Integrand size = 14, antiderivative size = 67 \[ \int (e x)^m \sinh \left (a+\frac {b}{x}\right ) \, dx=-\frac {1}{2} b e^a \left (-\frac {b}{x}\right )^m (e x)^m \Gamma \left (-1-m,-\frac {b}{x}\right )-\frac {1}{2} b e^{-a} \left (\frac {b}{x}\right )^m (e x)^m \Gamma \left (-1-m,\frac {b}{x}\right ) \] Output:
-1/2*b*exp(a)*(-b/x)^m*(e*x)^m*GAMMA(-1-m,-b/x)-1/2*b*(b/x)^m*(e*x)^m*GAMM A(-1-m,b/x)/exp(a)
Time = 0.07 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.91 \[ \int (e x)^m \sinh \left (a+\frac {b}{x}\right ) \, dx=-\frac {1}{2} b e^{-a} (e x)^m \left (e^{2 a} \left (-\frac {b}{x}\right )^m \Gamma \left (-1-m,-\frac {b}{x}\right )+\left (\frac {b}{x}\right )^m \Gamma \left (-1-m,\frac {b}{x}\right )\right ) \] Input:
Integrate[(e*x)^m*Sinh[a + b/x],x]
Output:
-1/2*(b*(e*x)^m*(E^(2*a)*(-(b/x))^m*Gamma[-1 - m, -(b/x)] + (b/x)^m*Gamma[ -1 - m, b/x]))/E^a
Result contains complex when optimal does not.
Time = 0.37 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.33, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {5873, 3042, 26, 3789, 2612}
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.
\(\displaystyle \int (e x)^m \sinh \left (a+\frac {b}{x}\right ) \, dx\) |
\(\Big \downarrow \) 5873 |
\(\displaystyle -\left (\frac {1}{x}\right )^m (e x)^m \int \left (\frac {1}{x}\right )^{-m-2} \sinh \left (a+\frac {b}{x}\right )d\frac {1}{x}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\left (\frac {1}{x}\right )^m (e x)^m \int -i \left (\frac {1}{x}\right )^{-m-2} \sin \left (i a+\frac {i b}{x}\right )d\frac {1}{x}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \left (\frac {1}{x}\right )^m (e x)^m \int \left (\frac {1}{x}\right )^{-m-2} \sin \left (i a+\frac {i b}{x}\right )d\frac {1}{x}\) |
\(\Big \downarrow \) 3789 |
\(\displaystyle i \left (\frac {1}{x}\right )^m (e x)^m \left (\frac {1}{2} i \int e^{a+\frac {b}{x}} \left (\frac {1}{x}\right )^{-m-2}d\frac {1}{x}-\frac {1}{2} i \int e^{-a-\frac {b}{x}} \left (\frac {1}{x}\right )^{-m-2}d\frac {1}{x}\right )\) |
\(\Big \downarrow \) 2612 |
\(\displaystyle i \left (\frac {1}{x}\right )^m (e x)^m \left (\frac {1}{2} i e^a b \left (\frac {1}{x}\right )^{-m} \left (-\frac {b}{x}\right )^m \Gamma \left (-m-1,-\frac {b}{x}\right )+\frac {1}{2} i e^{-a} b \left (\frac {1}{x}\right )^{-m} \left (\frac {b}{x}\right )^m \Gamma \left (-m-1,\frac {b}{x}\right )\right )\) |
Input:
Int[(e*x)^m*Sinh[a + b/x],x]
Output:
I*(x^(-1))^m*(e*x)^m*(((I/2)*b*E^a*(-(b/x))^m*Gamma[-1 - m, -(b/x)])/(x^(- 1))^m + ((I/2)*b*(b/x)^m*Gamma[-1 - m, b/x])/(E^a*(x^(-1))^m))
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[(F_)^((g_.)*((e_.) + (f_.)*(x_)))*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[(-F^(g*(e - c*(f/d))))*((c + d*x)^FracPart[m]/(d*((-f)*g*(Log[F]/d) )^(IntPart[m] + 1)*((-f)*g*Log[F]*((c + d*x)/d))^FracPart[m]))*Gamma[m + 1, ((-f)*g*(Log[F]/d))*(c + d*x)], x] /; FreeQ[{F, c, d, e, f, g, m}, x] && !IntegerQ[m]
Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[I /2 Int[(c + d*x)^m/E^(I*(e + f*x)), x], x] - Simp[I/2 Int[(c + d*x)^m*E ^(I*(e + f*x)), x], x] /; FreeQ[{c, d, e, f, m}, x]
Int[((e_.)*(x_))^(m_)*((a_.) + (b_.)*Sinh[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol] :> Simp[(-(e*x)^m)*(x^(-1))^m Subst[Int[(a + b*Sinh[c + d/x^n]) ^p/x^(m + 2), x], x, 1/x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IntegerQ[p ] && ILtQ[n, 0] && !RationalQ[m]
Result contains higher order function than in optimal. Order 5 vs. order 4.
Time = 0.24 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.04
method | result | size |
meijerg | \(\frac {\left (e x \right )^{m} b \operatorname {hypergeom}\left (\left [-\frac {m}{2}\right ], \left [\frac {3}{2}, 1-\frac {m}{2}\right ], \frac {b^{2}}{4 x^{2}}\right ) \cosh \left (a \right )}{m}+\frac {\left (e x \right )^{m} x \operatorname {hypergeom}\left (\left [-\frac {1}{2}-\frac {m}{2}\right ], \left [\frac {1}{2}, \frac {1}{2}-\frac {m}{2}\right ], \frac {b^{2}}{4 x^{2}}\right ) \sinh \left (a \right )}{1+m}\) | \(70\) |
Input:
int((e*x)^m*sinh(a+b/x),x,method=_RETURNVERBOSE)
Output:
(e*x)^m*b/m*hypergeom([-1/2*m],[3/2,1-1/2*m],1/4/x^2*b^2)*cosh(a)+(e*x)^m/ (1+m)*x*hypergeom([-1/2-1/2*m],[1/2,1/2-1/2*m],1/4/x^2*b^2)*sinh(a)
\[ \int (e x)^m \sinh \left (a+\frac {b}{x}\right ) \, dx=\int { \left (e x\right )^{m} \sinh \left (a + \frac {b}{x}\right ) \,d x } \] Input:
integrate((e*x)^m*sinh(a+b/x),x, algorithm="fricas")
Output:
integral((e*x)^m*sinh((a*x + b)/x), x)
\[ \int (e x)^m \sinh \left (a+\frac {b}{x}\right ) \, dx=\int \left (e x\right )^{m} \sinh {\left (a + \frac {b}{x} \right )}\, dx \] Input:
integrate((e*x)**m*sinh(a+b/x),x)
Output:
Integral((e*x)**m*sinh(a + b/x), x)
\[ \int (e x)^m \sinh \left (a+\frac {b}{x}\right ) \, dx=\int { \left (e x\right )^{m} \sinh \left (a + \frac {b}{x}\right ) \,d x } \] Input:
integrate((e*x)^m*sinh(a+b/x),x, algorithm="maxima")
Output:
integrate((e*x)^m*sinh(a + b/x), x)
\[ \int (e x)^m \sinh \left (a+\frac {b}{x}\right ) \, dx=\int { \left (e x\right )^{m} \sinh \left (a + \frac {b}{x}\right ) \,d x } \] Input:
integrate((e*x)^m*sinh(a+b/x),x, algorithm="giac")
Output:
integrate((e*x)^m*sinh(a + b/x), x)
Timed out. \[ \int (e x)^m \sinh \left (a+\frac {b}{x}\right ) \, dx=\int \mathrm {sinh}\left (a+\frac {b}{x}\right )\,{\left (e\,x\right )}^m \,d x \] Input:
int(sinh(a + b/x)*(e*x)^m,x)
Output:
int(sinh(a + b/x)*(e*x)^m, x)
\[ \int (e x)^m \sinh \left (a+\frac {b}{x}\right ) \, dx=\frac {e^{m} \left (x^{m} \cosh \left (\frac {a x +b}{x}\right ) b +x^{m} \sinh \left (\frac {a x +b}{x}\right ) m x +\left (\int \frac {x^{m} \sinh \left (\frac {a x +b}{x}\right )}{x^{2}}d x \right ) b^{2}\right )}{m \left (m +1\right )} \] Input:
int((e*x)^m*sinh(a+b/x),x)
Output:
(e**m*(x**m*cosh((a*x + b)/x)*b + x**m*sinh((a*x + b)/x)*m*x + int((x**m*s inh((a*x + b)/x))/x**2,x)*b**2))/(m*(m + 1))