\(\int (a \tanh ^3(x))^{3/2} \, dx\) [34]

Optimal result

Integrand size = 10, antiderivative size = 86 \[ \int \left (a \tanh ^3(x)\right )^{3/2} \, dx=-\frac {2}{3} a \sqrt {a \tanh ^3(x)}-\frac {a \arctan \left (\sqrt {\tanh (x)}\right ) \sqrt {a \tanh ^3(x)}}{\tanh ^{\frac {3}{2}}(x)}+\frac {a \text {arctanh}\left (\sqrt {\tanh (x)}\right ) \sqrt {a \tanh ^3(x)}}{\tanh ^{\frac {3}{2}}(x)}-\frac {2}{7} a \tanh ^2(x) \sqrt {a \tanh ^3(x)} \] Output:

-2/3*a*(a*tanh(x)^3)^(1/2)-a*arctan(tanh(x)^(1/2))*(a*tanh(x)^3)^(1/2)/tan 
h(x)^(3/2)+a*arctanh(tanh(x)^(1/2))*(a*tanh(x)^3)^(1/2)/tanh(x)^(3/2)-2/7* 
a*tanh(x)^2*(a*tanh(x)^3)^(1/2)
 

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.64 \[ \int \left (a \tanh ^3(x)\right )^{3/2} \, dx=-\frac {\left (a \tanh ^3(x)\right )^{3/2} \left (21 \arctan \left (\sqrt {\tanh (x)}\right )-21 \text {arctanh}\left (\sqrt {\tanh (x)}\right )+14 \tanh ^{\frac {3}{2}}(x)+6 \tanh ^{\frac {7}{2}}(x)\right )}{21 \tanh ^{\frac {9}{2}}(x)} \] Input:

Integrate[(a*Tanh[x]^3)^(3/2),x]
 

Output:

-1/21*((a*Tanh[x]^3)^(3/2)*(21*ArcTan[Sqrt[Tanh[x]]] - 21*ArcTanh[Sqrt[Tan 
h[x]]] + 14*Tanh[x]^(3/2) + 6*Tanh[x]^(7/2)))/Tanh[x]^(9/2)
 

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.74, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.300, Rules used = {3042, 4141, 3042, 3954, 3042, 3954, 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.

\(\displaystyle \int \left (a \tanh ^3(x)\right )^{3/2} \, dx\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4141

\(\displaystyle \frac {a \sqrt {a \tanh ^3(x)} \int \tanh ^{\frac {9}{2}}(x)dx}{\tanh ^{\frac {3}{2}}(x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {a \sqrt {a \tanh ^3(x)} \int (-i \tan (i x))^{9/2}dx}{\tanh ^{\frac {3}{2}}(x)}\)

\(\Big \downarrow \) 3954

\(\displaystyle \frac {a \sqrt {a \tanh ^3(x)} \left (\int \tanh ^{\frac {5}{2}}(x)dx-\frac {2}{7} \tanh ^{\frac {7}{2}}(x)\right )}{\tanh ^{\frac {3}{2}}(x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {a \sqrt {a \tanh ^3(x)} \left (-\frac {2}{7} \tanh ^{\frac {7}{2}}(x)+\int (-i \tan (i x))^{5/2}dx\right )}{\tanh ^{\frac {3}{2}}(x)}\)

\(\Big \downarrow \) 3954

\(\displaystyle \frac {a \sqrt {a \tanh ^3(x)} \left (\int \sqrt {\tanh (x)}dx-\frac {2}{7} \tanh ^{\frac {7}{2}}(x)-\frac {2}{3} \tanh ^{\frac {3}{2}}(x)\right )}{\tanh ^{\frac {3}{2}}(x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {a \sqrt {a \tanh ^3(x)} \left (\int \sqrt {-i \tan (i x)}dx-\frac {2}{7} \tanh ^{\frac {7}{2}}(x)-\frac {2}{3} \tanh ^{\frac {3}{2}}(x)\right )}{\tanh ^{\frac {3}{2}}(x)}\)

\(\Big \downarrow \) 3957

\(\displaystyle \frac {a \sqrt {a \tanh ^3(x)} \left (-\int -\frac {\sqrt {\tanh (x)}}{1-\tanh ^2(x)}d\tanh (x)-\frac {2}{7} \tanh ^{\frac {7}{2}}(x)-\frac {2}{3} \tanh ^{\frac {3}{2}}(x)\right )}{\tanh ^{\frac {3}{2}}(x)}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {a \sqrt {a \tanh ^3(x)} \left (\int \frac {\sqrt {\tanh (x)}}{1-\tanh ^2(x)}d\tanh (x)-\frac {2}{7} \tanh ^{\frac {7}{2}}(x)-\frac {2}{3} \tanh ^{\frac {3}{2}}(x)\right )}{\tanh ^{\frac {3}{2}}(x)}\)

\(\Big \downarrow \) 266

\(\displaystyle \frac {a \sqrt {a \tanh ^3(x)} \left (2 \int \frac {\tanh (x)}{1-\tanh ^2(x)}d\sqrt {\tanh (x)}-\frac {2}{7} \tanh ^{\frac {7}{2}}(x)-\frac {2}{3} \tanh ^{\frac {3}{2}}(x)\right )}{\tanh ^{\frac {3}{2}}(x)}\)

\(\Big \downarrow \) 827

\(\displaystyle \frac {a \sqrt {a \tanh ^3(x)} \left (2 \left (\frac {1}{2} \int \frac {1}{1-\tanh (x)}d\sqrt {\tanh (x)}-\frac {1}{2} \int \frac {1}{\tanh (x)+1}d\sqrt {\tanh (x)}\right )-\frac {2}{7} \tanh ^{\frac {7}{2}}(x)-\frac {2}{3} \tanh ^{\frac {3}{2}}(x)\right )}{\tanh ^{\frac {3}{2}}(x)}\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {a \sqrt {a \tanh ^3(x)} \left (2 \left (\frac {1}{2} \int \frac {1}{1-\tanh (x)}d\sqrt {\tanh (x)}-\frac {1}{2} \arctan \left (\sqrt {\tanh (x)}\right )\right )-\frac {2}{7} \tanh ^{\frac {7}{2}}(x)-\frac {2}{3} \tanh ^{\frac {3}{2}}(x)\right )}{\tanh ^{\frac {3}{2}}(x)}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {a \sqrt {a \tanh ^3(x)} \left (2 \left (\frac {1}{2} \text {arctanh}\left (\sqrt {\tanh (x)}\right )-\frac {1}{2} \arctan \left (\sqrt {\tanh (x)}\right )\right )-\frac {2}{7} \tanh ^{\frac {7}{2}}(x)-\frac {2}{3} \tanh ^{\frac {3}{2}}(x)\right )}{\tanh ^{\frac {3}{2}}(x)}\)

Input:

Int[(a*Tanh[x]^3)^(3/2),x]
 

Output:

(a*Sqrt[a*Tanh[x]^3]*(2*(-1/2*ArcTan[Sqrt[Tanh[x]]] + ArcTanh[Sqrt[Tanh[x] 
]]/2) - (2*Tanh[x]^(3/2))/3 - (2*Tanh[x]^(7/2))/7))/Tanh[x]^(3/2)
 

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 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 3954

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]
 

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 = 0.35 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.88

method	result	size
derivativedivides	\(\frac {\left (a \tanh \left (x \right )^{3}\right )^{\frac {3}{2}} \left (21 a^{\frac {7}{2}} \operatorname {arctanh}\left (\frac {\sqrt {a \tanh \left (x \right )}}{\sqrt {a}}\right )-21 a^{\frac {7}{2}} \arctan \left (\frac {\sqrt {a \tanh \left (x \right )}}{\sqrt {a}}\right )-6 \left (a \tanh \left (x \right )\right )^{\frac {7}{2}}-14 a^{2} \left (a \tanh \left (x \right )\right )^{\frac {3}{2}}\right )}{21 \tanh \left (x \right )^{3} \left (a \tanh \left (x \right )\right )^{\frac {3}{2}} a^{2}}\)	\(76\)
default	\(\frac {\left (a \tanh \left (x \right )^{3}\right )^{\frac {3}{2}} \left (21 a^{\frac {7}{2}} \operatorname {arctanh}\left (\frac {\sqrt {a \tanh \left (x \right )}}{\sqrt {a}}\right )-21 a^{\frac {7}{2}} \arctan \left (\frac {\sqrt {a \tanh \left (x \right )}}{\sqrt {a}}\right )-6 \left (a \tanh \left (x \right )\right )^{\frac {7}{2}}-14 a^{2} \left (a \tanh \left (x \right )\right )^{\frac {3}{2}}\right )}{21 \tanh \left (x \right )^{3} \left (a \tanh \left (x \right )\right )^{\frac {3}{2}} a^{2}}\)	\(76\)

Input:

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

Output:

1/21*(a*tanh(x)^3)^(3/2)*(21*a^(7/2)*arctanh((a*tanh(x))^(1/2)/a^(1/2))-21 
*a^(7/2)*arctan((a*tanh(x))^(1/2)/a^(1/2))-6*(a*tanh(x))^(7/2)-14*a^2*(a*t 
anh(x))^(3/2))/tanh(x)^3/(a*tanh(x))^(3/2)/a^2
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 615 vs. \(2 (66) = 132\).

Time = 0.12 (sec) , antiderivative size = 1267, normalized size of antiderivative = 14.73 \[ \int \left (a \tanh ^3(x)\right )^{3/2} \, dx=\text {Too large to display} \] Input:

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

Output:

[-1/84*(42*(a*cosh(x)^6 + 6*a*cosh(x)*sinh(x)^5 + a*sinh(x)^6 + 3*a*cosh(x 
)^4 + 3*(5*a*cosh(x)^2 + a)*sinh(x)^4 + 4*(5*a*cosh(x)^3 + 3*a*cosh(x))*si 
nh(x)^3 + 3*a*cosh(x)^2 + 3*(5*a*cosh(x)^4 + 6*a*cosh(x)^2 + a)*sinh(x)^2 
+ 6*(a*cosh(x)^5 + 2*a*cosh(x)^3 + a*cosh(x))*sinh(x) + a)*sqrt(-a)*arctan 
((cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 + 1)*sqrt(-a)*sqrt(a*sinh(x)/c 
osh(x))/(a*cosh(x)^2 + 2*a*cosh(x)*sinh(x) + a*sinh(x)^2)) - 21*(a*cosh(x) 
^6 + 6*a*cosh(x)*sinh(x)^5 + a*sinh(x)^6 + 3*a*cosh(x)^4 + 3*(5*a*cosh(x)^ 
2 + a)*sinh(x)^4 + 4*(5*a*cosh(x)^3 + 3*a*cosh(x))*sinh(x)^3 + 3*a*cosh(x) 
^2 + 3*(5*a*cosh(x)^4 + 6*a*cosh(x)^2 + a)*sinh(x)^2 + 6*(a*cosh(x)^5 + 2* 
a*cosh(x)^3 + a*cosh(x))*sinh(x) + a)*sqrt(-a)*log(-(a*cosh(x)^4 + 4*a*cos 
h(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*s 
inh(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)) + 16*(5*a*cosh(x)^6 + 30*a*cosh( 
x)*sinh(x)^5 + 5*a*sinh(x)^6 - a*cosh(x)^4 + (75*a*cosh(x)^2 - a)*sinh(x)^ 
4 + 4*(25*a*cosh(x)^3 - a*cosh(x))*sinh(x)^3 + a*cosh(x)^2 + (75*a*cosh(x) 
^4 - 6*a*cosh(x)^2 + a)*sinh(x)^2 + 2*(15*a*cosh(x)^5 - 2*a*cosh(x)^3 + a* 
cosh(x))*sinh(x) - 5*a)*sqrt(a*sinh(x)/cosh(x)))/(cosh(x)^6 + 6*cosh(x)*si 
nh(x)^5 + sinh(x)^6 + 3*(5*cosh(x)^2 + 1)*sinh(x)^4 + 3*cosh(x)^4 + 4*(5*c 
osh(x)^3 + 3*cosh(x))*sinh(x)^3 + 3*(5*cosh(x)^4 + 6*cosh(x)^2 + 1)*sin...
 

Sympy [F]

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

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

Output:

Integral((a*tanh(x)**3)**(3/2), x)
 

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 342 vs. \(2 (66) = 132\).

Time = 0.20 (sec) , antiderivative size = 342, normalized size of antiderivative = 3.98 \[ \int \left (a \tanh ^3(x)\right )^{3/2} \, dx=-\frac {1}{42} \, {\left (42 \, \sqrt {a} \arctan \left (-\frac {\sqrt {a} e^{\left (2 \, x\right )} - \sqrt {a e^{\left (4 \, x\right )} - a}}{\sqrt {a}}\right ) \mathrm {sgn}\left (e^{\left (4 \, x\right )} - 1\right ) + 21 \, \sqrt {a} \log \left ({\left | -\sqrt {a} e^{\left (2 \, x\right )} + \sqrt {a e^{\left (4 \, x\right )} - a} \right |}\right ) \mathrm {sgn}\left (e^{\left (4 \, x\right )} - 1\right ) + \frac {16 \, {\left (21 \, {\left (\sqrt {a} e^{\left (2 \, x\right )} - \sqrt {a e^{\left (4 \, x\right )} - a}\right )}^{6} a \mathrm {sgn}\left (e^{\left (4 \, x\right )} - 1\right ) + 42 \, {\left (\sqrt {a} e^{\left (2 \, x\right )} - \sqrt {a e^{\left (4 \, x\right )} - a}\right )}^{5} a^{\frac {3}{2}} \mathrm {sgn}\left (e^{\left (4 \, x\right )} - 1\right ) + 119 \, {\left (\sqrt {a} e^{\left (2 \, x\right )} - \sqrt {a e^{\left (4 \, x\right )} - a}\right )}^{4} a^{2} \mathrm {sgn}\left (e^{\left (4 \, x\right )} - 1\right ) + 56 \, {\left (\sqrt {a} e^{\left (2 \, x\right )} - \sqrt {a e^{\left (4 \, x\right )} - a}\right )}^{3} a^{\frac {5}{2}} \mathrm {sgn}\left (e^{\left (4 \, x\right )} - 1\right ) + 63 \, {\left (\sqrt {a} e^{\left (2 \, x\right )} - \sqrt {a e^{\left (4 \, x\right )} - a}\right )}^{2} a^{3} \mathrm {sgn}\left (e^{\left (4 \, x\right )} - 1\right ) + 14 \, {\left (\sqrt {a} e^{\left (2 \, x\right )} - \sqrt {a e^{\left (4 \, x\right )} - a}\right )} a^{\frac {7}{2}} \mathrm {sgn}\left (e^{\left (4 \, x\right )} - 1\right ) + 5 \, a^{4} \mathrm {sgn}\left (e^{\left (4 \, x\right )} - 1\right )\right )}}{{\left (\sqrt {a} e^{\left (2 \, x\right )} - \sqrt {a e^{\left (4 \, x\right )} - a} + \sqrt {a}\right )}^{7}}\right )} a \] Input:

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

Output:

-1/42*(42*sqrt(a)*arctan(-(sqrt(a)*e^(2*x) - sqrt(a*e^(4*x) - a))/sqrt(a)) 
*sgn(e^(4*x) - 1) + 21*sqrt(a)*log(abs(-sqrt(a)*e^(2*x) + sqrt(a*e^(4*x) - 
 a)))*sgn(e^(4*x) - 1) + 16*(21*(sqrt(a)*e^(2*x) - sqrt(a*e^(4*x) - a))^6* 
a*sgn(e^(4*x) - 1) + 42*(sqrt(a)*e^(2*x) - sqrt(a*e^(4*x) - a))^5*a^(3/2)* 
sgn(e^(4*x) - 1) + 119*(sqrt(a)*e^(2*x) - sqrt(a*e^(4*x) - a))^4*a^2*sgn(e 
^(4*x) - 1) + 56*(sqrt(a)*e^(2*x) - sqrt(a*e^(4*x) - a))^3*a^(5/2)*sgn(e^( 
4*x) - 1) + 63*(sqrt(a)*e^(2*x) - sqrt(a*e^(4*x) - a))^2*a^3*sgn(e^(4*x) - 
 1) + 14*(sqrt(a)*e^(2*x) - sqrt(a*e^(4*x) - a))*a^(7/2)*sgn(e^(4*x) - 1) 
+ 5*a^4*sgn(e^(4*x) - 1))/(sqrt(a)*e^(2*x) - sqrt(a*e^(4*x) - a) + sqrt(a) 
)^7)*a
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \left (a \tanh ^3(x)\right )^{3/2} \, dx=\sqrt {a}\, \left (\int \sqrt {\tanh \left (x \right )}\, \tanh \left (x \right )^{4}d x \right ) a \] Input:

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

Output:

sqrt(a)*int(sqrt(tanh(x))*tanh(x)**4,x)*a