3.166 \(\int \frac{\coth ^2(c+d x)}{(a+b \text{sech}^2(c+d x))^3} \, dx\)

Optimal. Leaf size=182 \[ -\frac{b^{3/2} \left (35 a^2+28 a b+8 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right )}{8 a^3 d (a+b)^{7/2}}-\frac{\left (8 a^2-11 a b-4 b^2\right ) \coth (c+d x)}{8 a^2 d (a+b)^3}-\frac{b (9 a+4 b) \coth (c+d x)}{8 a^2 d (a+b)^2 \left (a-b \tanh ^2(c+d x)+b\right )}+\frac{x}{a^3}-\frac{b \coth (c+d x)}{4 a d (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^2} \]

[Out]

x/a^3 - (b^(3/2)*(35*a^2 + 28*a*b + 8*b^2)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(8*a^3*(a + b)^(7/2)*
d) - ((8*a^2 - 11*a*b - 4*b^2)*Coth[c + d*x])/(8*a^2*(a + b)^3*d) - (b*Coth[c + d*x])/(4*a*(a + b)*d*(a + b -
b*Tanh[c + d*x]^2)^2) - (b*(9*a + 4*b)*Coth[c + d*x])/(8*a^2*(a + b)^2*d*(a + b - b*Tanh[c + d*x]^2))

________________________________________________________________________________________

Rubi [A]  time = 0.403154, antiderivative size = 182, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.348, Rules used = {4141, 1975, 472, 579, 583, 522, 206, 208} \[ -\frac{b^{3/2} \left (35 a^2+28 a b+8 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right )}{8 a^3 d (a+b)^{7/2}}-\frac{\left (8 a^2-11 a b-4 b^2\right ) \coth (c+d x)}{8 a^2 d (a+b)^3}-\frac{b (9 a+4 b) \coth (c+d x)}{8 a^2 d (a+b)^2 \left (a-b \tanh ^2(c+d x)+b\right )}+\frac{x}{a^3}-\frac{b \coth (c+d x)}{4 a d (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[Coth[c + d*x]^2/(a + b*Sech[c + d*x]^2)^3,x]

[Out]

x/a^3 - (b^(3/2)*(35*a^2 + 28*a*b + 8*b^2)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(8*a^3*(a + b)^(7/2)*
d) - ((8*a^2 - 11*a*b - 4*b^2)*Coth[c + d*x])/(8*a^2*(a + b)^3*d) - (b*Coth[c + d*x])/(4*a*(a + b)*d*(a + b -
b*Tanh[c + d*x]^2)^2) - (b*(9*a + 4*b)*Coth[c + d*x])/(8*a^2*(a + b)^2*d*(a + b - b*Tanh[c + d*x]^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 579

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_)*((e_) + (f_.)*(x_)^(n_)), x
_Symbol] :> -Simp[((b*e - a*f)*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*g*n*(b*c - a*d)*(p +
1)), x] + Dist[1/(a*n*(b*c - a*d)*(p + 1)), Int[(g*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f)*(
m + 1) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*f)*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d,
e, f, g, m, q}, x] && IGtQ[n, 0] && LtQ[p, -1]

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 208

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

Rubi steps

\begin{align*} \int \frac{\coth ^2(c+d x)}{\left (a+b \text{sech}^2(c+d x)\right )^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x^2 \left (1-x^2\right ) \left (a+b \left (1-x^2\right )\right )^3} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{x^2 \left (1-x^2\right ) \left (a+b-b x^2\right )^3} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=-\frac{b \coth (c+d x)}{4 a (a+b) d \left (a+b-b \tanh ^2(c+d x)\right )^2}-\frac{\operatorname{Subst}\left (\int \frac{-4 a+b-5 b x^2}{x^2 \left (1-x^2\right ) \left (a+b-b x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{4 a (a+b) d}\\ &=-\frac{b \coth (c+d x)}{4 a (a+b) d \left (a+b-b \tanh ^2(c+d x)\right )^2}-\frac{b (9 a+4 b) \coth (c+d x)}{8 a^2 (a+b)^2 d \left (a+b-b \tanh ^2(c+d x)\right )}+\frac{\operatorname{Subst}\left (\int \frac{8 a^2-11 a b-4 b^2+3 b (9 a+4 b) x^2}{x^2 \left (1-x^2\right ) \left (a+b-b x^2\right )} \, dx,x,\tanh (c+d x)\right )}{8 a^2 (a+b)^2 d}\\ &=-\frac{\left (8 a^2-11 a b-4 b^2\right ) \coth (c+d x)}{8 a^2 (a+b)^3 d}-\frac{b \coth (c+d x)}{4 a (a+b) d \left (a+b-b \tanh ^2(c+d x)\right )^2}-\frac{b (9 a+4 b) \coth (c+d x)}{8 a^2 (a+b)^2 d \left (a+b-b \tanh ^2(c+d x)\right )}-\frac{\operatorname{Subst}\left (\int \frac{-8 a^3-32 a^2 b-13 a b^2-4 b^3+b \left (8 a^2-11 a b-4 b^2\right ) x^2}{\left (1-x^2\right ) \left (a+b-b x^2\right )} \, dx,x,\tanh (c+d x)\right )}{8 a^2 (a+b)^3 d}\\ &=-\frac{\left (8 a^2-11 a b-4 b^2\right ) \coth (c+d x)}{8 a^2 (a+b)^3 d}-\frac{b \coth (c+d x)}{4 a (a+b) d \left (a+b-b \tanh ^2(c+d x)\right )^2}-\frac{b (9 a+4 b) \coth (c+d x)}{8 a^2 (a+b)^2 d \left (a+b-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^3 d}-\frac{\left (b^2 \left (35 a^2+28 a b+8 b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a+b-b x^2} \, dx,x,\tanh (c+d x)\right )}{8 a^3 (a+b)^3 d}\\ &=\frac{x}{a^3}-\frac{b^{3/2} \left (35 a^2+28 a b+8 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right )}{8 a^3 (a+b)^{7/2} d}-\frac{\left (8 a^2-11 a b-4 b^2\right ) \coth (c+d x)}{8 a^2 (a+b)^3 d}-\frac{b \coth (c+d x)}{4 a (a+b) d \left (a+b-b \tanh ^2(c+d x)\right )^2}-\frac{b (9 a+4 b) \coth (c+d x)}{8 a^2 (a+b)^2 d \left (a+b-b \tanh ^2(c+d x)\right )}\\ \end{align*}

Mathematica [C]  time = 7.03088, size = 2083, normalized size = 11.45 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Coth[c + d*x]^2/(a + b*Sech[c + d*x]^2)^3,x]

[Out]

((35*a^2 + 28*a*b + 8*b^2)*(a + 2*b + a*Cosh[2*c + 2*d*x])^3*Sech[c + d*x]^6*(((I/64)*b^2*ArcTan[Sech[d*x]*(((
-I/2)*Cosh[2*c])/(Sqrt[a + b]*Sqrt[b*Cosh[4*c] - b*Sinh[4*c]]) + ((I/2)*Sinh[2*c])/(Sqrt[a + b]*Sqrt[b*Cosh[4*
c] - b*Sinh[4*c]]))*(-(a*Sinh[d*x]) - 2*b*Sinh[d*x] + a*Sinh[2*c + d*x])]*Cosh[2*c])/(a^3*Sqrt[a + b]*d*Sqrt[b
*Cosh[4*c] - b*Sinh[4*c]]) - ((I/64)*b^2*ArcTan[Sech[d*x]*(((-I/2)*Cosh[2*c])/(Sqrt[a + b]*Sqrt[b*Cosh[4*c] -
b*Sinh[4*c]]) + ((I/2)*Sinh[2*c])/(Sqrt[a + b]*Sqrt[b*Cosh[4*c] - b*Sinh[4*c]]))*(-(a*Sinh[d*x]) - 2*b*Sinh[d*
x] + a*Sinh[2*c + d*x])]*Sinh[2*c])/(a^3*Sqrt[a + b]*d*Sqrt[b*Cosh[4*c] - b*Sinh[4*c]])))/((a + b)^3*(a + b*Se
ch[c + d*x]^2)^3) + ((a + 2*b + a*Cosh[2*c + 2*d*x])*Csch[c]*Csch[c + d*x]*Sech[2*c]*Sech[c + d*x]^6*(8*a^5*d*
x*Cosh[d*x] + 56*a^4*b*d*x*Cosh[d*x] + 184*a^3*b^2*d*x*Cosh[d*x] + 296*a^2*b^3*d*x*Cosh[d*x] + 224*a*b^4*d*x*C
osh[d*x] + 64*b^5*d*x*Cosh[d*x] - 12*a^5*d*x*Cosh[3*d*x] - 68*a^4*b*d*x*Cosh[3*d*x] - 132*a^3*b^2*d*x*Cosh[3*d
*x] - 108*a^2*b^3*d*x*Cosh[3*d*x] - 32*a*b^4*d*x*Cosh[3*d*x] - 8*a^5*d*x*Cosh[2*c - d*x] - 56*a^4*b*d*x*Cosh[2
*c - d*x] - 184*a^3*b^2*d*x*Cosh[2*c - d*x] - 296*a^2*b^3*d*x*Cosh[2*c - d*x] - 224*a*b^4*d*x*Cosh[2*c - d*x]
- 64*b^5*d*x*Cosh[2*c - d*x] - 8*a^5*d*x*Cosh[2*c + d*x] - 56*a^4*b*d*x*Cosh[2*c + d*x] - 184*a^3*b^2*d*x*Cosh
[2*c + d*x] - 296*a^2*b^3*d*x*Cosh[2*c + d*x] - 224*a*b^4*d*x*Cosh[2*c + d*x] - 64*b^5*d*x*Cosh[2*c + d*x] + 8
*a^5*d*x*Cosh[4*c + d*x] + 56*a^4*b*d*x*Cosh[4*c + d*x] + 184*a^3*b^2*d*x*Cosh[4*c + d*x] + 296*a^2*b^3*d*x*Co
sh[4*c + d*x] + 224*a*b^4*d*x*Cosh[4*c + d*x] + 64*b^5*d*x*Cosh[4*c + d*x] + 12*a^5*d*x*Cosh[2*c + 3*d*x] + 68
*a^4*b*d*x*Cosh[2*c + 3*d*x] + 132*a^3*b^2*d*x*Cosh[2*c + 3*d*x] + 108*a^2*b^3*d*x*Cosh[2*c + 3*d*x] + 32*a*b^
4*d*x*Cosh[2*c + 3*d*x] - 12*a^5*d*x*Cosh[4*c + 3*d*x] - 68*a^4*b*d*x*Cosh[4*c + 3*d*x] - 132*a^3*b^2*d*x*Cosh
[4*c + 3*d*x] - 108*a^2*b^3*d*x*Cosh[4*c + 3*d*x] - 32*a*b^4*d*x*Cosh[4*c + 3*d*x] + 12*a^5*d*x*Cosh[6*c + 3*d
*x] + 68*a^4*b*d*x*Cosh[6*c + 3*d*x] + 132*a^3*b^2*d*x*Cosh[6*c + 3*d*x] + 108*a^2*b^3*d*x*Cosh[6*c + 3*d*x] +
32*a*b^4*d*x*Cosh[6*c + 3*d*x] - 4*a^5*d*x*Cosh[2*c + 5*d*x] - 12*a^4*b*d*x*Cosh[2*c + 5*d*x] - 12*a^3*b^2*d*
x*Cosh[2*c + 5*d*x] - 4*a^2*b^3*d*x*Cosh[2*c + 5*d*x] + 4*a^5*d*x*Cosh[4*c + 5*d*x] + 12*a^4*b*d*x*Cosh[4*c +
5*d*x] + 12*a^3*b^2*d*x*Cosh[4*c + 5*d*x] + 4*a^2*b^3*d*x*Cosh[4*c + 5*d*x] - 4*a^5*d*x*Cosh[6*c + 5*d*x] - 12
*a^4*b*d*x*Cosh[6*c + 5*d*x] - 12*a^3*b^2*d*x*Cosh[6*c + 5*d*x] - 4*a^2*b^3*d*x*Cosh[6*c + 5*d*x] + 4*a^5*d*x*
Cosh[8*c + 5*d*x] + 12*a^4*b*d*x*Cosh[8*c + 5*d*x] + 12*a^3*b^2*d*x*Cosh[8*c + 5*d*x] + 4*a^2*b^3*d*x*Cosh[8*c
+ 5*d*x] - 32*a^5*Sinh[d*x] - 64*a^4*b*Sinh[d*x] - 30*a^2*b^3*Sinh[d*x] - 120*a*b^4*Sinh[d*x] - 48*b^5*Sinh[d
*x] + 32*a^5*Sinh[3*d*x] + 64*a^4*b*Sinh[3*d*x] + 26*a^3*b^2*Sinh[3*d*x] + 86*a^2*b^3*Sinh[3*d*x] + 32*a*b^4*S
inh[3*d*x] - 48*a^5*Sinh[2*c - d*x] - 128*a^4*b*Sinh[2*c - d*x] - 128*a^3*b^2*Sinh[2*c - d*x] - 30*a^2*b^3*Sin
h[2*c - d*x] - 120*a*b^4*Sinh[2*c - d*x] - 48*b^5*Sinh[2*c - d*x] + 48*a^5*Sinh[2*c + d*x] + 128*a^4*b*Sinh[2*
c + d*x] + 102*a^3*b^2*Sinh[2*c + d*x] - 86*a^2*b^3*Sinh[2*c + d*x] - 136*a*b^4*Sinh[2*c + d*x] - 48*b^5*Sinh[
2*c + d*x] - 32*a^5*Sinh[4*c + d*x] - 64*a^4*b*Sinh[4*c + d*x] + 26*a^3*b^2*Sinh[4*c + d*x] + 86*a^2*b^3*Sinh[
4*c + d*x] + 136*a*b^4*Sinh[4*c + d*x] + 48*b^5*Sinh[4*c + d*x] - 8*a^5*Sinh[2*c + 3*d*x] - 26*a^3*b^2*Sinh[2*
c + 3*d*x] - 86*a^2*b^3*Sinh[2*c + 3*d*x] - 32*a*b^4*Sinh[2*c + 3*d*x] + 32*a^5*Sinh[4*c + 3*d*x] + 64*a^4*b*S
inh[4*c + 3*d*x] - 13*a^3*b^2*Sinh[4*c + 3*d*x] - 36*a^2*b^3*Sinh[4*c + 3*d*x] - 16*a*b^4*Sinh[4*c + 3*d*x] -
8*a^5*Sinh[6*c + 3*d*x] + 13*a^3*b^2*Sinh[6*c + 3*d*x] + 36*a^2*b^3*Sinh[6*c + 3*d*x] + 16*a*b^4*Sinh[6*c + 3*
d*x] + 8*a^5*Sinh[2*c + 5*d*x] + 13*a^3*b^2*Sinh[2*c + 5*d*x] + 6*a^2*b^3*Sinh[2*c + 5*d*x] - 13*a^3*b^2*Sinh[
4*c + 5*d*x] - 6*a^2*b^3*Sinh[4*c + 5*d*x] + 8*a^5*Sinh[6*c + 5*d*x]))/(512*a^3*(a + b)^3*d*(a + b*Sech[c + d*
x]^2)^3)

________________________________________________________________________________________

Maple [B]  time = 0.12, size = 1442, normalized size = 7.9 \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*sech(d*x+c)^2)^3,x)

[Out]

-1/2/d/(a^3+3*a^2*b+3*a*b^2+b^3)*tanh(1/2*d*x+1/2*c)+1/d/a^3*ln(tanh(1/2*d*x+1/2*c)+1)-13/4/d*b^2/(a+b)^3/(tan
h(1/2*d*x+1/2*c)^4*a+b*tanh(1/2*d*x+1/2*c)^4+2*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b)^2*tanh(1
/2*d*x+1/2*c)^7-17/4/d*b^3/a/(a+b)^3/(tanh(1/2*d*x+1/2*c)^4*a+b*tanh(1/2*d*x+1/2*c)^4+2*tanh(1/2*d*x+1/2*c)^2*
a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b)^2*tanh(1/2*d*x+1/2*c)^7-1/d*b^4/a^2/(a+b)^3/(tanh(1/2*d*x+1/2*c)^4*a+b*tanh(1
/2*d*x+1/2*c)^4+2*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b)^2*tanh(1/2*d*x+1/2*c)^7-39/4/d*b^2/(a
+b)^3/(tanh(1/2*d*x+1/2*c)^4*a+b*tanh(1/2*d*x+1/2*c)^4+2*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b
)^2*tanh(1/2*d*x+1/2*c)^5-7/4/d*b^3/a/(a+b)^3/(tanh(1/2*d*x+1/2*c)^4*a+b*tanh(1/2*d*x+1/2*c)^4+2*tanh(1/2*d*x+
1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b)^2*tanh(1/2*d*x+1/2*c)^5+1/d*b^4/a^2/(a+b)^3/(tanh(1/2*d*x+1/2*c)^4*a
+b*tanh(1/2*d*x+1/2*c)^4+2*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b)^2*tanh(1/2*d*x+1/2*c)^5-39/4
/d*b^2/(a+b)^3/(tanh(1/2*d*x+1/2*c)^4*a+b*tanh(1/2*d*x+1/2*c)^4+2*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c
)^2*b+a+b)^2*tanh(1/2*d*x+1/2*c)^3-7/4/d*b^3/a/(a+b)^3/(tanh(1/2*d*x+1/2*c)^4*a+b*tanh(1/2*d*x+1/2*c)^4+2*tanh
(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b)^2*tanh(1/2*d*x+1/2*c)^3+1/d*b^4/a^2/(a+b)^3/(tanh(1/2*d*x+1
/2*c)^4*a+b*tanh(1/2*d*x+1/2*c)^4+2*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b)^2*tanh(1/2*d*x+1/2*
c)^3-13/4/d*b^2/(a+b)^3/(tanh(1/2*d*x+1/2*c)^4*a+b*tanh(1/2*d*x+1/2*c)^4+2*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*
d*x+1/2*c)^2*b+a+b)^2*tanh(1/2*d*x+1/2*c)-17/4/d*b^3/a/(a+b)^3/(tanh(1/2*d*x+1/2*c)^4*a+b*tanh(1/2*d*x+1/2*c)^
4+2*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b)^2*tanh(1/2*d*x+1/2*c)-1/d*b^4/a^2/(a+b)^3/(tanh(1/2
*d*x+1/2*c)^4*a+b*tanh(1/2*d*x+1/2*c)^4+2*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b)^2*tanh(1/2*d*
x+1/2*c)-35/16/d*b^(3/2)/a/(a+b)^(7/2)*ln((a+b)^(1/2)*tanh(1/2*d*x+1/2*c)^2+2*tanh(1/2*d*x+1/2*c)*b^(1/2)+(a+b
)^(1/2))+35/16/d*b^(3/2)/a/(a+b)^(7/2)*ln(-(a+b)^(1/2)*tanh(1/2*d*x+1/2*c)^2+2*tanh(1/2*d*x+1/2*c)*b^(1/2)-(a+
b)^(1/2))-7/4/d*b^(5/2)/a^2/(a+b)^(7/2)*ln((a+b)^(1/2)*tanh(1/2*d*x+1/2*c)^2+2*tanh(1/2*d*x+1/2*c)*b^(1/2)+(a+
b)^(1/2))+7/4/d*b^(5/2)/a^2/(a+b)^(7/2)*ln(-(a+b)^(1/2)*tanh(1/2*d*x+1/2*c)^2+2*tanh(1/2*d*x+1/2*c)*b^(1/2)-(a
+b)^(1/2))-1/2/d*b^(7/2)/a^3/(a+b)^(7/2)*ln((a+b)^(1/2)*tanh(1/2*d*x+1/2*c)^2+2*tanh(1/2*d*x+1/2*c)*b^(1/2)+(a
+b)^(1/2))+1/2/d*b^(7/2)/a^3/(a+b)^(7/2)*ln(-(a+b)^(1/2)*tanh(1/2*d*x+1/2*c)^2+2*tanh(1/2*d*x+1/2*c)*b^(1/2)-(
a+b)^(1/2))-1/2/d/(a+b)^3/tanh(1/2*d*x+1/2*c)-1/d/a^3*ln(tanh(1/2*d*x+1/2*c)-1)

________________________________________________________________________________________

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*sech(d*x+c)^2)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 3.81232, size = 27173, normalized size = 149.3 \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*sech(d*x+c)^2)^3,x, algorithm="fricas")

[Out]

[1/16*(16*(a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*d*x*cosh(d*x + c)^10 + 160*(a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^
3)*d*x*cosh(d*x + c)*sinh(d*x + c)^9 + 16*(a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*d*x*sinh(d*x + c)^10 - 4*(8*a^
5 - 13*a^3*b^2 - 36*a^2*b^3 - 16*a*b^4 - 4*(3*a^5 + 17*a^4*b + 33*a^3*b^2 + 27*a^2*b^3 + 8*a*b^4)*d*x)*cosh(d*
x + c)^8 - 4*(8*a^5 - 13*a^3*b^2 - 36*a^2*b^3 - 16*a*b^4 - 180*(a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*d*x*cosh(
d*x + c)^2 - 4*(3*a^5 + 17*a^4*b + 33*a^3*b^2 + 27*a^2*b^3 + 8*a*b^4)*d*x)*sinh(d*x + c)^8 + 32*(60*(a^5 + 3*a
^4*b + 3*a^3*b^2 + a^2*b^3)*d*x*cosh(d*x + c)^3 - (8*a^5 - 13*a^3*b^2 - 36*a^2*b^3 - 16*a*b^4 - 4*(3*a^5 + 17*
a^4*b + 33*a^3*b^2 + 27*a^2*b^3 + 8*a*b^4)*d*x)*cosh(d*x + c))*sinh(d*x + c)^7 - 8*(16*a^5 + 32*a^4*b - 13*a^3
*b^2 - 43*a^2*b^3 - 68*a*b^4 - 24*b^5 - 4*(a^5 + 7*a^4*b + 23*a^3*b^2 + 37*a^2*b^3 + 28*a*b^4 + 8*b^5)*d*x)*co
sh(d*x + c)^6 + 8*(420*(a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*d*x*cosh(d*x + c)^4 - 16*a^5 - 32*a^4*b + 13*a^3*
b^2 + 43*a^2*b^3 + 68*a*b^4 + 24*b^5 + 4*(a^5 + 7*a^4*b + 23*a^3*b^2 + 37*a^2*b^3 + 28*a*b^4 + 8*b^5)*d*x - 14
*(8*a^5 - 13*a^3*b^2 - 36*a^2*b^3 - 16*a*b^4 - 4*(3*a^5 + 17*a^4*b + 33*a^3*b^2 + 27*a^2*b^3 + 8*a*b^4)*d*x)*c
osh(d*x + c)^2)*sinh(d*x + c)^6 + 16*(252*(a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*d*x*cosh(d*x + c)^5 - 14*(8*a^
5 - 13*a^3*b^2 - 36*a^2*b^3 - 16*a*b^4 - 4*(3*a^5 + 17*a^4*b + 33*a^3*b^2 + 27*a^2*b^3 + 8*a*b^4)*d*x)*cosh(d*
x + c)^3 - 3*(16*a^5 + 32*a^4*b - 13*a^3*b^2 - 43*a^2*b^3 - 68*a*b^4 - 24*b^5 - 4*(a^5 + 7*a^4*b + 23*a^3*b^2
+ 37*a^2*b^3 + 28*a*b^4 + 8*b^5)*d*x)*cosh(d*x + c))*sinh(d*x + c)^5 - 32*a^5 - 52*a^3*b^2 - 24*a^2*b^3 - 8*(2
4*a^5 + 64*a^4*b + 64*a^3*b^2 + 15*a^2*b^3 + 60*a*b^4 + 24*b^5 + 4*(a^5 + 7*a^4*b + 23*a^3*b^2 + 37*a^2*b^3 +
28*a*b^4 + 8*b^5)*d*x)*cosh(d*x + c)^4 + 8*(420*(a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*d*x*cosh(d*x + c)^6 - 24
*a^5 - 64*a^4*b - 64*a^3*b^2 - 15*a^2*b^3 - 60*a*b^4 - 24*b^5 - 35*(8*a^5 - 13*a^3*b^2 - 36*a^2*b^3 - 16*a*b^4
- 4*(3*a^5 + 17*a^4*b + 33*a^3*b^2 + 27*a^2*b^3 + 8*a*b^4)*d*x)*cosh(d*x + c)^4 - 4*(a^5 + 7*a^4*b + 23*a^3*b
^2 + 37*a^2*b^3 + 28*a*b^4 + 8*b^5)*d*x - 15*(16*a^5 + 32*a^4*b - 13*a^3*b^2 - 43*a^2*b^3 - 68*a*b^4 - 24*b^5
- 4*(a^5 + 7*a^4*b + 23*a^3*b^2 + 37*a^2*b^3 + 28*a*b^4 + 8*b^5)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 32*(6
0*(a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*d*x*cosh(d*x + c)^7 - 7*(8*a^5 - 13*a^3*b^2 - 36*a^2*b^3 - 16*a*b^4 -
4*(3*a^5 + 17*a^4*b + 33*a^3*b^2 + 27*a^2*b^3 + 8*a*b^4)*d*x)*cosh(d*x + c)^5 - 5*(16*a^5 + 32*a^4*b - 13*a^3*
b^2 - 43*a^2*b^3 - 68*a*b^4 - 24*b^5 - 4*(a^5 + 7*a^4*b + 23*a^3*b^2 + 37*a^2*b^3 + 28*a*b^4 + 8*b^5)*d*x)*cos
h(d*x + c)^3 - (24*a^5 + 64*a^4*b + 64*a^3*b^2 + 15*a^2*b^3 + 60*a*b^4 + 24*b^5 + 4*(a^5 + 7*a^4*b + 23*a^3*b^
2 + 37*a^2*b^3 + 28*a*b^4 + 8*b^5)*d*x)*cosh(d*x + c))*sinh(d*x + c)^3 - 16*(a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b
^3)*d*x - 8*(16*a^5 + 32*a^4*b + 13*a^3*b^2 + 43*a^2*b^3 + 16*a*b^4 + 2*(3*a^5 + 17*a^4*b + 33*a^3*b^2 + 27*a^
2*b^3 + 8*a*b^4)*d*x)*cosh(d*x + c)^2 + 8*(90*(a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*d*x*cosh(d*x + c)^8 - 14*(
8*a^5 - 13*a^3*b^2 - 36*a^2*b^3 - 16*a*b^4 - 4*(3*a^5 + 17*a^4*b + 33*a^3*b^2 + 27*a^2*b^3 + 8*a*b^4)*d*x)*cos
h(d*x + c)^6 - 16*a^5 - 32*a^4*b - 13*a^3*b^2 - 43*a^2*b^3 - 16*a*b^4 - 15*(16*a^5 + 32*a^4*b - 13*a^3*b^2 - 4
3*a^2*b^3 - 68*a*b^4 - 24*b^5 - 4*(a^5 + 7*a^4*b + 23*a^3*b^2 + 37*a^2*b^3 + 28*a*b^4 + 8*b^5)*d*x)*cosh(d*x +
c)^4 - 2*(3*a^5 + 17*a^4*b + 33*a^3*b^2 + 27*a^2*b^3 + 8*a*b^4)*d*x - 6*(24*a^5 + 64*a^4*b + 64*a^3*b^2 + 15*
a^2*b^3 + 60*a*b^4 + 24*b^5 + 4*(a^5 + 7*a^4*b + 23*a^3*b^2 + 37*a^2*b^3 + 28*a*b^4 + 8*b^5)*d*x)*cosh(d*x + c
)^2)*sinh(d*x + c)^2 + ((35*a^4*b + 28*a^3*b^2 + 8*a^2*b^3)*cosh(d*x + c)^10 + 10*(35*a^4*b + 28*a^3*b^2 + 8*a
^2*b^3)*cosh(d*x + c)*sinh(d*x + c)^9 + (35*a^4*b + 28*a^3*b^2 + 8*a^2*b^3)*sinh(d*x + c)^10 + (105*a^4*b + 36
4*a^3*b^2 + 248*a^2*b^3 + 64*a*b^4)*cosh(d*x + c)^8 + (105*a^4*b + 364*a^3*b^2 + 248*a^2*b^3 + 64*a*b^4 + 45*(
35*a^4*b + 28*a^3*b^2 + 8*a^2*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^8 + 8*(15*(35*a^4*b + 28*a^3*b^2 + 8*a^2*b^3
)*cosh(d*x + c)^3 + (105*a^4*b + 364*a^3*b^2 + 248*a^2*b^3 + 64*a*b^4)*cosh(d*x + c))*sinh(d*x + c)^7 + 2*(35*
a^4*b + 168*a^3*b^2 + 400*a^2*b^3 + 256*a*b^4 + 64*b^5)*cosh(d*x + c)^6 + 2*(35*a^4*b + 168*a^3*b^2 + 400*a^2*
b^3 + 256*a*b^4 + 64*b^5 + 105*(35*a^4*b + 28*a^3*b^2 + 8*a^2*b^3)*cosh(d*x + c)^4 + 14*(105*a^4*b + 364*a^3*b
^2 + 248*a^2*b^3 + 64*a*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 4*(63*(35*a^4*b + 28*a^3*b^2 + 8*a^2*b^3)*cosh
(d*x + c)^5 + 14*(105*a^4*b + 364*a^3*b^2 + 248*a^2*b^3 + 64*a*b^4)*cosh(d*x + c)^3 + 3*(35*a^4*b + 168*a^3*b^
2 + 400*a^2*b^3 + 256*a*b^4 + 64*b^5)*cosh(d*x + c))*sinh(d*x + c)^5 - 35*a^4*b - 28*a^3*b^2 - 8*a^2*b^3 - 2*(
35*a^4*b + 168*a^3*b^2 + 400*a^2*b^3 + 256*a*b^4 + 64*b^5)*cosh(d*x + c)^4 + 2*(105*(35*a^4*b + 28*a^3*b^2 + 8
*a^2*b^3)*cosh(d*x + c)^6 - 35*a^4*b - 168*a^3*b^2 - 400*a^2*b^3 - 256*a*b^4 - 64*b^5 + 35*(105*a^4*b + 364*a^
3*b^2 + 248*a^2*b^3 + 64*a*b^4)*cosh(d*x + c)^4 + 15*(35*a^4*b + 168*a^3*b^2 + 400*a^2*b^3 + 256*a*b^4 + 64*b^
5)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 8*(15*(35*a^4*b + 28*a^3*b^2 + 8*a^2*b^3)*cosh(d*x + c)^7 + 7*(105*a^4*b
+ 364*a^3*b^2 + 248*a^2*b^3 + 64*a*b^4)*cosh(d*x + c)^5 + 5*(35*a^4*b + 168*a^3*b^2 + 400*a^2*b^3 + 256*a*b^4
+ 64*b^5)*cosh(d*x + c)^3 - (35*a^4*b + 168*a^3*b^2 + 400*a^2*b^3 + 256*a*b^4 + 64*b^5)*cosh(d*x + c))*sinh(d
*x + c)^3 - (105*a^4*b + 364*a^3*b^2 + 248*a^2*b^3 + 64*a*b^4)*cosh(d*x + c)^2 + (45*(35*a^4*b + 28*a^3*b^2 +
8*a^2*b^3)*cosh(d*x + c)^8 + 28*(105*a^4*b + 364*a^3*b^2 + 248*a^2*b^3 + 64*a*b^4)*cosh(d*x + c)^6 - 105*a^4*b
- 364*a^3*b^2 - 248*a^2*b^3 - 64*a*b^4 + 30*(35*a^4*b + 168*a^3*b^2 + 400*a^2*b^3 + 256*a*b^4 + 64*b^5)*cosh(
d*x + c)^4 - 12*(35*a^4*b + 168*a^3*b^2 + 400*a^2*b^3 + 256*a*b^4 + 64*b^5)*cosh(d*x + c)^2)*sinh(d*x + c)^2 +
2*(5*(35*a^4*b + 28*a^3*b^2 + 8*a^2*b^3)*cosh(d*x + c)^9 + 4*(105*a^4*b + 364*a^3*b^2 + 248*a^2*b^3 + 64*a*b^
4)*cosh(d*x + c)^7 + 6*(35*a^4*b + 168*a^3*b^2 + 400*a^2*b^3 + 256*a*b^4 + 64*b^5)*cosh(d*x + c)^5 - 4*(35*a^4
*b + 168*a^3*b^2 + 400*a^2*b^3 + 256*a*b^4 + 64*b^5)*cosh(d*x + c)^3 - (105*a^4*b + 364*a^3*b^2 + 248*a^2*b^3
+ 64*a*b^4)*cosh(d*x + c))*sinh(d*x + c))*sqrt(b/(a + b))*log((a^2*cosh(d*x + c)^4 + 4*a^2*cosh(d*x + c)*sinh(
d*x + c)^3 + a^2*sinh(d*x + c)^4 + 2*(a^2 + 2*a*b)*cosh(d*x + c)^2 + 2*(3*a^2*cosh(d*x + c)^2 + a^2 + 2*a*b)*s
inh(d*x + c)^2 + a^2 + 8*a*b + 8*b^2 + 4*(a^2*cosh(d*x + c)^3 + (a^2 + 2*a*b)*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
+ 3*a*b + 2*b^2)*sqrt(b/(a + b)))/(a*cosh(d*x + c)^4 + 4*a*cosh(d*x + c)*sinh(d*x + c)^3 + a*sinh(d*x + c)^4 +
2*(a + 2*b)*cosh(d*x + c)^2 + 2*(3*a*cosh(d*x + c)^2 + a + 2*b)*sinh(d*x + c)^2 + 4*(a*cosh(d*x + c)^3 + (a +
2*b)*cosh(d*x + c))*sinh(d*x + c) + a)) + 16*(10*(a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*d*x*cosh(d*x + c)^9 -
2*(8*a^5 - 13*a^3*b^2 - 36*a^2*b^3 - 16*a*b^4 - 4*(3*a^5 + 17*a^4*b + 33*a^3*b^2 + 27*a^2*b^3 + 8*a*b^4)*d*x)*
cosh(d*x + c)^7 - 3*(16*a^5 + 32*a^4*b - 13*a^3*b^2 - 43*a^2*b^3 - 68*a*b^4 - 24*b^5 - 4*(a^5 + 7*a^4*b + 23*a
^3*b^2 + 37*a^2*b^3 + 28*a*b^4 + 8*b^5)*d*x)*cosh(d*x + c)^5 - 2*(24*a^5 + 64*a^4*b + 64*a^3*b^2 + 15*a^2*b^3
+ 60*a*b^4 + 24*b^5 + 4*(a^5 + 7*a^4*b + 23*a^3*b^2 + 37*a^2*b^3 + 28*a*b^4 + 8*b^5)*d*x)*cosh(d*x + c)^3 - (1
6*a^5 + 32*a^4*b + 13*a^3*b^2 + 43*a^2*b^3 + 16*a*b^4 + 2*(3*a^5 + 17*a^4*b + 33*a^3*b^2 + 27*a^2*b^3 + 8*a*b^
4)*d*x)*cosh(d*x + c))*sinh(d*x + c))/((a^8 + 3*a^7*b + 3*a^6*b^2 + a^5*b^3)*d*cosh(d*x + c)^10 + 10*(a^8 + 3*
a^7*b + 3*a^6*b^2 + a^5*b^3)*d*cosh(d*x + c)*sinh(d*x + c)^9 + (a^8 + 3*a^7*b + 3*a^6*b^2 + a^5*b^3)*d*sinh(d*
x + c)^10 + (3*a^8 + 17*a^7*b + 33*a^6*b^2 + 27*a^5*b^3 + 8*a^4*b^4)*d*cosh(d*x + c)^8 + (45*(a^8 + 3*a^7*b +
3*a^6*b^2 + a^5*b^3)*d*cosh(d*x + c)^2 + (3*a^8 + 17*a^7*b + 33*a^6*b^2 + 27*a^5*b^3 + 8*a^4*b^4)*d)*sinh(d*x
+ c)^8 + 2*(a^8 + 7*a^7*b + 23*a^6*b^2 + 37*a^5*b^3 + 28*a^4*b^4 + 8*a^3*b^5)*d*cosh(d*x + c)^6 + 8*(15*(a^8 +
3*a^7*b + 3*a^6*b^2 + a^5*b^3)*d*cosh(d*x + c)^3 + (3*a^8 + 17*a^7*b + 33*a^6*b^2 + 27*a^5*b^3 + 8*a^4*b^4)*d
*cosh(d*x + c))*sinh(d*x + c)^7 + 2*(105*(a^8 + 3*a^7*b + 3*a^6*b^2 + a^5*b^3)*d*cosh(d*x + c)^4 + 14*(3*a^8 +
17*a^7*b + 33*a^6*b^2 + 27*a^5*b^3 + 8*a^4*b^4)*d*cosh(d*x + c)^2 + (a^8 + 7*a^7*b + 23*a^6*b^2 + 37*a^5*b^3
+ 28*a^4*b^4 + 8*a^3*b^5)*d)*sinh(d*x + c)^6 - 2*(a^8 + 7*a^7*b + 23*a^6*b^2 + 37*a^5*b^3 + 28*a^4*b^4 + 8*a^3
*b^5)*d*cosh(d*x + c)^4 + 4*(63*(a^8 + 3*a^7*b + 3*a^6*b^2 + a^5*b^3)*d*cosh(d*x + c)^5 + 14*(3*a^8 + 17*a^7*b
+ 33*a^6*b^2 + 27*a^5*b^3 + 8*a^4*b^4)*d*cosh(d*x + c)^3 + 3*(a^8 + 7*a^7*b + 23*a^6*b^2 + 37*a^5*b^3 + 28*a^
4*b^4 + 8*a^3*b^5)*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(105*(a^8 + 3*a^7*b + 3*a^6*b^2 + a^5*b^3)*d*cosh(d*x
+ c)^6 + 35*(3*a^8 + 17*a^7*b + 33*a^6*b^2 + 27*a^5*b^3 + 8*a^4*b^4)*d*cosh(d*x + c)^4 + 15*(a^8 + 7*a^7*b + 2
3*a^6*b^2 + 37*a^5*b^3 + 28*a^4*b^4 + 8*a^3*b^5)*d*cosh(d*x + c)^2 - (a^8 + 7*a^7*b + 23*a^6*b^2 + 37*a^5*b^3
+ 28*a^4*b^4 + 8*a^3*b^5)*d)*sinh(d*x + c)^4 - (3*a^8 + 17*a^7*b + 33*a^6*b^2 + 27*a^5*b^3 + 8*a^4*b^4)*d*cosh
(d*x + c)^2 + 8*(15*(a^8 + 3*a^7*b + 3*a^6*b^2 + a^5*b^3)*d*cosh(d*x + c)^7 + 7*(3*a^8 + 17*a^7*b + 33*a^6*b^2
+ 27*a^5*b^3 + 8*a^4*b^4)*d*cosh(d*x + c)^5 + 5*(a^8 + 7*a^7*b + 23*a^6*b^2 + 37*a^5*b^3 + 28*a^4*b^4 + 8*a^3
*b^5)*d*cosh(d*x + c)^3 - (a^8 + 7*a^7*b + 23*a^6*b^2 + 37*a^5*b^3 + 28*a^4*b^4 + 8*a^3*b^5)*d*cosh(d*x + c))*
sinh(d*x + c)^3 + (45*(a^8 + 3*a^7*b + 3*a^6*b^2 + a^5*b^3)*d*cosh(d*x + c)^8 + 28*(3*a^8 + 17*a^7*b + 33*a^6*
b^2 + 27*a^5*b^3 + 8*a^4*b^4)*d*cosh(d*x + c)^6 + 30*(a^8 + 7*a^7*b + 23*a^6*b^2 + 37*a^5*b^3 + 28*a^4*b^4 + 8
*a^3*b^5)*d*cosh(d*x + c)^4 - 12*(a^8 + 7*a^7*b + 23*a^6*b^2 + 37*a^5*b^3 + 28*a^4*b^4 + 8*a^3*b^5)*d*cosh(d*x
+ c)^2 - (3*a^8 + 17*a^7*b + 33*a^6*b^2 + 27*a^5*b^3 + 8*a^4*b^4)*d)*sinh(d*x + c)^2 - (a^8 + 3*a^7*b + 3*a^6
*b^2 + a^5*b^3)*d + 2*(5*(a^8 + 3*a^7*b + 3*a^6*b^2 + a^5*b^3)*d*cosh(d*x + c)^9 + 4*(3*a^8 + 17*a^7*b + 33*a^
6*b^2 + 27*a^5*b^3 + 8*a^4*b^4)*d*cosh(d*x + c)^7 + 6*(a^8 + 7*a^7*b + 23*a^6*b^2 + 37*a^5*b^3 + 28*a^4*b^4 +
8*a^3*b^5)*d*cosh(d*x + c)^5 - 4*(a^8 + 7*a^7*b + 23*a^6*b^2 + 37*a^5*b^3 + 28*a^4*b^4 + 8*a^3*b^5)*d*cosh(d*x
+ c)^3 - (3*a^8 + 17*a^7*b + 33*a^6*b^2 + 27*a^5*b^3 + 8*a^4*b^4)*d*cosh(d*x + c))*sinh(d*x + c)), 1/8*(8*(a^
5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*d*x*cosh(d*x + c)^10 + 80*(a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*d*x*cosh(d*
x + c)*sinh(d*x + c)^9 + 8*(a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*d*x*sinh(d*x + c)^10 - 2*(8*a^5 - 13*a^3*b^2
- 36*a^2*b^3 - 16*a*b^4 - 4*(3*a^5 + 17*a^4*b + 33*a^3*b^2 + 27*a^2*b^3 + 8*a*b^4)*d*x)*cosh(d*x + c)^8 - 2*(8
*a^5 - 13*a^3*b^2 - 36*a^2*b^3 - 16*a*b^4 - 180*(a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*d*x*cosh(d*x + c)^2 - 4*
(3*a^5 + 17*a^4*b + 33*a^3*b^2 + 27*a^2*b^3 + 8*a*b^4)*d*x)*sinh(d*x + c)^8 + 16*(60*(a^5 + 3*a^4*b + 3*a^3*b^
2 + a^2*b^3)*d*x*cosh(d*x + c)^3 - (8*a^5 - 13*a^3*b^2 - 36*a^2*b^3 - 16*a*b^4 - 4*(3*a^5 + 17*a^4*b + 33*a^3*
b^2 + 27*a^2*b^3 + 8*a*b^4)*d*x)*cosh(d*x + c))*sinh(d*x + c)^7 - 4*(16*a^5 + 32*a^4*b - 13*a^3*b^2 - 43*a^2*b
^3 - 68*a*b^4 - 24*b^5 - 4*(a^5 + 7*a^4*b + 23*a^3*b^2 + 37*a^2*b^3 + 28*a*b^4 + 8*b^5)*d*x)*cosh(d*x + c)^6 +
4*(420*(a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*d*x*cosh(d*x + c)^4 - 16*a^5 - 32*a^4*b + 13*a^3*b^2 + 43*a^2*b^
3 + 68*a*b^4 + 24*b^5 + 4*(a^5 + 7*a^4*b + 23*a^3*b^2 + 37*a^2*b^3 + 28*a*b^4 + 8*b^5)*d*x - 14*(8*a^5 - 13*a^
3*b^2 - 36*a^2*b^3 - 16*a*b^4 - 4*(3*a^5 + 17*a^4*b + 33*a^3*b^2 + 27*a^2*b^3 + 8*a*b^4)*d*x)*cosh(d*x + c)^2)
*sinh(d*x + c)^6 + 8*(252*(a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*d*x*cosh(d*x + c)^5 - 14*(8*a^5 - 13*a^3*b^2 -
36*a^2*b^3 - 16*a*b^4 - 4*(3*a^5 + 17*a^4*b + 33*a^3*b^2 + 27*a^2*b^3 + 8*a*b^4)*d*x)*cosh(d*x + c)^3 - 3*(16
*a^5 + 32*a^4*b - 13*a^3*b^2 - 43*a^2*b^3 - 68*a*b^4 - 24*b^5 - 4*(a^5 + 7*a^4*b + 23*a^3*b^2 + 37*a^2*b^3 + 2
8*a*b^4 + 8*b^5)*d*x)*cosh(d*x + c))*sinh(d*x + c)^5 - 16*a^5 - 26*a^3*b^2 - 12*a^2*b^3 - 4*(24*a^5 + 64*a^4*b
+ 64*a^3*b^2 + 15*a^2*b^3 + 60*a*b^4 + 24*b^5 + 4*(a^5 + 7*a^4*b + 23*a^3*b^2 + 37*a^2*b^3 + 28*a*b^4 + 8*b^5
)*d*x)*cosh(d*x + c)^4 + 4*(420*(a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*d*x*cosh(d*x + c)^6 - 24*a^5 - 64*a^4*b
- 64*a^3*b^2 - 15*a^2*b^3 - 60*a*b^4 - 24*b^5 - 35*(8*a^5 - 13*a^3*b^2 - 36*a^2*b^3 - 16*a*b^4 - 4*(3*a^5 + 17
*a^4*b + 33*a^3*b^2 + 27*a^2*b^3 + 8*a*b^4)*d*x)*cosh(d*x + c)^4 - 4*(a^5 + 7*a^4*b + 23*a^3*b^2 + 37*a^2*b^3
+ 28*a*b^4 + 8*b^5)*d*x - 15*(16*a^5 + 32*a^4*b - 13*a^3*b^2 - 43*a^2*b^3 - 68*a*b^4 - 24*b^5 - 4*(a^5 + 7*a^4
*b + 23*a^3*b^2 + 37*a^2*b^3 + 28*a*b^4 + 8*b^5)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 16*(60*(a^5 + 3*a^4*b
+ 3*a^3*b^2 + a^2*b^3)*d*x*cosh(d*x + c)^7 - 7*(8*a^5 - 13*a^3*b^2 - 36*a^2*b^3 - 16*a*b^4 - 4*(3*a^5 + 17*a^
4*b + 33*a^3*b^2 + 27*a^2*b^3 + 8*a*b^4)*d*x)*cosh(d*x + c)^5 - 5*(16*a^5 + 32*a^4*b - 13*a^3*b^2 - 43*a^2*b^3
- 68*a*b^4 - 24*b^5 - 4*(a^5 + 7*a^4*b + 23*a^3*b^2 + 37*a^2*b^3 + 28*a*b^4 + 8*b^5)*d*x)*cosh(d*x + c)^3 - (
24*a^5 + 64*a^4*b + 64*a^3*b^2 + 15*a^2*b^3 + 60*a*b^4 + 24*b^5 + 4*(a^5 + 7*a^4*b + 23*a^3*b^2 + 37*a^2*b^3 +
28*a*b^4 + 8*b^5)*d*x)*cosh(d*x + c))*sinh(d*x + c)^3 - 8*(a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*d*x - 4*(16*a
^5 + 32*a^4*b + 13*a^3*b^2 + 43*a^2*b^3 + 16*a*b^4 + 2*(3*a^5 + 17*a^4*b + 33*a^3*b^2 + 27*a^2*b^3 + 8*a*b^4)*
d*x)*cosh(d*x + c)^2 + 4*(90*(a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*d*x*cosh(d*x + c)^8 - 14*(8*a^5 - 13*a^3*b^
2 - 36*a^2*b^3 - 16*a*b^4 - 4*(3*a^5 + 17*a^4*b + 33*a^3*b^2 + 27*a^2*b^3 + 8*a*b^4)*d*x)*cosh(d*x + c)^6 - 16
*a^5 - 32*a^4*b - 13*a^3*b^2 - 43*a^2*b^3 - 16*a*b^4 - 15*(16*a^5 + 32*a^4*b - 13*a^3*b^2 - 43*a^2*b^3 - 68*a*
b^4 - 24*b^5 - 4*(a^5 + 7*a^4*b + 23*a^3*b^2 + 37*a^2*b^3 + 28*a*b^4 + 8*b^5)*d*x)*cosh(d*x + c)^4 - 2*(3*a^5
+ 17*a^4*b + 33*a^3*b^2 + 27*a^2*b^3 + 8*a*b^4)*d*x - 6*(24*a^5 + 64*a^4*b + 64*a^3*b^2 + 15*a^2*b^3 + 60*a*b^
4 + 24*b^5 + 4*(a^5 + 7*a^4*b + 23*a^3*b^2 + 37*a^2*b^3 + 28*a*b^4 + 8*b^5)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c
)^2 - ((35*a^4*b + 28*a^3*b^2 + 8*a^2*b^3)*cosh(d*x + c)^10 + 10*(35*a^4*b + 28*a^3*b^2 + 8*a^2*b^3)*cosh(d*x
+ c)*sinh(d*x + c)^9 + (35*a^4*b + 28*a^3*b^2 + 8*a^2*b^3)*sinh(d*x + c)^10 + (105*a^4*b + 364*a^3*b^2 + 248*a
^2*b^3 + 64*a*b^4)*cosh(d*x + c)^8 + (105*a^4*b + 364*a^3*b^2 + 248*a^2*b^3 + 64*a*b^4 + 45*(35*a^4*b + 28*a^3
*b^2 + 8*a^2*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^8 + 8*(15*(35*a^4*b + 28*a^3*b^2 + 8*a^2*b^3)*cosh(d*x + c)^3
+ (105*a^4*b + 364*a^3*b^2 + 248*a^2*b^3 + 64*a*b^4)*cosh(d*x + c))*sinh(d*x + c)^7 + 2*(35*a^4*b + 168*a^3*b
^2 + 400*a^2*b^3 + 256*a*b^4 + 64*b^5)*cosh(d*x + c)^6 + 2*(35*a^4*b + 168*a^3*b^2 + 400*a^2*b^3 + 256*a*b^4 +
64*b^5 + 105*(35*a^4*b + 28*a^3*b^2 + 8*a^2*b^3)*cosh(d*x + c)^4 + 14*(105*a^4*b + 364*a^3*b^2 + 248*a^2*b^3
+ 64*a*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 4*(63*(35*a^4*b + 28*a^3*b^2 + 8*a^2*b^3)*cosh(d*x + c)^5 + 14*
(105*a^4*b + 364*a^3*b^2 + 248*a^2*b^3 + 64*a*b^4)*cosh(d*x + c)^3 + 3*(35*a^4*b + 168*a^3*b^2 + 400*a^2*b^3 +
256*a*b^4 + 64*b^5)*cosh(d*x + c))*sinh(d*x + c)^5 - 35*a^4*b - 28*a^3*b^2 - 8*a^2*b^3 - 2*(35*a^4*b + 168*a^
3*b^2 + 400*a^2*b^3 + 256*a*b^4 + 64*b^5)*cosh(d*x + c)^4 + 2*(105*(35*a^4*b + 28*a^3*b^2 + 8*a^2*b^3)*cosh(d*
x + c)^6 - 35*a^4*b - 168*a^3*b^2 - 400*a^2*b^3 - 256*a*b^4 - 64*b^5 + 35*(105*a^4*b + 364*a^3*b^2 + 248*a^2*b
^3 + 64*a*b^4)*cosh(d*x + c)^4 + 15*(35*a^4*b + 168*a^3*b^2 + 400*a^2*b^3 + 256*a*b^4 + 64*b^5)*cosh(d*x + c)^
2)*sinh(d*x + c)^4 + 8*(15*(35*a^4*b + 28*a^3*b^2 + 8*a^2*b^3)*cosh(d*x + c)^7 + 7*(105*a^4*b + 364*a^3*b^2 +
248*a^2*b^3 + 64*a*b^4)*cosh(d*x + c)^5 + 5*(35*a^4*b + 168*a^3*b^2 + 400*a^2*b^3 + 256*a*b^4 + 64*b^5)*cosh(d
*x + c)^3 - (35*a^4*b + 168*a^3*b^2 + 400*a^2*b^3 + 256*a*b^4 + 64*b^5)*cosh(d*x + c))*sinh(d*x + c)^3 - (105*
a^4*b + 364*a^3*b^2 + 248*a^2*b^3 + 64*a*b^4)*cosh(d*x + c)^2 + (45*(35*a^4*b + 28*a^3*b^2 + 8*a^2*b^3)*cosh(d
*x + c)^8 + 28*(105*a^4*b + 364*a^3*b^2 + 248*a^2*b^3 + 64*a*b^4)*cosh(d*x + c)^6 - 105*a^4*b - 364*a^3*b^2 -
248*a^2*b^3 - 64*a*b^4 + 30*(35*a^4*b + 168*a^3*b^2 + 400*a^2*b^3 + 256*a*b^4 + 64*b^5)*cosh(d*x + c)^4 - 12*(
35*a^4*b + 168*a^3*b^2 + 400*a^2*b^3 + 256*a*b^4 + 64*b^5)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 2*(5*(35*a^4*b +
28*a^3*b^2 + 8*a^2*b^3)*cosh(d*x + c)^9 + 4*(105*a^4*b + 364*a^3*b^2 + 248*a^2*b^3 + 64*a*b^4)*cosh(d*x + c)^
7 + 6*(35*a^4*b + 168*a^3*b^2 + 400*a^2*b^3 + 256*a*b^4 + 64*b^5)*cosh(d*x + c)^5 - 4*(35*a^4*b + 168*a^3*b^2
+ 400*a^2*b^3 + 256*a*b^4 + 64*b^5)*cosh(d*x + c)^3 - (105*a^4*b + 364*a^3*b^2 + 248*a^2*b^3 + 64*a*b^4)*cosh(
d*x + c))*sinh(d*x + c))*sqrt(-b/(a + b))*arctan(1/2*(a*cosh(d*x + c)^2 + 2*a*cosh(d*x + c)*sinh(d*x + c) + a*
sinh(d*x + c)^2 + a + 2*b)*sqrt(-b/(a + b))/b) + 8*(10*(a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*d*x*cosh(d*x + c)
^9 - 2*(8*a^5 - 13*a^3*b^2 - 36*a^2*b^3 - 16*a*b^4 - 4*(3*a^5 + 17*a^4*b + 33*a^3*b^2 + 27*a^2*b^3 + 8*a*b^4)*
d*x)*cosh(d*x + c)^7 - 3*(16*a^5 + 32*a^4*b - 13*a^3*b^2 - 43*a^2*b^3 - 68*a*b^4 - 24*b^5 - 4*(a^5 + 7*a^4*b +
23*a^3*b^2 + 37*a^2*b^3 + 28*a*b^4 + 8*b^5)*d*x)*cosh(d*x + c)^5 - 2*(24*a^5 + 64*a^4*b + 64*a^3*b^2 + 15*a^2
*b^3 + 60*a*b^4 + 24*b^5 + 4*(a^5 + 7*a^4*b + 23*a^3*b^2 + 37*a^2*b^3 + 28*a*b^4 + 8*b^5)*d*x)*cosh(d*x + c)^3
- (16*a^5 + 32*a^4*b + 13*a^3*b^2 + 43*a^2*b^3 + 16*a*b^4 + 2*(3*a^5 + 17*a^4*b + 33*a^3*b^2 + 27*a^2*b^3 + 8
*a*b^4)*d*x)*cosh(d*x + c))*sinh(d*x + c))/((a^8 + 3*a^7*b + 3*a^6*b^2 + a^5*b^3)*d*cosh(d*x + c)^10 + 10*(a^8
+ 3*a^7*b + 3*a^6*b^2 + a^5*b^3)*d*cosh(d*x + c)*sinh(d*x + c)^9 + (a^8 + 3*a^7*b + 3*a^6*b^2 + a^5*b^3)*d*si
nh(d*x + c)^10 + (3*a^8 + 17*a^7*b + 33*a^6*b^2 + 27*a^5*b^3 + 8*a^4*b^4)*d*cosh(d*x + c)^8 + (45*(a^8 + 3*a^7
*b + 3*a^6*b^2 + a^5*b^3)*d*cosh(d*x + c)^2 + (3*a^8 + 17*a^7*b + 33*a^6*b^2 + 27*a^5*b^3 + 8*a^4*b^4)*d)*sinh
(d*x + c)^8 + 2*(a^8 + 7*a^7*b + 23*a^6*b^2 + 37*a^5*b^3 + 28*a^4*b^4 + 8*a^3*b^5)*d*cosh(d*x + c)^6 + 8*(15*(
a^8 + 3*a^7*b + 3*a^6*b^2 + a^5*b^3)*d*cosh(d*x + c)^3 + (3*a^8 + 17*a^7*b + 33*a^6*b^2 + 27*a^5*b^3 + 8*a^4*b
^4)*d*cosh(d*x + c))*sinh(d*x + c)^7 + 2*(105*(a^8 + 3*a^7*b + 3*a^6*b^2 + a^5*b^3)*d*cosh(d*x + c)^4 + 14*(3*
a^8 + 17*a^7*b + 33*a^6*b^2 + 27*a^5*b^3 + 8*a^4*b^4)*d*cosh(d*x + c)^2 + (a^8 + 7*a^7*b + 23*a^6*b^2 + 37*a^5
*b^3 + 28*a^4*b^4 + 8*a^3*b^5)*d)*sinh(d*x + c)^6 - 2*(a^8 + 7*a^7*b + 23*a^6*b^2 + 37*a^5*b^3 + 28*a^4*b^4 +
8*a^3*b^5)*d*cosh(d*x + c)^4 + 4*(63*(a^8 + 3*a^7*b + 3*a^6*b^2 + a^5*b^3)*d*cosh(d*x + c)^5 + 14*(3*a^8 + 17*
a^7*b + 33*a^6*b^2 + 27*a^5*b^3 + 8*a^4*b^4)*d*cosh(d*x + c)^3 + 3*(a^8 + 7*a^7*b + 23*a^6*b^2 + 37*a^5*b^3 +
28*a^4*b^4 + 8*a^3*b^5)*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(105*(a^8 + 3*a^7*b + 3*a^6*b^2 + a^5*b^3)*d*cosh
(d*x + c)^6 + 35*(3*a^8 + 17*a^7*b + 33*a^6*b^2 + 27*a^5*b^3 + 8*a^4*b^4)*d*cosh(d*x + c)^4 + 15*(a^8 + 7*a^7*
b + 23*a^6*b^2 + 37*a^5*b^3 + 28*a^4*b^4 + 8*a^3*b^5)*d*cosh(d*x + c)^2 - (a^8 + 7*a^7*b + 23*a^6*b^2 + 37*a^5
*b^3 + 28*a^4*b^4 + 8*a^3*b^5)*d)*sinh(d*x + c)^4 - (3*a^8 + 17*a^7*b + 33*a^6*b^2 + 27*a^5*b^3 + 8*a^4*b^4)*d
*cosh(d*x + c)^2 + 8*(15*(a^8 + 3*a^7*b + 3*a^6*b^2 + a^5*b^3)*d*cosh(d*x + c)^7 + 7*(3*a^8 + 17*a^7*b + 33*a^
6*b^2 + 27*a^5*b^3 + 8*a^4*b^4)*d*cosh(d*x + c)^5 + 5*(a^8 + 7*a^7*b + 23*a^6*b^2 + 37*a^5*b^3 + 28*a^4*b^4 +
8*a^3*b^5)*d*cosh(d*x + c)^3 - (a^8 + 7*a^7*b + 23*a^6*b^2 + 37*a^5*b^3 + 28*a^4*b^4 + 8*a^3*b^5)*d*cosh(d*x +
c))*sinh(d*x + c)^3 + (45*(a^8 + 3*a^7*b + 3*a^6*b^2 + a^5*b^3)*d*cosh(d*x + c)^8 + 28*(3*a^8 + 17*a^7*b + 33
*a^6*b^2 + 27*a^5*b^3 + 8*a^4*b^4)*d*cosh(d*x + c)^6 + 30*(a^8 + 7*a^7*b + 23*a^6*b^2 + 37*a^5*b^3 + 28*a^4*b^
4 + 8*a^3*b^5)*d*cosh(d*x + c)^4 - 12*(a^8 + 7*a^7*b + 23*a^6*b^2 + 37*a^5*b^3 + 28*a^4*b^4 + 8*a^3*b^5)*d*cos
h(d*x + c)^2 - (3*a^8 + 17*a^7*b + 33*a^6*b^2 + 27*a^5*b^3 + 8*a^4*b^4)*d)*sinh(d*x + c)^2 - (a^8 + 3*a^7*b +
3*a^6*b^2 + a^5*b^3)*d + 2*(5*(a^8 + 3*a^7*b + 3*a^6*b^2 + a^5*b^3)*d*cosh(d*x + c)^9 + 4*(3*a^8 + 17*a^7*b +
33*a^6*b^2 + 27*a^5*b^3 + 8*a^4*b^4)*d*cosh(d*x + c)^7 + 6*(a^8 + 7*a^7*b + 23*a^6*b^2 + 37*a^5*b^3 + 28*a^4*b
^4 + 8*a^3*b^5)*d*cosh(d*x + c)^5 - 4*(a^8 + 7*a^7*b + 23*a^6*b^2 + 37*a^5*b^3 + 28*a^4*b^4 + 8*a^3*b^5)*d*cos
h(d*x + c)^3 - (3*a^8 + 17*a^7*b + 33*a^6*b^2 + 27*a^5*b^3 + 8*a^4*b^4)*d*cosh(d*x + c))*sinh(d*x + c))]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(d*x+c)**2/(a+b*sech(d*x+c)**2)**3,x)

[Out]

Timed out

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Giac [B]  time = 2.58015, size = 543, normalized size = 2.98 \begin{align*} -\frac{\frac{{\left (35 \, a^{2} b^{2} e^{\left (2 \, c\right )} + 28 \, a b^{3} e^{\left (2 \, c\right )} + 8 \, b^{4} e^{\left (2 \, c\right )}\right )} \arctan \left (\frac{a e^{\left (2 \, d x + 2 \, c\right )} + a + 2 \, b}{2 \, \sqrt{-a b - b^{2}}}\right ) e^{\left (-2 \, c\right )}}{{\left (a^{6} + 3 \, a^{5} b + 3 \, a^{4} b^{2} + a^{3} b^{3}\right )} \sqrt{-a b - b^{2}}} - \frac{8 \, d x}{a^{3}} - \frac{2 \,{\left (13 \, a^{3} b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 36 \, a^{2} b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 16 \, a b^{4} e^{\left (6 \, d x + 6 \, c\right )} + 39 \, a^{3} b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 122 \, a^{2} b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 152 \, a b^{4} e^{\left (4 \, d x + 4 \, c\right )} + 48 \, b^{5} e^{\left (4 \, d x + 4 \, c\right )} + 39 \, a^{3} b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 92 \, a^{2} b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 32 \, a b^{4} e^{\left (2 \, d x + 2 \, c\right )} + 13 \, a^{3} b^{2} + 6 \, a^{2} b^{3}\right )}}{{\left (a^{6} + 3 \, a^{5} b + 3 \, a^{4} b^{2} + a^{3} b^{3}\right )}{\left (a e^{\left (4 \, d x + 4 \, c\right )} + 2 \, a e^{\left (2 \, d x + 2 \, c\right )} + 4 \, b e^{\left (2 \, d x + 2 \, c\right )} + a\right )}^{2}} + \frac{16}{{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )}{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}}}{8 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(d*x+c)^2/(a+b*sech(d*x+c)^2)^3,x, algorithm="giac")

[Out]

-1/8*((35*a^2*b^2*e^(2*c) + 28*a*b^3*e^(2*c) + 8*b^4*e^(2*c))*arctan(1/2*(a*e^(2*d*x + 2*c) + a + 2*b)/sqrt(-a
*b - b^2))*e^(-2*c)/((a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*sqrt(-a*b - b^2)) - 8*d*x/a^3 - 2*(13*a^3*b^2*e^(6*
d*x + 6*c) + 36*a^2*b^3*e^(6*d*x + 6*c) + 16*a*b^4*e^(6*d*x + 6*c) + 39*a^3*b^2*e^(4*d*x + 4*c) + 122*a^2*b^3*
e^(4*d*x + 4*c) + 152*a*b^4*e^(4*d*x + 4*c) + 48*b^5*e^(4*d*x + 4*c) + 39*a^3*b^2*e^(2*d*x + 2*c) + 92*a^2*b^3
*e^(2*d*x + 2*c) + 32*a*b^4*e^(2*d*x + 2*c) + 13*a^3*b^2 + 6*a^2*b^3)/((a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*(
a*e^(4*d*x + 4*c) + 2*a*e^(2*d*x + 2*c) + 4*b*e^(2*d*x + 2*c) + a)^2) + 16/((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*(e
^(2*d*x + 2*c) - 1)))/d