\(\int (e x)^m \sinh (a+\frac {b}{x}) \, dx\) [39]

Optimal result

Integrand size = 14, antiderivative size = 67 \[ \int (e x)^m \sinh \left (a+\frac {b}{x}\right ) \, dx=-\frac {1}{2} b e^a \left (-\frac {b}{x}\right )^m (e x)^m \Gamma \left (-1-m,-\frac {b}{x}\right )-\frac {1}{2} b e^{-a} \left (\frac {b}{x}\right )^m (e x)^m \Gamma \left (-1-m,\frac {b}{x}\right ) \]

[Out]

-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*GAMMA(-1-m,b/x)/exp(a)

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {5458, 3389, 2212} \[ \int (e x)^m \sinh \left (a+\frac {b}{x}\right ) \, dx=-\frac {1}{2} e^a b \left (-\frac {b}{x}\right )^m (e x)^m \Gamma \left (-m-1,-\frac {b}{x}\right )-\frac {1}{2} e^{-a} b \left (\frac {b}{x}\right )^m (e x)^m \Gamma \left (-m-1,\frac {b}{x}\right ) \]

[In]

Int[(e*x)^m*Sinh[a + b/x],x]

[Out]

-1/2*(b*E^a*(-(b/x))^m*(e*x)^m*Gamma[-1 - m, -(b/x)]) - (b*(b/x)^m*(e*x)^m*Gamma[-1 - m, b/x])/(2*E^a)

Rule 2212

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]

Rule 3389

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Dist[I/2, Int[(c + d*x)^m/E^(I*(e + f*x))
, x], x] - Dist[I/2, Int[(c + d*x)^m*E^(I*(e + f*x)), x], x] /; FreeQ[{c, d, e, f, m}, x]

Rule 5458

Int[((e_.)*(x_))^(m_)*((a_.) + (b_.)*Sinh[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol] :> Dist[(-(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] && Intege
rQ[p] && ILtQ[n, 0] &&  !RationalQ[m]

Rubi steps \begin{align*} \text {integral}& = -\left (\left (\left (\frac {1}{x}\right )^m (e x)^m\right ) \text {Subst}\left (\int x^{-2-m} \sinh (a+b x) \, dx,x,\frac {1}{x}\right )\right ) \\ & = -\left (\frac {1}{2} \left (\left (\frac {1}{x}\right )^m (e x)^m\right ) \text {Subst}\left (\int e^{-i (i a+i b x)} x^{-2-m} \, dx,x,\frac {1}{x}\right )\right )+\frac {1}{2} \left (\left (\frac {1}{x}\right )^m (e x)^m\right ) \text {Subst}\left (\int e^{i (i a+i b x)} x^{-2-m} \, dx,x,\frac {1}{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 )-\frac {1}{2} b e^{-a} \left (\frac {b}{x}\right )^m (e x)^m \Gamma \left (-1-m,\frac {b}{x}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.91 \[ \int (e x)^m \sinh \left (a+\frac {b}{x}\right ) \, dx=-\frac {1}{2} b e^{-a} (e x)^m \left (e^{2 a} \left (-\frac {b}{x}\right )^m \Gamma \left (-1-m,-\frac {b}{x}\right )+\left (\frac {b}{x}\right )^m \Gamma \left (-1-m,\frac {b}{x}\right )\right ) \]

[In]

Integrate[(e*x)^m*Sinh[a + b/x],x]

[Out]

-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

Maple [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4.

Time = 0.66 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.04

method	result	size
meijerg	\(\frac {\left (e x \right )^{m} b \operatorname {hypergeom}\left (\left [-\frac {m}{2}\right ], \left [\frac {3}{2}, 1-\frac {m}{2}\right ], \frac {b^{2}}{4 x^{2}}\right ) \cosh \left (a \right )}{m}+\frac {\left (e x \right )^{m} x \operatorname {hypergeom}\left (\left [-\frac {1}{2}-\frac {m}{2}\right ], \left [\frac {1}{2}, \frac {1}{2}-\frac {m}{2}\right ], \frac {b^{2}}{4 x^{2}}\right ) \sinh \left (a \right )}{1+m}\)	\(70\)

[In]

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

[Out]

(e*x)^m*b/m*hypergeom([-1/2*m],[3/2,1-1/2*m],1/4*b^2/x^2)*cosh(a)+(e*x)^m/(1+m)*x*hypergeom([-1/2-1/2*m],[1/2,
1/2-1/2*m],1/4*b^2/x^2)*sinh(a)

Fricas [F]

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

[In]

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

[Out]

integral((e*x)^m*sinh((a*x + b)/x), x)

Sympy [F]

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

[In]

integrate((e*x)**m*sinh(a+b/x),x)

[Out]

Integral((e*x)**m*sinh(a + b/x), x)

Maxima [F]

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

[In]

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

[Out]

integrate((e*x)^m*sinh(a + b/x), x)

Giac [F]

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

[In]

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

[Out]

integrate((e*x)^m*sinh(a + b/x), x)

Mupad [F(-1)]

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

[In]

int(sinh(a + b/x)*(e*x)^m,x)

[Out]

int(sinh(a + b/x)*(e*x)^m, x)