\(\int e^{c (a+b x)} \tanh ^2(a c+b c x)^{5/2} \, dx\) [245]

Optimal result

Integrand size = 25, antiderivative size = 311 \[ \int e^{c (a+b x)} \tanh ^2(a c+b c x)^{5/2} \, dx=\frac {e^{c (a+b x)} \coth (a c+b c x) \sqrt {\tanh ^2(a c+b c x)}}{b c}-\frac {4 e^{c (a+b x)} \coth (a c+b c x) \sqrt {\tanh ^2(a c+b c x)}}{b c \left (1+e^{2 c (a+b x)}\right )^4}+\frac {26 e^{c (a+b x)} \coth (a c+b c x) \sqrt {\tanh ^2(a c+b c x)}}{3 b c \left (1+e^{2 c (a+b x)}\right )^3}-\frac {55 e^{c (a+b x)} \coth (a c+b c x) \sqrt {\tanh ^2(a c+b c x)}}{6 b c \left (1+e^{2 c (a+b x)}\right )^2}+\frac {25 e^{c (a+b x)} \coth (a c+b c x) \sqrt {\tanh ^2(a c+b c x)}}{4 b c \left (1+e^{2 c (a+b x)}\right )}-\frac {15 \arctan \left (e^{c (a+b x)}\right ) \coth (a c+b c x) \sqrt {\tanh ^2(a c+b c x)}}{4 b c} \] Output:

exp(c*(b*x+a))*coth(b*c*x+a*c)*(tanh(b*c*x+a*c)^2)^(1/2)/b/c-4*exp(c*(b*x+ 
a))*coth(b*c*x+a*c)*(tanh(b*c*x+a*c)^2)^(1/2)/b/c/(1+exp(2*c*(b*x+a)))^4+2 
6/3*exp(c*(b*x+a))*coth(b*c*x+a*c)*(tanh(b*c*x+a*c)^2)^(1/2)/b/c/(1+exp(2* 
c*(b*x+a)))^3-55/6*exp(c*(b*x+a))*coth(b*c*x+a*c)*(tanh(b*c*x+a*c)^2)^(1/2 
)/b/c/(1+exp(2*c*(b*x+a)))^2+25/4*exp(c*(b*x+a))*coth(b*c*x+a*c)*(tanh(b*c 
*x+a*c)^2)^(1/2)/b/c/(1+exp(2*c*(b*x+a)))-15/4*arctan(exp(c*(b*x+a)))*coth 
(b*c*x+a*c)*(tanh(b*c*x+a*c)^2)^(1/2)/b/c
 

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.43 \[ \int e^{c (a+b x)} \tanh ^2(a c+b c x)^{5/2} \, dx=\frac {\left (e^{c (a+b x)} \left (33+157 e^{2 c (a+b x)}+187 e^{4 c (a+b x)}+123 e^{6 c (a+b x)}+12 e^{8 c (a+b x)}\right )-45 \left (1+e^{2 c (a+b x)}\right )^4 \arctan \left (e^{c (a+b x)}\right )\right ) \coth (c (a+b x)) \sqrt {\tanh ^2(c (a+b x))}}{12 b c \left (1+e^{2 c (a+b x)}\right )^4} \] Input:

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

Output:

((E^(c*(a + b*x))*(33 + 157*E^(2*c*(a + b*x)) + 187*E^(4*c*(a + b*x)) + 12 
3*E^(6*c*(a + b*x)) + 12*E^(8*c*(a + b*x))) - 45*(1 + E^(2*c*(a + b*x)))^4 
*ArcTan[E^(c*(a + b*x))])*Coth[c*(a + b*x)]*Sqrt[Tanh[c*(a + b*x)]^2])/(12 
*b*c*(1 + E^(2*c*(a + b*x)))^4)
 

Rubi [A] (verified)

Time = 1.52 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.52, 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.

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

\(\Big \downarrow \) 7271

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 300

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

\(\Big \downarrow \) 2009

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

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*ArcTan[E^(c*(a + b*x))])/4)*Coth[a*c + b*c*x]*Sqrt[Tanh[a*c + b*c*x 
]^2])/(b*c)
 

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 = 1.98 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.63

method	result	size
default	\(\frac {\operatorname {csgn}\left (\tanh \left (c \left (b x +a \right )\right )\right ) \left (\frac {\sinh \left (b c x +a c \right )^{5}}{\cosh \left (b c x +a c \right )^{4}}+\frac {5 \sinh \left (b c x +a c \right )^{3}}{\cosh \left (b c x +a c \right )^{4}}+\frac {5 \sinh \left (b c x +a c \right )}{\cosh \left (b c x +a c \right )^{4}}-5 \left (\frac {\operatorname {sech}\left (b c x +a c \right )^{3}}{4}+\frac {3 \,\operatorname {sech}\left (b c x +a c \right )}{8}\right ) \tanh \left (b c x +a c \right )-\frac {15 \arctan \left ({\mathrm e}^{b c x +a c}\right )}{4}+\frac {\sinh \left (b c x +a c \right )^{4}}{\cosh \left (b c x +a c \right )^{3}}+\frac {4 \sinh \left (b c x +a c \right )^{2}}{\cosh \left (b c x +a c \right )^{3}}+\frac {8}{3 \cosh \left (b c x +a c \right )^{3}}\right )}{c b}\)	\(195\)
risch	\(\frac {\left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right ) \sqrt {\frac {\left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right )^{2}}{\left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right )^{2}}}\, {\mathrm e}^{c \left (b x +a \right )}}{\left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right ) b c}+\frac {\sqrt {\frac {\left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right )^{2}}{\left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right )^{2}}}\, {\mathrm e}^{c \left (b x +a \right )} \left (75 \,{\mathrm e}^{6 c \left (b x +a \right )}+115 \,{\mathrm e}^{4 c \left (b x +a \right )}+109 \,{\mathrm e}^{2 c \left (b x +a \right )}+21\right )}{12 \left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right ) \left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right )^{3} c b}+\frac {15 i \left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right ) \sqrt {\frac {\left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right )^{2}}{\left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right )^{2}}}\, \ln \left ({\mathrm e}^{c \left (b x +a \right )}-i\right )}{8 \left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right ) c b}-\frac {15 i \left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right ) \sqrt {\frac {\left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right )^{2}}{\left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right )^{2}}}\, \ln \left ({\mathrm e}^{c \left (b x +a \right )}+i\right )}{8 \left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right ) c b}\)	\(324\)

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*(sinh(b*c*x+a*c)^5/cosh(b*c*x+a*c)^4+5*sinh(b*c* 
x+a*c)^3/cosh(b*c*x+a*c)^4+5*sinh(b*c*x+a*c)/cosh(b*c*x+a*c)^4-5*(1/4*sech 
(b*c*x+a*c)^3+3/8*sech(b*c*x+a*c))*tanh(b*c*x+a*c)-15/4*arctan(exp(b*c*x+a 
*c))+sinh(b*c*x+a*c)^4/cosh(b*c*x+a*c)^3+4*sinh(b*c*x+a*c)^2/cosh(b*c*x+a* 
c)^3+8/3/cosh(b*c*x+a*c)^3)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1226 vs. \(2 (281) = 562\).

Time = 0.10 (sec) , antiderivative size = 1226, normalized size of antiderivative = 3.94 \[ \int e^{c (a+b x)} \tanh ^2(a c+b c x)^{5/2} \, dx=\text {Too large to display} \] Input:

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

Output:

1/12*(12*cosh(b*c*x + a*c)^9 + 108*cosh(b*c*x + a*c)*sinh(b*c*x + a*c)^8 + 
 12*sinh(b*c*x + a*c)^9 + 3*(144*cosh(b*c*x + a*c)^2 + 41)*sinh(b*c*x + a* 
c)^7 + 123*cosh(b*c*x + a*c)^7 + 21*(48*cosh(b*c*x + a*c)^3 + 41*cosh(b*c* 
x + a*c))*sinh(b*c*x + a*c)^6 + (1512*cosh(b*c*x + a*c)^4 + 2583*cosh(b*c* 
x + a*c)^2 + 187)*sinh(b*c*x + a*c)^5 + 187*cosh(b*c*x + a*c)^5 + (1512*co 
sh(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 + (1008*cosh(b*c*x + a*c)^6 + 4305*cosh(b*c*x + a*c)^4 + 1 
870*cosh(b*c*x + a*c)^2 + 157)*sinh(b*c*x + a*c)^3 + 157*cosh(b*c*x + a*c) 
^3 + (432*cosh(b*c*x + a*c)^7 + 2583*cosh(b*c*x + a*c)^5 + 1870*cosh(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 + 1 
5*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)*arctan(cos 
h(b*c*x + a*c) + sinh(b*c*x + a*c)) + (108*cosh(b*c*x + a*c)^8 + 861*co...
 

Sympy [F(-1)]

Timed out. \[ \int e^{c (a+b x)} \tanh ^2(a c+b c x)^{5/2} \, dx=\text {Timed out} \] 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 = 145, normalized size of antiderivative = 0.47 \[ \int e^{c (a+b x)} \tanh ^2(a c+b c x)^{5/2} \, dx=-\frac {15 \, \arctan \left (e^{\left (b c x + a c\right )}\right )}{4 \, b c} + \frac {12 \, e^{\left (9 \, b c x + 9 \, a c\right )} + 123 \, e^{\left (7 \, b c x + 7 \, a c\right )} + 187 \, e^{\left (5 \, b c x + 5 \, a c\right )} + 157 \, e^{\left (3 \, b c x + 3 \, a c\right )} + 33 \, e^{\left (b c x + a c\right )}}{12 \, b c {\left (e^{\left (8 \, b c x + 8 \, a c\right )} + 4 \, e^{\left (6 \, b c x + 6 \, a c\right )} + 6 \, e^{\left (4 \, b c x + 4 \, a c\right )} + 4 \, e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )}} \] Input:

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

Output:

-15/4*arctan(e^(b*c*x + a*c))/(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) + 3 
3*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 = 0.13 (sec) , antiderivative size = 193, normalized size of antiderivative = 0.62 \[ \int e^{c (a+b x)} \tanh ^2(a c+b c x)^{5/2} \, dx=-\frac {15 \, \arctan \left (e^{\left (b c x + a c\right )}\right ) \mathrm {sgn}\left (e^{\left (2 \, b c x + 2 \, a c\right )} - 1\right )}{4 \, b c} + \frac {e^{\left (b c x + a c\right )} \mathrm {sgn}\left (e^{\left (2 \, b c x + 2 \, a c\right )} - 1\right )}{b c} + \frac {75 \, e^{\left (7 \, b c x + 7 \, a c\right )} \mathrm {sgn}\left (e^{\left (2 \, b c x + 2 \, a c\right )} - 1\right ) + 115 \, e^{\left (5 \, b c x + 5 \, a c\right )} \mathrm {sgn}\left (e^{\left (2 \, b c x + 2 \, a c\right )} - 1\right ) + 109 \, e^{\left (3 \, b c x + 3 \, a c\right )} \mathrm {sgn}\left (e^{\left (2 \, b c x + 2 \, a c\right )} - 1\right ) + 21 \, e^{\left (b c x + a c\right )} \mathrm {sgn}\left (e^{\left (2 \, b c x + 2 \, a c\right )} - 1\right )}{12 \, b c {\left (e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )}^{4}} \] Input:

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

Output:

-15/4*arctan(e^(b*c*x + a*c))*sgn(e^(2*b*c*x + 2*a*c) - 1)/(b*c) + e^(b*c* 
x + a*c)*sgn(e^(2*b*c*x + 2*a*c) - 1)/(b*c) + 1/12*(75*e^(7*b*c*x + 7*a*c) 
*sgn(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. \[ \int e^{c (a+b x)} \tanh ^2(a c+b c x)^{5/2} \, dx=\int {\mathrm {e}}^{c\,\left (a+b\,x\right )}\,{\left ({\mathrm {tanh}\left (a\,c+b\,c\,x\right )}^2\right )}^{5/2} \,d x \] 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 = 0.23 (sec) , antiderivative size = 248, normalized size of antiderivative = 0.80 \[ \int e^{c (a+b x)} \tanh ^2(a c+b c x)^{5/2} \, dx=\frac {-45 e^{8 b c x +8 a c} \mathit {atan} \left (e^{b c x +a c}\right )-180 e^{6 b c x +6 a c} \mathit {atan} \left (e^{b c x +a c}\right )-270 e^{4 b c x +4 a c} \mathit {atan} \left (e^{b c x +a c}\right )-180 e^{2 b c x +2 a c} \mathit {atan} \left (e^{b c x +a c}\right )-45 \mathit {atan} \left (e^{b c x +a c}\right )+12 e^{9 b c x +9 a c}+123 e^{7 b c x +7 a c}+187 e^{5 b c x +5 a c}+157 e^{3 b c x +3 a c}+33 e^{b c x +a c}}{12 b c \left (e^{8 b c x +8 a c}+4 e^{6 b c x +6 a c}+6 e^{4 b c x +4 a c}+4 e^{2 b c x +2 a c}+1\right )} \] Input:

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

Output:

( - 45*e**(8*a*c + 8*b*c*x)*atan(e**(a*c + b*c*x)) - 180*e**(6*a*c + 6*b*c 
*x)*atan(e**(a*c + b*c*x)) - 270*e**(4*a*c + 4*b*c*x)*atan(e**(a*c + b*c*x 
)) - 180*e**(2*a*c + 2*b*c*x)*atan(e**(a*c + b*c*x)) - 45*atan(e**(a*c + b 
*c*x)) + 12*e**(9*a*c + 9*b*c*x) + 123*e**(7*a*c + 7*b*c*x) + 187*e**(5*a* 
c + 5*b*c*x) + 157*e**(3*a*c + 3*b*c*x) + 33*e**(a*c + b*c*x))/(12*b*c*(e* 
*(8*a*c + 8*b*c*x) + 4*e**(6*a*c + 6*b*c*x) + 6*e**(4*a*c + 4*b*c*x) + 4*e 
**(2*a*c + 2*b*c*x) + 1))