\(\int \frac {(c+d x)^m}{(a+a \coth (e+f x))^2} \, dx\) [35]

Optimal result

Integrand size = 20, antiderivative size = 152 \[ \int \frac {(c+d x)^m}{(a+a \coth (e+f x))^2} \, dx=\frac {(c+d x)^{1+m}}{4 a^2 d (1+m)}+\frac {2^{-2-m} e^{-2 e+\frac {2 c f}{d}} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {2 f (c+d x)}{d}\right )}{a^2 f}-\frac {4^{-2-m} e^{-4 e+\frac {4 c f}{d}} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {4 f (c+d x)}{d}\right )}{a^2 f} \] Output:

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

Mathematica [A] (verified)

Time = 1.15 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.07 \[ \int \frac {(c+d x)^m}{(a+a \coth (e+f x))^2} \, dx=\frac {(c+d x)^m \text {csch}^2(e+f x) \left (\frac {4 e^{2 e} f (c+d x)}{d (1+m)}+2^{2-m} e^{\frac {2 c f}{d}} \left (f \left (\frac {c}{d}+x\right )\right )^{-m} \Gamma \left (1+m,\frac {2 f (c+d x)}{d}\right )-4^{-m} e^{-2 e+\frac {4 c f}{d}} \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {4 f (c+d x)}{d}\right )\right ) (\cosh (f x)+\sinh (f x))^2}{16 a^2 f (1+\coth (e+f x))^2} \] Input:

Integrate[(c + d*x)^m/(a + a*Coth[e + f*x])^2,x]
 

Output:

((c + d*x)^m*Csch[e + f*x]^2*((4*E^(2*e)*f*(c + d*x))/(d*(1 + m)) + (2^(2 
- m)*E^((2*c*f)/d)*Gamma[1 + m, (2*f*(c + d*x))/d])/(f*(c/d + x))^m - (E^( 
-2*e + (4*c*f)/d)*Gamma[1 + m, (4*f*(c + d*x))/d])/(4^m*((f*(c + d*x))/d)^ 
m))*(Cosh[f*x] + Sinh[f*x])^2)/(16*a^2*f*(1 + Coth[e + f*x])^2)
 

Rubi [A] (verified)

Time = 0.65 (sec) , antiderivative size = 152, 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, 4212, 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.

\(\displaystyle \int \frac {(c+d x)^m}{(a \coth (e+f x)+a)^2} \, dx\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4212

\(\displaystyle \int \left (\frac {e^{-4 e-4 f x} (c+d x)^m}{4 a^2}-\frac {e^{-2 e-2 f x} (c+d x)^m}{2 a^2}+\frac {(c+d x)^m}{4 a^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {2^{-m-2} e^{\frac {2 c f}{d}-2 e} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,\frac {2 f (c+d x)}{d}\right )}{a^2 f}-\frac {4^{-m-2} e^{\frac {4 c f}{d}-4 e} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,\frac {4 f (c+d x)}{d}\right )}{a^2 f}+\frac {(c+d x)^{m+1}}{4 a^2 d (m+1)}\)

Input:

Int[(c + d*x)^m/(a + a*Coth[e + f*x])^2,x]
 

Output:

(c + d*x)^(1 + m)/(4*a^2*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^2*f*((f*(c + d*x))/d)^m) - (4 
^(-2 - m)*E^(-4*e + (4*c*f)/d)*(c + d*x)^m*Gamma[1 + m, (4*f*(c + d*x))/d] 
)/(a^2*f*((f*(c + d*x))/d)^m)
 

Defintions of rubi rules used

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 4212

Int[((c_.) + (d_.)*(x_))^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), 
x_Symbol] :> Int[ExpandIntegrand[(c + d*x)^m, (1/(2*a) + E^(2*(a/b)*(e + f* 
x))/(2*a))^(-n), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[a^2 + b^2 
, 0] && ILtQ[n, 0]
 

Maple [F]

\[\int \frac {\left (d x +c \right )^{m}}{\left (a +a \coth \left (f x +e \right )\right )^{2}}d x\]

Input:

int((d*x+c)^m/(a+a*coth(f*x+e))^2,x)
 

Output:

int((d*x+c)^m/(a+a*coth(f*x+e))^2,x)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.63 \[ \int \frac {(c+d x)^m}{(a+a \coth (e+f x))^2} \, dx=-\frac {{\left (d m + d\right )} \cosh \left (\frac {d m \log \left (\frac {4 \, f}{d}\right ) + 4 \, d e - 4 \, c f}{d}\right ) \Gamma \left (m + 1, \frac {4 \, {\left (d f x + c f\right )}}{d}\right ) - 4 \, {\left (d m + d\right )} \cosh \left (\frac {d m \log \left (\frac {2 \, f}{d}\right ) + 2 \, d e - 2 \, c f}{d}\right ) \Gamma \left (m + 1, \frac {2 \, {\left (d f x + c f\right )}}{d}\right ) - {\left (d m + d\right )} \Gamma \left (m + 1, \frac {4 \, {\left (d f x + c f\right )}}{d}\right ) \sinh \left (\frac {d m \log \left (\frac {4 \, f}{d}\right ) + 4 \, d e - 4 \, c f}{d}\right ) + 4 \, {\left (d m + d\right )} \Gamma \left (m + 1, \frac {2 \, {\left (d f x + c f\right )}}{d}\right ) \sinh \left (\frac {d m \log \left (\frac {2 \, f}{d}\right ) + 2 \, d e - 2 \, c f}{d}\right ) - 4 \, {\left (d f x + c f\right )} \cosh \left (m \log \left (d x + c\right )\right ) - 4 \, {\left (d f x + c f\right )} \sinh \left (m \log \left (d x + c\right )\right )}{16 \, {\left (a^{2} d f m + a^{2} d f\right )}} \] Input:

integrate((d*x+c)^m/(a+a*coth(f*x+e))^2,x, algorithm="fricas")
 

Output:

-1/16*((d*m + d)*cosh((d*m*log(4*f/d) + 4*d*e - 4*c*f)/d)*gamma(m + 1, 4*( 
d*f*x + c*f)/d) - 4*(d*m + d)*cosh((d*m*log(2*f/d) + 2*d*e - 2*c*f)/d)*gam 
ma(m + 1, 2*(d*f*x + c*f)/d) - (d*m + d)*gamma(m + 1, 4*(d*f*x + c*f)/d)*s 
inh((d*m*log(4*f/d) + 4*d*e - 4*c*f)/d) + 4*(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) - 4*(d*f*x + c*f)*cos 
h(m*log(d*x + c)) - 4*(d*f*x + c*f)*sinh(m*log(d*x + c)))/(a^2*d*f*m + a^2 
*d*f)
 

Sympy [F]

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

integrate((d*x+c)**m/(a+a*coth(f*x+e))**2,x)
 

Output:

Integral((c + d*x)**m/(coth(e + f*x)**2 + 2*coth(e + f*x) + 1), x)/a**2
 

Maxima [F]

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

integrate((d*x+c)^m/(a+a*coth(f*x+e))^2,x, algorithm="maxima")
 

Output:

integrate((d*x + c)^m/(a*coth(f*x + e) + a)^2, x)
 

Giac [F]

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

integrate((d*x+c)^m/(a+a*coth(f*x+e))^2,x, algorithm="giac")
 

Output:

integrate((d*x + c)^m/(a*coth(f*x + e) + a)^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(c+d x)^m}{(a+a \coth (e+f x))^2} \, dx=\int \frac {{\left (c+d\,x\right )}^m}{{\left (a+a\,\mathrm {coth}\left (e+f\,x\right )\right )}^2} \,d x \] Input:

int((c + d*x)^m/(a + a*coth(e + f*x))^2,x)
 

Output:

int((c + d*x)^m/(a + a*coth(e + f*x))^2, x)
 

Reduce [F]

\[ \int \frac {(c+d x)^m}{(a+a \coth (e+f x))^2} \, dx=\frac {4 e^{4 f x +4 e} \left (d x +c \right )^{m} c f +4 e^{4 f x +4 e} \left (d x +c \right )^{m} d f x +4 e^{2 f x +2 e} \left (d x +c \right )^{m} d m +4 e^{2 f x +2 e} \left (d x +c \right )^{m} d -\left (d x +c \right )^{m} d m -\left (d x +c \right )^{m} d -4 e^{4 f x +6 e} \left (\int \frac {\left (d x +c \right )^{m}}{e^{2 f x +4 e} c +e^{2 f x +4 e} d x}d x \right ) d^{2} m^{2}-4 e^{4 f x +6 e} \left (\int \frac {\left (d x +c \right )^{m}}{e^{2 f x +4 e} c +e^{2 f x +4 e} d x}d x \right ) d^{2} m +e^{4 f x +4 e} \left (\int \frac {\left (d x +c \right )^{m}}{e^{4 f x +4 e} c +e^{4 f x +4 e} d x}d x \right ) d^{2} m^{2}+e^{4 f x +4 e} \left (\int \frac {\left (d x +c \right )^{m}}{e^{4 f x +4 e} c +e^{4 f x +4 e} d x}d x \right ) d^{2} m}{16 e^{4 f x +4 e} a^{2} d f \left (m +1\right )} \] Input:

int((d*x+c)^m/(a+a*coth(f*x+e))^2,x)
 

Output:

(4*e**(4*e + 4*f*x)*(c + d*x)**m*c*f + 4*e**(4*e + 4*f*x)*(c + d*x)**m*d*f 
*x + 4*e**(2*e + 2*f*x)*(c + d*x)**m*d*m + 4*e**(2*e + 2*f*x)*(c + d*x)**m 
*d - (c + d*x)**m*d*m - (c + d*x)**m*d - 4*e**(6*e + 4*f*x)*int((c + d*x)* 
*m/(e**(4*e + 2*f*x)*c + e**(4*e + 2*f*x)*d*x),x)*d**2*m**2 - 4*e**(6*e + 
4*f*x)*int((c + d*x)**m/(e**(4*e + 2*f*x)*c + e**(4*e + 2*f*x)*d*x),x)*d** 
2*m + e**(4*e + 4*f*x)*int((c + d*x)**m/(e**(4*e + 4*f*x)*c + e**(4*e + 4* 
f*x)*d*x),x)*d**2*m**2 + e**(4*e + 4*f*x)*int((c + d*x)**m/(e**(4*e + 4*f* 
x)*c + e**(4*e + 4*f*x)*d*x),x)*d**2*m)/(16*e**(4*e + 4*f*x)*a**2*d*f*(m + 
 1))