3.226 \(\int \frac {\tanh ^{-1}(\tanh (a+b x))^{3/2}}{x^{3/2}} \, dx\)
Optimal. Leaf size=81 \[ -\frac {2 \tanh ^{-1}(\tanh (a+b x))^{3/2}}{\sqrt {x}}+3 b \sqrt {x} \sqrt {\tanh ^{-1}(\tanh (a+b x))}-3 \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {\tanh ^{-1}(\tanh (a+b x))}}\right ) \left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \]
[Out]
-3*arctanh(b^(1/2)*x^(1/2)/arctanh(tanh(b*x+a))^(1/2))*(b*x-arctanh(tanh(b*x+a)))*b^(1/2)-2*arctanh(tanh(b*x+a
))^(3/2)/x^(1/2)+3*b*x^(1/2)*arctanh(tanh(b*x+a))^(1/2)
________________________________________________________________________________________
Rubi [A] time = 0.06, antiderivative size = 81, normalized size of antiderivative = 1.00,
number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used =
{2168, 2169, 2165} \[ -\frac {2 \tanh ^{-1}(\tanh (a+b x))^{3/2}}{\sqrt {x}}+3 b \sqrt {x} \sqrt {\tanh ^{-1}(\tanh (a+b x))}-3 \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {\tanh ^{-1}(\tanh (a+b x))}}\right ) \left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \]
Antiderivative was successfully verified.
[In]
Int[ArcTanh[Tanh[a + b*x]]^(3/2)/x^(3/2),x]
[Out]
-3*Sqrt[b]*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[ArcTanh[Tanh[a + b*x]]]]*(b*x - ArcTanh[Tanh[a + b*x]]) + 3*b*Sqrt[x
]*Sqrt[ArcTanh[Tanh[a + b*x]]] - (2*ArcTanh[Tanh[a + b*x]]^(3/2))/Sqrt[x]
Rule 2165
Int[1/(Sqrt[u_]*Sqrt[v_]), x_Symbol] :> With[{a = Simplify[D[u, x]], b = Simplify[D[v, x]]}, Simp[(2*ArcTanh[(
Rt[a*b, 2]*Sqrt[u])/(a*Sqrt[v])])/Rt[a*b, 2], x] /; NeQ[b*u - a*v, 0] && PosQ[a*b]] /; PiecewiseLinearQ[u, v,
x]
Rule 2168
Int[(u_)^(m_)*(v_)^(n_.), x_Symbol] :> With[{a = Simplify[D[u, x]], b = Simplify[D[v, x]]}, Simp[(u^(m + 1)*v^
n)/(a*(m + 1)), x] - Dist[(b*n)/(a*(m + 1)), Int[u^(m + 1)*v^(n - 1), x], x] /; NeQ[b*u - a*v, 0]] /; FreeQ[{m
, n}, x] && PiecewiseLinearQ[u, v, x] && NeQ[m, -1] && ((LtQ[m, -1] && GtQ[n, 0] && !(ILtQ[m + n, -2] && (Fra
ctionQ[m] || GeQ[2*n + m + 1, 0]))) || (IGtQ[n, 0] && IGtQ[m, 0] && LeQ[n, m]) || (IGtQ[n, 0] && !IntegerQ[m]
) || (ILtQ[m, 0] && !IntegerQ[n]))
Rule 2169
Int[(u_)^(m_)*(v_)^(n_.), x_Symbol] :> With[{a = Simplify[D[u, x]], b = Simplify[D[v, x]]}, Simp[(u^(m + 1)*v^
n)/(a*(m + n + 1)), x] - Dist[(n*(b*u - a*v))/(a*(m + n + 1)), Int[u^m*v^(n - 1), x], x] /; NeQ[b*u - a*v, 0]]
/; PiecewiseLinearQ[u, v, x] && NeQ[m + n + 2, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !In
tegerQ[n] || LtQ[0, m, n])) && !ILtQ[m + n, -2]
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(\tanh (a+b x))^{3/2}}{x^{3/2}} \, dx &=-\frac {2 \tanh ^{-1}(\tanh (a+b x))^{3/2}}{\sqrt {x}}+(3 b) \int \frac {\sqrt {\tanh ^{-1}(\tanh (a+b x))}}{\sqrt {x}} \, dx\\ &=3 b \sqrt {x} \sqrt {\tanh ^{-1}(\tanh (a+b x))}-\frac {2 \tanh ^{-1}(\tanh (a+b x))^{3/2}}{\sqrt {x}}-\frac {1}{2} \left (3 b \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )\right ) \int \frac {1}{\sqrt {x} \sqrt {\tanh ^{-1}(\tanh (a+b x))}} \, dx\\ &=-3 \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {\tanh ^{-1}(\tanh (a+b x))}}\right ) \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )+3 b \sqrt {x} \sqrt {\tanh ^{-1}(\tanh (a+b x))}-\frac {2 \tanh ^{-1}(\tanh (a+b x))^{3/2}}{\sqrt {x}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.06, size = 77, normalized size = 0.95 \[ \frac {\sqrt {\tanh ^{-1}(\tanh (a+b x))} \left (3 b x-2 \tanh ^{-1}(\tanh (a+b x))\right )}{\sqrt {x}}+3 \sqrt {b} \left (\tanh ^{-1}(\tanh (a+b x))-b x\right ) \log \left (\sqrt {b} \sqrt {\tanh ^{-1}(\tanh (a+b x))}+b \sqrt {x}\right ) \]
Antiderivative was successfully verified.
[In]
Integrate[ArcTanh[Tanh[a + b*x]]^(3/2)/x^(3/2),x]
[Out]
((3*b*x - 2*ArcTanh[Tanh[a + b*x]])*Sqrt[ArcTanh[Tanh[a + b*x]]])/Sqrt[x] + 3*Sqrt[b]*(-(b*x) + ArcTanh[Tanh[a
+ b*x]])*Log[b*Sqrt[x] + Sqrt[b]*Sqrt[ArcTanh[Tanh[a + b*x]]]]
________________________________________________________________________________________
fricas [A] time = 0.50, size = 109, normalized size = 1.35 \[ \left [\frac {3 \, a \sqrt {b} x \log \left (2 \, b x + 2 \, \sqrt {b x + a} \sqrt {b} \sqrt {x} + a\right ) + 2 \, \sqrt {b x + a} {\left (b x - 2 \, a\right )} \sqrt {x}}{2 \, x}, -\frac {3 \, a \sqrt {-b} x \arctan \left (\frac {\sqrt {b x + a} \sqrt {-b}}{b \sqrt {x}}\right ) - \sqrt {b x + a} {\left (b x - 2 \, a\right )} \sqrt {x}}{x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(arctanh(tanh(b*x+a))^(3/2)/x^(3/2),x, algorithm="fricas")
[Out]
[1/2*(3*a*sqrt(b)*x*log(2*b*x + 2*sqrt(b*x + a)*sqrt(b)*sqrt(x) + a) + 2*sqrt(b*x + a)*(b*x - 2*a)*sqrt(x))/x,
-(3*a*sqrt(-b)*x*arctan(sqrt(b*x + a)*sqrt(-b)/(b*sqrt(x))) - sqrt(b*x + a)*(b*x - 2*a)*sqrt(x))/x]
________________________________________________________________________________________
giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(arctanh(tanh(b*x+a))^(3/2)/x^(3/2),x, algorithm="giac")
[Out]
Exception raised: NotImplementedError >> Unable to parse Giac output: Warning, choosing root of [1,0,%%%{-4,[1
,0,0]%%%}+%%%{-4,[0,1,1]%%%}+%%%{-4,[0,1,0]%%%}+%%%{-4,[0,0,1]%%%},0,%%%{6,[2,0,0]%%%}+%%%{12,[1,1,1]%%%}+%%%{
4,[1,1,0]%%%}+%%%{4,[1,0,1]%%%}+%%%{6,[0,2,2]%%%}+%%%{4,[0,2,1]%%%}+%%%{6,[0,2,0]%%%}+%%%{4,[0,1,2]%%%}+%%%{4,
[0,1,1]%%%}+%%%{6,[0,0,2]%%%},0,%%%{-4,[3,0,0]%%%}+%%%{-12,[2,1,1]%%%}+%%%{4,[2,1,0]%%%}+%%%{4,[2,0,1]%%%}+%%%
{-12,[1,2,2]%%%}+%%%{8,[1,2,1]%%%}+%%%{4,[1,2,0]%%%}+%%%{8,[1,1,2]%%%}+%%%{-40,[1,1,1]%%%}+%%%{4,[1,0,2]%%%}+%
%%{-4,[0,3,3]%%%}+%%%{4,[0,3,2]%%%}+%%%{4,[0,3,1]%%%}+%%%{-4,[0,3,0]%%%}+%%%{4,[0,2,3]%%%}+%%%{-40,[0,2,2]%%%}
+%%%{4,[0,2,1]%%%}+%%%{4,[0,1,3]%%%}+%%%{4,[0,1,2]%%%}+%%%{-4,[0,0,3]%%%},0,%%%{1,[4,0,0]%%%}+%%%{4,[3,1,1]%%%
}+%%%{-4,[3,1,0]%%%}+%%%{-4,[3,0,1]%%%}+%%%{6,[2,2,2]%%%}+%%%{-12,[2,2,1]%%%}+%%%{6,[2,2,0]%%%}+%%%{-12,[2,1,2
]%%%}+%%%{4,[2,1,1]%%%}+%%%{6,[2,0,2]%%%}+%%%{4,[1,3,3]%%%}+%%%{-12,[1,3,2]%%%}+%%%{12,[1,3,1]%%%}+%%%{-4,[1,3
,0]%%%}+%%%{-12,[1,2,3]%%%}+%%%{8,[1,2,2]%%%}+%%%{4,[1,2,1]%%%}+%%%{12,[1,1,3]%%%}+%%%{4,[1,1,2]%%%}+%%%{-4,[1
,0,3]%%%}+%%%{1,[0,4,4]%%%}+%%%{-4,[0,4,3]%%%}+%%%{6,[0,4,2]%%%}+%%%{-4,[0,4,1]%%%}+%%%{1,[0,4,0]%%%}+%%%{-4,[
0,3,4]%%%}+%%%{4,[0,3,3]%%%}+%%%{4,[0,3,2]%%%}+%%%{-4,[0,3,1]%%%}+%%%{6,[0,2,4]%%%}+%%%{4,[0,2,3]%%%}+%%%{6,[0
,2,2]%%%}+%%%{-4,[0,1,4]%%%}+%%%{-4,[0,1,3]%%%}+%%%{1,[0,0,4]%%%}] at parameters values [0,85.3561567818,61.79
37478349]Warning, choosing root of [1,0,%%%{-4,[1,0,0]%%%}+%%%{-4,[0,1,1]%%%}+%%%{-4,[0,1,0]%%%}+%%%{-4,[0,0,1
]%%%},0,%%%{6,[2,0,0]%%%}+%%%{12,[1,1,1]%%%}+%%%{4,[1,1,0]%%%}+%%%{4,[1,0,1]%%%}+%%%{6,[0,2,2]%%%}+%%%{4,[0,2,
1]%%%}+%%%{6,[0,2,0]%%%}+%%%{4,[0,1,2]%%%}+%%%{4,[0,1,1]%%%}+%%%{6,[0,0,2]%%%},0,%%%{-4,[3,0,0]%%%}+%%%{-12,[2
,1,1]%%%}+%%%{4,[2,1,0]%%%}+%%%{4,[2,0,1]%%%}+%%%{-12,[1,2,2]%%%}+%%%{8,[1,2,1]%%%}+%%%{4,[1,2,0]%%%}+%%%{8,[1
,1,2]%%%}+%%%{-40,[1,1,1]%%%}+%%%{4,[1,0,2]%%%}+%%%{-4,[0,3,3]%%%}+%%%{4,[0,3,2]%%%}+%%%{4,[0,3,1]%%%}+%%%{-4,
[0,3,0]%%%}+%%%{4,[0,2,3]%%%}+%%%{-40,[0,2,2]%%%}+%%%{4,[0,2,1]%%%}+%%%{4,[0,1,3]%%%}+%%%{4,[0,1,2]%%%}+%%%{-4
,[0,0,3]%%%},0,%%%{1,[4,0,0]%%%}+%%%{4,[3,1,1]%%%}+%%%{-4,[3,1,0]%%%}+%%%{-4,[3,0,1]%%%}+%%%{6,[2,2,2]%%%}+%%%
{-12,[2,2,1]%%%}+%%%{6,[2,2,0]%%%}+%%%{-12,[2,1,2]%%%}+%%%{4,[2,1,1]%%%}+%%%{6,[2,0,2]%%%}+%%%{4,[1,3,3]%%%}+%
%%{-12,[1,3,2]%%%}+%%%{12,[1,3,1]%%%}+%%%{-4,[1,3,0]%%%}+%%%{-12,[1,2,3]%%%}+%%%{8,[1,2,2]%%%}+%%%{4,[1,2,1]%%
%}+%%%{12,[1,1,3]%%%}+%%%{4,[1,1,2]%%%}+%%%{-4,[1,0,3]%%%}+%%%{1,[0,4,4]%%%}+%%%{-4,[0,4,3]%%%}+%%%{6,[0,4,2]%
%%}+%%%{-4,[0,4,1]%%%}+%%%{1,[0,4,0]%%%}+%%%{-4,[0,3,4]%%%}+%%%{4,[0,3,3]%%%}+%%%{4,[0,3,2]%%%}+%%%{-4,[0,3,1]
%%%}+%%%{6,[0,2,4]%%%}+%%%{4,[0,2,3]%%%}+%%%{6,[0,2,2]%%%}+%%%{-4,[0,1,4]%%%}+%%%{-4,[0,1,3]%%%}+%%%{1,[0,0,4]
%%%}] at parameters values [0,71.707969239,78.6493344628]sqrt(2)/2/2*b/abs(b)*b^2/b*(2*(sqrt(2)*sqrt(a+b*x)*sq
rt(a+b*x)-3*sqrt(2)*a)*sqrt(a+b*x)*sqrt(-a*b+b*(a+b*x))/(-a*b+b*(a+b*x))-6*a*sqrt(2)/sqrt(b)*ln(abs(sqrt(-a*b+
b*(a+b*x))-sqrt(b)*sqrt(a+b*x))))
________________________________________________________________________________________
maple [B] time = 0.24, size = 280, normalized size = 3.46 \[ -\frac {2 \arctanh \left (\tanh \left (b x +a \right )\right )^{\frac {5}{2}}}{\left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x \right ) \sqrt {x}}+\frac {2 b \sqrt {x}\, \arctanh \left (\tanh \left (b x +a \right )\right )^{\frac {3}{2}}}{\arctanh \left (\tanh \left (b x +a \right )\right )-b x}+\frac {3 b a \sqrt {x}\, \sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}}{\arctanh \left (\tanh \left (b x +a \right )\right )-b x}+\frac {3 \sqrt {b}\, \ln \left (\sqrt {b}\, \sqrt {x}+\sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}\right ) a^{2}}{\arctanh \left (\tanh \left (b x +a \right )\right )-b x}+\frac {6 \sqrt {b}\, a \ln \left (\sqrt {b}\, \sqrt {x}+\sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}\right ) \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )}{\arctanh \left (\tanh \left (b x +a \right )\right )-b x}+\frac {3 b \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right ) \sqrt {x}\, \sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}}{\arctanh \left (\tanh \left (b x +a \right )\right )-b x}+\frac {3 \sqrt {b}\, \ln \left (\sqrt {b}\, \sqrt {x}+\sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}\right ) \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )^{2}}{\arctanh \left (\tanh \left (b x +a \right )\right )-b x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(arctanh(tanh(b*x+a))^(3/2)/x^(3/2),x)
[Out]
-2/(arctanh(tanh(b*x+a))-b*x)/x^(1/2)*arctanh(tanh(b*x+a))^(5/2)+2*b/(arctanh(tanh(b*x+a))-b*x)*x^(1/2)*arctan
h(tanh(b*x+a))^(3/2)+3*b/(arctanh(tanh(b*x+a))-b*x)*a*x^(1/2)*arctanh(tanh(b*x+a))^(1/2)+3*b^(1/2)/(arctanh(ta
nh(b*x+a))-b*x)*ln(b^(1/2)*x^(1/2)+arctanh(tanh(b*x+a))^(1/2))*a^2+6*b^(1/2)/(arctanh(tanh(b*x+a))-b*x)*a*ln(b
^(1/2)*x^(1/2)+arctanh(tanh(b*x+a))^(1/2))*(arctanh(tanh(b*x+a))-b*x-a)+3*b/(arctanh(tanh(b*x+a))-b*x)*(arctan
h(tanh(b*x+a))-b*x-a)*x^(1/2)*arctanh(tanh(b*x+a))^(1/2)+3*b^(1/2)/(arctanh(tanh(b*x+a))-b*x)*ln(b^(1/2)*x^(1/
2)+arctanh(tanh(b*x+a))^(1/2))*(arctanh(tanh(b*x+a))-b*x-a)^2
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{\frac {3}{2}}}{x^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(arctanh(tanh(b*x+a))^(3/2)/x^(3/2),x, algorithm="maxima")
[Out]
integrate(arctanh(tanh(b*x + a))^(3/2)/x^(3/2), x)
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^{3/2}}{x^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(atanh(tanh(a + b*x))^(3/2)/x^(3/2),x)
[Out]
int(atanh(tanh(a + b*x))^(3/2)/x^(3/2), x)
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {atanh}^{\frac {3}{2}}{\left (\tanh {\left (a + b x \right )} \right )}}{x^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(atanh(tanh(b*x+a))**(3/2)/x**(3/2),x)
[Out]
Integral(atanh(tanh(a + b*x))**(3/2)/x**(3/2), x)
________________________________________________________________________________________