Integrand size = 10, antiderivative size = 64 \[ \int \frac {1}{\sqrt {a \tanh ^3(x)}} \, dx=-\frac {2 \tanh (x)}{\sqrt {a \tanh ^3(x)}}-\frac {\arctan \left (\sqrt {\tanh (x)}\right ) \tanh ^{\frac {3}{2}}(x)}{\sqrt {a \tanh ^3(x)}}+\frac {\text {arctanh}\left (\sqrt {\tanh (x)}\right ) \tanh ^{\frac {3}{2}}(x)}{\sqrt {a \tanh ^3(x)}} \]
-2*tanh(x)/(a*tanh(x)^3)^(1/2)-arctan(tanh(x)^(1/2))*tanh(x)^(3/2)/(a*tanh (x)^3)^(1/2)+arctanh(tanh(x)^(1/2))*tanh(x)^(3/2)/(a*tanh(x)^3)^(1/2)
Time = 0.03 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.83 \[ \int \frac {1}{\sqrt {a \tanh ^3(x)}} \, dx=-\frac {\tanh (x) \left (2+\arctan \left (\sqrt [4]{\tanh ^2(x)}\right ) \sqrt [4]{\tanh ^2(x)}-\text {arctanh}\left (\sqrt [4]{\tanh ^2(x)}\right ) \sqrt [4]{\tanh ^2(x)}\right )}{\sqrt {a \tanh ^3(x)}} \]
-((Tanh[x]*(2 + ArcTan[(Tanh[x]^2)^(1/4)]*(Tanh[x]^2)^(1/4) - ArcTanh[(Tan h[x]^2)^(1/4)]*(Tanh[x]^2)^(1/4)))/Sqrt[a*Tanh[x]^3])
Time = 0.33 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.80, 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, 3955, 3042, 3957, 25, 266, 827, 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.
\(\displaystyle \int \frac {1}{\sqrt {a \tanh ^3(x)}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\sqrt {i a \tan (i x)^3}}dx\) |
\(\Big \downarrow \) 4141 |
\(\displaystyle \frac {\tanh ^{\frac {3}{2}}(x) \int \frac {1}{\tanh ^{\frac {3}{2}}(x)}dx}{\sqrt {a \tanh ^3(x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\tanh ^{\frac {3}{2}}(x) \int \frac {1}{(-i \tan (i x))^{3/2}}dx}{\sqrt {a \tanh ^3(x)}}\) |
\(\Big \downarrow \) 3955 |
\(\displaystyle \frac {\tanh ^{\frac {3}{2}}(x) \left (\int \sqrt {\tanh (x)}dx-\frac {2}{\sqrt {\tanh (x)}}\right )}{\sqrt {a \tanh ^3(x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\tanh ^{\frac {3}{2}}(x) \left (-\frac {2}{\sqrt {\tanh (x)}}+\int \sqrt {-i \tan (i x)}dx\right )}{\sqrt {a \tanh ^3(x)}}\) |
\(\Big \downarrow \) 3957 |
\(\displaystyle \frac {\tanh ^{\frac {3}{2}}(x) \left (-\int -\frac {\sqrt {\tanh (x)}}{1-\tanh ^2(x)}d\tanh (x)-\frac {2}{\sqrt {\tanh (x)}}\right )}{\sqrt {a \tanh ^3(x)}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\tanh ^{\frac {3}{2}}(x) \left (\int \frac {\sqrt {\tanh (x)}}{1-\tanh ^2(x)}d\tanh (x)-\frac {2}{\sqrt {\tanh (x)}}\right )}{\sqrt {a \tanh ^3(x)}}\) |
\(\Big \downarrow \) 266 |
\(\displaystyle \frac {\tanh ^{\frac {3}{2}}(x) \left (2 \int \frac {\tanh (x)}{1-\tanh ^2(x)}d\sqrt {\tanh (x)}-\frac {2}{\sqrt {\tanh (x)}}\right )}{\sqrt {a \tanh ^3(x)}}\) |
\(\Big \downarrow \) 827 |
\(\displaystyle \frac {\tanh ^{\frac {3}{2}}(x) \left (2 \left (\frac {1}{2} \int \frac {1}{1-\tanh (x)}d\sqrt {\tanh (x)}-\frac {1}{2} \int \frac {1}{\tanh (x)+1}d\sqrt {\tanh (x)}\right )-\frac {2}{\sqrt {\tanh (x)}}\right )}{\sqrt {a \tanh ^3(x)}}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {\tanh ^{\frac {3}{2}}(x) \left (2 \left (\frac {1}{2} \int \frac {1}{1-\tanh (x)}d\sqrt {\tanh (x)}-\frac {1}{2} \arctan \left (\sqrt {\tanh (x)}\right )\right )-\frac {2}{\sqrt {\tanh (x)}}\right )}{\sqrt {a \tanh ^3(x)}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {\tanh ^{\frac {3}{2}}(x) \left (2 \left (\frac {1}{2} \text {arctanh}\left (\sqrt {\tanh (x)}\right )-\frac {1}{2} \arctan \left (\sqrt {\tanh (x)}\right )\right )-\frac {2}{\sqrt {\tanh (x)}}\right )}{\sqrt {a \tanh ^3(x)}}\) |
((2*(-1/2*ArcTan[Sqrt[Tanh[x]]] + ArcTanh[Sqrt[Tanh[x]]]/2) - 2/Sqrt[Tanh[ x]])*Tanh[x]^(3/2))/Sqrt[a*Tanh[x]^3]
3.1.36.3.1 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[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2*b) Int[1/(r + s*x^2), x], x] - Simp[s/(2*b) 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*Tan[c + d*x] )^(n + 1)/(b*d*(n + 1)), x] - Simp[1/b^2 Int[(b*Tan[c + d*x])^(n + 2), x] , x] /; FreeQ[{b, c, d}, x] && LtQ[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 = 0.06 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.02
method | result | size |
derivativedivides | \(-\frac {\tanh \left (x \right ) \left (2 a^{\frac {5}{2}}-\operatorname {arctanh}\left (\frac {\sqrt {a \tanh \left (x \right )}}{\sqrt {a}}\right ) a^{2} \sqrt {a \tanh \left (x \right )}+\arctan \left (\frac {\sqrt {a \tanh \left (x \right )}}{\sqrt {a}}\right ) a^{2} \sqrt {a \tanh \left (x \right )}\right )}{\sqrt {a \tanh \left (x \right )^{3}}\, a^{\frac {5}{2}}}\) | \(65\) |
default | \(-\frac {\tanh \left (x \right ) \left (2 a^{\frac {5}{2}}-\operatorname {arctanh}\left (\frac {\sqrt {a \tanh \left (x \right )}}{\sqrt {a}}\right ) a^{2} \sqrt {a \tanh \left (x \right )}+\arctan \left (\frac {\sqrt {a \tanh \left (x \right )}}{\sqrt {a}}\right ) a^{2} \sqrt {a \tanh \left (x \right )}\right )}{\sqrt {a \tanh \left (x \right )^{3}}\, a^{\frac {5}{2}}}\) | \(65\) |
-tanh(x)*(2*a^(5/2)-arctanh((a*tanh(x))^(1/2)/a^(1/2))*a^2*(a*tanh(x))^(1/ 2)+arctan((a*tanh(x))^(1/2)/a^(1/2))*a^2*(a*tanh(x))^(1/2))/(a*tanh(x)^3)^ (1/2)/a^(5/2)
Leaf count of result is larger than twice the leaf count of optimal. 239 vs. \(2 (50) = 100\).
Time = 0.26 (sec) , antiderivative size = 516, normalized size of antiderivative = 8.06 \[ \int \frac {1}{\sqrt {a \tanh ^3(x)}} \, dx=\left [-\frac {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} \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 ) + {\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 1\right )} \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 ) + 8 \, {\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 1\right )} \sqrt {\frac {a \sinh \left (x\right )}{\cosh \left (x\right )}}}{4 \, {\left (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 {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} \arctan \left (\frac {{\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 1\right )} \sqrt {\frac {a \sinh \left (x\right )}{\cosh \left (x\right )}}}{\sqrt {a}}\right ) - {\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 1\right )} \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 ) + 8 \, {\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 1\right )} \sqrt {\frac {a \sinh \left (x\right )}{\cosh \left (x\right )}}}{4 \, {\left (a \cosh \left (x\right )^{2} + 2 \, a \cosh \left (x\right ) \sinh \left (x\right ) + a \sinh \left (x\right )^{2} - a\right )}}\right ] \]
[-1/4*(2*(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 - 1)*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)) + (cosh(x)^2 + 2* cosh(x)*sinh(x) + sinh(x)^2 - 1)*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*cosh(x)*sinh(x) + sinh(x)^2 + 1)*sqrt(-a)*sqrt(a*sinh(x )/cosh(x)) - 2*a)/(cosh(x)^4 + 4*cosh(x)^3*sinh(x) + 6*cosh(x)^2*sinh(x)^2 + 4*cosh(x)*sinh(x)^3 + sinh(x)^4)) + 8*(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 + 1)*sqrt(a*sinh(x)/cosh(x)))/(a*cosh(x)^2 + 2*a*cosh(x)*sinh(x) + a*sinh(x)^2 - a), -1/4*(2*(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 - 1)*sqrt(a)*arctan((cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 + 1)*sqrt(a*s inh(x)/cosh(x))/sqrt(a)) - (cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 - 1) *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*cosh(x)*sin h(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) + 8*(cosh(x)^ 2 + 2*cosh(x)*sinh(x) + sinh(x)^2 + 1)*sqrt(a*sinh(x)/cosh(x)))/(a*cosh(x) ^2 + 2*a*cosh(x)*sinh(x) + a*sinh(x)^2 - a)]
\[ \int \frac {1}{\sqrt {a \tanh ^3(x)}} \, dx=\int \frac {1}{\sqrt {a \tanh ^{3}{\left (x \right )}}}\, dx \]
\[ \int \frac {1}{\sqrt {a \tanh ^3(x)}} \, dx=\int { \frac {1}{\sqrt {a \tanh \left (x\right )^{3}}} \,d x } \]
Time = 0.28 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.64 \[ \int \frac {1}{\sqrt {a \tanh ^3(x)}} \, dx=\frac {4}{{\left (\sqrt {a} e^{\left (2 \, x\right )} - \sqrt {a e^{\left (4 \, x\right )} - a} - \sqrt {a}\right )} \mathrm {sgn}\left (e^{\left (4 \, x\right )} - 1\right )} \]
Timed out. \[ \int \frac {1}{\sqrt {a \tanh ^3(x)}} \, dx=\int \frac {1}{\sqrt {a\,{\mathrm {tanh}\left (x\right )}^3}} \,d x \]