Integrand size = 8, antiderivative size = 33 \[ \int \sinh \left (a+\frac {b}{x}\right ) \, dx=-b \cosh (a) \text {Chi}\left (\frac {b}{x}\right )+x \sinh \left (a+\frac {b}{x}\right )-b \sinh (a) \text {Shi}\left (\frac {b}{x}\right ) \] Output:
-b*cosh(a)*Chi(b/x)+x*sinh(a+b/x)-b*sinh(a)*Shi(b/x)
Time = 0.01 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00 \[ \int \sinh \left (a+\frac {b}{x}\right ) \, dx=-b \cosh (a) \text {Chi}\left (\frac {b}{x}\right )+x \sinh \left (a+\frac {b}{x}\right )-b \sinh (a) \text {Shi}\left (\frac {b}{x}\right ) \] Input:
Integrate[Sinh[a + b/x],x]
Output:
-(b*Cosh[a]*CoshIntegral[b/x]) + x*Sinh[a + b/x] - b*Sinh[a]*SinhIntegral[ b/x]
Result contains complex when optimal does not.
Time = 0.44 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.27, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.375, Rules used = {5825, 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.
\(\displaystyle \int \sinh \left (a+\frac {b}{x}\right ) \, dx\) |
\(\Big \downarrow \) 5825 |
\(\displaystyle -\int x^2 \sinh \left (a+\frac {b}{x}\right )d\frac {1}{x}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\int -i x^2 \sin \left (i a+\frac {i b}{x}\right )d\frac {1}{x}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \int x^2 \sin \left (i a+\frac {i b}{x}\right )d\frac {1}{x}\) |
\(\Big \downarrow \) 3778 |
\(\displaystyle i \left (i b \int x \cosh \left (a+\frac {b}{x}\right )d\frac {1}{x}-i x \sinh \left (a+\frac {b}{x}\right )\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle i \left (i b \int x \sin \left (i a+\frac {i b}{x}+\frac {\pi }{2}\right )d\frac {1}{x}-i x \sinh \left (a+\frac {b}{x}\right )\right )\) |
\(\Big \downarrow \) 3784 |
\(\displaystyle i \left (i b \left (\cosh (a) \int x \cosh \left (\frac {b}{x}\right )d\frac {1}{x}-i \sinh (a) \int i x \sinh \left (\frac {b}{x}\right )d\frac {1}{x}\right )-i x \sinh \left (a+\frac {b}{x}\right )\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \left (i b \left (\sinh (a) \int x \sinh \left (\frac {b}{x}\right )d\frac {1}{x}+\cosh (a) \int x \cosh \left (\frac {b}{x}\right )d\frac {1}{x}\right )-i x \sinh \left (a+\frac {b}{x}\right )\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle i \left (i b \left (\sinh (a) \int -i x \sin \left (\frac {i b}{x}\right )d\frac {1}{x}+\cosh (a) \int x \sin \left (\frac {i b}{x}+\frac {\pi }{2}\right )d\frac {1}{x}\right )-i x \sinh \left (a+\frac {b}{x}\right )\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \left (i b \left (\cosh (a) \int x \sin \left (\frac {i b}{x}+\frac {\pi }{2}\right )d\frac {1}{x}-i \sinh (a) \int x \sin \left (\frac {i b}{x}\right )d\frac {1}{x}\right )-i x \sinh \left (a+\frac {b}{x}\right )\right )\) |
\(\Big \downarrow \) 3779 |
\(\displaystyle i \left (i b \left (\sinh (a) \text {Shi}\left (\frac {b}{x}\right )+\cosh (a) \int x \sin \left (\frac {i b}{x}+\frac {\pi }{2}\right )d\frac {1}{x}\right )-i x \sinh \left (a+\frac {b}{x}\right )\right )\) |
\(\Big \downarrow \) 3782 |
\(\displaystyle i \left (i b \left (\cosh (a) \text {Chi}\left (\frac {b}{x}\right )+\sinh (a) \text {Shi}\left (\frac {b}{x}\right )\right )-i x \sinh \left (a+\frac {b}{x}\right )\right )\) |
Input:
Int[Sinh[a + b/x],x]
Output:
I*((-I)*x*Sinh[a + b/x] + I*b*(Cosh[a]*CoshIntegral[b/x] + Sinh[a]*SinhInt egral[b/x]))
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[((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]
Time = 0.20 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.70
method | result | size |
risch | \(\frac {{\mathrm e}^{-a} \operatorname {expIntegral}_{1}\left (\frac {b}{x}\right ) b}{2}-\frac {{\mathrm e}^{-\frac {a x +b}{x}} x}{2}+\frac {{\mathrm e}^{a} \operatorname {expIntegral}_{1}\left (-\frac {b}{x}\right ) b}{2}+\frac {{\mathrm e}^{\frac {a x +b}{x}} x}{2}\) | \(56\) |
meijerg | \(-\frac {\sqrt {\pi }\, \cosh \left (a \right ) b \left (\frac {4 \gamma -4-4 \ln \left (x \right )+4 \ln \left (i b \right )}{\sqrt {\pi }}+\frac {4}{\sqrt {\pi }}-\frac {4 x \sinh \left (\frac {b}{x}\right )}{\sqrt {\pi }\, b}+\frac {4 \,\operatorname {Chi}\left (\frac {b}{x}\right )-4 \ln \left (\frac {b}{x}\right )-4 \gamma }{\sqrt {\pi }}\right )}{4}-\frac {i \sqrt {\pi }\, \sinh \left (a \right ) b \left (\frac {4 i x \cosh \left (\frac {b}{x}\right )}{b \sqrt {\pi }}-\frac {4 i \operatorname {Shi}\left (\frac {b}{x}\right )}{\sqrt {\pi }}\right )}{4}\) | \(113\) |
Input:
int(sinh(a+b/x),x,method=_RETURNVERBOSE)
Output:
1/2*exp(-a)*Ei(1,b/x)*b-1/2*exp(-(a*x+b)/x)*x+1/2*exp(a)*Ei(1,-b/x)*b+1/2* exp((a*x+b)/x)*x
Time = 0.11 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.76 \[ \int \sinh \left (a+\frac {b}{x}\right ) \, dx=-\frac {1}{2} \, {\left (b {\rm Ei}\left (\frac {b}{x}\right ) + b {\rm Ei}\left (-\frac {b}{x}\right )\right )} \cosh \left (a\right ) - \frac {1}{2} \, {\left (b {\rm Ei}\left (\frac {b}{x}\right ) - b {\rm Ei}\left (-\frac {b}{x}\right )\right )} \sinh \left (a\right ) + x \sinh \left (\frac {a x + b}{x}\right ) \] Input:
integrate(sinh(a+b/x),x, algorithm="fricas")
Output:
-1/2*(b*Ei(b/x) + b*Ei(-b/x))*cosh(a) - 1/2*(b*Ei(b/x) - b*Ei(-b/x))*sinh( a) + x*sinh((a*x + b)/x)
\[ \int \sinh \left (a+\frac {b}{x}\right ) \, dx=\int \sinh {\left (a + \frac {b}{x} \right )}\, dx \] Input:
integrate(sinh(a+b/x),x)
Output:
Integral(sinh(a + b/x), x)
Time = 0.07 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.09 \[ \int \sinh \left (a+\frac {b}{x}\right ) \, dx=-\frac {1}{2} \, {\left ({\rm Ei}\left (-\frac {b}{x}\right ) e^{\left (-a\right )} + {\rm Ei}\left (\frac {b}{x}\right ) e^{a}\right )} b + x \sinh \left (a + \frac {b}{x}\right ) \] Input:
integrate(sinh(a+b/x),x, algorithm="maxima")
Output:
-1/2*(Ei(-b/x)*e^(-a) + Ei(b/x)*e^a)*b + x*sinh(a + b/x)
Leaf count of result is larger than twice the leaf count of optimal. 173 vs. \(2 (33) = 66\).
Time = 0.13 (sec) , antiderivative size = 173, normalized size of antiderivative = 5.24 \[ \int \sinh \left (a+\frac {b}{x}\right ) \, dx=-\frac {a b^{2} {\rm Ei}\left (a - \frac {a x + b}{x}\right ) e^{\left (-a\right )} - \frac {{\left (a x + b\right )} b^{2} {\rm Ei}\left (a - \frac {a x + b}{x}\right ) e^{\left (-a\right )}}{x} - b^{2} e^{\left (-\frac {a x + b}{x}\right )}}{2 \, {\left (a - \frac {a x + b}{x}\right )} b} - \frac {a b^{2} {\rm Ei}\left (-a + \frac {a x + b}{x}\right ) e^{a} - \frac {{\left (a x + b\right )} b^{2} {\rm Ei}\left (-a + \frac {a x + b}{x}\right ) e^{a}}{x} + b^{2} e^{\left (\frac {a x + b}{x}\right )}}{2 \, {\left (a - \frac {a x + b}{x}\right )} b} \] Input:
integrate(sinh(a+b/x),x, algorithm="giac")
Output:
-1/2*(a*b^2*Ei(a - (a*x + b)/x)*e^(-a) - (a*x + b)*b^2*Ei(a - (a*x + b)/x) *e^(-a)/x - b^2*e^(-(a*x + b)/x))/((a - (a*x + b)/x)*b) - 1/2*(a*b^2*Ei(-a + (a*x + b)/x)*e^a - (a*x + b)*b^2*Ei(-a + (a*x + b)/x)*e^a/x + b^2*e^((a *x + b)/x))/((a - (a*x + b)/x)*b)
Timed out. \[ \int \sinh \left (a+\frac {b}{x}\right ) \, dx=\int \mathrm {sinh}\left (a+\frac {b}{x}\right ) \,d x \] Input:
int(sinh(a + b/x),x)
Output:
int(sinh(a + b/x), x)
\[ \int \sinh \left (a+\frac {b}{x}\right ) \, dx=\frac {-e^{\frac {2 a x +b}{x}} \mathit {ei} \left (\frac {b}{x}\right ) b +e^{\frac {2 a x +2 b}{x}} x +e^{\frac {a x +b}{x}} \left (\int \frac {1}{e^{\frac {a x +b}{x}} x}d x \right ) b -x}{2 e^{\frac {a x +b}{x}}} \] Input:
int(sinh(a+b/x),x)
Output:
( - e**((2*a*x + b)/x)*ei(b/x)*b + e**((2*a*x + 2*b)/x)*x + e**((a*x + b)/ x)*int(1/(e**((a*x + b)/x)*x),x)*b - x)/(2*e**((a*x + b)/x))