\(\int \frac {e^{c (a+b x)}}{\tanh ^2(a c+b c x)^{5/2}} \, dx\) [250]

Optimal result

Integrand size = 25, antiderivative size = 319 \[ \int \frac {e^{c (a+b x)}}{\tanh ^2(a c+b c x)^{5/2}} \, dx=\frac {e^{c (a+b x)} \tanh (a c+b c x)}{b c \sqrt {\tanh ^2(a c+b c x)}}-\frac {4 e^{c (a+b x)} \tanh (a c+b c x)}{b c \left (1-e^{2 c (a+b x)}\right )^4 \sqrt {\tanh ^2(a c+b c x)}}+\frac {26 e^{c (a+b x)} \tanh (a c+b c x)}{3 b c \left (1-e^{2 c (a+b x)}\right )^3 \sqrt {\tanh ^2(a c+b c x)}}-\frac {55 e^{c (a+b x)} \tanh (a c+b c x)}{6 b c \left (1-e^{2 c (a+b x)}\right )^2 \sqrt {\tanh ^2(a c+b c x)}}+\frac {25 e^{c (a+b x)} \tanh (a c+b c x)}{4 b c \left (1-e^{2 c (a+b x)}\right ) \sqrt {\tanh ^2(a c+b c x)}}-\frac {15 \text {arctanh}\left (e^{c (a+b x)}\right ) \tanh (a c+b c x)}{4 b c \sqrt {\tanh ^2(a c+b c x)}} \] Output:

exp(c*(b*x+a))*tanh(b*c*x+a*c)/b/c/(tanh(b*c*x+a*c)^2)^(1/2)-4*exp(c*(b*x+ 
a))*tanh(b*c*x+a*c)/b/c/(1-exp(2*c*(b*x+a)))^4/(tanh(b*c*x+a*c)^2)^(1/2)+2 
6/3*exp(c*(b*x+a))*tanh(b*c*x+a*c)/b/c/(1-exp(2*c*(b*x+a)))^3/(tanh(b*c*x+ 
a*c)^2)^(1/2)-55/6*exp(c*(b*x+a))*tanh(b*c*x+a*c)/b/c/(1-exp(2*c*(b*x+a))) 
^2/(tanh(b*c*x+a*c)^2)^(1/2)+25/4*exp(c*(b*x+a))*tanh(b*c*x+a*c)/b/c/(1-ex 
p(2*c*(b*x+a)))/(tanh(b*c*x+a*c)^2)^(1/2)-15/4*arctanh(exp(c*(b*x+a)))*tan 
h(b*c*x+a*c)/b/c/(tanh(b*c*x+a*c)^2)^(1/2)
 

Mathematica [A] (warning: unable to verify)

Time = 11.48 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.51 \[ \int \frac {e^{c (a+b x)}}{\tanh ^2(a c+b c x)^{5/2}} \, dx=\frac {\left (66 e^{c (a+b x)}-314 e^{3 c (a+b x)}+374 e^{5 c (a+b x)}-246 e^{7 c (a+b x)}+24 e^{9 c (a+b x)}+45 \left (-1+e^{2 c (a+b x)}\right )^4 \log \left (1-e^{c (a+b x)}\right )-45 \left (-1+e^{2 c (a+b x)}\right )^4 \log \left (1+e^{c (a+b x)}\right )\right ) \tanh (c (a+b x))}{24 b c \left (-1+e^{2 c (a+b x)}\right )^4 \sqrt {\tanh ^2(c (a+b x))}} \] Input:

Integrate[E^(c*(a + b*x))/(Tanh[a*c + b*c*x]^2)^(5/2),x]
 

Output:

((66*E^(c*(a + b*x)) - 314*E^(3*c*(a + b*x)) + 374*E^(5*c*(a + b*x)) - 246 
*E^(7*c*(a + b*x)) + 24*E^(9*c*(a + b*x)) + 45*(-1 + E^(2*c*(a + b*x)))^4* 
Log[1 - E^(c*(a + b*x))] - 45*(-1 + E^(2*c*(a + b*x)))^4*Log[1 + E^(c*(a + 
 b*x))])*Tanh[c*(a + b*x)])/(24*b*c*(-1 + E^(2*c*(a + b*x)))^4*Sqrt[Tanh[c 
*(a + b*x)]^2])
 

Rubi [A] (verified)

Time = 1.76 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.53, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {7271, 2720, 25, 300, 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.

\(\displaystyle \int \frac {e^{c (a+b x)}}{\tanh ^2(a c+b c x)^{5/2}} \, dx\)

\(\Big \downarrow \) 7271

\(\displaystyle \frac {\tanh (a c+b c x) \int e^{c (a+b x)} \coth ^5(a c+b x c)dx}{\sqrt {\tanh ^2(a c+b c x)}}\)

\(\Big \downarrow \) 2720

\(\displaystyle \frac {\tanh (a c+b c x) \int -\frac {\left (1+e^{2 c (a+b x)}\right )^5}{\left (1-e^{2 c (a+b x)}\right )^5}de^{c (a+b x)}}{b c \sqrt {\tanh ^2(a c+b c x)}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\tanh (a c+b c x) \int \frac {\left (1+e^{2 c (a+b x)}\right )^5}{\left (1-e^{2 c (a+b x)}\right )^5}de^{c (a+b x)}}{b c \sqrt {\tanh ^2(a c+b c x)}}\)

\(\Big \downarrow \) 300

\(\displaystyle -\frac {\tanh (a c+b c x) \int \left (\frac {2 \left (1+10 e^{4 c (a+b x)}+5 e^{8 c (a+b x)}\right )}{\left (1-e^{2 c (a+b x)}\right )^5}-1\right )de^{c (a+b x)}}{b c \sqrt {\tanh ^2(a c+b c x)}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\left (-\frac {15}{4} \text {arctanh}\left (e^{c (a+b x)}\right )+e^{c (a+b x)}+\frac {25 e^{c (a+b x)}}{4 \left (1-e^{2 c (a+b x)}\right )}-\frac {55 e^{c (a+b x)}}{6 \left (1-e^{2 c (a+b x)}\right )^2}+\frac {26 e^{c (a+b x)}}{3 \left (1-e^{2 c (a+b x)}\right )^3}-\frac {4 e^{c (a+b x)}}{\left (1-e^{2 c (a+b x)}\right )^4}\right ) \tanh (a c+b c x)}{b c \sqrt {\tanh ^2(a c+b c x)}}\)

Input:

Int[E^(c*(a + b*x))/(Tanh[a*c + b*c*x]^2)^(5/2),x]
 

Output:

((E^(c*(a + b*x)) - (4*E^(c*(a + b*x)))/(1 - E^(2*c*(a + b*x)))^4 + (26*E^ 
(c*(a + b*x)))/(3*(1 - E^(2*c*(a + b*x)))^3) - (55*E^(c*(a + b*x)))/(6*(1 
- E^(2*c*(a + b*x)))^2) + (25*E^(c*(a + b*x)))/(4*(1 - E^(2*c*(a + b*x)))) 
 - (15*ArcTanh[E^(c*(a + b*x))])/4)*Tanh[a*c + b*c*x])/(b*c*Sqrt[Tanh[a*c 
+ b*c*x]^2])
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 300

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Int 
[PolynomialDivide[(a + b*x^2)^p, (c + d*x^2)^(-q), x], x] /; FreeQ[{a, b, c 
, d}, x] && NeQ[b*c - a*d, 0] && IGtQ[p, 0] && ILtQ[q, 0] && GeQ[p, -q]
 

rule 2009

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

rule 2720

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] 
   Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct 
ionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ 
[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) 
*(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
 

rule 7271

Int[(u_.)*((a_.)*(v_)^(m_.))^(p_), x_Symbol] :> Simp[a^IntPart[p]*((a*v^m)^ 
FracPart[p]/v^(m*FracPart[p]))   Int[u*v^(m*p), x], x] /; FreeQ[{a, m, p}, 
x] &&  !IntegerQ[p] &&  !FreeQ[v, x] &&  !(EqQ[a, 1] && EqQ[m, 1]) &&  !(Eq 
Q[v, x] && EqQ[m, 1])
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 1.83 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.61

method	result	size
default	\(\frac {\operatorname {csgn}\left (\tanh \left (c \left (b x +a \right )\right )\right ) \left (\frac {\cosh \left (b c x +a c \right )^{4}}{\sinh \left (b c x +a c \right )^{3}}-\frac {4 \cosh \left (b c x +a c \right )^{2}}{\sinh \left (b c x +a c \right )^{3}}+\frac {8}{3 \sinh \left (b c x +a c \right )^{3}}+\frac {\cosh \left (b c x +a c \right )^{5}}{\sinh \left (b c x +a c \right )^{4}}-\frac {5 \cosh \left (b c x +a c \right )^{3}}{\sinh \left (b c x +a c \right )^{4}}+\frac {5 \cosh \left (b c x +a c \right )}{\sinh \left (b c x +a c \right )^{4}}+5 \left (-\frac {\operatorname {csch}\left (b c x +a c \right )^{3}}{4}+\frac {3 \,\operatorname {csch}\left (b c x +a c \right )}{8}\right ) \coth \left (b c x +a c \right )-\frac {15 \,\operatorname {arctanh}\left ({\mathrm e}^{b c x +a c}\right )}{4}\right )}{c b}\)	\(195\)
risch	\(\frac {\left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right ) {\mathrm e}^{c \left (b x +a \right )}}{\sqrt {\frac {\left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right )^{2}}{\left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right )^{2}}}\, \left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right ) b c}-\frac {{\mathrm e}^{c \left (b x +a \right )} \left (75 \,{\mathrm e}^{6 c \left (b x +a \right )}-115 \,{\mathrm e}^{4 c \left (b x +a \right )}+109 \,{\mathrm e}^{2 c \left (b x +a \right )}-21\right )}{12 \left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right )^{3} \left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right ) \sqrt {\frac {\left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right )^{2}}{\left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right )^{2}}}\, c b}-\frac {15 \left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right ) \ln \left ({\mathrm e}^{c \left (b x +a \right )}+1\right )}{8 \sqrt {\frac {\left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right )^{2}}{\left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right )^{2}}}\, \left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right ) c b}+\frac {15 \left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right ) \ln \left ({\mathrm e}^{c \left (b x +a \right )}-1\right )}{8 \sqrt {\frac {\left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right )^{2}}{\left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right )^{2}}}\, \left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right ) c b}\)	\(320\)

Input:

int(exp(c*(b*x+a))/(tanh(b*c*x+a*c)^2)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

csgn(tanh(c*(b*x+a)))/c/b*(1/sinh(b*c*x+a*c)^3*cosh(b*c*x+a*c)^4-4/sinh(b* 
c*x+a*c)^3*cosh(b*c*x+a*c)^2+8/3/sinh(b*c*x+a*c)^3+cosh(b*c*x+a*c)^5/sinh( 
b*c*x+a*c)^4-5/sinh(b*c*x+a*c)^4*cosh(b*c*x+a*c)^3+5/sinh(b*c*x+a*c)^4*cos 
h(b*c*x+a*c)+5*(-1/4*csch(b*c*x+a*c)^3+3/8*csch(b*c*x+a*c))*coth(b*c*x+a*c 
)-15/4*arctanh(exp(b*c*x+a*c)))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1617 vs. \(2 (281) = 562\).

Time = 0.11 (sec) , antiderivative size = 1617, normalized size of antiderivative = 5.07 \[ \int \frac {e^{c (a+b x)}}{\tanh ^2(a c+b c x)^{5/2}} \, dx=\text {Too large to display} \] Input:

integrate(exp(c*(b*x+a))/(tanh(b*c*x+a*c)^2)^(5/2),x, algorithm="fricas")
 

Output:

1/24*(24*cosh(b*c*x + a*c)^9 + 216*cosh(b*c*x + a*c)*sinh(b*c*x + a*c)^8 + 
 24*sinh(b*c*x + a*c)^9 + 6*(144*cosh(b*c*x + a*c)^2 - 41)*sinh(b*c*x + a* 
c)^7 - 246*cosh(b*c*x + a*c)^7 + 42*(48*cosh(b*c*x + a*c)^3 - 41*cosh(b*c* 
x + a*c))*sinh(b*c*x + a*c)^6 + 2*(1512*cosh(b*c*x + a*c)^4 - 2583*cosh(b* 
c*x + a*c)^2 + 187)*sinh(b*c*x + a*c)^5 + 374*cosh(b*c*x + a*c)^5 + 2*(151 
2*cosh(b*c*x + a*c)^5 - 4305*cosh(b*c*x + a*c)^3 + 935*cosh(b*c*x + a*c))* 
sinh(b*c*x + a*c)^4 + 2*(1008*cosh(b*c*x + a*c)^6 - 4305*cosh(b*c*x + a*c) 
^4 + 1870*cosh(b*c*x + a*c)^2 - 157)*sinh(b*c*x + a*c)^3 - 314*cosh(b*c*x 
+ a*c)^3 + 2*(432*cosh(b*c*x + a*c)^7 - 2583*cosh(b*c*x + a*c)^5 + 1870*co 
sh(b*c*x + a*c)^3 - 471*cosh(b*c*x + a*c))*sinh(b*c*x + a*c)^2 - 45*(cosh( 
b*c*x + a*c)^8 + 8*cosh(b*c*x + a*c)*sinh(b*c*x + a*c)^7 + sinh(b*c*x + a* 
c)^8 + 4*(7*cosh(b*c*x + a*c)^2 - 1)*sinh(b*c*x + a*c)^6 - 4*cosh(b*c*x + 
a*c)^6 + 8*(7*cosh(b*c*x + a*c)^3 - 3*cosh(b*c*x + a*c))*sinh(b*c*x + a*c) 
^5 + 2*(35*cosh(b*c*x + a*c)^4 - 30*cosh(b*c*x + a*c)^2 + 3)*sinh(b*c*x + 
a*c)^4 + 6*cosh(b*c*x + a*c)^4 + 8*(7*cosh(b*c*x + a*c)^5 - 10*cosh(b*c*x 
+ a*c)^3 + 3*cosh(b*c*x + a*c))*sinh(b*c*x + a*c)^3 + 4*(7*cosh(b*c*x + a* 
c)^6 - 15*cosh(b*c*x + a*c)^4 + 9*cosh(b*c*x + a*c)^2 - 1)*sinh(b*c*x + a* 
c)^2 - 4*cosh(b*c*x + a*c)^2 + 8*(cosh(b*c*x + a*c)^7 - 3*cosh(b*c*x + a*c 
)^5 + 3*cosh(b*c*x + a*c)^3 - cosh(b*c*x + a*c))*sinh(b*c*x + a*c) + 1)*lo 
g(cosh(b*c*x + a*c) + sinh(b*c*x + a*c) + 1) + 45*(cosh(b*c*x + a*c)^8 ...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {e^{c (a+b x)}}{\tanh ^2(a c+b c x)^{5/2}} \, dx=\text {Timed out} \] Input:

integrate(exp(c*(b*x+a))/(tanh(b*c*x+a*c)**2)**(5/2),x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.52 \[ \int \frac {e^{c (a+b x)}}{\tanh ^2(a c+b c x)^{5/2}} \, dx=-\frac {15 \, \log \left (e^{\left (b c x + a c\right )} + 1\right )}{8 \, b c} + \frac {15 \, \log \left (e^{\left (b c x + a c\right )} - 1\right )}{8 \, b c} + \frac {12 \, e^{\left (9 \, b c x + 9 \, a c\right )} - 123 \, e^{\left (7 \, b c x + 7 \, a c\right )} + 187 \, e^{\left (5 \, b c x + 5 \, a c\right )} - 157 \, e^{\left (3 \, b c x + 3 \, a c\right )} + 33 \, e^{\left (b c x + a c\right )}}{12 \, b c {\left (e^{\left (8 \, b c x + 8 \, a c\right )} - 4 \, e^{\left (6 \, b c x + 6 \, a c\right )} + 6 \, e^{\left (4 \, b c x + 4 \, a c\right )} - 4 \, e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )}} \] Input:

integrate(exp(c*(b*x+a))/(tanh(b*c*x+a*c)^2)^(5/2),x, algorithm="maxima")
 

Output:

-15/8*log(e^(b*c*x + a*c) + 1)/(b*c) + 15/8*log(e^(b*c*x + a*c) - 1)/(b*c) 
 + 1/12*(12*e^(9*b*c*x + 9*a*c) - 123*e^(7*b*c*x + 7*a*c) + 187*e^(5*b*c*x 
 + 5*a*c) - 157*e^(3*b*c*x + 3*a*c) + 33*e^(b*c*x + a*c))/(b*c*(e^(8*b*c*x 
 + 8*a*c) - 4*e^(6*b*c*x + 6*a*c) + 6*e^(4*b*c*x + 4*a*c) - 4*e^(2*b*c*x + 
 2*a*c) + 1))
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 230, normalized size of antiderivative = 0.72 \[ \int \frac {e^{c (a+b x)}}{\tanh ^2(a c+b c x)^{5/2}} \, dx=\frac {e^{\left (b c x + a c\right )} \mathrm {sgn}\left (e^{\left (2 \, b c x + 2 \, a c\right )} - 1\right )}{b c} - \frac {15 \, \log \left (e^{\left (b c x + a c\right )} + 1\right ) \mathrm {sgn}\left (e^{\left (2 \, b c x + 2 \, a c\right )} - 1\right )}{8 \, b c} + \frac {15 \, \log \left ({\left | e^{\left (b c x + a c\right )} - 1 \right |}\right ) \mathrm {sgn}\left (e^{\left (2 \, b c x + 2 \, a c\right )} - 1\right )}{8 \, b c} - \frac {75 \, e^{\left (7 \, b c x + 7 \, a c\right )} \mathrm {sgn}\left (e^{\left (2 \, b c x + 2 \, a c\right )} - 1\right ) - 115 \, e^{\left (5 \, b c x + 5 \, a c\right )} \mathrm {sgn}\left (e^{\left (2 \, b c x + 2 \, a c\right )} - 1\right ) + 109 \, e^{\left (3 \, b c x + 3 \, a c\right )} \mathrm {sgn}\left (e^{\left (2 \, b c x + 2 \, a c\right )} - 1\right ) - 21 \, e^{\left (b c x + a c\right )} \mathrm {sgn}\left (e^{\left (2 \, b c x + 2 \, a c\right )} - 1\right )}{12 \, b c {\left (e^{\left (2 \, b c x + 2 \, a c\right )} - 1\right )}^{4}} \] Input:

integrate(exp(c*(b*x+a))/(tanh(b*c*x+a*c)^2)^(5/2),x, algorithm="giac")
 

Output:

e^(b*c*x + a*c)*sgn(e^(2*b*c*x + 2*a*c) - 1)/(b*c) - 15/8*log(e^(b*c*x + a 
*c) + 1)*sgn(e^(2*b*c*x + 2*a*c) - 1)/(b*c) + 15/8*log(abs(e^(b*c*x + a*c) 
 - 1))*sgn(e^(2*b*c*x + 2*a*c) - 1)/(b*c) - 1/12*(75*e^(7*b*c*x + 7*a*c)*s 
gn(e^(2*b*c*x + 2*a*c) - 1) - 115*e^(5*b*c*x + 5*a*c)*sgn(e^(2*b*c*x + 2*a 
*c) - 1) + 109*e^(3*b*c*x + 3*a*c)*sgn(e^(2*b*c*x + 2*a*c) - 1) - 21*e^(b* 
c*x + a*c)*sgn(e^(2*b*c*x + 2*a*c) - 1))/(b*c*(e^(2*b*c*x + 2*a*c) - 1)^4)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{c (a+b x)}}{\tanh ^2(a c+b c x)^{5/2}} \, dx=\int \frac {{\mathrm {e}}^{c\,\left (a+b\,x\right )}}{{\left ({\mathrm {tanh}\left (a\,c+b\,c\,x\right )}^2\right )}^{5/2}} \,d x \] Input:

int(exp(c*(a + b*x))/(tanh(a*c + b*c*x)^2)^(5/2),x)
 

Output:

int(exp(c*(a + b*x))/(tanh(a*c + b*c*x)^2)^(5/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 830, normalized size of antiderivative = 2.60 \[ \int \frac {e^{c (a+b x)}}{\tanh ^2(a c+b c x)^{5/2}} \, dx=\frac {-45 e^{6 b c x +6 a c} \mathrm {log}\left (e^{b c x +2 a c}+e^{a c}\right ) \tanh \left (b c x +a c \right )^{4}+45 e^{6 b c x +6 a c} \mathrm {log}\left (e^{b c x +2 a c}-e^{a c}\right ) \tanh \left (b c x +a c \right )^{4}+135 e^{4 b c x +4 a c} \mathrm {log}\left (e^{b c x +2 a c}+e^{a c}\right ) \tanh \left (b c x +a c \right )^{4}-135 e^{4 b c x +4 a c} \mathrm {log}\left (e^{b c x +2 a c}-e^{a c}\right ) \tanh \left (b c x +a c \right )^{4}-135 e^{2 b c x +2 a c} \mathrm {log}\left (e^{b c x +2 a c}+e^{a c}\right ) \tanh \left (b c x +a c \right )^{4}+135 e^{2 b c x +2 a c} \mathrm {log}\left (e^{b c x +2 a c}-e^{a c}\right ) \tanh \left (b c x +a c \right )^{4}+6 e^{b c x +a c}+24 e^{3 b c x +3 a c} \tanh \left (b c x +a c \right )+8 e^{b c x +a c} \tanh \left (b c x +a c \right )^{2}-8 e^{b c x +a c} \tanh \left (b c x +a c \right )+14 e^{7 b c x +7 a c} \tanh \left (b c x +a c \right )^{4}+16 e^{7 b c x +7 a c} \tanh \left (b c x +a c \right )^{3}-8 e^{7 b c x +7 a c} \tanh \left (b c x +a c \right )^{2}+8 e^{7 b c x +7 a c} \tanh \left (b c x +a c \right )-192 e^{5 b c x +5 a c} \tanh \left (b c x +a c \right )^{4}-48 e^{5 b c x +5 a c} \tanh \left (b c x +a c \right )^{3}+24 e^{5 b c x +5 a c} \tanh \left (b c x +a c \right )^{2}-24 e^{5 b c x +5 a c} \tanh \left (b c x +a c \right )+202 e^{3 b c x +3 a c} \tanh \left (b c x +a c \right )^{4}+48 e^{3 b c x +3 a c} \tanh \left (b c x +a c \right )^{3}-104 e^{b c x +a c} \tanh \left (b c x +a c \right )^{4}-16 e^{b c x +a c} \tanh \left (b c x +a c \right )^{3}+45 \,\mathrm {log}\left (e^{b c x +2 a c}+e^{a c}\right ) \tanh \left (b c x +a c \right )^{4}-45 \,\mathrm {log}\left (e^{b c x +2 a c}-e^{a c}\right ) \tanh \left (b c x +a c \right )^{4}-24 e^{3 b c x +3 a c} \tanh \left (b c x +a c \right )^{2}-6 e^{7 b c x +7 a c}+18 e^{5 b c x +5 a c}-18 e^{3 b c x +3 a c}}{24 \tanh \left (b c x +a c \right )^{4} b c \left (e^{6 b c x +6 a c}-3 e^{4 b c x +4 a c}+3 e^{2 b c x +2 a c}-1\right )} \] Input:

int(exp(c*(b*x+a))/(tanh(b*c*x+a*c)^2)^(5/2),x)
 

Output:

(14*e**(7*a*c + 7*b*c*x)*tanh(a*c + b*c*x)**4 + 16*e**(7*a*c + 7*b*c*x)*ta 
nh(a*c + b*c*x)**3 - 8*e**(7*a*c + 7*b*c*x)*tanh(a*c + b*c*x)**2 + 8*e**(7 
*a*c + 7*b*c*x)*tanh(a*c + b*c*x) - 6*e**(7*a*c + 7*b*c*x) - 45*e**(6*a*c 
+ 6*b*c*x)*log(e**(2*a*c + b*c*x) + e**(a*c))*tanh(a*c + b*c*x)**4 + 45*e* 
*(6*a*c + 6*b*c*x)*log(e**(2*a*c + b*c*x) - e**(a*c))*tanh(a*c + b*c*x)**4 
 - 192*e**(5*a*c + 5*b*c*x)*tanh(a*c + b*c*x)**4 - 48*e**(5*a*c + 5*b*c*x) 
*tanh(a*c + b*c*x)**3 + 24*e**(5*a*c + 5*b*c*x)*tanh(a*c + b*c*x)**2 - 24* 
e**(5*a*c + 5*b*c*x)*tanh(a*c + b*c*x) + 18*e**(5*a*c + 5*b*c*x) + 135*e** 
(4*a*c + 4*b*c*x)*log(e**(2*a*c + b*c*x) + e**(a*c))*tanh(a*c + b*c*x)**4 
- 135*e**(4*a*c + 4*b*c*x)*log(e**(2*a*c + b*c*x) - e**(a*c))*tanh(a*c + b 
*c*x)**4 + 202*e**(3*a*c + 3*b*c*x)*tanh(a*c + b*c*x)**4 + 48*e**(3*a*c + 
3*b*c*x)*tanh(a*c + b*c*x)**3 - 24*e**(3*a*c + 3*b*c*x)*tanh(a*c + b*c*x)* 
*2 + 24*e**(3*a*c + 3*b*c*x)*tanh(a*c + b*c*x) - 18*e**(3*a*c + 3*b*c*x) - 
 135*e**(2*a*c + 2*b*c*x)*log(e**(2*a*c + b*c*x) + e**(a*c))*tanh(a*c + b* 
c*x)**4 + 135*e**(2*a*c + 2*b*c*x)*log(e**(2*a*c + b*c*x) - e**(a*c))*tanh 
(a*c + b*c*x)**4 - 104*e**(a*c + b*c*x)*tanh(a*c + b*c*x)**4 - 16*e**(a*c 
+ b*c*x)*tanh(a*c + b*c*x)**3 + 8*e**(a*c + b*c*x)*tanh(a*c + b*c*x)**2 - 
8*e**(a*c + b*c*x)*tanh(a*c + b*c*x) + 6*e**(a*c + b*c*x) + 45*log(e**(2*a 
*c + b*c*x) + e**(a*c))*tanh(a*c + b*c*x)**4 - 45*log(e**(2*a*c + b*c*x) - 
 e**(a*c))*tanh(a*c + b*c*x)**4)/(24*tanh(a*c + b*c*x)**4*b*c*(e**(6*a*...