3.210 \(\int \frac{\coth ^2(x)}{(a+b \text{sech}^2(x))^{3/2}} \, dx\)
Optimal. Leaf size=88 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{a} \tanh (x)}{\sqrt{a-b \tanh ^2(x)+b}}\right )}{a^{3/2}}-\frac{(a-b) \coth (x) \sqrt{a-b \tanh ^2(x)+b}}{a (a+b)^2}-\frac{b \coth (x)}{a (a+b) \sqrt{a-b \tanh ^2(x)+b}} \]
[Out]
ArcTanh[(Sqrt[a]*Tanh[x])/Sqrt[a + b - b*Tanh[x]^2]]/a^(3/2) - (b*Coth[x])/(a*(a + b)*Sqrt[a + b - b*Tanh[x]^2
]) - ((a - b)*Coth[x]*Sqrt[a + b - b*Tanh[x]^2])/(a*(a + b)^2)
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Rubi [A] time = 0.274126, antiderivative size = 88, normalized size of antiderivative = 1.,
number of steps used = 7, number of rules used = 7, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.412, Rules used =
{4141, 1975, 472, 583, 12, 377, 206} \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{a} \tanh (x)}{\sqrt{a-b \tanh ^2(x)+b}}\right )}{a^{3/2}}-\frac{(a-b) \coth (x) \sqrt{a-b \tanh ^2(x)+b}}{a (a+b)^2}-\frac{b \coth (x)}{a (a+b) \sqrt{a-b \tanh ^2(x)+b}} \]
Antiderivative was successfully verified.
[In]
Int[Coth[x]^2/(a + b*Sech[x]^2)^(3/2),x]
[Out]
ArcTanh[(Sqrt[a]*Tanh[x])/Sqrt[a + b - b*Tanh[x]^2]]/a^(3/2) - (b*Coth[x])/(a*(a + b)*Sqrt[a + b - b*Tanh[x]^2
]) - ((a - b)*Coth[x]*Sqrt[a + b - b*Tanh[x]^2])/(a*(a + b)^2)
Rule 4141
Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_.)*((d_.)*tan[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> With[
{ff = FreeFactors[Tan[e + f*x], x]}, Dist[ff/f, Subst[Int[((d*ff*x)^m*(a + b*(1 + ff^2*x^2)^(n/2))^p)/(1 + ff^
2*x^2), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, d, e, f, m, p}, x] && IntegerQ[n/2] && (IntegerQ[m/2] ||
EqQ[n, 2])
Rule 1975
Int[(u_)^(p_.)*(v_)^(q_.)*((e_.)*(x_))^(m_.), x_Symbol] :> Int[(e*x)^m*ExpandToSum[u, x]^p*ExpandToSum[v, x]^q
, x] /; FreeQ[{e, m, p, q}, x] && BinomialQ[{u, v}, x] && EqQ[BinomialDegree[u, x] - BinomialDegree[v, x], 0]
&& !BinomialMatchQ[{u, v}, x]
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 12
Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]
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 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])
Rubi steps
\begin{align*} \int \frac{\coth ^2(x)}{\left (a+b \text{sech}^2(x)\right )^{3/2}} \, dx &=\operatorname{Subst}\left (\int \frac{1}{x^2 \left (1-x^2\right ) \left (a+b \left (1-x^2\right )\right )^{3/2}} \, dx,x,\tanh (x)\right )\\ &=\operatorname{Subst}\left (\int \frac{1}{x^2 \left (1-x^2\right ) \left (a+b-b x^2\right )^{3/2}} \, dx,x,\tanh (x)\right )\\ &=-\frac{b \coth (x)}{a (a+b) \sqrt{a+b-b \tanh ^2(x)}}-\frac{\operatorname{Subst}\left (\int \frac{-a+b-2 b x^2}{x^2 \left (1-x^2\right ) \sqrt{a+b-b x^2}} \, dx,x,\tanh (x)\right )}{a (a+b)}\\ &=-\frac{b \coth (x)}{a (a+b) \sqrt{a+b-b \tanh ^2(x)}}-\frac{(a-b) \coth (x) \sqrt{a+b-b \tanh ^2(x)}}{a (a+b)^2}+\frac{\operatorname{Subst}\left (\int \frac{(a+b)^2}{\left (1-x^2\right ) \sqrt{a+b-b x^2}} \, dx,x,\tanh (x)\right )}{a (a+b)^2}\\ &=-\frac{b \coth (x)}{a (a+b) \sqrt{a+b-b \tanh ^2(x)}}-\frac{(a-b) \coth (x) \sqrt{a+b-b \tanh ^2(x)}}{a (a+b)^2}+\frac{\operatorname{Subst}\left (\int \frac{1}{\left (1-x^2\right ) \sqrt{a+b-b x^2}} \, dx,x,\tanh (x)\right )}{a}\\ &=-\frac{b \coth (x)}{a (a+b) \sqrt{a+b-b \tanh ^2(x)}}-\frac{(a-b) \coth (x) \sqrt{a+b-b \tanh ^2(x)}}{a (a+b)^2}+\frac{\operatorname{Subst}\left (\int \frac{1}{1-a x^2} \, dx,x,\frac{\tanh (x)}{\sqrt{a+b-b \tanh ^2(x)}}\right )}{a}\\ &=\frac{\tanh ^{-1}\left (\frac{\sqrt{a} \tanh (x)}{\sqrt{a+b-b \tanh ^2(x)}}\right )}{a^{3/2}}-\frac{b \coth (x)}{a (a+b) \sqrt{a+b-b \tanh ^2(x)}}-\frac{(a-b) \coth (x) \sqrt{a+b-b \tanh ^2(x)}}{a (a+b)^2}\\ \end{align*}
Mathematica [A] time = 0.420422, size = 120, normalized size = 1.36 \[ \frac{\text{sech}^3(x) \left (\frac{\sqrt{2} (a \cosh (2 x)+a+2 b)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \sinh (x)}{\sqrt{a \cosh (2 x)+a+2 b}}\right )}{a^{3/2}}-\frac{(a \cosh (2 x)+a+2 b) \left (a \text{csch}(x) (a \cosh (2 x)+a+2 b)+2 b^2 \sinh (x)\right )}{a (a+b)^2}\right )}{4 \left (a+b \text{sech}^2(x)\right )^{3/2}} \]
Antiderivative was successfully verified.
[In]
Integrate[Coth[x]^2/(a + b*Sech[x]^2)^(3/2),x]
[Out]
(Sech[x]^3*((Sqrt[2]*ArcTanh[(Sqrt[2]*Sqrt[a]*Sinh[x])/Sqrt[a + 2*b + a*Cosh[2*x]]]*(a + 2*b + a*Cosh[2*x])^(3
/2))/a^(3/2) - ((a + 2*b + a*Cosh[2*x])*(a*(a + 2*b + a*Cosh[2*x])*Csch[x] + 2*b^2*Sinh[x]))/(a*(a + b)^2)))/(
4*(a + b*Sech[x]^2)^(3/2))
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Maple [F] time = 0.103, size = 0, normalized size = 0. \begin{align*} \int{ \left ({\rm coth} \left (x\right ) \right ) ^{2} \left ( a+b \left ({\rm sech} \left (x\right ) \right ) ^{2} \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(coth(x)^2/(a+b*sech(x)^2)^(3/2),x)
[Out]
int(coth(x)^2/(a+b*sech(x)^2)^(3/2),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\coth \left (x\right )^{2}}{{\left (b \operatorname{sech}\left (x\right )^{2} + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(coth(x)^2/(a+b*sech(x)^2)^(3/2),x, algorithm="maxima")
[Out]
integrate(coth(x)^2/(b*sech(x)^2 + a)^(3/2), x)
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Fricas [B] time = 4.42849, size = 10016, normalized size = 113.82 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(coth(x)^2/(a+b*sech(x)^2)^(3/2),x, algorithm="fricas")
[Out]
[1/4*(((a^3 + 2*a^2*b + a*b^2)*cosh(x)^6 + 6*(a^3 + 2*a^2*b + a*b^2)*cosh(x)*sinh(x)^5 + (a^3 + 2*a^2*b + a*b^
2)*sinh(x)^6 + (a^3 + 6*a^2*b + 9*a*b^2 + 4*b^3)*cosh(x)^4 + (a^3 + 6*a^2*b + 9*a*b^2 + 4*b^3 + 15*(a^3 + 2*a^
2*b + a*b^2)*cosh(x)^2)*sinh(x)^4 + 4*(5*(a^3 + 2*a^2*b + a*b^2)*cosh(x)^3 + (a^3 + 6*a^2*b + 9*a*b^2 + 4*b^3)
*cosh(x))*sinh(x)^3 - a^3 - 2*a^2*b - a*b^2 - (a^3 + 6*a^2*b + 9*a*b^2 + 4*b^3)*cosh(x)^2 + (15*(a^3 + 2*a^2*b
+ a*b^2)*cosh(x)^4 - a^3 - 6*a^2*b - 9*a*b^2 - 4*b^3 + 6*(a^3 + 6*a^2*b + 9*a*b^2 + 4*b^3)*cosh(x)^2)*sinh(x)
^2 + 2*(3*(a^3 + 2*a^2*b + a*b^2)*cosh(x)^5 + 2*(a^3 + 6*a^2*b + 9*a*b^2 + 4*b^3)*cosh(x)^3 - (a^3 + 6*a^2*b +
9*a*b^2 + 4*b^3)*cosh(x))*sinh(x))*sqrt(a)*log((a*b^2*cosh(x)^8 + 8*a*b^2*cosh(x)*sinh(x)^7 + a*b^2*sinh(x)^8
- 2*(a*b^2 - b^3)*cosh(x)^6 + 2*(14*a*b^2*cosh(x)^2 - a*b^2 + b^3)*sinh(x)^6 + 4*(14*a*b^2*cosh(x)^3 - 3*(a*b
^2 - b^3)*cosh(x))*sinh(x)^5 + (a^3 + 4*a^2*b + 9*a*b^2)*cosh(x)^4 + (70*a*b^2*cosh(x)^4 + a^3 + 4*a^2*b + 9*a
*b^2 - 30*(a*b^2 - b^3)*cosh(x)^2)*sinh(x)^4 + 4*(14*a*b^2*cosh(x)^5 - 10*(a*b^2 - b^3)*cosh(x)^3 + (a^3 + 4*a
^2*b + 9*a*b^2)*cosh(x))*sinh(x)^3 + a^3 + 2*(a^3 + 3*a^2*b)*cosh(x)^2 + 2*(14*a*b^2*cosh(x)^6 - 15*(a*b^2 - b
^3)*cosh(x)^4 + a^3 + 3*a^2*b + 3*(a^3 + 4*a^2*b + 9*a*b^2)*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 + 4*a*b)*cosh(x)^2 + (15*b^2*cosh(x)^4 - 18*b^2*cosh(x)^2 - a^2 - 4*a*b
)*sinh(x)^2 - a^2 + 2*(3*b^2*cosh(x)^5 - 6*b^2*cosh(x)^3 - (a^2 + 4*a*b)*cosh(x))*sinh(x))*sqrt(a)*sqrt((a*cos
h(x)^2 + a*sinh(x)^2 + a + 2*b)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2)) + 4*(2*a*b^2*cosh(x)^7 - 3*(a*b^2
- b^3)*cosh(x)^5 + (a^3 + 4*a^2*b + 9*a*b^2)*cosh(x)^3 + (a^3 + 3*a^2*b)*cosh(x))*sinh(x))/(cosh(x)^6 + 6*cos
h(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 + 2*a^2*b + a*b^2)*cosh(x)^6 + 6*(a^3 + 2*a^2*b + a*b^2)*cosh(x)*sinh(x)^5 + (a^3 + 2*
a^2*b + a*b^2)*sinh(x)^6 + (a^3 + 6*a^2*b + 9*a*b^2 + 4*b^3)*cosh(x)^4 + (a^3 + 6*a^2*b + 9*a*b^2 + 4*b^3 + 15
*(a^3 + 2*a^2*b + a*b^2)*cosh(x)^2)*sinh(x)^4 + 4*(5*(a^3 + 2*a^2*b + a*b^2)*cosh(x)^3 + (a^3 + 6*a^2*b + 9*a*
b^2 + 4*b^3)*cosh(x))*sinh(x)^3 - a^3 - 2*a^2*b - a*b^2 - (a^3 + 6*a^2*b + 9*a*b^2 + 4*b^3)*cosh(x)^2 + (15*(a
^3 + 2*a^2*b + a*b^2)*cosh(x)^4 - a^3 - 6*a^2*b - 9*a*b^2 - 4*b^3 + 6*(a^3 + 6*a^2*b + 9*a*b^2 + 4*b^3)*cosh(x
)^2)*sinh(x)^2 + 2*(3*(a^3 + 2*a^2*b + a*b^2)*cosh(x)^5 + 2*(a^3 + 6*a^2*b + 9*a*b^2 + 4*b^3)*cosh(x)^3 - (a^3
+ 6*a^2*b + 9*a*b^2 + 4*b^3)*cosh(x))*sinh(x))*sqrt(a)*log(-(a*cosh(x)^4 + 4*a*cosh(x)*sinh(x)^3 + a*sinh(x)^
4 + 2*(a + b)*cosh(x)^2 + 2*(3*a*cosh(x)^2 + a + b)*sinh(x)^2 + sqrt(2)*(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(
x)^2 + 1)*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*
(a*cosh(x)^3 + (a + b)*cosh(x))*sinh(x) + a)/(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2)) - 4*sqrt(2)*((a^3 +
a*b^2)*cosh(x)^4 + 4*(a^3 + a*b^2)*cosh(x)*sinh(x)^3 + (a^3 + a*b^2)*sinh(x)^4 + a^3 + a*b^2 + 2*(a^3 + 2*a^2*
b - a*b^2)*cosh(x)^2 + 2*(a^3 + 2*a^2*b - a*b^2 + 3*(a^3 + a*b^2)*cosh(x)^2)*sinh(x)^2 + 4*((a^3 + a*b^2)*cosh
(x)^3 + (a^3 + 2*a^2*b - a*b^2)*cosh(x))*sinh(x))*sqrt((a*cosh(x)^2 + a*sinh(x)^2 + a + 2*b)/(cosh(x)^2 - 2*co
sh(x)*sinh(x) + sinh(x)^2)))/((a^5 + 2*a^4*b + a^3*b^2)*cosh(x)^6 + 6*(a^5 + 2*a^4*b + a^3*b^2)*cosh(x)*sinh(x
)^5 + (a^5 + 2*a^4*b + a^3*b^2)*sinh(x)^6 - a^5 - 2*a^4*b - a^3*b^2 + (a^5 + 6*a^4*b + 9*a^3*b^2 + 4*a^2*b^3)*
cosh(x)^4 + (a^5 + 6*a^4*b + 9*a^3*b^2 + 4*a^2*b^3 + 15*(a^5 + 2*a^4*b + a^3*b^2)*cosh(x)^2)*sinh(x)^4 + 4*(5*
(a^5 + 2*a^4*b + a^3*b^2)*cosh(x)^3 + (a^5 + 6*a^4*b + 9*a^3*b^2 + 4*a^2*b^3)*cosh(x))*sinh(x)^3 - (a^5 + 6*a^
4*b + 9*a^3*b^2 + 4*a^2*b^3)*cosh(x)^2 - (a^5 + 6*a^4*b + 9*a^3*b^2 + 4*a^2*b^3 - 15*(a^5 + 2*a^4*b + a^3*b^2)
*cosh(x)^4 - 6*(a^5 + 6*a^4*b + 9*a^3*b^2 + 4*a^2*b^3)*cosh(x)^2)*sinh(x)^2 + 2*(3*(a^5 + 2*a^4*b + a^3*b^2)*c
osh(x)^5 + 2*(a^5 + 6*a^4*b + 9*a^3*b^2 + 4*a^2*b^3)*cosh(x)^3 - (a^5 + 6*a^4*b + 9*a^3*b^2 + 4*a^2*b^3)*cosh(
x))*sinh(x)), -1/2*(((a^3 + 2*a^2*b + a*b^2)*cosh(x)^6 + 6*(a^3 + 2*a^2*b + a*b^2)*cosh(x)*sinh(x)^5 + (a^3 +
2*a^2*b + a*b^2)*sinh(x)^6 + (a^3 + 6*a^2*b + 9*a*b^2 + 4*b^3)*cosh(x)^4 + (a^3 + 6*a^2*b + 9*a*b^2 + 4*b^3 +
15*(a^3 + 2*a^2*b + a*b^2)*cosh(x)^2)*sinh(x)^4 + 4*(5*(a^3 + 2*a^2*b + a*b^2)*cosh(x)^3 + (a^3 + 6*a^2*b + 9*
a*b^2 + 4*b^3)*cosh(x))*sinh(x)^3 - a^3 - 2*a^2*b - a*b^2 - (a^3 + 6*a^2*b + 9*a*b^2 + 4*b^3)*cosh(x)^2 + (15*
(a^3 + 2*a^2*b + a*b^2)*cosh(x)^4 - a^3 - 6*a^2*b - 9*a*b^2 - 4*b^3 + 6*(a^3 + 6*a^2*b + 9*a*b^2 + 4*b^3)*cosh
(x)^2)*sinh(x)^2 + 2*(3*(a^3 + 2*a^2*b + a*b^2)*cosh(x)^5 + 2*(a^3 + 6*a^2*b + 9*a*b^2 + 4*b^3)*cosh(x)^3 - (a
^3 + 6*a^2*b + 9*a*b^2 + 4*b^3)*cosh(x))*sinh(x))*sqrt(-a)*arctan(sqrt(2)*(b*cosh(x)^2 + 2*b*cosh(x)*sinh(x) +
b*sinh(x)^2 + a)*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))/(a*b*cosh(x)^4 + 4*a*b*cosh(x)*sinh(x)^3 + a*b*sinh(x)^4 - (a^2 + 3*a*b)*cosh(x)^2 + (6*a*b*cosh(x)^2 - a
^2 - 3*a*b)*sinh(x)^2 - a^2 + 2*(2*a*b*cosh(x)^3 - (a^2 + 3*a*b)*cosh(x))*sinh(x))) + ((a^3 + 2*a^2*b + a*b^2)
*cosh(x)^6 + 6*(a^3 + 2*a^2*b + a*b^2)*cosh(x)*sinh(x)^5 + (a^3 + 2*a^2*b + a*b^2)*sinh(x)^6 + (a^3 + 6*a^2*b
+ 9*a*b^2 + 4*b^3)*cosh(x)^4 + (a^3 + 6*a^2*b + 9*a*b^2 + 4*b^3 + 15*(a^3 + 2*a^2*b + a*b^2)*cosh(x)^2)*sinh(x
)^4 + 4*(5*(a^3 + 2*a^2*b + a*b^2)*cosh(x)^3 + (a^3 + 6*a^2*b + 9*a*b^2 + 4*b^3)*cosh(x))*sinh(x)^3 - a^3 - 2*
a^2*b - a*b^2 - (a^3 + 6*a^2*b + 9*a*b^2 + 4*b^3)*cosh(x)^2 + (15*(a^3 + 2*a^2*b + a*b^2)*cosh(x)^4 - a^3 - 6*
a^2*b - 9*a*b^2 - 4*b^3 + 6*(a^3 + 6*a^2*b + 9*a*b^2 + 4*b^3)*cosh(x)^2)*sinh(x)^2 + 2*(3*(a^3 + 2*a^2*b + a*b
^2)*cosh(x)^5 + 2*(a^3 + 6*a^2*b + 9*a*b^2 + 4*b^3)*cosh(x)^3 - (a^3 + 6*a^2*b + 9*a*b^2 + 4*b^3)*cosh(x))*sin
h(x))*sqrt(-a)*arctan(sqrt(2)*(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 + 1)*sqrt(-a)*sqrt((a*cosh(x)^2 + a*s
inh(x)^2 + a + 2*b)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2))/(a*cosh(x)^4 + 4*a*cosh(x)*sinh(x)^3 + a*sinh
(x)^4 + 2*(a + 2*b)*cosh(x)^2 + 2*(3*a*cosh(x)^2 + a + 2*b)*sinh(x)^2 + 4*(a*cosh(x)^3 + (a + 2*b)*cosh(x))*si
nh(x) + a)) + 2*sqrt(2)*((a^3 + a*b^2)*cosh(x)^4 + 4*(a^3 + a*b^2)*cosh(x)*sinh(x)^3 + (a^3 + a*b^2)*sinh(x)^4
+ a^3 + a*b^2 + 2*(a^3 + 2*a^2*b - a*b^2)*cosh(x)^2 + 2*(a^3 + 2*a^2*b - a*b^2 + 3*(a^3 + a*b^2)*cosh(x)^2)*s
inh(x)^2 + 4*((a^3 + a*b^2)*cosh(x)^3 + (a^3 + 2*a^2*b - a*b^2)*cosh(x))*sinh(x))*sqrt((a*cosh(x)^2 + a*sinh(x
)^2 + a + 2*b)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2)))/((a^5 + 2*a^4*b + a^3*b^2)*cosh(x)^6 + 6*(a^5 + 2
*a^4*b + a^3*b^2)*cosh(x)*sinh(x)^5 + (a^5 + 2*a^4*b + a^3*b^2)*sinh(x)^6 - a^5 - 2*a^4*b - a^3*b^2 + (a^5 + 6
*a^4*b + 9*a^3*b^2 + 4*a^2*b^3)*cosh(x)^4 + (a^5 + 6*a^4*b + 9*a^3*b^2 + 4*a^2*b^3 + 15*(a^5 + 2*a^4*b + a^3*b
^2)*cosh(x)^2)*sinh(x)^4 + 4*(5*(a^5 + 2*a^4*b + a^3*b^2)*cosh(x)^3 + (a^5 + 6*a^4*b + 9*a^3*b^2 + 4*a^2*b^3)*
cosh(x))*sinh(x)^3 - (a^5 + 6*a^4*b + 9*a^3*b^2 + 4*a^2*b^3)*cosh(x)^2 - (a^5 + 6*a^4*b + 9*a^3*b^2 + 4*a^2*b^
3 - 15*(a^5 + 2*a^4*b + a^3*b^2)*cosh(x)^4 - 6*(a^5 + 6*a^4*b + 9*a^3*b^2 + 4*a^2*b^3)*cosh(x)^2)*sinh(x)^2 +
2*(3*(a^5 + 2*a^4*b + a^3*b^2)*cosh(x)^5 + 2*(a^5 + 6*a^4*b + 9*a^3*b^2 + 4*a^2*b^3)*cosh(x)^3 - (a^5 + 6*a^4*
b + 9*a^3*b^2 + 4*a^2*b^3)*cosh(x))*sinh(x))]
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\coth ^{2}{\left (x \right )}}{\left (a + b \operatorname{sech}^{2}{\left (x \right )}\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(coth(x)**2/(a+b*sech(x)**2)**(3/2),x)
[Out]
Integral(coth(x)**2/(a + b*sech(x)**2)**(3/2), x)
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\coth \left (x\right )^{2}}{{\left (b \operatorname{sech}\left (x\right )^{2} + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(coth(x)^2/(a+b*sech(x)^2)^(3/2),x, algorithm="giac")
[Out]
integrate(coth(x)^2/(b*sech(x)^2 + a)^(3/2), x)