Integrand size = 11, antiderivative size = 45 \[ \int \tanh (x) (1+\tanh (x))^{3/2} \, dx=2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {1+\tanh (x)}}{\sqrt {2}}\right )-2 \sqrt {1+\tanh (x)}-\frac {2}{3} (1+\tanh (x))^{3/2} \] Output:
2*2^(1/2)*arctanh(1/2*(1+tanh(x))^(1/2)*2^(1/2))-2*(1+tanh(x))^(1/2)-2/3*( 1+tanh(x))^(3/2)
Time = 0.49 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.87 \[ \int \tanh (x) (1+\tanh (x))^{3/2} \, dx=2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {1+\tanh (x)}}{\sqrt {2}}\right )-\frac {2}{3} \sqrt {1+\tanh (x)} (4+\tanh (x)) \] Input:
Integrate[Tanh[x]*(1 + Tanh[x])^(3/2),x]
Output:
2*Sqrt[2]*ArcTanh[Sqrt[1 + Tanh[x]]/Sqrt[2]] - (2*Sqrt[1 + Tanh[x]]*(4 + T anh[x]))/3
Result contains complex when optimal does not.
Time = 0.35 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.24, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.727, Rules used = {3042, 26, 4010, 3042, 3959, 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.
\(\displaystyle \int \tanh (x) (\tanh (x)+1)^{3/2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -i (1-i \tan (i x))^{3/2} \tan (i x)dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -i \int (1-i \tan (i x))^{3/2} \tan (i x)dx\) |
\(\Big \downarrow \) 4010 |
\(\displaystyle -i \left (i \int (\tanh (x)+1)^{3/2}dx-\frac {2}{3} i (\tanh (x)+1)^{3/2}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -i \left (i \int (1-i \tan (i x))^{3/2}dx-\frac {2}{3} i (\tanh (x)+1)^{3/2}\right )\) |
\(\Big \downarrow \) 3959 |
\(\displaystyle -i \left (i \left (2 \int \sqrt {\tanh (x)+1}dx-2 \sqrt {\tanh (x)+1}\right )-\frac {2}{3} i (\tanh (x)+1)^{3/2}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -i \left (i \left (-2 \sqrt {\tanh (x)+1}+2 \int \sqrt {1-i \tan (i x)}dx\right )-\frac {2}{3} i (\tanh (x)+1)^{3/2}\right )\) |
\(\Big \downarrow \) 3961 |
\(\displaystyle -i \left (i \left (4 \int \frac {1}{1-\tanh (x)}d\sqrt {\tanh (x)+1}-2 \sqrt {\tanh (x)+1}\right )-\frac {2}{3} i (\tanh (x)+1)^{3/2}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -i \left (i \left (2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {\tanh (x)+1}}{\sqrt {2}}\right )-2 \sqrt {\tanh (x)+1}\right )-\frac {2}{3} i (\tanh (x)+1)^{3/2}\right )\) |
Input:
Int[Tanh[x]*(1 + Tanh[x])^(3/2),x]
Output:
(-I)*(((-2*I)/3)*(1 + Tanh[x])^(3/2) + I*(2*Sqrt[2]*ArcTanh[Sqrt[1 + Tanh[ x]]/Sqrt[2]] - 2*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[b*((a + b*Tan[c + d*x])^(n - 1)/(d*(n - 1))), x] + Simp[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] && GtQ[n , 1]
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[d*((a + b*Tan[e + f*x])^m/(f*m)), x] + Simp [(b*c + a*d)/b Int[(a + b*Tan[e + f*x])^m, x], x] /; FreeQ[{a, b, c, d, e , f, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && !LtQ[m, 0]
Time = 0.24 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.78
method | result | size |
derivativedivides | \(2 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {1+\tanh \left (x \right )}\, \sqrt {2}}{2}\right )-2 \sqrt {1+\tanh \left (x \right )}-\frac {2 \left (1+\tanh \left (x \right )\right )^{\frac {3}{2}}}{3}\) | \(35\) |
default | \(2 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {1+\tanh \left (x \right )}\, \sqrt {2}}{2}\right )-2 \sqrt {1+\tanh \left (x \right )}-\frac {2 \left (1+\tanh \left (x \right )\right )^{\frac {3}{2}}}{3}\) | \(35\) |
Input:
int(tanh(x)*(1+tanh(x))^(3/2),x,method=_RETURNVERBOSE)
Output:
2*2^(1/2)*arctanh(1/2*(1+tanh(x))^(1/2)*2^(1/2))-2*(1+tanh(x))^(1/2)-2/3*( 1+tanh(x))^(3/2)
Leaf count of result is larger than twice the leaf count of optimal. 206 vs. \(2 (34) = 68\).
Time = 0.09 (sec) , antiderivative size = 206, normalized size of antiderivative = 4.58 \[ \int \tanh (x) (1+\tanh (x))^{3/2} \, dx=\frac {3 \, {\left (\sqrt {2} \cosh \left (x\right )^{2} + 2 \, \sqrt {2} \cosh \left (x\right ) \sinh \left (x\right ) + \sqrt {2} \sinh \left (x\right )^{2} + \sqrt {2}\right )} \log \left (-2 \, \cosh \left (x\right )^{2} - 4 \, \cosh \left (x\right ) \sinh \left (x\right ) - 2 \, \sinh \left (x\right )^{2} - \frac {\sqrt {2} {\left (\sqrt {2} \cosh \left (x\right )^{3} + 3 \, \sqrt {2} \cosh \left (x\right ) \sinh \left (x\right )^{2} + \sqrt {2} \sinh \left (x\right )^{3} + {\left (3 \, \sqrt {2} \cosh \left (x\right )^{2} + \sqrt {2}\right )} \sinh \left (x\right ) + \sqrt {2} \cosh \left (x\right )\right )}}{\sqrt {\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 1}} - 1\right ) - \frac {2 \, \sqrt {2} {\left (5 \, \cosh \left (x\right )^{3} + 15 \, \cosh \left (x\right ) \sinh \left (x\right )^{2} + 5 \, \sinh \left (x\right )^{3} + 3 \, {\left (5 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right ) + 3 \, \cosh \left (x\right )\right )}}{\sqrt {\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 1}}}{3 \, {\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 1\right )}} \] Input:
integrate(tanh(x)*(1+tanh(x))^(3/2),x, algorithm="fricas")
Output:
1/3*(3*(sqrt(2)*cosh(x)^2 + 2*sqrt(2)*cosh(x)*sinh(x) + sqrt(2)*sinh(x)^2 + sqrt(2))*log(-2*cosh(x)^2 - 4*cosh(x)*sinh(x) - 2*sinh(x)^2 - sqrt(2)*(s qrt(2)*cosh(x)^3 + 3*sqrt(2)*cosh(x)*sinh(x)^2 + sqrt(2)*sinh(x)^3 + (3*sq rt(2)*cosh(x)^2 + sqrt(2))*sinh(x) + sqrt(2)*cosh(x))/sqrt(cosh(x)^2 + 2*c osh(x)*sinh(x) + sinh(x)^2 + 1) - 1) - 2*sqrt(2)*(5*cosh(x)^3 + 15*cosh(x) *sinh(x)^2 + 5*sinh(x)^3 + 3*(5*cosh(x)^2 + 1)*sinh(x) + 3*cosh(x))/sqrt(c osh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 + 1))/(cosh(x)^2 + 2*cosh(x)*sinh (x) + sinh(x)^2 + 1)
Time = 3.56 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.29 \[ \int \tanh (x) (1+\tanh (x))^{3/2} \, dx=- \sqrt {2} \left (\log {\left (\sqrt {\tanh {\left (x \right )} + 1} - \sqrt {2} \right )} - \log {\left (\sqrt {\tanh {\left (x \right )} + 1} + \sqrt {2} \right )}\right ) - \frac {2 \left (\tanh {\left (x \right )} + 1\right )^{\frac {3}{2}}}{3} - 2 \sqrt {\tanh {\left (x \right )} + 1} \] 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))) - 2*(tanh(x) + 1)**(3/2)/3 - 2*sqrt(tanh(x) + 1)
\[ \int \tanh (x) (1+\tanh (x))^{3/2} \, dx=\int { {\left (\tanh \left (x\right ) + 1\right )}^{\frac {3}{2}} \tanh \left (x\right ) \,d x } \] Input:
integrate(tanh(x)*(1+tanh(x))^(3/2),x, algorithm="maxima")
Output:
-2/3*sqrt(2)/(e^(-2*x) + 1)^(3/2) + integrate(2*sqrt(2)*e^(-x)/((e^(-x) + e^(-3*x))*(e^(-2*x) + 1)^(3/2)), x)
Leaf count of result is larger than twice the leaf count of optimal. 96 vs. \(2 (34) = 68\).
Time = 0.13 (sec) , antiderivative size = 96, normalized size of antiderivative = 2.13 \[ \int \tanh (x) (1+\tanh (x))^{3/2} \, dx=\frac {1}{3} \, \sqrt {2} {\left (\frac {2 \, {\left (9 \, {\left (\sqrt {e^{\left (4 \, x\right )} + e^{\left (2 \, x\right )}} - e^{\left (2 \, x\right )}\right )}^{2} - 12 \, \sqrt {e^{\left (4 \, x\right )} + e^{\left (2 \, x\right )}} + 12 \, e^{\left (2 \, x\right )} + 5\right )}}{{\left (\sqrt {e^{\left (4 \, x\right )} + e^{\left (2 \, x\right )}} - e^{\left (2 \, x\right )} - 1\right )}^{3}} - 3 \, \log \left (-2 \, \sqrt {e^{\left (4 \, x\right )} + e^{\left (2 \, x\right )}} + 2 \, e^{\left (2 \, x\right )} + 1\right )\right )} \] Input:
integrate(tanh(x)*(1+tanh(x))^(3/2),x, algorithm="giac")
Output:
1/3*sqrt(2)*(2*(9*(sqrt(e^(4*x) + e^(2*x)) - e^(2*x))^2 - 12*sqrt(e^(4*x) + e^(2*x)) + 12*e^(2*x) + 5)/(sqrt(e^(4*x) + e^(2*x)) - e^(2*x) - 1)^3 - 3 *log(-2*sqrt(e^(4*x) + e^(2*x)) + 2*e^(2*x) + 1))
Time = 0.14 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.76 \[ \int \tanh (x) (1+\tanh (x))^{3/2} \, dx=2\,\sqrt {2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\sqrt {\mathrm {tanh}\left (x\right )+1}}{2}\right )-2\,\sqrt {\mathrm {tanh}\left (x\right )+1}-\frac {2\,{\left (\mathrm {tanh}\left (x\right )+1\right )}^{3/2}}{3} \] Input:
int(tanh(x)*(tanh(x) + 1)^(3/2),x)
Output:
2*2^(1/2)*atanh((2^(1/2)*(tanh(x) + 1)^(1/2))/2) - 2*(tanh(x) + 1)^(1/2) - (2*(tanh(x) + 1)^(3/2))/3
\[ \int \tanh (x) (1+\tanh (x))^{3/2} \, dx=\int \sqrt {\tanh \left (x \right )+1}\, \tanh \left (x \right )^{2}d x +\int \sqrt {\tanh \left (x \right )+1}\, \tanh \left (x \right )d x \] Input:
int(tanh(x)*(1+tanh(x))^(3/2),x)
Output:
int(sqrt(tanh(x) + 1)*tanh(x)**2,x) + int(sqrt(tanh(x) + 1)*tanh(x),x)