Integrand size = 13, antiderivative size = 76 \[ \int \frac {\coth ^3(x)}{a+b \tanh (x)} \, dx=-\frac {b x}{a^2-b^2}+\frac {b \coth (x)}{a^2}-\frac {\coth ^2(x)}{2 a}+\frac {\left (a^2+b^2\right ) \log (\sinh (x))}{a^3}+\frac {b^4 \log (a \cosh (x)+b \sinh (x))}{a^3 \left (a^2-b^2\right )} \] Output:
-b*x/(a^2-b^2)+b*coth(x)/a^2-1/2*coth(x)^2/a+(a^2+b^2)*ln(sinh(x))/a^3+b^4 *ln(a*cosh(x)+b*sinh(x))/a^3/(a^2-b^2)
Time = 0.23 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00 \[ \int \frac {\coth ^3(x)}{a+b \tanh (x)} \, dx=\frac {b \coth (x)}{a^2}-\frac {\coth ^2(x)}{2 a}-\frac {\log (1-\coth (x))}{2 (a+b)}-\frac {\log (1+\coth (x))}{2 (a-b)}+\frac {b^4 \log (b+a \coth (x))}{a^3 \left (a^2-b^2\right )} \] Input:
Integrate[Coth[x]^3/(a + b*Tanh[x]),x]
Output:
(b*Coth[x])/a^2 - Coth[x]^2/(2*a) - Log[1 - Coth[x]]/(2*(a + b)) - Log[1 + Coth[x]]/(2*(a - b)) + (b^4*Log[b + a*Coth[x]])/(a^3*(a^2 - b^2))
Result contains complex when optimal does not.
Time = 0.78 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.43, number of steps used = 16, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.231, Rules used = {3042, 26, 4052, 27, 3042, 25, 4132, 25, 3042, 26, 4135, 26, 3042, 26, 3956, 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.
| \(\displaystyle \int \frac {\coth ^3(x)}{a+b \tanh (x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {i}{\tan (i x)^3 (a-i b \tan (i x))}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \int \frac {1}{\tan (i x)^3 (a-i b \tan (i x))}dx\) |
| \(\Big \downarrow \) 4052 |
| \(\displaystyle -i \left (-\frac {\int \frac {2 i \coth ^2(x) \left (-b \tanh ^2(x)-a \tanh (x)+b\right )}{a+b \tanh (x)}dx}{2 a}-\frac {i \coth ^2(x)}{2 a}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -i \left (-\frac {i \int \frac {\coth ^2(x) \left (-b \tanh ^2(x)-a \tanh (x)+b\right )}{a+b \tanh (x)}dx}{a}-\frac {i \coth ^2(x)}{2 a}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -i \left (-\frac {i \int -\frac {b \tan (i x)^2+i a \tan (i x)+b}{\tan (i x)^2 (a-i b \tan (i x))}dx}{a}-\frac {i \coth ^2(x)}{2 a}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -i \left (\frac {i \int \frac {b \tan (i x)^2+i a \tan (i x)+b}{\tan (i x)^2 (a-i b \tan (i x))}dx}{a}-\frac {i \coth ^2(x)}{2 a}\right )\) |
| \(\Big \downarrow \) 4132 |
| \(\displaystyle -i \left (\frac {i \left (\frac {b \coth (x)}{a}-\frac {\int -\frac {\coth (x) \left (a^2+b^2-b^2 \tanh ^2(x)\right )}{a+b \tanh (x)}dx}{a}\right )}{a}-\frac {i \coth ^2(x)}{2 a}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -i \left (\frac {i \left (\frac {\int \frac {\coth (x) \left (a^2+b^2-b^2 \tanh ^2(x)\right )}{a+b \tanh (x)}dx}{a}+\frac {b \coth (x)}{a}\right )}{a}-\frac {i \coth ^2(x)}{2 a}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -i \left (\frac {i \left (\frac {b \coth (x)}{a}+\frac {\int \frac {i \left (a^2+b^2+b^2 \tan (i x)^2\right )}{\tan (i x) (a-i b \tan (i x))}dx}{a}\right )}{a}-\frac {i \coth ^2(x)}{2 a}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \left (\frac {i \left (\frac {b \coth (x)}{a}+\frac {i \int \frac {a^2+b^2+b^2 \tan (i x)^2}{\tan (i x) (a-i b \tan (i x))}dx}{a}\right )}{a}-\frac {i \coth ^2(x)}{2 a}\right )\) |
| \(\Big \downarrow \) 4135 |
| \(\displaystyle -i \left (\frac {i \left (\frac {b \coth (x)}{a}+\frac {i \left (\frac {\left (a^2+b^2\right ) \int -i \coth (x)dx}{a}+\frac {b^4 \int -\frac {i (b+a \tanh (x))}{a+b \tanh (x)}dx}{a \left (a^2-b^2\right )}+\frac {i a^2 b x}{a^2-b^2}\right )}{a}\right )}{a}-\frac {i \coth ^2(x)}{2 a}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \left (\frac {i \left (\frac {b \coth (x)}{a}+\frac {i \left (-\frac {i \left (a^2+b^2\right ) \int \coth (x)dx}{a}-\frac {i b^4 \int \frac {b+a \tanh (x)}{a+b \tanh (x)}dx}{a \left (a^2-b^2\right )}+\frac {i a^2 b x}{a^2-b^2}\right )}{a}\right )}{a}-\frac {i \coth ^2(x)}{2 a}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -i \left (\frac {i \left (\frac {b \coth (x)}{a}+\frac {i \left (-\frac {i \left (a^2+b^2\right ) \int -i \tan \left (i x+\frac {\pi }{2}\right )dx}{a}-\frac {i b^4 \int \frac {b-i a \tan (i x)}{a-i b \tan (i x)}dx}{a \left (a^2-b^2\right )}+\frac {i a^2 b x}{a^2-b^2}\right )}{a}\right )}{a}-\frac {i \coth ^2(x)}{2 a}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \left (\frac {i \left (\frac {b \coth (x)}{a}+\frac {i \left (-\frac {\left (a^2+b^2\right ) \int \tan \left (i x+\frac {\pi }{2}\right )dx}{a}-\frac {i b^4 \int \frac {b-i a \tan (i x)}{a-i b \tan (i x)}dx}{a \left (a^2-b^2\right )}+\frac {i a^2 b x}{a^2-b^2}\right )}{a}\right )}{a}-\frac {i \coth ^2(x)}{2 a}\right )\) |
| \(\Big \downarrow \) 3956 |
| \(\displaystyle -i \left (\frac {i \left (\frac {b \coth (x)}{a}+\frac {i \left (-\frac {i b^4 \int \frac {b-i a \tan (i x)}{a-i b \tan (i x)}dx}{a \left (a^2-b^2\right )}+\frac {i a^2 b x}{a^2-b^2}-\frac {i \left (a^2+b^2\right ) \log (\sinh (x))}{a}\right )}{a}\right )}{a}-\frac {i \coth ^2(x)}{2 a}\right )\) |
| \(\Big \downarrow \) 4013 |
| \(\displaystyle -i \left (\frac {i \left (\frac {b \coth (x)}{a}+\frac {i \left (\frac {i a^2 b x}{a^2-b^2}-\frac {i \left (a^2+b^2\right ) \log (\sinh (x))}{a}-\frac {i b^4 \log (a \cosh (x)+b \sinh (x))}{a \left (a^2-b^2\right )}\right )}{a}\right )}{a}-\frac {i \coth ^2(x)}{2 a}\right )\) |
Input:
Int[Coth[x]^3/(a + b*Tanh[x]),x]
Output:
(-I)*(((-1/2*I)*Coth[x]^2)/a + (I*((b*Coth[x])/a + (I*((I*a^2*b*x)/(a^2 - b^2) - (I*(a^2 + b^2)*Log[Sinh[x]])/a - (I*b^4*Log[a*Cosh[x] + b*Sinh[x]]) /(a*(a^2 - b^2))))/a))/a)
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_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d *x], x]]/d, x] /; FreeQ[{c, d}, x]
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[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b^2*(a + b*Tan[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^(n + 1)/(f*(m + 1)*(a^2 + b^2)*(b*c - a*d))), x] + Simp[1 /((m + 1)*(a^2 + b^2)*(b*c - a*d)) Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[a*(b*c - a*d)*(m + 1) - b^2*d*(m + n + 2) - b*(b*c - a*d)*(m + 1)*Tan[e + f*x] - b^2*d*(m + n + 2)*Tan[e + f*x]^2, x], x], x] / ; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && IntegerQ[2*m] && LtQ[m, -1] && (LtQ[n, 0] || Integ erQ[m]) && !(ILtQ[n, -1] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(A*b^2 - a*(b*B - a*C))*(a + b*Tan[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 + b^2))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^2 + b^2)) Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[A*(a*(b*c - a*d)*(m + 1) - b^2*d* (m + n + 2)) + (b*B - a*C)*(b*c*(m + 1) + a*d*(n + 1)) - (m + 1)*(b*c - a*d )*(A*b - a*B - b*C)*Tan[e + f*x] - d*(A*b^2 - a*(b*B - a*C))*(m + n + 2)*Ta n[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ [b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, -1] && !(ILtQ[n, -1] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))
Int[((A_.) + (C_.)*tan[(e_.) + (f_.)*(x_)]^2)/(((a_) + (b_.)*tan[(e_.) + (f _.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])), x_Symbol] :> Simp[(a*( A*c - c*C) - b*(A*d - C*d))*(x/((a^2 + b^2)*(c^2 + d^2))), x] + (Simp[(A*b^ 2 + a^2*C)/((b*c - a*d)*(a^2 + b^2)) Int[(b - a*Tan[e + f*x])/(a + b*Tan[ e + f*x]), x], x] - Simp[(c^2*C + A*d^2)/((b*c - a*d)*(c^2 + d^2)) Int[(d - c*Tan[e + f*x])/(c + d*Tan[e + f*x]), x], x]) /; FreeQ[{a, b, c, d, e, f , A, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
Time = 0.54 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.21
| method | result | size |
| parallelrisch | \(\frac {2 \ln \left (a +b \tanh \left (x \right )\right ) b^{4}-2 \ln \left (1-\tanh \left (x \right )\right ) a^{4}+\left (2 a^{4}-2 b^{4}\right ) \ln \left (\tanh \left (x \right )\right )-2 \left (a +b \right ) \left (\frac {\coth \left (x \right )^{2} a \left (a -b \right )}{2}-b \coth \left (x \right ) \left (a -b \right )+a^{2} x \right ) a}{2 a^{5}-2 a^{3} b^{2}}\) | \(92\) |
| derivativedivides | \(\frac {b^{4} \ln \left (a +b \tanh \left (x \right )\right )}{a^{3} \left (a +b \right ) \left (a -b \right )}-\frac {\ln \left (\tanh \left (x \right )-1\right )}{2 b +2 a}+\frac {b}{a^{2} \tanh \left (x \right )}-\frac {\left (-a^{2}-b^{2}\right ) \ln \left (\tanh \left (x \right )\right )}{a^{3}}-\frac {1}{2 a \tanh \left (x \right )^{2}}-\frac {\ln \left (1+\tanh \left (x \right )\right )}{2 a -2 b}\) | \(97\) |
| default | \(\frac {b^{4} \ln \left (a +b \tanh \left (x \right )\right )}{a^{3} \left (a +b \right ) \left (a -b \right )}-\frac {\ln \left (\tanh \left (x \right )-1\right )}{2 b +2 a}+\frac {b}{a^{2} \tanh \left (x \right )}-\frac {\left (-a^{2}-b^{2}\right ) \ln \left (\tanh \left (x \right )\right )}{a^{3}}-\frac {1}{2 a \tanh \left (x \right )^{2}}-\frac {\ln \left (1+\tanh \left (x \right )\right )}{2 a -2 b}\) | \(97\) |
| risch | \(\frac {x}{a +b}-\frac {2 x}{a}-\frac {2 x \,b^{2}}{a^{3}}-\frac {2 x \,b^{4}}{a^{3} \left (a^{2}-b^{2}\right )}-\frac {2 \left ({\mathrm e}^{2 x} a -{\mathrm e}^{2 x} b +b \right )}{\left ({\mathrm e}^{2 x}-1\right )^{2} a^{2}}+\frac {\ln \left ({\mathrm e}^{2 x}-1\right )}{a}+\frac {\ln \left ({\mathrm e}^{2 x}-1\right ) b^{2}}{a^{3}}+\frac {b^{4} \ln \left ({\mathrm e}^{2 x}+\frac {a -b}{a +b}\right )}{a^{3} \left (a^{2}-b^{2}\right )}\) | \(132\) |
Input:
int(coth(x)^3/(a+b*tanh(x)),x,method=_RETURNVERBOSE)
Output:
(2*ln(a+b*tanh(x))*b^4-2*ln(1-tanh(x))*a^4+(2*a^4-2*b^4)*ln(tanh(x))-2*(a+ b)*(1/2*coth(x)^2*a*(a-b)-b*coth(x)*(a-b)+a^2*x)*a)/(2*a^5-2*a^3*b^2)
Leaf count of result is larger than twice the leaf count of optimal. 641 vs. \(2 (74) = 148\).
Time = 0.12 (sec) , antiderivative size = 641, normalized size of antiderivative = 8.43 \[ \int \frac {\coth ^3(x)}{a+b \tanh (x)} \, dx =\text {Too large to display} \] Input:
integrate(coth(x)^3/(a+b*tanh(x)),x, algorithm="fricas")
Output:
-((a^4 + a^3*b)*x*cosh(x)^4 + 4*(a^4 + a^3*b)*x*cosh(x)*sinh(x)^3 + (a^4 + a^3*b)*x*sinh(x)^4 + 2*a^3*b - 2*a*b^3 + 2*(a^4 - a^3*b - a^2*b^2 + a*b^3 - (a^4 + a^3*b)*x)*cosh(x)^2 + 2*(a^4 - a^3*b - a^2*b^2 + a*b^3 + 3*(a^4 + a^3*b)*x*cosh(x)^2 - (a^4 + a^3*b)*x)*sinh(x)^2 + (a^4 + a^3*b)*x - (b^4 *cosh(x)^4 + 4*b^4*cosh(x)*sinh(x)^3 + b^4*sinh(x)^4 - 2*b^4*cosh(x)^2 + b ^4 + 2*(3*b^4*cosh(x)^2 - b^4)*sinh(x)^2 + 4*(b^4*cosh(x)^3 - b^4*cosh(x)) *sinh(x))*log(2*(a*cosh(x) + b*sinh(x))/(cosh(x) - sinh(x))) - ((a^4 - b^4 )*cosh(x)^4 + 4*(a^4 - b^4)*cosh(x)*sinh(x)^3 + (a^4 - b^4)*sinh(x)^4 + a^ 4 - b^4 - 2*(a^4 - b^4)*cosh(x)^2 - 2*(a^4 - b^4 - 3*(a^4 - b^4)*cosh(x)^2 )*sinh(x)^2 + 4*((a^4 - b^4)*cosh(x)^3 - (a^4 - b^4)*cosh(x))*sinh(x))*log (2*sinh(x)/(cosh(x) - sinh(x))) + 4*((a^4 + a^3*b)*x*cosh(x)^3 + (a^4 - a^ 3*b - a^2*b^2 + a*b^3 - (a^4 + a^3*b)*x)*cosh(x))*sinh(x))/(a^5 - a^3*b^2 + (a^5 - a^3*b^2)*cosh(x)^4 + 4*(a^5 - a^3*b^2)*cosh(x)*sinh(x)^3 + (a^5 - a^3*b^2)*sinh(x)^4 - 2*(a^5 - a^3*b^2)*cosh(x)^2 - 2*(a^5 - a^3*b^2 - 3*( a^5 - a^3*b^2)*cosh(x)^2)*sinh(x)^2 + 4*((a^5 - a^3*b^2)*cosh(x)^3 - (a^5 - a^3*b^2)*cosh(x))*sinh(x))
\[ \int \frac {\coth ^3(x)}{a+b \tanh (x)} \, dx=\int \frac {\coth ^{3}{\left (x \right )}}{a + b \tanh {\left (x \right )}}\, dx \] Input:
integrate(coth(x)**3/(a+b*tanh(x)),x)
Output:
Integral(coth(x)**3/(a + b*tanh(x)), x)
Time = 0.04 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.59 \[ \int \frac {\coth ^3(x)}{a+b \tanh (x)} \, dx=\frac {b^{4} \log \left (-{\left (a - b\right )} e^{\left (-2 \, x\right )} - a - b\right )}{a^{5} - a^{3} b^{2}} + \frac {2 \, {\left ({\left (a + b\right )} e^{\left (-2 \, x\right )} - b\right )}}{2 \, a^{2} e^{\left (-2 \, x\right )} - a^{2} e^{\left (-4 \, x\right )} - a^{2}} + \frac {x}{a + b} + \frac {{\left (a^{2} + b^{2}\right )} \log \left (e^{\left (-x\right )} + 1\right )}{a^{3}} + \frac {{\left (a^{2} + b^{2}\right )} \log \left (e^{\left (-x\right )} - 1\right )}{a^{3}} \] Input:
integrate(coth(x)^3/(a+b*tanh(x)),x, algorithm="maxima")
Output:
b^4*log(-(a - b)*e^(-2*x) - a - b)/(a^5 - a^3*b^2) + 2*((a + b)*e^(-2*x) - b)/(2*a^2*e^(-2*x) - a^2*e^(-4*x) - a^2) + x/(a + b) + (a^2 + b^2)*log(e^ (-x) + 1)/a^3 + (a^2 + b^2)*log(e^(-x) - 1)/a^3
Time = 0.12 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.28 \[ \int \frac {\coth ^3(x)}{a+b \tanh (x)} \, dx=\frac {b^{4} \log \left ({\left | a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} + a - b \right |}\right )}{a^{5} - a^{3} b^{2}} - \frac {x}{a - b} + \frac {{\left (a^{2} + b^{2}\right )} \log \left ({\left | e^{\left (2 \, x\right )} - 1 \right |}\right )}{a^{3}} - \frac {2 \, {\left (a b + {\left (a^{2} - a b\right )} e^{\left (2 \, x\right )}\right )}}{a^{3} {\left (e^{\left (2 \, x\right )} - 1\right )}^{2}} \] Input:
integrate(coth(x)^3/(a+b*tanh(x)),x, algorithm="giac")
Output:
b^4*log(abs(a*e^(2*x) + b*e^(2*x) + a - b))/(a^5 - a^3*b^2) - x/(a - b) + (a^2 + b^2)*log(abs(e^(2*x) - 1))/a^3 - 2*(a*b + (a^2 - a*b)*e^(2*x))/(a^3 *(e^(2*x) - 1)^2)
Time = 2.45 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.46 \[ \int \frac {\coth ^3(x)}{a+b \tanh (x)} \, dx=\frac {\ln \left ({\mathrm {e}}^{2\,x}-1\right )\,\left (a^2+b^2\right )}{a^3}-\frac {x}{a-b}-\frac {2}{a\,\left ({\mathrm {e}}^{4\,x}-2\,{\mathrm {e}}^{2\,x}+1\right )}+\frac {b^4\,\ln \left (a-b+a\,{\mathrm {e}}^{2\,x}+b\,{\mathrm {e}}^{2\,x}\right )}{a^5-a^3\,b^2}-\frac {2\,\left (a^2-b^2\right )}{a^2\,\left (a+b\right )\,\left ({\mathrm {e}}^{2\,x}-1\right )} \] Input:
int(coth(x)^3/(a + b*tanh(x)),x)
Output:
(log(exp(2*x) - 1)*(a^2 + b^2))/a^3 - x/(a - b) - 2/(a*(exp(4*x) - 2*exp(2 *x) + 1)) + (b^4*log(a - b + a*exp(2*x) + b*exp(2*x)))/(a^5 - a^3*b^2) - ( 2*(a^2 - b^2))/(a^2*(a + b)*(exp(2*x) - 1))
\[ \int \frac {\coth ^3(x)}{a+b \tanh (x)} \, dx=\int \frac {\coth \left (x \right )^{3}}{\tanh \left (x \right ) b +a}d x \] Input:
int(coth(x)^3/(a+b*tanh(x)),x)
Output:
int(coth(x)^3/(a+b*tanh(x)),x)