3.42 \(\int \frac {\tanh ^2(x)}{(a+b \coth ^2(x))^{3/2}} \, dx\)
Optimal. Leaf size=85 \[ -\frac {(a+2 b) \tanh (x) \sqrt {a+b \coth ^2(x)}}{a^2 (a+b)}+\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b} \coth (x)}{\sqrt {a+b \coth ^2(x)}}\right )}{(a+b)^{3/2}}+\frac {b \tanh (x)}{a (a+b) \sqrt {a+b \coth ^2(x)}} \]
[Out]
arctanh(coth(x)*(a+b)^(1/2)/(a+b*coth(x)^2)^(1/2))/(a+b)^(3/2)+b*tanh(x)/a/(a+b)/(a+b*coth(x)^2)^(1/2)-(a+2*b)
*(a+b*coth(x)^2)^(1/2)*tanh(x)/a^2/(a+b)
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Rubi [A] time = 0.16, antiderivative size = 85, 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 =
{3670, 472, 583, 12, 377, 206} \[ -\frac {(a+2 b) \tanh (x) \sqrt {a+b \coth ^2(x)}}{a^2 (a+b)}+\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b} \coth (x)}{\sqrt {a+b \coth ^2(x)}}\right )}{(a+b)^{3/2}}+\frac {b \tanh (x)}{a (a+b) \sqrt {a+b \coth ^2(x)}} \]
Antiderivative was successfully verified.
[In]
Int[Tanh[x]^2/(a + b*Coth[x]^2)^(3/2),x]
[Out]
ArcTanh[(Sqrt[a + b]*Coth[x])/Sqrt[a + b*Coth[x]^2]]/(a + b)^(3/2) + (b*Tanh[x])/(a*(a + b)*Sqrt[a + b*Coth[x]
^2]) - ((a + 2*b)*Sqrt[a + b*Coth[x]^2]*Tanh[x])/(a^2*(a + b))
Rule 12
Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]
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 377
Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]
Rule 472
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[(b*(e*x
)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*e*n*(b*c - a*d)*(p + 1)), x] + Dist[1/(a*n*(b*c - a*d)*(
p + 1)), Int[(e*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*b*(m + 1) + n*(b*c - a*d)*(p + 1) + d*b*(m + n*(
p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, m, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p
, -1] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]
Rule 583
Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[(e*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*c*g*(m + 1)), x] + Dist[1/(a*c*
g^n*(m + 1)), Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + n + 1) - e
*n*(b*c*p + a*d*q) - b*e*d*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] &&
IGtQ[n, 0] && LtQ[m, -1]
Rule 3670
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 ^2(x)}{\left (a+b \coth ^2(x)\right )^{3/2}} \, dx &=\operatorname {Subst}\left (\int \frac {1}{x^2 \left (1-x^2\right ) \left (a+b x^2\right )^{3/2}} \, dx,x,\coth (x)\right )\\ &=\frac {b \tanh (x)}{a (a+b) \sqrt {a+b \coth ^2(x)}}-\frac {\operatorname {Subst}\left (\int \frac {-a-2 b+2 b x^2}{x^2 \left (1-x^2\right ) \sqrt {a+b x^2}} \, dx,x,\coth (x)\right )}{a (a+b)}\\ &=\frac {b \tanh (x)}{a (a+b) \sqrt {a+b \coth ^2(x)}}-\frac {(a+2 b) \sqrt {a+b \coth ^2(x)} \tanh (x)}{a^2 (a+b)}+\frac {\operatorname {Subst}\left (\int \frac {a^2}{\left (1-x^2\right ) \sqrt {a+b x^2}} \, dx,x,\coth (x)\right )}{a^2 (a+b)}\\ &=\frac {b \tanh (x)}{a (a+b) \sqrt {a+b \coth ^2(x)}}-\frac {(a+2 b) \sqrt {a+b \coth ^2(x)} \tanh (x)}{a^2 (a+b)}+\frac {\operatorname {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {a+b x^2}} \, dx,x,\coth (x)\right )}{a+b}\\ &=\frac {b \tanh (x)}{a (a+b) \sqrt {a+b \coth ^2(x)}}-\frac {(a+2 b) \sqrt {a+b \coth ^2(x)} \tanh (x)}{a^2 (a+b)}+\frac {\operatorname {Subst}\left (\int \frac {1}{1-(a+b) x^2} \, dx,x,\frac {\coth (x)}{\sqrt {a+b \coth ^2(x)}}\right )}{a+b}\\ &=\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b} \coth (x)}{\sqrt {a+b \coth ^2(x)}}\right )}{(a+b)^{3/2}}+\frac {b \tanh (x)}{a (a+b) \sqrt {a+b \coth ^2(x)}}-\frac {(a+2 b) \sqrt {a+b \coth ^2(x)} \tanh (x)}{a^2 (a+b)}\\ \end {align*}
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Mathematica [A] time = 6.01, size = 149, normalized size = 1.75 \[ \frac {1}{2} \sqrt {\text {csch}^2(x) ((a+b) \cosh (2 x)-a+b)} \left (\frac {2 \sinh (x) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a+b} \cosh (x)}{\sqrt {(a+b) \cosh (2 x)-a+b}}\right )}{(a+b)^{3/2} \sqrt {(a+b) \cosh (2 x)-a+b}}-\frac {\sqrt {2} \tanh (x) \left (\left (a^2+2 a b+2 b^2\right ) \cosh (2 x)-a^2+2 b^2\right )}{a^2 (a+b) ((a+b) \cosh (2 x)-a+b)}\right ) \]
Warning: Unable to verify antiderivative.
[In]
Integrate[Tanh[x]^2/(a + b*Coth[x]^2)^(3/2),x]
[Out]
(Sqrt[(-a + b + (a + b)*Cosh[2*x])*Csch[x]^2]*((2*ArcTanh[(Sqrt[2]*Sqrt[a + b]*Cosh[x])/Sqrt[-a + b + (a + b)*
Cosh[2*x]]]*Sinh[x])/((a + b)^(3/2)*Sqrt[-a + b + (a + b)*Cosh[2*x]]) - (Sqrt[2]*(-a^2 + 2*b^2 + (a^2 + 2*a*b
+ 2*b^2)*Cosh[2*x])*Tanh[x])/(a^2*(a + b)*(-a + b + (a + b)*Cosh[2*x]))))/2
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fricas [B] time = 0.82, size = 3931, normalized size = 46.25 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(tanh(x)^2/(a+b*coth(x)^2)^(3/2),x, algorithm="fricas")
[Out]
[1/4*(((a^3 + a^2*b)*cosh(x)^6 + 6*(a^3 + a^2*b)*cosh(x)*sinh(x)^5 + (a^3 + a^2*b)*sinh(x)^6 - (a^3 - 3*a^2*b)
*cosh(x)^4 - (a^3 - 3*a^2*b - 15*(a^3 + a^2*b)*cosh(x)^2)*sinh(x)^4 + 4*(5*(a^3 + a^2*b)*cosh(x)^3 - (a^3 - 3*
a^2*b)*cosh(x))*sinh(x)^3 + a^3 + a^2*b - (a^3 - 3*a^2*b)*cosh(x)^2 + (15*(a^3 + a^2*b)*cosh(x)^4 - a^3 + 3*a^
2*b - 6*(a^3 - 3*a^2*b)*cosh(x)^2)*sinh(x)^2 + 2*(3*(a^3 + a^2*b)*cosh(x)^5 - 2*(a^3 - 3*a^2*b)*cosh(x)^3 - (a
^3 - 3*a^2*b)*cosh(x))*sinh(x))*sqrt(a + b)*log(((a*b^2 + b^3)*cosh(x)^8 + 8*(a*b^2 + b^3)*cosh(x)*sinh(x)^7 +
(a*b^2 + b^3)*sinh(x)^8 + 2*(a*b^2 + 2*b^3)*cosh(x)^6 + 2*(a*b^2 + 2*b^3 + 14*(a*b^2 + b^3)*cosh(x)^2)*sinh(x
)^6 + 4*(14*(a*b^2 + b^3)*cosh(x)^3 + 3*(a*b^2 + 2*b^3)*cosh(x))*sinh(x)^5 + (a^3 - a^2*b + 4*a*b^2 + 6*b^3)*c
osh(x)^4 + (70*(a*b^2 + b^3)*cosh(x)^4 + a^3 - a^2*b + 4*a*b^2 + 6*b^3 + 30*(a*b^2 + 2*b^3)*cosh(x)^2)*sinh(x)
^4 + 4*(14*(a*b^2 + b^3)*cosh(x)^5 + 10*(a*b^2 + 2*b^3)*cosh(x)^3 + (a^3 - a^2*b + 4*a*b^2 + 6*b^3)*cosh(x))*s
inh(x)^3 + a^3 + 3*a^2*b + 3*a*b^2 + b^3 - 2*(a^3 - 3*a*b^2 - 2*b^3)*cosh(x)^2 + 2*(14*(a*b^2 + b^3)*cosh(x)^6
+ 15*(a*b^2 + 2*b^3)*cosh(x)^4 - a^3 + 3*a*b^2 + 2*b^3 + 3*(a^3 - a^2*b + 4*a*b^2 + 6*b^3)*cosh(x)^2)*sinh(x)
^2 + sqrt(2)*(b^2*cosh(x)^6 + 6*b^2*cosh(x)*sinh(x)^5 + b^2*sinh(x)^6 + 3*b^2*cosh(x)^4 + 3*(5*b^2*cosh(x)^2 +
b^2)*sinh(x)^4 + 4*(5*b^2*cosh(x)^3 + 3*b^2*cosh(x))*sinh(x)^3 - (a^2 - 2*a*b - 3*b^2)*cosh(x)^2 + (15*b^2*co
sh(x)^4 + 18*b^2*cosh(x)^2 - a^2 + 2*a*b + 3*b^2)*sinh(x)^2 + a^2 + 2*a*b + b^2 + 2*(3*b^2*cosh(x)^5 + 6*b^2*c
osh(x)^3 - (a^2 - 2*a*b - 3*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*b^2 + b^3)*cosh(x)^7 + 3*(a*b^2 + 2*b^3)*cosh(x)^
5 + (a^3 - a^2*b + 4*a*b^2 + 6*b^3)*cosh(x)^3 - (a^3 - 3*a*b^2 - 2*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)) + ((a^3 + a^2*b)*cosh(x)^6 + 6*(a^3 + a^2*b)*cosh(x)*sinh(x)^5 + (a^3 + a^2*b)*sinh(x)^6 - (a^3
- 3*a^2*b)*cosh(x)^4 - (a^3 - 3*a^2*b - 15*(a^3 + a^2*b)*cosh(x)^2)*sinh(x)^4 + 4*(5*(a^3 + a^2*b)*cosh(x)^3 -
(a^3 - 3*a^2*b)*cosh(x))*sinh(x)^3 + a^3 + a^2*b - (a^3 - 3*a^2*b)*cosh(x)^2 + (15*(a^3 + a^2*b)*cosh(x)^4 -
a^3 + 3*a^2*b - 6*(a^3 - 3*a^2*b)*cosh(x)^2)*sinh(x)^2 + 2*(3*(a^3 + a^2*b)*cosh(x)^5 - 2*(a^3 - 3*a^2*b)*cosh
(x)^3 - (a^3 - 3*a^2*b)*cosh(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*a*cosh(x)^2 + 2*(3*(a + b)*cosh(x)^2 - a)*sinh(x)^2 + sqrt(2)*(cosh(x)^2 + 2*cosh(x)*sin
h(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 - a*cosh(x))*sinh(x) + a + b)/(cosh(x)^2 + 2*cosh(x)*sinh(x) + si
nh(x)^2)) - 4*sqrt(2)*((a^3 + 3*a^2*b + 4*a*b^2 + 2*b^3)*cosh(x)^4 + 4*(a^3 + 3*a^2*b + 4*a*b^2 + 2*b^3)*cosh(
x)*sinh(x)^3 + (a^3 + 3*a^2*b + 4*a*b^2 + 2*b^3)*sinh(x)^4 + a^3 + 3*a^2*b + 4*a*b^2 + 2*b^3 - 2*(a^3 + a^2*b
- 2*a*b^2 - 2*b^3)*cosh(x)^2 - 2*(a^3 + a^2*b - 2*a*b^2 - 2*b^3 - 3*(a^3 + 3*a^2*b + 4*a*b^2 + 2*b^3)*cosh(x)^
2)*sinh(x)^2 + 4*((a^3 + 3*a^2*b + 4*a*b^2 + 2*b^3)*cosh(x)^3 - (a^3 + a^2*b - 2*a*b^2 - 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 +
3*a^4*b + 3*a^3*b^2 + a^2*b^3)*cosh(x)^6 + 6*(a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*cosh(x)*sinh(x)^5 + (a^5 +
3*a^4*b + 3*a^3*b^2 + a^2*b^3)*sinh(x)^6 + a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3 - (a^5 - a^4*b - 5*a^3*b^2 - 3
*a^2*b^3)*cosh(x)^4 - (a^5 - a^4*b - 5*a^3*b^2 - 3*a^2*b^3 - 15*(a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*cosh(x)^
2)*sinh(x)^4 + 4*(5*(a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*cosh(x)^3 - (a^5 - a^4*b - 5*a^3*b^2 - 3*a^2*b^3)*co
sh(x))*sinh(x)^3 - (a^5 - a^4*b - 5*a^3*b^2 - 3*a^2*b^3)*cosh(x)^2 - (a^5 - a^4*b - 5*a^3*b^2 - 3*a^2*b^3 - 15
*(a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*cosh(x)^4 + 6*(a^5 - a^4*b - 5*a^3*b^2 - 3*a^2*b^3)*cosh(x)^2)*sinh(x)^
2 + 2*(3*(a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*cosh(x)^5 - 2*(a^5 - a^4*b - 5*a^3*b^2 - 3*a^2*b^3)*cosh(x)^3 -
(a^5 - a^4*b - 5*a^3*b^2 - 3*a^2*b^3)*cosh(x))*sinh(x)), -1/2*(((a^3 + a^2*b)*cosh(x)^6 + 6*(a^3 + a^2*b)*cos
h(x)*sinh(x)^5 + (a^3 + a^2*b)*sinh(x)^6 - (a^3 - 3*a^2*b)*cosh(x)^4 - (a^3 - 3*a^2*b - 15*(a^3 + a^2*b)*cosh(
x)^2)*sinh(x)^4 + 4*(5*(a^3 + a^2*b)*cosh(x)^3 - (a^3 - 3*a^2*b)*cosh(x))*sinh(x)^3 + a^3 + a^2*b - (a^3 - 3*a
^2*b)*cosh(x)^2 + (15*(a^3 + a^2*b)*cosh(x)^4 - a^3 + 3*a^2*b - 6*(a^3 - 3*a^2*b)*cosh(x)^2)*sinh(x)^2 + 2*(3*
(a^3 + a^2*b)*cosh(x)^5 - 2*(a^3 - 3*a^2*b)*cosh(x)^3 - (a^3 - 3*a^2*b)*cosh(x))*sinh(x))*sqrt(-a - b)*arctan(
sqrt(2)*(b*cosh(x)^2 + 2*b*cosh(x)*sinh(x) + b*sinh(x)^2 + a + b)*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))/((a*b + b^2)*cosh(x)^4 + 4*(a*b + b^2)*cosh
(x)*sinh(x)^3 + (a*b + b^2)*sinh(x)^4 - (a^2 - a*b - 2*b^2)*cosh(x)^2 + (6*(a*b + b^2)*cosh(x)^2 - a^2 + a*b +
2*b^2)*sinh(x)^2 + a^2 + 2*a*b + b^2 + 2*(2*(a*b + b^2)*cosh(x)^3 - (a^2 - a*b - 2*b^2)*cosh(x))*sinh(x))) +
((a^3 + a^2*b)*cosh(x)^6 + 6*(a^3 + a^2*b)*cosh(x)*sinh(x)^5 + (a^3 + a^2*b)*sinh(x)^6 - (a^3 - 3*a^2*b)*cosh(
x)^4 - (a^3 - 3*a^2*b - 15*(a^3 + a^2*b)*cosh(x)^2)*sinh(x)^4 + 4*(5*(a^3 + a^2*b)*cosh(x)^3 - (a^3 - 3*a^2*b)
*cosh(x))*sinh(x)^3 + a^3 + a^2*b - (a^3 - 3*a^2*b)*cosh(x)^2 + (15*(a^3 + a^2*b)*cosh(x)^4 - a^3 + 3*a^2*b -
6*(a^3 - 3*a^2*b)*cosh(x)^2)*sinh(x)^2 + 2*(3*(a^3 + a^2*b)*cosh(x)^5 - 2*(a^3 - 3*a^2*b)*cosh(x)^3 - (a^3 - 3
*a^2*b)*cosh(x))*sinh(x))*sqrt(-a - b)*arctan(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))/((a + b
)*cosh(x)^4 + 4*(a + b)*cosh(x)*sinh(x)^3 + (a + b)*sinh(x)^4 - 2*(a - b)*cosh(x)^2 + 2*(3*(a + b)*cosh(x)^2 -
a + b)*sinh(x)^2 + 4*((a + b)*cosh(x)^3 - (a - b)*cosh(x))*sinh(x) + a + b)) + 2*sqrt(2)*((a^3 + 3*a^2*b + 4*
a*b^2 + 2*b^3)*cosh(x)^4 + 4*(a^3 + 3*a^2*b + 4*a*b^2 + 2*b^3)*cosh(x)*sinh(x)^3 + (a^3 + 3*a^2*b + 4*a*b^2 +
2*b^3)*sinh(x)^4 + a^3 + 3*a^2*b + 4*a*b^2 + 2*b^3 - 2*(a^3 + a^2*b - 2*a*b^2 - 2*b^3)*cosh(x)^2 - 2*(a^3 + a^
2*b - 2*a*b^2 - 2*b^3 - 3*(a^3 + 3*a^2*b + 4*a*b^2 + 2*b^3)*cosh(x)^2)*sinh(x)^2 + 4*((a^3 + 3*a^2*b + 4*a*b^2
+ 2*b^3)*cosh(x)^3 - (a^3 + a^2*b - 2*a*b^2 - 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 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*cosh(x)^6
+ 6*(a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*cosh(x)*sinh(x)^5 + (a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*sinh(x)^6
+ a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3 - (a^5 - a^4*b - 5*a^3*b^2 - 3*a^2*b^3)*cosh(x)^4 - (a^5 - a^4*b - 5*a^3
*b^2 - 3*a^2*b^3 - 15*(a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*cosh(x)^2)*sinh(x)^4 + 4*(5*(a^5 + 3*a^4*b + 3*a^3
*b^2 + a^2*b^3)*cosh(x)^3 - (a^5 - a^4*b - 5*a^3*b^2 - 3*a^2*b^3)*cosh(x))*sinh(x)^3 - (a^5 - a^4*b - 5*a^3*b^
2 - 3*a^2*b^3)*cosh(x)^2 - (a^5 - a^4*b - 5*a^3*b^2 - 3*a^2*b^3 - 15*(a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*cos
h(x)^4 + 6*(a^5 - a^4*b - 5*a^3*b^2 - 3*a^2*b^3)*cosh(x)^2)*sinh(x)^2 + 2*(3*(a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*
b^3)*cosh(x)^5 - 2*(a^5 - a^4*b - 5*a^3*b^2 - 3*a^2*b^3)*cosh(x)^3 - (a^5 - a^4*b - 5*a^3*b^2 - 3*a^2*b^3)*cos
h(x))*sinh(x))]
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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(tanh(x)^2/(a+b*coth(x)^2)^(3/2),x, algorithm="giac")
[Out]
Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Warn
ing, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Check [ab
s(t_nostep-1)]Evaluation time: 1.22Error: Bad Argument Type
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maple [F] time = 0.48, size = 0, normalized size = 0.00 \[ \int \frac {\tanh ^{2}\relax (x )}{\left (a +b \left (\coth ^{2}\relax (x )\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(tanh(x)^2/(a+b*coth(x)^2)^(3/2),x)
[Out]
int(tanh(x)^2/(a+b*coth(x)^2)^(3/2),x)
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\tanh \relax (x)^{2}}{{\left (b \coth \relax (x)^{2} + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(tanh(x)^2/(a+b*coth(x)^2)^(3/2),x, algorithm="maxima")
[Out]
integrate(tanh(x)^2/(b*coth(x)^2 + a)^(3/2), x)
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {tanh}\relax (x)}^2}{{\left (b\,{\mathrm {coth}\relax (x)}^2+a\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(tanh(x)^2/(a + b*coth(x)^2)^(3/2),x)
[Out]
int(tanh(x)^2/(a + b*coth(x)^2)^(3/2), x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\tanh ^{2}{\relax (x )}}{\left (a + b \coth ^{2}{\relax (x )}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(tanh(x)**2/(a+b*coth(x)**2)**(3/2),x)
[Out]
Integral(tanh(x)**2/(a + b*coth(x)**2)**(3/2), x)
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