\(\int \tanh ^2(x) (a+b \tanh ^2(x))^{3/2} \, dx\) [220]

Optimal result

Integrand size = 17, antiderivative size = 123 \[ \int \tanh ^2(x) \left (a+b \tanh ^2(x)\right )^{3/2} \, dx=-\frac {\left (3 a^2+12 a b+8 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \tanh (x)}{\sqrt {a+b \tanh ^2(x)}}\right )}{8 \sqrt {b}}+(a+b)^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b} \tanh (x)}{\sqrt {a+b \tanh ^2(x)}}\right )-\frac {1}{8} (5 a+4 b) \tanh (x) \sqrt {a+b \tanh ^2(x)}-\frac {1}{4} b \tanh ^3(x) \sqrt {a+b \tanh ^2(x)} \] Output:

-1/8*(3*a^2+12*a*b+8*b^2)*arctanh(b^(1/2)*tanh(x)/(a+b*tanh(x)^2)^(1/2))/b 
^(1/2)+(a+b)^(3/2)*arctanh((a+b)^(1/2)*tanh(x)/(a+b*tanh(x)^2)^(1/2))-1/8* 
(5*a+4*b)*tanh(x)*(a+b*tanh(x)^2)^(1/2)-1/4*b*tanh(x)^3*(a+b*tanh(x)^2)^(1 
/2)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 4.99 (sec) , antiderivative size = 247, normalized size of antiderivative = 2.01 \[ \int \tanh ^2(x) \left (a+b \tanh ^2(x)\right )^{3/2} \, dx=\frac {\left (4 \sqrt {2} a (5 a+4 b) \sqrt {\frac {(a-b+(a+b) \cosh (2 x)) \text {csch}^2(x)}{b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {(a-b+(a+b) \cosh (2 x)) \text {csch}^2(x)}{b}}}{\sqrt {2}}\right ),1\right )-32 \sqrt {2} a (a+b) \sqrt {\frac {(a-b+(a+b) \cosh (2 x)) \text {csch}^2(x)}{b}} \operatorname {EllipticPi}\left (\frac {b}{a+b},\arcsin \left (\frac {\sqrt {\frac {(a-b+(a+b) \cosh (2 x)) \text {csch}^2(x)}{b}}}{\sqrt {2}}\right ),1\right )-\left (15 a^2+5 a b+2 b^2+4 \left (5 a^2+4 a b-2 b^2\right ) \cosh (2 x)+\left (5 a^2+11 a b+6 b^2\right ) \cosh (4 x)\right ) \text {sech}^4(x)\right ) \tanh (x)}{32 \sqrt {2} \sqrt {(a-b+(a+b) \cosh (2 x)) \text {sech}^2(x)}} \] Input:

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

Output:

((4*Sqrt[2]*a*(5*a + 4*b)*Sqrt[((a - b + (a + b)*Cosh[2*x])*Csch[x]^2)/b]* 
EllipticF[ArcSin[Sqrt[((a - b + (a + b)*Cosh[2*x])*Csch[x]^2)/b]/Sqrt[2]], 
 1] - 32*Sqrt[2]*a*(a + b)*Sqrt[((a - b + (a + b)*Cosh[2*x])*Csch[x]^2)/b] 
*EllipticPi[b/(a + b), ArcSin[Sqrt[((a - b + (a + b)*Cosh[2*x])*Csch[x]^2) 
/b]/Sqrt[2]], 1] - (15*a^2 + 5*a*b + 2*b^2 + 4*(5*a^2 + 4*a*b - 2*b^2)*Cos 
h[2*x] + (5*a^2 + 11*a*b + 6*b^2)*Cosh[4*x])*Sech[x]^4)*Tanh[x])/(32*Sqrt[ 
2]*Sqrt[(a - b + (a + b)*Cosh[2*x])*Sech[x]^2])
 

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.07, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.765, Rules used = {3042, 25, 4153, 25, 379, 25, 444, 27, 398, 224, 219, 291, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 4153

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 379

\(\displaystyle -\frac {1}{4} \int -\frac {\tanh ^2(x) \left (b (5 a+4 b) \tanh ^2(x)+a (4 a+3 b)\right )}{\left (1-\tanh ^2(x)\right ) \sqrt {b \tanh ^2(x)+a}}d\tanh (x)-\frac {1}{4} b \tanh ^3(x) \sqrt {a+b \tanh ^2(x)}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{4} \int \frac {\tanh ^2(x) \left (b (5 a+4 b) \tanh ^2(x)+a (4 a+3 b)\right )}{\left (1-\tanh ^2(x)\right ) \sqrt {b \tanh ^2(x)+a}}d\tanh (x)-\frac {1}{4} b \tanh ^3(x) \sqrt {a+b \tanh ^2(x)}\)

\(\Big \downarrow \) 444

\(\displaystyle \frac {1}{4} \left (\frac {\int \frac {b \left (\left (3 a^2+12 b a+8 b^2\right ) \tanh ^2(x)+a (5 a+4 b)\right )}{\left (1-\tanh ^2(x)\right ) \sqrt {b \tanh ^2(x)+a}}d\tanh (x)}{2 b}-\frac {1}{2} (5 a+4 b) \tanh (x) \sqrt {a+b \tanh ^2(x)}\right )-\frac {1}{4} b \tanh ^3(x) \sqrt {a+b \tanh ^2(x)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{4} \left (\frac {1}{2} \int \frac {\left (3 a^2+12 b a+8 b^2\right ) \tanh ^2(x)+a (5 a+4 b)}{\left (1-\tanh ^2(x)\right ) \sqrt {b \tanh ^2(x)+a}}d\tanh (x)-\frac {1}{2} (5 a+4 b) \tanh (x) \sqrt {a+b \tanh ^2(x)}\right )-\frac {1}{4} b \tanh ^3(x) \sqrt {a+b \tanh ^2(x)}\)

\(\Big \downarrow \) 398

\(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (8 (a+b)^2 \int \frac {1}{\left (1-\tanh ^2(x)\right ) \sqrt {b \tanh ^2(x)+a}}d\tanh (x)-\left (3 a^2+12 a b+8 b^2\right ) \int \frac {1}{\sqrt {b \tanh ^2(x)+a}}d\tanh (x)\right )-\frac {1}{2} (5 a+4 b) \tanh (x) \sqrt {a+b \tanh ^2(x)}\right )-\frac {1}{4} b \tanh ^3(x) \sqrt {a+b \tanh ^2(x)}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (8 (a+b)^2 \int \frac {1}{\left (1-\tanh ^2(x)\right ) \sqrt {b \tanh ^2(x)+a}}d\tanh (x)-\left (3 a^2+12 a b+8 b^2\right ) \int \frac {1}{1-\frac {b \tanh ^2(x)}{b \tanh ^2(x)+a}}d\frac {\tanh (x)}{\sqrt {b \tanh ^2(x)+a}}\right )-\frac {1}{2} (5 a+4 b) \tanh (x) \sqrt {a+b \tanh ^2(x)}\right )-\frac {1}{4} b \tanh ^3(x) \sqrt {a+b \tanh ^2(x)}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (8 (a+b)^2 \int \frac {1}{\left (1-\tanh ^2(x)\right ) \sqrt {b \tanh ^2(x)+a}}d\tanh (x)-\frac {\left (3 a^2+12 a b+8 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \tanh (x)}{\sqrt {a+b \tanh ^2(x)}}\right )}{\sqrt {b}}\right )-\frac {1}{2} (5 a+4 b) \tanh (x) \sqrt {a+b \tanh ^2(x)}\right )-\frac {1}{4} b \tanh ^3(x) \sqrt {a+b \tanh ^2(x)}\)

\(\Big \downarrow \) 291

\(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (8 (a+b)^2 \int \frac {1}{1-\frac {(a+b) \tanh ^2(x)}{b \tanh ^2(x)+a}}d\frac {\tanh (x)}{\sqrt {b \tanh ^2(x)+a}}-\frac {\left (3 a^2+12 a b+8 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \tanh (x)}{\sqrt {a+b \tanh ^2(x)}}\right )}{\sqrt {b}}\right )-\frac {1}{2} (5 a+4 b) \tanh (x) \sqrt {a+b \tanh ^2(x)}\right )-\frac {1}{4} b \tanh ^3(x) \sqrt {a+b \tanh ^2(x)}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (8 (a+b)^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b} \tanh (x)}{\sqrt {a+b \tanh ^2(x)}}\right )-\frac {\left (3 a^2+12 a b+8 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \tanh (x)}{\sqrt {a+b \tanh ^2(x)}}\right )}{\sqrt {b}}\right )-\frac {1}{2} (5 a+4 b) \tanh (x) \sqrt {a+b \tanh ^2(x)}\right )-\frac {1}{4} b \tanh ^3(x) \sqrt {a+b \tanh ^2(x)}\)

Input:

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

Output:

-1/4*(b*Tanh[x]^3*Sqrt[a + b*Tanh[x]^2]) + ((-(((3*a^2 + 12*a*b + 8*b^2)*A 
rcTanh[(Sqrt[b]*Tanh[x])/Sqrt[a + b*Tanh[x]^2]])/Sqrt[b]) + 8*(a + b)^(3/2 
)*ArcTanh[(Sqrt[a + b]*Tanh[x])/Sqrt[a + b*Tanh[x]^2]])/2 - ((5*a + 4*b)*T 
anh[x]*Sqrt[a + b*Tanh[x]^2])/2)/4
 

Defintions of rubi rules used

rule 25

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

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

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 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 379

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ 
), x_Symbol] :> Simp[d*(e*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 
1)/(b*e*(m + 2*(p + q) + 1))), x] + Simp[1/(b*(m + 2*(p + q) + 1))   Int[(e 
*x)^m*(a + b*x^2)^p*(c + d*x^2)^(q - 2)*Simp[c*((b*c - a*d)*(m + 1) + b*c*2 
*(p + q)) + (d*(b*c - a*d)*(m + 1) + d*2*(q - 1)*(b*c - a*d) + b*c*d*2*(p + 
 q))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b*c - a*d, 0 
] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, e, m, 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 444

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q 
_.)*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[f*g*(g*x)^(m - 1)*(a + b*x^2)^ 
(p + 1)*((c + d*x^2)^(q + 1)/(b*d*(m + 2*(p + q + 1) + 1))), x] - Simp[g^2/ 
(b*d*(m + 2*(p + q + 1) + 1))   Int[(g*x)^(m - 2)*(a + b*x^2)^p*(c + d*x^2) 
^q*Simp[a*f*c*(m - 1) + (a*f*d*(m + 2*q + 1) + b*(f*c*(m + 2*p + 1) - e*d*( 
m + 2*(p + q + 1) + 1)))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, 
q}, x] && GtQ[m, 1]
 

rule 3042

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

rule 4153

Int[((d_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((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[(d*ff*(x/c))^m*((a + b*(ff*x)^n)^p/(c^2 + f 
f^2*x^2)), x], x, c*(Tan[e + f*x]/ff)], x]] /; FreeQ[{a, b, c, d, e, f, m, 
n, p}, x] && (IGtQ[p, 0] || EqQ[n, 2] || EqQ[n, 4] || (IntegerQ[p] && Ratio 
nalQ[n]))
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(528\) vs. \(2(101)=202\).

Time = 0.05 (sec) , antiderivative size = 529, normalized size of antiderivative = 4.30

method	result	size
derivativedivides	\(-\frac {\tanh \left (x \right ) \left (a +b \tanh \left (x \right )^{2}\right )^{\frac {3}{2}}}{4}-\frac {3 a \left (\frac {\tanh \left (x \right ) \sqrt {a +b \tanh \left (x \right )^{2}}}{2}+\frac {a \ln \left (\sqrt {b}\, \tanh \left (x \right )+\sqrt {a +b \tanh \left (x \right )^{2}}\right )}{2 \sqrt {b}}\right )}{4}-\frac {\left (b \left (\tanh \left (x \right )-1\right )^{2}+2 b \left (\tanh \left (x \right )-1\right )+a +b \right )^{\frac {3}{2}}}{6}-\frac {b \left (\frac {\left (2 b \left (\tanh \left (x \right )-1\right )+2 b \right ) \sqrt {b \left (\tanh \left (x \right )-1\right )^{2}+2 b \left (\tanh \left (x \right )-1\right )+a +b}}{4 b}+\frac {\left (4 b \left (a +b \right )-4 b^{2}\right ) \ln \left (\frac {b \left (\tanh \left (x \right )-1\right )+b}{\sqrt {b}}+\sqrt {b \left (\tanh \left (x \right )-1\right )^{2}+2 b \left (\tanh \left (x \right )-1\right )+a +b}\right )}{8 b^{\frac {3}{2}}}\right )}{2}-\frac {\left (a +b \right ) \left (\sqrt {b \left (\tanh \left (x \right )-1\right )^{2}+2 b \left (\tanh \left (x \right )-1\right )+a +b}+\sqrt {b}\, \ln \left (\frac {b \left (\tanh \left (x \right )-1\right )+b}{\sqrt {b}}+\sqrt {b \left (\tanh \left (x \right )-1\right )^{2}+2 b \left (\tanh \left (x \right )-1\right )+a +b}\right )-\sqrt {a +b}\, \ln \left (\frac {2 a +2 b +2 b \left (\tanh \left (x \right )-1\right )+2 \sqrt {a +b}\, \sqrt {b \left (\tanh \left (x \right )-1\right )^{2}+2 b \left (\tanh \left (x \right )-1\right )+a +b}}{\tanh \left (x \right )-1}\right )\right )}{2}+\frac {\left (b \left (\tanh \left (x \right )+1\right )^{2}-2 b \left (\tanh \left (x \right )+1\right )+a +b \right )^{\frac {3}{2}}}{6}-\frac {b \left (\frac {\left (2 b \left (\tanh \left (x \right )+1\right )-2 b \right ) \sqrt {b \left (\tanh \left (x \right )+1\right )^{2}-2 b \left (\tanh \left (x \right )+1\right )+a +b}}{4 b}+\frac {\left (4 b \left (a +b \right )-4 b^{2}\right ) \ln \left (\frac {b \left (\tanh \left (x \right )+1\right )-b}{\sqrt {b}}+\sqrt {b \left (\tanh \left (x \right )+1\right )^{2}-2 b \left (\tanh \left (x \right )+1\right )+a +b}\right )}{8 b^{\frac {3}{2}}}\right )}{2}+\frac {\left (a +b \right ) \left (\sqrt {b \left (\tanh \left (x \right )+1\right )^{2}-2 b \left (\tanh \left (x \right )+1\right )+a +b}-\sqrt {b}\, \ln \left (\frac {b \left (\tanh \left (x \right )+1\right )-b}{\sqrt {b}}+\sqrt {b \left (\tanh \left (x \right )+1\right )^{2}-2 b \left (\tanh \left (x \right )+1\right )+a +b}\right )-\sqrt {a +b}\, \ln \left (\frac {2 a +2 b -2 b \left (\tanh \left (x \right )+1\right )+2 \sqrt {a +b}\, \sqrt {b \left (\tanh \left (x \right )+1\right )^{2}-2 b \left (\tanh \left (x \right )+1\right )+a +b}}{\tanh \left (x \right )+1}\right )\right )}{2}\)	\(529\)
default	\(-\frac {\tanh \left (x \right ) \left (a +b \tanh \left (x \right )^{2}\right )^{\frac {3}{2}}}{4}-\frac {3 a \left (\frac {\tanh \left (x \right ) \sqrt {a +b \tanh \left (x \right )^{2}}}{2}+\frac {a \ln \left (\sqrt {b}\, \tanh \left (x \right )+\sqrt {a +b \tanh \left (x \right )^{2}}\right )}{2 \sqrt {b}}\right )}{4}-\frac {\left (b \left (\tanh \left (x \right )-1\right )^{2}+2 b \left (\tanh \left (x \right )-1\right )+a +b \right )^{\frac {3}{2}}}{6}-\frac {b \left (\frac {\left (2 b \left (\tanh \left (x \right )-1\right )+2 b \right ) \sqrt {b \left (\tanh \left (x \right )-1\right )^{2}+2 b \left (\tanh \left (x \right )-1\right )+a +b}}{4 b}+\frac {\left (4 b \left (a +b \right )-4 b^{2}\right ) \ln \left (\frac {b \left (\tanh \left (x \right )-1\right )+b}{\sqrt {b}}+\sqrt {b \left (\tanh \left (x \right )-1\right )^{2}+2 b \left (\tanh \left (x \right )-1\right )+a +b}\right )}{8 b^{\frac {3}{2}}}\right )}{2}-\frac {\left (a +b \right ) \left (\sqrt {b \left (\tanh \left (x \right )-1\right )^{2}+2 b \left (\tanh \left (x \right )-1\right )+a +b}+\sqrt {b}\, \ln \left (\frac {b \left (\tanh \left (x \right )-1\right )+b}{\sqrt {b}}+\sqrt {b \left (\tanh \left (x \right )-1\right )^{2}+2 b \left (\tanh \left (x \right )-1\right )+a +b}\right )-\sqrt {a +b}\, \ln \left (\frac {2 a +2 b +2 b \left (\tanh \left (x \right )-1\right )+2 \sqrt {a +b}\, \sqrt {b \left (\tanh \left (x \right )-1\right )^{2}+2 b \left (\tanh \left (x \right )-1\right )+a +b}}{\tanh \left (x \right )-1}\right )\right )}{2}+\frac {\left (b \left (\tanh \left (x \right )+1\right )^{2}-2 b \left (\tanh \left (x \right )+1\right )+a +b \right )^{\frac {3}{2}}}{6}-\frac {b \left (\frac {\left (2 b \left (\tanh \left (x \right )+1\right )-2 b \right ) \sqrt {b \left (\tanh \left (x \right )+1\right )^{2}-2 b \left (\tanh \left (x \right )+1\right )+a +b}}{4 b}+\frac {\left (4 b \left (a +b \right )-4 b^{2}\right ) \ln \left (\frac {b \left (\tanh \left (x \right )+1\right )-b}{\sqrt {b}}+\sqrt {b \left (\tanh \left (x \right )+1\right )^{2}-2 b \left (\tanh \left (x \right )+1\right )+a +b}\right )}{8 b^{\frac {3}{2}}}\right )}{2}+\frac {\left (a +b \right ) \left (\sqrt {b \left (\tanh \left (x \right )+1\right )^{2}-2 b \left (\tanh \left (x \right )+1\right )+a +b}-\sqrt {b}\, \ln \left (\frac {b \left (\tanh \left (x \right )+1\right )-b}{\sqrt {b}}+\sqrt {b \left (\tanh \left (x \right )+1\right )^{2}-2 b \left (\tanh \left (x \right )+1\right )+a +b}\right )-\sqrt {a +b}\, \ln \left (\frac {2 a +2 b -2 b \left (\tanh \left (x \right )+1\right )+2 \sqrt {a +b}\, \sqrt {b \left (\tanh \left (x \right )+1\right )^{2}-2 b \left (\tanh \left (x \right )+1\right )+a +b}}{\tanh \left (x \right )+1}\right )\right )}{2}\)	\(529\)

Input:

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

Output:

-1/4*tanh(x)*(a+b*tanh(x)^2)^(3/2)-3/4*a*(1/2*tanh(x)*(a+b*tanh(x)^2)^(1/2 
)+1/2*a/b^(1/2)*ln(b^(1/2)*tanh(x)+(a+b*tanh(x)^2)^(1/2)))-1/6*(b*(tanh(x) 
-1)^2+2*b*(tanh(x)-1)+a+b)^(3/2)-1/2*b*(1/4*(2*b*(tanh(x)-1)+2*b)/b*(b*(ta 
nh(x)-1)^2+2*b*(tanh(x)-1)+a+b)^(1/2)+1/8*(4*b*(a+b)-4*b^2)/b^(3/2)*ln((b* 
(tanh(x)-1)+b)/b^(1/2)+(b*(tanh(x)-1)^2+2*b*(tanh(x)-1)+a+b)^(1/2)))-1/2*( 
a+b)*((b*(tanh(x)-1)^2+2*b*(tanh(x)-1)+a+b)^(1/2)+b^(1/2)*ln((b*(tanh(x)-1 
)+b)/b^(1/2)+(b*(tanh(x)-1)^2+2*b*(tanh(x)-1)+a+b)^(1/2))-(a+b)^(1/2)*ln(( 
2*a+2*b+2*b*(tanh(x)-1)+2*(a+b)^(1/2)*(b*(tanh(x)-1)^2+2*b*(tanh(x)-1)+a+b 
)^(1/2))/(tanh(x)-1)))+1/6*(b*(tanh(x)+1)^2-2*b*(tanh(x)+1)+a+b)^(3/2)-1/2 
*b*(1/4*(2*b*(tanh(x)+1)-2*b)/b*(b*(tanh(x)+1)^2-2*b*(tanh(x)+1)+a+b)^(1/2 
)+1/8*(4*b*(a+b)-4*b^2)/b^(3/2)*ln((b*(tanh(x)+1)-b)/b^(1/2)+(b*(tanh(x)+1 
)^2-2*b*(tanh(x)+1)+a+b)^(1/2)))+1/2*(a+b)*((b*(tanh(x)+1)^2-2*b*(tanh(x)+ 
1)+a+b)^(1/2)-b^(1/2)*ln((b*(tanh(x)+1)-b)/b^(1/2)+(b*(tanh(x)+1)^2-2*b*(t 
anh(x)+1)+a+b)^(1/2))-(a+b)^(1/2)*ln((2*a+2*b-2*b*(tanh(x)+1)+2*(a+b)^(1/2 
)*(b*(tanh(x)+1)^2-2*b*(tanh(x)+1)+a+b)^(1/2))/(tanh(x)+1)))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2167 vs. \(2 (101) = 202\).

Time = 0.43 (sec) , antiderivative size = 10046, normalized size of antiderivative = 81.67 \[ \int \tanh ^2(x) \left (a+b \tanh ^2(x)\right )^{3/2} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 949 vs. \(2 (101) = 202\).

Time = 0.98 (sec) , antiderivative size = 949, normalized size of antiderivative = 7.72 \[ \int \tanh ^2(x) \left (a+b \tanh ^2(x)\right )^{3/2} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/2*(a + b)^(3/2)*log(abs(-sqrt(a + b)*e^(2*x) + sqrt(a*e^(4*x) + b*e^(4* 
x) + 2*a*e^(2*x) - 2*b*e^(2*x) + a + b) + sqrt(a + b))) + 1/2*(a + b)^(3/2 
)*log(abs(-sqrt(a + b)*e^(2*x) + sqrt(a*e^(4*x) + b*e^(4*x) + 2*a*e^(2*x) 
- 2*b*e^(2*x) + a + b) - sqrt(a + b))) - 1/4*(3*a^2 + 12*a*b + 8*b^2)*arct 
an(-1/2*(sqrt(a + b)*e^(2*x) - sqrt(a*e^(4*x) + b*e^(4*x) + 2*a*e^(2*x) - 
2*b*e^(2*x) + a + b) + sqrt(a + b))/sqrt(-b))/sqrt(-b) - 1/2*(a^2 + 2*a*b 
+ b^2)*log(abs(-(sqrt(a + b)*e^(2*x) - sqrt(a*e^(4*x) + b*e^(4*x) + 2*a*e^ 
(2*x) - 2*b*e^(2*x) + a + b))*(a + b) - sqrt(a + b)*(a - b)))/sqrt(a + b) 
- 1/2*((5*a^2 + 20*a*b + 16*b^2)*(sqrt(a + b)*e^(2*x) - sqrt(a*e^(4*x) + b 
*e^(4*x) + 2*a*e^(2*x) - 2*b*e^(2*x) + a + b))^7 + (35*a^2 + 76*a*b + 16*b 
^2)*(sqrt(a + b)*e^(2*x) - sqrt(a*e^(4*x) + b*e^(4*x) + 2*a*e^(2*x) - 2*b* 
e^(2*x) + a + b))^6*sqrt(a + b) + (105*a^3 + 153*a^2*b - 28*a*b^2 - 48*b^3 
)*(sqrt(a + b)*e^(2*x) - sqrt(a*e^(4*x) + b*e^(4*x) + 2*a*e^(2*x) - 2*b*e^ 
(2*x) + a + b))^5 + (175*a^3 - 25*a^2*b - 260*a*b^2 - 176*b^3)*(sqrt(a + b 
)*e^(2*x) - sqrt(a*e^(4*x) + b*e^(4*x) + 2*a*e^(2*x) - 2*b*e^(2*x) + a + b 
))^4*sqrt(a + b) + (175*a^4 - 110*a^3*b - 417*a^2*b^2 + 60*a*b^3 + 304*b^4 
)*(sqrt(a + b)*e^(2*x) - sqrt(a*e^(4*x) + b*e^(4*x) + 2*a*e^(2*x) - 2*b*e^ 
(2*x) + a + b))^3 + (105*a^4 - 210*a^3*b - 55*a^2*b^2 + 484*a*b^3 + 48*b^4 
)*(sqrt(a + b)*e^(2*x) - sqrt(a*e^(4*x) + b*e^(4*x) + 2*a*e^(2*x) - 2*b*e^ 
(2*x) + a + b))^2*sqrt(a + b) + (35*a^5 - 79*a^4*b + 53*a^3*b^2 + 195*a...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int(sqrt(tanh(x)**2*b + a)*tanh(x)**4,x)*b + int(sqrt(tanh(x)**2*b + a)*ta 
nh(x)**2,x)*a