Integrand size = 11, antiderivative size = 50 \[ \int \frac {1}{(a \coth (x)+b \text {csch}(x))^3} \, dx=\frac {a^2-b^2}{2 a^3 (b+a \cosh (x))^2}+\frac {2 b}{a^3 (b+a \cosh (x))}+\frac {\log (b+a \cosh (x))}{a^3} \]
Time = 0.15 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.54 \[ \int \frac {1}{(a \coth (x)+b \text {csch}(x))^3} \, dx=\frac {a^2+3 b^2+a^2 \log (b+a \cosh (x))+2 b^2 \log (b+a \cosh (x))+a^2 \cosh (2 x) \log (b+a \cosh (x))+4 a b \cosh (x) (1+\log (b+a \cosh (x)))}{2 a^3 (b+a \cosh (x))^2} \]
(a^2 + 3*b^2 + a^2*Log[b + a*Cosh[x]] + 2*b^2*Log[b + a*Cosh[x]] + a^2*Cos h[2*x]*Log[b + a*Cosh[x]] + 4*a*b*Cosh[x]*(1 + Log[b + a*Cosh[x]]))/(2*a^3 *(b + a*Cosh[x])^2)
Time = 0.34 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.94, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.818, Rules used = {3042, 4892, 26, 26, 3042, 26, 3147, 476, 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.
\(\displaystyle \int \frac {1}{(a \coth (x)+b \text {csch}(x))^3} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{(i a \cot (i x)+i b \csc (i x))^3}dx\) |
\(\Big \downarrow \) 4892 |
\(\displaystyle \int -\frac {i \sinh ^3(x)}{(i a \cosh (x)+i b)^3}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -i \int \frac {i \sinh ^3(x)}{(b+a \cosh (x))^3}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \int \frac {\sinh ^3(x)}{(a \cosh (x)+b)^3}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {i \cos \left (-\frac {\pi }{2}+i x\right )^3}{\left (b-a \sin \left (-\frac {\pi }{2}+i x\right )\right )^3}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \int \frac {\cos \left (i x-\frac {\pi }{2}\right )^3}{\left (b-a \sin \left (i x-\frac {\pi }{2}\right )\right )^3}dx\) |
\(\Big \downarrow \) 3147 |
\(\displaystyle -\frac {\int \frac {a^2-a^2 \cosh ^2(x)}{(b+a \cosh (x))^3}d(a \cosh (x))}{a^3}\) |
\(\Big \downarrow \) 476 |
\(\displaystyle -\frac {\int \left (\frac {2 b}{(b+a \cosh (x))^2}+\frac {1}{-b-a \cosh (x)}+\frac {a^2-b^2}{(b+a \cosh (x))^3}\right )d(a \cosh (x))}{a^3}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {-\frac {a^2-b^2}{2 (a \cosh (x)+b)^2}-\frac {2 b}{a \cosh (x)+b}-\log (a \cosh (x)+b)}{a^3}\) |
3.7.51.3.1 Defintions of rubi rules used
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[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ ExpandIntegrand[(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[p, 0]
Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m _.), x_Symbol] :> Simp[1/(b^p*f) Subst[Int[(a + x)^m*(b^2 - x^2)^((p - 1) /2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && IntegerQ[(p - 1)/2] && NeQ[a^2 - b^2, 0]
Int[(cot[(c_.) + (d_.)*(x_)]^(n_.)*(a_.) + csc[(c_.) + (d_.)*(x_)]^(n_.)*(b _.))^(p_)*(u_.), x_Symbol] :> Int[ActivateTrig[u]*Csc[c + d*x]^(n*p)*(b + a *Cos[c + d*x]^n)^p, x] /; FreeQ[{a, b, c, d}, x] && IntegersQ[n, p]
Time = 10.25 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.50
method | result | size |
risch | \(-\frac {x}{a^{3}}+\frac {2 \,{\mathrm e}^{x} \left (2 b \,{\mathrm e}^{2 x} a +a^{2} {\mathrm e}^{x}+3 b^{2} {\mathrm e}^{x}+2 a b \right )}{a^{3} \left (a \,{\mathrm e}^{2 x}+2 \,{\mathrm e}^{x} b +a \right )^{2}}+\frac {\ln \left ({\mathrm e}^{2 x}+\frac {2 b \,{\mathrm e}^{x}}{a}+1\right )}{a^{3}}\) | \(75\) |
default | \(\frac {\frac {2 a^{2} \left (a +b \right )}{\left (a -b \right ) \left (\tanh \left (\frac {x}{2}\right )^{2} a -\tanh \left (\frac {x}{2}\right )^{2} b +a +b \right )^{2}}-\frac {2 a}{\tanh \left (\frac {x}{2}\right )^{2} a -\tanh \left (\frac {x}{2}\right )^{2} b +a +b}+\ln \left (\tanh \left (\frac {x}{2}\right )^{2} a -\tanh \left (\frac {x}{2}\right )^{2} b +a +b \right )}{a^{3}}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{a^{3}}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{a^{3}}\) | \(114\) |
-x/a^3+2/a^3*exp(x)*(2*b*exp(2*x)*a+a^2*exp(x)+3*b^2*exp(x)+2*a*b)/(a*exp( 2*x)+2*exp(x)*b+a)^2+1/a^3*ln(exp(2*x)+2*b/a*exp(x)+1)
Leaf count of result is larger than twice the leaf count of optimal. 521 vs. \(2 (48) = 96\).
Time = 0.25 (sec) , antiderivative size = 521, normalized size of antiderivative = 10.42 \[ \int \frac {1}{(a \coth (x)+b \text {csch}(x))^3} \, dx=-\frac {a^{2} x \cosh \left (x\right )^{4} + a^{2} x \sinh \left (x\right )^{4} + 4 \, {\left (a b x - a b\right )} \cosh \left (x\right )^{3} + 4 \, {\left (a^{2} x \cosh \left (x\right ) + a b x - a b\right )} \sinh \left (x\right )^{3} + a^{2} x - 2 \, {\left (a^{2} + 3 \, b^{2} - {\left (a^{2} + 2 \, b^{2}\right )} x\right )} \cosh \left (x\right )^{2} + 2 \, {\left (3 \, a^{2} x \cosh \left (x\right )^{2} - a^{2} - 3 \, b^{2} + {\left (a^{2} + 2 \, b^{2}\right )} x + 6 \, {\left (a b x - a b\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} + 4 \, {\left (a b x - a b\right )} \cosh \left (x\right ) - {\left (a^{2} \cosh \left (x\right )^{4} + a^{2} \sinh \left (x\right )^{4} + 4 \, a b \cosh \left (x\right )^{3} + 4 \, {\left (a^{2} \cosh \left (x\right ) + a b\right )} \sinh \left (x\right )^{3} + 4 \, a b \cosh \left (x\right ) + 2 \, {\left (a^{2} + 2 \, b^{2}\right )} \cosh \left (x\right )^{2} + 2 \, {\left (3 \, a^{2} \cosh \left (x\right )^{2} + 6 \, a b \cosh \left (x\right ) + a^{2} + 2 \, b^{2}\right )} \sinh \left (x\right )^{2} + a^{2} + 4 \, {\left (a^{2} \cosh \left (x\right )^{3} + 3 \, a b \cosh \left (x\right )^{2} + a b + {\left (a^{2} + 2 \, b^{2}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )\right )} \log \left (\frac {2 \, {\left (a \cosh \left (x\right ) + b\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) + 4 \, {\left (a^{2} x \cosh \left (x\right )^{3} + a b x + 3 \, {\left (a b x - a b\right )} \cosh \left (x\right )^{2} - a b - {\left (a^{2} + 3 \, b^{2} - {\left (a^{2} + 2 \, b^{2}\right )} x\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )}{a^{5} \cosh \left (x\right )^{4} + a^{5} \sinh \left (x\right )^{4} + 4 \, a^{4} b \cosh \left (x\right )^{3} + 4 \, a^{4} b \cosh \left (x\right ) + a^{5} + 4 \, {\left (a^{5} \cosh \left (x\right ) + a^{4} b\right )} \sinh \left (x\right )^{3} + 2 \, {\left (a^{5} + 2 \, a^{3} b^{2}\right )} \cosh \left (x\right )^{2} + 2 \, {\left (3 \, a^{5} \cosh \left (x\right )^{2} + 6 \, a^{4} b \cosh \left (x\right ) + a^{5} + 2 \, a^{3} b^{2}\right )} \sinh \left (x\right )^{2} + 4 \, {\left (a^{5} \cosh \left (x\right )^{3} + 3 \, a^{4} b \cosh \left (x\right )^{2} + a^{4} b + {\left (a^{5} + 2 \, a^{3} b^{2}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )} \]
-(a^2*x*cosh(x)^4 + a^2*x*sinh(x)^4 + 4*(a*b*x - a*b)*cosh(x)^3 + 4*(a^2*x *cosh(x) + a*b*x - a*b)*sinh(x)^3 + a^2*x - 2*(a^2 + 3*b^2 - (a^2 + 2*b^2) *x)*cosh(x)^2 + 2*(3*a^2*x*cosh(x)^2 - a^2 - 3*b^2 + (a^2 + 2*b^2)*x + 6*( a*b*x - a*b)*cosh(x))*sinh(x)^2 + 4*(a*b*x - a*b)*cosh(x) - (a^2*cosh(x)^4 + a^2*sinh(x)^4 + 4*a*b*cosh(x)^3 + 4*(a^2*cosh(x) + a*b)*sinh(x)^3 + 4*a *b*cosh(x) + 2*(a^2 + 2*b^2)*cosh(x)^2 + 2*(3*a^2*cosh(x)^2 + 6*a*b*cosh(x ) + a^2 + 2*b^2)*sinh(x)^2 + a^2 + 4*(a^2*cosh(x)^3 + 3*a*b*cosh(x)^2 + a* b + (a^2 + 2*b^2)*cosh(x))*sinh(x))*log(2*(a*cosh(x) + b)/(cosh(x) - sinh( x))) + 4*(a^2*x*cosh(x)^3 + a*b*x + 3*(a*b*x - a*b)*cosh(x)^2 - a*b - (a^2 + 3*b^2 - (a^2 + 2*b^2)*x)*cosh(x))*sinh(x))/(a^5*cosh(x)^4 + a^5*sinh(x) ^4 + 4*a^4*b*cosh(x)^3 + 4*a^4*b*cosh(x) + a^5 + 4*(a^5*cosh(x) + a^4*b)*s inh(x)^3 + 2*(a^5 + 2*a^3*b^2)*cosh(x)^2 + 2*(3*a^5*cosh(x)^2 + 6*a^4*b*co sh(x) + a^5 + 2*a^3*b^2)*sinh(x)^2 + 4*(a^5*cosh(x)^3 + 3*a^4*b*cosh(x)^2 + a^4*b + (a^5 + 2*a^3*b^2)*cosh(x))*sinh(x))
\[ \int \frac {1}{(a \coth (x)+b \text {csch}(x))^3} \, dx=\int \frac {1}{\left (a \coth {\left (x \right )} + b \operatorname {csch}{\left (x \right )}\right )^{3}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 111 vs. \(2 (48) = 96\).
Time = 0.20 (sec) , antiderivative size = 111, normalized size of antiderivative = 2.22 \[ \int \frac {1}{(a \coth (x)+b \text {csch}(x))^3} \, dx=\frac {2 \, {\left (2 \, a b e^{\left (-x\right )} + 2 \, a b e^{\left (-3 \, x\right )} + {\left (a^{2} + 3 \, b^{2}\right )} e^{\left (-2 \, x\right )}\right )}}{4 \, a^{4} b e^{\left (-x\right )} + 4 \, a^{4} b e^{\left (-3 \, x\right )} + a^{5} e^{\left (-4 \, x\right )} + a^{5} + 2 \, {\left (a^{5} + 2 \, a^{3} b^{2}\right )} e^{\left (-2 \, x\right )}} + \frac {x}{a^{3}} + \frac {\log \left (2 \, b e^{\left (-x\right )} + a e^{\left (-2 \, x\right )} + a\right )}{a^{3}} \]
2*(2*a*b*e^(-x) + 2*a*b*e^(-3*x) + (a^2 + 3*b^2)*e^(-2*x))/(4*a^4*b*e^(-x) + 4*a^4*b*e^(-3*x) + a^5*e^(-4*x) + a^5 + 2*(a^5 + 2*a^3*b^2)*e^(-2*x)) + x/a^3 + log(2*b*e^(-x) + a*e^(-2*x) + a)/a^3
Time = 0.26 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.32 \[ \int \frac {1}{(a \coth (x)+b \text {csch}(x))^3} \, dx=\frac {\log \left ({\left | a {\left (e^{\left (-x\right )} + e^{x}\right )} + 2 \, b \right |}\right )}{a^{3}} - \frac {3 \, a {\left (e^{\left (-x\right )} + e^{x}\right )}^{2} + 4 \, b {\left (e^{\left (-x\right )} + e^{x}\right )} - 4 \, a}{2 \, {\left (a {\left (e^{\left (-x\right )} + e^{x}\right )} + 2 \, b\right )}^{2} a^{2}} \]
log(abs(a*(e^(-x) + e^x) + 2*b))/a^3 - 1/2*(3*a*(e^(-x) + e^x)^2 + 4*b*(e^ (-x) + e^x) - 4*a)/((a*(e^(-x) + e^x) + 2*b)^2*a^2)
Timed out. \[ \int \frac {1}{(a \coth (x)+b \text {csch}(x))^3} \, dx=\int \frac {1}{{\left (\frac {b}{\mathrm {sinh}\left (x\right )}+a\,\mathrm {coth}\left (x\right )\right )}^3} \,d x \]