Integrand size = 23, antiderivative size = 110 \[ \int \frac {\coth ^3(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\frac {b^3}{2 a^2 (a+b)^2 d \left (b+a \cosh ^2(c+d x)\right )}-\frac {\text {csch}^2(c+d x)}{2 (a+b)^2 d}+\frac {b^2 (3 a+b) \log \left (b+a \cosh ^2(c+d x)\right )}{2 a^2 (a+b)^3 d}+\frac {(a+3 b) \log (\sinh (c+d x))}{(a+b)^3 d} \] Output:
1/2*b^3/a^2/(a+b)^2/d/(b+a*cosh(d*x+c)^2)-1/2*csch(d*x+c)^2/(a+b)^2/d+1/2* b^2*(3*a+b)*ln(b+a*cosh(d*x+c)^2)/a^2/(a+b)^3/d+(a+3*b)*ln(sinh(d*x+c))/(a +b)^3/d
Time = 0.86 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.18 \[ \int \frac {\coth ^3(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\frac {(a+2 b+a \cosh (2 (c+d x)))^2 \text {sech}^4(c+d x) \left (-\left ((a+b) \text {csch}^2(c+d x)\right )+2 (a+3 b) \log (\sinh (c+d x))+\frac {b^2 (3 a+b) \log \left (a+b+a \sinh ^2(c+d x)\right )}{a^2}+\frac {b^3 (a+b)}{a^2 \left (a+b+a \sinh ^2(c+d x)\right )}\right )}{8 (a+b)^3 d \left (a+b \text {sech}^2(c+d x)\right )^2} \] Input:
Integrate[Coth[c + d*x]^3/(a + b*Sech[c + d*x]^2)^2,x]
Output:
((a + 2*b + a*Cosh[2*(c + d*x)])^2*Sech[c + d*x]^4*(-((a + b)*Csch[c + d*x ]^2) + 2*(a + 3*b)*Log[Sinh[c + d*x]] + (b^2*(3*a + b)*Log[a + b + a*Sinh[ c + d*x]^2])/a^2 + (b^3*(a + b))/(a^2*(a + b + a*Sinh[c + d*x]^2))))/(8*(a + b)^3*d*(a + b*Sech[c + d*x]^2)^2)
Time = 0.40 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.98, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3042, 26, 4626, 354, 99, 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.
| \(\displaystyle \int \frac {\coth ^3(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {i}{\tan (i c+i d x)^3 \left (a+b \sec (i c+i d x)^2\right )^2}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \int \frac {1}{\left (b \sec (i c+i d x)^2+a\right )^2 \tan (i c+i d x)^3}dx\) |
| \(\Big \downarrow \) 4626 |
| \(\displaystyle \frac {\int \frac {\cosh ^7(c+d x)}{\left (1-\cosh ^2(c+d x)\right )^2 \left (a \cosh ^2(c+d x)+b\right )^2}d\cosh (c+d x)}{d}\) |
| \(\Big \downarrow \) 354 |
| \(\displaystyle \frac {\int \frac {\cosh ^6(c+d x)}{\left (1-\cosh ^2(c+d x)\right )^2 \left (a \cosh ^2(c+d x)+b\right )^2}d\cosh ^2(c+d x)}{2 d}\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \frac {\int \left (-\frac {b^3}{a (a+b)^2 \left (a \cosh ^2(c+d x)+b\right )^2}+\frac {(3 a+b) b^2}{a (a+b)^3 \left (a \cosh ^2(c+d x)+b\right )}+\frac {a+3 b}{(a+b)^3 \left (\cosh ^2(c+d x)-1\right )}+\frac {1}{(a+b)^2 \left (\cosh ^2(c+d x)-1\right )^2}\right )d\cosh ^2(c+d x)}{2 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {b^3}{a^2 (a+b)^2 \left (a \cosh ^2(c+d x)+b\right )}+\frac {b^2 (3 a+b) \log \left (a \cosh ^2(c+d x)+b\right )}{a^2 (a+b)^3}+\frac {1}{(a+b)^2 \left (1-\cosh ^2(c+d x)\right )}+\frac {(a+3 b) \log \left (1-\cosh ^2(c+d x)\right )}{(a+b)^3}}{2 d}\) |
Input:
Int[Coth[c + d*x]^3/(a + b*Sech[c + d*x]^2)^2,x]
Output:
(1/((a + b)^2*(1 - Cosh[c + d*x]^2)) + b^3/(a^2*(a + b)^2*(b + a*Cosh[c + d*x]^2)) + ((a + 3*b)*Log[1 - Cosh[c + d*x]^2])/(a + b)^3 + (b^2*(3*a + b) *Log[b + a*Cosh[c + d*x]^2])/(a^2*(a + b)^3))/(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_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | | (GtQ[m, 0] && GeQ[n, -1]))
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[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_.)*tan[(e_.) + (f_.)*(x_ )]^(m_.), x_Symbol] :> Module[{ff = FreeFactors[Cos[e + f*x], x]}, Simp[-(f *ff^(m + n*p - 1))^(-1) Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*((b + a*(ff* x)^n)^p/x^(m + n*p)), x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, n} , x] && IntegerQ[(m - 1)/2] && IntegerQ[n] && IntegerQ[p]
Leaf count of result is larger than twice the leaf count of optimal. \(259\) vs. \(2(104)=208\).
Time = 34.79 (sec) , antiderivative size = 260, normalized size of antiderivative = 2.36
| method | result | size |
| derivativedivides | \(\frac {-\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{8 \left (a^{2}+2 a b +b^{2}\right )}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a^{2}}-\frac {1}{8 \left (a +b \right )^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (4 a +12 b \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 \left (a +b \right )^{3}}+\frac {b^{2} \left (-\frac {2 a b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} b +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b}+\frac {\left (3 a +b \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} b +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b \right )}{2}\right )}{a^{2} \left (a +b \right )^{3}}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a^{2}}}{d}\) | \(260\) |
| default | \(\frac {-\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{8 \left (a^{2}+2 a b +b^{2}\right )}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a^{2}}-\frac {1}{8 \left (a +b \right )^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (4 a +12 b \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 \left (a +b \right )^{3}}+\frac {b^{2} \left (-\frac {2 a b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} b +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b}+\frac {\left (3 a +b \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} b +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b \right )}{2}\right )}{a^{2} \left (a +b \right )^{3}}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a^{2}}}{d}\) | \(260\) |
| risch | \(\frac {x}{a^{2}}-\frac {2 a x}{a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}}-\frac {2 a c}{d \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}-\frac {6 b x}{a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}}-\frac {6 b c}{d \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}-\frac {6 b^{2} x}{a \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}-\frac {6 b^{2} c}{a d \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}-\frac {2 b^{3} x}{a^{2} \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}-\frac {2 b^{3} c}{a^{2} d \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}-\frac {2 \,{\mathrm e}^{2 d x +2 c} \left (a^{3} {\mathrm e}^{4 d x +4 c}-b^{3} {\mathrm e}^{4 d x +4 c}+2 a^{3} {\mathrm e}^{2 d x +2 c}+4 a^{2} b \,{\mathrm e}^{2 d x +2 c}+2 b^{3} {\mathrm e}^{2 d x +2 c}+a^{3}-b^{3}\right )}{d \left (a +b \right )^{2} \left ({\mathrm e}^{2 d x +2 c}-1\right )^{2} a^{2} \left ({\mathrm e}^{4 d x +4 c} a +2 a \,{\mathrm e}^{2 d x +2 c}+4 b \,{\mathrm e}^{2 d x +2 c}+a \right )}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}-1\right ) a}{d \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {3 \ln \left ({\mathrm e}^{2 d x +2 c}-1\right ) b}{d \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {3 b^{2} \ln \left ({\mathrm e}^{4 d x +4 c}+\frac {2 \left (a +2 b \right ) {\mathrm e}^{2 d x +2 c}}{a}+1\right )}{2 a d \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {b^{3} \ln \left ({\mathrm e}^{4 d x +4 c}+\frac {2 \left (a +2 b \right ) {\mathrm e}^{2 d x +2 c}}{a}+1\right )}{2 a^{2} d \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}\) | \(595\) |
Input:
int(coth(d*x+c)^3/(a+b*sech(d*x+c)^2)^2,x,method=_RETURNVERBOSE)
Output:
1/d*(-1/8*tanh(1/2*d*x+1/2*c)^2/(a^2+2*a*b+b^2)-1/a^2*ln(tanh(1/2*d*x+1/2* c)-1)-1/8/(a+b)^2/tanh(1/2*d*x+1/2*c)^2+1/4/(a+b)^3*(4*a+12*b)*ln(tanh(1/2 *d*x+1/2*c))+b^2/a^2/(a+b)^3*(-2*a*b*tanh(1/2*d*x+1/2*c)^2/(tanh(1/2*d*x+1 /2*c)^4*a+tanh(1/2*d*x+1/2*c)^4*b+2*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x +1/2*c)^2*b+a+b)+1/2*(3*a+b)*ln(tanh(1/2*d*x+1/2*c)^4*a+tanh(1/2*d*x+1/2*c )^4*b+2*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b))-1/a^2*ln(t anh(1/2*d*x+1/2*c)+1))
Leaf count of result is larger than twice the leaf count of optimal. 3624 vs. \(2 (104) = 208\).
Time = 0.68 (sec) , antiderivative size = 3624, normalized size of antiderivative = 32.95 \[ \int \frac {\coth ^3(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\text {Too large to display} \] Input:
integrate(coth(d*x+c)^3/(a+b*sech(d*x+c)^2)^2,x, algorithm="fricas")
Output:
Too large to include
\[ \int \frac {\coth ^3(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\int \frac {\coth ^{3}{\left (c + d x \right )}}{\left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right )^{2}}\, dx \] Input:
integrate(coth(d*x+c)**3/(a+b*sech(d*x+c)**2)**2,x)
Output:
Integral(coth(c + d*x)**3/(a + b*sech(c + d*x)**2)**2, x)
Leaf count of result is larger than twice the leaf count of optimal. 384 vs. \(2 (104) = 208\).
Time = 0.09 (sec) , antiderivative size = 384, normalized size of antiderivative = 3.49 \[ \int \frac {\coth ^3(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\frac {{\left (3 \, a b^{2} + b^{3}\right )} \log \left (2 \, {\left (a + 2 \, b\right )} e^{\left (-2 \, d x - 2 \, c\right )} + a e^{\left (-4 \, d x - 4 \, c\right )} + a\right )}{2 \, {\left (a^{5} + 3 \, a^{4} b + 3 \, a^{3} b^{2} + a^{2} b^{3}\right )} d} + \frac {{\left (a + 3 \, b\right )} \log \left (e^{\left (-d x - c\right )} + 1\right )}{{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} d} + \frac {{\left (a + 3 \, b\right )} \log \left (e^{\left (-d x - c\right )} - 1\right )}{{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} d} - \frac {2 \, {\left ({\left (a^{3} - b^{3}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + 2 \, {\left (a^{3} + 2 \, a^{2} b + b^{3}\right )} e^{\left (-4 \, d x - 4 \, c\right )} + {\left (a^{3} - b^{3}\right )} e^{\left (-6 \, d x - 6 \, c\right )}\right )}}{{\left (a^{5} + 2 \, a^{4} b + a^{3} b^{2} + 4 \, {\left (a^{4} b + 2 \, a^{3} b^{2} + a^{2} b^{3}\right )} e^{\left (-2 \, d x - 2 \, c\right )} - 2 \, {\left (a^{5} + 6 \, a^{4} b + 9 \, a^{3} b^{2} + 4 \, a^{2} b^{3}\right )} e^{\left (-4 \, d x - 4 \, c\right )} + 4 \, {\left (a^{4} b + 2 \, a^{3} b^{2} + a^{2} b^{3}\right )} e^{\left (-6 \, d x - 6 \, c\right )} + {\left (a^{5} + 2 \, a^{4} b + a^{3} b^{2}\right )} e^{\left (-8 \, d x - 8 \, c\right )}\right )} d} + \frac {d x + c}{a^{2} d} \] Input:
integrate(coth(d*x+c)^3/(a+b*sech(d*x+c)^2)^2,x, algorithm="maxima")
Output:
1/2*(3*a*b^2 + b^3)*log(2*(a + 2*b)*e^(-2*d*x - 2*c) + a*e^(-4*d*x - 4*c) + a)/((a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*d) + (a + 3*b)*log(e^(-d*x - c ) + 1)/((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d) + (a + 3*b)*log(e^(-d*x - c) - 1)/((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d) - 2*((a^3 - b^3)*e^(-2*d*x - 2*c) + 2*(a^3 + 2*a^2*b + b^3)*e^(-4*d*x - 4*c) + (a^3 - b^3)*e^(-6*d*x - 6*c))/ ((a^5 + 2*a^4*b + a^3*b^2 + 4*(a^4*b + 2*a^3*b^2 + a^2*b^3)*e^(-2*d*x - 2* c) - 2*(a^5 + 6*a^4*b + 9*a^3*b^2 + 4*a^2*b^3)*e^(-4*d*x - 4*c) + 4*(a^4*b + 2*a^3*b^2 + a^2*b^3)*e^(-6*d*x - 6*c) + (a^5 + 2*a^4*b + a^3*b^2)*e^(-8 *d*x - 8*c))*d) + (d*x + c)/(a^2*d)
Exception generated. \[ \int \frac {\coth ^3(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(coth(d*x+c)^3/(a+b*sech(d*x+c)^2)^2,x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Limit: Max order reached or unable to make series expansion Error: Bad Argument Value
Timed out. \[ \int \frac {\coth ^3(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^4\,{\mathrm {coth}\left (c+d\,x\right )}^3}{{\left (a\,{\mathrm {cosh}\left (c+d\,x\right )}^2+b\right )}^2} \,d x \] Input:
int(coth(c + d*x)^3/(a + b/cosh(c + d*x)^2)^2,x)
Output:
int((cosh(c + d*x)^4*coth(c + d*x)^3)/(b + a*cosh(c + d*x)^2)^2, x)
Time = 0.39 (sec) , antiderivative size = 2792, normalized size of antiderivative = 25.38 \[ \int \frac {\coth ^3(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx =\text {Too large to display} \] Input:
int(coth(d*x+c)^3/(a+b*sech(d*x+c)^2)^2,x)
Output:
(2*e**(8*c + 8*d*x)*log(e**(c + d*x) - 1)*a**4*b + 6*e**(8*c + 8*d*x)*log( e**(c + d*x) - 1)*a**3*b**2 + 2*e**(8*c + 8*d*x)*log(e**(c + d*x) + 1)*a** 4*b + 6*e**(8*c + 8*d*x)*log(e**(c + d*x) + 1)*a**3*b**2 + 3*e**(8*c + 8*d *x)*log( - sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b) + e**(c + d*x)*sqrt(a))*a **2*b**3 + e**(8*c + 8*d*x)*log( - sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b) + e**(c + d*x)*sqrt(a))*a*b**4 + 3*e**(8*c + 8*d*x)*log(sqrt(2*sqrt(b)*sqrt (a + b) - a - 2*b) + e**(c + d*x)*sqrt(a))*a**2*b**3 + e**(8*c + 8*d*x)*lo g(sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b) + e**(c + d*x)*sqrt(a))*a*b**4 + 3 *e**(8*c + 8*d*x)*log(2*sqrt(b)*sqrt(a + b) + e**(2*c + 2*d*x)*a + a + 2*b )*a**2*b**3 + e**(8*c + 8*d*x)*log(2*sqrt(b)*sqrt(a + b) + e**(2*c + 2*d*x )*a + a + 2*b)*a*b**4 + e**(8*c + 8*d*x)*a**5 - 2*e**(8*c + 8*d*x)*a**4*b* d*x + e**(8*c + 8*d*x)*a**4*b - 6*e**(8*c + 8*d*x)*a**3*b**2*d*x - 6*e**(8 *c + 8*d*x)*a**2*b**3*d*x - e**(8*c + 8*d*x)*a**2*b**3 - 2*e**(8*c + 8*d*x )*a*b**4*d*x - e**(8*c + 8*d*x)*a*b**4 + 8*e**(6*c + 6*d*x)*log(e**(c + d* x) - 1)*a**3*b**2 + 24*e**(6*c + 6*d*x)*log(e**(c + d*x) - 1)*a**2*b**3 + 8*e**(6*c + 6*d*x)*log(e**(c + d*x) + 1)*a**3*b**2 + 24*e**(6*c + 6*d*x)*l og(e**(c + d*x) + 1)*a**2*b**3 + 12*e**(6*c + 6*d*x)*log( - sqrt(2*sqrt(b) *sqrt(a + b) - a - 2*b) + e**(c + d*x)*sqrt(a))*a*b**4 + 4*e**(6*c + 6*d*x )*log( - sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b) + e**(c + d*x)*sqrt(a))*b** 5 + 12*e**(6*c + 6*d*x)*log(sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b) + e**...