3.1.43 \(\int \frac {\coth ^3(x)}{(a+b \coth ^2(x))^{5/2}} \, dx\) [43]

Optimal. Leaf size=74 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \coth ^2(x)}}{\sqrt {a+b}}\right )}{(a+b)^{5/2}}+\frac {a}{3 b (a+b) \left (a+b \coth ^2(x)\right )^{3/2}}-\frac {1}{(a+b)^2 \sqrt {a+b \coth ^2(x)}} \]

[Out]

arctanh((a+b*coth(x)^2)^(1/2)/(a+b)^(1/2))/(a+b)^(5/2)+1/3*a/b/(a+b)/(a+b*coth(x)^2)^(3/2)-1/(a+b)^2/(a+b*coth
(x)^2)^(1/2)

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Rubi [A]
time = 0.11, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {3751, 457, 79, 53, 65, 214} \begin {gather*} \frac {a}{3 b (a+b) \left (a+b \coth ^2(x)\right )^{3/2}}-\frac {1}{(a+b)^2 \sqrt {a+b \coth ^2(x)}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \coth ^2(x)}}{\sqrt {a+b}}\right )}{(a+b)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Coth[x]^3/(a + b*Coth[x]^2)^(5/2),x]

[Out]

ArcTanh[Sqrt[a + b*Coth[x]^2]/Sqrt[a + b]]/(a + b)^(5/2) + a/(3*b*(a + b)*(a + b*Coth[x]^2)^(3/2)) - 1/((a + b
)^2*Sqrt[a + b*Coth[x]^2])

Rule 53

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n + 1
)/((b*c - a*d)*(m + 1))), x] - Dist[d*((m + n + 2)/((b*c - a*d)*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 79

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(-(b*e - a*f
))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1
) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e,
f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || L
tQ[p, n]))))

Rule 214

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

Rule 457

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 3751

Int[((d_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_.), x_Symbol]
:> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[c*(ff/f), Subst[Int[(d*ff*(x/c))^m*((a + b*(ff*x)^n)^p/(c^2
+ ff^2*x^2)), x], x, c*(Tan[e + f*x]/ff)], x]] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && (IGtQ[p, 0] || EqQ
[n, 2] || EqQ[n, 4] || (IntegerQ[p] && RationalQ[n]))

Rubi steps

\begin {align*} \int \frac {\coth ^3(x)}{\left (a+b \coth ^2(x)\right )^{5/2}} \, dx &=\text {Subst}\left (\int \frac {x^3}{\left (1-x^2\right ) \left (a+b x^2\right )^{5/2}} \, dx,x,\coth (x)\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \frac {x}{(1-x) (a+b x)^{5/2}} \, dx,x,\coth ^2(x)\right )\\ &=\frac {a}{3 b (a+b) \left (a+b \coth ^2(x)\right )^{3/2}}+\frac {\text {Subst}\left (\int \frac {1}{(1-x) (a+b x)^{3/2}} \, dx,x,\coth ^2(x)\right )}{2 (a+b)}\\ &=\frac {a}{3 b (a+b) \left (a+b \coth ^2(x)\right )^{3/2}}-\frac {1}{(a+b)^2 \sqrt {a+b \coth ^2(x)}}+\frac {\text {Subst}\left (\int \frac {1}{(1-x) \sqrt {a+b x}} \, dx,x,\coth ^2(x)\right )}{2 (a+b)^2}\\ &=\frac {a}{3 b (a+b) \left (a+b \coth ^2(x)\right )^{3/2}}-\frac {1}{(a+b)^2 \sqrt {a+b \coth ^2(x)}}+\frac {\text {Subst}\left (\int \frac {1}{1+\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \coth ^2(x)}\right )}{b (a+b)^2}\\ &=\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \coth ^2(x)}}{\sqrt {a+b}}\right )}{(a+b)^{5/2}}+\frac {a}{3 b (a+b) \left (a+b \coth ^2(x)\right )^{3/2}}-\frac {1}{(a+b)^2 \sqrt {a+b \coth ^2(x)}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 0.08, size = 63, normalized size = 0.85 \begin {gather*} \frac {a (a+b)-3 b \left (a+b \coth ^2(x)\right ) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {a+b \coth ^2(x)}{a+b}\right )}{3 b (a+b)^2 \left (a+b \coth ^2(x)\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Coth[x]^3/(a + b*Coth[x]^2)^(5/2),x]

[Out]

(a*(a + b) - 3*b*(a + b*Coth[x]^2)*Hypergeometric2F1[-1/2, 1, 1/2, (a + b*Coth[x]^2)/(a + b)])/(3*b*(a + b)^2*
(a + b*Coth[x]^2)^(3/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(564\) vs. \(2(62)=124\).
time = 0.72, size = 565, normalized size = 7.64 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(x)^3/(a+b*coth(x)^2)^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/3/b/(a+b*coth(x)^2)^(3/2)-1/6/(a+b)/(b*(coth(x)-1)^2+2*b*(coth(x)-1)+a+b)^(3/2)+1/2*b/(a+b)*(2/3*(2*b*(coth(
x)-1)+2*b)/(4*b*(a+b)-4*b^2)/(b*(coth(x)-1)^2+2*b*(coth(x)-1)+a+b)^(3/2)+16/3*b/(4*b*(a+b)-4*b^2)^2*(2*b*(coth
(x)-1)+2*b)/(b*(coth(x)-1)^2+2*b*(coth(x)-1)+a+b)^(1/2))-1/2/(a+b)*(1/(a+b)/(b*(coth(x)-1)^2+2*b*(coth(x)-1)+a
+b)^(1/2)-2*b/(a+b)*(2*b*(coth(x)-1)+2*b)/(4*b*(a+b)-4*b^2)/(b*(coth(x)-1)^2+2*b*(coth(x)-1)+a+b)^(1/2)-1/(a+b
)^(3/2)*ln((2*a+2*b+2*b*(coth(x)-1)+2*(a+b)^(1/2)*(b*(coth(x)-1)^2+2*b*(coth(x)-1)+a+b)^(1/2))/(coth(x)-1)))-1
/6/(a+b)/(b*(1+coth(x))^2-2*b*(1+coth(x))+a+b)^(3/2)-1/2*b/(a+b)*(2/3*(2*b*(1+coth(x))-2*b)/(4*b*(a+b)-4*b^2)/
(b*(1+coth(x))^2-2*b*(1+coth(x))+a+b)^(3/2)+16/3*b/(4*b*(a+b)-4*b^2)^2*(2*b*(1+coth(x))-2*b)/(b*(1+coth(x))^2-
2*b*(1+coth(x))+a+b)^(1/2))-1/2/(a+b)*(1/(a+b)/(b*(1+coth(x))^2-2*b*(1+coth(x))+a+b)^(1/2)+2*b/(a+b)*(2*b*(1+c
oth(x))-2*b)/(4*b*(a+b)-4*b^2)/(b*(1+coth(x))^2-2*b*(1+coth(x))+a+b)^(1/2)-1/(a+b)^(3/2)*ln((2*a+2*b-2*b*(1+co
th(x))+2*(a+b)^(1/2)*(b*(1+coth(x))^2-2*b*(1+coth(x))+a+b)^(1/2))/(1+coth(x))))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)^3/(a+b*coth(x)^2)^(5/2),x, algorithm="maxima")

[Out]

integrate(coth(x)^3/(b*coth(x)^2 + a)^(5/2), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2964 vs. \(2 (62) = 124\).
time = 0.71, size = 6560, normalized size = 88.65 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)^3/(a+b*coth(x)^2)^(5/2),x, algorithm="fricas")

[Out]

[1/12*(3*((a^2*b + 2*a*b^2 + b^3)*cosh(x)^8 + 8*(a^2*b + 2*a*b^2 + b^3)*cosh(x)*sinh(x)^7 + (a^2*b + 2*a*b^2 +
b^3)*sinh(x)^8 - 4*(a^2*b - b^3)*cosh(x)^6 - 4*(a^2*b - b^3 - 7*(a^2*b + 2*a*b^2 + b^3)*cosh(x)^2)*sinh(x)^6
+ 8*(7*(a^2*b + 2*a*b^2 + b^3)*cosh(x)^3 - 3*(a^2*b - b^3)*cosh(x))*sinh(x)^5 + 2*(3*a^2*b - 2*a*b^2 + 3*b^3)*
cosh(x)^4 + 2*(35*(a^2*b + 2*a*b^2 + b^3)*cosh(x)^4 + 3*a^2*b - 2*a*b^2 + 3*b^3 - 30*(a^2*b - b^3)*cosh(x)^2)*
sinh(x)^4 + 8*(7*(a^2*b + 2*a*b^2 + b^3)*cosh(x)^5 - 10*(a^2*b - b^3)*cosh(x)^3 + (3*a^2*b - 2*a*b^2 + 3*b^3)*
cosh(x))*sinh(x)^3 + a^2*b + 2*a*b^2 + b^3 - 4*(a^2*b - b^3)*cosh(x)^2 + 4*(7*(a^2*b + 2*a*b^2 + b^3)*cosh(x)^
6 - 15*(a^2*b - b^3)*cosh(x)^4 - a^2*b + b^3 + 3*(3*a^2*b - 2*a*b^2 + 3*b^3)*cosh(x)^2)*sinh(x)^2 + 8*((a^2*b
+ 2*a*b^2 + b^3)*cosh(x)^7 - 3*(a^2*b - b^3)*cosh(x)^5 + (3*a^2*b - 2*a*b^2 + 3*b^3)*cosh(x)^3 - (a^2*b - b^3)
*cosh(x))*sinh(x))*sqrt(a + b)*log(-((a^3 + a^2*b)*cosh(x)^8 + 8*(a^3 + a^2*b)*cosh(x)*sinh(x)^7 + (a^3 + a^2*
b)*sinh(x)^8 - 2*(2*a^3 + a^2*b)*cosh(x)^6 - 2*(2*a^3 + a^2*b - 14*(a^3 + a^2*b)*cosh(x)^2)*sinh(x)^6 + 4*(14*
(a^3 + a^2*b)*cosh(x)^3 - 3*(2*a^3 + a^2*b)*cosh(x))*sinh(x)^5 + (6*a^3 + 4*a^2*b - a*b^2 + b^3)*cosh(x)^4 + (
70*(a^3 + a^2*b)*cosh(x)^4 + 6*a^3 + 4*a^2*b - a*b^2 + b^3 - 30*(2*a^3 + a^2*b)*cosh(x)^2)*sinh(x)^4 + 4*(14*(
a^3 + a^2*b)*cosh(x)^5 - 10*(2*a^3 + a^2*b)*cosh(x)^3 + (6*a^3 + 4*a^2*b - a*b^2 + b^3)*cosh(x))*sinh(x)^3 + a
^3 + 3*a^2*b + 3*a*b^2 + b^3 - 2*(2*a^3 + 3*a^2*b - b^3)*cosh(x)^2 + 2*(14*(a^3 + a^2*b)*cosh(x)^6 - 15*(2*a^3
+ a^2*b)*cosh(x)^4 - 2*a^3 - 3*a^2*b + b^3 + 3*(6*a^3 + 4*a^2*b - a*b^2 + b^3)*cosh(x)^2)*sinh(x)^2 + sqrt(2)
*(a^2*cosh(x)^6 + 6*a^2*cosh(x)*sinh(x)^5 + a^2*sinh(x)^6 - 3*a^2*cosh(x)^4 + 3*(5*a^2*cosh(x)^2 - a^2)*sinh(x
)^4 + 4*(5*a^2*cosh(x)^3 - 3*a^2*cosh(x))*sinh(x)^3 + (3*a^2 + 2*a*b - b^2)*cosh(x)^2 + (15*a^2*cosh(x)^4 - 18
*a^2*cosh(x)^2 + 3*a^2 + 2*a*b - b^2)*sinh(x)^2 - a^2 - 2*a*b - b^2 + 2*(3*a^2*cosh(x)^5 - 6*a^2*cosh(x)^3 + (
3*a^2 + 2*a*b - b^2)*cosh(x))*sinh(x))*sqrt(a + b)*sqrt(((a + b)*cosh(x)^2 + (a + b)*sinh(x)^2 - a + b)/(cosh(
x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2)) + 4*(2*(a^3 + a^2*b)*cosh(x)^7 - 3*(2*a^3 + a^2*b)*cosh(x)^5 + (6*a^3 +
4*a^2*b - a*b^2 + b^3)*cosh(x)^3 - (2*a^3 + 3*a^2*b - b^3)*cosh(x))*sinh(x))/(cosh(x)^6 + 6*cosh(x)^5*sinh(x)
+ 15*cosh(x)^4*sinh(x)^2 + 20*cosh(x)^3*sinh(x)^3 + 15*cosh(x)^2*sinh(x)^4 + 6*cosh(x)*sinh(x)^5 + sinh(x)^6)
) + 3*((a^2*b + 2*a*b^2 + b^3)*cosh(x)^8 + 8*(a^2*b + 2*a*b^2 + b^3)*cosh(x)*sinh(x)^7 + (a^2*b + 2*a*b^2 + b^
3)*sinh(x)^8 - 4*(a^2*b - b^3)*cosh(x)^6 - 4*(a^2*b - b^3 - 7*(a^2*b + 2*a*b^2 + b^3)*cosh(x)^2)*sinh(x)^6 + 8
*(7*(a^2*b + 2*a*b^2 + b^3)*cosh(x)^3 - 3*(a^2*b - b^3)*cosh(x))*sinh(x)^5 + 2*(3*a^2*b - 2*a*b^2 + 3*b^3)*cos
h(x)^4 + 2*(35*(a^2*b + 2*a*b^2 + b^3)*cosh(x)^4 + 3*a^2*b - 2*a*b^2 + 3*b^3 - 30*(a^2*b - b^3)*cosh(x)^2)*sin
h(x)^4 + 8*(7*(a^2*b + 2*a*b^2 + b^3)*cosh(x)^5 - 10*(a^2*b - b^3)*cosh(x)^3 + (3*a^2*b - 2*a*b^2 + 3*b^3)*cos
h(x))*sinh(x)^3 + a^2*b + 2*a*b^2 + b^3 - 4*(a^2*b - b^3)*cosh(x)^2 + 4*(7*(a^2*b + 2*a*b^2 + b^3)*cosh(x)^6 -
15*(a^2*b - b^3)*cosh(x)^4 - a^2*b + b^3 + 3*(3*a^2*b - 2*a*b^2 + 3*b^3)*cosh(x)^2)*sinh(x)^2 + 8*((a^2*b + 2
*a*b^2 + b^3)*cosh(x)^7 - 3*(a^2*b - b^3)*cosh(x)^5 + (3*a^2*b - 2*a*b^2 + 3*b^3)*cosh(x)^3 - (a^2*b - b^3)*co
sh(x))*sinh(x))*sqrt(a + b)*log(((a + b)*cosh(x)^4 + 4*(a + b)*cosh(x)*sinh(x)^3 + (a + b)*sinh(x)^4 + 2*b*cos
h(x)^2 + 2*(3*(a + b)*cosh(x)^2 + b)*sinh(x)^2 + sqrt(2)*(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 + 1)*sqrt(
a + b)*sqrt(((a + b)*cosh(x)^2 + (a + b)*sinh(x)^2 - a + b)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2)) + 4*(
(a + b)*cosh(x)^3 + b*cosh(x))*sinh(x) + a + b)/(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2)) + 4*sqrt(2)*((a^3
- a^2*b - 5*a*b^2 - 3*b^3)*cosh(x)^6 + 6*(a^3 - a^2*b - 5*a*b^2 - 3*b^3)*cosh(x)*sinh(x)^5 + (a^3 - a^2*b - 5
*a*b^2 - 3*b^3)*sinh(x)^6 - 3*(a^3 - a^2*b - a*b^2 + b^3)*cosh(x)^4 - 3*(a^3 - a^2*b - a*b^2 + b^3 - 5*(a^3 -
a^2*b - 5*a*b^2 - 3*b^3)*cosh(x)^2)*sinh(x)^4 + 4*(5*(a^3 - a^2*b - 5*a*b^2 - 3*b^3)*cosh(x)^3 - 3*(a^3 - a^2*
b - a*b^2 + b^3)*cosh(x))*sinh(x)^3 - a^3 + a^2*b + 5*a*b^2 + 3*b^3 + 3*(a^3 - a^2*b - a*b^2 + b^3)*cosh(x)^2
+ 3*(5*(a^3 - a^2*b - 5*a*b^2 - 3*b^3)*cosh(x)^4 + a^3 - a^2*b - a*b^2 + b^3 - 6*(a^3 - a^2*b - a*b^2 + b^3)*c
osh(x)^2)*sinh(x)^2 + 6*((a^3 - a^2*b - 5*a*b^2 - 3*b^3)*cosh(x)^5 - 2*(a^3 - a^2*b - a*b^2 + b^3)*cosh(x)^3 +
(a^3 - a^2*b - a*b^2 + b^3)*cosh(x))*sinh(x))*sqrt(((a + b)*cosh(x)^2 + (a + b)*sinh(x)^2 - a + b)/(cosh(x)^2
- 2*cosh(x)*sinh(x) + sinh(x)^2)))/((a^5*b + 5*a^4*b^2 + 10*a^3*b^3 + 10*a^2*b^4 + 5*a*b^5 + b^6)*cosh(x)^8 +
8*(a^5*b + 5*a^4*b^2 + 10*a^3*b^3 + 10*a^2*b^4 + 5*a*b^5 + b^6)*cosh(x)*sinh(x)^7 + (a^5*b + 5*a^4*b^2 + 10*a
^3*b^3 + 10*a^2*b^4 + 5*a*b^5 + b^6)*sinh(x)^8 - 4*(a^5*b + 3*a^4*b^2 + 2*a^3*b^3 - 2*a^2*b^4 - 3*a*b^5 - b^6)
*cosh(x)^6 - 4*(a^5*b + 3*a^4*b^2 + 2*a^3*b^3 - 2*a^2*b^4 - 3*a*b^5 - b^6 - 7*(a^5*b + 5*a^4*b^2 + 10*a^3*b^3
+ 10*a^2*b^4 + 5*a*b^5 + b^6)*cosh(x)^2)*sinh(x...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\coth ^{3}{\left (x \right )}}{\left (a + b \coth ^{2}{\left (x \right )}\right )^{\frac {5}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)**3/(a+b*coth(x)**2)**(5/2),x)

[Out]

Integral(coth(x)**3/(a + b*coth(x)**2)**(5/2), x)

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)^3/(a+b*coth(x)^2)^(5/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Warning, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Ch
eck [abs(ex

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Mupad [B]
time = 3.34, size = 82, normalized size = 1.11 \begin {gather*} \frac {\mathrm {atanh}\left (\frac {\sqrt {b\,{\mathrm {coth}\left (x\right )}^2+a}\,\left (2\,a^2+4\,a\,b+2\,b^2\right )}{2\,{\left (a+b\right )}^{5/2}}\right )}{{\left (a+b\right )}^{5/2}}+\frac {\frac {a}{3\,\left (a+b\right )}-\frac {b\,\left (b\,{\mathrm {coth}\left (x\right )}^2+a\right )}{{\left (a+b\right )}^2}}{b\,{\left (b\,{\mathrm {coth}\left (x\right )}^2+a\right )}^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(x)^3/(a + b*coth(x)^2)^(5/2),x)

[Out]

atanh(((a + b*coth(x)^2)^(1/2)*(4*a*b + 2*a^2 + 2*b^2))/(2*(a + b)^(5/2)))/(a + b)^(5/2) + (a/(3*(a + b)) - (b
*(a + b*coth(x)^2))/(a + b)^2)/(b*(a + b*coth(x)^2)^(3/2))

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