Integrand size = 10, antiderivative size = 34 \[ \int \left (-1+\text {sech}^2(x)\right )^{3/2} \, dx=-\coth (x) \log (\cosh (x)) \sqrt {-\tanh ^2(x)}+\frac {1}{2} \tanh (x) \sqrt {-\tanh ^2(x)} \]
Time = 0.02 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.79 \[ \int \left (-1+\text {sech}^2(x)\right )^{3/2} \, dx=\frac {1}{2} \left (1-2 \coth ^2(x) \log (\cosh (x))\right ) \tanh (x) \sqrt {-\tanh ^2(x)} \]
Result contains complex when optimal does not.
Time = 0.33 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.100, Rules used = {3042, 4609, 3042, 4141, 3042, 26, 3954, 26, 3042, 26, 3956}
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.
\(\displaystyle \int \left (\text {sech}^2(x)-1\right )^{3/2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \left (-1+\sec (i x)^2\right )^{3/2}dx\) |
\(\Big \downarrow \) 4609 |
\(\displaystyle \int \left (-\tanh ^2(x)\right )^{3/2}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \left (\tan (i x)^2\right )^{3/2}dx\) |
\(\Big \downarrow \) 4141 |
\(\displaystyle \sqrt {-\tanh ^2(x)} (-\coth (x)) \int \tanh ^3(x)dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \sqrt {-\tanh ^2(x)} (-\coth (x)) \int i \tan (i x)^3dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -i \sqrt {-\tanh ^2(x)} \coth (x) \int \tan (i x)^3dx\) |
\(\Big \downarrow \) 3954 |
\(\displaystyle -i \sqrt {-\tanh ^2(x)} \coth (x) \left (\frac {1}{2} i \tanh ^2(x)-\int i \tanh (x)dx\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -i \sqrt {-\tanh ^2(x)} \coth (x) \left (\frac {1}{2} i \tanh ^2(x)-i \int \tanh (x)dx\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -i \sqrt {-\tanh ^2(x)} \coth (x) \left (\frac {1}{2} i \tanh ^2(x)-i \int -i \tan (i x)dx\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -i \sqrt {-\tanh ^2(x)} \coth (x) \left (\frac {1}{2} i \tanh ^2(x)-\int \tan (i x)dx\right )\) |
\(\Big \downarrow \) 3956 |
\(\displaystyle -i \sqrt {-\tanh ^2(x)} \coth (x) \left (\frac {1}{2} i \tanh ^2(x)-i \log (\cosh (x))\right )\) |
3.2.73.3.1 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[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((b*Tan[c + d *x])^(n - 1)/(d*(n - 1))), x] - Simp[b^2 Int[(b*Tan[c + d*x])^(n - 2), x] , x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]
Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d *x], x]]/d, x] /; FreeQ[{c, d}, x]
Int[(u_.)*((b_.)*tan[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[(b*ff^n)^IntPart[p]*((b*Tan[e + f*x]^ n)^FracPart[p]/(Tan[e + f*x]/ff)^(n*FracPart[p])) Int[ActivateTrig[u]*(Ta n[e + f*x]/ff)^(n*p), x], x]] /; FreeQ[{b, e, f, n, p}, x] && !IntegerQ[p] && IntegerQ[n] && (EqQ[u, 1] || MatchQ[u, ((d_.)*(trig_)[e + f*x])^(m_.) / ; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig]])
Int[(u_.)*((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Int[A ctivateTrig[u*(b*tan[e + f*x]^2)^p], x] /; FreeQ[{a, b, e, f, p}, x] && EqQ [a + b, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(94\) vs. \(2(28)=56\).
Time = 0.22 (sec) , antiderivative size = 95, normalized size of antiderivative = 2.79
method | result | size |
risch | \(-\frac {\sqrt {-\frac {\left ({\mathrm e}^{2 x}-1\right )^{2}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, \left ({\mathrm e}^{4 x} \ln \left (1+{\mathrm e}^{2 x}\right )-{\mathrm e}^{4 x} x +2 \,{\mathrm e}^{2 x} \ln \left (1+{\mathrm e}^{2 x}\right )-2 \,{\mathrm e}^{2 x} x +2 \,{\mathrm e}^{2 x}+\ln \left (1+{\mathrm e}^{2 x}\right )-x \right )}{\left ({\mathrm e}^{2 x}-1\right ) \left (1+{\mathrm e}^{2 x}\right )}\) | \(95\) |
-(-(exp(2*x)-1)^2/(1+exp(2*x))^2)^(1/2)*(exp(4*x)*ln(1+exp(2*x))-exp(4*x)* x+2*exp(2*x)*ln(1+exp(2*x))-2*exp(2*x)*x+2*exp(2*x)+ln(1+exp(2*x))-x)/(exp (2*x)-1)/(1+exp(2*x))
Leaf count of result is larger than twice the leaf count of optimal. 404 vs. \(2 (28) = 56\).
Time = 0.27 (sec) , antiderivative size = 404, normalized size of antiderivative = 11.88 \[ \int \left (-1+\text {sech}^2(x)\right )^{3/2} \, dx=\frac {{\left (x \cosh \left (x\right )^{4} + {\left (x e^{\left (2 \, x\right )} + x\right )} \sinh \left (x\right )^{4} + 4 \, {\left (x \cosh \left (x\right ) e^{\left (2 \, x\right )} + x \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 2 \, {\left (x - 1\right )} \cosh \left (x\right )^{2} + 2 \, {\left (3 \, x \cosh \left (x\right )^{2} + {\left (3 \, x \cosh \left (x\right )^{2} + x - 1\right )} e^{\left (2 \, x\right )} + x - 1\right )} \sinh \left (x\right )^{2} + {\left (x \cosh \left (x\right )^{4} + 2 \, {\left (x - 1\right )} \cosh \left (x\right )^{2} + x\right )} e^{\left (2 \, x\right )} - {\left ({\left (e^{\left (2 \, x\right )} + 1\right )} \sinh \left (x\right )^{4} + \cosh \left (x\right )^{4} + 4 \, {\left (\cosh \left (x\right ) e^{\left (2 \, x\right )} + \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 2 \, {\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 )^{2} + 2 \, \cosh \left (x\right )^{2} + {\left (\cosh \left (x\right )^{4} + 2 \, \cosh \left (x\right )^{2} + 1\right )} e^{\left (2 \, x\right )} + 4 \, {\left (\cosh \left (x\right )^{3} + {\left (\cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} e^{\left (2 \, x\right )} + \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1\right )} \log \left (\frac {2 \, \cosh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) + 4 \, {\left (x \cosh \left (x\right )^{3} + {\left (x - 1\right )} \cosh \left (x\right ) + {\left (x \cosh \left (x\right )^{3} + {\left (x - 1\right )} \cosh \left (x\right )\right )} e^{\left (2 \, x\right )}\right )} \sinh \left (x\right ) + x\right )} \sqrt {-\frac {e^{\left (4 \, x\right )} - 2 \, e^{\left (2 \, x\right )} + 1}{e^{\left (4 \, x\right )} + 2 \, e^{\left (2 \, x\right )} + 1}}}{{\left (e^{\left (2 \, x\right )} - 1\right )} \sinh \left (x\right )^{4} - \cosh \left (x\right )^{4} + 4 \, {\left (\cosh \left (x\right ) e^{\left (2 \, x\right )} - \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} - 2 \, {\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 )^{2} - 2 \, \cosh \left (x\right )^{2} + {\left (\cosh \left (x\right )^{4} + 2 \, \cosh \left (x\right )^{2} + 1\right )} e^{\left (2 \, x\right )} - 4 \, {\left (\cosh \left (x\right )^{3} - {\left (\cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} e^{\left (2 \, x\right )} + \cosh \left (x\right )\right )} \sinh \left (x\right ) - 1} \]
(x*cosh(x)^4 + (x*e^(2*x) + x)*sinh(x)^4 + 4*(x*cosh(x)*e^(2*x) + x*cosh(x ))*sinh(x)^3 + 2*(x - 1)*cosh(x)^2 + 2*(3*x*cosh(x)^2 + (3*x*cosh(x)^2 + x - 1)*e^(2*x) + x - 1)*sinh(x)^2 + (x*cosh(x)^4 + 2*(x - 1)*cosh(x)^2 + x) *e^(2*x) - ((e^(2*x) + 1)*sinh(x)^4 + cosh(x)^4 + 4*(cosh(x)*e^(2*x) + cos h(x))*sinh(x)^3 + 2*(3*cosh(x)^2 + (3*cosh(x)^2 + 1)*e^(2*x) + 1)*sinh(x)^ 2 + 2*cosh(x)^2 + (cosh(x)^4 + 2*cosh(x)^2 + 1)*e^(2*x) + 4*(cosh(x)^3 + ( cosh(x)^3 + cosh(x))*e^(2*x) + cosh(x))*sinh(x) + 1)*log(2*cosh(x)/(cosh(x ) - sinh(x))) + 4*(x*cosh(x)^3 + (x - 1)*cosh(x) + (x*cosh(x)^3 + (x - 1)* cosh(x))*e^(2*x))*sinh(x) + x)*sqrt(-(e^(4*x) - 2*e^(2*x) + 1)/(e^(4*x) + 2*e^(2*x) + 1))/((e^(2*x) - 1)*sinh(x)^4 - cosh(x)^4 + 4*(cosh(x)*e^(2*x) - cosh(x))*sinh(x)^3 - 2*(3*cosh(x)^2 - (3*cosh(x)^2 + 1)*e^(2*x) + 1)*sin h(x)^2 - 2*cosh(x)^2 + (cosh(x)^4 + 2*cosh(x)^2 + 1)*e^(2*x) - 4*(cosh(x)^ 3 - (cosh(x)^3 + cosh(x))*e^(2*x) + cosh(x))*sinh(x) - 1)
\[ \int \left (-1+\text {sech}^2(x)\right )^{3/2} \, dx=\int \left (\operatorname {sech}^{2}{\left (x \right )} - 1\right )^{\frac {3}{2}}\, dx \]
Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.97 \[ \int \left (-1+\text {sech}^2(x)\right )^{3/2} \, dx=i \, x + \frac {2 i \, e^{\left (-2 \, x\right )}}{2 \, e^{\left (-2 \, x\right )} + e^{\left (-4 \, x\right )} + 1} + i \, \log \left (e^{\left (-2 \, x\right )} + 1\right ) \]
Result contains complex when optimal does not.
Time = 0.29 (sec) , antiderivative size = 83, normalized size of antiderivative = 2.44 \[ \int \left (-1+\text {sech}^2(x)\right )^{3/2} \, dx=-i \, x \mathrm {sgn}\left (-e^{\left (4 \, x\right )} + 1\right ) + i \, \log \left (e^{\left (2 \, x\right )} + 1\right ) \mathrm {sgn}\left (-e^{\left (4 \, x\right )} + 1\right ) - \frac {i \, {\left (3 \, e^{\left (4 \, x\right )} \mathrm {sgn}\left (-e^{\left (4 \, x\right )} + 1\right ) + 2 \, e^{\left (2 \, x\right )} \mathrm {sgn}\left (-e^{\left (4 \, x\right )} + 1\right ) + 3 \, \mathrm {sgn}\left (-e^{\left (4 \, x\right )} + 1\right )\right )}}{2 \, {\left (e^{\left (2 \, x\right )} + 1\right )}^{2}} \]
-I*x*sgn(-e^(4*x) + 1) + I*log(e^(2*x) + 1)*sgn(-e^(4*x) + 1) - 1/2*I*(3*e ^(4*x)*sgn(-e^(4*x) + 1) + 2*e^(2*x)*sgn(-e^(4*x) + 1) + 3*sgn(-e^(4*x) + 1))/(e^(2*x) + 1)^2
Timed out. \[ \int \left (-1+\text {sech}^2(x)\right )^{3/2} \, dx=\int {\left (\frac {1}{{\mathrm {cosh}\left (x\right )}^2}-1\right )}^{3/2} \,d x \]