Integrand size = 15, antiderivative size = 124 \[ \int \tanh (x) \left (a+b \tanh ^4(x)\right )^{3/2} \, dx=-\frac {1}{4} \sqrt {b} (3 a+2 b) \text {arctanh}\left (\frac {\sqrt {b} \tanh ^2(x)}{\sqrt {a+b \tanh ^4(x)}}\right )+\frac {1}{2} (a+b)^{3/2} \text {arctanh}\left (\frac {a+b \tanh ^2(x)}{\sqrt {a+b} \sqrt {a+b \tanh ^4(x)}}\right )-\frac {1}{4} \left (2 (a+b)+b \tanh ^2(x)\right ) \sqrt {a+b \tanh ^4(x)}-\frac {1}{6} \left (a+b \tanh ^4(x)\right )^{3/2} \]
1/2*(a+b)^(3/2)*arctanh((a+b*tanh(x)^2)/(a+b)^(1/2)/(a+b*tanh(x)^4)^(1/2)) -1/4*(3*a+2*b)*arctanh(b^(1/2)*tanh(x)^2/(a+b*tanh(x)^4)^(1/2))*b^(1/2)-1/ 4*(a+b*tanh(x)^4)^(1/2)*(2*a+2*b+b*tanh(x)^2)-1/6*(a+b*tanh(x)^4)^(3/2)
Time = 4.55 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.34 \[ \int \tanh (x) \left (a+b \tanh ^4(x)\right )^{3/2} \, dx=\frac {1}{12} \left (-6 \sqrt {b} (a+b) \text {arctanh}\left (\frac {\sqrt {b} \tanh ^2(x)}{\sqrt {a+b \tanh ^4(x)}}\right )+6 (a+b)^{3/2} \text {arctanh}\left (\frac {a+b \tanh ^2(x)}{\sqrt {a+b} \sqrt {a+b \tanh ^4(x)}}\right )-\sqrt {a+b \tanh ^4(x)} \left (8 a+6 b+3 b \tanh ^2(x)+2 b \tanh ^4(x)\right )-\frac {3 \sqrt {a} \sqrt {b} \text {arcsinh}\left (\frac {\sqrt {b} \tanh ^2(x)}{\sqrt {a}}\right ) \sqrt {a+b \tanh ^4(x)}}{\sqrt {1+\frac {b \tanh ^4(x)}{a}}}\right ) \]
(-6*Sqrt[b]*(a + b)*ArcTanh[(Sqrt[b]*Tanh[x]^2)/Sqrt[a + b*Tanh[x]^4]] + 6 *(a + b)^(3/2)*ArcTanh[(a + b*Tanh[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Tanh[x]^4 ])] - Sqrt[a + b*Tanh[x]^4]*(8*a + 6*b + 3*b*Tanh[x]^2 + 2*b*Tanh[x]^4) - (3*Sqrt[a]*Sqrt[b]*ArcSinh[(Sqrt[b]*Tanh[x]^2)/Sqrt[a]]*Sqrt[a + b*Tanh[x] ^4])/Sqrt[1 + (b*Tanh[x]^4)/a])/12
Time = 0.40 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.06, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.933, Rules used = {3042, 26, 4153, 26, 1577, 493, 25, 682, 27, 719, 224, 219, 488, 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.
| \(\displaystyle \int \tanh (x) \left (a+b \tanh ^4(x)\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -i \tan (i x) \left (a+b \tan (i x)^4\right )^{3/2}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \int \tan (i x) \left (b \tan (i x)^4+a\right )^{3/2}dx\) |
| \(\Big \downarrow \) 4153 |
| \(\displaystyle -i \int \frac {i \tanh (x) \left (b \tanh ^4(x)+a\right )^{3/2}}{1-\tanh ^2(x)}d\tanh (x)\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \int \frac {\tanh (x) \left (a+b \tanh ^4(x)\right )^{3/2}}{1-\tanh ^2(x)}d\tanh (x)\) |
| \(\Big \downarrow \) 1577 |
| \(\displaystyle \frac {1}{2} \int \frac {\left (b \tanh ^4(x)+a\right )^{3/2}}{1-\tanh ^2(x)}d\tanh ^2(x)\) |
| \(\Big \downarrow \) 493 |
| \(\displaystyle \frac {1}{2} \left (-\int -\frac {\left (b \tanh ^2(x)+a\right ) \sqrt {b \tanh ^4(x)+a}}{1-\tanh ^2(x)}d\tanh ^2(x)-\frac {1}{3} \left (a+b \tanh ^4(x)\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} \left (\int \frac {\left (b \tanh ^2(x)+a\right ) \sqrt {b \tanh ^4(x)+a}}{1-\tanh ^2(x)}d\tanh ^2(x)-\frac {1}{3} \left (a+b \tanh ^4(x)\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 682 |
| \(\displaystyle \frac {1}{2} \left (\frac {\int \frac {b \left (b (3 a+2 b) \tanh ^2(x)+a (2 a+b)\right )}{\left (1-\tanh ^2(x)\right ) \sqrt {b \tanh ^4(x)+a}}d\tanh ^2(x)}{2 b}-\frac {1}{3} \left (a+b \tanh ^4(x)\right )^{3/2}-\frac {1}{2} \left (2 (a+b)+b \tanh ^2(x)\right ) \sqrt {a+b \tanh ^4(x)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{2} \int \frac {b (3 a+2 b) \tanh ^2(x)+a (2 a+b)}{\left (1-\tanh ^2(x)\right ) \sqrt {b \tanh ^4(x)+a}}d\tanh ^2(x)-\frac {1}{3} \left (a+b \tanh ^4(x)\right )^{3/2}-\frac {1}{2} \left (2 (a+b)+b \tanh ^2(x)\right ) \sqrt {a+b \tanh ^4(x)}\right )\) |
| \(\Big \downarrow \) 719 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{2} \left (2 (a+b)^2 \int \frac {1}{\left (1-\tanh ^2(x)\right ) \sqrt {b \tanh ^4(x)+a}}d\tanh ^2(x)-b (3 a+2 b) \int \frac {1}{\sqrt {b \tanh ^4(x)+a}}d\tanh ^2(x)\right )-\frac {1}{3} \left (a+b \tanh ^4(x)\right )^{3/2}-\frac {1}{2} \left (2 (a+b)+b \tanh ^2(x)\right ) \sqrt {a+b \tanh ^4(x)}\right )\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{2} \left (2 (a+b)^2 \int \frac {1}{\left (1-\tanh ^2(x)\right ) \sqrt {b \tanh ^4(x)+a}}d\tanh ^2(x)-b (3 a+2 b) \int \frac {1}{1-b \tanh ^4(x)}d\frac {\tanh ^2(x)}{\sqrt {b \tanh ^4(x)+a}}\right )-\frac {1}{3} \left (a+b \tanh ^4(x)\right )^{3/2}-\frac {1}{2} \left (2 (a+b)+b \tanh ^2(x)\right ) \sqrt {a+b \tanh ^4(x)}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{2} \left (2 (a+b)^2 \int \frac {1}{\left (1-\tanh ^2(x)\right ) \sqrt {b \tanh ^4(x)+a}}d\tanh ^2(x)-\sqrt {b} (3 a+2 b) \text {arctanh}\left (\frac {\sqrt {b} \tanh ^2(x)}{\sqrt {a+b \tanh ^4(x)}}\right )\right )-\frac {1}{3} \left (a+b \tanh ^4(x)\right )^{3/2}-\frac {1}{2} \left (2 (a+b)+b \tanh ^2(x)\right ) \sqrt {a+b \tanh ^4(x)}\right )\) |
| \(\Big \downarrow \) 488 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{2} \left (-2 (a+b)^2 \int \frac {1}{-\tanh ^4(x)+a+b}d\frac {-b \tanh ^2(x)-a}{\sqrt {b \tanh ^4(x)+a}}-\sqrt {b} (3 a+2 b) \text {arctanh}\left (\frac {\sqrt {b} \tanh ^2(x)}{\sqrt {a+b \tanh ^4(x)}}\right )\right )-\frac {1}{3} \left (a+b \tanh ^4(x)\right )^{3/2}-\frac {1}{2} \left (2 (a+b)+b \tanh ^2(x)\right ) \sqrt {a+b \tanh ^4(x)}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{2} \left (-2 (a+b)^{3/2} \text {arctanh}\left (\frac {-a-b \tanh ^2(x)}{\sqrt {a+b} \sqrt {a+b \tanh ^4(x)}}\right )-\sqrt {b} (3 a+2 b) \text {arctanh}\left (\frac {\sqrt {b} \tanh ^2(x)}{\sqrt {a+b \tanh ^4(x)}}\right )\right )-\frac {1}{3} \left (a+b \tanh ^4(x)\right )^{3/2}-\frac {1}{2} \left (2 (a+b)+b \tanh ^2(x)\right ) \sqrt {a+b \tanh ^4(x)}\right )\) |
((-(Sqrt[b]*(3*a + 2*b)*ArcTanh[(Sqrt[b]*Tanh[x]^2)/Sqrt[a + b*Tanh[x]^4]] ) - 2*(a + b)^(3/2)*ArcTanh[(-a - b*Tanh[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Tan h[x]^4])])/2 - ((2*(a + b) + b*Tanh[x]^2)*Sqrt[a + b*Tanh[x]^4])/2 - (a + b*Tanh[x]^4)^(3/2)/3)/2
3.3.59.3.1 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_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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])
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]
Int[1/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> -Subst[ Int[1/(b*c^2 + a*d^2 - x^2), x], x, (a*d - b*c*x)/Sqrt[a + b*x^2]] /; FreeQ [{a, b, c, d}, x]
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ (c + d*x)^(n + 1)*((a + b*x^2)^p/(d*(n + 2*p + 1))), x] + Simp[2*(p/(d*(n + 2*p + 1))) Int[(c + d*x)^n*(a + b*x^2)^(p - 1)*(a*d - b*c*x), x], x] /; FreeQ[{a, b, c, d, n}, x] && GtQ[p, 0] && NeQ[n + 2*p + 1, 0] && ( !Rationa lQ[n] || LtQ[n, 1]) && !ILtQ[n + 2*p, 0] && IntQuadraticQ[a, 0, b, c, d, n , p, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*c*d*(2*p + 1) + g*c*e*(m + 2*p + 1)*x)*((a + c*x^2)^p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] + Simp[2*(p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))) Int[(d + e*x) ^m*(a + c*x^2)^(p - 1)*Simp[f*a*c*e^2*(m + 2*p + 2) + a*c*d*e*g*m - (c^2*f* d*e*(m + 2*p + 2) - g*(c^2*d^2*(2*p + 1) + a*c*e^2*(m + 2*p + 1)))*x, x], x ], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] || ! RationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) && !ILtQ[m + 2*p, 0] && (Intege rQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[g/e Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Simp[(e*f - d*g)/e Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && !IGtQ[m, 0]
Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Simp[1/2 Subst[Int[(d + e*x)^q*(a + c*x^2)^p, x], x, x^2], x] /; Free Q[{a, c, d, e, p, q}, x]
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]))
Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.90 (sec) , antiderivative size = 620, normalized size of antiderivative = 5.00
| method | result | size |
| derivativedivides | \(-\frac {b \tanh \left (x \right )^{4} \sqrt {a +b \tanh \left (x \right )^{4}}}{6}-\frac {b \tanh \left (x \right )^{2} \sqrt {a +b \tanh \left (x \right )^{4}}}{4}-\frac {2 \sqrt {a +b \tanh \left (x \right )^{4}}\, a}{3}-\frac {b \sqrt {a +b \tanh \left (x \right )^{4}}}{2}-\frac {\left (\frac {5}{3} a b +b^{2}\right ) \sqrt {1-\frac {i \sqrt {b}\, \tanh \left (x \right )^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, \tanh \left (x \right )^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (\tanh \left (x \right ) \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{2 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {a +b \tanh \left (x \right )^{4}}}-\frac {3 \ln \left (2 \sqrt {b}\, \tanh \left (x \right )^{2}+2 \sqrt {a +b \tanh \left (x \right )^{4}}\right ) \sqrt {b}\, a}{4}-\frac {\ln \left (2 \sqrt {b}\, \tanh \left (x \right )^{2}+2 \sqrt {a +b \tanh \left (x \right )^{4}}\right ) b^{\frac {3}{2}}}{2}-\frac {i \left (-\frac {7}{5} a b -b^{2}\right ) \sqrt {a}\, \sqrt {1-\frac {i \sqrt {b}\, \tanh \left (x \right )^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, \tanh \left (x \right )^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (\tanh \left (x \right ) \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (\tanh \left (x \right ) \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{2 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {a +b \tanh \left (x \right )^{4}}\, \sqrt {b}}+\frac {a^{2} \operatorname {arctanh}\left (\frac {2 b \tanh \left (x \right )^{2}+2 a}{2 \sqrt {a +b}\, \sqrt {a +b \tanh \left (x \right )^{4}}}\right )}{2 \sqrt {a +b}}+\frac {a b \,\operatorname {arctanh}\left (\frac {2 b \tanh \left (x \right )^{2}+2 a}{2 \sqrt {a +b}\, \sqrt {a +b \tanh \left (x \right )^{4}}}\right )}{\sqrt {a +b}}+\frac {b^{2} \operatorname {arctanh}\left (\frac {2 b \tanh \left (x \right )^{2}+2 a}{2 \sqrt {a +b}\, \sqrt {a +b \tanh \left (x \right )^{4}}}\right )}{2 \sqrt {a +b}}-\frac {\left (-\frac {5}{3} a b -b^{2}\right ) \sqrt {1-\frac {i \sqrt {b}\, \tanh \left (x \right )^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, \tanh \left (x \right )^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (\tanh \left (x \right ) \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{2 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {a +b \tanh \left (x \right )^{4}}}-\frac {i \left (\frac {7}{5} a b +b^{2}\right ) \sqrt {a}\, \sqrt {1-\frac {i \sqrt {b}\, \tanh \left (x \right )^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, \tanh \left (x \right )^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (\tanh \left (x \right ) \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (\tanh \left (x \right ) \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{2 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {a +b \tanh \left (x \right )^{4}}\, \sqrt {b}}\) | \(620\) |
| default | \(-\frac {b \tanh \left (x \right )^{4} \sqrt {a +b \tanh \left (x \right )^{4}}}{6}-\frac {b \tanh \left (x \right )^{2} \sqrt {a +b \tanh \left (x \right )^{4}}}{4}-\frac {2 \sqrt {a +b \tanh \left (x \right )^{4}}\, a}{3}-\frac {b \sqrt {a +b \tanh \left (x \right )^{4}}}{2}-\frac {\left (\frac {5}{3} a b +b^{2}\right ) \sqrt {1-\frac {i \sqrt {b}\, \tanh \left (x \right )^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, \tanh \left (x \right )^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (\tanh \left (x \right ) \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{2 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {a +b \tanh \left (x \right )^{4}}}-\frac {3 \ln \left (2 \sqrt {b}\, \tanh \left (x \right )^{2}+2 \sqrt {a +b \tanh \left (x \right )^{4}}\right ) \sqrt {b}\, a}{4}-\frac {\ln \left (2 \sqrt {b}\, \tanh \left (x \right )^{2}+2 \sqrt {a +b \tanh \left (x \right )^{4}}\right ) b^{\frac {3}{2}}}{2}-\frac {i \left (-\frac {7}{5} a b -b^{2}\right ) \sqrt {a}\, \sqrt {1-\frac {i \sqrt {b}\, \tanh \left (x \right )^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, \tanh \left (x \right )^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (\tanh \left (x \right ) \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (\tanh \left (x \right ) \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{2 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {a +b \tanh \left (x \right )^{4}}\, \sqrt {b}}+\frac {a^{2} \operatorname {arctanh}\left (\frac {2 b \tanh \left (x \right )^{2}+2 a}{2 \sqrt {a +b}\, \sqrt {a +b \tanh \left (x \right )^{4}}}\right )}{2 \sqrt {a +b}}+\frac {a b \,\operatorname {arctanh}\left (\frac {2 b \tanh \left (x \right )^{2}+2 a}{2 \sqrt {a +b}\, \sqrt {a +b \tanh \left (x \right )^{4}}}\right )}{\sqrt {a +b}}+\frac {b^{2} \operatorname {arctanh}\left (\frac {2 b \tanh \left (x \right )^{2}+2 a}{2 \sqrt {a +b}\, \sqrt {a +b \tanh \left (x \right )^{4}}}\right )}{2 \sqrt {a +b}}-\frac {\left (-\frac {5}{3} a b -b^{2}\right ) \sqrt {1-\frac {i \sqrt {b}\, \tanh \left (x \right )^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, \tanh \left (x \right )^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (\tanh \left (x \right ) \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{2 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {a +b \tanh \left (x \right )^{4}}}-\frac {i \left (\frac {7}{5} a b +b^{2}\right ) \sqrt {a}\, \sqrt {1-\frac {i \sqrt {b}\, \tanh \left (x \right )^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, \tanh \left (x \right )^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (\tanh \left (x \right ) \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (\tanh \left (x \right ) \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{2 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {a +b \tanh \left (x \right )^{4}}\, \sqrt {b}}\) | \(620\) |
-1/6*b*tanh(x)^4*(a+b*tanh(x)^4)^(1/2)-1/4*b*tanh(x)^2*(a+b*tanh(x)^4)^(1/ 2)-2/3*(a+b*tanh(x)^4)^(1/2)*a-1/2*b*(a+b*tanh(x)^4)^(1/2)-1/2*(5/3*a*b+b^ 2)/(I/a^(1/2)*b^(1/2))^(1/2)*(1-I/a^(1/2)*b^(1/2)*tanh(x)^2)^(1/2)*(1+I/a^ (1/2)*b^(1/2)*tanh(x)^2)^(1/2)/(a+b*tanh(x)^4)^(1/2)*EllipticF(tanh(x)*(I/ a^(1/2)*b^(1/2))^(1/2),I)-3/4*ln(2*b^(1/2)*tanh(x)^2+2*(a+b*tanh(x)^4)^(1/ 2))*b^(1/2)*a-1/2*ln(2*b^(1/2)*tanh(x)^2+2*(a+b*tanh(x)^4)^(1/2))*b^(3/2)- 1/2*I*(-7/5*a*b-b^2)*a^(1/2)/(I/a^(1/2)*b^(1/2))^(1/2)*(1-I/a^(1/2)*b^(1/2 )*tanh(x)^2)^(1/2)*(1+I/a^(1/2)*b^(1/2)*tanh(x)^2)^(1/2)/(a+b*tanh(x)^4)^( 1/2)/b^(1/2)*(EllipticF(tanh(x)*(I/a^(1/2)*b^(1/2))^(1/2),I)-EllipticE(tan h(x)*(I/a^(1/2)*b^(1/2))^(1/2),I))+1/2*a^2/(a+b)^(1/2)*arctanh(1/2*(2*b*ta nh(x)^2+2*a)/(a+b)^(1/2)/(a+b*tanh(x)^4)^(1/2))+a*b/(a+b)^(1/2)*arctanh(1/ 2*(2*b*tanh(x)^2+2*a)/(a+b)^(1/2)/(a+b*tanh(x)^4)^(1/2))+1/2*b^2/(a+b)^(1/ 2)*arctanh(1/2*(2*b*tanh(x)^2+2*a)/(a+b)^(1/2)/(a+b*tanh(x)^4)^(1/2))-1/2* (-5/3*a*b-b^2)/(I/a^(1/2)*b^(1/2))^(1/2)*(1-I/a^(1/2)*b^(1/2)*tanh(x)^2)^( 1/2)*(1+I/a^(1/2)*b^(1/2)*tanh(x)^2)^(1/2)/(a+b*tanh(x)^4)^(1/2)*EllipticF (tanh(x)*(I/a^(1/2)*b^(1/2))^(1/2),I)-1/2*I*(7/5*a*b+b^2)*a^(1/2)/(I/a^(1/ 2)*b^(1/2))^(1/2)*(1-I/a^(1/2)*b^(1/2)*tanh(x)^2)^(1/2)*(1+I/a^(1/2)*b^(1/ 2)*tanh(x)^2)^(1/2)/(a+b*tanh(x)^4)^(1/2)/b^(1/2)*(EllipticF(tanh(x)*(I/a^ (1/2)*b^(1/2))^(1/2),I)-EllipticE(tanh(x)*(I/a^(1/2)*b^(1/2))^(1/2),I))
Leaf count of result is larger than twice the leaf count of optimal. 2646 vs. \(2 (101) = 202\).
Time = 0.55 (sec) , antiderivative size = 11528, normalized size of antiderivative = 92.97 \[ \int \tanh (x) \left (a+b \tanh ^4(x)\right )^{3/2} \, dx=\text {Too large to display} \]
\[ \int \tanh (x) \left (a+b \tanh ^4(x)\right )^{3/2} \, dx=\int \left (a + b \tanh ^{4}{\left (x \right )}\right )^{\frac {3}{2}} \tanh {\left (x \right )}\, dx \]
\[ \int \tanh (x) \left (a+b \tanh ^4(x)\right )^{3/2} \, dx=\int { {\left (b \tanh \left (x\right )^{4} + a\right )}^{\frac {3}{2}} \tanh \left (x\right ) \,d x } \]
\[ \int \tanh (x) \left (a+b \tanh ^4(x)\right )^{3/2} \, dx=\int { {\left (b \tanh \left (x\right )^{4} + a\right )}^{\frac {3}{2}} \tanh \left (x\right ) \,d x } \]
Timed out. \[ \int \tanh (x) \left (a+b \tanh ^4(x)\right )^{3/2} \, dx=\int \mathrm {tanh}\left (x\right )\,{\left (b\,{\mathrm {tanh}\left (x\right )}^4+a\right )}^{3/2} \,d x \]