Integrand size = 11, antiderivative size = 49 \[ \int \frac {\tanh (x)}{(1+\tanh (x))^{3/2}} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {1+\tanh (x)}}{\sqrt {2}}\right )}{2 \sqrt {2}}+\frac {1}{3 (1+\tanh (x))^{3/2}}-\frac {1}{2 \sqrt {1+\tanh (x)}} \] Output:
1/4*2^(1/2)*arctanh(1/2*(1+tanh(x))^(1/2)*2^(1/2))+1/3/(1+tanh(x))^(3/2)-1 /2/(1+tanh(x))^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.39 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.73 \[ \int \frac {\tanh (x)}{(1+\tanh (x))^{3/2}} \, dx=\frac {2-3 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},\frac {1}{2} (1+\tanh (x))\right ) (1+\tanh (x))}{6 (1+\tanh (x))^{3/2}} \] Input:
Integrate[Tanh[x]/(1 + Tanh[x])^(3/2),x]
Output:
(2 - 3*Hypergeometric2F1[-1/2, 1, 1/2, (1 + Tanh[x])/2]*(1 + Tanh[x]))/(6* (1 + Tanh[x])^(3/2))
Result contains complex when optimal does not.
Time = 0.34 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.16, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.727, Rules used = {3042, 26, 4009, 3042, 3960, 3042, 3961, 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.
\(\displaystyle \int \frac {\tanh (x)}{(\tanh (x)+1)^{3/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\frac {i \tan (i x)}{(1-i \tan (i x))^{3/2}}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -i \int \frac {\tan (i x)}{(1-i \tan (i x))^{3/2}}dx\) |
\(\Big \downarrow \) 4009 |
\(\displaystyle -i \left (\frac {1}{2} i \int \frac {1}{\sqrt {\tanh (x)+1}}dx+\frac {i}{3 (\tanh (x)+1)^{3/2}}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -i \left (\frac {1}{2} i \int \frac {1}{\sqrt {1-i \tan (i x)}}dx+\frac {i}{3 (\tanh (x)+1)^{3/2}}\right )\) |
\(\Big \downarrow \) 3960 |
\(\displaystyle -i \left (\frac {1}{2} i \left (\frac {1}{2} \int \sqrt {\tanh (x)+1}dx-\frac {1}{\sqrt {\tanh (x)+1}}\right )+\frac {i}{3 (\tanh (x)+1)^{3/2}}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -i \left (\frac {1}{2} i \left (-\frac {1}{\sqrt {\tanh (x)+1}}+\frac {1}{2} \int \sqrt {1-i \tan (i x)}dx\right )+\frac {i}{3 (\tanh (x)+1)^{3/2}}\right )\) |
\(\Big \downarrow \) 3961 |
\(\displaystyle -i \left (\frac {1}{2} i \left (\int \frac {1}{1-\tanh (x)}d\sqrt {\tanh (x)+1}-\frac {1}{\sqrt {\tanh (x)+1}}\right )+\frac {i}{3 (\tanh (x)+1)^{3/2}}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -i \left (\frac {1}{2} i \left (\frac {\text {arctanh}\left (\frac {\sqrt {\tanh (x)+1}}{\sqrt {2}}\right )}{\sqrt {2}}-\frac {1}{\sqrt {\tanh (x)+1}}\right )+\frac {i}{3 (\tanh (x)+1)^{3/2}}\right )\) |
Input:
Int[Tanh[x]/(1 + Tanh[x])^(3/2),x]
Output:
(-I)*((I/3)/(1 + Tanh[x])^(3/2) + (I/2)*(ArcTanh[Sqrt[1 + Tanh[x]]/Sqrt[2] ]/Sqrt[2] - 1/Sqrt[1 + Tanh[x]]))
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[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[a*((a + b*Tan[c + d*x])^n/(2*b*d*n)), x] + Simp[1/(2*a) Int[(a + b*Tan[c + d*x])^ (n + 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 + b^2, 0] && LtQ[n, 0]
Int[Sqrt[(a_) + (b_.)*tan[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[-2*(b/d) Subst[Int[1/(2*a - x^2), x], x, Sqrt[a + b*Tan[c + d*x]]], x] /; FreeQ[{a , b, c, d}, x] && EqQ[a^2 + b^2, 0]
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-(b*c - a*d))*((a + b*Tan[e + f*x])^m/(2*a *f*m)), x] + Simp[(b*c + a*d)/(2*a*b) Int[(a + b*Tan[e + f*x])^(m + 1), x ], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2 , 0] && LtQ[m, 0]
Time = 0.22 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.71
method | result | size |
derivativedivides | \(\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {1+\tanh \left (x \right )}\, \sqrt {2}}{2}\right )}{4}+\frac {1}{3 \left (1+\tanh \left (x \right )\right )^{\frac {3}{2}}}-\frac {1}{2 \sqrt {1+\tanh \left (x \right )}}\) | \(35\) |
default | \(\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {1+\tanh \left (x \right )}\, \sqrt {2}}{2}\right )}{4}+\frac {1}{3 \left (1+\tanh \left (x \right )\right )^{\frac {3}{2}}}-\frac {1}{2 \sqrt {1+\tanh \left (x \right )}}\) | \(35\) |
Input:
int(tanh(x)/(1+tanh(x))^(3/2),x,method=_RETURNVERBOSE)
Output:
1/4*2^(1/2)*arctanh(1/2*(1+tanh(x))^(1/2)*2^(1/2))+1/3/(1+tanh(x))^(3/2)-1 /2/(1+tanh(x))^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 240 vs. \(2 (34) = 68\).
Time = 0.09 (sec) , antiderivative size = 240, normalized size of antiderivative = 4.90 \[ \int \frac {\tanh (x)}{(1+\tanh (x))^{3/2}} \, dx=\frac {3 \, {\left (\sqrt {2} \cosh \left (x\right )^{3} + 3 \, \sqrt {2} \cosh \left (x\right )^{2} \sinh \left (x\right ) + 3 \, \sqrt {2} \cosh \left (x\right ) \sinh \left (x\right )^{2} + \sqrt {2} \sinh \left (x\right )^{3}\right )} \log \left (-2 \, \cosh \left (x\right )^{2} - 4 \, \cosh \left (x\right ) \sinh \left (x\right ) - 2 \, \sinh \left (x\right )^{2} - \frac {\sqrt {2} {\left (\sqrt {2} \cosh \left (x\right )^{3} + 3 \, \sqrt {2} \cosh \left (x\right ) \sinh \left (x\right )^{2} + \sqrt {2} \sinh \left (x\right )^{3} + {\left (3 \, \sqrt {2} \cosh \left (x\right )^{2} + \sqrt {2}\right )} \sinh \left (x\right ) + \sqrt {2} \cosh \left (x\right )\right )}}{\sqrt {\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 1}} - 1\right ) - \frac {2 \, \sqrt {2} {\left (2 \, \cosh \left (x\right )^{4} + 8 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + 2 \, \sinh \left (x\right )^{4} + {\left (12 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{2} + \cosh \left (x\right )^{2} + 2 \, {\left (4 \, \cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} \sinh \left (x\right ) - 1\right )}}{\sqrt {\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 1}}}{24 \, {\left (\cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right )^{2} \sinh \left (x\right ) + 3 \, \cosh \left (x\right ) \sinh \left (x\right )^{2} + \sinh \left (x\right )^{3}\right )}} \] Input:
integrate(tanh(x)/(1+tanh(x))^(3/2),x, algorithm="fricas")
Output:
1/24*(3*(sqrt(2)*cosh(x)^3 + 3*sqrt(2)*cosh(x)^2*sinh(x) + 3*sqrt(2)*cosh( x)*sinh(x)^2 + sqrt(2)*sinh(x)^3)*log(-2*cosh(x)^2 - 4*cosh(x)*sinh(x) - 2 *sinh(x)^2 - sqrt(2)*(sqrt(2)*cosh(x)^3 + 3*sqrt(2)*cosh(x)*sinh(x)^2 + sq rt(2)*sinh(x)^3 + (3*sqrt(2)*cosh(x)^2 + sqrt(2))*sinh(x) + sqrt(2)*cosh(x ))/sqrt(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 + 1) - 1) - 2*sqrt(2)*(2 *cosh(x)^4 + 8*cosh(x)*sinh(x)^3 + 2*sinh(x)^4 + (12*cosh(x)^2 + 1)*sinh(x )^2 + cosh(x)^2 + 2*(4*cosh(x)^3 + cosh(x))*sinh(x) - 1)/sqrt(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 + 1))/(cosh(x)^3 + 3*cosh(x)^2*sinh(x) + 3*c osh(x)*sinh(x)^2 + sinh(x)^3)
Time = 4.73 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.22 \[ \int \frac {\tanh (x)}{(1+\tanh (x))^{3/2}} \, dx=- \frac {\sqrt {2} \left (\log {\left (\sqrt {\tanh {\left (x \right )} + 1} - \sqrt {2} \right )} - \log {\left (\sqrt {\tanh {\left (x \right )} + 1} + \sqrt {2} \right )}\right )}{8} - \frac {1}{2 \sqrt {\tanh {\left (x \right )} + 1}} + \frac {1}{3 \left (\tanh {\left (x \right )} + 1\right )^{\frac {3}{2}}} \] Input:
integrate(tanh(x)/(1+tanh(x))**(3/2),x)
Output:
-sqrt(2)*(log(sqrt(tanh(x) + 1) - sqrt(2)) - log(sqrt(tanh(x) + 1) + sqrt( 2)))/8 - 1/(2*sqrt(tanh(x) + 1)) + 1/(3*(tanh(x) + 1)**(3/2))
\[ \int \frac {\tanh (x)}{(1+\tanh (x))^{3/2}} \, dx=\int { \frac {\tanh \left (x\right )}{{\left (\tanh \left (x\right ) + 1\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(tanh(x)/(1+tanh(x))^(3/2),x, algorithm="maxima")
Output:
1/12*sqrt(2)*(e^(-2*x) + 1)^(3/2) + integrate(1/2*e^(-x)/(sqrt(2)*e^(-x)/( e^(-2*x) + 1)^(3/2) + sqrt(2)*e^(-3*x)/(e^(-2*x) + 1)^(3/2)), x)
Leaf count of result is larger than twice the leaf count of optimal. 73 vs. \(2 (34) = 68\).
Time = 0.12 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.49 \[ \int \frac {\tanh (x)}{(1+\tanh (x))^{3/2}} \, dx=-\frac {1}{24} \, \sqrt {2} {\left (\frac {2 \, {\left (3 \, \sqrt {e^{\left (4 \, x\right )} + e^{\left (2 \, x\right )}} - 3 \, e^{\left (2 \, x\right )} - 1\right )}}{{\left (\sqrt {e^{\left (4 \, x\right )} + e^{\left (2 \, x\right )}} - e^{\left (2 \, x\right )}\right )}^{3}} + 3 \, \log \left (-2 \, \sqrt {e^{\left (4 \, x\right )} + e^{\left (2 \, x\right )}} + 2 \, e^{\left (2 \, x\right )} + 1\right )\right )} \] Input:
integrate(tanh(x)/(1+tanh(x))^(3/2),x, algorithm="giac")
Output:
-1/24*sqrt(2)*(2*(3*sqrt(e^(4*x) + e^(2*x)) - 3*e^(2*x) - 1)/(sqrt(e^(4*x) + e^(2*x)) - e^(2*x))^3 + 3*log(-2*sqrt(e^(4*x) + e^(2*x)) + 2*e^(2*x) + 1))
Time = 2.13 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.65 \[ \int \frac {\tanh (x)}{(1+\tanh (x))^{3/2}} \, dx=\frac {\sqrt {2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\sqrt {\mathrm {tanh}\left (x\right )+1}}{2}\right )}{4}-\frac {\frac {\mathrm {tanh}\left (x\right )}{2}+\frac {1}{6}}{{\left (\mathrm {tanh}\left (x\right )+1\right )}^{3/2}} \] Input:
int(tanh(x)/(tanh(x) + 1)^(3/2),x)
Output:
(2^(1/2)*atanh((2^(1/2)*(tanh(x) + 1)^(1/2))/2))/4 - (tanh(x)/2 + 1/6)/(ta nh(x) + 1)^(3/2)
\[ \int \frac {\tanh (x)}{(1+\tanh (x))^{3/2}} \, dx=\int \frac {\sqrt {\tanh \left (x \right )+1}\, \tanh \left (x \right )}{\tanh \left (x \right )^{2}+2 \tanh \left (x \right )+1}d x \] Input:
int(tanh(x)/(1+tanh(x))^(3/2),x)
Output:
int((sqrt(tanh(x) + 1)*tanh(x))/(tanh(x)**2 + 2*tanh(x) + 1),x)