\(\int \frac {(c+d x)^m}{(a+a \tanh (e+f x))^3} \, dx\) [52]

Optimal result

Integrand size = 20, antiderivative size = 224 \[ \int \frac {(c+d x)^m}{(a+a \tanh (e+f x))^3} \, dx=\frac {(c+d x)^{1+m}}{8 a^3 d (1+m)}-\frac {3\ 2^{-4-m} e^{-2 e+\frac {2 c f}{d}} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {2 f (c+d x)}{d}\right )}{a^3 f}-\frac {3\ 2^{-5-2 m} e^{-4 e+\frac {4 c f}{d}} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {4 f (c+d x)}{d}\right )}{a^3 f}-\frac {2^{-4-m} 3^{-1-m} e^{-6 e+\frac {6 c f}{d}} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {6 f (c+d x)}{d}\right )}{a^3 f} \] Output:

1/8*(d*x+c)^(1+m)/a^3/d/(1+m)-3*2^(-4-m)*exp(-2*e+2*c*f/d)*(d*x+c)^m*GAMMA 
(1+m,2*f*(d*x+c)/d)/a^3/f/((f*(d*x+c)/d)^m)-3*2^(-5-2*m)*exp(-4*e+4*c*f/d) 
*(d*x+c)^m*GAMMA(1+m,4*f*(d*x+c)/d)/a^3/f/((f*(d*x+c)/d)^m)-2^(-4-m)*3^(-1 
-m)*exp(-6*e+6*c*f/d)*(d*x+c)^m*GAMMA(1+m,6*f*(d*x+c)/d)/a^3/f/((f*(d*x+c) 
/d)^m)
 

Mathematica [A] (verified)

Time = 1.27 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.03 \[ \int \frac {(c+d x)^m}{(a+a \tanh (e+f x))^3} \, dx=\frac {2^{-5-2 m} 3^{-1-m} e^{-3 e} \left (f \left (\frac {c}{d}+x\right )\right )^{-m} (c+d x)^m \left (12^{1+m} e^{6 e} f \left (f \left (\frac {c}{d}+x\right )\right )^m (c+d x)-2^{1+m} 3^{2+m} d e^{4 e+\frac {2 c f}{d}} (1+m) \Gamma \left (1+m,\frac {2 f (c+d x)}{d}\right )-3^{2+m} d e^{2 e+\frac {4 c f}{d}} (1+m) \Gamma \left (1+m,\frac {4 f (c+d x)}{d}\right )-2^{1+m} d e^{\frac {6 c f}{d}} (1+m) \Gamma \left (1+m,\frac {6 f (c+d x)}{d}\right )\right ) \text {sech}^3(e+f x) (\cosh (f x)+\sinh (f x))^3}{a^3 d f (1+m) (1+\tanh (e+f x))^3} \] Input:

Integrate[(c + d*x)^m/(a + a*Tanh[e + f*x])^3,x]
 

Output:

(2^(-5 - 2*m)*3^(-1 - m)*(c + d*x)^m*(12^(1 + m)*E^(6*e)*f*(f*(c/d + x))^m 
*(c + d*x) - 2^(1 + m)*3^(2 + m)*d*E^(4*e + (2*c*f)/d)*(1 + m)*Gamma[1 + m 
, (2*f*(c + d*x))/d] - 3^(2 + m)*d*E^(2*e + (4*c*f)/d)*(1 + m)*Gamma[1 + m 
, (4*f*(c + d*x))/d] - 2^(1 + m)*d*E^((6*c*f)/d)*(1 + m)*Gamma[1 + m, (6*f 
*(c + d*x))/d])*Sech[e + f*x]^3*(Cosh[f*x] + Sinh[f*x])^3)/(a^3*d*E^(3*e)* 
f*(1 + m)*(f*(c/d + x))^m*(1 + Tanh[e + f*x])^3)
 

Rubi [A] (verified)

Time = 0.58 (sec) , antiderivative size = 224, 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.

\(\displaystyle \int \frac {(c+d x)^m}{(a \tanh (e+f x)+a)^3} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {(c+d x)^m}{(a-i a \tan (i e+i f x))^3}dx\)

\(\Big \downarrow \) 4212

\(\displaystyle \int \left (\frac {e^{-6 e-6 f x} (c+d x)^m}{8 a^3}+\frac {3 e^{-4 e-4 f x} (c+d x)^m}{8 a^3}+\frac {3 e^{-2 e-2 f x} (c+d x)^m}{8 a^3}+\frac {(c+d x)^m}{8 a^3}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {3\ 2^{-m-4} e^{\frac {2 c f}{d}-2 e} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,\frac {2 f (c+d x)}{d}\right )}{a^3 f}-\frac {3\ 2^{-2 m-5} e^{\frac {4 c f}{d}-4 e} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,\frac {4 f (c+d x)}{d}\right )}{a^3 f}-\frac {2^{-m-4} 3^{-m-1} e^{\frac {6 c f}{d}-6 e} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,\frac {6 f (c+d x)}{d}\right )}{a^3 f}+\frac {(c+d x)^{m+1}}{8 a^3 d (m+1)}\)

Input:

Int[(c + d*x)^m/(a + a*Tanh[e + f*x])^3,x]
 

Output:

(c + d*x)^(1 + m)/(8*a^3*d*(1 + m)) - (3*2^(-4 - m)*E^(-2*e + (2*c*f)/d)*( 
c + d*x)^m*Gamma[1 + m, (2*f*(c + d*x))/d])/(a^3*f*((f*(c + d*x))/d)^m) - 
(3*2^(-5 - 2*m)*E^(-4*e + (4*c*f)/d)*(c + d*x)^m*Gamma[1 + m, (4*f*(c + d* 
x))/d])/(a^3*f*((f*(c + d*x))/d)^m) - (2^(-4 - m)*3^(-1 - m)*E^(-6*e + (6* 
c*f)/d)*(c + d*x)^m*Gamma[1 + m, (6*f*(c + d*x))/d])/(a^3*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]

\[\int \frac {\left (d x +c \right )^{m}}{\left (a +\tanh \left (f x +e \right ) a \right )^{3}}d x\]

Input:

int((d*x+c)^m/(a+tanh(f*x+e)*a)^3,x)
 

Output:

int((d*x+c)^m/(a+tanh(f*x+e)*a)^3,x)
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 345, normalized size of antiderivative = 1.54 \[ \int \frac {(c+d x)^m}{(a+a \tanh (e+f x))^3} \, dx=-\frac {2 \, {\left (d m + d\right )} \cosh \left (\frac {d m \log \left (\frac {6 \, f}{d}\right ) + 6 \, d e - 6 \, c f}{d}\right ) \Gamma \left (m + 1, \frac {6 \, {\left (d f x + c f\right )}}{d}\right ) + 9 \, {\left (d m + d\right )} \cosh \left (\frac {d m \log \left (\frac {4 \, f}{d}\right ) + 4 \, d e - 4 \, c f}{d}\right ) \Gamma \left (m + 1, \frac {4 \, {\left (d f x + c f\right )}}{d}\right ) + 18 \, {\left (d m + d\right )} \cosh \left (\frac {d m \log \left (\frac {2 \, f}{d}\right ) + 2 \, d e - 2 \, c f}{d}\right ) \Gamma \left (m + 1, \frac {2 \, {\left (d f x + c f\right )}}{d}\right ) - 2 \, {\left (d m + d\right )} \Gamma \left (m + 1, \frac {6 \, {\left (d f x + c f\right )}}{d}\right ) \sinh \left (\frac {d m \log \left (\frac {6 \, f}{d}\right ) + 6 \, d e - 6 \, c f}{d}\right ) - 9 \, {\left (d m + d\right )} \Gamma \left (m + 1, \frac {4 \, {\left (d f x + c f\right )}}{d}\right ) \sinh \left (\frac {d m \log \left (\frac {4 \, f}{d}\right ) + 4 \, d e - 4 \, c f}{d}\right ) - 18 \, {\left (d m + d\right )} \Gamma \left (m + 1, \frac {2 \, {\left (d f x + c f\right )}}{d}\right ) \sinh \left (\frac {d m \log \left (\frac {2 \, f}{d}\right ) + 2 \, d e - 2 \, c f}{d}\right ) - 12 \, {\left (d f x + c f\right )} \cosh \left (m \log \left (d x + c\right )\right ) - 12 \, {\left (d f x + c f\right )} \sinh \left (m \log \left (d x + c\right )\right )}{96 \, {\left (a^{3} d f m + a^{3} d f\right )}} \] Input:

integrate((d*x+c)^m/(a+a*tanh(f*x+e))^3,x, algorithm="fricas")
 

Output:

-1/96*(2*(d*m + d)*cosh((d*m*log(6*f/d) + 6*d*e - 6*c*f)/d)*gamma(m + 1, 6 
*(d*f*x + c*f)/d) + 9*(d*m + d)*cosh((d*m*log(4*f/d) + 4*d*e - 4*c*f)/d)*g 
amma(m + 1, 4*(d*f*x + c*f)/d) + 18*(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) - 2*(d*m + d)*gamma(m + 1, 6* 
(d*f*x + c*f)/d)*sinh((d*m*log(6*f/d) + 6*d*e - 6*c*f)/d) - 9*(d*m + d)*ga 
mma(m + 1, 4*(d*f*x + c*f)/d)*sinh((d*m*log(4*f/d) + 4*d*e - 4*c*f)/d) - 1 
8*(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) - 12*(d*f*x + c*f)*cosh(m*log(d*x + c)) - 12*(d*f*x + c*f)*sinh 
(m*log(d*x + c)))/(a^3*d*f*m + a^3*d*f)
 

Sympy [F]

\[ \int \frac {(c+d x)^m}{(a+a \tanh (e+f x))^3} \, dx=\frac {\int \frac {\left (c + d x\right )^{m}}{\tanh ^{3}{\left (e + f x \right )} + 3 \tanh ^{2}{\left (e + f x \right )} + 3 \tanh {\left (e + f x \right )} + 1}\, dx}{a^{3}} \] Input:

integrate((d*x+c)**m/(a+a*tanh(f*x+e))**3,x)
 

Output:

Integral((c + d*x)**m/(tanh(e + f*x)**3 + 3*tanh(e + f*x)**2 + 3*tanh(e + 
f*x) + 1), x)/a**3
 

Maxima [F]

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

integrate((d*x+c)^m/(a+a*tanh(f*x+e))^3,x, algorithm="maxima")
 

Output:

integrate((d*x + c)^m/(a*tanh(f*x + e) + a)^3, x)
 

Giac [F]

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

integrate((d*x+c)^m/(a+a*tanh(f*x+e))^3,x, algorithm="giac")
 

Output:

integrate((d*x + c)^m/(a*tanh(f*x + e) + a)^3, x)
 

Mupad [F(-1)]

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

int((c + d*x)^m/(a + a*tanh(e + f*x))^3,x)
 

Output:

int((c + d*x)^m/(a + a*tanh(e + f*x))^3, x)
 

Reduce [F]

\[ \int \frac {(c+d x)^m}{(a+a \tanh (e+f x))^3} \, dx=\text {too large to display} \] Input:

int((d*x+c)^m/(a+a*tanh(f*x+e))^3,x)
 

Output:

(12*e**(6*e + 6*f*x)*(c + d*x)**m*tanh(e + f*x)**2*c*f + 12*e**(6*e + 6*f* 
x)*(c + d*x)**m*tanh(e + f*x)**2*d*f*x + 18*e**(6*e + 6*f*x)*(c + d*x)**m* 
tanh(e + f*x)**2*d*m + 18*e**(6*e + 6*f*x)*(c + d*x)**m*tanh(e + f*x)**2*d 
 + 24*e**(6*e + 6*f*x)*(c + d*x)**m*tanh(e + f*x)*c*f + 24*e**(6*e + 6*f*x 
)*(c + d*x)**m*tanh(e + f*x)*d*f*x + 12*e**(6*e + 6*f*x)*(c + d*x)**m*tanh 
(e + f*x)*d*m + 12*e**(6*e + 6*f*x)*(c + d*x)**m*tanh(e + f*x)*d + 12*e**( 
6*e + 6*f*x)*(c + d*x)**m*c*f + 12*e**(6*e + 6*f*x)*(c + d*x)**m*d*f*x - 3 
0*e**(6*e + 6*f*x)*(c + d*x)**m*d*m - 30*e**(6*e + 6*f*x)*(c + d*x)**m*d + 
 6*e**(4*e + 4*f*x)*(c + d*x)**m*tanh(e + f*x)**2*d*m + 6*e**(4*e + 4*f*x) 
*(c + d*x)**m*tanh(e + f*x)**2*d + 12*e**(4*e + 4*f*x)*(c + d*x)**m*tanh(e 
 + f*x)*d*m + 12*e**(4*e + 4*f*x)*(c + d*x)**m*tanh(e + f*x)*d + 6*e**(4*e 
 + 4*f*x)*(c + d*x)**m*d*m + 6*e**(4*e + 4*f*x)*(c + d*x)**m*d - 3*e**(2*e 
 + 2*f*x)*(c + d*x)**m*tanh(e + f*x)**2*d*m - 3*e**(2*e + 2*f*x)*(c + d*x) 
**m*tanh(e + f*x)**2*d - 6*e**(2*e + 2*f*x)*(c + d*x)**m*tanh(e + f*x)*d*m 
 - 6*e**(2*e + 2*f*x)*(c + d*x)**m*tanh(e + f*x)*d - 3*e**(2*e + 2*f*x)*(c 
 + d*x)**m*d*m - 3*e**(2*e + 2*f*x)*(c + d*x)**m*d - 2*(c + d*x)**m*tanh(e 
 + f*x)**2*d*m - 2*(c + d*x)**m*tanh(e + f*x)**2*d - 4*(c + d*x)**m*tanh(e 
 + f*x)*d*m - 4*(c + d*x)**m*tanh(e + f*x)*d - 2*(c + d*x)**m*d*m - 2*(c + 
 d*x)**m*d + 18*e**(10*e + 6*f*x)*int((c + d*x)**m/(e**(6*e + 2*f*x)*c + e 
**(6*e + 2*f*x)*d*x),x)*tanh(e + f*x)**2*d**2*m**2 + 18*e**(10*e + 6*f*...