3.1.42 \(\int \frac {\tanh ^2(x)}{(a+b \coth ^2(x))^{3/2}} \, dx\) [42]
Optimal. Leaf size=85 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {a+b} \coth (x)}{\sqrt {a+b \coth ^2(x)}}\right )}{(a+b)^{3/2}}+\frac {b \tanh (x)}{a (a+b) \sqrt {a+b \coth ^2(x)}}-\frac {(a+2 b) \sqrt {a+b \coth ^2(x)} \tanh (x)}{a^2 (a+b)} \]
[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 = 0.11, 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 = {3751, 483, 597,
12, 385, 212} \begin {gather*} -\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)}} \end {gather*}
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 212
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
Rule 385
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 483
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 597
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 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 ^2(x)}{\left (a+b \coth ^2(x)\right )^{3/2}} \, dx &=\text {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 {\text {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 {\text {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 {\text {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 {\text {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 [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 5.00, size = 240, normalized size = 2.82 \begin {gather*} \frac {\sinh ^3(x) \left (8 a (a+b) \cosh (x) \left (2 a^2+5 a b \coth ^2(x)+3 b^2 \coth ^4(x)\right ) \, _2F_1\left (2,2;\frac {7}{2};\frac {(a+b) \cosh ^2(x)}{a}\right )+8 a (a+b) \cosh (x) \left (a+b \coth ^2(x)\right )^2 \, _3F_2\left (2,2,2;1,\frac {7}{2};\frac {(a+b) \cosh ^2(x)}{a}\right )+\frac {15 \left (a+b \coth ^2(x)\right ) \left (3 a^2+12 a b \coth ^2(x)+8 b^2 \coth ^4(x)\right ) \sinh ^3(x) \left (-\text {ArcSin}\left (\sqrt {\frac {(a+b) \cosh ^2(x)}{a}}\right ) \left (a+b \coth ^2(x)\right )-a \text {csch}^2(x) \sqrt {-\frac {(a+b) \cosh ^2(x) \left (a+b \coth ^2(x)\right ) \sinh ^2(x)}{a^2}}\right ) \tanh (x)}{\left (-\frac {(a+b) \cosh ^2(x) \left (a+b \coth ^2(x)\right ) \sinh ^2(x)}{a^2}\right )^{3/2}}\right )}{15 a^5 \sqrt {a+b \coth ^2(x)}} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
Integrate[Tanh[x]^2/(a + b*Coth[x]^2)^(3/2),x]
[Out]
(Sinh[x]^3*(8*a*(a + b)*Cosh[x]*(2*a^2 + 5*a*b*Coth[x]^2 + 3*b^2*Coth[x]^4)*Hypergeometric2F1[2, 2, 7/2, ((a +
b)*Cosh[x]^2)/a] + 8*a*(a + b)*Cosh[x]*(a + b*Coth[x]^2)^2*HypergeometricPFQ[{2, 2, 2}, {1, 7/2}, ((a + b)*Co
sh[x]^2)/a] + (15*(a + b*Coth[x]^2)*(3*a^2 + 12*a*b*Coth[x]^2 + 8*b^2*Coth[x]^4)*Sinh[x]^3*(-(ArcSin[Sqrt[((a
+ b)*Cosh[x]^2)/a]]*(a + b*Coth[x]^2)) - a*Csch[x]^2*Sqrt[-(((a + b)*Cosh[x]^2*(a + b*Coth[x]^2)*Sinh[x]^2)/a^
2)])*Tanh[x])/(-(((a + b)*Cosh[x]^2*(a + b*Coth[x]^2)*Sinh[x]^2)/a^2))^(3/2)))/(15*a^5*Sqrt[a + b*Coth[x]^2])
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Maple [F]
time = 2.18, size = 0, normalized size = 0.00 \[\int \frac {\tanh ^{2}\left (x \right )}{\left (a +b \left (\coth ^{2}\left (x \right )\right )\right )^{\frac {3}{2}}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1686 vs.
\(2 (75) = 150\).
time = 0.59, size = 3931, normalized size = 46.25 \begin {gather*} \text {Too large to display} \end {gather*}
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 - ...
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\tanh ^{2}{\left (x \right )}}{\left (a + b \coth ^{2}{\left (x \right )}\right )^{\frac {3}{2}}}\, dx \end {gather*}
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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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)^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,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 )}^2}{{\left (b\,{\mathrm {coth}\left (x\right )}^2+a\right )}^{3/2}} \,d x \end {gather*}
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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