Integrand size = 12, antiderivative size = 29 \[ \int \left (1-\text {sech}^2(x)\right )^{3/2} \, dx=\coth (x) \log (\cosh (x)) \sqrt {\tanh ^2(x)}-\frac {1}{2} \coth (x) \tanh ^2(x)^{3/2} \] Output:
coth(x)*ln(cosh(x))*(tanh(x)^2)^(1/2)-1/2*coth(x)*(tanh(x)^2)^(3/2)
Time = 0.02 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.86 \[ \int \left (1-\text {sech}^2(x)\right )^{3/2} \, dx=\frac {1}{2} (2 \coth (x) \log (\cosh (x))+\text {csch}(x) \text {sech}(x)) \sqrt {\tanh ^2(x)} \] Input:
Integrate[(1 - Sech[x]^2)^(3/2),x]
Output:
((2*Coth[x]*Log[Cosh[x]] + Csch[x]*Sech[x])*Sqrt[Tanh[x]^2])/2
Result contains complex when optimal does not.
Time = 0.33 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.10, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.917, 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.
\(\displaystyle \int \left (1-\text {sech}^2(x)\right )^{3/2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \left (1-\sec (i x)^2\right )^{3/2}dx\) |
\(\Big \downarrow \) 4609 |
\(\displaystyle \int \tanh ^2(x)^{3/2}dx\) |
\(\Big \downarrow \) 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 |
\(\displaystyle i \sqrt {\tanh ^2(x)} \coth (x) \left (\frac {1}{2} i \tanh ^2(x)-i \log (\cosh (x))\right )\) |
Input:
Int[(1 - Sech[x]^2)^(3/2),x]
Output:
I*Coth[x]*Sqrt[Tanh[x]^2]*((-I)*Log[Cosh[x]] + (I/2)*Tanh[x]^2)
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]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.26 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.90
method | result | size |
default | \(\operatorname {csgn}\left (\tanh \left (x \right )\right ) \left (-\frac {\tanh \left (x \right )^{2}}{2}-\frac {\ln \left (-1+\tanh \left (x \right )\right )}{2}-\frac {\ln \left (\tanh \left (x \right )+1\right )}{2}\right )\) | \(26\) |
risch | \(\frac {\sqrt {\frac {\left ({\mathrm e}^{2 x}-1\right )^{2}}{\left ({\mathrm e}^{2 x}+1\right )^{2}}}\, \left ({\mathrm e}^{4 x} \ln \left ({\mathrm e}^{2 x}+1\right )-{\mathrm e}^{4 x} x +2 \,{\mathrm e}^{2 x} \ln \left ({\mathrm e}^{2 x}+1\right )-2 \,{\mathrm e}^{2 x} x +2 \,{\mathrm e}^{2 x}+\ln \left ({\mathrm e}^{2 x}+1\right )-x \right )}{\left ({\mathrm e}^{2 x}-1\right ) \left ({\mathrm e}^{2 x}+1\right )}\) | \(93\) |
Input:
int((1-sech(x)^2)^(3/2),x,method=_RETURNVERBOSE)
Output:
csgn(tanh(x))*(-1/2*tanh(x)^2-1/2*ln(-1+tanh(x))-1/2*ln(tanh(x)+1))
Leaf count of result is larger than twice the leaf count of optimal. 183 vs. \(2 (23) = 46\).
Time = 0.08 (sec) , antiderivative size = 183, normalized size of antiderivative = 6.31 \[ \int \left (1-\text {sech}^2(x)\right )^{3/2} \, dx=-\frac {x \cosh \left (x\right )^{4} + 4 \, x \cosh \left (x\right ) \sinh \left (x\right )^{3} + x \sinh \left (x\right )^{4} + 2 \, {\left (x - 1\right )} \cosh \left (x\right )^{2} + 2 \, {\left (3 \, x \cosh \left (x\right )^{2} + x - 1\right )} \sinh \left (x\right )^{2} - {\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 2 \, {\left (3 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{2} + 2 \, \cosh \left (x\right )^{2} + 4 \, {\left (\cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1\right )} \log \left (\frac {2 \, \cosh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) + 4 \, {\left (x \cosh \left (x\right )^{3} + {\left (x - 1\right )} \cosh \left (x\right )\right )} \sinh \left (x\right ) + x}{\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 2 \, {\left (3 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{2} + 2 \, \cosh \left (x\right )^{2} + 4 \, {\left (\cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1} \] Input:
integrate((1-sech(x)^2)^(3/2),x, algorithm="fricas")
Output:
-(x*cosh(x)^4 + 4*x*cosh(x)*sinh(x)^3 + x*sinh(x)^4 + 2*(x - 1)*cosh(x)^2 + 2*(3*x*cosh(x)^2 + x - 1)*sinh(x)^2 - (cosh(x)^4 + 4*cosh(x)*sinh(x)^3 + sinh(x)^4 + 2*(3*cosh(x)^2 + 1)*sinh(x)^2 + 2*cosh(x)^2 + 4*(cosh(x)^3 + cosh(x))*sinh(x) + 1)*log(2*cosh(x)/(cosh(x) - sinh(x))) + 4*(x*cosh(x)^3 + (x - 1)*cosh(x))*sinh(x) + x)/(cosh(x)^4 + 4*cosh(x)*sinh(x)^3 + sinh(x) ^4 + 2*(3*cosh(x)^2 + 1)*sinh(x)^2 + 2*cosh(x)^2 + 4*(cosh(x)^3 + cosh(x)) *sinh(x) + 1)
\[ \int \left (1-\text {sech}^2(x)\right )^{3/2} \, dx=\int \left (1 - \operatorname {sech}^{2}{\left (x \right )}\right )^{\frac {3}{2}}\, dx \] Input:
integrate((1-sech(x)**2)**(3/2),x)
Output:
Integral((1 - sech(x)**2)**(3/2), x)
Time = 0.12 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.14 \[ \int \left (1-\text {sech}^2(x)\right )^{3/2} \, dx=-x - \frac {2 \, e^{\left (-2 \, x\right )}}{2 \, e^{\left (-2 \, x\right )} + e^{\left (-4 \, x\right )} + 1} - \log \left (e^{\left (-2 \, x\right )} + 1\right ) \] Input:
integrate((1-sech(x)^2)^(3/2),x, algorithm="maxima")
Output:
-x - 2*e^(-2*x)/(2*e^(-2*x) + e^(-4*x) + 1) - log(e^(-2*x) + 1)
Leaf count of result is larger than twice the leaf count of optimal. 72 vs. \(2 (23) = 46\).
Time = 0.11 (sec) , antiderivative size = 72, normalized size of antiderivative = 2.48 \[ \int \left (1-\text {sech}^2(x)\right )^{3/2} \, dx=-x \mathrm {sgn}\left (e^{\left (4 \, x\right )} - 1\right ) + \log \left (e^{\left (2 \, x\right )} + 1\right ) \mathrm {sgn}\left (e^{\left (4 \, x\right )} - 1\right ) - \frac {3 \, e^{\left (4 \, x\right )} \mathrm {sgn}\left (e^{\left (4 \, x\right )} - 1\right ) + 2 \, e^{\left (2 \, x\right )} \mathrm {sgn}\left (e^{\left (4 \, x\right )} - 1\right ) + 3 \, \mathrm {sgn}\left (e^{\left (4 \, x\right )} - 1\right )}{2 \, {\left (e^{\left (2 \, x\right )} + 1\right )}^{2}} \] Input:
integrate((1-sech(x)^2)^(3/2),x, algorithm="giac")
Output:
-x*sgn(e^(4*x) - 1) + log(e^(2*x) + 1)*sgn(e^(4*x) - 1) - 1/2*(3*e^(4*x)*s gn(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. \[ \int \left (1-\text {sech}^2(x)\right )^{3/2} \, dx=\int {\left (1-\frac {1}{{\mathrm {cosh}\left (x\right )}^2}\right )}^{3/2} \,d x \] Input:
int((1 - 1/cosh(x)^2)^(3/2),x)
Output:
int((1 - 1/cosh(x)^2)^(3/2), x)
\[ \int \left (1-\text {sech}^2(x)\right )^{3/2} \, dx=\int \sqrt {-\mathrm {sech}\left (x \right )^{2}+1}d x -\left (\int \sqrt {-\mathrm {sech}\left (x \right )^{2}+1}\, \mathrm {sech}\left (x \right )^{2}d x \right ) \] Input:
int((1-sech(x)^2)^(3/2),x)
Output:
int(sqrt( - sech(x)**2 + 1),x) - int(sqrt( - sech(x)**2 + 1)*sech(x)**2,x)