Integrand size = 10, antiderivative size = 35 \[ \int \left (-1+\tanh ^2(x)\right )^{3/2} \, dx=\frac {1}{2} \text {arctanh}\left (\frac {\tanh (x)}{\sqrt {-\text {sech}^2(x)}}\right )-\frac {1}{2} \sqrt {-\text {sech}^2(x)} \tanh (x) \]
Time = 0.02 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.66 \[ \int \left (-1+\tanh ^2(x)\right )^{3/2} \, dx=-\frac {1}{2} \sqrt {-\text {sech}^2(x)} (\arctan (\sinh (x)) \cosh (x)+\tanh (x)) \]
Time = 0.25 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {3042, 4140, 3042, 4610, 211, 224, 219}
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.
\(\displaystyle \int \left (\tanh ^2(x)-1\right )^{3/2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \left (-1-\tan (i x)^2\right )^{3/2}dx\) |
\(\Big \downarrow \) 4140 |
\(\displaystyle \int \left (-\text {sech}^2(x)\right )^{3/2}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \left (-\sec (i x)^2\right )^{3/2}dx\) |
\(\Big \downarrow \) 4610 |
\(\displaystyle -\int \sqrt {\tanh ^2(x)-1}d\tanh (x)\) |
\(\Big \downarrow \) 211 |
\(\displaystyle \frac {1}{2} \int \frac {1}{\sqrt {\tanh ^2(x)-1}}d\tanh (x)-\frac {1}{2} \tanh (x) \sqrt {\tanh ^2(x)-1}\) |
\(\Big \downarrow \) 224 |
\(\displaystyle \frac {1}{2} \int \frac {1}{1-\frac {\tanh ^2(x)}{\tanh ^2(x)-1}}d\frac {\tanh (x)}{\sqrt {\tanh ^2(x)-1}}-\frac {1}{2} \tanh (x) \sqrt {\tanh ^2(x)-1}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{2} \text {arctanh}\left (\frac {\tanh (x)}{\sqrt {\tanh ^2(x)-1}}\right )-\frac {1}{2} \tanh (x) \sqrt {\tanh ^2(x)-1}\) |
3.3.5.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 )), x] + Simp[2*a*(p/(2*p + 1)) Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ {a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
Int[(u_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Int[A ctivateTrig[u*(a*sec[e + f*x]^2)^p], x] /; FreeQ[{a, b, e, f, p}, x] && EqQ [a, b]
Int[((b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> With[{ff = FreeFac tors[Tan[e + f*x], x]}, Simp[b*(ff/f) Subst[Int[(b + b*ff^2*x^2)^(p - 1), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{b, e, f, p}, x] && !IntegerQ[p]
Time = 0.08 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.80
method | result | size |
derivativedivides | \(-\frac {\tanh \left (x \right ) \sqrt {\tanh \left (x \right )^{2}-1}}{2}+\frac {\ln \left (\tanh \left (x \right )+\sqrt {\tanh \left (x \right )^{2}-1}\right )}{2}\) | \(28\) |
default | \(-\frac {\tanh \left (x \right ) \sqrt {\tanh \left (x \right )^{2}-1}}{2}+\frac {\ln \left (\tanh \left (x \right )+\sqrt {\tanh \left (x \right )^{2}-1}\right )}{2}\) | \(28\) |
risch | \(-\frac {\sqrt {-\frac {{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, \left ({\mathrm e}^{2 x}-1\right )}{1+{\mathrm e}^{2 x}}-\frac {i \sqrt {-\frac {{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, {\mathrm e}^{-x} \left (1+{\mathrm e}^{2 x}\right ) \ln \left ({\mathrm e}^{x}+i\right )}{2}+\frac {i \sqrt {-\frac {{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, {\mathrm e}^{-x} \left (1+{\mathrm e}^{2 x}\right ) \ln \left ({\mathrm e}^{x}-i\right )}{2}\) | \(104\) |
Leaf count of result is larger than twice the leaf count of optimal. 385 vs. \(2 (27) = 54\).
Time = 0.30 (sec) , antiderivative size = 385, normalized size of antiderivative = 11.00 \[ \int \left (-1+\tanh ^2(x)\right )^{3/2} \, dx=\frac {{\left (4 \, \cosh \left (x\right ) e^{x} \sinh \left (x\right )^{3} + e^{x} \sinh \left (x\right )^{4} + 2 \, {\left (3 \, \cosh \left (x\right )^{2} + 1\right )} e^{x} \sinh \left (x\right )^{2} + 4 \, {\left (\cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} e^{x} \sinh \left (x\right ) + {\left (\cosh \left (x\right )^{4} + 2 \, \cosh \left (x\right )^{2} + 1\right )} e^{x}\right )} \log \left ({\left (\cosh \left (x\right ) e^{x} + e^{x} \sinh \left (x\right ) + \sqrt {-\frac {e^{\left (2 \, x\right )}}{e^{\left (4 \, x\right )} + 2 \, e^{\left (2 \, x\right )} + 1}} {\left (e^{\left (2 \, x\right )} + 1\right )}\right )} e^{\left (-x\right )}\right ) - {\left (4 \, \cosh \left (x\right ) e^{x} \sinh \left (x\right )^{3} + e^{x} \sinh \left (x\right )^{4} + 2 \, {\left (3 \, \cosh \left (x\right )^{2} + 1\right )} e^{x} \sinh \left (x\right )^{2} + 4 \, {\left (\cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} e^{x} \sinh \left (x\right ) + {\left (\cosh \left (x\right )^{4} + 2 \, \cosh \left (x\right )^{2} + 1\right )} e^{x}\right )} \log \left ({\left (\cosh \left (x\right ) e^{x} + e^{x} \sinh \left (x\right ) - \sqrt {-\frac {e^{\left (2 \, x\right )}}{e^{\left (4 \, x\right )} + 2 \, e^{\left (2 \, x\right )} + 1}} {\left (e^{\left (2 \, x\right )} + 1\right )}\right )} e^{\left (-x\right )}\right ) - 2 \, {\left ({\left (e^{\left (2 \, x\right )} + 1\right )} \sinh \left (x\right )^{3} + \cosh \left (x\right )^{3} + 3 \, {\left (\cosh \left (x\right ) e^{\left (2 \, x\right )} + \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} + {\left (\cosh \left (x\right )^{3} - \cosh \left (x\right )\right )} e^{\left (2 \, x\right )} + {\left (3 \, \cosh \left (x\right )^{2} + {\left (3 \, \cosh \left (x\right )^{2} - 1\right )} e^{\left (2 \, x\right )} - 1\right )} \sinh \left (x\right ) - \cosh \left (x\right )\right )} \sqrt {-\frac {e^{\left (2 \, x\right )}}{e^{\left (4 \, x\right )} + 2 \, e^{\left (2 \, x\right )} + 1}}}{2 \, {\left (4 \, \cosh \left (x\right ) e^{x} \sinh \left (x\right )^{3} + e^{x} \sinh \left (x\right )^{4} + 2 \, {\left (3 \, \cosh \left (x\right )^{2} + 1\right )} e^{x} \sinh \left (x\right )^{2} + 4 \, {\left (\cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} e^{x} \sinh \left (x\right ) + {\left (\cosh \left (x\right )^{4} + 2 \, \cosh \left (x\right )^{2} + 1\right )} e^{x}\right )}} \]
1/2*((4*cosh(x)*e^x*sinh(x)^3 + e^x*sinh(x)^4 + 2*(3*cosh(x)^2 + 1)*e^x*si nh(x)^2 + 4*(cosh(x)^3 + cosh(x))*e^x*sinh(x) + (cosh(x)^4 + 2*cosh(x)^2 + 1)*e^x)*log((cosh(x)*e^x + e^x*sinh(x) + sqrt(-e^(2*x)/(e^(4*x) + 2*e^(2* x) + 1))*(e^(2*x) + 1))*e^(-x)) - (4*cosh(x)*e^x*sinh(x)^3 + e^x*sinh(x)^4 + 2*(3*cosh(x)^2 + 1)*e^x*sinh(x)^2 + 4*(cosh(x)^3 + cosh(x))*e^x*sinh(x) + (cosh(x)^4 + 2*cosh(x)^2 + 1)*e^x)*log((cosh(x)*e^x + e^x*sinh(x) - sqr t(-e^(2*x)/(e^(4*x) + 2*e^(2*x) + 1))*(e^(2*x) + 1))*e^(-x)) - 2*((e^(2*x) + 1)*sinh(x)^3 + cosh(x)^3 + 3*(cosh(x)*e^(2*x) + cosh(x))*sinh(x)^2 + (c osh(x)^3 - cosh(x))*e^(2*x) + (3*cosh(x)^2 + (3*cosh(x)^2 - 1)*e^(2*x) - 1 )*sinh(x) - cosh(x))*sqrt(-e^(2*x)/(e^(4*x) + 2*e^(2*x) + 1)))/(4*cosh(x)* e^x*sinh(x)^3 + e^x*sinh(x)^4 + 2*(3*cosh(x)^2 + 1)*e^x*sinh(x)^2 + 4*(cos h(x)^3 + cosh(x))*e^x*sinh(x) + (cosh(x)^4 + 2*cosh(x)^2 + 1)*e^x)
\[ \int \left (-1+\tanh ^2(x)\right )^{3/2} \, dx=\int \left (\tanh ^{2}{\left (x \right )} - 1\right )^{\frac {3}{2}}\, dx \]
Result contains complex when optimal does not.
Time = 0.31 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.91 \[ \int \left (-1+\tanh ^2(x)\right )^{3/2} \, dx=\frac {-i \, e^{\left (3 \, x\right )} + i \, e^{x}}{e^{\left (4 \, x\right )} + 2 \, e^{\left (2 \, x\right )} + 1} - i \, \arctan \left (e^{x}\right ) \]
Time = 0.28 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.17 \[ \int \left (-1+\tanh ^2(x)\right )^{3/2} \, dx=\frac {\sqrt {-e^{\left (2 \, x\right )}} + \frac {1}{\sqrt {-e^{\left (2 \, x\right )}}}}{{\left (\sqrt {-e^{\left (2 \, x\right )}} + \frac {1}{\sqrt {-e^{\left (2 \, x\right )}}}\right )}^{2} - 4} \]
Time = 1.82 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.77 \[ \int \left (-1+\tanh ^2(x)\right )^{3/2} \, dx=\frac {\ln \left (\mathrm {tanh}\left (x\right )+\sqrt {{\mathrm {tanh}\left (x\right )}^2-1}\right )}{2}-\frac {\mathrm {tanh}\left (x\right )\,\sqrt {{\mathrm {tanh}\left (x\right )}^2-1}}{2} \]