\(\int \frac {(c+d x)^m}{a+a \coth (e+f x)} \, dx\) [34]

Optimal result

Integrand size = 20, antiderivative size = 88 \[ \int \frac {(c+d x)^m}{a+a \coth (e+f x)} \, dx=\frac {(c+d x)^{1+m}}{2 a d (1+m)}+\frac {2^{-2-m} e^{-2 e+\frac {2 c f}{d}} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {2 f (c+d x)}{d}\right )}{a f} \] Output:

1/2*(d*x+c)^(1+m)/a/d/(1+m)+2^(-2-m)*exp(-2*e+2*c*f/d)*(d*x+c)^m*GAMMA(1+m 
,2*f*(d*x+c)/d)/a/f/((f*(d*x+c)/d)^m)
 

Mathematica [A] (verified)

Time = 0.83 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.31 \[ \int \frac {(c+d x)^m}{a+a \coth (e+f x)} \, dx=\frac {\text {csch}(e+f x) \left (\frac {e^e f (c+d x)^{1+m}}{d (1+m)}+2^{-1-m} e^{-e+\frac {2 c f}{d}} (c+d x)^m \left (\frac {c f}{d}+f x\right )^{-m} \Gamma \left (1+m,2 \left (\frac {c f}{d}+f x\right )\right )\right ) (\cosh (f x)+\sinh (f x))}{2 f (a+a \coth (e+f x))} \] Input:

Integrate[(c + d*x)^m/(a + a*Coth[e + f*x]),x]
 

Output:

(Csch[e + f*x]*((E^e*f*(c + d*x)^(1 + m))/(d*(1 + m)) + (2^(-1 - m)*E^(-e 
+ (2*c*f)/d)*(c + d*x)^m*Gamma[1 + m, 2*((c*f)/d + f*x)])/((c*f)/d + f*x)^ 
m)*(Cosh[f*x] + Sinh[f*x]))/(2*f*(a + a*Coth[e + f*x]))
 

Rubi [A] (verified)

Time = 0.59 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {3042, 4210, 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.

\(\displaystyle \int \frac {(c+d x)^m}{a \coth (e+f x)+a} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {(c+d x)^m}{a-i a \tan \left (i e+i f x+\frac {\pi }{2}\right )}dx\)

\(\Big \downarrow \) 4210

\(\displaystyle \frac {(c+d x)^{m+1}}{2 a d (m+1)}+\frac {\int e^{i (2 i e+2 i f x+\pi )} (c+d x)^mdx}{2 a}\)

\(\Big \downarrow \) 2612

\(\displaystyle \frac {2^{-m-2} e^{\frac {2 c f}{d}-2 e} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,\frac {2 f (c+d x)}{d}\right )}{a f}+\frac {(c+d x)^{m+1}}{2 a d (m+1)}\)

Input:

Int[(c + d*x)^m/(a + a*Coth[e + f*x]),x]
 

Output:

(c + d*x)^(1 + m)/(2*a*d*(1 + m)) + (2^(-2 - m)*E^(-2*e + (2*c*f)/d)*(c + 
d*x)^m*Gamma[1 + m, (2*f*(c + d*x))/d])/(a*f*((f*(c + d*x))/d)^m)
 

Defintions of rubi rules used

rule 2612

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 3042

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

rule 4210

Int[((c_.) + (d_.)*(x_))^(m_)/((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Sym 
bol] :> Simp[(c + d*x)^(m + 1)/(2*a*d*(m + 1)), x] + Simp[1/(2*a)   Int[(c 
+ d*x)^m*E^(2*(a/b)*(e + f*x)), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] & 
& EqQ[a^2 + b^2, 0] &&  !IntegerQ[m]
 

Maple [F]

\[\int \frac {\left (d x +c \right )^{m}}{a +a \coth \left (f x +e \right )}d x\]

Input:

int((d*x+c)^m/(a+a*coth(f*x+e)),x)
 

Output:

int((d*x+c)^m/(a+a*coth(f*x+e)),x)
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.68 \[ \int \frac {(c+d x)^m}{a+a \coth (e+f x)} \, dx=\frac {{\left (d m + d\right )} \cosh \left (\frac {d m \log \left (\frac {2 \, f}{d}\right ) + 2 \, d e - 2 \, c f}{d}\right ) \Gamma \left (m + 1, \frac {2 \, {\left (d f x + c f\right )}}{d}\right ) - {\left (d m + d\right )} \Gamma \left (m + 1, \frac {2 \, {\left (d f x + c f\right )}}{d}\right ) \sinh \left (\frac {d m \log \left (\frac {2 \, f}{d}\right ) + 2 \, d e - 2 \, c f}{d}\right ) + 2 \, {\left (d f x + c f\right )} \cosh \left (m \log \left (d x + c\right )\right ) + 2 \, {\left (d f x + c f\right )} \sinh \left (m \log \left (d x + c\right )\right )}{4 \, {\left (a d f m + a d f\right )}} \] Input:

integrate((d*x+c)^m/(a+a*coth(f*x+e)),x, algorithm="fricas")
 

Output:

1/4*((d*m + d)*cosh((d*m*log(2*f/d) + 2*d*e - 2*c*f)/d)*gamma(m + 1, 2*(d* 
f*x + c*f)/d) - (d*m + d)*gamma(m + 1, 2*(d*f*x + c*f)/d)*sinh((d*m*log(2* 
f/d) + 2*d*e - 2*c*f)/d) + 2*(d*f*x + c*f)*cosh(m*log(d*x + c)) + 2*(d*f*x 
 + c*f)*sinh(m*log(d*x + c)))/(a*d*f*m + a*d*f)
 

Sympy [F]

\[ \int \frac {(c+d x)^m}{a+a \coth (e+f x)} \, dx=\frac {\int \frac {\left (c + d x\right )^{m}}{\coth {\left (e + f x \right )} + 1}\, dx}{a} \] Input:

integrate((d*x+c)**m/(a+a*coth(f*x+e)),x)
 

Output:

Integral((c + d*x)**m/(coth(e + f*x) + 1), x)/a
 

Maxima [F]

\[ \int \frac {(c+d x)^m}{a+a \coth (e+f x)} \, dx=\int { \frac {{\left (d x + c\right )}^{m}}{a \coth \left (f x + e\right ) + a} \,d x } \] Input:

integrate((d*x+c)^m/(a+a*coth(f*x+e)),x, algorithm="maxima")
 

Output:

integrate((d*x + c)^m/(a*coth(f*x + e) + a), x)
 

Giac [F]

\[ \int \frac {(c+d x)^m}{a+a \coth (e+f x)} \, dx=\int { \frac {{\left (d x + c\right )}^{m}}{a \coth \left (f x + e\right ) + a} \,d x } \] Input:

integrate((d*x+c)^m/(a+a*coth(f*x+e)),x, algorithm="giac")
 

Output:

integrate((d*x + c)^m/(a*coth(f*x + e) + a), x)
 

Mupad [F(-1)]

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

int((c + d*x)^m/(a + a*coth(e + f*x)),x)
 

Output:

int((c + d*x)^m/(a + a*coth(e + f*x)), x)
 

Reduce [F]

\[ \int \frac {(c+d x)^m}{a+a \coth (e+f x)} \, dx=\frac {2 e^{2 f x +2 e} \left (d x +c \right )^{m} c f +2 e^{2 f x +2 e} \left (d x +c \right )^{m} d f x +\left (d x +c \right )^{m} d m +\left (d x +c \right )^{m} d -e^{2 f x +2 e} \left (\int \frac {\left (d x +c \right )^{m}}{e^{2 f x +2 e} c +e^{2 f x +2 e} d x}d x \right ) d^{2} m^{2}-e^{2 f x +2 e} \left (\int \frac {\left (d x +c \right )^{m}}{e^{2 f x +2 e} c +e^{2 f x +2 e} d x}d x \right ) d^{2} m}{4 e^{2 f x +2 e} a d f \left (m +1\right )} \] Input:

int((d*x+c)^m/(a+a*coth(f*x+e)),x)
 

Output:

(2*e**(2*e + 2*f*x)*(c + d*x)**m*c*f + 2*e**(2*e + 2*f*x)*(c + d*x)**m*d*f 
*x + (c + d*x)**m*d*m + (c + d*x)**m*d - e**(2*e + 2*f*x)*int((c + d*x)**m 
/(e**(2*e + 2*f*x)*c + e**(2*e + 2*f*x)*d*x),x)*d**2*m**2 - e**(2*e + 2*f* 
x)*int((c + d*x)**m/(e**(2*e + 2*f*x)*c + e**(2*e + 2*f*x)*d*x),x)*d**2*m) 
/(4*e**(2*e + 2*f*x)*a*d*f*(m + 1))