\(\int \frac {1}{\sqrt {a \tanh ^3(x)}} \, dx\) [36]

Optimal result

Integrand size = 10, antiderivative size = 64 \[ \int \frac {1}{\sqrt {a \tanh ^3(x)}} \, dx=-\frac {2 \tanh (x)}{\sqrt {a \tanh ^3(x)}}-\frac {\arctan \left (\sqrt {\tanh (x)}\right ) \tanh ^{\frac {3}{2}}(x)}{\sqrt {a \tanh ^3(x)}}+\frac {\text {arctanh}\left (\sqrt {\tanh (x)}\right ) \tanh ^{\frac {3}{2}}(x)}{\sqrt {a \tanh ^3(x)}} \] Output:

-2*tanh(x)/(a*tanh(x)^3)^(1/2)-arctan(tanh(x)^(1/2))*tanh(x)^(3/2)/(a*tanh 
(x)^3)^(1/2)+arctanh(tanh(x)^(1/2))*tanh(x)^(3/2)/(a*tanh(x)^3)^(1/2)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.83 \[ \int \frac {1}{\sqrt {a \tanh ^3(x)}} \, dx=-\frac {\tanh (x) \left (2+\arctan \left (\sqrt [4]{\tanh ^2(x)}\right ) \sqrt [4]{\tanh ^2(x)}-\text {arctanh}\left (\sqrt [4]{\tanh ^2(x)}\right ) \sqrt [4]{\tanh ^2(x)}\right )}{\sqrt {a \tanh ^3(x)}} \] Input:

Integrate[1/Sqrt[a*Tanh[x]^3],x]
 

Output:

-((Tanh[x]*(2 + ArcTan[(Tanh[x]^2)^(1/4)]*(Tanh[x]^2)^(1/4) - ArcTanh[(Tan 
h[x]^2)^(1/4)]*(Tanh[x]^2)^(1/4)))/Sqrt[a*Tanh[x]^3])
 

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.80, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.100, Rules used = {3042, 4141, 3042, 3955, 3042, 3957, 25, 266, 827, 216, 219}

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.

\(\displaystyle \int \frac {1}{\sqrt {a \tanh ^3(x)}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{\sqrt {i a \tan (i x)^3}}dx\)

\(\Big \downarrow \) 4141

\(\displaystyle \frac {\tanh ^{\frac {3}{2}}(x) \int \frac {1}{\tanh ^{\frac {3}{2}}(x)}dx}{\sqrt {a \tanh ^3(x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\tanh ^{\frac {3}{2}}(x) \int \frac {1}{(-i \tan (i x))^{3/2}}dx}{\sqrt {a \tanh ^3(x)}}\)

\(\Big \downarrow \) 3955

\(\displaystyle \frac {\tanh ^{\frac {3}{2}}(x) \left (\int \sqrt {\tanh (x)}dx-\frac {2}{\sqrt {\tanh (x)}}\right )}{\sqrt {a \tanh ^3(x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\tanh ^{\frac {3}{2}}(x) \left (-\frac {2}{\sqrt {\tanh (x)}}+\int \sqrt {-i \tan (i x)}dx\right )}{\sqrt {a \tanh ^3(x)}}\)

\(\Big \downarrow \) 3957

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 266

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

\(\Big \downarrow \) 827

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

\(\Big \downarrow \) 216

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

\(\Big \downarrow \) 219

\(\displaystyle \frac {\tanh ^{\frac {3}{2}}(x) \left (2 \left (\frac {1}{2} \text {arctanh}\left (\sqrt {\tanh (x)}\right )-\frac {1}{2} \arctan \left (\sqrt {\tanh (x)}\right )\right )-\frac {2}{\sqrt {\tanh (x)}}\right )}{\sqrt {a \tanh ^3(x)}}\)

Input:

Int[1/Sqrt[a*Tanh[x]^3],x]
 

Output:

((2*(-1/2*ArcTan[Sqrt[Tanh[x]]] + ArcTanh[Sqrt[Tanh[x]]]/2) - 2/Sqrt[Tanh[ 
x]])*Tanh[x]^(3/2))/Sqrt[a*Tanh[x]^3]
 

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 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 266

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De 
nominator[m]}, Simp[k/c   Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) 
^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I 
ntBinomialQ[a, b, c, 2, m, p, x]
 

rule 827

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 
 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2*b)   Int[1/(r + s*x^2), x], 
x] - Simp[s/(2*b)   Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !GtQ 
[a/b, 0]
 

rule 3042

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

rule 3955

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

rule 3957

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

rule 4141

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]])
 

Maple [A] (verified)

Time = 0.34 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.02

method	result	size
derivativedivides	\(-\frac {\tanh \left (x \right ) \left (2 a^{\frac {5}{2}}+\arctan \left (\frac {\sqrt {a \tanh \left (x \right )}}{\sqrt {a}}\right ) a^{2} \sqrt {a \tanh \left (x \right )}-\operatorname {arctanh}\left (\frac {\sqrt {a \tanh \left (x \right )}}{\sqrt {a}}\right ) a^{2} \sqrt {a \tanh \left (x \right )}\right )}{\sqrt {a \tanh \left (x \right )^{3}}\, a^{\frac {5}{2}}}\)	\(65\)
default	\(-\frac {\tanh \left (x \right ) \left (2 a^{\frac {5}{2}}+\arctan \left (\frac {\sqrt {a \tanh \left (x \right )}}{\sqrt {a}}\right ) a^{2} \sqrt {a \tanh \left (x \right )}-\operatorname {arctanh}\left (\frac {\sqrt {a \tanh \left (x \right )}}{\sqrt {a}}\right ) a^{2} \sqrt {a \tanh \left (x \right )}\right )}{\sqrt {a \tanh \left (x \right )^{3}}\, a^{\frac {5}{2}}}\)	\(65\)

Input:

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

Output:

-tanh(x)*(2*a^(5/2)+arctan((a*tanh(x))^(1/2)/a^(1/2))*a^2*(a*tanh(x))^(1/2 
)-arctanh((a*tanh(x))^(1/2)/a^(1/2))*a^2*(a*tanh(x))^(1/2))/(a*tanh(x)^3)^ 
(1/2)/a^(5/2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 239 vs. \(2 (50) = 100\).

Time = 0.11 (sec) , antiderivative size = 514, normalized size of antiderivative = 8.03 \[ \int \frac {1}{\sqrt {a \tanh ^3(x)}} \, dx =\text {Too large to display} \] Input:

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

Output:

[-1/4*(2*(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 - 1)*sqrt(-a)*arctan(( 
cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 + 1)*sqrt(-a)*sqrt(a*sinh(x)/cos 
h(x))/(a*cosh(x)^2 + 2*a*cosh(x)*sinh(x) + a*sinh(x)^2)) + (cosh(x)^2 + 2* 
cosh(x)*sinh(x) + sinh(x)^2 - 1)*sqrt(-a)*log(-(a*cosh(x)^4 + 4*a*cosh(x)^ 
3*sinh(x) + 6*a*cosh(x)^2*sinh(x)^2 + 4*a*cosh(x)*sinh(x)^3 + a*sinh(x)^4 
+ 2*(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 + 1)*sqrt(-a)*sqrt(a*sinh(x 
)/cosh(x)) - 2*a)/(cosh(x)^4 + 4*cosh(x)^3*sinh(x) + 6*cosh(x)^2*sinh(x)^2 
 + 4*cosh(x)*sinh(x)^3 + sinh(x)^4)) + 8*(cosh(x)^2 + 2*cosh(x)*sinh(x) + 
sinh(x)^2 + 1)*sqrt(a*sinh(x)/cosh(x)))/(a*cosh(x)^2 + 2*a*cosh(x)*sinh(x) 
 + a*sinh(x)^2 - a), -1/4*(2*(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 - 
1)*sqrt(a)*arctan((cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 + 1)*sqrt(a*s 
inh(x)/cosh(x))/sqrt(a)) - (cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 - 1) 
*sqrt(a)*log(2*a*cosh(x)^4 + 8*a*cosh(x)^3*sinh(x) + 12*a*cosh(x)^2*sinh(x 
)^2 + 8*a*cosh(x)*sinh(x)^3 + 2*a*sinh(x)^4 + 2*(cosh(x)^4 + 4*cosh(x)*sin 
h(x)^3 + sinh(x)^4 + (6*cosh(x)^2 + 1)*sinh(x)^2 + cosh(x)^2 + 2*(2*cosh(x 
)^3 + cosh(x))*sinh(x))*sqrt(a)*sqrt(a*sinh(x)/cosh(x)) - a) + 8*(cosh(x)^ 
2 + 2*cosh(x)*sinh(x) + sinh(x)^2 + 1)*sqrt(a*sinh(x)/cosh(x)))/(a*cosh(x) 
^2 + 2*a*cosh(x)*sinh(x) + a*sinh(x)^2 - a)]
 

Sympy [F]

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

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

Output:

Integral(1/sqrt(a*tanh(x)**3), x)
 

Maxima [F]

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

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

Output:

integrate(1/sqrt(a*tanh(x)^3), x)
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.64 \[ \int \frac {1}{\sqrt {a \tanh ^3(x)}} \, dx=\frac {4}{{\left (\sqrt {a} e^{\left (2 \, x\right )} - \sqrt {a e^{\left (4 \, x\right )} - a} - \sqrt {a}\right )} \mathrm {sgn}\left (e^{\left (4 \, x\right )} - 1\right )} \] Input:

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

Output:

4/((sqrt(a)*e^(2*x) - sqrt(a*e^(4*x) - a) - sqrt(a))*sgn(e^(4*x) - 1))
 

Mupad [F(-1)]

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

int(1/(a*tanh(x)^3)^(1/2),x)
 

Output:

int(1/(a*tanh(x)^3)^(1/2), x)
 

Reduce [F]

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

int(1/(a*tanh(x)^3)^(1/2),x)
 

Output:

(sqrt(a)*int(sqrt(tanh(x))/tanh(x)**2,x))/a