Integrand size = 23, antiderivative size = 98 \[ \int \frac {\tanh ^3(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\frac {\log (\cosh (c+d x))}{(a+b)^3 d}+\frac {\log \left (a+b \tanh ^2(c+d x)\right )}{2 (a+b)^3 d}+\frac {a}{4 b (a+b) d \left (a+b \tanh ^2(c+d x)\right )^2}-\frac {1}{2 (a+b)^2 d \left (a+b \tanh ^2(c+d x)\right )} \] Output:
ln(cosh(d*x+c))/(a+b)^3/d+1/2*ln(a+b*tanh(d*x+c)^2)/(a+b)^3/d+1/4*a/b/(a+b )/d/(a+b*tanh(d*x+c)^2)^2-1/2/(a+b)^2/d/(a+b*tanh(d*x+c)^2)
Time = 0.33 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.82 \[ \int \frac {\tanh ^3(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\frac {4 \log (\cosh (c+d x))+2 \log \left (a+b \tanh ^2(c+d x)\right )+\frac {a (a+b)^2}{b \left (a+b \tanh ^2(c+d x)\right )^2}-\frac {2 (a+b)}{a+b \tanh ^2(c+d x)}}{4 (a+b)^3 d} \] Input:
Integrate[Tanh[c + d*x]^3/(a + b*Tanh[c + d*x]^2)^3,x]
Output:
(4*Log[Cosh[c + d*x]] + 2*Log[a + b*Tanh[c + d*x]^2] + (a*(a + b)^2)/(b*(a + b*Tanh[c + d*x]^2)^2) - (2*(a + b))/(a + b*Tanh[c + d*x]^2))/(4*(a + b) ^3*d)
Time = 0.34 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.97, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {3042, 26, 4153, 26, 354, 86, 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.
| \(\displaystyle \int \frac {\tanh ^3(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {i \tan (i c+i d x)^3}{\left (a-b \tan (i c+i d x)^2\right )^3}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \int \frac {\tan (i c+i d x)^3}{\left (a-b \tan (i c+i d x)^2\right )^3}dx\) |
| \(\Big \downarrow \) 4153 |
| \(\displaystyle \frac {i \int -\frac {i \tanh ^3(c+d x)}{\left (1-\tanh ^2(c+d x)\right ) \left (b \tanh ^2(c+d x)+a\right )^3}d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {\int \frac {\tanh ^3(c+d x)}{\left (1-\tanh ^2(c+d x)\right ) \left (b \tanh ^2(c+d x)+a\right )^3}d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 354 |
| \(\displaystyle \frac {\int \frac {\tanh ^2(c+d x)}{\left (1-\tanh ^2(c+d x)\right ) \left (b \tanh ^2(c+d x)+a\right )^3}d\tanh ^2(c+d x)}{2 d}\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle \frac {\int \left (-\frac {a}{(a+b) \left (b \tanh ^2(c+d x)+a\right )^3}-\frac {1}{(a+b)^3 \left (\tanh ^2(c+d x)-1\right )}+\frac {b}{(a+b)^3 \left (b \tanh ^2(c+d x)+a\right )}+\frac {b}{(a+b)^2 \left (b \tanh ^2(c+d x)+a\right )^2}\right )d\tanh ^2(c+d x)}{2 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {a}{2 b (a+b) \left (a+b \tanh ^2(c+d x)\right )^2}-\frac {1}{(a+b)^2 \left (a+b \tanh ^2(c+d x)\right )}-\frac {\log \left (1-\tanh ^2(c+d x)\right )}{(a+b)^3}+\frac {\log \left (a+b \tanh ^2(c+d x)\right )}{(a+b)^3}}{2 d}\) |
Input:
Int[Tanh[c + d*x]^3/(a + b*Tanh[c + d*x]^2)^3,x]
Output:
(-(Log[1 - Tanh[c + d*x]^2]/(a + b)^3) + Log[a + b*Tanh[c + d*x]^2]/(a + b )^3 + a/(2*b*(a + b)*(a + b*Tanh[c + d*x]^2)^2) - 1/((a + b)^2*(a + b*Tanh [c + d*x]^2)))/(2*d)
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_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ .), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 ] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S ymbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x , x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ [(m - 1)/2]
Int[((d_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_.), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[c*(ff/f) Subst[Int[(d*ff*(x/c))^m*((a + b*(ff*x)^n)^p/(c^2 + f f^2*x^2)), x], x, c*(Tan[e + f*x]/ff)], x]] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && (IGtQ[p, 0] || EqQ[n, 2] || EqQ[n, 4] || (IntegerQ[p] && Ratio nalQ[n]))
Time = 0.12 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.17
| method | result | size |
| derivativedivides | \(\frac {-\frac {-\frac {-a -b}{a +b \tanh \left (d x +c \right )^{2}}-\ln \left (a +b \tanh \left (d x +c \right )^{2}\right )-\frac {a \left (a^{2}+2 a b +b^{2}\right )}{2 b \left (a +b \tanh \left (d x +c \right )^{2}\right )^{2}}}{2 \left (a +b \right )^{3}}-\frac {\ln \left (1+\tanh \left (d x +c \right )\right )}{2 \left (a +b \right )^{3}}-\frac {\ln \left (-1+\tanh \left (d x +c \right )\right )}{2 \left (a +b \right )^{3}}}{d}\) | \(115\) |
| default | \(\frac {-\frac {-\frac {-a -b}{a +b \tanh \left (d x +c \right )^{2}}-\ln \left (a +b \tanh \left (d x +c \right )^{2}\right )-\frac {a \left (a^{2}+2 a b +b^{2}\right )}{2 b \left (a +b \tanh \left (d x +c \right )^{2}\right )^{2}}}{2 \left (a +b \right )^{3}}-\frac {\ln \left (1+\tanh \left (d x +c \right )\right )}{2 \left (a +b \right )^{3}}-\frac {\ln \left (-1+\tanh \left (d x +c \right )\right )}{2 \left (a +b \right )^{3}}}{d}\) | \(115\) |
| parallelrisch | \(\frac {a^{3} b -a \,b^{3}-8 \ln \left (1-\tanh \left (d x +c \right )\right ) \tanh \left (d x +c \right )^{2} a \,b^{3}-4 b^{4} \tanh \left (d x +c \right )^{4} x d -8 x \tanh \left (d x +c \right )^{2} a \,b^{3} d +2 \ln \left (a +b \tanh \left (d x +c \right )^{2}\right ) \tanh \left (d x +c \right )^{4} b^{4}-4 a^{2} b^{2} d x -2 \tanh \left (d x +c \right )^{2} a \,b^{3}-4 \ln \left (1-\tanh \left (d x +c \right )\right ) \tanh \left (d x +c \right )^{4} b^{4}+4 \ln \left (a +b \tanh \left (d x +c \right )^{2}\right ) \tanh \left (d x +c \right )^{2} a \,b^{3}+2 \ln \left (a +b \tanh \left (d x +c \right )^{2}\right ) a^{2} b^{2}-2 \tanh \left (d x +c \right )^{2} b^{4}-4 \ln \left (1-\tanh \left (d x +c \right )\right ) a^{2} b^{2}}{4 \left (a +b \tanh \left (d x +c \right )^{2}\right )^{2} \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right ) b^{2} d}\) | \(266\) |
| risch | \(-\frac {x}{a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}}-\frac {2 c}{d \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {2 \,{\mathrm e}^{2 d x +2 c} \left (a^{2} {\mathrm e}^{4 d x +4 c}-b^{2} {\mathrm e}^{4 d x +4 c}+2 a^{2} {\mathrm e}^{2 d x +2 c}-2 a b \,{\mathrm e}^{2 d x +2 c}+2 b^{2} {\mathrm e}^{2 d x +2 c}+a^{2}-b^{2}\right )}{\left (a \,{\mathrm e}^{4 d x +4 c}+b \,{\mathrm e}^{4 d x +4 c}+2 a \,{\mathrm e}^{2 d x +2 c}-2 b \,{\mathrm e}^{2 d x +2 c}+a +b \right )^{2} d \left (a^{2}+2 a b +b^{2}\right ) \left (a +b \right )}+\frac {\ln \left ({\mathrm e}^{4 d x +4 c}+\frac {2 \left (a -b \right ) {\mathrm e}^{2 d x +2 c}}{a +b}+1\right )}{2 d \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}\) | \(272\) |
Input:
int(tanh(d*x+c)^3/(a+b*tanh(d*x+c)^2)^3,x,method=_RETURNVERBOSE)
Output:
1/d*(-1/2/(a+b)^3*(-(-a-b)/(a+b*tanh(d*x+c)^2)-ln(a+b*tanh(d*x+c)^2)-1/2*a *(a^2+2*a*b+b^2)/b/(a+b*tanh(d*x+c)^2)^2)-1/2/(a+b)^3*ln(1+tanh(d*x+c))-1/ 2/(a+b)^3*ln(-1+tanh(d*x+c)))
Leaf count of result is larger than twice the leaf count of optimal. 2611 vs. \(2 (92) = 184\).
Time = 0.14 (sec) , antiderivative size = 2611, normalized size of antiderivative = 26.64 \[ \int \frac {\tanh ^3(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\text {Too large to display} \] Input:
integrate(tanh(d*x+c)^3/(a+b*tanh(d*x+c)^2)^3,x, algorithm="fricas")
Output:
-1/2*(2*(a^2 + 2*a*b + b^2)*d*x*cosh(d*x + c)^8 + 16*(a^2 + 2*a*b + b^2)*d *x*cosh(d*x + c)*sinh(d*x + c)^7 + 2*(a^2 + 2*a*b + b^2)*d*x*sinh(d*x + c) ^8 + 4*(2*(a^2 - b^2)*d*x - a^2 + b^2)*cosh(d*x + c)^6 + 4*(14*(a^2 + 2*a* b + b^2)*d*x*cosh(d*x + c)^2 + 2*(a^2 - b^2)*d*x - a^2 + b^2)*sinh(d*x + c )^6 + 8*(14*(a^2 + 2*a*b + b^2)*d*x*cosh(d*x + c)^3 + 3*(2*(a^2 - b^2)*d*x - a^2 + b^2)*cosh(d*x + c))*sinh(d*x + c)^5 + 4*((3*a^2 - 2*a*b + 3*b^2)* d*x - 2*a^2 + 2*a*b - 2*b^2)*cosh(d*x + c)^4 + 4*(35*(a^2 + 2*a*b + b^2)*d *x*cosh(d*x + c)^4 + (3*a^2 - 2*a*b + 3*b^2)*d*x + 15*(2*(a^2 - b^2)*d*x - a^2 + b^2)*cosh(d*x + c)^2 - 2*a^2 + 2*a*b - 2*b^2)*sinh(d*x + c)^4 + 16* (7*(a^2 + 2*a*b + b^2)*d*x*cosh(d*x + c)^5 + 5*(2*(a^2 - b^2)*d*x - a^2 + b^2)*cosh(d*x + c)^3 + ((3*a^2 - 2*a*b + 3*b^2)*d*x - 2*a^2 + 2*a*b - 2*b^ 2)*cosh(d*x + c))*sinh(d*x + c)^3 + 2*(a^2 + 2*a*b + b^2)*d*x + 4*(2*(a^2 - b^2)*d*x - a^2 + b^2)*cosh(d*x + c)^2 + 4*(14*(a^2 + 2*a*b + b^2)*d*x*co sh(d*x + c)^6 + 15*(2*(a^2 - b^2)*d*x - a^2 + b^2)*cosh(d*x + c)^4 + 2*(a^ 2 - b^2)*d*x + 6*((3*a^2 - 2*a*b + 3*b^2)*d*x - 2*a^2 + 2*a*b - 2*b^2)*cos h(d*x + c)^2 - a^2 + b^2)*sinh(d*x + c)^2 - ((a^2 + 2*a*b + b^2)*cosh(d*x + c)^8 + 8*(a^2 + 2*a*b + b^2)*cosh(d*x + c)*sinh(d*x + c)^7 + (a^2 + 2*a* b + b^2)*sinh(d*x + c)^8 + 4*(a^2 - b^2)*cosh(d*x + c)^6 + 4*(7*(a^2 + 2*a *b + b^2)*cosh(d*x + c)^2 + a^2 - b^2)*sinh(d*x + c)^6 + 8*(7*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^3 + 3*(a^2 - b^2)*cosh(d*x + c))*sinh(d*x + c)^5 ...
Timed out. \[ \int \frac {\tanh ^3(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\text {Timed out} \] Input:
integrate(tanh(d*x+c)**3/(a+b*tanh(d*x+c)**2)**3,x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 384 vs. \(2 (92) = 184\).
Time = 0.08 (sec) , antiderivative size = 384, normalized size of antiderivative = 3.92 \[ \int \frac {\tanh ^3(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\frac {d x + c}{{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} d} + \frac {2 \, {\left ({\left (a^{2} - b^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + 2 \, {\left (a^{2} - a b + b^{2}\right )} e^{\left (-4 \, d x - 4 \, c\right )} + {\left (a^{2} - b^{2}\right )} e^{\left (-6 \, d x - 6 \, c\right )}\right )}}{{\left (a^{5} + 5 \, a^{4} b + 10 \, a^{3} b^{2} + 10 \, a^{2} b^{3} + 5 \, a b^{4} + b^{5} + 4 \, {\left (a^{5} + 3 \, a^{4} b + 2 \, a^{3} b^{2} - 2 \, a^{2} b^{3} - 3 \, a b^{4} - b^{5}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + 2 \, {\left (3 \, a^{5} + 7 \, a^{4} b + 6 \, a^{3} b^{2} + 6 \, a^{2} b^{3} + 7 \, a b^{4} + 3 \, b^{5}\right )} e^{\left (-4 \, d x - 4 \, c\right )} + 4 \, {\left (a^{5} + 3 \, a^{4} b + 2 \, a^{3} b^{2} - 2 \, a^{2} b^{3} - 3 \, a b^{4} - b^{5}\right )} e^{\left (-6 \, d x - 6 \, c\right )} + {\left (a^{5} + 5 \, a^{4} b + 10 \, a^{3} b^{2} + 10 \, a^{2} b^{3} + 5 \, a b^{4} + b^{5}\right )} e^{\left (-8 \, d x - 8 \, c\right )}\right )} d} + \frac {\log \left (2 \, {\left (a - b\right )} e^{\left (-2 \, d x - 2 \, c\right )} + {\left (a + b\right )} e^{\left (-4 \, d x - 4 \, c\right )} + a + b\right )}{2 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} d} \] Input:
integrate(tanh(d*x+c)^3/(a+b*tanh(d*x+c)^2)^3,x, algorithm="maxima")
Output:
(d*x + c)/((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d) + 2*((a^2 - b^2)*e^(-2*d*x - 2*c) + 2*(a^2 - a*b + b^2)*e^(-4*d*x - 4*c) + (a^2 - b^2)*e^(-6*d*x - 6*c ))/((a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5 + 4*(a^5 + 3* a^4*b + 2*a^3*b^2 - 2*a^2*b^3 - 3*a*b^4 - b^5)*e^(-2*d*x - 2*c) + 2*(3*a^5 + 7*a^4*b + 6*a^3*b^2 + 6*a^2*b^3 + 7*a*b^4 + 3*b^5)*e^(-4*d*x - 4*c) + 4 *(a^5 + 3*a^4*b + 2*a^3*b^2 - 2*a^2*b^3 - 3*a*b^4 - b^5)*e^(-6*d*x - 6*c) + (a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*e^(-8*d*x - 8* c))*d) + 1/2*log(2*(a - b)*e^(-2*d*x - 2*c) + (a + b)*e^(-4*d*x - 4*c) + a + b)/((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d)
Leaf count of result is larger than twice the leaf count of optimal. 245 vs. \(2 (92) = 184\).
Time = 0.28 (sec) , antiderivative size = 245, normalized size of antiderivative = 2.50 \[ \int \frac {\tanh ^3(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\frac {\frac {2 \, \log \left ({\left | a {\left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )}\right )} + b {\left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )}\right )} + 2 \, a - 2 \, b \right |}\right )}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}} - \frac {3 \, a {\left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )}\right )}^{2} + 3 \, b {\left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )}\right )}^{2} + 4 \, a {\left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )}\right )} - 4 \, b {\left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )}\right )} - 4 \, a - 4 \, b}{{\left (a^{2} + 2 \, a b + b^{2}\right )} {\left (a {\left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )}\right )} + b {\left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )}\right )} + 2 \, a - 2 \, b\right )}^{2}}}{4 \, d} \] Input:
integrate(tanh(d*x+c)^3/(a+b*tanh(d*x+c)^2)^3,x, algorithm="giac")
Output:
1/4*(2*log(abs(a*(e^(2*d*x + 2*c) + e^(-2*d*x - 2*c)) + b*(e^(2*d*x + 2*c) + e^(-2*d*x - 2*c)) + 2*a - 2*b))/(a^3 + 3*a^2*b + 3*a*b^2 + b^3) - (3*a* (e^(2*d*x + 2*c) + e^(-2*d*x - 2*c))^2 + 3*b*(e^(2*d*x + 2*c) + e^(-2*d*x - 2*c))^2 + 4*a*(e^(2*d*x + 2*c) + e^(-2*d*x - 2*c)) - 4*b*(e^(2*d*x + 2*c ) + e^(-2*d*x - 2*c)) - 4*a - 4*b)/((a^2 + 2*a*b + b^2)*(a*(e^(2*d*x + 2*c ) + e^(-2*d*x - 2*c)) + b*(e^(2*d*x + 2*c) + e^(-2*d*x - 2*c)) + 2*a - 2*b )^2))/d
Time = 2.84 (sec) , antiderivative size = 397, normalized size of antiderivative = 4.05 \[ \int \frac {\tanh ^3(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=-\frac {-a^3+b^3\,\left (2\,{\mathrm {tanh}\left (c+d\,x\right )}^2+{\mathrm {tanh}\left (c+d\,x\right )}^4\,\mathrm {atan}\left (\frac {a\,{\mathrm {tanh}\left (c+d\,x\right )}^2\,1{}\mathrm {i}+b\,{\mathrm {tanh}\left (c+d\,x\right )}^2\,1{}\mathrm {i}}{2\,a-a\,{\mathrm {tanh}\left (c+d\,x\right )}^2+b\,{\mathrm {tanh}\left (c+d\,x\right )}^2}\right )\,4{}\mathrm {i}\right )+a\,b^2\,\left (2\,{\mathrm {tanh}\left (c+d\,x\right )}^2+1+{\mathrm {tanh}\left (c+d\,x\right )}^2\,\mathrm {atan}\left (\frac {a\,{\mathrm {tanh}\left (c+d\,x\right )}^2\,1{}\mathrm {i}+b\,{\mathrm {tanh}\left (c+d\,x\right )}^2\,1{}\mathrm {i}}{2\,a-a\,{\mathrm {tanh}\left (c+d\,x\right )}^2+b\,{\mathrm {tanh}\left (c+d\,x\right )}^2}\right )\,8{}\mathrm {i}\right )+a^2\,b\,\mathrm {atan}\left (\frac {a\,{\mathrm {tanh}\left (c+d\,x\right )}^2\,1{}\mathrm {i}+b\,{\mathrm {tanh}\left (c+d\,x\right )}^2\,1{}\mathrm {i}}{2\,a-a\,{\mathrm {tanh}\left (c+d\,x\right )}^2+b\,{\mathrm {tanh}\left (c+d\,x\right )}^2}\right )\,4{}\mathrm {i}}{4\,d\,a^5\,b+8\,d\,a^4\,b^2\,{\mathrm {tanh}\left (c+d\,x\right )}^2+12\,d\,a^4\,b^2+4\,d\,a^3\,b^3\,{\mathrm {tanh}\left (c+d\,x\right )}^4+24\,d\,a^3\,b^3\,{\mathrm {tanh}\left (c+d\,x\right )}^2+12\,d\,a^3\,b^3+12\,d\,a^2\,b^4\,{\mathrm {tanh}\left (c+d\,x\right )}^4+24\,d\,a^2\,b^4\,{\mathrm {tanh}\left (c+d\,x\right )}^2+4\,d\,a^2\,b^4+12\,d\,a\,b^5\,{\mathrm {tanh}\left (c+d\,x\right )}^4+8\,d\,a\,b^5\,{\mathrm {tanh}\left (c+d\,x\right )}^2+4\,d\,b^6\,{\mathrm {tanh}\left (c+d\,x\right )}^4} \] Input:
int(tanh(c + d*x)^3/(a + b*tanh(c + d*x)^2)^3,x)
Output:
-(b^3*(tanh(c + d*x)^4*atan((a*tanh(c + d*x)^2*1i + b*tanh(c + d*x)^2*1i)/ (2*a - a*tanh(c + d*x)^2 + b*tanh(c + d*x)^2))*4i + 2*tanh(c + d*x)^2) - a ^3 + a*b^2*(tanh(c + d*x)^2*atan((a*tanh(c + d*x)^2*1i + b*tanh(c + d*x)^2 *1i)/(2*a - a*tanh(c + d*x)^2 + b*tanh(c + d*x)^2))*8i + 2*tanh(c + d*x)^2 + 1) + a^2*b*atan((a*tanh(c + d*x)^2*1i + b*tanh(c + d*x)^2*1i)/(2*a - a* tanh(c + d*x)^2 + b*tanh(c + d*x)^2))*4i)/(4*a^2*b^4*d + 12*a^3*b^3*d + 12 *a^4*b^2*d + 4*b^6*d*tanh(c + d*x)^4 + 4*a^5*b*d + 24*a^2*b^4*d*tanh(c + d *x)^2 + 24*a^3*b^3*d*tanh(c + d*x)^2 + 8*a^4*b^2*d*tanh(c + d*x)^2 + 12*a^ 2*b^4*d*tanh(c + d*x)^4 + 4*a^3*b^3*d*tanh(c + d*x)^4 + 8*a*b^5*d*tanh(c + d*x)^2 + 12*a*b^5*d*tanh(c + d*x)^4)
Time = 0.27 (sec) , antiderivative size = 10556, normalized size of antiderivative = 107.71 \[ \int \frac {\tanh ^3(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx =\text {Too large to display} \] Input:
int(tanh(d*x+c)^3/(a+b*tanh(d*x+c)^2)^3,x)
Output:
(2*e**(8*c + 8*d*x)*log(e**(2*c + 2*d*x)*sqrt(a + b) + sqrt(a + b) - 2*e** (c + d*x)*sqrt(b))*tanh(c + d*x)**4*a**3*b**3 + 2*e**(8*c + 8*d*x)*log(e** (2*c + 2*d*x)*sqrt(a + b) + sqrt(a + b) - 2*e**(c + d*x)*sqrt(b))*tanh(c + d*x)**4*a**2*b**4 - 2*e**(8*c + 8*d*x)*log(e**(2*c + 2*d*x)*sqrt(a + b) + sqrt(a + b) - 2*e**(c + d*x)*sqrt(b))*tanh(c + d*x)**4*a*b**5 - 2*e**(8*c + 8*d*x)*log(e**(2*c + 2*d*x)*sqrt(a + b) + sqrt(a + b) - 2*e**(c + d*x)* sqrt(b))*tanh(c + d*x)**4*b**6 + 4*e**(8*c + 8*d*x)*log(e**(2*c + 2*d*x)*s qrt(a + b) + sqrt(a + b) - 2*e**(c + d*x)*sqrt(b))*tanh(c + d*x)**2*a**4*b **2 + 4*e**(8*c + 8*d*x)*log(e**(2*c + 2*d*x)*sqrt(a + b) + sqrt(a + b) - 2*e**(c + d*x)*sqrt(b))*tanh(c + d*x)**2*a**3*b**3 - 4*e**(8*c + 8*d*x)*lo g(e**(2*c + 2*d*x)*sqrt(a + b) + sqrt(a + b) - 2*e**(c + d*x)*sqrt(b))*tan h(c + d*x)**2*a**2*b**4 - 4*e**(8*c + 8*d*x)*log(e**(2*c + 2*d*x)*sqrt(a + b) + sqrt(a + b) - 2*e**(c + d*x)*sqrt(b))*tanh(c + d*x)**2*a*b**5 + 2*e* *(8*c + 8*d*x)*log(e**(2*c + 2*d*x)*sqrt(a + b) + sqrt(a + b) - 2*e**(c + d*x)*sqrt(b))*a**5*b + 2*e**(8*c + 8*d*x)*log(e**(2*c + 2*d*x)*sqrt(a + b) + sqrt(a + b) - 2*e**(c + d*x)*sqrt(b))*a**4*b**2 - 2*e**(8*c + 8*d*x)*lo g(e**(2*c + 2*d*x)*sqrt(a + b) + sqrt(a + b) - 2*e**(c + d*x)*sqrt(b))*a** 3*b**3 - 2*e**(8*c + 8*d*x)*log(e**(2*c + 2*d*x)*sqrt(a + b) + sqrt(a + b) - 2*e**(c + d*x)*sqrt(b))*a**2*b**4 + 2*e**(8*c + 8*d*x)*log(e**(2*c + 2* d*x)*sqrt(a + b) + sqrt(a + b) + 2*e**(c + d*x)*sqrt(b))*tanh(c + d*x)*...