\(\int (e x)^m \sinh ^3(a+\frac {b}{x}) \, dx\) [37]

Optimal result

Integrand size = 16, antiderivative size = 146 \[ \int (e x)^m \sinh ^3\left (a+\frac {b}{x}\right ) \, dx=-\frac {1}{8} 3^{1+m} b e^{3 a} \left (-\frac {b}{x}\right )^m (e x)^m \Gamma \left (-1-m,-\frac {3 b}{x}\right )+\frac {3}{8} b e^a \left (-\frac {b}{x}\right )^m (e x)^m \Gamma \left (-1-m,-\frac {b}{x}\right )+\frac {3}{8} b e^{-a} \left (\frac {b}{x}\right )^m (e x)^m \Gamma \left (-1-m,\frac {b}{x}\right )-\frac {1}{8} 3^{1+m} b e^{-3 a} \left (\frac {b}{x}\right )^m (e x)^m \Gamma \left (-1-m,\frac {3 b}{x}\right ) \] Output:

-1/8*3^(1+m)*b*exp(3*a)*(-b/x)^m*(e*x)^m*GAMMA(-1-m,-3*b/x)+3/8*b*exp(a)*( 
-b/x)^m*(e*x)^m*GAMMA(-1-m,-b/x)+3/8*b*(b/x)^m*(e*x)^m*GAMMA(-1-m,b/x)/exp 
(a)-1/8*3^(1+m)*b*(b/x)^m*(e*x)^m*GAMMA(-1-m,3*b/x)/exp(3*a)
 

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.79 \[ \int (e x)^m \sinh ^3\left (a+\frac {b}{x}\right ) \, dx=-\frac {3}{8} b e^{-3 a} (e x)^m \left (3^m e^{6 a} \left (-\frac {b}{x}\right )^m \Gamma \left (-1-m,-\frac {3 b}{x}\right )-e^{4 a} \left (-\frac {b}{x}\right )^m \Gamma \left (-1-m,-\frac {b}{x}\right )+\left (\frac {b}{x}\right )^m \left (-e^{2 a} \Gamma \left (-1-m,\frac {b}{x}\right )+3^m \Gamma \left (-1-m,\frac {3 b}{x}\right )\right )\right ) \] Input:

Integrate[(e*x)^m*Sinh[a + b/x]^3,x]
 

Output:

(-3*b*(e*x)^m*(3^m*E^(6*a)*(-(b/x))^m*Gamma[-1 - m, (-3*b)/x] - E^(4*a)*(- 
(b/x))^m*Gamma[-1 - m, -(b/x)] + (b/x)^m*(-(E^(2*a)*Gamma[-1 - m, b/x]) + 
3^m*Gamma[-1 - m, (3*b)/x])))/(8*E^(3*a))
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.50 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.21, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {5873, 3042, 26, 3793, 2009}

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.

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

\(\Big \downarrow \) 5873

\(\displaystyle -\left (\frac {1}{x}\right )^m (e x)^m \int \left (\frac {1}{x}\right )^{-m-2} \sinh ^3\left (a+\frac {b}{x}\right )d\frac {1}{x}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\left (\frac {1}{x}\right )^m (e x)^m \int i \left (\frac {1}{x}\right )^{-m-2} \sin \left (i a+\frac {i b}{x}\right )^3d\frac {1}{x}\)

\(\Big \downarrow \) 26

\(\displaystyle -i \left (\frac {1}{x}\right )^m (e x)^m \int \left (\frac {1}{x}\right )^{-m-2} \sin \left (i a+\frac {i b}{x}\right )^3d\frac {1}{x}\)

\(\Big \downarrow \) 3793

\(\displaystyle -i \left (\frac {1}{x}\right )^m (e x)^m \int \left (\frac {3}{4} i \left (\frac {1}{x}\right )^{-m-2} \sinh \left (a+\frac {b}{x}\right )-\frac {1}{4} i \left (\frac {1}{x}\right )^{-m-2} \sinh \left (3 a+\frac {3 b}{x}\right )\right )d\frac {1}{x}\)

\(\Big \downarrow \) 2009

\(\displaystyle -i \left (\frac {1}{x}\right )^m (e x)^m \left (-\frac {1}{8} i e^{3 a} b 3^{m+1} \left (\frac {1}{x}\right )^{-m} \left (-\frac {b}{x}\right )^m \Gamma \left (-m-1,-\frac {3 b}{x}\right )+\frac {3}{8} i e^a b \left (\frac {1}{x}\right )^{-m} \left (-\frac {b}{x}\right )^m \Gamma \left (-m-1,-\frac {b}{x}\right )+\frac {3}{8} i e^{-a} b \left (\frac {1}{x}\right )^{-m} \left (\frac {b}{x}\right )^m \Gamma \left (-m-1,\frac {b}{x}\right )-\frac {1}{8} i e^{-3 a} b 3^{m+1} \left (\frac {1}{x}\right )^{-m} \left (\frac {b}{x}\right )^m \Gamma \left (-m-1,\frac {3 b}{x}\right )\right )\)

Input:

Int[(e*x)^m*Sinh[a + b/x]^3,x]
 

Output:

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

Defintions of rubi rules used

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3793

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In 
t[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 5873

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]
 

Maple [F]

\[\int \left (e x \right )^{m} \sinh \left (a +\frac {b}{x}\right )^{3}d x\]

Input:

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

Output:

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

Fricas [F]

\[ \int (e x)^m \sinh ^3\left (a+\frac {b}{x}\right ) \, dx=\int { \left (e x\right )^{m} \sinh \left (a + \frac {b}{x}\right )^{3} \,d x } \] Input:

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

Output:

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

Sympy [F]

\[ \int (e x)^m \sinh ^3\left (a+\frac {b}{x}\right ) \, dx=\int \left (e x\right )^{m} \sinh ^{3}{\left (a + \frac {b}{x} \right )}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int (e x)^m \sinh ^3\left (a+\frac {b}{x}\right ) \, dx=\int { \left (e x\right )^{m} \sinh \left (a + \frac {b}{x}\right )^{3} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int (e x)^m \sinh ^3\left (a+\frac {b}{x}\right ) \, dx=\int { \left (e x\right )^{m} \sinh \left (a + \frac {b}{x}\right )^{3} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int (e x)^m \sinh ^3\left (a+\frac {b}{x}\right ) \, dx=\int {\mathrm {sinh}\left (a+\frac {b}{x}\right )}^3\,{\left (e\,x\right )}^m \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int (e x)^m \sinh ^3\left (a+\frac {b}{x}\right ) \, dx=\frac {e^{m} \left (3 x^{m} e^{\frac {6 a x +6 b}{x}} b +x^{m} e^{\frac {6 a x +6 b}{x}} m x -3 x^{m} e^{\frac {4 a x +4 b}{x}} b -3 x^{m} e^{\frac {4 a x +4 b}{x}} m x +3 e^{\frac {3 a x +3 b}{x}} \left (\int \frac {x^{m}}{e^{\frac {a x +b}{x}} x^{2}}d x \right ) b^{2}-9 e^{\frac {3 a x +3 b}{x}} \left (\int \frac {x^{m}}{e^{\frac {3 a x +3 b}{x}} x^{2}}d x \right ) b^{2}-3 e^{\frac {3 a x +3 b}{x}} \left (\int \frac {x^{m} e^{\frac {a x +b}{x}}}{x^{2}}d x \right ) b^{2}+9 e^{\frac {3 a x +3 b}{x}} \left (\int \frac {x^{m} e^{\frac {3 a x +3 b}{x}}}{x^{2}}d x \right ) b^{2}-3 x^{m} e^{\frac {2 a x +2 b}{x}} b +3 x^{m} e^{\frac {2 a x +2 b}{x}} m x +3 x^{m} b -x^{m} m x \right )}{8 e^{\frac {3 a x +3 b}{x}} m \left (m +1\right )} \] Input:

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

Output:

(e**m*(3*x**m*e**((6*a*x + 6*b)/x)*b + x**m*e**((6*a*x + 6*b)/x)*m*x - 3*x 
**m*e**((4*a*x + 4*b)/x)*b - 3*x**m*e**((4*a*x + 4*b)/x)*m*x + 3*e**((3*a* 
x + 3*b)/x)*int(x**m/(e**((a*x + b)/x)*x**2),x)*b**2 - 9*e**((3*a*x + 3*b) 
/x)*int(x**m/(e**((3*a*x + 3*b)/x)*x**2),x)*b**2 - 3*e**((3*a*x + 3*b)/x)* 
int((x**m*e**((a*x + b)/x))/x**2,x)*b**2 + 9*e**((3*a*x + 3*b)/x)*int((x** 
m*e**((3*a*x + 3*b)/x))/x**2,x)*b**2 - 3*x**m*e**((2*a*x + 2*b)/x)*b + 3*x 
**m*e**((2*a*x + 2*b)/x)*m*x + 3*x**m*b - x**m*m*x))/(8*e**((3*a*x + 3*b)/ 
x)*m*(m + 1))