\(\int \sinh (a+b x) \tanh ^3(c+b x) \, dx\) [34]

Optimal result

Integrand size = 15, antiderivative size = 72 \[ \int \sinh (a+b x) \tanh ^3(c+b x) \, dx=-\frac {3 \arctan (\sinh (c+b x)) \cosh (a-c)}{2 b}+\frac {\text {sech}(c+b x) \sinh (a-c)}{b}+\frac {\sinh (a+b x)}{b}+\frac {\cosh (a-c) \text {sech}(c+b x) \tanh (c+b x)}{2 b} \] Output:

-3/2*arctan(sinh(b*x+c))*cosh(a-c)/b+sech(b*x+c)*sinh(a-c)/b+sinh(b*x+a)/b 
+1/2*cosh(a-c)*sech(b*x+c)*tanh(b*x+c)/b
 

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.97 \[ \int \sinh (a+b x) \tanh ^3(c+b x) \, dx=\frac {-12 \arctan \left (\sinh (c)+\cosh (c) \tanh \left (\frac {b x}{2}\right )\right ) \cosh (a-c)+\text {sech}^2(c+b x) (2 \sinh (a-2 c-b x)+5 \sinh (a+b x)+\sinh (a+2 c+3 b x))}{4 b} \] Input:

Integrate[Sinh[a + b*x]*Tanh[c + b*x]^3,x]
 

Output:

(-12*ArcTan[Sinh[c] + Cosh[c]*Tanh[(b*x)/2]]*Cosh[a - c] + Sech[c + b*x]^2 
*(2*Sinh[a - 2*c - b*x] + 5*Sinh[a + b*x] + Sinh[a + 2*c + 3*b*x]))/(4*b)
 

Rubi [A] (verified)

Time = 0.76 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.19, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {6154, 3042, 25, 3091, 3042, 4257, 6157, 3042, 26, 3086, 24, 6154, 3042, 3117, 4257}

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.

\(\displaystyle \int \sinh (a+b x) \tanh ^3(b x+c) \, dx\)

\(\Big \downarrow \) 6154

\(\displaystyle \int \cosh (a+b x) \tanh ^2(c+b x)dx-\cosh (a-c) \int \text {sech}(c+b x) \tanh ^2(c+b x)dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \cosh (a+b x) \tanh ^2(c+b x)dx-\cosh (a-c) \int -\sec (i c+i b x) \tan (i c+i b x)^2dx\)

\(\Big \downarrow \) 25

\(\displaystyle \int \cosh (a+b x) \tanh ^2(c+b x)dx+\cosh (a-c) \int \sec (i c+i b x) \tan (i c+i b x)^2dx\)

\(\Big \downarrow \) 3091

\(\displaystyle \int \cosh (a+b x) \tanh ^2(c+b x)dx+\cosh (a-c) \left (\frac {\tanh (b x+c) \text {sech}(b x+c)}{2 b}-\frac {1}{2} \int \text {sech}(c+b x)dx\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \cosh (a+b x) \tanh ^2(c+b x)dx+\cosh (a-c) \left (\frac {\tanh (b x+c) \text {sech}(b x+c)}{2 b}-\frac {1}{2} \int \csc \left (i c+i b x+\frac {\pi }{2}\right )dx\right )\)

\(\Big \downarrow \) 4257

\(\displaystyle \int \cosh (a+b x) \tanh ^2(c+b x)dx+\cosh (a-c) \left (\frac {\tanh (b x+c) \text {sech}(b x+c)}{2 b}-\frac {\arctan (\sinh (b x+c))}{2 b}\right )\)

\(\Big \downarrow \) 6157

\(\displaystyle \int \sinh (a+b x) \tanh (c+b x)dx-\sinh (a-c) \int \text {sech}(c+b x) \tanh (c+b x)dx+\cosh (a-c) \left (\frac {\tanh (b x+c) \text {sech}(b x+c)}{2 b}-\frac {\arctan (\sinh (b x+c))}{2 b}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \sinh (a+b x) \tanh (c+b x)dx-\sinh (a-c) \int -i \sec (i c+i b x) \tan (i c+i b x)dx+\cosh (a-c) \left (\frac {\tanh (b x+c) \text {sech}(b x+c)}{2 b}-\frac {\arctan (\sinh (b x+c))}{2 b}\right )\)

\(\Big \downarrow \) 26

\(\displaystyle \int \sinh (a+b x) \tanh (c+b x)dx+i \sinh (a-c) \int \sec (i c+i b x) \tan (i c+i b x)dx+\cosh (a-c) \left (\frac {\tanh (b x+c) \text {sech}(b x+c)}{2 b}-\frac {\arctan (\sinh (b x+c))}{2 b}\right )\)

\(\Big \downarrow \) 3086

\(\displaystyle \int \sinh (a+b x) \tanh (c+b x)dx+\frac {\sinh (a-c) \int 1d\text {sech}(c+b x)}{b}+\cosh (a-c) \left (\frac {\tanh (b x+c) \text {sech}(b x+c)}{2 b}-\frac {\arctan (\sinh (b x+c))}{2 b}\right )\)

\(\Big \downarrow \) 24

\(\displaystyle \int \sinh (a+b x) \tanh (c+b x)dx+\cosh (a-c) \left (\frac {\tanh (b x+c) \text {sech}(b x+c)}{2 b}-\frac {\arctan (\sinh (b x+c))}{2 b}\right )+\frac {\sinh (a-c) \text {sech}(b x+c)}{b}\)

\(\Big \downarrow \) 6154

\(\displaystyle -\cosh (a-c) \int \text {sech}(c+b x)dx+\int \cosh (a+b x)dx+\cosh (a-c) \left (\frac {\tanh (b x+c) \text {sech}(b x+c)}{2 b}-\frac {\arctan (\sinh (b x+c))}{2 b}\right )+\frac {\sinh (a-c) \text {sech}(b x+c)}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\cosh (a-c) \int \csc \left (i c+i b x+\frac {\pi }{2}\right )dx+\int \sin \left (i a+i b x+\frac {\pi }{2}\right )dx+\cosh (a-c) \left (\frac {\tanh (b x+c) \text {sech}(b x+c)}{2 b}-\frac {\arctan (\sinh (b x+c))}{2 b}\right )+\frac {\sinh (a-c) \text {sech}(b x+c)}{b}\)

\(\Big \downarrow \) 3117

\(\displaystyle -\cosh (a-c) \int \csc \left (i c+i b x+\frac {\pi }{2}\right )dx+\cosh (a-c) \left (\frac {\tanh (b x+c) \text {sech}(b x+c)}{2 b}-\frac {\arctan (\sinh (b x+c))}{2 b}\right )+\frac {\sinh (a-c) \text {sech}(b x+c)}{b}+\frac {\sinh (a+b x)}{b}\)

\(\Big \downarrow \) 4257

\(\displaystyle -\frac {\cosh (a-c) \arctan (\sinh (b x+c))}{b}+\cosh (a-c) \left (\frac {\tanh (b x+c) \text {sech}(b x+c)}{2 b}-\frac {\arctan (\sinh (b x+c))}{2 b}\right )+\frac {\sinh (a-c) \text {sech}(b x+c)}{b}+\frac {\sinh (a+b x)}{b}\)

Input:

Int[Sinh[a + b*x]*Tanh[c + b*x]^3,x]
 

Output:

-((ArcTan[Sinh[c + b*x]]*Cosh[a - c])/b) + (Sech[c + b*x]*Sinh[a - c])/b + 
 Sinh[a + b*x]/b + Cosh[a - c]*(-1/2*ArcTan[Sinh[c + b*x]]/b + (Sech[c + b 
*x]*Tanh[c + b*x])/(2*b))
 

Defintions of rubi rules used

rule 24

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]
 

rule 25

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

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3086

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( 
n_.), x_Symbol] :> Simp[a/f   Subst[Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2 
), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2 
] &&  !(IntegerQ[m/2] && LtQ[0, m, n + 1])
 

rule 3091

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( 
n_), x_Symbol] :> Simp[b*(a*Sec[e + f*x])^m*((b*Tan[e + f*x])^(n - 1)/(f*(m 
 + n - 1))), x] - Simp[b^2*((n - 1)/(m + n - 1))   Int[(a*Sec[e + f*x])^m*( 
b*Tan[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && GtQ[n, 1] & 
& NeQ[m + n - 1, 0] && IntegersQ[2*m, 2*n]
 

rule 3117

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; 
 FreeQ[{c, d}, x]
 

rule 4257

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] 
 /; FreeQ[{c, d}, x]
 

rule 6154

Int[Sinh[v_]*Tanh[w_]^(n_.), x_Symbol] :> Int[Cosh[v]*Tanh[w]^(n - 1), x] - 
 Simp[Cosh[v - w]   Int[Sech[w]*Tanh[w]^(n - 1), x], x] /; GtQ[n, 0] && NeQ 
[w, v] && FreeQ[v - w, x]
 

rule 6157

Int[Cosh[v_]*Tanh[w_]^(n_.), x_Symbol] :> Int[Sinh[v]*Tanh[w]^(n - 1), x] - 
 Simp[Sinh[v - w]   Int[Sech[w]*Tanh[w]^(n - 1), x], x] /; GtQ[n, 0] && NeQ 
[w, v] && FreeQ[v - w, x]
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.14 (sec) , antiderivative size = 240, normalized size of antiderivative = 3.33

method	result	size
risch	\(\frac {{\mathrm e}^{b x +a}}{2 b}-\frac {{\mathrm e}^{-b x -a}}{2 b}+\frac {{\mathrm e}^{b x +a} \left (3 \,{\mathrm e}^{2 b x +4 a +2 c}-{\mathrm e}^{2 b x +2 a +4 c}+{\mathrm e}^{4 a}-3 \,{\mathrm e}^{2 a +2 c}\right )}{2 b \left ({\mathrm e}^{2 b x +2 a +2 c}+{\mathrm e}^{2 a}\right )^{2}}+\frac {3 i \ln \left ({\mathrm e}^{b x +a}-i {\mathrm e}^{a -c}\right ) {\mathrm e}^{-a -c} {\mathrm e}^{2 a}}{4 b}+\frac {3 i \ln \left ({\mathrm e}^{b x +a}-i {\mathrm e}^{a -c}\right ) {\mathrm e}^{-a -c} {\mathrm e}^{2 c}}{4 b}-\frac {3 i \ln \left ({\mathrm e}^{b x +a}+i {\mathrm e}^{a -c}\right ) {\mathrm e}^{-a -c} {\mathrm e}^{2 a}}{4 b}-\frac {3 i \ln \left ({\mathrm e}^{b x +a}+i {\mathrm e}^{a -c}\right ) {\mathrm e}^{-a -c} {\mathrm e}^{2 c}}{4 b}\)	\(240\)

Input:

int(sinh(b*x+a)*tanh(b*x+c)^3,x,method=_RETURNVERBOSE)
                                                                                    
                                                                                    
 

Output:

1/2/b*exp(b*x+a)-1/2/b*exp(-b*x-a)+1/2*exp(b*x+a)*(3*exp(2*b*x+4*a+2*c)-ex 
p(2*b*x+2*a+4*c)+exp(4*a)-3*exp(2*a+2*c))/b/(exp(2*b*x+2*a+2*c)+exp(2*a))^ 
2+3/4*I*ln(exp(b*x+a)-I*exp(a-c))/b*exp(-a-c)*exp(2*a)+3/4*I*ln(exp(b*x+a) 
-I*exp(a-c))/b*exp(-a-c)*exp(2*c)-3/4*I*ln(exp(b*x+a)+I*exp(a-c))/b*exp(-a 
-c)*exp(2*a)-3/4*I*ln(exp(b*x+a)+I*exp(a-c))/b*exp(-a-c)*exp(2*c)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1737 vs. \(2 (68) = 136\).

Time = 0.12 (sec) , antiderivative size = 1737, normalized size of antiderivative = 24.12 \[ \int \sinh (a+b x) \tanh ^3(c+b x) \, dx=\text {Too large to display} \] Input:

integrate(sinh(b*x+a)*tanh(b*x+c)^3,x, algorithm="fricas")
 

Output:

1/2*(cosh(b*x + c)^6*cosh(-a + c)^2 + (cosh(-a + c)^2 - 2*cosh(-a + c)*sin 
h(-a + c) + sinh(-a + c)^2)*sinh(b*x + c)^6 + 6*(cosh(b*x + c)*cosh(-a + c 
)^2 - 2*cosh(b*x + c)*cosh(-a + c)*sinh(-a + c) + cosh(b*x + c)*sinh(-a + 
c)^2)*sinh(b*x + c)^5 + (5*cosh(-a + c)^2 - 2)*cosh(b*x + c)^4 + (15*cosh( 
b*x + c)^2*cosh(-a + c)^2 + 5*(3*cosh(b*x + c)^2 + 1)*sinh(-a + c)^2 + 5*c 
osh(-a + c)^2 - 10*(3*cosh(b*x + c)^2*cosh(-a + c) + cosh(-a + c))*sinh(-a 
 + c) - 2)*sinh(b*x + c)^4 + 4*(5*cosh(b*x + c)^3*cosh(-a + c)^2 + 5*(cosh 
(b*x + c)^3 + cosh(b*x + c))*sinh(-a + c)^2 + (5*cosh(-a + c)^2 - 2)*cosh( 
b*x + c) - 10*(cosh(b*x + c)^3*cosh(-a + c) + cosh(b*x + c)*cosh(-a + c))* 
sinh(-a + c))*sinh(b*x + c)^3 + (2*cosh(-a + c)^2 - 5)*cosh(b*x + c)^2 + ( 
15*cosh(b*x + c)^4*cosh(-a + c)^2 + 6*(5*cosh(-a + c)^2 - 2)*cosh(b*x + c) 
^2 + (15*cosh(b*x + c)^4 + 30*cosh(b*x + c)^2 + 2)*sinh(-a + c)^2 + 2*cosh 
(-a + c)^2 - 2*(15*cosh(b*x + c)^4*cosh(-a + c) + 30*cosh(b*x + c)^2*cosh( 
-a + c) + 2*cosh(-a + c))*sinh(-a + c) - 5)*sinh(b*x + c)^2 + (cosh(b*x + 
c)^6 + 5*cosh(b*x + c)^4 + 2*cosh(b*x + c)^2)*sinh(-a + c)^2 - 3*((cosh(-a 
 + c)^2 + 1)*cosh(b*x + c)^5 + (cosh(-a + c)^2 - 2*cosh(-a + c)*sinh(-a + 
c) + sinh(-a + c)^2 + 1)*sinh(b*x + c)^5 - 5*(2*cosh(b*x + c)*cosh(-a + c) 
*sinh(-a + c) - cosh(b*x + c)*sinh(-a + c)^2 - (cosh(-a + c)^2 + 1)*cosh(b 
*x + c))*sinh(b*x + c)^4 + 2*(cosh(-a + c)^2 + 1)*cosh(b*x + c)^3 + 2*(5*( 
cosh(-a + c)^2 + 1)*cosh(b*x + c)^2 + (5*cosh(b*x + c)^2 + 1)*sinh(-a +...
 

Sympy [F]

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

integrate(sinh(b*x+a)*tanh(b*x+c)**3,x)
 

Output:

Integral(sinh(a + b*x)*tanh(b*x + c)**3, x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 149 vs. \(2 (68) = 136\).

Time = 0.14 (sec) , antiderivative size = 149, normalized size of antiderivative = 2.07 \[ \int \sinh (a+b x) \tanh ^3(c+b x) \, dx=\frac {3 \, {\left (e^{\left (2 \, a\right )} + e^{\left (2 \, c\right )}\right )} \arctan \left (e^{\left (-b x - c\right )}\right ) e^{\left (-a - c\right )}}{2 \, b} - \frac {e^{\left (-b x - a\right )}}{2 \, b} + \frac {{\left (5 \, e^{\left (2 \, a + 2 \, c\right )} - e^{\left (4 \, c\right )}\right )} e^{\left (-2 \, b x - 2 \, a\right )} + {\left (2 \, e^{\left (4 \, a\right )} - 3 \, e^{\left (2 \, a + 2 \, c\right )}\right )} e^{\left (-4 \, b x - 4 \, a\right )} + e^{\left (4 \, c\right )}}{2 \, b {\left (e^{\left (-b x - a + 4 \, c\right )} + 2 \, e^{\left (-3 \, b x - a + 2 \, c\right )} + e^{\left (-5 \, b x - a\right )}\right )}} \] Input:

integrate(sinh(b*x+a)*tanh(b*x+c)^3,x, algorithm="maxima")
 

Output:

3/2*(e^(2*a) + e^(2*c))*arctan(e^(-b*x - c))*e^(-a - c)/b - 1/2*e^(-b*x - 
a)/b + 1/2*((5*e^(2*a + 2*c) - e^(4*c))*e^(-2*b*x - 2*a) + (2*e^(4*a) - 3* 
e^(2*a + 2*c))*e^(-4*b*x - 4*a) + e^(4*c))/(b*(e^(-b*x - a + 4*c) + 2*e^(- 
3*b*x - a + 2*c) + e^(-5*b*x - a)))
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.67 \[ \int \sinh (a+b x) \tanh ^3(c+b x) \, dx=-\frac {3 \, {\left (e^{\left (2 \, a\right )} + e^{\left (2 \, c\right )}\right )} \arctan \left (e^{\left (b x + c\right )}\right ) e^{\left (-a - c\right )} - \frac {3 \, e^{\left (3 \, b x + 5 \, a + 2 \, c\right )} - e^{\left (3 \, b x + 3 \, a + 4 \, c\right )} + e^{\left (b x + 5 \, a\right )} - 3 \, e^{\left (b x + 3 \, a + 2 \, c\right )}}{{\left (e^{\left (2 \, b x + 2 \, a + 2 \, c\right )} + e^{\left (2 \, a\right )}\right )}^{2}} - e^{\left (b x + a\right )} + e^{\left (-b x - a\right )}}{2 \, b} \] Input:

integrate(sinh(b*x+a)*tanh(b*x+c)^3,x, algorithm="giac")
 

Output:

-1/2*(3*(e^(2*a) + e^(2*c))*arctan(e^(b*x + c))*e^(-a - c) - (3*e^(3*b*x + 
 5*a + 2*c) - e^(3*b*x + 3*a + 4*c) + e^(b*x + 5*a) - 3*e^(b*x + 3*a + 2*c 
))/(e^(2*b*x + 2*a + 2*c) + e^(2*a))^2 - e^(b*x + a) + e^(-b*x - a))/b
 

Mupad [F(-1)]

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

int(sinh(a + b*x)*tanh(c + b*x)^3,x)
 

Output:

int(sinh(a + b*x)*tanh(c + b*x)^3, x)
 

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.86 \[ \int \sinh (a+b x) \tanh ^3(c+b x) \, dx=\frac {-6 e^{b x +2 a} \mathit {atan} \left (e^{b x +c}\right )-6 e^{b x +2 c} \mathit {atan} \left (e^{b x +c}\right )-2 e^{b x +a +c} \cosh \left (b x +a \right ) \tanh \left (b x +c \right )+3 e^{2 b x +2 a +c}-2 e^{b x +a +c} \sinh \left (b x +a \right ) \tanh \left (b x +c \right )^{2}+2 e^{b x +a +c} \sinh \left (b x +a \right )-3 e^{c}}{4 e^{b x +a +c} b} \] Input:

int(sinh(b*x+a)*tanh(b*x+c)^3,x)
 

Output:

( - 6*e**(2*a + b*x)*atan(e**(b*x + c)) - 6*e**(b*x + 2*c)*atan(e**(b*x + 
c)) - 2*e**(a + b*x + c)*cosh(a + b*x)*tanh(b*x + c) + 3*e**(2*a + 2*b*x + 
 c) - 2*e**(a + b*x + c)*sinh(a + b*x)*tanh(b*x + c)**2 + 2*e**(a + b*x + 
c)*sinh(a + b*x) - 3*e**c)/(4*e**(a + b*x + c)*b)