\(\int (a+b \tanh ^2(x))^{3/2} \, dx\) [222]

Optimal result

Integrand size = 12, antiderivative size = 88 \[ \int \left (a+b \tanh ^2(x)\right )^{3/2} \, dx=-\frac {1}{2} \sqrt {b} (3 a+2 b) \text {arctanh}\left (\frac {\sqrt {b} \tanh (x)}{\sqrt {a+b \tanh ^2(x)}}\right )+(a+b)^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b} \tanh (x)}{\sqrt {a+b \tanh ^2(x)}}\right )-\frac {1}{2} b \tanh (x) \sqrt {a+b \tanh ^2(x)} \] Output:

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

Mathematica [A] (verified)

Time = 0.43 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.25 \[ \int \left (a+b \tanh ^2(x)\right )^{3/2} \, dx=\frac {1}{2} \left (-2 (-a-b)^{3/2} \arctan \left (\frac {\sqrt {b} \text {sech}^2(x)+\tanh (x) \sqrt {a+b \tanh ^2(x)}}{\sqrt {-a-b}}\right )+\sqrt {b} (3 a+2 b) \log \left (-\sqrt {b} \tanh (x)+\sqrt {a+b \tanh ^2(x)}\right )-b \tanh (x) \sqrt {a+b \tanh ^2(x)}\right ) \] Input:

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

Output:

(-2*(-a - b)^(3/2)*ArcTan[(Sqrt[b]*Sech[x]^2 + Tanh[x]*Sqrt[a + b*Tanh[x]^ 
2])/Sqrt[-a - b]] + Sqrt[b]*(3*a + 2*b)*Log[-(Sqrt[b]*Tanh[x]) + Sqrt[a + 
b*Tanh[x]^2]] - b*Tanh[x]*Sqrt[a + b*Tanh[x]^2])/2
 

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.05, 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, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4144

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

\(\Big \downarrow \) 318

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 398

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], 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 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. \(472\) vs. \(2(70)=140\).

Time = 0.04 (sec) , antiderivative size = 473, normalized size of antiderivative = 5.38

method	result	size
derivativedivides	\(-\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}\)	\(473\)
default	\(-\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}\)	\(473\)

Input:

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

Output:

-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*(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*(tan 
h(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) 
))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 865 vs. \(2 (70) = 140\).

Time = 0.28 (sec) , antiderivative size = 4841, normalized size of antiderivative = 55.01 \[ \int \left (a+b \tanh ^2(x)\right )^{3/2} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 584 vs. \(2 (70) = 140\).

Time = 0.61 (sec) , antiderivative size = 584, normalized size of antiderivative = 6.64 \[ \int \left (a+b \tanh ^2(x)\right )^{3/2} \, dx =\text {Too large to display} \] Input:

integrate((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))) - (3*a*b + 2*b^2)*arctan(-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(ab 
s(-(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) - 2*((a*b + 2 
*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))^3 + (3*a*b - 2*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))^2*sqrt(a + b) + (3*a 
^2*b - 3*a*b^2 - 2*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)) + (a^2*b - a*b^2 + 2*b^3)*sqrt(a + b 
))/((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 + 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) + a - 3*b)^2
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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