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

Optimal. Leaf size=67 \[ -\frac{1}{2} e^a b \left (-\frac{b}{x}\right )^m (e x)^m \text{Gamma}\left (-m-1,-\frac{b}{x}\right )-\frac{1}{2} e^{-a} b \left (\frac{b}{x}\right )^m (e x)^m \text{Gamma}\left (-m-1,\frac{b}{x}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0875178, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {5350, 3308, 2181} \[ -\frac{1}{2} e^a b \left (-\frac{b}{x}\right )^m (e x)^m \text{Gamma}\left (-m-1,-\frac{b}{x}\right )-\frac{1}{2} e^{-a} b \left (\frac{b}{x}\right )^m (e x)^m \text{Gamma}\left (-m-1,\frac{b}{x}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(e*x)^m*Sinh[a + b/x],x]

[Out]

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

Rule 5350

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] && IntegerQ
[p] && ILtQ[n, 0] &&  !RationalQ[m]

Rule 3308

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 2181

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> -Simp[(F^(g*(e - (c*f)/d))*(c +
d*x)^FracPart[m]*Gamma[m + 1, (-((f*g*Log[F])/d))*(c + d*x)])/(d*(-((f*g*Log[F])/d))^(IntPart[m] + 1)*(-((f*g*
Log[F]*(c + d*x))/d))^FracPart[m]), x] /; FreeQ[{F, c, d, e, f, g, m}, x] &&  !IntegerQ[m]

Rubi steps

\begin{align*} \int (e x)^m \sinh \left (a+\frac{b}{x}\right ) \, dx &=-\left (\left (\left (\frac{1}{x}\right )^m (e x)^m\right ) \operatorname{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 ) \operatorname{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 ) \operatorname{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]  time = 0.0752309, size = 63, normalized size = 0.94 \[ -\frac{1}{2} b (e x)^m \left ((\sinh (a)+\cosh (a)) \left (-\frac{b}{x}\right )^m \text{Gamma}\left (-m-1,-\frac{b}{x}\right )+(\cosh (a)-\sinh (a)) \left (\frac{b}{x}\right )^m \text{Gamma}\left (-m-1,\frac{b}{x}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(e*x)^m*Sinh[a + b/x],x]

[Out]

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

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Maple [C]  time = 0.034, size = 70, normalized size = 1. \begin{align*}{\frac{ \left ( ex \right ) ^{m}b\cosh \left ( a \right ) }{m}{\mbox{$_1$F$_2$}(-{\frac{m}{2}};\,{\frac{3}{2}},1-{\frac{m}{2}};\,{\frac{{b}^{2}}{4\,{x}^{2}}})}}+{\frac{ \left ( ex \right ) ^{m}x\sinh \left ( a \right ) }{1+m}{\mbox{$_1$F$_2$}(-{\frac{1}{2}}-{\frac{m}{2}};\,{\frac{1}{2}},{\frac{1}{2}}-{\frac{m}{2}};\,{\frac{{b}^{2}}{4\,{x}^{2}}})}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(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)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (e x\right )^{m} \sinh \left (a + \frac{b}{x}\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((e*x)^m*sinh(a + b/x), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\left (e x\right )^{m} \sinh \left (\frac{a x + b}{x}\right ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (e x\right )^{m} \sinh{\left (a + \frac{b}{x} \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((e*x)**m*sinh(a + b/x), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (e x\right )^{m} \sinh \left (a + \frac{b}{x}\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((e*x)^m*sinh(a + b/x), x)