) e^{\left (-a\right )} - \frac {{\left (a x^{2} + b\right )} b^{2} {\rm Ei}\left (a - \frac {a x^{2} + b}{x^{2}}\right ) e^{\left (-a\right )}}{x^{2}} - b^{2} e^{\left (-\frac {a x^{2} + b}{x^{2}}\right )}}{4 \, {\left (a - \frac {a x^{2} + b}{x^{2}}\right )} b} - \frac {a b^{2} {\rm Ei}\left (-a + \frac {a x^{2} + b}{x^{2}}\right ) e^{a} - \frac {{\left (a x^{2} + b\right )} b^{2} {\rm Ei}\left (-a + \frac {a x^{2} + b}{x^{2}}\right ) e^{a}}{x^{2}} + b^{2} e^{\left (\frac {a x^{2} + b}{x^{2}}\right )}}{4 \, {\left (a - \frac {a x^{2} + b}{x^{2}}\right )} b} \]
-1/4*(a*b^2*Ei(a - (a*x^2 + b)/x^2)*e^(-a) - (a*x^2 + b)*b^2*Ei(a - (a*x^2 + b)/x^2)*e^(-a)/x^2 - b^2*e^(-(a*x^2 + b)/x^2))/((a - (a*x^2 + b)/x^2)*b ) - 1/4*(a*b^2*Ei(-a + (a*x^2 + b)/x^2)*e^a - (a*x^2 + b)*b^2*Ei(-a + (a*x ^2 + b)/x^2)*e^a/x^2 + b^2*e^((a*x^2 + b)/x^2))/((a - (a*x^2 + b)/x^2)*b)
Timed out. \[ \int x \sinh \left (a+\frac {b}{x^2}\right ) \, dx=\int x\,\mathrm {sinh}\left (a+\frac {b}{x^2}\right ) \,d x \]