Integrand size = 10, antiderivative size = 63 \[ \int \sqrt {a \tanh ^3(x)} \, dx=-2 \coth (x) \sqrt {a \tanh ^3(x)}+\frac {\arctan \left (\sqrt {\tanh (x)}\right ) \sqrt {a \tanh ^3(x)}}{\tanh ^{\frac {3}{2}}(x)}+\frac {\text {arctanh}\left (\sqrt {\tanh (x)}\right ) \sqrt {a \tanh ^3(x)}}{\tanh ^{\frac {3}{2}}(x)} \]
-2*coth(x)*(a*tanh(x)^3)^(1/2)+arctan(tanh(x)^(1/2))*(a*tanh(x)^3)^(1/2)/t anh(x)^(3/2)+arctanh(tanh(x)^(1/2))*(a*tanh(x)^3)^(1/2)/tanh(x)^(3/2)
Time = 0.03 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.63 \[ \int \sqrt {a \tanh ^3(x)} \, dx=\frac {\left (\arctan \left (\sqrt {\tanh (x)}\right )+\text {arctanh}\left (\sqrt {\tanh (x)}\right )-2 \sqrt {\tanh (x)}\right ) \sqrt {a \tanh ^3(x)}}{\tanh ^{\frac {3}{2}}(x)} \]
((ArcTan[Sqrt[Tanh[x]]] + ArcTanh[Sqrt[Tanh[x]]] - 2*Sqrt[Tanh[x]])*Sqrt[a *Tanh[x]^3])/Tanh[x]^(3/2)
Time = 0.32 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.81, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.100, Rules used = {3042, 4141, 3042, 3954, 3042, 3957, 25, 266, 756, 216, 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.
| \(\displaystyle \int \sqrt {a \tanh ^3(x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sqrt {i a \tan (i x)^3}dx\) |
| \(\Big \downarrow \) 4141 |
| \(\displaystyle \frac {\sqrt {a \tanh ^3(x)} \int \tanh ^{\frac {3}{2}}(x)dx}{\tanh ^{\frac {3}{2}}(x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sqrt {a \tanh ^3(x)} \int (-i \tan (i x))^{3/2}dx}{\tanh ^{\frac {3}{2}}(x)}\) |
| \(\Big \downarrow \) 3954 |
| \(\displaystyle \frac {\sqrt {a \tanh ^3(x)} \left (\int \frac {1}{\sqrt {\tanh (x)}}dx-2 \sqrt {\tanh (x)}\right )}{\tanh ^{\frac {3}{2}}(x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sqrt {a \tanh ^3(x)} \left (-2 \sqrt {\tanh (x)}+\int \frac {1}{\sqrt {-i \tan (i x)}}dx\right )}{\tanh ^{\frac {3}{2}}(x)}\) |
| \(\Big \downarrow \) 3957 |
| \(\displaystyle \frac {\sqrt {a \tanh ^3(x)} \left (-\int -\frac {1}{\sqrt {\tanh (x)} \left (1-\tanh ^2(x)\right )}d\tanh (x)-2 \sqrt {\tanh (x)}\right )}{\tanh ^{\frac {3}{2}}(x)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {a \tanh ^3(x)} \left (\int \frac {1}{\sqrt {\tanh (x)} \left (1-\tanh ^2(x)\right )}d\tanh (x)-2 \sqrt {\tanh (x)}\right )}{\tanh ^{\frac {3}{2}}(x)}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {\sqrt {a \tanh ^3(x)} \left (2 \int \frac {1}{1-\tanh ^2(x)}d\sqrt {\tanh (x)}-2 \sqrt {\tanh (x)}\right )}{\tanh ^{\frac {3}{2}}(x)}\) |
| \(\Big \downarrow \) 756 |
| \(\displaystyle \frac {\sqrt {a \tanh ^3(x)} \left (2 \left (\frac {1}{2} \int \frac {1}{1-\tanh (x)}d\sqrt {\tanh (x)}+\frac {1}{2} \int \frac {1}{\tanh (x)+1}d\sqrt {\tanh (x)}\right )-2 \sqrt {\tanh (x)}\right )}{\tanh ^{\frac {3}{2}}(x)}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {\sqrt {a \tanh ^3(x)} \left (2 \left (\frac {1}{2} \int \frac {1}{1-\tanh (x)}d\sqrt {\tanh (x)}+\frac {1}{2} \arctan \left (\sqrt {\tanh (x)}\right )\right )-2 \sqrt {\tanh (x)}\right )}{\tanh ^{\frac {3}{2}}(x)}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\sqrt {a \tanh ^3(x)} \left (2 \left (\frac {1}{2} \arctan \left (\sqrt {\tanh (x)}\right )+\frac {1}{2} \text {arctanh}\left (\sqrt {\tanh (x)}\right )\right )-2 \sqrt {\tanh (x)}\right )}{\tanh ^{\frac {3}{2}}(x)}\) |
((2*(ArcTan[Sqrt[Tanh[x]]]/2 + ArcTanh[Sqrt[Tanh[x]]]/2) - 2*Sqrt[Tanh[x]] )*Sqrt[a*Tanh[x]^3])/Tanh[x]^(3/2)
3.1.35.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
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[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De nominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) ^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I ntBinomialQ[a, b, c, 2, m, p, x]
Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2 ]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a) Int[1/(r - s*x^2), x], x] + Simp[r/(2*a) Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ[a /b, 0]
Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((b*Tan[c + d *x])^(n - 1)/(d*(n - 1))), x] - Simp[b^2 Int[(b*Tan[c + d*x])^(n - 2), x] , x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]
Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b/d Subst[Int [x^n/(b^2 + x^2), x], x, b*Tan[c + d*x]], x] /; FreeQ[{b, c, d, n}, x] && !IntegerQ[n]
Int[(u_.)*((b_.)*tan[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[(b*ff^n)^IntPart[p]*((b*Tan[e + f*x]^ n)^FracPart[p]/(Tan[e + f*x]/ff)^(n*FracPart[p])) Int[ActivateTrig[u]*(Ta n[e + f*x]/ff)^(n*p), x], x]] /; FreeQ[{b, e, f, n, p}, x] && !IntegerQ[p] && IntegerQ[n] && (EqQ[u, 1] || MatchQ[u, ((d_.)*(trig_)[e + f*x])^(m_.) / ; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig]])
Time = 0.04 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.94
| method | result | size |
| derivativedivides | \(\frac {\sqrt {a \tanh \left (x \right )^{3}}\, \left (-2 \sqrt {a \tanh \left (x \right )}+\sqrt {a}\, \operatorname {arctanh}\left (\frac {\sqrt {a \tanh \left (x \right )}}{\sqrt {a}}\right )+\sqrt {a}\, \arctan \left (\frac {\sqrt {a \tanh \left (x \right )}}{\sqrt {a}}\right )\right )}{\tanh \left (x \right ) \sqrt {a \tanh \left (x \right )}}\) | \(59\) |
| default | \(\frac {\sqrt {a \tanh \left (x \right )^{3}}\, \left (-2 \sqrt {a \tanh \left (x \right )}+\sqrt {a}\, \operatorname {arctanh}\left (\frac {\sqrt {a \tanh \left (x \right )}}{\sqrt {a}}\right )+\sqrt {a}\, \arctan \left (\frac {\sqrt {a \tanh \left (x \right )}}{\sqrt {a}}\right )\right )}{\tanh \left (x \right ) \sqrt {a \tanh \left (x \right )}}\) | \(59\) |
(a*tanh(x)^3)^(1/2)/tanh(x)/(a*tanh(x))^(1/2)*(-2*(a*tanh(x))^(1/2)+a^(1/2 )*arctanh((a*tanh(x))^(1/2)/a^(1/2))+a^(1/2)*arctan((a*tanh(x))^(1/2)/a^(1 /2)))
Leaf count of result is larger than twice the leaf count of optimal. 173 vs. \(2 (49) = 98\).
Time = 0.27 (sec) , antiderivative size = 376, normalized size of antiderivative = 5.97 \[ \int \sqrt {a \tanh ^3(x)} \, dx=\left [-\frac {1}{2} \, \sqrt {-a} \arctan \left (\frac {{\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}\right )} \sqrt {-a} \sqrt {\frac {a \sinh \left (x\right )}{\cosh \left (x\right )}}}{a \cosh \left (x\right )^{2} + 2 \, a \cosh \left (x\right ) \sinh \left (x\right ) + a \sinh \left (x\right )^{2} - a}\right ) + \frac {1}{4} \, \sqrt {-a} \log \left (-\frac {a \cosh \left (x\right )^{4} + 4 \, a \cosh \left (x\right )^{3} \sinh \left (x\right ) + 6 \, a \cosh \left (x\right )^{2} \sinh \left (x\right )^{2} + 4 \, a \cosh \left (x\right ) \sinh \left (x\right )^{3} + a \sinh \left (x\right )^{4} + 2 \, {\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 1\right )} \sqrt {-a} \sqrt {\frac {a \sinh \left (x\right )}{\cosh \left (x\right )}} - 2 \, a}{\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right )^{3} \sinh \left (x\right ) + 6 \, \cosh \left (x\right )^{2} \sinh \left (x\right )^{2} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4}}\right ) - 2 \, \sqrt {\frac {a \sinh \left (x\right )}{\cosh \left (x\right )}}, -\frac {1}{2} \, \sqrt {a} \arctan \left (\frac {\sqrt {a} \sqrt {\frac {a \sinh \left (x\right )}{\cosh \left (x\right )}}}{a \cosh \left (x\right )^{2} + 2 \, a \cosh \left (x\right ) \sinh \left (x\right ) + a \sinh \left (x\right )^{2} - a}\right ) + \frac {1}{4} \, \sqrt {a} \log \left (2 \, a \cosh \left (x\right )^{4} + 8 \, a \cosh \left (x\right )^{3} \sinh \left (x\right ) + 12 \, a \cosh \left (x\right )^{2} \sinh \left (x\right )^{2} + 8 \, a \cosh \left (x\right ) \sinh \left (x\right )^{3} + 2 \, a \sinh \left (x\right )^{4} + 2 \, {\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + {\left (6 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{2} + \cosh \left (x\right )^{2} + 2 \, {\left (2 \, \cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} \sinh \left (x\right )\right )} \sqrt {a} \sqrt {\frac {a \sinh \left (x\right )}{\cosh \left (x\right )}} - a\right ) - 2 \, \sqrt {\frac {a \sinh \left (x\right )}{\cosh \left (x\right )}}\right ] \]
[-1/2*sqrt(-a)*arctan((cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2)*sqrt(-a) *sqrt(a*sinh(x)/cosh(x))/(a*cosh(x)^2 + 2*a*cosh(x)*sinh(x) + a*sinh(x)^2 - a)) + 1/4*sqrt(-a)*log(-(a*cosh(x)^4 + 4*a*cosh(x)^3*sinh(x) + 6*a*cosh( x)^2*sinh(x)^2 + 4*a*cosh(x)*sinh(x)^3 + a*sinh(x)^4 + 2*(cosh(x)^2 + 2*co sh(x)*sinh(x) + sinh(x)^2 + 1)*sqrt(-a)*sqrt(a*sinh(x)/cosh(x)) - 2*a)/(co sh(x)^4 + 4*cosh(x)^3*sinh(x) + 6*cosh(x)^2*sinh(x)^2 + 4*cosh(x)*sinh(x)^ 3 + sinh(x)^4)) - 2*sqrt(a*sinh(x)/cosh(x)), -1/2*sqrt(a)*arctan(sqrt(a)*s qrt(a*sinh(x)/cosh(x))/(a*cosh(x)^2 + 2*a*cosh(x)*sinh(x) + a*sinh(x)^2 - a)) + 1/4*sqrt(a)*log(2*a*cosh(x)^4 + 8*a*cosh(x)^3*sinh(x) + 12*a*cosh(x) ^2*sinh(x)^2 + 8*a*cosh(x)*sinh(x)^3 + 2*a*sinh(x)^4 + 2*(cosh(x)^4 + 4*co sh(x)*sinh(x)^3 + sinh(x)^4 + (6*cosh(x)^2 + 1)*sinh(x)^2 + cosh(x)^2 + 2* (2*cosh(x)^3 + cosh(x))*sinh(x))*sqrt(a)*sqrt(a*sinh(x)/cosh(x)) - a) - 2* sqrt(a*sinh(x)/cosh(x))]
\[ \int \sqrt {a \tanh ^3(x)} \, dx=\int \sqrt {a \tanh ^{3}{\left (x \right )}}\, dx \]
\[ \int \sqrt {a \tanh ^3(x)} \, dx=\int { \sqrt {a \tanh \left (x\right )^{3}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 115 vs. \(2 (49) = 98\).
Time = 0.30 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.83 \[ \int \sqrt {a \tanh ^3(x)} \, dx=\sqrt {a} \arctan \left (-\frac {\sqrt {a} e^{\left (2 \, x\right )} - \sqrt {a e^{\left (4 \, x\right )} - a}}{\sqrt {a}}\right ) \mathrm {sgn}\left (e^{\left (4 \, x\right )} - 1\right ) - \frac {1}{2} \, \sqrt {a} \log \left ({\left | -\sqrt {a} e^{\left (2 \, x\right )} + \sqrt {a e^{\left (4 \, x\right )} - a} \right |}\right ) \mathrm {sgn}\left (e^{\left (4 \, x\right )} - 1\right ) - \frac {4 \, a \mathrm {sgn}\left (e^{\left (4 \, x\right )} - 1\right )}{\sqrt {a} e^{\left (2 \, x\right )} - \sqrt {a e^{\left (4 \, x\right )} - a} + \sqrt {a}} \]
sqrt(a)*arctan(-(sqrt(a)*e^(2*x) - sqrt(a*e^(4*x) - a))/sqrt(a))*sgn(e^(4* x) - 1) - 1/2*sqrt(a)*log(abs(-sqrt(a)*e^(2*x) + sqrt(a*e^(4*x) - a)))*sgn (e^(4*x) - 1) - 4*a*sgn(e^(4*x) - 1)/(sqrt(a)*e^(2*x) - sqrt(a*e^(4*x) - a ) + sqrt(a))
Timed out. \[ \int \sqrt {a \tanh ^3(x)} \, dx=\int \sqrt {a\,{\mathrm {tanh}\left (x\right )}^3} \,d x \]