Integrand size = 10, antiderivative size = 29 \[ \int \left (1+\text {csch}^2(x)\right )^{3/2} \, dx=-\frac {1}{2} \coth ^2(x)^{3/2} \tanh (x)+\sqrt {\coth ^2(x)} \log (\sinh (x)) \tanh (x) \] Output:
-1/2*(coth(x)^2)^(3/2)*tanh(x)+(coth(x)^2)^(1/2)*ln(sinh(x))*tanh(x)
Time = 0.02 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.83 \[ \int \left (1+\text {csch}^2(x)\right )^{3/2} \, dx=-\frac {1}{2} \sqrt {\coth ^2(x)} \left (\text {csch}^2(x)-2 \log (\sinh (x))\right ) \tanh (x) \] Input:
Integrate[(1 + Csch[x]^2)^(3/2),x]
Output:
-1/2*(Sqrt[Coth[x]^2]*(Csch[x]^2 - 2*Log[Sinh[x]])*Tanh[x])
Result contains complex when optimal does not.
Time = 0.35 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.10, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 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.
\(\displaystyle \int \left (\text {csch}^2(x)+1\right )^{3/2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \left (1-\sec \left (\frac {\pi }{2}+i x\right )^2\right )^{3/2}dx\) |
\(\Big \downarrow \) 4609 |
\(\displaystyle \int \coth ^2(x)^{3/2}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \left (-\tan \left (\frac {\pi }{2}+i x\right )^2\right )^{3/2}dx\) |
\(\Big \downarrow \) 4141 |
\(\displaystyle \tanh (x) \sqrt {\coth ^2(x)} \int \coth ^3(x)dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \tanh (x) \sqrt {\coth ^2(x)} \int i \tan \left (i x+\frac {\pi }{2}\right )^3dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \tanh (x) \sqrt {\coth ^2(x)} \int \tan \left (i x+\frac {\pi }{2}\right )^3dx\) |
\(\Big \downarrow \) 3954 |
\(\displaystyle i \tanh (x) \sqrt {\coth ^2(x)} \left (\frac {1}{2} i \coth ^2(x)-\int i \coth (x)dx\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \tanh (x) \sqrt {\coth ^2(x)} \left (\frac {1}{2} i \coth ^2(x)-i \int \coth (x)dx\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle i \tanh (x) \sqrt {\coth ^2(x)} \left (\frac {1}{2} i \coth ^2(x)-i \int -i \tan \left (i x+\frac {\pi }{2}\right )dx\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \tanh (x) \sqrt {\coth ^2(x)} \left (\frac {1}{2} i \coth ^2(x)-\int \tan \left (i x+\frac {\pi }{2}\right )dx\right )\) |
\(\Big \downarrow \) 3956 |
\(\displaystyle i \tanh (x) \sqrt {\coth ^2(x)} \left (\frac {1}{2} i \coth ^2(x)-i \log (\sinh (x))\right )\) |
Input:
Int[(1 + Csch[x]^2)^(3/2),x]
Output:
I*Sqrt[Coth[x]^2]*((I/2)*Coth[x]^2 - I*Log[Sinh[x]])*Tanh[x]
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 = 0.28 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.90
method | result | size |
default | \(\operatorname {csgn}\left (\coth \left (x \right )\right ) \left (-\frac {\coth \left (x \right )^{2}}{2}-\frac {\ln \left (\coth \left (x \right )-1\right )}{2}-\frac {\ln \left (1+\coth \left (x \right )\right )}{2}\right )\) | \(26\) |
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+csch(x)^2)^(3/2),x,method=_RETURNVERBOSE)
Output:
csgn(coth(x))*(-1/2*coth(x)^2-1/2*ln(coth(x)-1)-1/2*ln(1+coth(x)))
Leaf count of result is larger than twice the leaf count of optimal. 190 vs. \(2 (23) = 46\).
Time = 0.09 (sec) , antiderivative size = 190, normalized size of antiderivative = 6.55 \[ \int \left (1+\text {csch}^2(x)\right )^{3/2} \, dx=-\frac {x \cosh \left (x\right )^{4} + 4 \, x \cosh \left (x\right ) \sinh \left (x\right )^{3} + x \sinh \left (x\right )^{4} - 2 \, {\left (x - 1\right )} \cosh \left (x\right )^{2} + 2 \, {\left (3 \, x \cosh \left (x\right )^{2} - x + 1\right )} \sinh \left (x\right )^{2} - {\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 2 \, {\left (3 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{2} - 2 \, \cosh \left (x\right )^{2} + 4 \, {\left (\cosh \left (x\right )^{3} - \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1\right )} \log \left (\frac {2 \, \sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) + 4 \, {\left (x \cosh \left (x\right )^{3} - {\left (x - 1\right )} \cosh \left (x\right )\right )} \sinh \left (x\right ) + x}{\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 2 \, {\left (3 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{2} - 2 \, \cosh \left (x\right )^{2} + 4 \, {\left (\cosh \left (x\right )^{3} - \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1} \] Input:
integrate((1+csch(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*sinh(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)
\[ \int \left (1+\text {csch}^2(x)\right )^{3/2} \, dx=\int \left (\operatorname {csch}^{2}{\left (x \right )} + 1\right )^{\frac {3}{2}}\, dx \] Input:
integrate((1+csch(x)**2)**(3/2),x)
Output:
Integral((csch(x)**2 + 1)**(3/2), x)
Time = 0.11 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.52 \[ \int \left (1+\text {csch}^2(x)\right )^{3/2} \, dx=-x - \frac {2 \, e^{\left (-2 \, x\right )}}{2 \, e^{\left (-2 \, x\right )} - e^{\left (-4 \, x\right )} - 1} - \log \left (e^{\left (-x\right )} + 1\right ) - \log \left (e^{\left (-x\right )} - 1\right ) \] Input:
integrate((1+csch(x)^2)^(3/2),x, algorithm="maxima")
Output:
-x - 2*e^(-2*x)/(2*e^(-2*x) - e^(-4*x) - 1) - log(e^(-x) + 1) - log(e^(-x) - 1)
Leaf count of result is larger than twice the leaf count of optimal. 73 vs. \(2 (23) = 46\).
Time = 0.11 (sec) , antiderivative size = 73, normalized size of antiderivative = 2.52 \[ \int \left (1+\text {csch}^2(x)\right )^{3/2} \, dx=-x \mathrm {sgn}\left (e^{\left (4 \, x\right )} - 1\right ) + \log \left ({\left | e^{\left (2 \, x\right )} - 1 \right |}\right ) \mathrm {sgn}\left (e^{\left (4 \, x\right )} - 1\right ) - \frac {3 \, e^{\left (4 \, x\right )} \mathrm {sgn}\left (e^{\left (4 \, x\right )} - 1\right ) - 2 \, e^{\left (2 \, x\right )} \mathrm {sgn}\left (e^{\left (4 \, x\right )} - 1\right ) + 3 \, \mathrm {sgn}\left (e^{\left (4 \, x\right )} - 1\right )}{2 \, {\left (e^{\left (2 \, x\right )} - 1\right )}^{2}} \] Input:
integrate((1+csch(x)^2)^(3/2),x, algorithm="giac")
Output:
-x*sgn(e^(4*x) - 1) + log(abs(e^(2*x) - 1))*sgn(e^(4*x) - 1) - 1/2*(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. \[ \int \left (1+\text {csch}^2(x)\right )^{3/2} \, dx=\int {\left (\frac {1}{{\mathrm {sinh}\left (x\right )}^2}+1\right )}^{3/2} \,d x \] Input:
int((1/sinh(x)^2 + 1)^(3/2),x)
Output:
int((1/sinh(x)^2 + 1)^(3/2), x)
\[ \int \left (1+\text {csch}^2(x)\right )^{3/2} \, dx=\int \sqrt {\mathrm {csch}\left (x \right )^{2}+1}d x +\int \sqrt {\mathrm {csch}\left (x \right )^{2}+1}\, \mathrm {csch}\left (x \right )^{2}d x \] Input:
int((1+csch(x)^2)^(3/2),x)
Output:
int(sqrt(csch(x)**2 + 1),x) + int(sqrt(csch(x)**2 + 1)*csch(x)**2,x)