3.3.18 \(\int \frac {\coth (x)}{(a+b \text {sech}^2(x))^{5/2}} \, dx\) [218]

Optimal. Leaf size=109 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}^2(x)}}{\sqrt {a}}\right )}{a^{5/2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}^2(x)}}{\sqrt {a+b}}\right )}{(a+b)^{5/2}}-\frac {b}{3 a (a+b) \left (a+b \text {sech}^2(x)\right )^{3/2}}-\frac {b (2 a+b)}{a^2 (a+b)^2 \sqrt {a+b \text {sech}^2(x)}} \]

[Out]

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

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Rubi [A]
time = 0.14, antiderivative size = 109, 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 = {4224, 457, 87, 157, 162, 65, 214} \begin {gather*} \frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}^2(x)}}{\sqrt {a}}\right )}{a^{5/2}}-\frac {b (2 a+b)}{a^2 (a+b)^2 \sqrt {a+b \text {sech}^2(x)}}-\frac {b}{3 a (a+b) \left (a+b \text {sech}^2(x)\right )^{3/2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}^2(x)}}{\sqrt {a+b}}\right )}{(a+b)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

ArcTanh[Sqrt[a + b*Sech[x]^2]/Sqrt[a]]/a^(5/2) - ArcTanh[Sqrt[a + b*Sech[x]^2]/Sqrt[a + b]]/(a + b)^(5/2) - b/
(3*a*(a + b)*(a + b*Sech[x]^2)^(3/2)) - (b*(2*a + b))/(a^2*(a + b)^2*Sqrt[a + b*Sech[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 4224

Int[((a_) + (b_.)*((c_.)*sec[(e_.) + (f_.)*(x_)])^(n_))^(p_.)*tan[(e_.) + (f_.)*(x_)]^(m_.), x_Symbol] :> With
[{ff = FreeFactors[Sec[e + f*x], x]}, Dist[1/f, Subst[Int[(-1 + ff^2*x^2)^((m - 1)/2)*((a + b*(c*ff*x)^n)^p/x)
, x], x, Sec[e + f*x]/ff], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && IntegerQ[(m - 1)/2] && (GtQ[m, 0] || EqQ[
n, 2] || EqQ[n, 4] || IGtQ[p, 0] || IntegersQ[2*n, p])

Rubi steps

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

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(242\) vs. \(2(109)=218\).
time = 0.73, size = 242, normalized size = 2.22 \begin {gather*} \frac {\left (-\frac {2 b \cosh (x) (a+2 b+a \cosh (2 x)) \left (7 a^2+16 a b+6 b^2+a (7 a+4 b) \cosh (2 x)\right )}{3 a^2 (a+b)^2}-\frac {(a+2 b+a \cosh (2 x))^{5/2} \left (\sqrt {a} \left (a^2-2 a b-b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a+b} \cosh (x)}{\sqrt {a+2 b+a \cosh (2 x)}}\right )+(a+b)^2 \left (\sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {2 a+2 b} \cosh (x)}{\sqrt {a+2 b+a \cosh (2 x)}}\right )-2 \sqrt {a+b} \log \left (\sqrt {2} \sqrt {a} \cosh (x)+\sqrt {a+2 b+a \cosh (2 x)}\right )\right )\right )}{\sqrt {2} a^{5/2} (a+b)^{5/2}}\right ) \text {sech}^5(x)}{8 \left (a+b \text {sech}^2(x)\right )^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(((-2*b*Cosh[x]*(a + 2*b + a*Cosh[2*x])*(7*a^2 + 16*a*b + 6*b^2 + a*(7*a + 4*b)*Cosh[2*x]))/(3*a^2*(a + b)^2)
- ((a + 2*b + a*Cosh[2*x])^(5/2)*(Sqrt[a]*(a^2 - 2*a*b - b^2)*ArcTanh[(Sqrt[2]*Sqrt[a + b]*Cosh[x])/Sqrt[a + 2
*b + a*Cosh[2*x]]] + (a + b)^2*(Sqrt[a]*ArcTanh[(Sqrt[2*a + 2*b]*Cosh[x])/Sqrt[a + 2*b + a*Cosh[2*x]]] - 2*Sqr
t[a + b]*Log[Sqrt[2]*Sqrt[a]*Cosh[x] + Sqrt[a + 2*b + a*Cosh[2*x]]])))/(Sqrt[2]*a^(5/2)*(a + b)^(5/2)))*Sech[x
]^5)/(8*(a + b*Sech[x]^2)^(5/2))

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Maple [F]
time = 1.64, size = 0, normalized size = 0.00 \[\int \frac {\coth \left (x \right )}{\left (a +b \mathrm {sech}\left (x \right )^{2}\right )^{\frac {5}{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

integrate(coth(x)/(b*sech(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. 4236 vs. \(2 (91) = 182\).
time = 0.95, size = 18563, normalized size = 170.30 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/12*(3*((a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*cosh(x)^8 + 8*(a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*cosh(x)*si
nh(x)^7 + (a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*sinh(x)^8 + 4*(a^5 + 5*a^4*b + 9*a^3*b^2 + 7*a^2*b^3 + 2*a*b^4
)*cosh(x)^6 + 4*(a^5 + 5*a^4*b + 9*a^3*b^2 + 7*a^2*b^3 + 2*a*b^4 + 7*(a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*cos
h(x)^2)*sinh(x)^6 + 8*(7*(a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*cosh(x)^3 + 3*(a^5 + 5*a^4*b + 9*a^3*b^2 + 7*a^
2*b^3 + 2*a*b^4)*cosh(x))*sinh(x)^5 + a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3 + 2*(3*a^5 + 17*a^4*b + 41*a^3*b^2 +
51*a^2*b^3 + 32*a*b^4 + 8*b^5)*cosh(x)^4 + 2*(3*a^5 + 17*a^4*b + 41*a^3*b^2 + 51*a^2*b^3 + 32*a*b^4 + 8*b^5 +
35*(a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*cosh(x)^4 + 30*(a^5 + 5*a^4*b + 9*a^3*b^2 + 7*a^2*b^3 + 2*a*b^4)*cos
h(x)^2)*sinh(x)^4 + 8*(7*(a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*cosh(x)^5 + 10*(a^5 + 5*a^4*b + 9*a^3*b^2 + 7*a
^2*b^3 + 2*a*b^4)*cosh(x)^3 + (3*a^5 + 17*a^4*b + 41*a^3*b^2 + 51*a^2*b^3 + 32*a*b^4 + 8*b^5)*cosh(x))*sinh(x)
^3 + 4*(a^5 + 5*a^4*b + 9*a^3*b^2 + 7*a^2*b^3 + 2*a*b^4)*cosh(x)^2 + 4*(7*(a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3
)*cosh(x)^6 + a^5 + 5*a^4*b + 9*a^3*b^2 + 7*a^2*b^3 + 2*a*b^4 + 15*(a^5 + 5*a^4*b + 9*a^3*b^2 + 7*a^2*b^3 + 2*
a*b^4)*cosh(x)^4 + 3*(3*a^5 + 17*a^4*b + 41*a^3*b^2 + 51*a^2*b^3 + 32*a*b^4 + 8*b^5)*cosh(x)^2)*sinh(x)^2 + 8*
((a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*cosh(x)^7 + 3*(a^5 + 5*a^4*b + 9*a^3*b^2 + 7*a^2*b^3 + 2*a*b^4)*cosh(x)
^5 + (3*a^5 + 17*a^4*b + 41*a^3*b^2 + 51*a^2*b^3 + 32*a*b^4 + 8*b^5)*cosh(x)^3 + (a^5 + 5*a^4*b + 9*a^3*b^2 +
7*a^2*b^3 + 2*a*b^4)*cosh(x))*sinh(x))*sqrt(a)*log(((a^3 + 2*a^2*b + a*b^2)*cosh(x)^8 + 8*(a^3 + 2*a^2*b + a*b
^2)*cosh(x)*sinh(x)^7 + (a^3 + 2*a^2*b + a*b^2)*sinh(x)^8 + 2*(2*a^3 + 5*a^2*b + 4*a*b^2 + b^3)*cosh(x)^6 + 2*
(2*a^3 + 5*a^2*b + 4*a*b^2 + b^3 + 14*(a^3 + 2*a^2*b + a*b^2)*cosh(x)^2)*sinh(x)^6 + 4*(14*(a^3 + 2*a^2*b + a*
b^2)*cosh(x)^3 + 3*(2*a^3 + 5*a^2*b + 4*a*b^2 + b^3)*cosh(x))*sinh(x)^5 + (6*a^3 + 14*a^2*b + 9*a*b^2)*cosh(x)
^4 + (70*(a^3 + 2*a^2*b + a*b^2)*cosh(x)^4 + 6*a^3 + 14*a^2*b + 9*a*b^2 + 30*(2*a^3 + 5*a^2*b + 4*a*b^2 + b^3)
*cosh(x)^2)*sinh(x)^4 + 4*(14*(a^3 + 2*a^2*b + a*b^2)*cosh(x)^5 + 10*(2*a^3 + 5*a^2*b + 4*a*b^2 + b^3)*cosh(x)
^3 + (6*a^3 + 14*a^2*b + 9*a*b^2)*cosh(x))*sinh(x)^3 + a^3 + 2*(2*a^3 + 3*a^2*b)*cosh(x)^2 + 2*(14*(a^3 + 2*a^
2*b + a*b^2)*cosh(x)^6 + 15*(2*a^3 + 5*a^2*b + 4*a*b^2 + b^3)*cosh(x)^4 + 2*a^3 + 3*a^2*b + 3*(6*a^3 + 14*a^2*
b + 9*a*b^2)*cosh(x)^2)*sinh(x)^2 + sqrt(2)*((a^2 + 2*a*b + b^2)*cosh(x)^6 + 6*(a^2 + 2*a*b + b^2)*cosh(x)*sin
h(x)^5 + (a^2 + 2*a*b + b^2)*sinh(x)^6 + 3*(a^2 + 2*a*b + b^2)*cosh(x)^4 + 3*(5*(a^2 + 2*a*b + b^2)*cosh(x)^2
+ a^2 + 2*a*b + b^2)*sinh(x)^4 + 4*(5*(a^2 + 2*a*b + b^2)*cosh(x)^3 + 3*(a^2 + 2*a*b + b^2)*cosh(x))*sinh(x)^3
+ (3*a^2 + 4*a*b)*cosh(x)^2 + (15*(a^2 + 2*a*b + b^2)*cosh(x)^4 + 18*(a^2 + 2*a*b + b^2)*cosh(x)^2 + 3*a^2 +
4*a*b)*sinh(x)^2 + a^2 + 2*(3*(a^2 + 2*a*b + b^2)*cosh(x)^5 + 6*(a^2 + 2*a*b + b^2)*cosh(x)^3 + (3*a^2 + 4*a*b
)*cosh(x))*sinh(x))*sqrt(a)*sqrt((a*cosh(x)^2 + a*sinh(x)^2 + a + 2*b)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x
)^2)) + 4*(2*(a^3 + 2*a^2*b + a*b^2)*cosh(x)^7 + 3*(2*a^3 + 5*a^2*b + 4*a*b^2 + b^3)*cosh(x)^5 + (6*a^3 + 14*a
^2*b + 9*a*b^2)*cosh(x)^3 + (2*a^3 + 3*a^2*b)*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*cos
h(x)^8 + 8*a^5*cosh(x)*sinh(x)^7 + a^5*sinh(x)^8 + 4*(a^5 + 2*a^4*b)*cosh(x)^6 + 4*(7*a^5*cosh(x)^2 + a^5 + 2*
a^4*b)*sinh(x)^6 + 8*(7*a^5*cosh(x)^3 + 3*(a^5 + 2*a^4*b)*cosh(x))*sinh(x)^5 + a^5 + 2*(3*a^5 + 8*a^4*b + 8*a^
3*b^2)*cosh(x)^4 + 2*(35*a^5*cosh(x)^4 + 3*a^5 + 8*a^4*b + 8*a^3*b^2 + 30*(a^5 + 2*a^4*b)*cosh(x)^2)*sinh(x)^4
+ 8*(7*a^5*cosh(x)^5 + 10*(a^5 + 2*a^4*b)*cosh(x)^3 + (3*a^5 + 8*a^4*b + 8*a^3*b^2)*cosh(x))*sinh(x)^3 + 4*(a
^5 + 2*a^4*b)*cosh(x)^2 + 4*(7*a^5*cosh(x)^6 + a^5 + 2*a^4*b + 15*(a^5 + 2*a^4*b)*cosh(x)^4 + 3*(3*a^5 + 8*a^4
*b + 8*a^3*b^2)*cosh(x)^2)*sinh(x)^2 + 8*(a^5*cosh(x)^7 + 3*(a^5 + 2*a^4*b)*cosh(x)^5 + (3*a^5 + 8*a^4*b + 8*a
^3*b^2)*cosh(x)^3 + (a^5 + 2*a^4*b)*cosh(x))*sinh(x))*sqrt(a + b)*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 + 3*b)*cosh(x)^2 + 2*(3*(2*a + b)*cosh(x)^2 + 2*a + 3*b)*sinh(x)^2
- 2*sqrt(2)*(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 + 1)*sqrt(a + b)*sqrt((a*cosh(x)^2 + a*sinh(x)^2 + a +
2*b)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2)) + 4*((2*a + b)*cosh(x)^3 + (2*a + 3*b)*cosh(x))*sinh(x) + 2
*a + b)/(cosh(x)^4 + 4*cosh(x)*sinh(x)^3 + sinh(x)^4 + 2*(3*cosh(x)^2 - 1)*sinh(x)^2 - 2*cosh(x)^2 + 4*(cosh(x
)^3 - cosh(x))*sinh(x) + 1)) + 3*((a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*cosh(x)^8 + 8*(a^5 + 3*a^4*b + 3*a^3*b
^2 + a^2*b^3)*cosh(x)*sinh(x)^7 + (a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*sinh(x)^8 + 4*(a^5 + 5*a^4*b + 9*a^3*b
^2 + 7*a^2*b^3 + 2*a*b^4)*cosh(x)^6 + 4*(a^5 + 5*a^4*b + 9*a^3*b^2 + 7*a^2*b^3 + 2*a*b^4 + 7*(a^5 + 3*a^4*b +
3*a^3*b^2 + a^2*b^3)*cosh(x)^2)*sinh(x)^6 + 8*(...

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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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)/(a+b*sech(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:Error: Bad Argument Type

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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