3.13 \(\int \frac {1}{(b \coth (c+d x))^{2/3}} \, dx\)

Optimal. Leaf size=218 \[ -\frac {\log \left (b^{2/3}-\sqrt [3]{b} \sqrt [3]{b \coth (c+d x)}+(b \coth (c+d x))^{2/3}\right )}{4 b^{2/3} d}+\frac {\log \left (b^{2/3}+\sqrt [3]{b} \sqrt [3]{b \coth (c+d x)}+(b \coth (c+d x))^{2/3}\right )}{4 b^{2/3} d}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b \coth (c+d x)}}{\sqrt [3]{b}}}{\sqrt {3}}\right )}{2 b^{2/3} d}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{b \coth (c+d x)}}{\sqrt [3]{b}}+1}{\sqrt {3}}\right )}{2 b^{2/3} d}+\frac {\tanh ^{-1}\left (\frac {\sqrt [3]{b \coth (c+d x)}}{\sqrt [3]{b}}\right )}{b^{2/3} d} \]

[Out]

arctanh((b*coth(d*x+c))^(1/3)/b^(1/3))/b^(2/3)/d-1/4*ln(b^(2/3)-b^(1/3)*(b*coth(d*x+c))^(1/3)+(b*coth(d*x+c))^
(2/3))/b^(2/3)/d+1/4*ln(b^(2/3)+b^(1/3)*(b*coth(d*x+c))^(1/3)+(b*coth(d*x+c))^(2/3))/b^(2/3)/d-1/2*arctan(1/3*
(1-2*(b*coth(d*x+c))^(1/3)/b^(1/3))*3^(1/2))*3^(1/2)/b^(2/3)/d+1/2*arctan(1/3*(1+2*(b*coth(d*x+c))^(1/3)/b^(1/
3))*3^(1/2))*3^(1/2)/b^(2/3)/d

________________________________________________________________________________________

Rubi [A]  time = 0.24, antiderivative size = 218, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {3476, 329, 210, 634, 618, 204, 628, 206} \[ -\frac {\log \left (b^{2/3}-\sqrt [3]{b} \sqrt [3]{b \coth (c+d x)}+(b \coth (c+d x))^{2/3}\right )}{4 b^{2/3} d}+\frac {\log \left (b^{2/3}+\sqrt [3]{b} \sqrt [3]{b \coth (c+d x)}+(b \coth (c+d x))^{2/3}\right )}{4 b^{2/3} d}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b \coth (c+d x)}}{\sqrt [3]{b}}}{\sqrt {3}}\right )}{2 b^{2/3} d}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{b \coth (c+d x)}}{\sqrt [3]{b}}+1}{\sqrt {3}}\right )}{2 b^{2/3} d}+\frac {\tanh ^{-1}\left (\frac {\sqrt [3]{b \coth (c+d x)}}{\sqrt [3]{b}}\right )}{b^{2/3} d} \]

Antiderivative was successfully verified.

[In]

Int[(b*Coth[c + d*x])^(-2/3),x]

[Out]

-(Sqrt[3]*ArcTan[(1 - (2*(b*Coth[c + d*x])^(1/3))/b^(1/3))/Sqrt[3]])/(2*b^(2/3)*d) + (Sqrt[3]*ArcTan[(1 + (2*(
b*Coth[c + d*x])^(1/3))/b^(1/3))/Sqrt[3]])/(2*b^(2/3)*d) + ArcTanh[(b*Coth[c + d*x])^(1/3)/b^(1/3)]/(b^(2/3)*d
) - Log[b^(2/3) - b^(1/3)*(b*Coth[c + d*x])^(1/3) + (b*Coth[c + d*x])^(2/3)]/(4*b^(2/3)*d) + Log[b^(2/3) + b^(
1/3)*(b*Coth[c + d*x])^(1/3) + (b*Coth[c + d*x])^(2/3)]/(4*b^(2/3)*d)

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 210

Int[((a_) + (b_.)*(x_)^(n_))^(-1), x_Symbol] :> Module[{r = Numerator[Rt[-(a/b), n]], s = Denominator[Rt[-(a/b
), n]], k, u}, Simp[u = Int[(r - s*Cos[(2*k*Pi)/n]*x)/(r^2 - 2*r*s*Cos[(2*k*Pi)/n]*x + s^2*x^2), x] + Int[(r +
s*Cos[(2*k*Pi)/n]*x)/(r^2 + 2*r*s*Cos[(2*k*Pi)/n]*x + s^2*x^2), x]; (2*r^2*Int[1/(r^2 - s^2*x^2), x])/(a*n) +
Dist[(2*r)/(a*n), Sum[u, {k, 1, (n - 2)/4}], x], x]] /; FreeQ[{a, b}, x] && IGtQ[(n - 2)/4, 0] && NegQ[a/b]

Rule 329

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/c^n)^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 3476

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[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]

Rubi steps

\begin {align*} \int \frac {1}{(b \coth (c+d x))^{2/3}} \, dx &=-\frac {b \operatorname {Subst}\left (\int \frac {1}{x^{2/3} \left (-b^2+x^2\right )} \, dx,x,b \coth (c+d x)\right )}{d}\\ &=-\frac {(3 b) \operatorname {Subst}\left (\int \frac {1}{-b^2+x^6} \, dx,x,\sqrt [3]{b \coth (c+d x)}\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\sqrt [3]{b}-\frac {x}{2}}{b^{2/3}-\sqrt [3]{b} x+x^2} \, dx,x,\sqrt [3]{b \coth (c+d x)}\right )}{b^{2/3} d}+\frac {\operatorname {Subst}\left (\int \frac {\sqrt [3]{b}+\frac {x}{2}}{b^{2/3}+\sqrt [3]{b} x+x^2} \, dx,x,\sqrt [3]{b \coth (c+d x)}\right )}{b^{2/3} d}+\frac {\operatorname {Subst}\left (\int \frac {1}{b^{2/3}-x^2} \, dx,x,\sqrt [3]{b \coth (c+d x)}\right )}{\sqrt [3]{b} d}\\ &=\frac {\tanh ^{-1}\left (\frac {\sqrt [3]{b \coth (c+d x)}}{\sqrt [3]{b}}\right )}{b^{2/3} d}-\frac {\operatorname {Subst}\left (\int \frac {-\sqrt [3]{b}+2 x}{b^{2/3}-\sqrt [3]{b} x+x^2} \, dx,x,\sqrt [3]{b \coth (c+d x)}\right )}{4 b^{2/3} d}+\frac {\operatorname {Subst}\left (\int \frac {\sqrt [3]{b}+2 x}{b^{2/3}+\sqrt [3]{b} x+x^2} \, dx,x,\sqrt [3]{b \coth (c+d x)}\right )}{4 b^{2/3} d}+\frac {3 \operatorname {Subst}\left (\int \frac {1}{b^{2/3}-\sqrt [3]{b} x+x^2} \, dx,x,\sqrt [3]{b \coth (c+d x)}\right )}{4 \sqrt [3]{b} d}+\frac {3 \operatorname {Subst}\left (\int \frac {1}{b^{2/3}+\sqrt [3]{b} x+x^2} \, dx,x,\sqrt [3]{b \coth (c+d x)}\right )}{4 \sqrt [3]{b} d}\\ &=\frac {\tanh ^{-1}\left (\frac {\sqrt [3]{b \coth (c+d x)}}{\sqrt [3]{b}}\right )}{b^{2/3} d}-\frac {\log \left (b^{2/3}-\sqrt [3]{b} \sqrt [3]{b \coth (c+d x)}+(b \coth (c+d x))^{2/3}\right )}{4 b^{2/3} d}+\frac {\log \left (b^{2/3}+\sqrt [3]{b} \sqrt [3]{b \coth (c+d x)}+(b \coth (c+d x))^{2/3}\right )}{4 b^{2/3} d}+\frac {3 \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b \coth (c+d x)}}{\sqrt [3]{b}}\right )}{2 b^{2/3} d}-\frac {3 \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{b \coth (c+d x)}}{\sqrt [3]{b}}\right )}{2 b^{2/3} d}\\ &=-\frac {\sqrt {3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b \coth (c+d x)}}{\sqrt [3]{b}}}{\sqrt {3}}\right )}{2 b^{2/3} d}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{b \coth (c+d x)}}{\sqrt [3]{b}}}{\sqrt {3}}\right )}{2 b^{2/3} d}+\frac {\tanh ^{-1}\left (\frac {\sqrt [3]{b \coth (c+d x)}}{\sqrt [3]{b}}\right )}{b^{2/3} d}-\frac {\log \left (b^{2/3}-\sqrt [3]{b} \sqrt [3]{b \coth (c+d x)}+(b \coth (c+d x))^{2/3}\right )}{4 b^{2/3} d}+\frac {\log \left (b^{2/3}+\sqrt [3]{b} \sqrt [3]{b \coth (c+d x)}+(b \coth (c+d x))^{2/3}\right )}{4 b^{2/3} d}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 0.03, size = 36, normalized size = 0.17 \[ \frac {3 \sqrt [3]{b \coth (c+d x)} \, _2F_1\left (\frac {1}{6},1;\frac {7}{6};\coth ^2(c+d x)\right )}{b d} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*Coth[c + d*x])^(-2/3),x]

[Out]

(3*(b*Coth[c + d*x])^(1/3)*Hypergeometric2F1[1/6, 1, 7/6, Coth[c + d*x]^2])/(b*d)

________________________________________________________________________________________

fricas [B]  time = 0.46, size = 356, normalized size = 1.63 \[ \frac {2 \, \sqrt {3} b \sqrt {-\left (-b^{2}\right )^{\frac {1}{3}}} \arctan \left (-\frac {\sqrt {3} \left (-b^{2}\right )^{\frac {1}{3}} b \sqrt {-\left (-b^{2}\right )^{\frac {1}{3}}} - 2 \, \sqrt {3} \left (-b^{2}\right )^{\frac {2}{3}} \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {1}{3}} \sqrt {-\left (-b^{2}\right )^{\frac {1}{3}}}}{3 \, b^{2}}\right ) + 2 \, \sqrt {3} {\left (b^{2}\right )}^{\frac {1}{6}} b \arctan \left (-\frac {\sqrt {3} {\left (b^{2}\right )}^{\frac {1}{6}} {\left ({\left (b^{2}\right )}^{\frac {1}{3}} b - 2 \, {\left (b^{2}\right )}^{\frac {2}{3}} \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {1}{3}}\right )}}{3 \, b^{2}}\right ) + \left (-b^{2}\right )^{\frac {2}{3}} \log \left (b \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {2}{3}} - \left (-b^{2}\right )^{\frac {1}{3}} b + \left (-b^{2}\right )^{\frac {2}{3}} \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {1}{3}}\right ) - {\left (b^{2}\right )}^{\frac {2}{3}} \log \left (b \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {2}{3}} + {\left (b^{2}\right )}^{\frac {1}{3}} b - {\left (b^{2}\right )}^{\frac {2}{3}} \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {1}{3}}\right ) - 2 \, \left (-b^{2}\right )^{\frac {2}{3}} \log \left (b \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {1}{3}} - \left (-b^{2}\right )^{\frac {2}{3}}\right ) + 2 \, {\left (b^{2}\right )}^{\frac {2}{3}} \log \left (b \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {1}{3}} + {\left (b^{2}\right )}^{\frac {2}{3}}\right )}{4 \, b^{2} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*coth(d*x+c))^(2/3),x, algorithm="fricas")

[Out]

1/4*(2*sqrt(3)*b*sqrt(-(-b^2)^(1/3))*arctan(-1/3*(sqrt(3)*(-b^2)^(1/3)*b*sqrt(-(-b^2)^(1/3)) - 2*sqrt(3)*(-b^2
)^(2/3)*(b*cosh(d*x + c)/sinh(d*x + c))^(1/3)*sqrt(-(-b^2)^(1/3)))/b^2) + 2*sqrt(3)*(b^2)^(1/6)*b*arctan(-1/3*
sqrt(3)*(b^2)^(1/6)*((b^2)^(1/3)*b - 2*(b^2)^(2/3)*(b*cosh(d*x + c)/sinh(d*x + c))^(1/3))/b^2) + (-b^2)^(2/3)*
log(b*(b*cosh(d*x + c)/sinh(d*x + c))^(2/3) - (-b^2)^(1/3)*b + (-b^2)^(2/3)*(b*cosh(d*x + c)/sinh(d*x + c))^(1
/3)) - (b^2)^(2/3)*log(b*(b*cosh(d*x + c)/sinh(d*x + c))^(2/3) + (b^2)^(1/3)*b - (b^2)^(2/3)*(b*cosh(d*x + c)/
sinh(d*x + c))^(1/3)) - 2*(-b^2)^(2/3)*log(b*(b*cosh(d*x + c)/sinh(d*x + c))^(1/3) - (-b^2)^(2/3)) + 2*(b^2)^(
2/3)*log(b*(b*cosh(d*x + c)/sinh(d*x + c))^(1/3) + (b^2)^(2/3)))/(b^2*d)

________________________________________________________________________________________

giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*coth(d*x+c))^(2/3),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Mini
mal poly. in rootof must be fraction free Error: Bad Argument ValueMinimal poly. in rootof must be fraction fr
ee Error: Bad Argument ValueMinimal poly. in rootof must be fraction free Error: Bad Argument ValueMinimal pol
y. in rootof must be fraction free Error: Bad Argument ValueMinimal poly. in rootof must be fraction free Erro
r: Bad Argument ValueMinimal poly. in rootof must be fraction free Error: Bad Argument ValueMinimal poly. in r
ootof must be fraction free Error: Bad Argument ValueMinimal poly. in rootof must be fraction free Error: Bad
Argument ValueMinimal poly. in rootof must be fraction free Error: Bad Argument ValueMinimal poly. in rootof m
ust be fraction free Error: Bad Argument ValueMinimal poly. in rootof must be fraction free Error: Bad Argumen
t ValueMinimal poly. in rootof must be fraction free Error: Bad Argument ValueMinimal poly. in rootof must be
fraction free Error: Bad Argument ValueMinimal poly. in rootof must be fraction free Error: Bad Argument Value
Minimal poly. in rootof must be fraction free Error: Bad Argument ValueMinimal poly. in rootof must be fractio
n free Error: Bad Argument ValueMinimal poly. in rootof must be fraction free Error: Bad Argument ValueMinimal
poly. in rootof must be fraction free Error: Bad Argument ValueMinimal poly. in rootof must be fraction free
Error: Bad Argument ValueMinimal poly. in rootof must be fraction free Error: Bad Argument ValueMinimal poly.
in rootof must be fraction free Error: Bad Argument ValueMinimal poly. in rootof must be fraction free Error:
Bad Argument ValueMinimal poly. in rootof must be fraction free Error: Bad Argument ValueMinimal poly. in root
of must be fraction free Error: Bad Argument ValueMinimal poly. in rootof must be fraction free Error: Bad Arg
ument ValueMinimal poly. in rootof must be fraction free Error: Bad Argument ValueMinimal poly. in rootof must
be fraction free Error: Bad Argument ValueMinimal poly. in rootof must be fraction free Error: Bad Argument V
alueUnable to build a single algebraic extension for simplifying.Trying rational simplification only. This mig
ht return a wrong answer if simplifying 0/0!Unable to build a single algebraic extension for simplifying.Tryin
g rational simplification only. This might return a wrong answer if simplifying 0/0!Unable to build a single a
lgebraic extension for simplifying.Trying rational simplification only. This might return a wrong answer if si
mplifying 0/0!Unable to build a single algebraic extension for simplifying.Trying rational simplification only
. This might return a wrong answer if simplifying 0/0!Unable to build a single algebraic extension for simplif
ying.Trying rational simplification only. This might return a wrong answer if simplifying 0/0!Unable to build
a single algebraic extension for simplifying.Trying rational simplification only. This might return a wrong an
swer if simplifying 0/0!Unable to build a single algebraic extension for simplifying.Trying rational simplific
ation only. This might return a wrong answer if simplifying 0/0!Unable to build a single algebraic extension f
or simplifying.Trying rational simplification only. This might return a wrong answer if simplifying 0/0!Unable
to build a single algebraic extension for simplifying.Trying rational simplification only. This might return
a wrong answer if simplifying 0/0!Unable to build a single algebraic extension for simplifying.Trying rational
simplification only. This might return a wrong answer if simplifying 0/0!Unable to build a single algebraic e
xtension for simplifying.Trying rational simplification only. This might return a wrong answer if simplifying
0/0!Unable to build a single algebraic extension for simplifying.Trying rational simplification only. This mig
ht return a wrong answer if simplifying 0/0!Unable to build a single algebraic extension for simplifying.Tryin
g rational simplification only. This might return a wrong answer if simplifying 0/0!Unable to build a single a
lgebraic extension for simplifying.Trying rational simplification only. This might return a wrong answer if si
mplifying 0/0!Unable to build a single algebraic extension for simplifying.Trying rational simplification only
. This might return a wrong answer if simplifying 0/0!Unable to build a single algebraic extension for simplif
ying.Trying rational simplification only. This might return a wrong answer if simplifying 0/0!Unable to build
a single algebraic extension for simplifying.Trying rational simplification only. This might return a wrong an
swer if simplifying 0/0!Unable to build a single algebraic extension for simplifying.Trying rational simplific
ation only. This might return a wrong answer if simplifying 0/0!Unable to build a single algebraic extension f
or simplifying.Trying rational simplification only. This might return a wrong answer if simplifying 0/0!Unable
to build a single algebraic extension for simplifying.Trying rational simplification only. This might return
a wrong answer if simplifying 0/0!Unable to build a single algebraic extension for simplifying.Trying rational
simplification only. This might return a wrong answer if simplifying 0/0!Unable to build a single algebraic e
xtension for simplifying.Trying rational simplification only. This might return a wrong answer if simplifying
0/0!Unable to build a single algebraic extension for simplifying.Trying rational simplification only. This mig
ht return a wrong answer if simplifying 0/0!Unable to build a single algebraic extension for simplifying.Tryin
g rational simplification only. This might return a wrong answer if simplifying 0/0!Unable to build a single a
lgebraic extension for simplifying.Trying rational simplification only. This might return a wrong answer if si
mplifying 0/0!Unable to build a single algebraic extension for simplifying.Trying rational simplification only
. This might return a wrong answer if simplifying 0/0!Unable to build a single algebraic extension for simplif
ying.Trying rational simplification only. This might return a wrong answer if simplifying 0/0!Unable to build
a single algebraic extension for simplifying.Trying rational simplification only. This might return a wrong an
swer if simplifying 0/0!Unable to build a single algebraic extension for simplifying.Trying rational simplific
ation only. This might return a wrong answer if simplifying 0/0!Unable to build a single algebraic extension f
or simplifying.Trying rational simplification only. This might return a wrong answer if simplifying 0/0!Unable
to build a single algebraic extension for simplifying.Trying rational simplification only. This might return
a wrong answer if simplifying 0/0!Unable to build a single algebraic extension for simplifying.Trying rational
simplification only. This might return a wrong answer if simplifying 0/0!Unable to build a single algebraic e
xtension for simplifying.Trying rational simplification only. This might return a wrong answer if simplifying
0/0!Unable to build a single algebraic extension for simplifying.Trying rational simplification only. This mig
ht return a wrong answer if simplifying 0/0!Unable to build a single algebraic extension for simplifying.Tryin
g rational simplification only. This might return a wrong answer if simplifying 0/0!Unable to build a single a
lgebraic extension for simplifying.Trying rational simplification only. This might return a wrong answer if si
mplifying 0/0!Unable to build a single algebraic extension for simplifying.Trying rational simplification only
. This might return a wrong answer if simplifying 0/0!Unable to build a single algebraic extension for simplif
ying.Trying rational simplification only. This might return a wrong answer if simplifying 0/0!Unable to build
a single algebraic extension for simplifying.Trying rational simplification only. This might return a wrong an
swer if simplifying 0/0!Unable to build a single algebraic extension for simplifying.Trying rational simplific
ation only. This might return a wrong answer if simplifying 0/0!Unable to build a single algebraic extension f
or simplifying.Trying rational simplification only. This might return a wrong answer if simplifying 0/0!Unable
to build a single algebraic extension for simplifying.Trying rational simplification only. This might return
a wrong answer if simplifying 0/0!Unable to build a single algebraic extension for simplifying.Trying rational
simplification only. This might return a wrong answer if simplifying 0/0!Evaluation time: 1.62Done

________________________________________________________________________________________

maple [A]  time = 0.09, size = 193, normalized size = 0.89 \[ -\frac {\ln \left (\left (b \coth \left (d x +c \right )\right )^{\frac {1}{3}}-b^{\frac {1}{3}}\right )}{2 b^{\frac {2}{3}} d}+\frac {\ln \left (b^{\frac {2}{3}}+b^{\frac {1}{3}} \left (b \coth \left (d x +c \right )\right )^{\frac {1}{3}}+\left (b \coth \left (d x +c \right )\right )^{\frac {2}{3}}\right )}{4 b^{\frac {2}{3}} d}+\frac {\arctan \left (\frac {\left (1+\frac {2 \left (b \coth \left (d x +c \right )\right )^{\frac {1}{3}}}{b^{\frac {1}{3}}}\right ) \sqrt {3}}{3}\right ) \sqrt {3}}{2 b^{\frac {2}{3}} d}+\frac {\ln \left (\left (b \coth \left (d x +c \right )\right )^{\frac {1}{3}}+b^{\frac {1}{3}}\right )}{2 b^{\frac {2}{3}} d}-\frac {\ln \left (b^{\frac {2}{3}}-b^{\frac {1}{3}} \left (b \coth \left (d x +c \right )\right )^{\frac {1}{3}}+\left (b \coth \left (d x +c \right )\right )^{\frac {2}{3}}\right )}{4 b^{\frac {2}{3}} d}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b \coth \left (d x +c \right )\right )^{\frac {1}{3}}}{b^{\frac {1}{3}}}-1\right )}{3}\right )}{2 b^{\frac {2}{3}} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*coth(d*x+c))^(2/3),x)

[Out]

-1/2/b^(2/3)/d*ln((b*coth(d*x+c))^(1/3)-b^(1/3))+1/4*ln(b^(2/3)+b^(1/3)*(b*coth(d*x+c))^(1/3)+(b*coth(d*x+c))^
(2/3))/b^(2/3)/d+1/2*arctan(1/3*(1+2*(b*coth(d*x+c))^(1/3)/b^(1/3))*3^(1/2))*3^(1/2)/b^(2/3)/d+1/2/b^(2/3)/d*l
n((b*coth(d*x+c))^(1/3)+b^(1/3))-1/4*ln(b^(2/3)-b^(1/3)*(b*coth(d*x+c))^(1/3)+(b*coth(d*x+c))^(2/3))/b^(2/3)/d
+1/2/b^(2/3)/d*3^(1/2)*arctan(1/3*3^(1/2)*(2*(b*coth(d*x+c))^(1/3)/b^(1/3)-1))

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b \coth \left (d x + c\right )\right )^{\frac {2}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*coth(d*x+c))^(2/3),x, algorithm="maxima")

[Out]

integrate((b*coth(d*x + c))^(-2/3), x)

________________________________________________________________________________________

mupad [B]  time = 1.39, size = 197, normalized size = 0.90 \[ \frac {\mathrm {atanh}\left (\frac {{\left (b\,\mathrm {coth}\left (c+d\,x\right )\right )}^{1/3}}{b^{1/3}}\right )}{b^{2/3}\,d}-\frac {\mathrm {atan}\left (\frac {b^{10/3}\,{\left (b\,\mathrm {coth}\left (c+d\,x\right )\right )}^{1/3}\,243{}\mathrm {i}}{-243\,b^{11/3}+\sqrt {3}\,b^{11/3}\,243{}\mathrm {i}}-\frac {243\,\sqrt {3}\,b^{10/3}\,{\left (b\,\mathrm {coth}\left (c+d\,x\right )\right )}^{1/3}}{-243\,b^{11/3}+\sqrt {3}\,b^{11/3}\,243{}\mathrm {i}}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2\,b^{2/3}\,d}-\frac {\mathrm {atan}\left (\frac {b^{10/3}\,{\left (b\,\mathrm {coth}\left (c+d\,x\right )\right )}^{1/3}\,243{}\mathrm {i}}{243\,b^{11/3}+\sqrt {3}\,b^{11/3}\,243{}\mathrm {i}}+\frac {243\,\sqrt {3}\,b^{10/3}\,{\left (b\,\mathrm {coth}\left (c+d\,x\right )\right )}^{1/3}}{243\,b^{11/3}+\sqrt {3}\,b^{11/3}\,243{}\mathrm {i}}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2\,b^{2/3}\,d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*coth(c + d*x))^(2/3),x)

[Out]

atanh((b*coth(c + d*x))^(1/3)/b^(1/3))/(b^(2/3)*d) - (atan((b^(10/3)*(b*coth(c + d*x))^(1/3)*243i)/(3^(1/2)*b^
(11/3)*243i - 243*b^(11/3)) - (243*3^(1/2)*b^(10/3)*(b*coth(c + d*x))^(1/3))/(3^(1/2)*b^(11/3)*243i - 243*b^(1
1/3)))*(3^(1/2)*1i + 1)*1i)/(2*b^(2/3)*d) - (atan((b^(10/3)*(b*coth(c + d*x))^(1/3)*243i)/(3^(1/2)*b^(11/3)*24
3i + 243*b^(11/3)) + (243*3^(1/2)*b^(10/3)*(b*coth(c + d*x))^(1/3))/(3^(1/2)*b^(11/3)*243i + 243*b^(11/3)))*(3
^(1/2)*1i - 1)*1i)/(2*b^(2/3)*d)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b \coth {\left (c + d x \right )}\right )^{\frac {2}{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*coth(d*x+c))**(2/3),x)

[Out]

Integral((b*coth(c + d*x))**(-2/3), x)

________________________________________________________________________________________