3.37 \(\int (e x)^m \sinh ^3(a+\frac{b}{x}) \, dx\)

Optimal. Leaf size=146 \[ -\frac{1}{8} e^{3 a} b 3^{m+1} \left (-\frac{b}{x}\right )^m (e x)^m \text{Gamma}\left (-m-1,-\frac{3 b}{x}\right )+\frac{3}{8} e^a b \left (-\frac{b}{x}\right )^m (e x)^m \text{Gamma}\left (-m-1,-\frac{b}{x}\right )+\frac{3}{8} e^{-a} b \left (\frac{b}{x}\right )^m (e x)^m \text{Gamma}\left (-m-1,\frac{b}{x}\right )-\frac{1}{8} e^{-3 a} b 3^{m+1} \left (\frac{b}{x}\right )^m (e x)^m \text{Gamma}\left (-m-1,\frac{3 b}{x}\right ) \]

[Out]

-(3^(1 + m)*b*E^(3*a)*(-(b/x))^m*(e*x)^m*Gamma[-1 - m, (-3*b)/x])/8 + (3*b*E^a*(-(b/x))^m*(e*x)^m*Gamma[-1 - m
, -(b/x)])/8 + (3*b*(b/x)^m*(e*x)^m*Gamma[-1 - m, b/x])/(8*E^a) - (3^(1 + m)*b*(b/x)^m*(e*x)^m*Gamma[-1 - m, (
3*b)/x])/(8*E^(3*a))

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Rubi [A]  time = 0.250817, antiderivative size = 146, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {5350, 3312, 3308, 2181} \[ -\frac{1}{8} e^{3 a} b 3^{m+1} \left (-\frac{b}{x}\right )^m (e x)^m \text{Gamma}\left (-m-1,-\frac{3 b}{x}\right )+\frac{3}{8} e^a b \left (-\frac{b}{x}\right )^m (e x)^m \text{Gamma}\left (-m-1,-\frac{b}{x}\right )+\frac{3}{8} e^{-a} b \left (\frac{b}{x}\right )^m (e x)^m \text{Gamma}\left (-m-1,\frac{b}{x}\right )-\frac{1}{8} e^{-3 a} b 3^{m+1} \left (\frac{b}{x}\right )^m (e x)^m \text{Gamma}\left (-m-1,\frac{3 b}{x}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(e*x)^m*Sinh[a + b/x]^3,x]

[Out]

-(3^(1 + m)*b*E^(3*a)*(-(b/x))^m*(e*x)^m*Gamma[-1 - m, (-3*b)/x])/8 + (3*b*E^a*(-(b/x))^m*(e*x)^m*Gamma[-1 - m
, -(b/x)])/8 + (3*b*(b/x)^m*(e*x)^m*Gamma[-1 - m, b/x])/(8*E^a) - (3^(1 + m)*b*(b/x)^m*(e*x)^m*Gamma[-1 - m, (
3*b)/x])/(8*E^(3*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 3312

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin
[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1])
)

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 ^3\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 ^3(a+b x) \, dx,x,\frac{1}{x}\right )\right )\\ &=-\left (\left (i \left (\frac{1}{x}\right )^m (e x)^m\right ) \operatorname{Subst}\left (\int \left (\frac{3}{4} i x^{-2-m} \sinh (a+b x)-\frac{1}{4} i x^{-2-m} \sinh (3 a+3 b x)\right ) \, dx,x,\frac{1}{x}\right )\right )\\ &=-\left (\frac{1}{4} \left (\left (\frac{1}{x}\right )^m (e x)^m\right ) \operatorname{Subst}\left (\int x^{-2-m} \sinh (3 a+3 b x) \, dx,x,\frac{1}{x}\right )\right )+\frac{1}{4} \left (3 \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 )\\ &=-\left (\frac{1}{8} \left (\left (\frac{1}{x}\right )^m (e x)^m\right ) \operatorname{Subst}\left (\int e^{-i (3 i a+3 i b x)} x^{-2-m} \, dx,x,\frac{1}{x}\right )\right )+\frac{1}{8} \left (\left (\frac{1}{x}\right )^m (e x)^m\right ) \operatorname{Subst}\left (\int e^{i (3 i a+3 i b x)} x^{-2-m} \, dx,x,\frac{1}{x}\right )+\frac{1}{8} \left (3 \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}{8} \left (3 \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}{8} 3^{1+m} b e^{3 a} \left (-\frac{b}{x}\right )^m (e x)^m \Gamma \left (-1-m,-\frac{3 b}{x}\right )+\frac{3}{8} b e^a \left (-\frac{b}{x}\right )^m (e x)^m \Gamma \left (-1-m,-\frac{b}{x}\right )+\frac{3}{8} b e^{-a} \left (\frac{b}{x}\right )^m (e x)^m \Gamma \left (-1-m,\frac{b}{x}\right )-\frac{1}{8} 3^{1+m} b e^{-3 a} \left (\frac{b}{x}\right )^m (e x)^m \Gamma \left (-1-m,\frac{3 b}{x}\right )\\ \end{align*}

Mathematica [F]  time = 180.001, size = 0, normalized size = 0. \[ \text{\$Aborted} \]

Verification is Not applicable to the result.

[In]

Integrate[(e*x)^m*Sinh[a + b/x]^3,x]

[Out]

$Aborted

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Maple [F]  time = 0.075, size = 0, normalized size = 0. \begin{align*} \int \left ( ex \right ) ^{m} \left ( \sinh \left ( a+{\frac{b}{x}} \right ) \right ) ^{3}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x)^m*sinh(a+b/x)^3,x)

[Out]

int((e*x)^m*sinh(a+b/x)^3,x)

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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 )^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^m*sinh(a+b/x)^3,x, algorithm="maxima")

[Out]

integrate((e*x)^m*sinh(a + b/x)^3, 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 )^{3}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^m*sinh(a+b/x)^3,x, algorithm="fricas")

[Out]

integral((e*x)^m*sinh((a*x + b)/x)^3, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (e x\right )^{m} \sinh ^{3}{\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)**3,x)

[Out]

Integral((e*x)**m*sinh(a + b/x)**3, 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 )^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^m*sinh(a+b/x)^3,x, algorithm="giac")

[Out]

integrate((e*x)^m*sinh(a + b/x)^3, x)