3.187 \(\int \frac{\coth ^2(c+d x)}{(a+b \tanh ^2(c+d x))^2} \, dx\)

Optimal. Leaf size=119 \[ -\frac{b^{3/2} (5 a+3 b) \tan ^{-1}\left (\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right )}{2 a^{5/2} d (a+b)^2}-\frac{(2 a+3 b) \coth (c+d x)}{2 a^2 d (a+b)}+\frac{b \coth (c+d x)}{2 a d (a+b) \left (a+b \tanh ^2(c+d x)\right )}+\frac{x}{(a+b)^2} \]

[Out]

x/(a + b)^2 - (b^(3/2)*(5*a + 3*b)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/(2*a^(5/2)*(a + b)^2*d) - ((2*a +
3*b)*Coth[c + d*x])/(2*a^2*(a + b)*d) + (b*Coth[c + d*x])/(2*a*(a + b)*d*(a + b*Tanh[c + d*x]^2))

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Rubi [A]  time = 0.194075, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {3670, 472, 583, 522, 206, 205} \[ -\frac{b^{3/2} (5 a+3 b) \tan ^{-1}\left (\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right )}{2 a^{5/2} d (a+b)^2}-\frac{(2 a+3 b) \coth (c+d x)}{2 a^2 d (a+b)}+\frac{b \coth (c+d x)}{2 a d (a+b) \left (a+b \tanh ^2(c+d x)\right )}+\frac{x}{(a+b)^2} \]

Antiderivative was successfully verified.

[In]

Int[Coth[c + d*x]^2/(a + b*Tanh[c + d*x]^2)^2,x]

[Out]

x/(a + b)^2 - (b^(3/2)*(5*a + 3*b)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/(2*a^(5/2)*(a + b)^2*d) - ((2*a +
3*b)*Coth[c + d*x])/(2*a^2*(a + b)*d) + (b*Coth[c + d*x])/(2*a*(a + b)*d*(a + b*Tanh[c + d*x]^2))

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]))

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 522

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Dist[(b*e - a*f
)/(b*c - a*d), Int[1/(a + b*x^n), x], x] - Dist[(d*e - c*f)/(b*c - a*d), Int[1/(c + d*x^n), x], x] /; FreeQ[{a
, b, c, d, e, f, n}, 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 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

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

Mathematica [A]  time = 1.55879, size = 111, normalized size = 0.93 \[ -\frac{\frac{b^{3/2} (5 a+3 b) \tan ^{-1}\left (\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right )}{a^{5/2} (a+b)^2}+\frac{b^2 \sinh (2 (c+d x))}{a^2 (a+b) ((a+b) \cosh (2 (c+d x))+a-b)}+\frac{2 \coth (c+d x)}{a^2}-\frac{2 (c+d x)}{(a+b)^2}}{2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((-2*(c + d*x))/(a + b)^2 + (b^(3/2)*(5*a + 3*b)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/(a^(5/2)*(a + b)^2)
+ (2*Coth[c + d*x])/a^2 + (b^2*Sinh[2*(c + d*x)])/(a^2*(a + b)*(a - b + (a + b)*Cosh[2*(c + d*x)])))/(2*d)

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Maple [B]  time = 0.108, size = 1061, normalized size = 8.9 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/d/a^2*tanh(1/2*d*x+1/2*c)+1/d/(a+b)^2*ln(tanh(1/2*d*x+1/2*c)+1)-1/d*b^2/(a+b)^2/(tanh(1/2*d*x+1/2*c)^4*a+
2*tanh(1/2*d*x+1/2*c)^2*a+4*tanh(1/2*d*x+1/2*c)^2*b+a)/a*tanh(1/2*d*x+1/2*c)^3-1/d*b^3/(a+b)^2/a^2/(tanh(1/2*d
*x+1/2*c)^4*a+2*tanh(1/2*d*x+1/2*c)^2*a+4*tanh(1/2*d*x+1/2*c)^2*b+a)*tanh(1/2*d*x+1/2*c)^3-1/d*b^2/(a+b)^2/(ta
nh(1/2*d*x+1/2*c)^4*a+2*tanh(1/2*d*x+1/2*c)^2*a+4*tanh(1/2*d*x+1/2*c)^2*b+a)/a*tanh(1/2*d*x+1/2*c)-1/d*b^3/(a+
b)^2/a^2/(tanh(1/2*d*x+1/2*c)^4*a+2*tanh(1/2*d*x+1/2*c)^2*a+4*tanh(1/2*d*x+1/2*c)^2*b+a)*tanh(1/2*d*x+1/2*c)+5
/2/d*b^2/(a+b)^2/(b*(a+b))^(1/2)/((2*(b*(a+b))^(1/2)-a-2*b)*a)^(1/2)*arctanh(a*tanh(1/2*d*x+1/2*c)/((2*(b*(a+b
))^(1/2)-a-2*b)*a)^(1/2))-5/2/d*b^2/(a+b)^2/a/((2*(b*(a+b))^(1/2)-a-2*b)*a)^(1/2)*arctanh(a*tanh(1/2*d*x+1/2*c
)/((2*(b*(a+b))^(1/2)-a-2*b)*a)^(1/2))+4/d*b^3/(a+b)^2/a/(b*(a+b))^(1/2)/((2*(b*(a+b))^(1/2)-a-2*b)*a)^(1/2)*a
rctanh(a*tanh(1/2*d*x+1/2*c)/((2*(b*(a+b))^(1/2)-a-2*b)*a)^(1/2))+5/2/d*b^2/(a+b)^2/(b*(a+b))^(1/2)/((2*(b*(a+
b))^(1/2)+a+2*b)*a)^(1/2)*arctan(a*tanh(1/2*d*x+1/2*c)/((2*(b*(a+b))^(1/2)+a+2*b)*a)^(1/2))+5/2/d*b^2/(a+b)^2/
a/((2*(b*(a+b))^(1/2)+a+2*b)*a)^(1/2)*arctan(a*tanh(1/2*d*x+1/2*c)/((2*(b*(a+b))^(1/2)+a+2*b)*a)^(1/2))+4/d*b^
3/(a+b)^2/a/(b*(a+b))^(1/2)/((2*(b*(a+b))^(1/2)+a+2*b)*a)^(1/2)*arctan(a*tanh(1/2*d*x+1/2*c)/((2*(b*(a+b))^(1/
2)+a+2*b)*a)^(1/2))-3/2/d*b^3/(a+b)^2/a^2/((2*(b*(a+b))^(1/2)-a-2*b)*a)^(1/2)*arctanh(a*tanh(1/2*d*x+1/2*c)/((
2*(b*(a+b))^(1/2)-a-2*b)*a)^(1/2))+3/2/d*b^4/(a+b)^2/a^2/(b*(a+b))^(1/2)/((2*(b*(a+b))^(1/2)-a-2*b)*a)^(1/2)*a
rctanh(a*tanh(1/2*d*x+1/2*c)/((2*(b*(a+b))^(1/2)-a-2*b)*a)^(1/2))+3/2/d*b^3/(a+b)^2/a^2/((2*(b*(a+b))^(1/2)+a+
2*b)*a)^(1/2)*arctan(a*tanh(1/2*d*x+1/2*c)/((2*(b*(a+b))^(1/2)+a+2*b)*a)^(1/2))+3/2/d*b^4/(a+b)^2/a^2/(b*(a+b)
)^(1/2)/((2*(b*(a+b))^(1/2)+a+2*b)*a)^(1/2)*arctan(a*tanh(1/2*d*x+1/2*c)/((2*(b*(a+b))^(1/2)+a+2*b)*a)^(1/2))-
1/2/d/a^2/tanh(1/2*d*x+1/2*c)-1/d/(a+b)^2*ln(tanh(1/2*d*x+1/2*c)-1)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(d*x+c)^2/(a+b*tanh(d*x+c)^2)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.73135, size = 8676, normalized size = 72.91 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*(4*(a^3 + a^2*b)*d*x*cosh(d*x + c)^6 + 24*(a^3 + a^2*b)*d*x*cosh(d*x + c)*sinh(d*x + c)^5 + 4*(a^3 + a^2*
b)*d*x*sinh(d*x + c)^6 - 4*(2*a^3 + 6*a^2*b + 5*a*b^2 + 3*b^3 - (a^3 - 3*a^2*b)*d*x)*cosh(d*x + c)^4 + 4*(15*(
a^3 + a^2*b)*d*x*cosh(d*x + c)^2 - 2*a^3 - 6*a^2*b - 5*a*b^2 - 3*b^3 + (a^3 - 3*a^2*b)*d*x)*sinh(d*x + c)^4 +
16*(5*(a^3 + a^2*b)*d*x*cosh(d*x + c)^3 - (2*a^3 + 6*a^2*b + 5*a*b^2 + 3*b^3 - (a^3 - 3*a^2*b)*d*x)*cosh(d*x +
c))*sinh(d*x + c)^3 - 8*a^3 - 24*a^2*b - 28*a*b^2 - 12*b^3 - 4*(a^3 + a^2*b)*d*x - 4*(4*a^3 + 4*a^2*b - 4*a*b
^2 - 6*b^3 + (a^3 - 3*a^2*b)*d*x)*cosh(d*x + c)^2 + 4*(15*(a^3 + a^2*b)*d*x*cosh(d*x + c)^4 - 4*a^3 - 4*a^2*b
+ 4*a*b^2 + 6*b^3 - (a^3 - 3*a^2*b)*d*x - 6*(2*a^3 + 6*a^2*b + 5*a*b^2 + 3*b^3 - (a^3 - 3*a^2*b)*d*x)*cosh(d*x
+ c)^2)*sinh(d*x + c)^2 + ((5*a^2*b + 8*a*b^2 + 3*b^3)*cosh(d*x + c)^6 + 6*(5*a^2*b + 8*a*b^2 + 3*b^3)*cosh(d
*x + c)*sinh(d*x + c)^5 + (5*a^2*b + 8*a*b^2 + 3*b^3)*sinh(d*x + c)^6 + (5*a^2*b - 12*a*b^2 - 9*b^3)*cosh(d*x
+ c)^4 + (5*a^2*b - 12*a*b^2 - 9*b^3 + 15*(5*a^2*b + 8*a*b^2 + 3*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 4*(5*
(5*a^2*b + 8*a*b^2 + 3*b^3)*cosh(d*x + c)^3 + (5*a^2*b - 12*a*b^2 - 9*b^3)*cosh(d*x + c))*sinh(d*x + c)^3 - 5*
a^2*b - 8*a*b^2 - 3*b^3 - (5*a^2*b - 12*a*b^2 - 9*b^3)*cosh(d*x + c)^2 + (15*(5*a^2*b + 8*a*b^2 + 3*b^3)*cosh(
d*x + c)^4 - 5*a^2*b + 12*a*b^2 + 9*b^3 + 6*(5*a^2*b - 12*a*b^2 - 9*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 2*
(3*(5*a^2*b + 8*a*b^2 + 3*b^3)*cosh(d*x + c)^5 + 2*(5*a^2*b - 12*a*b^2 - 9*b^3)*cosh(d*x + c)^3 - (5*a^2*b - 1
2*a*b^2 - 9*b^3)*cosh(d*x + c))*sinh(d*x + c))*sqrt(-b/a)*log(((a^2 + 2*a*b + b^2)*cosh(d*x + c)^4 + 4*(a^2 +
2*a*b + b^2)*cosh(d*x + c)*sinh(d*x + c)^3 + (a^2 + 2*a*b + b^2)*sinh(d*x + c)^4 + 2*(a^2 - b^2)*cosh(d*x + c)
^2 + 2*(3*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^2 + a^2 - b^2)*sinh(d*x + c)^2 + a^2 - 6*a*b + b^2 + 4*((a^2 + 2*a
*b + b^2)*cosh(d*x + c)^3 + (a^2 - b^2)*cosh(d*x + c))*sinh(d*x + c) - 4*((a^2 + a*b)*cosh(d*x + c)^2 + 2*(a^2
+ a*b)*cosh(d*x + c)*sinh(d*x + c) + (a^2 + a*b)*sinh(d*x + c)^2 + a^2 - a*b)*sqrt(-b/a))/((a + b)*cosh(d*x +
c)^4 + 4*(a + b)*cosh(d*x + c)*sinh(d*x + c)^3 + (a + b)*sinh(d*x + c)^4 + 2*(a - b)*cosh(d*x + c)^2 + 2*(3*(
a + b)*cosh(d*x + c)^2 + a - b)*sinh(d*x + c)^2 + 4*((a + b)*cosh(d*x + c)^3 + (a - b)*cosh(d*x + c))*sinh(d*x
+ c) + a + b)) + 8*(3*(a^3 + a^2*b)*d*x*cosh(d*x + c)^5 - 2*(2*a^3 + 6*a^2*b + 5*a*b^2 + 3*b^3 - (a^3 - 3*a^2
*b)*d*x)*cosh(d*x + c)^3 - (4*a^3 + 4*a^2*b - 4*a*b^2 - 6*b^3 + (a^3 - 3*a^2*b)*d*x)*cosh(d*x + c))*sinh(d*x +
c))/((a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*d*cosh(d*x + c)^6 + 6*(a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*d*cosh
(d*x + c)*sinh(d*x + c)^5 + (a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*d*sinh(d*x + c)^6 + (a^5 - a^4*b - 5*a^3*b^2
- 3*a^2*b^3)*d*cosh(d*x + c)^4 + (15*(a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*d*cosh(d*x + c)^2 + (a^5 - a^4*b -
5*a^3*b^2 - 3*a^2*b^3)*d)*sinh(d*x + c)^4 - (a^5 - a^4*b - 5*a^3*b^2 - 3*a^2*b^3)*d*cosh(d*x + c)^2 + 4*(5*(a
^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*d*cosh(d*x + c)^3 + (a^5 - a^4*b - 5*a^3*b^2 - 3*a^2*b^3)*d*cosh(d*x + c))
*sinh(d*x + c)^3 + (15*(a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*d*cosh(d*x + c)^4 + 6*(a^5 - a^4*b - 5*a^3*b^2 -
3*a^2*b^3)*d*cosh(d*x + c)^2 - (a^5 - a^4*b - 5*a^3*b^2 - 3*a^2*b^3)*d)*sinh(d*x + c)^2 - (a^5 + 3*a^4*b + 3*a
^3*b^2 + a^2*b^3)*d + 2*(3*(a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*d*cosh(d*x + c)^5 + 2*(a^5 - a^4*b - 5*a^3*b^
2 - 3*a^2*b^3)*d*cosh(d*x + c)^3 - (a^5 - a^4*b - 5*a^3*b^2 - 3*a^2*b^3)*d*cosh(d*x + c))*sinh(d*x + c)), 1/2*
(2*(a^3 + a^2*b)*d*x*cosh(d*x + c)^6 + 12*(a^3 + a^2*b)*d*x*cosh(d*x + c)*sinh(d*x + c)^5 + 2*(a^3 + a^2*b)*d*
x*sinh(d*x + c)^6 - 2*(2*a^3 + 6*a^2*b + 5*a*b^2 + 3*b^3 - (a^3 - 3*a^2*b)*d*x)*cosh(d*x + c)^4 + 2*(15*(a^3 +
a^2*b)*d*x*cosh(d*x + c)^2 - 2*a^3 - 6*a^2*b - 5*a*b^2 - 3*b^3 + (a^3 - 3*a^2*b)*d*x)*sinh(d*x + c)^4 + 8*(5*
(a^3 + a^2*b)*d*x*cosh(d*x + c)^3 - (2*a^3 + 6*a^2*b + 5*a*b^2 + 3*b^3 - (a^3 - 3*a^2*b)*d*x)*cosh(d*x + c))*s
inh(d*x + c)^3 - 4*a^3 - 12*a^2*b - 14*a*b^2 - 6*b^3 - 2*(a^3 + a^2*b)*d*x - 2*(4*a^3 + 4*a^2*b - 4*a*b^2 - 6*
b^3 + (a^3 - 3*a^2*b)*d*x)*cosh(d*x + c)^2 + 2*(15*(a^3 + a^2*b)*d*x*cosh(d*x + c)^4 - 4*a^3 - 4*a^2*b + 4*a*b
^2 + 6*b^3 - (a^3 - 3*a^2*b)*d*x - 6*(2*a^3 + 6*a^2*b + 5*a*b^2 + 3*b^3 - (a^3 - 3*a^2*b)*d*x)*cosh(d*x + c)^2
)*sinh(d*x + c)^2 - ((5*a^2*b + 8*a*b^2 + 3*b^3)*cosh(d*x + c)^6 + 6*(5*a^2*b + 8*a*b^2 + 3*b^3)*cosh(d*x + c)
*sinh(d*x + c)^5 + (5*a^2*b + 8*a*b^2 + 3*b^3)*sinh(d*x + c)^6 + (5*a^2*b - 12*a*b^2 - 9*b^3)*cosh(d*x + c)^4
+ (5*a^2*b - 12*a*b^2 - 9*b^3 + 15*(5*a^2*b + 8*a*b^2 + 3*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 4*(5*(5*a^2*
b + 8*a*b^2 + 3*b^3)*cosh(d*x + c)^3 + (5*a^2*b - 12*a*b^2 - 9*b^3)*cosh(d*x + c))*sinh(d*x + c)^3 - 5*a^2*b -
8*a*b^2 - 3*b^3 - (5*a^2*b - 12*a*b^2 - 9*b^3)*cosh(d*x + c)^2 + (15*(5*a^2*b + 8*a*b^2 + 3*b^3)*cosh(d*x + c
)^4 - 5*a^2*b + 12*a*b^2 + 9*b^3 + 6*(5*a^2*b - 12*a*b^2 - 9*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 2*(3*(5*a
^2*b + 8*a*b^2 + 3*b^3)*cosh(d*x + c)^5 + 2*(5*a^2*b - 12*a*b^2 - 9*b^3)*cosh(d*x + c)^3 - (5*a^2*b - 12*a*b^2
- 9*b^3)*cosh(d*x + c))*sinh(d*x + c))*sqrt(b/a)*arctan(1/2*((a + b)*cosh(d*x + c)^2 + 2*(a + b)*cosh(d*x + c
)*sinh(d*x + c) + (a + b)*sinh(d*x + c)^2 + a - b)*sqrt(b/a)/b) + 4*(3*(a^3 + a^2*b)*d*x*cosh(d*x + c)^5 - 2*(
2*a^3 + 6*a^2*b + 5*a*b^2 + 3*b^3 - (a^3 - 3*a^2*b)*d*x)*cosh(d*x + c)^3 - (4*a^3 + 4*a^2*b - 4*a*b^2 - 6*b^3
+ (a^3 - 3*a^2*b)*d*x)*cosh(d*x + c))*sinh(d*x + c))/((a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*d*cosh(d*x + c)^6
+ 6*(a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*d*cosh(d*x + c)*sinh(d*x + c)^5 + (a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b
^3)*d*sinh(d*x + c)^6 + (a^5 - a^4*b - 5*a^3*b^2 - 3*a^2*b^3)*d*cosh(d*x + c)^4 + (15*(a^5 + 3*a^4*b + 3*a^3*b
^2 + a^2*b^3)*d*cosh(d*x + c)^2 + (a^5 - a^4*b - 5*a^3*b^2 - 3*a^2*b^3)*d)*sinh(d*x + c)^4 - (a^5 - a^4*b - 5*
a^3*b^2 - 3*a^2*b^3)*d*cosh(d*x + c)^2 + 4*(5*(a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*d*cosh(d*x + c)^3 + (a^5 -
a^4*b - 5*a^3*b^2 - 3*a^2*b^3)*d*cosh(d*x + c))*sinh(d*x + c)^3 + (15*(a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*d
*cosh(d*x + c)^4 + 6*(a^5 - a^4*b - 5*a^3*b^2 - 3*a^2*b^3)*d*cosh(d*x + c)^2 - (a^5 - a^4*b - 5*a^3*b^2 - 3*a^
2*b^3)*d)*sinh(d*x + c)^2 - (a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*d + 2*(3*(a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^
3)*d*cosh(d*x + c)^5 + 2*(a^5 - a^4*b - 5*a^3*b^2 - 3*a^2*b^3)*d*cosh(d*x + c)^3 - (a^5 - a^4*b - 5*a^3*b^2 -
3*a^2*b^3)*d*cosh(d*x + c))*sinh(d*x + c))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\coth ^{2}{\left (c + d x \right )}}{\left (a + b \tanh ^{2}{\left (c + d x \right )}\right )^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(d*x+c)**2/(a+b*tanh(d*x+c)**2)**2,x)

[Out]

Integral(coth(c + d*x)**2/(a + b*tanh(c + d*x)**2)**2, x)

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Giac [B]  time = 1.25187, size = 464, normalized size = 3.9 \begin{align*} -\frac{{\left (5 \, a b^{2} + 3 \, b^{3}\right )} \arctan \left (\frac{a e^{\left (2 \, d x + 2 \, c\right )} + b e^{\left (2 \, d x + 2 \, c\right )} + a - b}{2 \, \sqrt{a b}}\right )}{2 \,{\left (a^{4} d + 2 \, a^{3} b d + a^{2} b^{2} d\right )} \sqrt{a b}} + \frac{d x + c}{a^{2} d + 2 \, a b d + b^{2} d} - \frac{2 \, a^{3} e^{\left (4 \, d x + 4 \, c\right )} + 6 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} + 5 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 3 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 4 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} + 4 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} - 4 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} - 6 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a^{3} + 6 \, a^{2} b + 7 \, a b^{2} + 3 \, b^{3}}{{\left (a^{4} d + 2 \, a^{3} b d + a^{2} b^{2} d\right )}{\left (a e^{\left (6 \, d x + 6 \, c\right )} + b e^{\left (6 \, d x + 6 \, c\right )} + a e^{\left (4 \, d x + 4 \, c\right )} - 3 \, b e^{\left (4 \, d x + 4 \, c\right )} - a e^{\left (2 \, d x + 2 \, c\right )} + 3 \, b e^{\left (2 \, d x + 2 \, c\right )} - a - b\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(d*x+c)^2/(a+b*tanh(d*x+c)^2)^2,x, algorithm="giac")

[Out]

-1/2*(5*a*b^2 + 3*b^3)*arctan(1/2*(a*e^(2*d*x + 2*c) + b*e^(2*d*x + 2*c) + a - b)/sqrt(a*b))/((a^4*d + 2*a^3*b
*d + a^2*b^2*d)*sqrt(a*b)) + (d*x + c)/(a^2*d + 2*a*b*d + b^2*d) - (2*a^3*e^(4*d*x + 4*c) + 6*a^2*b*e^(4*d*x +
4*c) + 5*a*b^2*e^(4*d*x + 4*c) + 3*b^3*e^(4*d*x + 4*c) + 4*a^3*e^(2*d*x + 2*c) + 4*a^2*b*e^(2*d*x + 2*c) - 4*
a*b^2*e^(2*d*x + 2*c) - 6*b^3*e^(2*d*x + 2*c) + 2*a^3 + 6*a^2*b + 7*a*b^2 + 3*b^3)/((a^4*d + 2*a^3*b*d + a^2*b
^2*d)*(a*e^(6*d*x + 6*c) + b*e^(6*d*x + 6*c) + a*e^(4*d*x + 4*c) - 3*b*e^(4*d*x + 4*c) - a*e^(2*d*x + 2*c) + 3
*b*e^(2*d*x + 2*c) - a - b))