Integrand size = 13, antiderivative size = 144 \[ \int \frac {\tanh ^2(x)}{(a+b \sinh (x))^2} \, dx=-\frac {2 a^3 \text {arctanh}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{5/2}}+\frac {4 a b^2 \text {arctanh}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{5/2}}-\frac {2 a b \text {sech}(x)}{\left (a^2+b^2\right )^2}-\frac {a^2 b \cosh (x)}{\left (a^2+b^2\right )^2 (a+b \sinh (x))}-\frac {\left (a^2-b^2\right ) \tanh (x)}{\left (a^2+b^2\right )^2} \]
-2*a^3*arctanh((b-a*tanh(1/2*x))/(a^2+b^2)^(1/2))/(a^2+b^2)^(5/2)+4*a*b^2* arctanh((b-a*tanh(1/2*x))/(a^2+b^2)^(1/2))/(a^2+b^2)^(5/2)-2*a*b*sech(x)/( a^2+b^2)^2-a^2*b*cosh(x)/(a^2+b^2)^2/(a+b*sinh(x))-(a^2-b^2)*tanh(x)/(a^2+ b^2)^2
Time = 0.24 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.69 \[ \int \frac {\tanh ^2(x)}{(a+b \sinh (x))^2} \, dx=\frac {\frac {2 a \left (a^2-2 b^2\right ) \arctan \left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-a^2-b^2}}-2 a b \text {sech}(x)-\frac {a^2 b \cosh (x)}{a+b \sinh (x)}+\left (-a^2+b^2\right ) \tanh (x)}{\left (a^2+b^2\right )^2} \]
((2*a*(a^2 - 2*b^2)*ArcTan[(b - a*Tanh[x/2])/Sqrt[-a^2 - b^2]])/Sqrt[-a^2 - b^2] - 2*a*b*Sech[x] - (a^2*b*Cosh[x])/(a + b*Sinh[x]) + (-a^2 + b^2)*Ta nh[x])/(a^2 + b^2)^2
Time = 0.49 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {3042, 25, 3210, 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.
| \(\displaystyle \int \frac {\tanh ^2(x)}{(a+b \sinh (x))^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {\tan (i x)^2}{(a-i b \sin (i x))^2}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {\tan (i x)^2}{(a-i b \sin (i x))^2}dx\) |
| \(\Big \downarrow \) 3210 |
| \(\displaystyle -\int \left (-\frac {a^2}{\left (a^2+b^2\right ) (a+b \sinh (x))^2}+\frac {2 b^2 a}{\left (a^2+b^2\right )^2 (a+b \sinh (x))}+\frac {\text {sech}^2(x) \left (a^2 \left (1-\frac {b^2}{a^2}\right )-2 a b \sinh (x)\right )}{\left (a^2+b^2\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {4 a b^2 \text {arctanh}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{5/2}}-\frac {\left (a^2-b^2\right ) \tanh (x)}{\left (a^2+b^2\right )^2}-\frac {2 a b \text {sech}(x)}{\left (a^2+b^2\right )^2}-\frac {a^2 b \cosh (x)}{\left (a^2+b^2\right )^2 (a+b \sinh (x))}-\frac {2 a^3 \text {arctanh}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{5/2}}\) |
(-2*a^3*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(5/2) + (4 *a*b^2*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(5/2) - (2* a*b*Sech[x])/(a^2 + b^2)^2 - (a^2*b*Cosh[x])/((a^2 + b^2)^2*(a + b*Sinh[x] )) - ((a^2 - b^2)*Tanh[x])/(a^2 + b^2)^2
3.3.38.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*tan[(e_.) + (f_.)*(x_)]^(p_ ), x_Symbol] :> Int[ExpandIntegrand[Sin[e + f*x]^p*((a + b*Sin[e + f*x])^m/ (1 - Sin[e + f*x]^2)^(p/2)), x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[m, p/2]
Time = 0.86 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.99
| method | result | size |
| default | \(-\frac {2 a \left (\frac {-b^{2} \tanh \left (\frac {x}{2}\right )-a b}{\tanh \left (\frac {x}{2}\right )^{2} a -2 b \tanh \left (\frac {x}{2}\right )-a}-\frac {\left (a^{2}-2 b^{2}\right ) \operatorname {arctanh}\left (\frac {2 a \tanh \left (\frac {x}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{\sqrt {a^{2}+b^{2}}}\right )}{\left (a^{2}+b^{2}\right )^{2}}+\frac {2 \left (-a^{2}+b^{2}\right ) \tanh \left (\frac {x}{2}\right )-4 a b}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (1+\tanh \left (\frac {x}{2}\right )^{2}\right )}\) | \(142\) |
| risch | \(\frac {2 a^{3} {\mathrm e}^{3 x}-4 a \,b^{2} {\mathrm e}^{3 x}-8 a^{2} b \,{\mathrm e}^{2 x}-2 b^{3} {\mathrm e}^{2 x}+6 a^{3} {\mathrm e}^{x}-4 a^{2} b +2 b^{3}}{\left (b \,{\mathrm e}^{2 x}+2 \,{\mathrm e}^{x} a -b \right ) \left (1+{\mathrm e}^{2 x}\right ) \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {a^{3} \ln \left ({\mathrm e}^{x}+\frac {\left (a^{2}+b^{2}\right )^{\frac {5}{2}} a -a^{6}-3 a^{4} b^{2}-3 a^{2} b^{4}-b^{6}}{b \left (a^{2}+b^{2}\right )^{\frac {5}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {5}{2}}}-\frac {2 b^{2} a \ln \left ({\mathrm e}^{x}+\frac {\left (a^{2}+b^{2}\right )^{\frac {5}{2}} a -a^{6}-3 a^{4} b^{2}-3 a^{2} b^{4}-b^{6}}{b \left (a^{2}+b^{2}\right )^{\frac {5}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {5}{2}}}-\frac {a^{3} \ln \left ({\mathrm e}^{x}+\frac {\left (a^{2}+b^{2}\right )^{\frac {5}{2}} a +a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}{b \left (a^{2}+b^{2}\right )^{\frac {5}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {5}{2}}}+\frac {2 b^{2} a \ln \left ({\mathrm e}^{x}+\frac {\left (a^{2}+b^{2}\right )^{\frac {5}{2}} a +a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}{b \left (a^{2}+b^{2}\right )^{\frac {5}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {5}{2}}}\) | \(369\) |
-2*a/(a^2+b^2)^2*((-b^2*tanh(1/2*x)-a*b)/(tanh(1/2*x)^2*a-2*b*tanh(1/2*x)- a)-(a^2-2*b^2)/(a^2+b^2)^(1/2)*arctanh(1/2*(2*a*tanh(1/2*x)-2*b)/(a^2+b^2) ^(1/2)))+2/(a^4+2*a^2*b^2+b^4)*((-a^2+b^2)*tanh(1/2*x)-2*a*b)/(1+tanh(1/2* x)^2)
Leaf count of result is larger than twice the leaf count of optimal. 900 vs. \(2 (136) = 272\).
Time = 0.29 (sec) , antiderivative size = 900, normalized size of antiderivative = 6.25 \[ \int \frac {\tanh ^2(x)}{(a+b \sinh (x))^2} \, dx=\text {Too large to display} \]
(4*a^4*b + 2*a^2*b^3 - 2*b^5 - 2*(a^5 - a^3*b^2 - 2*a*b^4)*cosh(x)^3 - 2*( a^5 - a^3*b^2 - 2*a*b^4)*sinh(x)^3 + 2*(4*a^4*b + 5*a^2*b^3 + b^5)*cosh(x) ^2 + 2*(4*a^4*b + 5*a^2*b^3 + b^5 - 3*(a^5 - a^3*b^2 - 2*a*b^4)*cosh(x))*s inh(x)^2 + ((a^3*b - 2*a*b^3)*cosh(x)^4 + (a^3*b - 2*a*b^3)*sinh(x)^4 - a^ 3*b + 2*a*b^3 + 2*(a^4 - 2*a^2*b^2)*cosh(x)^3 + 2*(a^4 - 2*a^2*b^2 + 2*(a^ 3*b - 2*a*b^3)*cosh(x))*sinh(x)^3 + 6*((a^3*b - 2*a*b^3)*cosh(x)^2 + (a^4 - 2*a^2*b^2)*cosh(x))*sinh(x)^2 + 2*(a^4 - 2*a^2*b^2)*cosh(x) + 2*(a^4 - 2 *a^2*b^2 + 2*(a^3*b - 2*a*b^3)*cosh(x)^3 + 3*(a^4 - 2*a^2*b^2)*cosh(x)^2)* sinh(x))*sqrt(a^2 + b^2)*log((b^2*cosh(x)^2 + b^2*sinh(x)^2 + 2*a*b*cosh(x ) + 2*a^2 + b^2 + 2*(b^2*cosh(x) + a*b)*sinh(x) + 2*sqrt(a^2 + b^2)*(b*cos h(x) + b*sinh(x) + a))/(b*cosh(x)^2 + b*sinh(x)^2 + 2*a*cosh(x) + 2*(b*cos h(x) + a)*sinh(x) - b)) - 6*(a^5 + a^3*b^2)*cosh(x) - 2*(3*a^5 + 3*a^3*b^2 + 3*(a^5 - a^3*b^2 - 2*a*b^4)*cosh(x)^2 - 2*(4*a^4*b + 5*a^2*b^3 + b^5)*c osh(x))*sinh(x))/(a^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7 - (a^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7)*cosh(x)^4 - (a^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7)*sinh (x)^4 - 2*(a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*cosh(x)^3 - 2*(a^7 + 3*a^5 *b^2 + 3*a^3*b^4 + a*b^6 + 2*(a^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7)*cosh(x) )*sinh(x)^3 - 6*((a^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7)*cosh(x)^2 + (a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*cosh(x))*sinh(x)^2 - 2*(a^7 + 3*a^5*b^2 + 3 *a^3*b^4 + a*b^6)*cosh(x) - 2*(a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6 + 2*...
\[ \int \frac {\tanh ^2(x)}{(a+b \sinh (x))^2} \, dx=\int \frac {\tanh ^{2}{\left (x \right )}}{\left (a + b \sinh {\left (x \right )}\right )^{2}}\, dx \]
Time = 0.33 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.55 \[ \int \frac {\tanh ^2(x)}{(a+b \sinh (x))^2} \, dx=\frac {{\left (a^{2} - 2 \, b^{2}\right )} a \log \left (\frac {b e^{\left (-x\right )} - a - \sqrt {a^{2} + b^{2}}}{b e^{\left (-x\right )} - a + \sqrt {a^{2} + b^{2}}}\right )}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \sqrt {a^{2} + b^{2}}} - \frac {2 \, {\left (3 \, a^{3} e^{\left (-x\right )} + 2 \, a^{2} b - b^{3} + {\left (4 \, a^{2} b + b^{3}\right )} e^{\left (-2 \, x\right )} + {\left (a^{3} - 2 \, a b^{2}\right )} e^{\left (-3 \, x\right )}\right )}}{a^{4} b + 2 \, a^{2} b^{3} + b^{5} + 2 \, {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} e^{\left (-x\right )} + 2 \, {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} e^{\left (-3 \, x\right )} - {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} e^{\left (-4 \, x\right )}} \]
(a^2 - 2*b^2)*a*log((b*e^(-x) - a - sqrt(a^2 + b^2))/(b*e^(-x) - a + sqrt( a^2 + b^2)))/((a^4 + 2*a^2*b^2 + b^4)*sqrt(a^2 + b^2)) - 2*(3*a^3*e^(-x) + 2*a^2*b - b^3 + (4*a^2*b + b^3)*e^(-2*x) + (a^3 - 2*a*b^2)*e^(-3*x))/(a^4 *b + 2*a^2*b^3 + b^5 + 2*(a^5 + 2*a^3*b^2 + a*b^4)*e^(-x) + 2*(a^5 + 2*a^3 *b^2 + a*b^4)*e^(-3*x) - (a^4*b + 2*a^2*b^3 + b^5)*e^(-4*x))
Time = 0.32 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.26 \[ \int \frac {\tanh ^2(x)}{(a+b \sinh (x))^2} \, dx=\frac {{\left (a^{3} - 2 \, a b^{2}\right )} \log \left (\frac {{\left | 2 \, b e^{x} + 2 \, a - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, b e^{x} + 2 \, a + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \sqrt {a^{2} + b^{2}}} + \frac {2 \, {\left (a^{3} e^{\left (3 \, x\right )} - 2 \, a b^{2} e^{\left (3 \, x\right )} - 4 \, a^{2} b e^{\left (2 \, x\right )} - b^{3} e^{\left (2 \, x\right )} + 3 \, a^{3} e^{x} - 2 \, a^{2} b + b^{3}\right )}}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} {\left (b e^{\left (4 \, x\right )} + 2 \, a e^{\left (3 \, x\right )} + 2 \, a e^{x} - b\right )}} \]
(a^3 - 2*a*b^2)*log(abs(2*b*e^x + 2*a - 2*sqrt(a^2 + b^2))/abs(2*b*e^x + 2 *a + 2*sqrt(a^2 + b^2)))/((a^4 + 2*a^2*b^2 + b^4)*sqrt(a^2 + b^2)) + 2*(a^ 3*e^(3*x) - 2*a*b^2*e^(3*x) - 4*a^2*b*e^(2*x) - b^3*e^(2*x) + 3*a^3*e^x - 2*a^2*b + b^3)/((a^4 + 2*a^2*b^2 + b^4)*(b*e^(4*x) + 2*a*e^(3*x) + 2*a*e^x - b))
Time = 1.89 (sec) , antiderivative size = 377, normalized size of antiderivative = 2.62 \[ \int \frac {\tanh ^2(x)}{(a+b \sinh (x))^2} \, dx=\frac {\frac {2\,\left (a^2\,b^9-2\,a^4\,b^7\right )}{b^3\,\left (a^3+a\,b^2\right )\,\left (a^3\,b^3+a\,b^5\right )}-\frac {2\,{\mathrm {e}}^{2\,x}\,\left (4\,a^4\,b^7+a^2\,b^9\right )}{b^3\,\left (a^3+a\,b^2\right )\,\left (a^3\,b^3+a\,b^5\right )}+\frac {6\,a^5\,b^3\,{\mathrm {e}}^x}{\left (a^3+a\,b^2\right )\,\left (a^3\,b^3+a\,b^5\right )}-\frac {2\,a\,{\mathrm {e}}^{3\,x}\,\left (2\,a^2\,b^9-a^4\,b^7\right )}{b^4\,\left (a^3+a\,b^2\right )\,\left (a^3\,b^3+a\,b^5\right )}}{2\,a\,{\mathrm {e}}^x-b+2\,a\,{\mathrm {e}}^{3\,x}+b\,{\mathrm {e}}^{4\,x}}-\frac {a\,\ln \left (\frac {2\,{\mathrm {e}}^x\,\left (2\,a\,b^2-a^3\right )}{b\,{\left (a^2+b^2\right )}^2}-\frac {2\,a\,\left (a^2-2\,b^2\right )\,\left (b-a\,{\mathrm {e}}^x\right )}{b\,{\left (a^2+b^2\right )}^{5/2}}\right )\,\left (a^2-2\,b^2\right )}{{\left (a^2+b^2\right )}^{5/2}}+\frac {a\,\ln \left (\frac {2\,{\mathrm {e}}^x\,\left (2\,a\,b^2-a^3\right )}{b\,{\left (a^2+b^2\right )}^2}+\frac {2\,a\,\left (a^2-2\,b^2\right )\,\left (b-a\,{\mathrm {e}}^x\right )}{b\,{\left (a^2+b^2\right )}^{5/2}}\right )\,\left (a^2-2\,b^2\right )}{{\left (a^2+b^2\right )}^{5/2}} \]
((2*(a^2*b^9 - 2*a^4*b^7))/(b^3*(a*b^2 + a^3)*(a*b^5 + a^3*b^3)) - (2*exp( 2*x)*(a^2*b^9 + 4*a^4*b^7))/(b^3*(a*b^2 + a^3)*(a*b^5 + a^3*b^3)) + (6*a^5 *b^3*exp(x))/((a*b^2 + a^3)*(a*b^5 + a^3*b^3)) - (2*a*exp(3*x)*(2*a^2*b^9 - a^4*b^7))/(b^4*(a*b^2 + a^3)*(a*b^5 + a^3*b^3)))/(2*a*exp(x) - b + 2*a*e xp(3*x) + b*exp(4*x)) - (a*log((2*exp(x)*(2*a*b^2 - a^3))/(b*(a^2 + b^2)^2 ) - (2*a*(a^2 - 2*b^2)*(b - a*exp(x)))/(b*(a^2 + b^2)^(5/2)))*(a^2 - 2*b^2 ))/(a^2 + b^2)^(5/2) + (a*log((2*exp(x)*(2*a*b^2 - a^3))/(b*(a^2 + b^2)^2) + (2*a*(a^2 - 2*b^2)*(b - a*exp(x)))/(b*(a^2 + b^2)^(5/2)))*(a^2 - 2*b^2) )/(a^2 + b^2)^(5/2)