Integrand size = 15, antiderivative size = 89 \[ \int \tanh (x) \sqrt {a+b \tanh ^4(x)} \, dx=-\frac {1}{2} \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \tanh ^2(x)}{\sqrt {a+b \tanh ^4(x)}}\right )+\frac {1}{2} \sqrt {a+b} \text {arctanh}\left (\frac {a+b \tanh ^2(x)}{\sqrt {a+b} \sqrt {a+b \tanh ^4(x)}}\right )-\frac {1}{2} \sqrt {a+b \tanh ^4(x)} \]
-1/2*arctanh(b^(1/2)*tanh(x)^2/(a+b*tanh(x)^4)^(1/2))*b^(1/2)+1/2*arctanh( (a+b*tanh(x)^2)/(a+b)^(1/2)/(a+b*tanh(x)^4)^(1/2))*(a+b)^(1/2)-1/2*(a+b*ta nh(x)^4)^(1/2)
Time = 0.08 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.97 \[ \int \tanh (x) \sqrt {a+b \tanh ^4(x)} \, dx=\frac {1}{2} \left (-\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \tanh ^2(x)}{\sqrt {a+b \tanh ^4(x)}}\right )+\sqrt {a+b} \text {arctanh}\left (\frac {a+b \tanh ^2(x)}{\sqrt {a+b} \sqrt {a+b \tanh ^4(x)}}\right )-\sqrt {a+b \tanh ^4(x)}\right ) \]
(-(Sqrt[b]*ArcTanh[(Sqrt[b]*Tanh[x]^2)/Sqrt[a + b*Tanh[x]^4]]) + Sqrt[a + b]*ArcTanh[(a + b*Tanh[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Tanh[x]^4])] - Sqrt[a + b*Tanh[x]^4])/2
Time = 0.33 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.01, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {3042, 26, 4153, 26, 1577, 493, 25, 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.
\(\displaystyle \int \tanh (x) \sqrt {a+b \tanh ^4(x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -i \tan (i x) \sqrt {a+b \tan (i x)^4}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -i \int \tan (i x) \sqrt {b \tan (i x)^4+a}dx\) |
\(\Big \downarrow \) 4153 |
\(\displaystyle -i \int \frac {i \tanh (x) \sqrt {b \tanh ^4(x)+a}}{1-\tanh ^2(x)}d\tanh (x)\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \int \frac {\tanh (x) \sqrt {a+b \tanh ^4(x)}}{1-\tanh ^2(x)}d\tanh (x)\) |
\(\Big \downarrow \) 1577 |
\(\displaystyle \frac {1}{2} \int \frac {\sqrt {b \tanh ^4(x)+a}}{1-\tanh ^2(x)}d\tanh ^2(x)\) |
\(\Big \downarrow \) 493 |
\(\displaystyle \frac {1}{2} \left (-\int -\frac {b \tanh ^2(x)+a}{\left (1-\tanh ^2(x)\right ) \sqrt {b \tanh ^4(x)+a}}d\tanh ^2(x)-\sqrt {a+b \tanh ^4(x)}\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {1}{2} \left (\int \frac {b \tanh ^2(x)+a}{\left (1-\tanh ^2(x)\right ) \sqrt {b \tanh ^4(x)+a}}d\tanh ^2(x)-\sqrt {a+b \tanh ^4(x)}\right )\) |
\(\Big \downarrow \) 719 |
\(\displaystyle \frac {1}{2} \left (-b \int \frac {1}{\sqrt {b \tanh ^4(x)+a}}d\tanh ^2(x)+(a+b) \int \frac {1}{\left (1-\tanh ^2(x)\right ) \sqrt {b \tanh ^4(x)+a}}d\tanh ^2(x)-\sqrt {a+b \tanh ^4(x)}\right )\) |
\(\Big \downarrow \) 224 |
\(\displaystyle \frac {1}{2} \left (-b \int \frac {1}{1-b \tanh ^4(x)}d\frac {\tanh ^2(x)}{\sqrt {b \tanh ^4(x)+a}}+(a+b) \int \frac {1}{\left (1-\tanh ^2(x)\right ) \sqrt {b \tanh ^4(x)+a}}d\tanh ^2(x)-\sqrt {a+b \tanh ^4(x)}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{2} \left ((a+b) \int \frac {1}{\left (1-\tanh ^2(x)\right ) \sqrt {b \tanh ^4(x)+a}}d\tanh ^2(x)-\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \tanh ^2(x)}{\sqrt {a+b \tanh ^4(x)}}\right )-\sqrt {a+b \tanh ^4(x)}\right )\) |
\(\Big \downarrow \) 488 |
\(\displaystyle \frac {1}{2} \left (-(a+b) \int \frac {1}{-\tanh ^4(x)+a+b}d\frac {-b \tanh ^2(x)-a}{\sqrt {b \tanh ^4(x)+a}}-\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \tanh ^2(x)}{\sqrt {a+b \tanh ^4(x)}}\right )-\sqrt {a+b \tanh ^4(x)}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{2} \left (-\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \tanh ^2(x)}{\sqrt {a+b \tanh ^4(x)}}\right )-\sqrt {a+b} \text {arctanh}\left (\frac {-a-b \tanh ^2(x)}{\sqrt {a+b} \sqrt {a+b \tanh ^4(x)}}\right )-\sqrt {a+b \tanh ^4(x)}\right )\) |
(-(Sqrt[b]*ArcTanh[(Sqrt[b]*Tanh[x]^2)/Sqrt[a + b*Tanh[x]^4]]) - Sqrt[a + b]*ArcTanh[(-a - b*Tanh[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Tanh[x]^4])] - Sqrt[ a + b*Tanh[x]^4])/2
3.3.60.3.1 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_)^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[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]))
Time = 0.71 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.30
method | result | size |
derivativedivides | \(-\frac {\sqrt {a +b \tanh \left (x \right )^{4}}}{2}-\frac {\sqrt {b}\, \ln \left (2 \sqrt {b}\, \tanh \left (x \right )^{2}+2 \sqrt {a +b \tanh \left (x \right )^{4}}\right )}{2}+\frac {b \,\operatorname {arctanh}\left (\frac {2 b \tanh \left (x \right )^{2}+2 a}{2 \sqrt {a +b}\, \sqrt {a +b \tanh \left (x \right )^{4}}}\right )}{2 \sqrt {a +b}}+\frac {a \,\operatorname {arctanh}\left (\frac {2 b \tanh \left (x \right )^{2}+2 a}{2 \sqrt {a +b}\, \sqrt {a +b \tanh \left (x \right )^{4}}}\right )}{2 \sqrt {a +b}}\) | \(116\) |
default | \(-\frac {\sqrt {a +b \tanh \left (x \right )^{4}}}{2}-\frac {\sqrt {b}\, \ln \left (2 \sqrt {b}\, \tanh \left (x \right )^{2}+2 \sqrt {a +b \tanh \left (x \right )^{4}}\right )}{2}+\frac {b \,\operatorname {arctanh}\left (\frac {2 b \tanh \left (x \right )^{2}+2 a}{2 \sqrt {a +b}\, \sqrt {a +b \tanh \left (x \right )^{4}}}\right )}{2 \sqrt {a +b}}+\frac {a \,\operatorname {arctanh}\left (\frac {2 b \tanh \left (x \right )^{2}+2 a}{2 \sqrt {a +b}\, \sqrt {a +b \tanh \left (x \right )^{4}}}\right )}{2 \sqrt {a +b}}\) | \(116\) |
-1/2*(a+b*tanh(x)^4)^(1/2)-1/2*b^(1/2)*ln(2*b^(1/2)*tanh(x)^2+2*(a+b*tanh( x)^4)^(1/2))+1/2*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*a/(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))
Leaf count of result is larger than twice the leaf count of optimal. 1048 vs. \(2 (69) = 138\).
Time = 0.44 (sec) , antiderivative size = 5136, normalized size of antiderivative = 57.71 \[ \int \tanh (x) \sqrt {a+b \tanh ^4(x)} \, dx=\text {Too large to display} \]
\[ \int \tanh (x) \sqrt {a+b \tanh ^4(x)} \, dx=\int \sqrt {a + b \tanh ^{4}{\left (x \right )}} \tanh {\left (x \right )}\, dx \]
\[ \int \tanh (x) \sqrt {a+b \tanh ^4(x)} \, dx=\int { \sqrt {b \tanh \left (x\right )^{4} + a} \tanh \left (x\right ) \,d x } \]
\[ \int \tanh (x) \sqrt {a+b \tanh ^4(x)} \, dx=\int { \sqrt {b \tanh \left (x\right )^{4} + a} \tanh \left (x\right ) \,d x } \]
Timed out. \[ \int \tanh (x) \sqrt {a+b \tanh ^4(x)} \, dx=\int \mathrm {tanh}\left (x\right )\,\sqrt {b\,{\mathrm {tanh}\left (x\right )}^4+a} \,d x \]