\(\int e^{c (a+b x)} \coth ^3(d+e x) \, dx\) [244]

Optimal result

Integrand size = 18, antiderivative size = 161 \[ \int e^{c (a+b x)} \coth ^3(d+e x) \, dx=\frac {e^{c (a+b x)}}{b c}-\frac {6 e^{c (a+b x)} \operatorname {Hypergeometric2F1}\left (1,\frac {b c}{2 e},1+\frac {b c}{2 e},e^{2 (d+e x)}\right )}{b c}+\frac {12 e^{c (a+b x)} \operatorname {Hypergeometric2F1}\left (2,\frac {b c}{2 e},1+\frac {b c}{2 e},e^{2 (d+e x)}\right )}{b c}-\frac {8 e^{c (a+b x)} \operatorname {Hypergeometric2F1}\left (3,\frac {b c}{2 e},1+\frac {b c}{2 e},e^{2 (d+e x)}\right )}{b c} \] Output:

exp(c*(b*x+a))/b/c-6*exp(c*(b*x+a))*hypergeom([1, 1/2*b*c/e],[1+1/2*b*c/e] 
,exp(2*e*x+2*d))/b/c+12*exp(c*(b*x+a))*hypergeom([2, 1/2*b*c/e],[1+1/2*b*c 
/e],exp(2*e*x+2*d))/b/c-8*exp(c*(b*x+a))*hypergeom([3, 1/2*b*c/e],[1+1/2*b 
*c/e],exp(2*e*x+2*d))/b/c
 

Mathematica [A] (verified)

Time = 1.12 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.30 \[ \int e^{c (a+b x)} \coth ^3(d+e x) \, dx=\frac {e^{c (a+b x)} \coth (d)}{b c}-\frac {e^{c (a+b x)} \text {csch}^2(d+e x)}{2 e}+\frac {\left (b^2 c^2+2 e^2\right ) e^{a c+2 d+b c x} \left (b c e^{2 e x} \operatorname {Hypergeometric2F1}\left (1,1+\frac {b c}{2 e},2+\frac {b c}{2 e},e^{2 (d+e x)}\right )-(b c+2 e) \operatorname {Hypergeometric2F1}\left (1,\frac {b c}{2 e},1+\frac {b c}{2 e},e^{2 (d+e x)}\right )\right )}{b c e^2 (b c+2 e) \left (-1+e^{2 d}\right )}+\frac {b c e^{c (a+b x)} \text {csch}(d) \text {csch}(d+e x) \sinh (e x)}{2 e^2} \] Input:

Integrate[E^(c*(a + b*x))*Coth[d + e*x]^3,x]
 

Output:

(E^(c*(a + b*x))*Coth[d])/(b*c) - (E^(c*(a + b*x))*Csch[d + e*x]^2)/(2*e) 
+ ((b^2*c^2 + 2*e^2)*E^(a*c + 2*d + b*c*x)*(b*c*E^(2*e*x)*Hypergeometric2F 
1[1, 1 + (b*c)/(2*e), 2 + (b*c)/(2*e), E^(2*(d + e*x))] - (b*c + 2*e)*Hype 
rgeometric2F1[1, (b*c)/(2*e), 1 + (b*c)/(2*e), E^(2*(d + e*x))]))/(b*c*e^2 
*(b*c + 2*e)*(-1 + E^(2*d))) + (b*c*E^(c*(a + b*x))*Csch[d]*Csch[d + e*x]* 
Sinh[e*x])/(2*e^2)
 

Rubi [A] (verified)

Time = 0.69 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {6008, 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.

\(\displaystyle \int e^{c (a+b x)} \coth ^3(d+e x) \, dx\)

\(\Big \downarrow \) 6008

\(\displaystyle \int \left (\frac {6 e^{c (a+b x)}}{e^{2 (d+e x)}-1}+\frac {12 e^{c (a+b x)}}{\left (e^{2 (d+e x)}-1\right )^2}+\frac {8 e^{c (a+b x)}}{\left (e^{2 (d+e x)}-1\right )^3}+e^{c (a+b x)}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {6 e^{c (a+b x)} \operatorname {Hypergeometric2F1}\left (1,\frac {b c}{2 e},\frac {b c}{2 e}+1,e^{2 (d+e x)}\right )}{b c}+\frac {12 e^{c (a+b x)} \operatorname {Hypergeometric2F1}\left (2,\frac {b c}{2 e},\frac {b c}{2 e}+1,e^{2 (d+e x)}\right )}{b c}-\frac {8 e^{c (a+b x)} \operatorname {Hypergeometric2F1}\left (3,\frac {b c}{2 e},\frac {b c}{2 e}+1,e^{2 (d+e x)}\right )}{b c}+\frac {e^{c (a+b x)}}{b c}\)

Input:

Int[E^(c*(a + b*x))*Coth[d + e*x]^3,x]
 

Output:

E^(c*(a + b*x))/(b*c) - (6*E^(c*(a + b*x))*Hypergeometric2F1[1, (b*c)/(2*e 
), 1 + (b*c)/(2*e), E^(2*(d + e*x))])/(b*c) + (12*E^(c*(a + b*x))*Hypergeo 
metric2F1[2, (b*c)/(2*e), 1 + (b*c)/(2*e), E^(2*(d + e*x))])/(b*c) - (8*E^ 
(c*(a + b*x))*Hypergeometric2F1[3, (b*c)/(2*e), 1 + (b*c)/(2*e), E^(2*(d + 
 e*x))])/(b*c)
 

Defintions of rubi rules used

rule 2009

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

rule 6008

Int[Coth[(d_.) + (e_.)*(x_)]^(n_.)*(F_)^((c_.)*((a_.) + (b_.)*(x_))), x_Sym 
bol] :> Int[ExpandIntegrand[F^(c*(a + b*x))*((1 + E^(2*(d + e*x)))^n/(-1 + 
E^(2*(d + e*x)))^n), x], x] /; FreeQ[{F, a, b, c, d, e}, x] && IntegerQ[n]
 

Maple [F]

\[\int {\mathrm e}^{c \left (b x +a \right )} \coth \left (e x +d \right )^{3}d x\]

Input:

int(exp(c*(b*x+a))*coth(e*x+d)^3,x)
 

Output:

int(exp(c*(b*x+a))*coth(e*x+d)^3,x)
 

Fricas [F]

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

integrate(exp(c*(b*x+a))*coth(e*x+d)^3,x, algorithm="fricas")
 

Output:

integral(coth(e*x + d)^3*e^(b*c*x + a*c), x)
 

Sympy [F]

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

integrate(exp(c*(b*x+a))*coth(e*x+d)**3,x)
 

Output:

exp(a*c)*Integral(exp(b*c*x)*coth(d + e*x)**3, x)
 

Maxima [F]

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

integrate(exp(c*(b*x+a))*coth(e*x+d)^3,x, algorithm="maxima")
 

Output:

48*(b^2*c^2*e*e^(a*c) + 2*e^3*e^(a*c))*integrate(e^(b*c*x)/(b^3*c^3 - 12*b 
^2*c^2*e + 44*b*c*e^2 - 48*e^3 + (b^3*c^3*e^(8*d) - 12*b^2*c^2*e*e^(8*d) + 
 44*b*c*e^2*e^(8*d) - 48*e^3*e^(8*d))*e^(8*e*x) - 4*(b^3*c^3*e^(6*d) - 12* 
b^2*c^2*e*e^(6*d) + 44*b*c*e^2*e^(6*d) - 48*e^3*e^(6*d))*e^(6*e*x) + 6*(b^ 
3*c^3*e^(4*d) - 12*b^2*c^2*e*e^(4*d) + 44*b*c*e^2*e^(4*d) - 48*e^3*e^(4*d) 
)*e^(4*e*x) - 4*(b^3*c^3*e^(2*d) - 12*b^2*c^2*e*e^(2*d) + 44*b*c*e^2*e^(2* 
d) - 48*e^3*e^(2*d))*e^(2*e*x)), x) - (b^3*c^3*e^(a*c) + 36*b^2*c^2*e*e^(a 
*c) + 44*b*c*e^2*e^(a*c) + 48*e^3*e^(a*c) + (b^3*c^3*e^(a*c + 6*d) - 12*b^ 
2*c^2*e*e^(a*c + 6*d) + 44*b*c*e^2*e^(a*c + 6*d) - 48*e^3*e^(a*c + 6*d))*e 
^(6*e*x) + 3*(b^3*c^3*e^(a*c + 4*d) - 8*b^2*c^2*e*e^(a*c + 4*d) + 4*b*c*e^ 
2*e^(a*c + 4*d) + 48*e^3*e^(a*c + 4*d))*e^(4*e*x) + 3*(b^3*c^3*e^(a*c + 2* 
d) - 28*b*c*e^2*e^(a*c + 2*d) - 48*e^3*e^(a*c + 2*d))*e^(2*e*x))*e^(b*c*x) 
/(b^4*c^4 - 12*b^3*c^3*e + 44*b^2*c^2*e^2 - 48*b*c*e^3 - (b^4*c^4*e^(6*d) 
- 12*b^3*c^3*e*e^(6*d) + 44*b^2*c^2*e^2*e^(6*d) - 48*b*c*e^3*e^(6*d))*e^(6 
*e*x) + 3*(b^4*c^4*e^(4*d) - 12*b^3*c^3*e*e^(4*d) + 44*b^2*c^2*e^2*e^(4*d) 
 - 48*b*c*e^3*e^(4*d))*e^(4*e*x) - 3*(b^4*c^4*e^(2*d) - 12*b^3*c^3*e*e^(2* 
d) + 44*b^2*c^2*e^2*e^(2*d) - 48*b*c*e^3*e^(2*d))*e^(2*e*x))
 

Giac [F]

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

integrate(exp(c*(b*x+a))*coth(e*x+d)^3,x, algorithm="giac")
 

Output:

integrate(coth(e*x + d)^3*e^((b*x + a)*c), x)
 

Mupad [F(-1)]

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

int(coth(d + e*x)^3*exp(c*(a + b*x)),x)
 

Output:

int(coth(d + e*x)^3*exp(c*(a + b*x)), x)
 

Reduce [F]

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

int(exp(c*(b*x+a))*coth(e*x+d)^3,x)
 

Output:

(e**(a*c)*(e**(b*c*x + 4*d + 4*e*x)*b**2*c**2 - 6*e**(b*c*x + 4*d + 4*e*x) 
*b*c*e + 8*e**(b*c*x + 4*d + 4*e*x)*e**2 + 4*e**(b*c*x + 2*d + 2*e*x)*b**2 
*c**2 - 12*e**(b*c*x + 2*d + 2*e*x)*b*c*e - 16*e**(b*c*x + 2*d + 2*e*x)*e* 
*2 + 7*e**(b*c*x)*b**2*c**2 + 6*e**(b*c*x)*b*c*e + 8*e**(b*c*x)*e**2 + 8*e 
**(4*d + 4*e*x)*int(e**(b*c*x)/(e**(6*d + 6*e*x)*b**2*c**2 - 6*e**(6*d + 6 
*e*x)*b*c*e + 8*e**(6*d + 6*e*x)*e**2 - 3*e**(4*d + 4*e*x)*b**2*c**2 + 18* 
e**(4*d + 4*e*x)*b*c*e - 24*e**(4*d + 4*e*x)*e**2 + 3*e**(2*d + 2*e*x)*b** 
2*c**2 - 18*e**(2*d + 2*e*x)*b*c*e + 24*e**(2*d + 2*e*x)*e**2 - b**2*c**2 
+ 6*b*c*e - 8*e**2),x)*b**5*c**5 - 48*e**(4*d + 4*e*x)*int(e**(b*c*x)/(e** 
(6*d + 6*e*x)*b**2*c**2 - 6*e**(6*d + 6*e*x)*b*c*e + 8*e**(6*d + 6*e*x)*e* 
*2 - 3*e**(4*d + 4*e*x)*b**2*c**2 + 18*e**(4*d + 4*e*x)*b*c*e - 24*e**(4*d 
 + 4*e*x)*e**2 + 3*e**(2*d + 2*e*x)*b**2*c**2 - 18*e**(2*d + 2*e*x)*b*c*e 
+ 24*e**(2*d + 2*e*x)*e**2 - b**2*c**2 + 6*b*c*e - 8*e**2),x)*b**4*c**4*e 
+ 80*e**(4*d + 4*e*x)*int(e**(b*c*x)/(e**(6*d + 6*e*x)*b**2*c**2 - 6*e**(6 
*d + 6*e*x)*b*c*e + 8*e**(6*d + 6*e*x)*e**2 - 3*e**(4*d + 4*e*x)*b**2*c**2 
 + 18*e**(4*d + 4*e*x)*b*c*e - 24*e**(4*d + 4*e*x)*e**2 + 3*e**(2*d + 2*e* 
x)*b**2*c**2 - 18*e**(2*d + 2*e*x)*b*c*e + 24*e**(2*d + 2*e*x)*e**2 - b**2 
*c**2 + 6*b*c*e - 8*e**2),x)*b**3*c**3*e**2 - 96*e**(4*d + 4*e*x)*int(e**( 
b*c*x)/(e**(6*d + 6*e*x)*b**2*c**2 - 6*e**(6*d + 6*e*x)*b*c*e + 8*e**(6*d 
+ 6*e*x)*e**2 - 3*e**(4*d + 4*e*x)*b**2*c**2 + 18*e**(4*d + 4*e*x)*b*c*...