Integrand size = 13, antiderivative size = 57 \[ \int \frac {\tanh ^3(x)}{a+b \cosh (x)} \, dx=\frac {\left (a^2-b^2\right ) \log (\cosh (x))}{a^3}-\frac {\left (a^2-b^2\right ) \log (a+b \cosh (x))}{a^3}-\frac {b \text {sech}(x)}{a^2}+\frac {\text {sech}^2(x)}{2 a} \]
Time = 0.10 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.81 \[ \int \frac {\tanh ^3(x)}{a+b \cosh (x)} \, dx=\frac {2 \left (a^2-b^2\right ) (\log (\cosh (x))-\log (a+b \cosh (x)))-2 a b \text {sech}(x)+a^2 \text {sech}^2(x)}{2 a^3} \]
Time = 0.29 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.04, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {3042, 26, 3200, 522, 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.
| \(\displaystyle \int \frac {\tanh ^3(x)}{a+b \cosh (x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {i}{\tan \left (-\frac {\pi }{2}+i x\right )^3 \left (a-b \sin \left (-\frac {\pi }{2}+i x\right )\right )}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \int \frac {1}{\left (a-b \sin \left (i x-\frac {\pi }{2}\right )\right ) \tan \left (i x-\frac {\pi }{2}\right )^3}dx\) |
| \(\Big \downarrow \) 3200 |
| \(\displaystyle -\int \frac {\left (b^2-b^2 \cosh ^2(x)\right ) \text {sech}^3(x)}{b^3 (a+b \cosh (x))}d(b \cosh (x))\) |
| \(\Big \downarrow \) 522 |
| \(\displaystyle -\int \left (\frac {\text {sech}^3(x)}{a b}-\frac {\text {sech}^2(x)}{a^2}+\frac {\left (b^2-a^2\right ) \text {sech}(x)}{a^3 b}+\frac {a^2-b^2}{a^3 (a+b \cosh (x))}\right )d(b \cosh (x))\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {b \text {sech}(x)}{a^2}+\frac {\left (a^2-b^2\right ) \log (b \cosh (x))}{a^3}-\frac {\left (a^2-b^2\right ) \log (a+b \cosh (x))}{a^3}+\frac {\text {sech}^2(x)}{2 a}\) |
((a^2 - b^2)*Log[b*Cosh[x]])/a^3 - ((a^2 - b^2)*Log[a + b*Cosh[x]])/a^3 - (b*Sech[x])/a^2 + Sech[x]^2/(2*a)
3.2.80.3.1 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[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_. ), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*tan[(e_.) + (f_.)*(x_)]^(p _.), x_Symbol] :> Simp[1/f Subst[Int[(x^p*(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] && NeQ[a^2 - b ^2, 0] && IntegerQ[(p + 1)/2]
Time = 0.34 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.67
| method | result | size |
| default | \(\frac {\frac {2 a^{2}}{\left (1+\tanh \left (\frac {x}{2}\right )^{2}\right )^{2}}+\left (a^{2}-b^{2}\right ) \ln \left (1+\tanh \left (\frac {x}{2}\right )^{2}\right )-\frac {2 a \left (a +b \right )}{1+\tanh \left (\frac {x}{2}\right )^{2}}}{a^{3}}-\frac {\left (a -b \right ) \left (a +b \right ) \ln \left (\tanh \left (\frac {x}{2}\right )^{2} a -\tanh \left (\frac {x}{2}\right )^{2} b -a -b \right )}{a^{3}}\) | \(95\) |
| risch | \(\frac {2 \,{\mathrm e}^{x} \left (-b \,{\mathrm e}^{2 x}+a \,{\mathrm e}^{x}-b \right )}{\left (1+{\mathrm e}^{2 x}\right )^{2} a^{2}}+\frac {\ln \left (1+{\mathrm e}^{2 x}\right )}{a}-\frac {\ln \left (1+{\mathrm e}^{2 x}\right ) b^{2}}{a^{3}}-\frac {\ln \left ({\mathrm e}^{2 x}+\frac {2 a \,{\mathrm e}^{x}}{b}+1\right )}{a}+\frac {\ln \left ({\mathrm e}^{2 x}+\frac {2 a \,{\mathrm e}^{x}}{b}+1\right ) b^{2}}{a^{3}}\) | \(100\) |
1/a^3*(2*a^2/(1+tanh(1/2*x)^2)^2+(a^2-b^2)*ln(1+tanh(1/2*x)^2)-2*a*(a+b)/( 1+tanh(1/2*x)^2))-(a-b)*(a+b)/a^3*ln(tanh(1/2*x)^2*a-tanh(1/2*x)^2*b-a-b)
Leaf count of result is larger than twice the leaf count of optimal. 450 vs. \(2 (55) = 110\).
Time = 0.27 (sec) , antiderivative size = 450, normalized size of antiderivative = 7.89 \[ \int \frac {\tanh ^3(x)}{a+b \cosh (x)} \, dx=-\frac {2 \, a b \cosh \left (x\right )^{3} + 2 \, a b \sinh \left (x\right )^{3} - 2 \, a^{2} \cosh \left (x\right )^{2} + 2 \, a b \cosh \left (x\right ) + 2 \, {\left (3 \, a b \cosh \left (x\right ) - a^{2}\right )} \sinh \left (x\right )^{2} + {\left ({\left (a^{2} - b^{2}\right )} \cosh \left (x\right )^{4} + 4 \, {\left (a^{2} - b^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right )^{3} + {\left (a^{2} - b^{2}\right )} \sinh \left (x\right )^{4} + 2 \, {\left (a^{2} - b^{2}\right )} \cosh \left (x\right )^{2} + 2 \, {\left (3 \, {\left (a^{2} - b^{2}\right )} \cosh \left (x\right )^{2} + a^{2} - b^{2}\right )} \sinh \left (x\right )^{2} + a^{2} - b^{2} + 4 \, {\left ({\left (a^{2} - b^{2}\right )} \cosh \left (x\right )^{3} + {\left (a^{2} - b^{2}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )\right )} \log \left (\frac {2 \, {\left (b \cosh \left (x\right ) + a\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) - {\left ({\left (a^{2} - b^{2}\right )} \cosh \left (x\right )^{4} + 4 \, {\left (a^{2} - b^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right )^{3} + {\left (a^{2} - b^{2}\right )} \sinh \left (x\right )^{4} + 2 \, {\left (a^{2} - b^{2}\right )} \cosh \left (x\right )^{2} + 2 \, {\left (3 \, {\left (a^{2} - b^{2}\right )} \cosh \left (x\right )^{2} + a^{2} - b^{2}\right )} \sinh \left (x\right )^{2} + a^{2} - b^{2} + 4 \, {\left ({\left (a^{2} - b^{2}\right )} \cosh \left (x\right )^{3} + {\left (a^{2} - b^{2}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )\right )} \log \left (\frac {2 \, \cosh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) + 2 \, {\left (3 \, a b \cosh \left (x\right )^{2} - 2 \, a^{2} \cosh \left (x\right ) + a b\right )} \sinh \left (x\right )}{a^{3} \cosh \left (x\right )^{4} + 4 \, a^{3} \cosh \left (x\right ) \sinh \left (x\right )^{3} + a^{3} \sinh \left (x\right )^{4} + 2 \, a^{3} \cosh \left (x\right )^{2} + a^{3} + 2 \, {\left (3 \, a^{3} \cosh \left (x\right )^{2} + a^{3}\right )} \sinh \left (x\right )^{2} + 4 \, {\left (a^{3} \cosh \left (x\right )^{3} + a^{3} \cosh \left (x\right )\right )} \sinh \left (x\right )} \]
-(2*a*b*cosh(x)^3 + 2*a*b*sinh(x)^3 - 2*a^2*cosh(x)^2 + 2*a*b*cosh(x) + 2* (3*a*b*cosh(x) - a^2)*sinh(x)^2 + ((a^2 - b^2)*cosh(x)^4 + 4*(a^2 - b^2)*c osh(x)*sinh(x)^3 + (a^2 - b^2)*sinh(x)^4 + 2*(a^2 - b^2)*cosh(x)^2 + 2*(3* (a^2 - b^2)*cosh(x)^2 + a^2 - b^2)*sinh(x)^2 + a^2 - b^2 + 4*((a^2 - b^2)* cosh(x)^3 + (a^2 - b^2)*cosh(x))*sinh(x))*log(2*(b*cosh(x) + a)/(cosh(x) - sinh(x))) - ((a^2 - b^2)*cosh(x)^4 + 4*(a^2 - b^2)*cosh(x)*sinh(x)^3 + (a ^2 - b^2)*sinh(x)^4 + 2*(a^2 - b^2)*cosh(x)^2 + 2*(3*(a^2 - b^2)*cosh(x)^2 + a^2 - b^2)*sinh(x)^2 + a^2 - b^2 + 4*((a^2 - b^2)*cosh(x)^3 + (a^2 - b^ 2)*cosh(x))*sinh(x))*log(2*cosh(x)/(cosh(x) - sinh(x))) + 2*(3*a*b*cosh(x) ^2 - 2*a^2*cosh(x) + a*b)*sinh(x))/(a^3*cosh(x)^4 + 4*a^3*cosh(x)*sinh(x)^ 3 + a^3*sinh(x)^4 + 2*a^3*cosh(x)^2 + a^3 + 2*(3*a^3*cosh(x)^2 + a^3)*sinh (x)^2 + 4*(a^3*cosh(x)^3 + a^3*cosh(x))*sinh(x))
\[ \int \frac {\tanh ^3(x)}{a+b \cosh (x)} \, dx=\int \frac {\tanh ^{3}{\left (x \right )}}{a + b \cosh {\left (x \right )}}\, dx \]
Time = 0.28 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.68 \[ \int \frac {\tanh ^3(x)}{a+b \cosh (x)} \, dx=-\frac {2 \, {\left (b e^{\left (-x\right )} - a e^{\left (-2 \, x\right )} + b e^{\left (-3 \, x\right )}\right )}}{2 \, a^{2} e^{\left (-2 \, x\right )} + a^{2} e^{\left (-4 \, x\right )} + a^{2}} - \frac {{\left (a^{2} - b^{2}\right )} \log \left (2 \, a e^{\left (-x\right )} + b e^{\left (-2 \, x\right )} + b\right )}{a^{3}} + \frac {{\left (a^{2} - b^{2}\right )} \log \left (e^{\left (-2 \, x\right )} + 1\right )}{a^{3}} \]
-2*(b*e^(-x) - a*e^(-2*x) + b*e^(-3*x))/(2*a^2*e^(-2*x) + a^2*e^(-4*x) + a ^2) - (a^2 - b^2)*log(2*a*e^(-x) + b*e^(-2*x) + b)/a^3 + (a^2 - b^2)*log(e ^(-2*x) + 1)/a^3
Leaf count of result is larger than twice the leaf count of optimal. 115 vs. \(2 (55) = 110\).
Time = 0.27 (sec) , antiderivative size = 115, normalized size of antiderivative = 2.02 \[ \int \frac {\tanh ^3(x)}{a+b \cosh (x)} \, dx=\frac {{\left (a^{2} - b^{2}\right )} \log \left (e^{\left (-x\right )} + e^{x}\right )}{a^{3}} - \frac {{\left (a^{2} b - b^{3}\right )} \log \left ({\left | b {\left (e^{\left (-x\right )} + e^{x}\right )} + 2 \, a \right |}\right )}{a^{3} b} - \frac {3 \, a^{2} {\left (e^{\left (-x\right )} + e^{x}\right )}^{2} - 3 \, b^{2} {\left (e^{\left (-x\right )} + e^{x}\right )}^{2} + 4 \, a b {\left (e^{\left (-x\right )} + e^{x}\right )} - 4 \, a^{2}}{2 \, a^{3} {\left (e^{\left (-x\right )} + e^{x}\right )}^{2}} \]
(a^2 - b^2)*log(e^(-x) + e^x)/a^3 - (a^2*b - b^3)*log(abs(b*(e^(-x) + e^x) + 2*a))/(a^3*b) - 1/2*(3*a^2*(e^(-x) + e^x)^2 - 3*b^2*(e^(-x) + e^x)^2 + 4*a*b*(e^(-x) + e^x) - 4*a^2)/(a^3*(e^(-x) + e^x)^2)
Time = 2.42 (sec) , antiderivative size = 1221, normalized size of antiderivative = 21.42 \[ \int \frac {\tanh ^3(x)}{a+b \cosh (x)} \, dx=\frac {\frac {2}{a}-\frac {2\,b\,{\mathrm {e}}^x}{a^2}}{{\mathrm {e}}^{2\,x}+1}-\frac {2}{a\,\left (2\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{4\,x}+1\right )}+\frac {\left (2\,\mathrm {atan}\left (\left (4\,a^4\,b^3\,{\left (a^2-b^2\right )}^2\,\sqrt {-a^6}-4\,a^6\,b\,{\left (a^2-b^2\right )}^2\,\sqrt {-a^6}\right )\,\left ({\mathrm {e}}^x\,\left (\frac {1}{16\,a^4\,b^2\,{\left (a^2-b^2\right )}^3\,\sqrt {{\left (a^2-b^2\right )}^2}}-\frac {{\left (a^2-2\,b^2\right )}^2}{16\,a^8\,b^2\,{\left (a^2-b^2\right )}^3\,\sqrt {{\left (a^2-b^2\right )}^2}}\right )+\frac {1}{8\,a^5\,b\,{\left (a^2-b^2\right )}^3\,\sqrt {{\left (a^2-b^2\right )}^2}}+\frac {a^2-2\,b^2}{8\,a^7\,b\,{\left (a^2-b^2\right )}^3\,\sqrt {{\left (a^2-b^2\right )}^2}}\right )\right )+2\,\mathrm {atan}\left (\frac {a^2\,\sqrt {-a^6}\,\sqrt {a^4-2\,a^2\,b^2+b^4}-2\,b^2\,\sqrt {-a^6}\,\sqrt {a^4-2\,a^2\,b^2+b^4}}{2\,a^3\,{\left (a^2-b^2\right )}^2}+\frac {\left (a^7-a^5\,b^2\right )\,\sqrt {-a^6}}{2\,a^6\,\left (a^2-b^2\right )\,\sqrt {{\left (a^2-b^2\right )}^2}}+\frac {a^6\,b^2\,{\mathrm {e}}^{3\,x}\,\left (\frac {2\,\left (a^7-a^5\,b^2\right )\,\sqrt {a^4-2\,a^2\,b^2+b^4}}{a^{11}\,b^3\,\left (a^2-b^2\right )\,\sqrt {{\left (a^2-b^2\right )}^2}}-\frac {2\,\left (a^2-2\,b^2\right )\,\left (a^2\,\sqrt {-a^6}\,\sqrt {a^4-2\,a^2\,b^2+b^4}-2\,b^2\,\sqrt {-a^6}\,\sqrt {a^4-2\,a^2\,b^2+b^4}\right )\,\sqrt {a^4-2\,a^2\,b^2+b^4}}{a^{10}\,b^3\,{\left (a^2-b^2\right )}^2\,\sqrt {-a^6}}\right )\,\sqrt {-a^6}}{8\,\sqrt {a^4-2\,a^2\,b^2+b^4}}-\frac {a^6\,b^2\,{\mathrm {e}}^x\,\sqrt {-a^6}\,\left (\frac {8\,\left (a^4-2\,a^2\,b^2+b^4\right )}{a^8\,b\,{\left (a^2-b^2\right )}^2}-\frac {4\,\left (2\,a^6\,b-2\,a^4\,b^3\right )\,\sqrt {a^4-2\,a^2\,b^2+b^4}}{a^{12}\,b^2\,\left (a^2-b^2\right )\,\sqrt {{\left (a^2-b^2\right )}^2}}-\frac {2\,\left (a^7-a^5\,b^2\right )\,\sqrt {a^4-2\,a^2\,b^2+b^4}}{a^{11}\,b^3\,\left (a^2-b^2\right )\,\sqrt {{\left (a^2-b^2\right )}^2}}+\frac {2\,\left (a^2-2\,b^2\right )\,\left (a^2\,\sqrt {-a^6}\,\sqrt {a^4-2\,a^2\,b^2+b^4}-2\,b^2\,\sqrt {-a^6}\,\sqrt {a^4-2\,a^2\,b^2+b^4}\right )\,\sqrt {a^4-2\,a^2\,b^2+b^4}}{a^{10}\,b^3\,{\left (a^2-b^2\right )}^2\,\sqrt {-a^6}}\right )}{8\,\sqrt {a^4-2\,a^2\,b^2+b^4}}+\frac {a^6\,b^2\,{\mathrm {e}}^{2\,x}\,\sqrt {-a^6}\,\left (\frac {4\,\left (a^2-2\,b^2\right )\,\left (a^4-2\,a^2\,b^2+b^4\right )}{a^9\,b^2\,{\left (a^2-b^2\right )}^2}+\frac {4\,\left (a^2\,\sqrt {-a^6}\,\sqrt {a^4-2\,a^2\,b^2+b^4}-2\,b^2\,\sqrt {-a^6}\,\sqrt {a^4-2\,a^2\,b^2+b^4}\right )\,\sqrt {a^4-2\,a^2\,b^2+b^4}}{a^9\,b^2\,{\left (a^2-b^2\right )}^2\,\sqrt {-a^6}}+\frac {2\,\left (2\,a^6\,b-2\,a^4\,b^3\right )\,\sqrt {a^4-2\,a^2\,b^2+b^4}}{a^{11}\,b^3\,\left (a^2-b^2\right )\,\sqrt {{\left (a^2-b^2\right )}^2}}+\frac {4\,\left (a^7-a^5\,b^2\right )\,\sqrt {a^4-2\,a^2\,b^2+b^4}}{a^{12}\,b^2\,\left (a^2-b^2\right )\,\sqrt {{\left (a^2-b^2\right )}^2}}\right )}{8\,\sqrt {a^4-2\,a^2\,b^2+b^4}}\right )\right )\,\sqrt {a^4-2\,a^2\,b^2+b^4}}{\sqrt {-a^6}} \]
(2/a - (2*b*exp(x))/a^2)/(exp(2*x) + 1) - 2/(a*(2*exp(2*x) + exp(4*x) + 1) ) + ((2*atan((4*a^4*b^3*(a^2 - b^2)^2*(-a^6)^(1/2) - 4*a^6*b*(a^2 - b^2)^2 *(-a^6)^(1/2))*(exp(x)*(1/(16*a^4*b^2*(a^2 - b^2)^3*((a^2 - b^2)^2)^(1/2)) - (a^2 - 2*b^2)^2/(16*a^8*b^2*(a^2 - b^2)^3*((a^2 - b^2)^2)^(1/2))) + 1/( 8*a^5*b*(a^2 - b^2)^3*((a^2 - b^2)^2)^(1/2)) + (a^2 - 2*b^2)/(8*a^7*b*(a^2 - b^2)^3*((a^2 - b^2)^2)^(1/2)))) + 2*atan((a^2*(-a^6)^(1/2)*(a^4 + b^4 - 2*a^2*b^2)^(1/2) - 2*b^2*(-a^6)^(1/2)*(a^4 + b^4 - 2*a^2*b^2)^(1/2))/(2*a ^3*(a^2 - b^2)^2) + ((a^7 - a^5*b^2)*(-a^6)^(1/2))/(2*a^6*(a^2 - b^2)*((a^ 2 - b^2)^2)^(1/2)) + (a^6*b^2*exp(3*x)*((2*(a^7 - a^5*b^2)*(a^4 + b^4 - 2* a^2*b^2)^(1/2))/(a^11*b^3*(a^2 - b^2)*((a^2 - b^2)^2)^(1/2)) - (2*(a^2 - 2 *b^2)*(a^2*(-a^6)^(1/2)*(a^4 + b^4 - 2*a^2*b^2)^(1/2) - 2*b^2*(-a^6)^(1/2) *(a^4 + b^4 - 2*a^2*b^2)^(1/2))*(a^4 + b^4 - 2*a^2*b^2)^(1/2))/(a^10*b^3*( a^2 - b^2)^2*(-a^6)^(1/2)))*(-a^6)^(1/2))/(8*(a^4 + b^4 - 2*a^2*b^2)^(1/2) ) - (a^6*b^2*exp(x)*(-a^6)^(1/2)*((8*(a^4 + b^4 - 2*a^2*b^2))/(a^8*b*(a^2 - b^2)^2) - (4*(2*a^6*b - 2*a^4*b^3)*(a^4 + b^4 - 2*a^2*b^2)^(1/2))/(a^12* b^2*(a^2 - b^2)*((a^2 - b^2)^2)^(1/2)) - (2*(a^7 - a^5*b^2)*(a^4 + b^4 - 2 *a^2*b^2)^(1/2))/(a^11*b^3*(a^2 - b^2)*((a^2 - b^2)^2)^(1/2)) + (2*(a^2 - 2*b^2)*(a^2*(-a^6)^(1/2)*(a^4 + b^4 - 2*a^2*b^2)^(1/2) - 2*b^2*(-a^6)^(1/2 )*(a^4 + b^4 - 2*a^2*b^2)^(1/2))*(a^4 + b^4 - 2*a^2*b^2)^(1/2))/(a^10*b^3* (a^2 - b^2)^2*(-a^6)^(1/2))))/(8*(a^4 + b^4 - 2*a^2*b^2)^(1/2)) + (a^6*...