\(\int (-1-\text {csch}^2(x))^{3/2} \, dx\) [25]

Optimal result

Integrand size = 12, antiderivative size = 34 \[ \int \left (-1-\text {csch}^2(x)\right )^{3/2} \, dx=\frac {1}{2} \coth (x) \sqrt {-\coth ^2(x)}-\sqrt {-\coth ^2(x)} \log (\sinh (x)) \tanh (x) \]

[Out]

1/2*coth(x)*(-coth(x)^2)^(1/2)-ln(sinh(x))*(-coth(x)^2)^(1/2)*tanh(x)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4206, 3739, 3554, 3556} \[ \int \left (-1-\text {csch}^2(x)\right )^{3/2} \, dx=\frac {1}{2} \coth (x) \sqrt {-\coth ^2(x)}-\tanh (x) \sqrt {-\coth ^2(x)} \log (\sinh (x)) \]

[In]

Int[(-1 - Csch[x]^2)^(3/2),x]

[Out]

(Coth[x]*Sqrt[-Coth[x]^2])/2 - Sqrt[-Coth[x]^2]*Log[Sinh[x]]*Tanh[x]

Rule 3554

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((b*Tan[c + d*x])^(n - 1)/(d*(n - 1))), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]

Rule 3556

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rule 3739

Int[(u_.)*((b_.)*tan[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Di
st[(b*ff^n)^IntPart[p]*((b*Tan[e + f*x]^n)^FracPart[p]/(Tan[e + f*x]/ff)^(n*FracPart[p])), Int[ActivateTrig[u]
*(Tan[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
]])

Rule 4206

Int[(u_.)*((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Int[ActivateTrig[u*(b*tan[e + f*x]^2)^p]
, x] /; FreeQ[{a, b, e, f, p}, x] && EqQ[a + b, 0]

Rubi steps \begin{align*} \text {integral}& = \int \left (-\coth ^2(x)\right )^{3/2} \, dx \\ & = -\left (\left (\sqrt {-\coth ^2(x)} \tanh (x)\right ) \int \coth ^3(x) \, dx\right ) \\ & = \frac {1}{2} \coth (x) \sqrt {-\coth ^2(x)}-\left (\sqrt {-\coth ^2(x)} \tanh (x)\right ) \int \coth (x) \, dx \\ & = \frac {1}{2} \coth (x) \sqrt {-\coth ^2(x)}-\sqrt {-\coth ^2(x)} \log (\sinh (x)) \tanh (x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.88 \[ \int \left (-1-\text {csch}^2(x)\right )^{3/2} \, dx=\frac {1}{2} \sqrt {-\coth ^2(x)} \left (\coth ^2(x)-2 (\log (\cosh (x))+\log (\tanh (x)))\right ) \tanh (x) \]

[In]

Integrate[(-1 - Csch[x]^2)^(3/2),x]

[Out]

(Sqrt[-Coth[x]^2]*(Coth[x]^2 - 2*(Log[Cosh[x]] + Log[Tanh[x]]))*Tanh[x])/2

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(94\) vs. \(2(28)=56\).

Time = 0.26 (sec) , antiderivative size = 95, normalized size of antiderivative = 2.79

method	result	size
risch	\(-\frac {\sqrt {-\frac {\left (1+{\mathrm e}^{2 x}\right )^{2}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}\, \left ({\mathrm e}^{4 x} \ln \left ({\mathrm e}^{2 x}-1\right )-{\mathrm e}^{4 x} x -2 \,{\mathrm e}^{2 x} \ln \left ({\mathrm e}^{2 x}-1\right )+2 \,{\mathrm e}^{2 x} x -2 \,{\mathrm e}^{2 x}+\ln \left ({\mathrm e}^{2 x}-1\right )-x \right )}{\left (1+{\mathrm e}^{2 x}\right ) \left ({\mathrm e}^{2 x}-1\right )}\)	\(95\)

[In]

int((-1-csch(x)^2)^(3/2),x,method=_RETURNVERBOSE)

[Out]

-(-(1+exp(2*x))^2/(exp(2*x)-1)^2)^(1/2)*(exp(4*x)*ln(exp(2*x)-1)-exp(4*x)*x-2*exp(2*x)*ln(exp(2*x)-1)+2*exp(2*
x)*x-2*exp(2*x)+ln(exp(2*x)-1)-x)/(1+exp(2*x))/(exp(2*x)-1)

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.25 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.74 \[ \int \left (-1-\text {csch}^2(x)\right )^{3/2} \, dx=\frac {i \, x e^{\left (4 \, x\right )} - 2 \, {\left (i \, x - i\right )} e^{\left (2 \, x\right )} + {\left (-i \, e^{\left (4 \, x\right )} + 2 i \, e^{\left (2 \, x\right )} - i\right )} \log \left (e^{\left (2 \, x\right )} - 1\right ) + i \, x}{e^{\left (4 \, x\right )} - 2 \, e^{\left (2 \, x\right )} + 1} \]

[In]

integrate((-1-csch(x)^2)^(3/2),x, algorithm="fricas")

[Out]

(I*x*e^(4*x) - 2*(I*x - I)*e^(2*x) + (-I*e^(4*x) + 2*I*e^(2*x) - I)*log(e^(2*x) - 1) + I*x)/(e^(4*x) - 2*e^(2*
x) + 1)

Sympy [F]

\[ \int \left (-1-\text {csch}^2(x)\right )^{3/2} \, dx=\int \left (- \operatorname {csch}^{2}{\left (x \right )} - 1\right )^{\frac {3}{2}}\, dx \]

[In]

integrate((-1-csch(x)**2)**(3/2),x)

[Out]

Integral((-csch(x)**2 - 1)**(3/2), x)

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.27 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.29 \[ \int \left (-1-\text {csch}^2(x)\right )^{3/2} \, dx=i \, x + \frac {2 i \, e^{\left (-2 \, x\right )}}{2 \, e^{\left (-2 \, x\right )} - e^{\left (-4 \, x\right )} - 1} + i \, \log \left (e^{\left (-x\right )} + 1\right ) + i \, \log \left (e^{\left (-x\right )} - 1\right ) \]

[In]

integrate((-1-csch(x)^2)^(3/2),x, algorithm="maxima")

[Out]

I*x + 2*I*e^(-2*x)/(2*e^(-2*x) - e^(-4*x) - 1) + I*log(e^(-x) + 1) + I*log(e^(-x) - 1)

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.28 (sec) , antiderivative size = 84, normalized size of antiderivative = 2.47 \[ \int \left (-1-\text {csch}^2(x)\right )^{3/2} \, dx=-i \, x \mathrm {sgn}\left (-e^{\left (4 \, x\right )} + 1\right ) + i \, \log \left ({\left | e^{\left (2 \, x\right )} - 1 \right |}\right ) \mathrm {sgn}\left (-e^{\left (4 \, x\right )} + 1\right ) - \frac {i \, {\left (3 \, e^{\left (4 \, x\right )} \mathrm {sgn}\left (-e^{\left (4 \, x\right )} + 1\right ) - 2 \, e^{\left (2 \, x\right )} \mathrm {sgn}\left (-e^{\left (4 \, x\right )} + 1\right ) + 3 \, \mathrm {sgn}\left (-e^{\left (4 \, x\right )} + 1\right )\right )}}{2 \, {\left (e^{\left (2 \, x\right )} - 1\right )}^{2}} \]

[In]

integrate((-1-csch(x)^2)^(3/2),x, algorithm="giac")

[Out]

-I*x*sgn(-e^(4*x) + 1) + I*log(abs(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

Mupad [F(-1)]

Timed out. \[ \int \left (-1-\text {csch}^2(x)\right )^{3/2} \, dx=\int {\left (-\frac {1}{{\mathrm {sinh}\left (x\right )}^2}-1\right )}^{3/2} \,d x \]

[In]

int((- 1/sinh(x)^2 - 1)^(3/2),x)

[Out]

int((- 1/sinh(x)^2 - 1)^(3/2), x)