Integrand size = 12, antiderivative size = 129 \[ \int \frac {1}{(a+b \tanh (c+d x))^3} \, dx=\frac {a \left (a^2+3 b^2\right ) x}{\left (a^2-b^2\right )^3}-\frac {b \left (3 a^2+b^2\right ) \log (a \cosh (c+d x)+b \sinh (c+d x))}{\left (a^2-b^2\right )^3 d}+\frac {b}{2 \left (a^2-b^2\right ) d (a+b \tanh (c+d x))^2}+\frac {2 a b}{\left (a^2-b^2\right )^2 d (a+b \tanh (c+d x))} \] Output:
a*(a^2+3*b^2)*x/(a^2-b^2)^3-b*(3*a^2+b^2)*ln(a*cosh(d*x+c)+b*sinh(d*x+c))/ (a^2-b^2)^3/d+1/2*b/(a^2-b^2)/d/(a+b*tanh(d*x+c))^2+2*a*b/(a^2-b^2)^2/d/(a +b*tanh(d*x+c))
Time = 1.56 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.95 \[ \int \frac {1}{(a+b \tanh (c+d x))^3} \, dx=\frac {-\frac {\log (1-\tanh (c+d x))}{(a+b)^3}+\frac {\log (1+\tanh (c+d x))}{(a-b)^3}+\frac {b \left (-2 \left (3 a^2+b^2\right ) \log (a+b \tanh (c+d x))+\frac {\left (a^2-b^2\right ) \left (5 a^2-b^2+4 a b \tanh (c+d x)\right )}{(a+b \tanh (c+d x))^2}\right )}{\left (a^2-b^2\right )^3}}{2 d} \] Input:
Integrate[(a + b*Tanh[c + d*x])^(-3),x]
Output:
(-(Log[1 - Tanh[c + d*x]]/(a + b)^3) + Log[1 + Tanh[c + d*x]]/(a - b)^3 + (b*(-2*(3*a^2 + b^2)*Log[a + b*Tanh[c + d*x]] + ((a^2 - b^2)*(5*a^2 - b^2 + 4*a*b*Tanh[c + d*x]))/(a + b*Tanh[c + d*x])^2))/(a^2 - b^2)^3)/(2*d)
Time = 0.73 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.20, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {3042, 3964, 3042, 4012, 3042, 4014, 26, 3042, 4013}
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.
| \(\displaystyle \int \frac {1}{(a+b \tanh (c+d x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(a-i b \tan (i c+i d x))^3}dx\) |
| \(\Big \downarrow \) 3964 |
| \(\displaystyle \frac {\int \frac {a-b \tanh (c+d x)}{(a+b \tanh (c+d x))^2}dx}{a^2-b^2}+\frac {b}{2 d \left (a^2-b^2\right ) (a+b \tanh (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {b}{2 d \left (a^2-b^2\right ) (a+b \tanh (c+d x))^2}+\frac {\int \frac {a+i b \tan (i c+i d x)}{(a-i b \tan (i c+i d x))^2}dx}{a^2-b^2}\) |
| \(\Big \downarrow \) 4012 |
| \(\displaystyle \frac {\frac {\int \frac {a^2-2 b \tanh (c+d x) a+b^2}{a+b \tanh (c+d x)}dx}{a^2-b^2}+\frac {2 a b}{d \left (a^2-b^2\right ) (a+b \tanh (c+d x))}}{a^2-b^2}+\frac {b}{2 d \left (a^2-b^2\right ) (a+b \tanh (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {b}{2 d \left (a^2-b^2\right ) (a+b \tanh (c+d x))^2}+\frac {\frac {2 a b}{d \left (a^2-b^2\right ) (a+b \tanh (c+d x))}+\frac {\int \frac {a^2+2 i b \tan (i c+i d x) a+b^2}{a-i b \tan (i c+i d x)}dx}{a^2-b^2}}{a^2-b^2}\) |
| \(\Big \downarrow \) 4014 |
| \(\displaystyle \frac {b}{2 d \left (a^2-b^2\right ) (a+b \tanh (c+d x))^2}+\frac {\frac {2 a b}{d \left (a^2-b^2\right ) (a+b \tanh (c+d x))}+\frac {\frac {a x \left (a^2+3 b^2\right )}{a^2-b^2}-\frac {i b \left (3 a^2+b^2\right ) \int -\frac {i (b+a \tanh (c+d x))}{a+b \tanh (c+d x)}dx}{a^2-b^2}}{a^2-b^2}}{a^2-b^2}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {\frac {\frac {a x \left (a^2+3 b^2\right )}{a^2-b^2}-\frac {b \left (3 a^2+b^2\right ) \int \frac {b+a \tanh (c+d x)}{a+b \tanh (c+d x)}dx}{a^2-b^2}}{a^2-b^2}+\frac {2 a b}{d \left (a^2-b^2\right ) (a+b \tanh (c+d x))}}{a^2-b^2}+\frac {b}{2 d \left (a^2-b^2\right ) (a+b \tanh (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {b}{2 d \left (a^2-b^2\right ) (a+b \tanh (c+d x))^2}+\frac {\frac {2 a b}{d \left (a^2-b^2\right ) (a+b \tanh (c+d x))}+\frac {\frac {a x \left (a^2+3 b^2\right )}{a^2-b^2}-\frac {b \left (3 a^2+b^2\right ) \int \frac {b-i a \tan (i c+i d x)}{a-i b \tan (i c+i d x)}dx}{a^2-b^2}}{a^2-b^2}}{a^2-b^2}\) |
| \(\Big \downarrow \) 4013 |
| \(\displaystyle \frac {b}{2 d \left (a^2-b^2\right ) (a+b \tanh (c+d x))^2}+\frac {\frac {2 a b}{d \left (a^2-b^2\right ) (a+b \tanh (c+d x))}+\frac {\frac {a x \left (a^2+3 b^2\right )}{a^2-b^2}-\frac {b \left (3 a^2+b^2\right ) \log (a \cosh (c+d x)+b \sinh (c+d x))}{d \left (a^2-b^2\right )}}{a^2-b^2}}{a^2-b^2}\) |
Input:
Int[(a + b*Tanh[c + d*x])^(-3),x]
Output:
b/(2*(a^2 - b^2)*d*(a + b*Tanh[c + d*x])^2) + (((a*(a^2 + 3*b^2)*x)/(a^2 - b^2) - (b*(3*a^2 + b^2)*Log[a*Cosh[c + d*x] + b*Sinh[c + d*x]])/((a^2 - b ^2)*d))/(a^2 - b^2) + (2*a*b)/((a^2 - b^2)*d*(a + b*Tanh[c + d*x])))/(a^2 - b^2)
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((a + b*Tan[c + d*x])^(n + 1)/(d*(n + 1)*(a^2 + b^2))), x] + Simp[1/(a^2 + b^2) Int[(a - b*Tan[c + d*x])*(a + b*Tan[c + d*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && LtQ[n, -1]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*c - a*d)*((a + b*Tan[e + f*x])^(m + 1)/ (f*(m + 1)*(a^2 + b^2))), x] + Simp[1/(a^2 + b^2) Int[(a + b*Tan[e + f*x] )^(m + 1)*Simp[a*c + b*d - (b*c - a*d)*Tan[e + f*x], x], x], x] /; FreeQ[{a , b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && LtQ[m, -1 ]
Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_) + (b_.)*tan[(e_.) + (f_.)* (x_)]), x_Symbol] :> Simp[(c/(b*f))*Log[RemoveContent[a*Cos[e + f*x] + b*Si n[e + f*x], x]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && EqQ[a*c + b*d, 0]
Int[((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_.) + (b_.)*tan[(e_.) + (f_. )*(x_)]), x_Symbol] :> Simp[(a*c + b*d)*(x/(a^2 + b^2)), x] + Simp[(b*c - a *d)/(a^2 + b^2) Int[(b - a*Tan[e + f*x])/(a + b*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && N eQ[a*c + b*d, 0]
Time = 0.30 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.01
| method | result | size |
| derivativedivides | \(\frac {\frac {b}{2 \left (a -b \right ) \left (a +b \right ) \left (a +b \tanh \left (d x +c \right )\right )^{2}}+\frac {2 a b}{\left (a +b \right )^{2} \left (a -b \right )^{2} \left (a +b \tanh \left (d x +c \right )\right )}-\frac {b \left (3 a^{2}+b^{2}\right ) \ln \left (a +b \tanh \left (d x +c \right )\right )}{\left (a +b \right )^{3} \left (a -b \right )^{3}}+\frac {\ln \left (\tanh \left (d x +c \right )+1\right )}{2 \left (a -b \right )^{3}}-\frac {\ln \left (\tanh \left (d x +c \right )-1\right )}{2 \left (a +b \right )^{3}}}{d}\) | \(130\) |
| default | \(\frac {\frac {b}{2 \left (a -b \right ) \left (a +b \right ) \left (a +b \tanh \left (d x +c \right )\right )^{2}}+\frac {2 a b}{\left (a +b \right )^{2} \left (a -b \right )^{2} \left (a +b \tanh \left (d x +c \right )\right )}-\frac {b \left (3 a^{2}+b^{2}\right ) \ln \left (a +b \tanh \left (d x +c \right )\right )}{\left (a +b \right )^{3} \left (a -b \right )^{3}}+\frac {\ln \left (\tanh \left (d x +c \right )+1\right )}{2 \left (a -b \right )^{3}}-\frac {\ln \left (\tanh \left (d x +c \right )-1\right )}{2 \left (a +b \right )^{3}}}{d}\) | \(130\) |
| risch | \(\frac {x}{a^{3}+3 a^{2} b +3 b^{2} a +b^{3}}+\frac {6 b \,a^{2} x}{a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}}+\frac {2 b^{3} x}{a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}}+\frac {6 b \,a^{2} c}{d \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right )}+\frac {2 b^{3} c}{d \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right )}+\frac {2 b^{2} \left (3 \,{\mathrm e}^{2 d x +2 c} a^{2}+2 \,{\mathrm e}^{2 d x +2 c} b a -b^{2} {\mathrm e}^{2 d x +2 c}+3 a^{2}-3 a b \right )}{\left (a -b \right )^{2} d \left (a^{3}+3 a^{2} b +3 b^{2} a +b^{3}\right ) \left ({\mathrm e}^{2 d x +2 c} a +{\mathrm e}^{2 d x +2 c} b +a -b \right )^{2}}-\frac {3 b \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a -b}{a +b}\right ) a^{2}}{d \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right )}-\frac {b^{3} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a -b}{a +b}\right )}{d \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right )}\) | \(396\) |
| parallelrisch | \(-\frac {-2 a^{5} b^{2} d x -6 a^{4} b^{3} d x -2 x \tanh \left (d x +c \right )^{2} b^{7} d -6 \ln \left (1-\tanh \left (d x +c \right )\right ) \tanh \left (d x +c \right )^{2} a^{2} b^{5}+6 \ln \left (a +b \tanh \left (d x +c \right )\right ) \tanh \left (d x +c \right )^{2} a^{2} b^{5}-12 \ln \left (1-\tanh \left (d x +c \right )\right ) \tanh \left (d x +c \right ) a^{3} b^{4}-4 \ln \left (1-\tanh \left (d x +c \right )\right ) \tanh \left (d x +c \right ) a \,b^{6}+12 \ln \left (a +b \tanh \left (d x +c \right )\right ) \tanh \left (d x +c \right ) a^{3} b^{4}+4 \ln \left (a +b \tanh \left (d x +c \right )\right ) \tanh \left (d x +c \right ) a \,b^{6}-6 a^{3} b^{4} d x -2 a^{2} b^{5} d x -b^{7}-5 a^{4} b^{3}+6 a^{2} b^{5}-4 \tanh \left (d x +c \right ) a^{3} b^{4}-4 x \tanh \left (d x +c \right ) a \,b^{6} d +2 \ln \left (a +b \tanh \left (d x +c \right )\right ) a^{2} b^{5}+4 \tanh \left (d x +c \right ) a \,b^{6}-6 \ln \left (1-\tanh \left (d x +c \right )\right ) a^{4} b^{3}-2 \ln \left (1-\tanh \left (d x +c \right )\right ) \tanh \left (d x +c \right )^{2} b^{7}-4 x \tanh \left (d x +c \right ) a^{4} b^{3} d -12 x \tanh \left (d x +c \right ) a^{3} b^{4} d -12 x \tanh \left (d x +c \right ) a^{2} b^{5} d -2 x \tanh \left (d x +c \right )^{2} a^{3} b^{4} d -6 x \tanh \left (d x +c \right )^{2} a^{2} b^{5} d -6 x \tanh \left (d x +c \right )^{2} a \,b^{6} d +2 \ln \left (a +b \tanh \left (d x +c \right )\right ) \tanh \left (d x +c \right )^{2} b^{7}-2 \ln \left (1-\tanh \left (d x +c \right )\right ) a^{2} b^{5}+6 \ln \left (a +b \tanh \left (d x +c \right )\right ) a^{4} b^{3}}{2 \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \left (a +b \tanh \left (d x +c \right )\right )^{2} b^{2} d}\) | \(539\) |
Input:
int(1/(a+b*tanh(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
1/d*(1/2*b/(a-b)/(a+b)/(a+b*tanh(d*x+c))^2+2*a*b/(a+b)^2/(a-b)^2/(a+b*tanh (d*x+c))-b*(3*a^2+b^2)/(a+b)^3/(a-b)^3*ln(a+b*tanh(d*x+c))+1/2/(a-b)^3*ln( tanh(d*x+c)+1)-1/2/(a+b)^3*ln(tanh(d*x+c)-1))
Leaf count of result is larger than twice the leaf count of optimal. 1427 vs. \(2 (127) = 254\).
Time = 0.11 (sec) , antiderivative size = 1427, normalized size of antiderivative = 11.06 \[ \int \frac {1}{(a+b \tanh (c+d x))^3} \, dx=\text {Too large to display} \] Input:
integrate(1/(a+b*tanh(d*x+c))^3,x, algorithm="fricas")
Output:
((a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*d*x*cosh(d*x + c)^4 + 4*(a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*d*x*cos h(d*x + c)*sinh(d*x + c)^3 + (a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5* a*b^4 + b^5)*d*x*sinh(d*x + c)^4 + 6*a^3*b^2 - 12*a^2*b^3 + 6*a*b^4 + (a^5 + a^4*b - 2*a^3*b^2 - 2*a^2*b^3 + a*b^4 + b^5)*d*x + 2*(3*a^3*b^2 - a^2*b ^3 - 3*a*b^4 + b^5 + (a^5 + 3*a^4*b + 2*a^3*b^2 - 2*a^2*b^3 - 3*a*b^4 - b^ 5)*d*x)*cosh(d*x + c)^2 + 2*(3*a^3*b^2 - a^2*b^3 - 3*a*b^4 + b^5 + 3*(a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*d*x*cosh(d*x + c)^2 + (a^5 + 3*a^4*b + 2*a^3*b^2 - 2*a^2*b^3 - 3*a*b^4 - b^5)*d*x)*sinh(d*x + c )^2 - (3*a^4*b - 6*a^3*b^2 + 4*a^2*b^3 - 2*a*b^4 + b^5 + (3*a^4*b + 6*a^3* b^2 + 4*a^2*b^3 + 2*a*b^4 + b^5)*cosh(d*x + c)^4 + 4*(3*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + 2*a*b^4 + b^5)*cosh(d*x + c)*sinh(d*x + c)^3 + (3*a^4*b + 6* a^3*b^2 + 4*a^2*b^3 + 2*a*b^4 + b^5)*sinh(d*x + c)^4 + 2*(3*a^4*b - 2*a^2* b^3 - b^5)*cosh(d*x + c)^2 + 2*(3*a^4*b - 2*a^2*b^3 - b^5 + 3*(3*a^4*b + 6 *a^3*b^2 + 4*a^2*b^3 + 2*a*b^4 + b^5)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4 *((3*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + 2*a*b^4 + b^5)*cosh(d*x + c)^3 + (3*a ^4*b - 2*a^2*b^3 - b^5)*cosh(d*x + c))*sinh(d*x + c))*log(2*(a*cosh(d*x + c) + b*sinh(d*x + c))/(cosh(d*x + c) - sinh(d*x + c))) + 4*((a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*d*x*cosh(d*x + c)^3 + (3*a^3*b ^2 - a^2*b^3 - 3*a*b^4 + b^5 + (a^5 + 3*a^4*b + 2*a^3*b^2 - 2*a^2*b^3 -...
Leaf count of result is larger than twice the leaf count of optimal. 6412 vs. \(2 (107) = 214\).
Time = 12.43 (sec) , antiderivative size = 6412, normalized size of antiderivative = 49.71 \[ \int \frac {1}{(a+b \tanh (c+d x))^3} \, dx=\text {Too large to display} \] Input:
integrate(1/(a+b*tanh(d*x+c))**3,x)
Output:
Piecewise((zoo*x/tanh(c)**3, Eq(a, 0) & Eq(b, 0) & Eq(d, 0)), (x/a**3, Eq( b, 0)), (-3*d*x*tanh(c + d*x)**3/(24*b**3*d*tanh(c + d*x)**3 - 72*b**3*d*t anh(c + d*x)**2 + 72*b**3*d*tanh(c + d*x) - 24*b**3*d) + 9*d*x*tanh(c + d* x)**2/(24*b**3*d*tanh(c + d*x)**3 - 72*b**3*d*tanh(c + d*x)**2 + 72*b**3*d *tanh(c + d*x) - 24*b**3*d) - 9*d*x*tanh(c + d*x)/(24*b**3*d*tanh(c + d*x) **3 - 72*b**3*d*tanh(c + d*x)**2 + 72*b**3*d*tanh(c + d*x) - 24*b**3*d) + 3*d*x/(24*b**3*d*tanh(c + d*x)**3 - 72*b**3*d*tanh(c + d*x)**2 + 72*b**3*d *tanh(c + d*x) - 24*b**3*d) + 3*tanh(c + d*x)**2/(24*b**3*d*tanh(c + d*x)* *3 - 72*b**3*d*tanh(c + d*x)**2 + 72*b**3*d*tanh(c + d*x) - 24*b**3*d) - 9 *tanh(c + d*x)/(24*b**3*d*tanh(c + d*x)**3 - 72*b**3*d*tanh(c + d*x)**2 + 72*b**3*d*tanh(c + d*x) - 24*b**3*d) + 10/(24*b**3*d*tanh(c + d*x)**3 - 72 *b**3*d*tanh(c + d*x)**2 + 72*b**3*d*tanh(c + d*x) - 24*b**3*d), Eq(a, -b) ), (3*d*x*tanh(c + d*x)**3/(24*b**3*d*tanh(c + d*x)**3 + 72*b**3*d*tanh(c + d*x)**2 + 72*b**3*d*tanh(c + d*x) + 24*b**3*d) + 9*d*x*tanh(c + d*x)**2/ (24*b**3*d*tanh(c + d*x)**3 + 72*b**3*d*tanh(c + d*x)**2 + 72*b**3*d*tanh( c + d*x) + 24*b**3*d) + 9*d*x*tanh(c + d*x)/(24*b**3*d*tanh(c + d*x)**3 + 72*b**3*d*tanh(c + d*x)**2 + 72*b**3*d*tanh(c + d*x) + 24*b**3*d) + 3*d*x/ (24*b**3*d*tanh(c + d*x)**3 + 72*b**3*d*tanh(c + d*x)**2 + 72*b**3*d*tanh( c + d*x) + 24*b**3*d) - 3*tanh(c + d*x)**2/(24*b**3*d*tanh(c + d*x)**3 + 7 2*b**3*d*tanh(c + d*x)**2 + 72*b**3*d*tanh(c + d*x) + 24*b**3*d) - 9*ta...
Leaf count of result is larger than twice the leaf count of optimal. 325 vs. \(2 (127) = 254\).
Time = 0.08 (sec) , antiderivative size = 325, normalized size of antiderivative = 2.52 \[ \int \frac {1}{(a+b \tanh (c+d x))^3} \, dx=-\frac {{\left (3 \, a^{2} b + b^{3}\right )} \log \left (-{\left (a - b\right )} e^{\left (-2 \, d x - 2 \, c\right )} - a - b\right )}{{\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} d} - \frac {2 \, {\left (3 \, a^{2} b^{2} + 3 \, a b^{3} + {\left (3 \, a^{2} b^{2} - 2 \, a b^{3} - b^{4}\right )} e^{\left (-2 \, d x - 2 \, c\right )}\right )}}{{\left (a^{7} + a^{6} b - 3 \, a^{5} b^{2} - 3 \, a^{4} b^{3} + 3 \, a^{3} b^{4} + 3 \, a^{2} b^{5} - a b^{6} - b^{7} + 2 \, {\left (a^{7} - a^{6} b - 3 \, a^{5} b^{2} + 3 \, a^{4} b^{3} + 3 \, a^{3} b^{4} - 3 \, a^{2} b^{5} - a b^{6} + b^{7}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + {\left (a^{7} - 3 \, a^{6} b + a^{5} b^{2} + 5 \, a^{4} b^{3} - 5 \, a^{3} b^{4} - a^{2} b^{5} + 3 \, a b^{6} - b^{7}\right )} e^{\left (-4 \, d x - 4 \, c\right )}\right )} d} + \frac {d x + c}{{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} d} \] Input:
integrate(1/(a+b*tanh(d*x+c))^3,x, algorithm="maxima")
Output:
-(3*a^2*b + b^3)*log(-(a - b)*e^(-2*d*x - 2*c) - a - b)/((a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*d) - 2*(3*a^2*b^2 + 3*a*b^3 + (3*a^2*b^2 - 2*a*b^3 - b^ 4)*e^(-2*d*x - 2*c))/((a^7 + a^6*b - 3*a^5*b^2 - 3*a^4*b^3 + 3*a^3*b^4 + 3 *a^2*b^5 - a*b^6 - b^7 + 2*(a^7 - a^6*b - 3*a^5*b^2 + 3*a^4*b^3 + 3*a^3*b^ 4 - 3*a^2*b^5 - a*b^6 + b^7)*e^(-2*d*x - 2*c) + (a^7 - 3*a^6*b + a^5*b^2 + 5*a^4*b^3 - 5*a^3*b^4 - a^2*b^5 + 3*a*b^6 - b^7)*e^(-4*d*x - 4*c))*d) + ( d*x + c)/((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d)
Time = 0.14 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.59 \[ \int \frac {1}{(a+b \tanh (c+d x))^3} \, dx=-\frac {\frac {{\left (3 \, a^{2} b + b^{3}\right )} \log \left ({\left | -a e^{\left (2 \, d x + 2 \, c\right )} - b e^{\left (2 \, d x + 2 \, c\right )} - a + b \right |}\right )}{a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}} - \frac {d x + c}{a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}} - \frac {2 \, {\left ({\left (3 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right )} e^{\left (2 \, d x + 2 \, c\right )} + \frac {3 \, {\left (a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4}\right )}}{a + b}\right )}}{{\left (a e^{\left (2 \, d x + 2 \, c\right )} + b e^{\left (2 \, d x + 2 \, c\right )} + a - b\right )}^{2} {\left (a + b\right )}^{2} {\left (a - b\right )}^{3}}}{d} \] Input:
integrate(1/(a+b*tanh(d*x+c))^3,x, algorithm="giac")
Output:
-((3*a^2*b + b^3)*log(abs(-a*e^(2*d*x + 2*c) - b*e^(2*d*x + 2*c) - a + b)) /(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6) - (d*x + c)/(a^3 - 3*a^2*b + 3*a*b^2 - b^3) - 2*((3*a^2*b^2 - 4*a*b^3 + b^4)*e^(2*d*x + 2*c) + 3*(a^3*b^2 - 2*a ^2*b^3 + a*b^4)/(a + b))/((a*e^(2*d*x + 2*c) + b*e^(2*d*x + 2*c) + a - b)^ 2*(a + b)^2*(a - b)^3))/d
Time = 2.99 (sec) , antiderivative size = 304, normalized size of antiderivative = 2.36 \[ \int \frac {1}{(a+b \tanh (c+d x))^3} \, dx=\frac {\mathrm {tanh}\left (c+d\,x\right )\,\left (\frac {1}{a\,d}-\frac {a^4+a^2\,b^2}{a\,d\,\left (a^4-2\,a^2\,b^2+b^4\right )}\right )+\frac {a^2\,x}{\left (a+b\right )\,\left (a^2+2\,a\,b+b^2\right )}+\frac {b^2\,x\,{\mathrm {tanh}\left (c+d\,x\right )}^2}{a^3+3\,a^2\,b+3\,a\,b^2+b^3}+\frac {{\mathrm {tanh}\left (c+d\,x\right )}^2\,\left (\frac {b^5}{2}-\frac {5\,a^2\,b^3}{2}\right )}{a^2\,d\,\left (a^4-2\,a^2\,b^2+b^4\right )}+\frac {2\,a\,b\,x\,\mathrm {tanh}\left (c+d\,x\right )}{a^3+3\,a^2\,b+3\,a\,b^2+b^3}}{a^2+2\,a\,b\,\mathrm {tanh}\left (c+d\,x\right )+b^2\,{\mathrm {tanh}\left (c+d\,x\right )}^2}-\frac {\ln \left (a+b\,\mathrm {tanh}\left (c+d\,x\right )\right )\,\left (3\,a^2\,b+b^3\right )}{d\,\left (a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6\right )}+\frac {\ln \left (\mathrm {tanh}\left (c+d\,x\right )+1\right )\,\left (3\,a^2\,b+b^3\right )}{d\,{\left (a^2-b^2\right )}^3} \] Input:
int(1/(a + b*tanh(c + d*x))^3,x)
Output:
(tanh(c + d*x)*(1/(a*d) - (a^4 + a^2*b^2)/(a*d*(a^4 + b^4 - 2*a^2*b^2))) + (a^2*x)/((a + b)*(2*a*b + a^2 + b^2)) + (b^2*x*tanh(c + d*x)^2)/(3*a*b^2 + 3*a^2*b + a^3 + b^3) + (tanh(c + d*x)^2*(b^5/2 - (5*a^2*b^3)/2))/(a^2*d* (a^4 + b^4 - 2*a^2*b^2)) + (2*a*b*x*tanh(c + d*x))/(3*a*b^2 + 3*a^2*b + a^ 3 + b^3))/(a^2 + b^2*tanh(c + d*x)^2 + 2*a*b*tanh(c + d*x)) - (log(a + b*t anh(c + d*x))*(3*a^2*b + b^3))/(d*(a^6 - b^6 + 3*a^2*b^4 - 3*a^4*b^2)) + ( log(tanh(c + d*x) + 1)*(3*a^2*b + b^3))/(d*(a^2 - b^2)^3)
Time = 0.26 (sec) , antiderivative size = 1186, normalized size of antiderivative = 9.19 \[ \int \frac {1}{(a+b \tanh (c+d x))^3} \, dx =\text {Too large to display} \] Input:
int(1/(a+b*tanh(d*x+c))^3,x)
Output:
( - 3*e**(4*c + 4*d*x)*log(e**(2*c + 2*d*x)*a + e**(2*c + 2*d*x)*b + a - b )*a**4*b - 6*e**(4*c + 4*d*x)*log(e**(2*c + 2*d*x)*a + e**(2*c + 2*d*x)*b + a - b)*a**3*b**2 - 4*e**(4*c + 4*d*x)*log(e**(2*c + 2*d*x)*a + e**(2*c + 2*d*x)*b + a - b)*a**2*b**3 - 2*e**(4*c + 4*d*x)*log(e**(2*c + 2*d*x)*a + e**(2*c + 2*d*x)*b + a - b)*a*b**4 - e**(4*c + 4*d*x)*log(e**(2*c + 2*d*x )*a + e**(2*c + 2*d*x)*b + a - b)*b**5 + e**(4*c + 4*d*x)*a**5*d*x + 5*e** (4*c + 4*d*x)*a**4*b*d*x + 10*e**(4*c + 4*d*x)*a**3*b**2*d*x - 3*e**(4*c + 4*d*x)*a**3*b**2 + 10*e**(4*c + 4*d*x)*a**2*b**3*d*x - 5*e**(4*c + 4*d*x) *a**2*b**3 + 5*e**(4*c + 4*d*x)*a*b**4*d*x - e**(4*c + 4*d*x)*a*b**4 + e** (4*c + 4*d*x)*b**5*d*x + e**(4*c + 4*d*x)*b**5 - 6*e**(2*c + 2*d*x)*log(e* *(2*c + 2*d*x)*a + e**(2*c + 2*d*x)*b + a - b)*a**4*b + 4*e**(2*c + 2*d*x) *log(e**(2*c + 2*d*x)*a + e**(2*c + 2*d*x)*b + a - b)*a**2*b**3 + 2*e**(2* c + 2*d*x)*log(e**(2*c + 2*d*x)*a + e**(2*c + 2*d*x)*b + a - b)*b**5 + 2*e **(2*c + 2*d*x)*a**5*d*x + 6*e**(2*c + 2*d*x)*a**4*b*d*x + 4*e**(2*c + 2*d *x)*a**3*b**2*d*x - 4*e**(2*c + 2*d*x)*a**2*b**3*d*x - 6*e**(2*c + 2*d*x)* a*b**4*d*x - 2*e**(2*c + 2*d*x)*b**5*d*x - 3*log(e**(2*c + 2*d*x)*a + e**( 2*c + 2*d*x)*b + a - b)*a**4*b + 6*log(e**(2*c + 2*d*x)*a + e**(2*c + 2*d* x)*b + a - b)*a**3*b**2 - 4*log(e**(2*c + 2*d*x)*a + e**(2*c + 2*d*x)*b + a - b)*a**2*b**3 + 2*log(e**(2*c + 2*d*x)*a + e**(2*c + 2*d*x)*b + a - b)* a*b**4 - log(e**(2*c + 2*d*x)*a + e**(2*c + 2*d*x)*b + a - b)*b**5 + a*...