3.1.46 \(\int \frac {\tanh (x)}{(a+b \coth ^2(x))^{5/2}} \, dx\) [46]

Optimal. Leaf size=108 \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \coth ^2(x)}}{\sqrt {a}}\right )}{a^{5/2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \coth ^2(x)}}{\sqrt {a+b}}\right )}{(a+b)^{5/2}}+\frac {b}{3 a (a+b) \left (a+b \coth ^2(x)\right )^{3/2}}+\frac {b (2 a+b)}{a^2 (a+b)^2 \sqrt {a+b \coth ^2(x)}} \]

[Out]

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

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Rubi [A]
time = 0.15, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {3751, 457, 87, 157, 162, 65, 214} \begin {gather*} -\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \coth ^2(x)}}{\sqrt {a}}\right )}{a^{5/2}}+\frac {b (2 a+b)}{a^2 (a+b)^2 \sqrt {a+b \coth ^2(x)}}+\frac {b}{3 a (a+b) \left (a+b \coth ^2(x)\right )^{3/2}}+\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[Tanh[x]/(a + b*Coth[x]^2)^(5/2),x]

[Out]

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

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 87

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

Rule 157

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f
))), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegersQ[2*m, 2*n, 2*p]

Rule 162

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :>
Dist[(b*g - a*h)/(b*c - a*d), Int[(e + f*x)^p/(a + b*x), x], x] - Dist[(d*g - c*h)/(b*c - a*d), Int[(e + f*x)
^p/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]

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 {\tanh (x)}{\left (a+b \coth ^2(x)\right )^{5/2}} \, dx &=\text {Subst}\left (\int \frac {1}{x \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 {1}{(1-x) x (a+b x)^{5/2}} \, dx,x,\coth ^2(x)\right )\\ &=\frac {b}{3 a (a+b) \left (a+b \coth ^2(x)\right )^{3/2}}-\frac {\text {Subst}\left (\int \frac {-a-b+b x}{(1-x) x (a+b x)^{3/2}} \, dx,x,\coth ^2(x)\right )}{2 a (a+b)}\\ &=\frac {b}{3 a (a+b) \left (a+b \coth ^2(x)\right )^{3/2}}+\frac {b (2 a+b)}{a^2 (a+b)^2 \sqrt {a+b \coth ^2(x)}}-\frac {\text {Subst}\left (\int \frac {-\frac {1}{2} (a+b)^2+\frac {1}{2} b (2 a+b) x}{(1-x) x \sqrt {a+b x}} \, dx,x,\coth ^2(x)\right )}{a^2 (a+b)^2}\\ &=\frac {b}{3 a (a+b) \left (a+b \coth ^2(x)\right )^{3/2}}+\frac {b (2 a+b)}{a^2 (a+b)^2 \sqrt {a+b \coth ^2(x)}}+\frac {\text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\coth ^2(x)\right )}{2 a^2}+\frac {\text {Subst}\left (\int \frac {1}{(1-x) \sqrt {a+b x}} \, dx,x,\coth ^2(x)\right )}{2 (a+b)^2}\\ &=\frac {b}{3 a (a+b) \left (a+b \coth ^2(x)\right )^{3/2}}+\frac {b (2 a+b)}{a^2 (a+b)^2 \sqrt {a+b \coth ^2(x)}}+\frac {\text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \coth ^2(x)}\right )}{a^2 b}+\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}}\right )}{a^{5/2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \coth ^2(x)}}{\sqrt {a+b}}\right )}{(a+b)^{5/2}}+\frac {b}{3 a (a+b) \left (a+b \coth ^2(x)\right )^{3/2}}+\frac {b (2 a+b)}{a^2 (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 = 0.05, size = 73, normalized size = 0.68 \begin {gather*} \frac {-a \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\frac {a+b \coth ^2(x)}{a+b}\right )+(a+b) \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};1+\frac {b \coth ^2(x)}{a}\right )}{3 a (a+b) \left (a+b \coth ^2(x)\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 2.08, size = 0, normalized size = 0.00 \[\int \frac {\tanh \left (x \right )}{\left (a +b \left (\coth ^{2}\left (x \right )\right )\right )^{\frac {5}{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

integrate(tanh(x)/(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. 4452 vs. \(2 (90) = 180\).
time = 1.22, size = 19199, normalized size = 177.77 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/12*(3*((a^5 + 2*a^4*b + a^3*b^2)*cosh(x)^8 + 8*(a^5 + 2*a^4*b + a^3*b^2)*cosh(x)*sinh(x)^7 + (a^5 + 2*a^4*b
+ a^3*b^2)*sinh(x)^8 - 4*(a^5 - a^3*b^2)*cosh(x)^6 - 4*(a^5 - a^3*b^2 - 7*(a^5 + 2*a^4*b + a^3*b^2)*cosh(x)^2
)*sinh(x)^6 + 8*(7*(a^5 + 2*a^4*b + a^3*b^2)*cosh(x)^3 - 3*(a^5 - a^3*b^2)*cosh(x))*sinh(x)^5 + a^5 + 2*a^4*b
+ a^3*b^2 + 2*(3*a^5 - 2*a^4*b + 3*a^3*b^2)*cosh(x)^4 + 2*(3*a^5 - 2*a^4*b + 3*a^3*b^2 + 35*(a^5 + 2*a^4*b + a
^3*b^2)*cosh(x)^4 - 30*(a^5 - a^3*b^2)*cosh(x)^2)*sinh(x)^4 + 8*(7*(a^5 + 2*a^4*b + a^3*b^2)*cosh(x)^5 - 10*(a
^5 - a^3*b^2)*cosh(x)^3 + (3*a^5 - 2*a^4*b + 3*a^3*b^2)*cosh(x))*sinh(x)^3 - 4*(a^5 - a^3*b^2)*cosh(x)^2 + 4*(
7*(a^5 + 2*a^4*b + a^3*b^2)*cosh(x)^6 - a^5 + a^3*b^2 - 15*(a^5 - a^3*b^2)*cosh(x)^4 + 3*(3*a^5 - 2*a^4*b + 3*
a^3*b^2)*cosh(x)^2)*sinh(x)^2 + 8*((a^5 + 2*a^4*b + a^3*b^2)*cosh(x)^7 - 3*(a^5 - a^3*b^2)*cosh(x)^5 + (3*a^5
- 2*a^4*b + 3*a^3*b^2)*cosh(x)^3 - (a^5 - a^3*b^2)*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)*cos
h(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)) + 6*((a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 +
b^5)*cosh(x)^8 + 8*(a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*cosh(x)*sinh(x)^7 + (a^5 + 5*a^4*
b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*sinh(x)^8 - 4*(a^5 + 3*a^4*b + 2*a^3*b^2 - 2*a^2*b^3 - 3*a*b^4 -
b^5)*cosh(x)^6 - 4*(a^5 + 3*a^4*b + 2*a^3*b^2 - 2*a^2*b^3 - 3*a*b^4 - b^5 - 7*(a^5 + 5*a^4*b + 10*a^3*b^2 + 10
*a^2*b^3 + 5*a*b^4 + b^5)*cosh(x)^2)*sinh(x)^6 + 8*(7*(a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5
)*cosh(x)^3 - 3*(a^5 + 3*a^4*b + 2*a^3*b^2 - 2*a^2*b^3 - 3*a*b^4 - b^5)*cosh(x))*sinh(x)^5 + a^5 + 5*a^4*b + 1
0*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5 + 2*(3*a^5 + 7*a^4*b + 6*a^3*b^2 + 6*a^2*b^3 + 7*a*b^4 + 3*b^5)*cosh(x)
^4 + 2*(3*a^5 + 7*a^4*b + 6*a^3*b^2 + 6*a^2*b^3 + 7*a*b^4 + 3*b^5 + 35*(a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^
3 + 5*a*b^4 + b^5)*cosh(x)^4 - 30*(a^5 + 3*a^4*b + 2*a^3*b^2 - 2*a^2*b^3 - 3*a*b^4 - b^5)*cosh(x)^2)*sinh(x)^4
+ 8*(7*(a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*cosh(x)^5 - 10*(a^5 + 3*a^4*b + 2*a^3*b^2 -
2*a^2*b^3 - 3*a*b^4 - b^5)*cosh(x)^3 + (3*a^5 + 7*a^4*b + 6*a^3*b^2 + 6*a^2*b^3 + 7*a*b^4 + 3*b^5)*cosh(x))*si
nh(x)^3 - 4*(a^5 + 3*a^4*b + 2*a^3*b^2 - 2*a^2*b^3 - 3*a*b^4 - b^5)*cosh(x)^2 + 4*(7*(a^5 + 5*a^4*b + 10*a^3*b
^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*cosh(x)^6 - a^5 - 3*a^4*b - 2*a^3*b^2 + 2*a^2*b^3 + 3*a*b^4 + b^5 - 15*(a^5 +
3*a^4*b + 2*a^3*b^2 - 2*a^2*b^3 - 3*a*b^4 - b^5)*cosh(x)^4 + 3*(3*a^5 + 7*a^4*b + 6*a^3*b^2 + 6*a^2*b^3 + 7*a
*b^4 + 3*b^5)*cosh(x)^2)*sinh(x)^2 + 8*((a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*cosh(x)^7 -
3*(a^5 + 3*a^4*b + 2*a^3*b^2 - 2*a^2*b^3 - 3*a*b^4 - b^5)*cosh(x)^5 + (3*a^5 + 7*a^4*b + 6*a^3*b^2 + 6*a^2*b^3
+ 7*a*b^4 + 3*b^5)*cosh(x)^3 - (a^5 + 3*a^4*b + 2*a^3*b^2 - 2*a^2*b^3 - 3*a*b^4 - b^5)*cosh(x))*sinh(x))*sqrt
(a)*log(-((2*a + b)*cosh(x)^4 + 4*(2*a + b)*cosh(x)*sinh(x)^3 + (2*a + b)*sinh(x)^4 - 2*(2*a - b)*cosh(x)^2 +
2*(3*(2*a + b)*cosh(x)^2 - 2*a + b)*sinh(x)^2 - 2*sqrt(2)*(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 - 1)*sqrt
(a)*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 + b)*cosh(x)^3 - (2*a - b)*cosh(x))*sinh(x) + 2*a + b)/(cosh(x)^4 + 4*cosh(x)*sinh(x)^3 + sinh(x)^4 + 2*(3*c
osh(x)^2 + 1)*sinh(x)^2 + 2*cosh(x)^2 + 4*(cosh(x)^3 + cosh(x))*sinh(x) + 1)) + 3*((a^5 + 2*a^4*b + a^3*b^2)*c
osh(x)^8 + 8*(a^5 + 2*a^4*b + a^3*b^2)*cosh(x)*sinh(x)^7 + (a^5 + 2*a^4*b + a^3*b^2)*sinh(x)^8 - 4*(a^5 - a^3*
b^2)*cosh(x)^6 - 4*(a^5 - a^3*b^2 - 7*(a^5 + 2*...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\tanh {\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(tanh(x)/(a+b*coth(x)**2)**(5/2),x)

[Out]

Integral(tanh(x)/(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(tanh(x)/(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 [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\mathrm {tanh}\left (x\right )}{{\left (b\,{\mathrm {coth}\left (x\right )}^2+a\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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