Integrand size = 15, antiderivative size = 63 \[ \int \tanh (x) \left (a+b \tanh ^2(x)\right )^{3/2} \, dx=(a+b)^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b \tanh ^2(x)}}{\sqrt {a+b}}\right )-(a+b) \sqrt {a+b \tanh ^2(x)}-\frac {1}{3} \left (a+b \tanh ^2(x)\right )^{3/2} \]
(a+b)^(3/2)*arctanh((a+b*tanh(x)^2)^(1/2)/(a+b)^(1/2))-(a+b)*(a+b*tanh(x)^ 2)^(1/2)-1/3*(a+b*tanh(x)^2)^(3/2)
Time = 0.19 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.94 \[ \int \tanh (x) \left (a+b \tanh ^2(x)\right )^{3/2} \, dx=(a+b)^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b \tanh ^2(x)}}{\sqrt {a+b}}\right )-\frac {1}{3} \sqrt {a+b \tanh ^2(x)} \left (4 a+3 b+b \tanh ^2(x)\right ) \]
(a + b)^(3/2)*ArcTanh[Sqrt[a + b*Tanh[x]^2]/Sqrt[a + b]] - (Sqrt[a + b*Tan h[x]^2]*(4*a + 3*b + b*Tanh[x]^2))/3
Time = 0.27 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.11, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {3042, 26, 4153, 26, 353, 60, 60, 73, 221}
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.
| \(\displaystyle \int \tanh (x) \left (a+b \tanh ^2(x)\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -i \tan (i x) \left (a-b \tan (i x)^2\right )^{3/2}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \int \tan (i x) \left (a-b \tan (i x)^2\right )^{3/2}dx\) |
| \(\Big \downarrow \) 4153 |
| \(\displaystyle -i \int \frac {i \tanh (x) \left (b \tanh ^2(x)+a\right )^{3/2}}{1-\tanh ^2(x)}d\tanh (x)\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \int \frac {\tanh (x) \left (a+b \tanh ^2(x)\right )^{3/2}}{1-\tanh ^2(x)}d\tanh (x)\) |
| \(\Big \downarrow \) 353 |
| \(\displaystyle \frac {1}{2} \int \frac {\left (b \tanh ^2(x)+a\right )^{3/2}}{1-\tanh ^2(x)}d\tanh ^2(x)\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {1}{2} \left ((a+b) \int \frac {\sqrt {b \tanh ^2(x)+a}}{1-\tanh ^2(x)}d\tanh ^2(x)-\frac {2}{3} \left (a+b \tanh ^2(x)\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {1}{2} \left ((a+b) \left ((a+b) \int \frac {1}{\left (1-\tanh ^2(x)\right ) \sqrt {b \tanh ^2(x)+a}}d\tanh ^2(x)-2 \sqrt {a+b \tanh ^2(x)}\right )-\frac {2}{3} \left (a+b \tanh ^2(x)\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{2} \left ((a+b) \left (\frac {2 (a+b) \int \frac {1}{\frac {a+b}{b}-\frac {\tanh ^4(x)}{b}}d\sqrt {b \tanh ^2(x)+a}}{b}-2 \sqrt {a+b \tanh ^2(x)}\right )-\frac {2}{3} \left (a+b \tanh ^2(x)\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {1}{2} \left ((a+b) \left (2 \sqrt {a+b} \text {arctanh}\left (\frac {\sqrt {a+b \tanh ^2(x)}}{\sqrt {a+b}}\right )-2 \sqrt {a+b \tanh ^2(x)}\right )-\frac {2}{3} \left (a+b \tanh ^2(x)\right )^{3/2}\right )\) |
((-2*(a + b*Tanh[x]^2)^(3/2))/3 + (a + b)*(2*Sqrt[a + b]*ArcTanh[Sqrt[a + b*Tanh[x]^2]/Sqrt[a + b]] - 2*Sqrt[a + b*Tanh[x]^2]))/2
3.3.21.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_Symbol] :> Simp[1/2 Subst[Int[(a + b*x)^p*(c + d*x)^q, x], x, x^2], x] /; FreeQ[ {a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0]
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]))
Leaf count of result is larger than twice the leaf count of optimal. \(472\) vs. \(2(51)=102\).
Time = 0.06 (sec) , antiderivative size = 473, normalized size of antiderivative = 7.51
| 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 \left (a +b \right ) b -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 (1+\tanh \left (x \right )\right )^{2}-2 b \left (1+\tanh \left (x \right )\right )+a +b \right )^{\frac {3}{2}}}{6}+\frac {b \left (\frac {\left (2 b \left (1+\tanh \left (x \right )\right )-2 b \right ) \sqrt {b \left (1+\tanh \left (x \right )\right )^{2}-2 b \left (1+\tanh \left (x \right )\right )+a +b}}{4 b}+\frac {\left (4 \left (a +b \right ) b -4 b^{2}\right ) \ln \left (\frac {b \left (1+\tanh \left (x \right )\right )-b}{\sqrt {b}}+\sqrt {b \left (1+\tanh \left (x \right )\right )^{2}-2 b \left (1+\tanh \left (x \right )\right )+a +b}\right )}{8 b^{\frac {3}{2}}}\right )}{2}-\frac {\left (a +b \right ) \left (\sqrt {b \left (1+\tanh \left (x \right )\right )^{2}-2 b \left (1+\tanh \left (x \right )\right )+a +b}-\sqrt {b}\, \ln \left (\frac {b \left (1+\tanh \left (x \right )\right )-b}{\sqrt {b}}+\sqrt {b \left (1+\tanh \left (x \right )\right )^{2}-2 b \left (1+\tanh \left (x \right )\right )+a +b}\right )-\sqrt {a +b}\, \ln \left (\frac {2 a +2 b -2 b \left (1+\tanh \left (x \right )\right )+2 \sqrt {a +b}\, \sqrt {b \left (1+\tanh \left (x \right )\right )^{2}-2 b \left (1+\tanh \left (x \right )\right )+a +b}}{1+\tanh \left (x \right )}\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 \left (a +b \right ) b -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 (1+\tanh \left (x \right )\right )^{2}-2 b \left (1+\tanh \left (x \right )\right )+a +b \right )^{\frac {3}{2}}}{6}+\frac {b \left (\frac {\left (2 b \left (1+\tanh \left (x \right )\right )-2 b \right ) \sqrt {b \left (1+\tanh \left (x \right )\right )^{2}-2 b \left (1+\tanh \left (x \right )\right )+a +b}}{4 b}+\frac {\left (4 \left (a +b \right ) b -4 b^{2}\right ) \ln \left (\frac {b \left (1+\tanh \left (x \right )\right )-b}{\sqrt {b}}+\sqrt {b \left (1+\tanh \left (x \right )\right )^{2}-2 b \left (1+\tanh \left (x \right )\right )+a +b}\right )}{8 b^{\frac {3}{2}}}\right )}{2}-\frac {\left (a +b \right ) \left (\sqrt {b \left (1+\tanh \left (x \right )\right )^{2}-2 b \left (1+\tanh \left (x \right )\right )+a +b}-\sqrt {b}\, \ln \left (\frac {b \left (1+\tanh \left (x \right )\right )-b}{\sqrt {b}}+\sqrt {b \left (1+\tanh \left (x \right )\right )^{2}-2 b \left (1+\tanh \left (x \right )\right )+a +b}\right )-\sqrt {a +b}\, \ln \left (\frac {2 a +2 b -2 b \left (1+\tanh \left (x \right )\right )+2 \sqrt {a +b}\, \sqrt {b \left (1+\tanh \left (x \right )\right )^{2}-2 b \left (1+\tanh \left (x \right )\right )+a +b}}{1+\tanh \left (x \right )}\right )\right )}{2}\) | \(473\) |
-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*(a+b)*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*(1+tanh(x))^2-2*b*(1+tanh(x) )+a+b)^(3/2)+1/2*b*(1/4*(2*b*(1+tanh(x))-2*b)/b*(b*(1+tanh(x))^2-2*b*(1+ta nh(x))+a+b)^(1/2)+1/8*(4*(a+b)*b-4*b^2)/b^(3/2)*ln((b*(1+tanh(x))-b)/b^(1/ 2)+(b*(1+tanh(x))^2-2*b*(1+tanh(x))+a+b)^(1/2)))-1/2*(a+b)*((b*(1+tanh(x)) ^2-2*b*(1+tanh(x))+a+b)^(1/2)-b^(1/2)*ln((b*(1+tanh(x))-b)/b^(1/2)+(b*(1+t anh(x))^2-2*b*(1+tanh(x))+a+b)^(1/2))-(a+b)^(1/2)*ln((2*a+2*b-2*b*(1+tanh( x))+2*(a+b)^(1/2)*(b*(1+tanh(x))^2-2*b*(1+tanh(x))+a+b)^(1/2))/(1+tanh(x)) ))
Leaf count of result is larger than twice the leaf count of optimal. 910 vs. \(2 (51) = 102\).
Time = 0.37 (sec) , antiderivative size = 2385, normalized size of antiderivative = 37.86 \[ \int \tanh (x) \left (a+b \tanh ^2(x)\right )^{3/2} \, dx=\text {Too large to display} \]
[1/12*(3*((a + b)*cosh(x)^6 + 6*(a + b)*cosh(x)*sinh(x)^5 + (a + b)*sinh(x )^6 + 3*(a + b)*cosh(x)^4 + 3*(5*(a + b)*cosh(x)^2 + a + b)*sinh(x)^4 + 4* (5*(a + b)*cosh(x)^3 + 3*(a + b)*cosh(x))*sinh(x)^3 + 3*(a + b)*cosh(x)^2 + 3*(5*(a + b)*cosh(x)^4 + 6*(a + b)*cosh(x)^2 + a + b)*sinh(x)^2 + 6*((a + b)*cosh(x)^5 + 2*(a + b)*cosh(x)^3 + (a + b)*cosh(x))*sinh(x) + a + b)*s qrt(a + b)*log(((a^3 + a^2*b)*cosh(x)^8 + 8*(a^3 + a^2*b)*cosh(x)*sinh(x)^ 7 + (a^3 + a^2*b)*sinh(x)^8 + 2*(2*a^3 + a^2*b)*cosh(x)^6 + 2*(2*a^3 + a^2 *b + 14*(a^3 + a^2*b)*cosh(x)^2)*sinh(x)^6 + 4*(14*(a^3 + a^2*b)*cosh(x)^3 + 3*(2*a^3 + a^2*b)*cosh(x))*sinh(x)^5 + (6*a^3 + 4*a^2*b - a*b^2 + b^3)* cosh(x)^4 + (70*(a^3 + a^2*b)*cosh(x)^4 + 6*a^3 + 4*a^2*b - a*b^2 + b^3 + 30*(2*a^3 + a^2*b)*cosh(x)^2)*sinh(x)^4 + 4*(14*(a^3 + a^2*b)*cosh(x)^5 + 10*(2*a^3 + a^2*b)*cosh(x)^3 + (6*a^3 + 4*a^2*b - a*b^2 + b^3)*cosh(x))*si nh(x)^3 + a^3 + 3*a^2*b + 3*a*b^2 + b^3 + 2*(2*a^3 + 3*a^2*b - b^3)*cosh(x )^2 + 2*(14*(a^3 + a^2*b)*cosh(x)^6 + 15*(2*a^3 + a^2*b)*cosh(x)^4 + 2*a^3 + 3*a^2*b - b^3 + 3*(6*a^3 + 4*a^2*b - a*b^2 + b^3)*cosh(x)^2)*sinh(x)^2 + sqrt(2)*(a^2*cosh(x)^6 + 6*a^2*cosh(x)*sinh(x)^5 + a^2*sinh(x)^6 + 3*a^2 *cosh(x)^4 + 3*(5*a^2*cosh(x)^2 + a^2)*sinh(x)^4 + 4*(5*a^2*cosh(x)^3 + 3* a^2*cosh(x))*sinh(x)^3 + (3*a^2 + 2*a*b - b^2)*cosh(x)^2 + (15*a^2*cosh(x) ^4 + 18*a^2*cosh(x)^2 + 3*a^2 + 2*a*b - b^2)*sinh(x)^2 + a^2 + 2*a*b + b^2 + 2*(3*a^2*cosh(x)^5 + 6*a^2*cosh(x)^3 + (3*a^2 + 2*a*b - b^2)*cosh(x)...
Time = 6.97 (sec) , antiderivative size = 165, normalized size of antiderivative = 2.62 \[ \int \tanh (x) \left (a+b \tanh ^2(x)\right )^{3/2} \, dx=- a \left (\begin {cases} \frac {2 \left (\frac {b \sqrt {a + b \tanh ^{2}{\left (x \right )}}}{2} + \frac {b \left (a + b\right ) \operatorname {atan}{\left (\frac {\sqrt {a + b \tanh ^{2}{\left (x \right )}}}{\sqrt {- a - b}} \right )}}{2 \sqrt {- a - b}}\right )}{b} & \text {for}\: b \neq 0 \\\frac {\sqrt {a} \log {\left (2 \tanh ^{2}{\left (x \right )} - 2 \right )}}{2} & \text {otherwise} \end {cases}\right ) - b \left (\begin {cases} \frac {2 \left (\frac {b^{2} \sqrt {a + b \tanh ^{2}{\left (x \right )}}}{2} + \frac {b^{2} \left (a + b\right ) \operatorname {atan}{\left (\frac {\sqrt {a + b \tanh ^{2}{\left (x \right )}}}{\sqrt {- a - b}} \right )}}{2 \sqrt {- a - b}} + \frac {b \left (a + b \tanh ^{2}{\left (x \right )}\right )^{\frac {3}{2}}}{6}\right )}{b^{2}} & \text {for}\: b \neq 0 \\\sqrt {a} \left (\frac {\log {\left (\tanh ^{2}{\left (x \right )} - 1 \right )}}{2} + \frac {\tanh ^{2}{\left (x \right )}}{2}\right ) & \text {otherwise} \end {cases}\right ) \]
-a*Piecewise((2*(b*sqrt(a + b*tanh(x)**2)/2 + b*(a + b)*atan(sqrt(a + b*ta nh(x)**2)/sqrt(-a - b))/(2*sqrt(-a - b)))/b, Ne(b, 0)), (sqrt(a)*log(2*tan h(x)**2 - 2)/2, True)) - b*Piecewise((2*(b**2*sqrt(a + b*tanh(x)**2)/2 + b **2*(a + b)*atan(sqrt(a + b*tanh(x)**2)/sqrt(-a - b))/(2*sqrt(-a - b)) + b *(a + b*tanh(x)**2)**(3/2)/6)/b**2, Ne(b, 0)), (sqrt(a)*(log(tanh(x)**2 - 1)/2 + tanh(x)**2/2), True))
\[ \int \tanh (x) \left (a+b \tanh ^2(x)\right )^{3/2} \, dx=\int { {\left (b \tanh \left (x\right )^{2} + a\right )}^{\frac {3}{2}} \tanh \left (x\right ) \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 662 vs. \(2 (51) = 102\).
Time = 1.02 (sec) , antiderivative size = 662, normalized size of antiderivative = 10.51 \[ \int \tanh (x) \left (a+b \tanh ^2(x)\right )^{3/2} \, dx=\frac {1}{2} \, {\left (a + b\right )}^{\frac {3}{2}} \log \left ({\left | -\sqrt {a + b} e^{\left (2 \, x\right )} + \sqrt {a e^{\left (4 \, x\right )} + b e^{\left (4 \, x\right )} + 2 \, a e^{\left (2 \, x\right )} - 2 \, b e^{\left (2 \, x\right )} + a + b} + \sqrt {a + b} \right |}\right ) - \frac {1}{2} \, {\left (a + b\right )}^{\frac {3}{2}} \log \left ({\left | -\sqrt {a + b} e^{\left (2 \, x\right )} + \sqrt {a e^{\left (4 \, x\right )} + b e^{\left (4 \, x\right )} + 2 \, a e^{\left (2 \, x\right )} - 2 \, b e^{\left (2 \, x\right )} + a + b} - \sqrt {a + b} \right |}\right ) - \frac {{\left (a^{2} + 2 \, a b + b^{2}\right )} \log \left ({\left | -{\left (\sqrt {a + b} e^{\left (2 \, x\right )} - \sqrt {a e^{\left (4 \, x\right )} + b e^{\left (4 \, x\right )} + 2 \, a e^{\left (2 \, x\right )} - 2 \, b e^{\left (2 \, x\right )} + a + b}\right )} {\left (a + b\right )} - \sqrt {a + b} {\left (a - b\right )} \right |}\right )}{2 \, \sqrt {a + b}} - \frac {8 \, {\left (3 \, {\left (a b + b^{2}\right )} {\left (\sqrt {a + b} e^{\left (2 \, x\right )} - \sqrt {a e^{\left (4 \, x\right )} + b e^{\left (4 \, x\right )} + 2 \, a e^{\left (2 \, x\right )} - 2 \, b e^{\left (2 \, x\right )} + a + b}\right )}^{5} + 3 \, {\left (3 \, a b + b^{2}\right )} {\left (\sqrt {a + b} e^{\left (2 \, x\right )} - \sqrt {a e^{\left (4 \, x\right )} + b e^{\left (4 \, x\right )} + 2 \, a e^{\left (2 \, x\right )} - 2 \, b e^{\left (2 \, x\right )} + a + b}\right )}^{4} \sqrt {a + b} + 2 \, {\left (3 \, a^{2} b - 6 \, a b^{2} - 5 \, b^{3}\right )} {\left (\sqrt {a + b} e^{\left (2 \, x\right )} - \sqrt {a e^{\left (4 \, x\right )} + b e^{\left (4 \, x\right )} + 2 \, a e^{\left (2 \, x\right )} - 2 \, b e^{\left (2 \, x\right )} + a + b}\right )}^{3} - 6 \, {\left (a^{2} b + 4 \, a b^{2} + 3 \, b^{3}\right )} {\left (\sqrt {a + b} e^{\left (2 \, x\right )} - \sqrt {a e^{\left (4 \, x\right )} + b e^{\left (4 \, x\right )} + 2 \, a e^{\left (2 \, x\right )} - 2 \, b e^{\left (2 \, x\right )} + a + b}\right )}^{2} \sqrt {a + b} - 3 \, {\left (3 \, a^{3} b + a^{2} b^{2} - 15 \, a b^{3} - 13 \, b^{4}\right )} {\left (\sqrt {a + b} e^{\left (2 \, x\right )} - \sqrt {a e^{\left (4 \, x\right )} + b e^{\left (4 \, x\right )} + 2 \, a e^{\left (2 \, x\right )} - 2 \, b e^{\left (2 \, x\right )} + a + b}\right )} - {\left (3 \, a^{3} b - 9 \, a^{2} b^{2} + 5 \, a b^{3} + 17 \, b^{4}\right )} \sqrt {a + b}\right )}}{3 \, {\left ({\left (\sqrt {a + b} e^{\left (2 \, x\right )} - \sqrt {a e^{\left (4 \, x\right )} + b e^{\left (4 \, x\right )} + 2 \, a e^{\left (2 \, x\right )} - 2 \, b e^{\left (2 \, x\right )} + a + b}\right )}^{2} + 2 \, {\left (\sqrt {a + b} e^{\left (2 \, x\right )} - \sqrt {a e^{\left (4 \, x\right )} + b e^{\left (4 \, x\right )} + 2 \, a e^{\left (2 \, x\right )} - 2 \, b e^{\left (2 \, x\right )} + a + b}\right )} \sqrt {a + b} + a - 3 \, b\right )}^{3}} \]
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))) - 1/2*(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) - 8/3*(3*(a*b + 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))^5 + 3*(3*a*b + 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))^4*sqrt(a + b) + 2*(3*a ^2*b - 6*a*b^2 - 5*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))^3 - 6*(a^2*b + 4*a*b^2 + 3*b^3)*(sqr t(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*(3*a^3*b + a^2*b^2 - 15*a*b^3 - 13*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*a^3*b - 9*a^2*b^2 + 5*a*b^3 + 17*b^4)*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)^3
Time = 4.10 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.02 \[ \int \tanh (x) \left (a+b \tanh ^2(x)\right )^{3/2} \, dx=\mathrm {atanh}\left (\frac {{\left (a+b\right )}^{3/2}\,\sqrt {b\,{\mathrm {tanh}\left (x\right )}^2+a}}{a^2+2\,a\,b+b^2}\right )\,{\left (a+b\right )}^{3/2}-\left (a+b\right )\,\sqrt {b\,{\mathrm {tanh}\left (x\right )}^2+a}-\frac {{\left (b\,{\mathrm {tanh}\left (x\right )}^2+a\right )}^{3/2}}{3} \]