\(\int (-1-\tanh ^2(x))^{3/2} \, dx\) [228]

Optimal result

Integrand size = 12, antiderivative size = 67 \[ \int \left (-1-\tanh ^2(x)\right )^{3/2} \, dx=-\frac {5}{2} \arctan \left (\frac {\tanh (x)}{\sqrt {-1-\tanh ^2(x)}}\right )+2 \sqrt {2} \arctan \left (\frac {\sqrt {2} \tanh (x)}{\sqrt {-1-\tanh ^2(x)}}\right )+\frac {1}{2} \tanh (x) \sqrt {-1-\tanh ^2(x)} \] Output:

-5/2*arctan(tanh(x)/(-1-tanh(x)^2)^(1/2))+2*2^(1/2)*arctan(2^(1/2)*tanh(x) 
/(-1-tanh(x)^2)^(1/2))+1/2*tanh(x)*(-1-tanh(x)^2)^(1/2)
 

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.13 \[ \int \left (-1-\tanh ^2(x)\right )^{3/2} \, dx=-\frac {\left (-4 \sqrt {2} \text {arcsinh}\left (\sqrt {2} \sinh (x)\right ) \cosh ^3(x)+5 \text {arctanh}\left (\frac {\sinh (x)}{\sqrt {\cosh (2 x)}}\right ) \cosh ^3(x)+\cosh (x) \sqrt {\cosh (2 x)} \sinh (x)\right ) \left (-1-\tanh ^2(x)\right )^{3/2}}{2 \cosh ^{\frac {3}{2}}(2 x)} \] Input:

Integrate[(-1 - Tanh[x]^2)^(3/2),x]
 

Output:

-1/2*((-4*Sqrt[2]*ArcSinh[Sqrt[2]*Sinh[x]]*Cosh[x]^3 + 5*ArcTanh[Sinh[x]/S 
qrt[Cosh[2*x]]]*Cosh[x]^3 + Cosh[x]*Sqrt[Cosh[2*x]]*Sinh[x])*(-1 - Tanh[x] 
^2)^(3/2))/Cosh[2*x]^(3/2)
 

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.04, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {3042, 4144, 318, 25, 398, 224, 216, 291, 216}

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

\(\displaystyle \int \left (-1+\tan (i x)^2\right )^{3/2}dx\)

\(\Big \downarrow \) 4144

\(\displaystyle \int \frac {\left (-\tanh ^2(x)-1\right )^{3/2}}{1-\tanh ^2(x)}d\tanh (x)\)

\(\Big \downarrow \) 318

\(\displaystyle \frac {1}{2} \tanh (x) \sqrt {-\tanh ^2(x)-1}-\frac {1}{2} \int -\frac {5 \tanh ^2(x)+3}{\sqrt {-\tanh ^2(x)-1} \left (1-\tanh ^2(x)\right )}d\tanh (x)\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{2} \int \frac {5 \tanh ^2(x)+3}{\sqrt {-\tanh ^2(x)-1} \left (1-\tanh ^2(x)\right )}d\tanh (x)+\frac {1}{2} \sqrt {-\tanh ^2(x)-1} \tanh (x)\)

\(\Big \downarrow \) 398

\(\displaystyle \frac {1}{2} \left (8 \int \frac {1}{\sqrt {-\tanh ^2(x)-1} \left (1-\tanh ^2(x)\right )}d\tanh (x)-5 \int \frac {1}{\sqrt {-\tanh ^2(x)-1}}d\tanh (x)\right )+\frac {1}{2} \sqrt {-\tanh ^2(x)-1} \tanh (x)\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {1}{2} \left (8 \int \frac {1}{\sqrt {-\tanh ^2(x)-1} \left (1-\tanh ^2(x)\right )}d\tanh (x)-5 \int \frac {1}{\frac {\tanh ^2(x)}{-\tanh ^2(x)-1}+1}d\frac {\tanh (x)}{\sqrt {-\tanh ^2(x)-1}}\right )+\frac {1}{2} \sqrt {-\tanh ^2(x)-1} \tanh (x)\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {1}{2} \left (8 \int \frac {1}{\sqrt {-\tanh ^2(x)-1} \left (1-\tanh ^2(x)\right )}d\tanh (x)-5 \arctan \left (\frac {\tanh (x)}{\sqrt {-\tanh ^2(x)-1}}\right )\right )+\frac {1}{2} \sqrt {-\tanh ^2(x)-1} \tanh (x)\)

\(\Big \downarrow \) 291

\(\displaystyle \frac {1}{2} \left (8 \int \frac {1}{\frac {2 \tanh ^2(x)}{-\tanh ^2(x)-1}+1}d\frac {\tanh (x)}{\sqrt {-\tanh ^2(x)-1}}-5 \arctan \left (\frac {\tanh (x)}{\sqrt {-\tanh ^2(x)-1}}\right )\right )+\frac {1}{2} \sqrt {-\tanh ^2(x)-1} \tanh (x)\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {1}{2} \left (4 \sqrt {2} \arctan \left (\frac {\sqrt {2} \tanh (x)}{\sqrt {-\tanh ^2(x)-1}}\right )-5 \arctan \left (\frac {\tanh (x)}{\sqrt {-\tanh ^2(x)-1}}\right )\right )+\frac {1}{2} \tanh (x) \sqrt {-\tanh ^2(x)-1}\)

Input:

Int[(-1 - Tanh[x]^2)^(3/2),x]
 

Output:

(-5*ArcTan[Tanh[x]/Sqrt[-1 - Tanh[x]^2]] + 4*Sqrt[2]*ArcTan[(Sqrt[2]*Tanh[ 
x])/Sqrt[-1 - Tanh[x]^2]])/2 + (Tanh[x]*Sqrt[-1 - Tanh[x]^2])/2
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 216

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A 
rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a 
, 0] || GtQ[b, 0])
 

rule 224

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]
 

rule 291

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Subst 
[Int[1/(c - (b*c - a*d)*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b, c, 
d}, x] && NeQ[b*c - a*d, 0]
 

rule 318

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim 
p[d*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 1)/(b*(2*(p + q) + 1))), x] + S 
imp[1/(b*(2*(p + q) + 1))   Int[(a + b*x^2)^p*(c + d*x^2)^(q - 2)*Simp[c*(b 
*c*(2*(p + q) + 1) - a*d) + d*(b*c*(2*(p + 2*q - 1) + 1) - a*d*(2*(q - 1) + 
 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b*c - a*d, 0] && G 
tQ[q, 1] && NeQ[2*(p + q) + 1, 0] &&  !IGtQ[p, 1] && IntBinomialQ[a, b, c, 
d, 2, p, q, x]
 

rule 398

Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]) 
, x_Symbol] :> Simp[f/b   Int[1/Sqrt[c + d*x^2], x], x] + Simp[(b*e - a*f)/ 
b   Int[1/((a + b*x^2)*Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f} 
, x]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4144

Int[((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_), x_Symbol] :> 
With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[c*(ff/f)   Subst[Int[(a + b* 
(ff*x)^n)^p/(c^2 + ff^2*x^2), x], x, c*(Tan[e + f*x]/ff)], x]] /; FreeQ[{a, 
 b, c, e, f, n, p}, x] && (IntegersQ[n, p] || IGtQ[p, 0] || EqQ[n^2, 4] || 
EqQ[n^2, 16])
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(210\) vs. \(2(53)=106\).

Time = 0.05 (sec) , antiderivative size = 211, normalized size of antiderivative = 3.15

method	result	size
derivativedivides	\(-\frac {\left (-\left (\tanh \left (x \right )-1\right )^{2}-2 \tanh \left (x \right )\right )^{\frac {3}{2}}}{6}+\frac {\tanh \left (x \right ) \sqrt {-\left (\tanh \left (x \right )-1\right )^{2}-2 \tanh \left (x \right )}}{4}-\frac {5 \arctan \left (\frac {\tanh \left (x \right )}{\sqrt {-\left (\tanh \left (x \right )-1\right )^{2}-2 \tanh \left (x \right )}}\right )}{4}+\sqrt {-\left (\tanh \left (x \right )-1\right )^{2}-2 \tanh \left (x \right )}-\sqrt {2}\, \arctan \left (\frac {\left (-2-2 \tanh \left (x \right )\right ) \sqrt {2}}{4 \sqrt {-\left (\tanh \left (x \right )-1\right )^{2}-2 \tanh \left (x \right )}}\right )+\frac {\left (-\left (\tanh \left (x \right )+1\right )^{2}+2 \tanh \left (x \right )\right )^{\frac {3}{2}}}{6}+\frac {\tanh \left (x \right ) \sqrt {-\left (\tanh \left (x \right )+1\right )^{2}+2 \tanh \left (x \right )}}{4}-\frac {5 \arctan \left (\frac {\tanh \left (x \right )}{\sqrt {-\left (\tanh \left (x \right )+1\right )^{2}+2 \tanh \left (x \right )}}\right )}{4}-\sqrt {-\left (\tanh \left (x \right )+1\right )^{2}+2 \tanh \left (x \right )}+\sqrt {2}\, \arctan \left (\frac {\left (-2+2 \tanh \left (x \right )\right ) \sqrt {2}}{4 \sqrt {-\left (\tanh \left (x \right )+1\right )^{2}+2 \tanh \left (x \right )}}\right )\)	\(211\)
default	\(-\frac {\left (-\left (\tanh \left (x \right )-1\right )^{2}-2 \tanh \left (x \right )\right )^{\frac {3}{2}}}{6}+\frac {\tanh \left (x \right ) \sqrt {-\left (\tanh \left (x \right )-1\right )^{2}-2 \tanh \left (x \right )}}{4}-\frac {5 \arctan \left (\frac {\tanh \left (x \right )}{\sqrt {-\left (\tanh \left (x \right )-1\right )^{2}-2 \tanh \left (x \right )}}\right )}{4}+\sqrt {-\left (\tanh \left (x \right )-1\right )^{2}-2 \tanh \left (x \right )}-\sqrt {2}\, \arctan \left (\frac {\left (-2-2 \tanh \left (x \right )\right ) \sqrt {2}}{4 \sqrt {-\left (\tanh \left (x \right )-1\right )^{2}-2 \tanh \left (x \right )}}\right )+\frac {\left (-\left (\tanh \left (x \right )+1\right )^{2}+2 \tanh \left (x \right )\right )^{\frac {3}{2}}}{6}+\frac {\tanh \left (x \right ) \sqrt {-\left (\tanh \left (x \right )+1\right )^{2}+2 \tanh \left (x \right )}}{4}-\frac {5 \arctan \left (\frac {\tanh \left (x \right )}{\sqrt {-\left (\tanh \left (x \right )+1\right )^{2}+2 \tanh \left (x \right )}}\right )}{4}-\sqrt {-\left (\tanh \left (x \right )+1\right )^{2}+2 \tanh \left (x \right )}+\sqrt {2}\, \arctan \left (\frac {\left (-2+2 \tanh \left (x \right )\right ) \sqrt {2}}{4 \sqrt {-\left (\tanh \left (x \right )+1\right )^{2}+2 \tanh \left (x \right )}}\right )\)	\(211\)

Input:

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

Output:

-1/6*(-(tanh(x)-1)^2-2*tanh(x))^(3/2)+1/4*tanh(x)*(-(tanh(x)-1)^2-2*tanh(x 
))^(1/2)-5/4*arctan(tanh(x)/(-(tanh(x)-1)^2-2*tanh(x))^(1/2))+(-(tanh(x)-1 
)^2-2*tanh(x))^(1/2)-2^(1/2)*arctan(1/4*(-2-2*tanh(x))*2^(1/2)/(-(tanh(x)- 
1)^2-2*tanh(x))^(1/2))+1/6*(-(tanh(x)+1)^2+2*tanh(x))^(3/2)+1/4*tanh(x)*(- 
(tanh(x)+1)^2+2*tanh(x))^(1/2)-5/4*arctan(tanh(x)/(-(tanh(x)+1)^2+2*tanh(x 
))^(1/2))-(-(tanh(x)+1)^2+2*tanh(x))^(1/2)+2^(1/2)*arctan(1/4*(-2+2*tanh(x 
))*2^(1/2)/(-(tanh(x)+1)^2+2*tanh(x))^(1/2))
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.09 (sec) , antiderivative size = 361, normalized size of antiderivative = 5.39 \[ \int \left (-1-\tanh ^2(x)\right )^{3/2} \, dx=\frac {2 \, {\left (\sqrt {-2} e^{\left (4 \, x\right )} + 2 \, \sqrt {-2} e^{\left (2 \, x\right )} + \sqrt {-2}\right )} \log \left (2 \, {\left (\sqrt {-2} \sqrt {-2 \, e^{\left (4 \, x\right )} - 2} + 2 \, e^{\left (2 \, x\right )} + 2\right )} e^{\left (-2 \, x\right )}\right ) - 2 \, {\left (\sqrt {-2} e^{\left (4 \, x\right )} + 2 \, \sqrt {-2} e^{\left (2 \, x\right )} + \sqrt {-2}\right )} \log \left (-2 \, {\left (\sqrt {-2} \sqrt {-2 \, e^{\left (4 \, x\right )} - 2} - 2 \, e^{\left (2 \, x\right )} - 2\right )} e^{\left (-2 \, x\right )}\right ) - 5 \, {\left (i \, e^{\left (4 \, x\right )} + 2 i \, e^{\left (2 \, x\right )} + i\right )} \log \left (-4 \, {\left (i \, \sqrt {-2 \, e^{\left (4 \, x\right )} - 2} + e^{\left (2 \, x\right )} - 1\right )} e^{\left (-2 \, x\right )}\right ) - 5 \, {\left (-i \, e^{\left (4 \, x\right )} - 2 i \, e^{\left (2 \, x\right )} - i\right )} \log \left (-4 \, {\left (-i \, \sqrt {-2 \, e^{\left (4 \, x\right )} - 2} + e^{\left (2 \, x\right )} - 1\right )} e^{\left (-2 \, x\right )}\right ) - 2 \, {\left (\sqrt {-2} e^{\left (4 \, x\right )} + 2 \, \sqrt {-2} e^{\left (2 \, x\right )} + \sqrt {-2}\right )} \log \left (4 \, {\left (\sqrt {-2 \, e^{\left (4 \, x\right )} - 2} {\left (e^{\left (2 \, x\right )} - 2\right )} + \sqrt {-2} e^{\left (4 \, x\right )} - \sqrt {-2} e^{\left (2 \, x\right )} + 2 \, \sqrt {-2}\right )} e^{\left (-4 \, x\right )}\right ) + 2 \, {\left (\sqrt {-2} e^{\left (4 \, x\right )} + 2 \, \sqrt {-2} e^{\left (2 \, x\right )} + \sqrt {-2}\right )} \log \left (4 \, {\left (\sqrt {-2 \, e^{\left (4 \, x\right )} - 2} {\left (e^{\left (2 \, x\right )} - 2\right )} - \sqrt {-2} e^{\left (4 \, x\right )} + \sqrt {-2} e^{\left (2 \, x\right )} - 2 \, \sqrt {-2}\right )} e^{\left (-4 \, x\right )}\right ) + 2 \, \sqrt {-2 \, e^{\left (4 \, x\right )} - 2} {\left (e^{\left (2 \, x\right )} - 1\right )}}{4 \, {\left (e^{\left (4 \, x\right )} + 2 \, e^{\left (2 \, x\right )} + 1\right )}} \] Input:

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

Output:

1/4*(2*(sqrt(-2)*e^(4*x) + 2*sqrt(-2)*e^(2*x) + sqrt(-2))*log(2*(sqrt(-2)* 
sqrt(-2*e^(4*x) - 2) + 2*e^(2*x) + 2)*e^(-2*x)) - 2*(sqrt(-2)*e^(4*x) + 2* 
sqrt(-2)*e^(2*x) + sqrt(-2))*log(-2*(sqrt(-2)*sqrt(-2*e^(4*x) - 2) - 2*e^( 
2*x) - 2)*e^(-2*x)) - 5*(I*e^(4*x) + 2*I*e^(2*x) + I)*log(-4*(I*sqrt(-2*e^ 
(4*x) - 2) + e^(2*x) - 1)*e^(-2*x)) - 5*(-I*e^(4*x) - 2*I*e^(2*x) - I)*log 
(-4*(-I*sqrt(-2*e^(4*x) - 2) + e^(2*x) - 1)*e^(-2*x)) - 2*(sqrt(-2)*e^(4*x 
) + 2*sqrt(-2)*e^(2*x) + sqrt(-2))*log(4*(sqrt(-2*e^(4*x) - 2)*(e^(2*x) - 
2) + sqrt(-2)*e^(4*x) - sqrt(-2)*e^(2*x) + 2*sqrt(-2))*e^(-4*x)) + 2*(sqrt 
(-2)*e^(4*x) + 2*sqrt(-2)*e^(2*x) + sqrt(-2))*log(4*(sqrt(-2*e^(4*x) - 2)* 
(e^(2*x) - 2) - sqrt(-2)*e^(4*x) + sqrt(-2)*e^(2*x) - 2*sqrt(-2))*e^(-4*x) 
) + 2*sqrt(-2*e^(4*x) - 2)*(e^(2*x) - 1))/(e^(4*x) + 2*e^(2*x) + 1)
 

Sympy [F]

\[ \int \left (-1-\tanh ^2(x)\right )^{3/2} \, dx=\int \left (- \tanh ^{2}{\left (x \right )} - 1\right )^{\frac {3}{2}}\, dx \] Input:

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

Output:

Integral((-tanh(x)**2 - 1)**(3/2), x)
 

Maxima [F]

\[ \int \left (-1-\tanh ^2(x)\right )^{3/2} \, dx=\int { {\left (-\tanh \left (x\right )^{2} - 1\right )}^{\frac {3}{2}} \,d x } \] Input:

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

Output:

integrate((-tanh(x)^2 - 1)^(3/2), x)
 

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.13 (sec) , antiderivative size = 204, normalized size of antiderivative = 3.04 \[ \int \left (-1-\tanh ^2(x)\right )^{3/2} \, dx=-\frac {1}{4} \, \sqrt {2} {\left (-5 i \, \sqrt {2} \log \left (\frac {\sqrt {2} - \sqrt {e^{\left (4 \, x\right )} + 1} + e^{\left (2 \, x\right )} + 1}{\sqrt {2} + \sqrt {e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )} - 1}\right ) - \frac {4 \, {\left (-3 i \, {\left (\sqrt {e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )}\right )}^{3} + i \, {\left (\sqrt {e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )}\right )}^{2} + i \, \sqrt {e^{\left (4 \, x\right )} + 1} - i \, e^{\left (2 \, x\right )} + i\right )}}{{\left ({\left (\sqrt {e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )}\right )}^{2} - 2 \, \sqrt {e^{\left (4 \, x\right )} + 1} + 2 \, e^{\left (2 \, x\right )} - 1\right )}^{2}} - 4 i \, \log \left (\sqrt {e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )} + 1\right ) - 4 i \, \log \left (\sqrt {e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )}\right ) + 4 i \, \log \left (-\sqrt {e^{\left (4 \, x\right )} + 1} + e^{\left (2 \, x\right )} + 1\right )\right )} \] Input:

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

Output:

-1/4*sqrt(2)*(-5*I*sqrt(2)*log((sqrt(2) - sqrt(e^(4*x) + 1) + e^(2*x) + 1) 
/(sqrt(2) + sqrt(e^(4*x) + 1) - e^(2*x) - 1)) - 4*(-3*I*(sqrt(e^(4*x) + 1) 
 - e^(2*x))^3 + I*(sqrt(e^(4*x) + 1) - e^(2*x))^2 + I*sqrt(e^(4*x) + 1) - 
I*e^(2*x) + I)/((sqrt(e^(4*x) + 1) - e^(2*x))^2 - 2*sqrt(e^(4*x) + 1) + 2* 
e^(2*x) - 1)^2 - 4*I*log(sqrt(e^(4*x) + 1) - e^(2*x) + 1) - 4*I*log(sqrt(e 
^(4*x) + 1) - e^(2*x)) + 4*I*log(-sqrt(e^(4*x) + 1) + e^(2*x) + 1))
 

Mupad [F(-1)]

Timed out. \[ \int \left (-1-\tanh ^2(x)\right )^{3/2} \, dx=\int {\left (-{\mathrm {tanh}\left (x\right )}^2-1\right )}^{3/2} \,d x \] Input:

int((- tanh(x)^2 - 1)^(3/2),x)
 

Output:

int((- tanh(x)^2 - 1)^(3/2), x)
 

Reduce [F]

\[ \int \left (-1-\tanh ^2(x)\right )^{3/2} \, dx=-i \left (\int \sqrt {\tanh \left (x \right )^{2}+1}d x +\int \sqrt {\tanh \left (x \right )^{2}+1}\, \tanh \left (x \right )^{2}d x \right ) \] Input:

int((-1-tanh(x)^2)^(3/2),x)
 

Output:

 - i*(int(sqrt(tanh(x)**2 + 1),x) + int(sqrt(tanh(x)**2 + 1)*tanh(x)**2,x) 
)