3.2.66 \(\int \frac {\coth ^2(c+d x)}{(a+b \text {sech}^2(c+d x))^3} \, dx\) [166]

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

[Out]

x/a^3-1/8*b^(3/2)*(35*a^2+28*a*b+8*b^2)*arctanh(b^(1/2)*tanh(d*x+c)/(a+b)^(1/2))/a^3/(a+b)^(7/2)/d-1/8*(8*a^2-
11*a*b-4*b^2)*coth(d*x+c)/a^2/(a+b)^3/d-1/4*b*coth(d*x+c)/a/(a+b)/d/(a+b-b*tanh(d*x+c)^2)^2-1/8*b*(9*a+4*b)*co
th(d*x+c)/a^2/(a+b)^2/d/(a+b-b*tanh(d*x+c)^2)

________________________________________________________________________________________

Rubi [A]
time = 0.28, antiderivative size = 182, normalized size of antiderivative = 1.00, 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 = {4226, 2000, 483, 593, 597, 536, 212, 214} \begin {gather*} \frac {x}{a^3}-\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 {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 {b \coth (c+d x)}{4 a d (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^2} \end {gather*}

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 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 214

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

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 536

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 593

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 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 2000

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 4226

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])

Rubi steps

\begin {align*} \int \frac {\coth ^2(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx &=\frac {\text {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 {\text {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 {\text {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 {\text {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 {\text {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 {\text {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 ) \text {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 [B] Leaf count is larger than twice the leaf count of optimal. \(1769\) vs. \(2(182)=364\).
time = 6.07, size = 1769, normalized size = 9.72 \begin {gather*} \frac {(a+2 b+a \cosh (2 (c+d x))) \text {sech}^6(c+d x) \left (-\frac {8 b^2 \left (35 a^2+28 a b+8 b^2\right ) \tanh ^{-1}\left (\frac {\text {sech}(d x) (\cosh (2 c)-\sinh (2 c)) ((a+2 b) \sinh (d x)-a \sinh (2 c+d x))}{2 \sqrt {a+b} \sqrt {b (\cosh (c)-\sinh (c))^4}}\right ) (a+2 b+a \cosh (2 (c+d x)))^2 (\cosh (2 c)-\sinh (2 c))}{\sqrt {a+b} \sqrt {b (\cosh (c)-\sinh (c))^4}}+\text {csch}(c) \text {csch}(c+d x) \text {sech}(2 c) \left (8 (a+b)^3 \left (a^2+4 a b+8 b^2\right ) d x \cosh (d x)-4 a (a+b)^3 (3 a+8 b) 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 \cosh (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 \sinh (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 \sinh (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 \sinh (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)\right )\right )}{512 a^3 (a+b)^3 d \left (a+b \text {sech}^2(c+d x)\right )^3} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]^6*((-8*b^2*(35*a^2 + 28*a*b + 8*b^2)*ArcTanh[(Sech[d*x]*(Cosh[2
*c] - Sinh[2*c])*((a + 2*b)*Sinh[d*x] - a*Sinh[2*c + d*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cosh[c] - Sinh[c])^4])]*(a
+ 2*b + a*Cosh[2*(c + d*x)])^2*(Cosh[2*c] - Sinh[2*c]))/(Sqrt[a + b]*Sqrt[b*(Cosh[c] - Sinh[c])^4]) + Csch[c]*
Csch[c + d*x]*Sech[2*c]*(8*(a + b)^3*(a^2 + 4*a*b + 8*b^2)*d*x*Cosh[d*x] - 4*a*(a + b)^3*(3*a + 8*b)*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*Cosh[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] + 1
2*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*Sinh[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*Sinh[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*Si
nh[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*S
inh[2*c + 3*d*x] + 32*a^5*Sinh[4*c + 3*d*x] + 64*a^4*b*Sinh[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*Sin
h[6*c + 5*d*x])))/(512*a^3*(a + b)^3*d*(a + b*Sech[c + d*x]^2)^3)

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(393\) vs. \(2(166)=332\).
time = 3.10, size = 394, normalized size = 2.16 Too large to display

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,method=_RETURNVERBOSE)

[Out]

1/d*(-1/2/(a^3+3*a^2*b+3*a*b^2+b^3)*tanh(1/2*d*x+1/2*c)+2*b^2/(a+b)^3/a^3*(((-13/8*a^3-17/8*a^2*b-1/2*a*b^2)*t
anh(1/2*d*x+1/2*c)^7-1/8*(39*a^2+7*a*b-4*b^2)*a*tanh(1/2*d*x+1/2*c)^5-1/8*(39*a^2+7*a*b-4*b^2)*a*tanh(1/2*d*x+
1/2*c)^3+(-13/8*a^3-17/8*a^2*b-1/2*a*b^2)*tanh(1/2*d*x+1/2*c))/(a*tanh(1/2*d*x+1/2*c)^4+b*tanh(1/2*d*x+1/2*c)^
4+2*a*tanh(1/2*d*x+1/2*c)^2-2*b*tanh(1/2*d*x+1/2*c)^2+a+b)^2+1/8*(35*a^2+28*a*b+8*b^2)*(-1/4/b^(1/2)/(a+b)^(1/
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))+1/4/b^(1/2)/(a+b)^(1/2)*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/a^3*ln(tanh(1/2*d*x+1/2*c)-1)
-1/2/(a+b)^3/tanh(1/2*d*x+1/2*c)+1/a^3*ln(tanh(1/2*d*x+1/2*c)+1))

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1971 vs. \(2 (172) = 344\).
time = 0.68, size = 1971, normalized size = 10.83 \begin {gather*} \text {Too large to display} \end {gather*}

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]

1/4*(3*a^2*b + 3*a*b^2 + b^3)*log(a*e^(4*d*x + 4*c) + 2*(a + 2*b)*e^(2*d*x + 2*c) + a)/((a^6 + 3*a^5*b + 3*a^4
*b^2 + a^3*b^3)*d) - 1/4*(3*a^2*b + 3*a*b^2 + b^3)*log(2*(a + 2*b)*e^(-2*d*x - 2*c) + a*e^(-4*d*x - 4*c) + a)/
((a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*d) - 1/64*(15*a^3*b + 70*a^2*b^2 + 56*a*b^3 + 16*b^4)*log((a*e^(2*d*x +
2*c) + a + 2*b - 2*sqrt((a + b)*b))/(a*e^(2*d*x + 2*c) + a + 2*b + 2*sqrt((a + b)*b)))/((a^6 + 3*a^5*b + 3*a^
4*b^2 + a^3*b^3)*sqrt((a + b)*b)*d) + 1/64*(15*a^3*b + 70*a^2*b^2 + 56*a*b^3 + 16*b^4)*log((a*e^(-2*d*x - 2*c)
+ a + 2*b - 2*sqrt((a + b)*b))/(a*e^(-2*d*x - 2*c) + a + 2*b + 2*sqrt((a + b)*b)))/((a^6 + 3*a^5*b + 3*a^4*b^
2 + a^3*b^3)*sqrt((a + b)*b)*d) - 15/32*b*log((a*e^(-2*d*x - 2*c) + a + 2*b - 2*sqrt((a + b)*b))/(a*e^(-2*d*x
- 2*c) + a + 2*b + 2*sqrt((a + b)*b)))/((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*sqrt((a + b)*b)*d) + 1/16*(8*a^5 + 9*a
^4*b + 28*a^3*b^2 + 12*a^2*b^3 + (8*a^5 - 9*a^4*b - 98*a^3*b^2 - 160*a^2*b^3 - 64*a*b^4)*e^(8*d*x + 8*c) + 2*(
16*a^5 + 23*a^4*b - 77*a^3*b^2 - 246*a^2*b^3 - 288*a*b^4 - 96*b^5)*e^(6*d*x + 6*c) + 2*(24*a^5 + 64*a^4*b + 99
*a^3*b^2 + 190*a^2*b^3 + 272*a*b^4 + 96*b^5)*e^(4*d*x + 4*c) + 2*(16*a^5 + 41*a^4*b + 77*a^3*b^2 + 130*a^2*b^3
+ 48*a*b^4)*e^(2*d*x + 2*c))/((a^8 + 3*a^7*b + 3*a^6*b^2 + a^5*b^3 - (a^8 + 3*a^7*b + 3*a^6*b^2 + a^5*b^3)*e^
(10*d*x + 10*c) - (3*a^8 + 17*a^7*b + 33*a^6*b^2 + 27*a^5*b^3 + 8*a^4*b^4)*e^(8*d*x + 8*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)*e^(6*d*x + 6*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)*e^(4*d*x + 4*c) + (3*a^8 + 17*a^7*b + 33*a^6*b^2 + 27*a^5*b^3 + 8*a^4*b^4)*e^(2*d
*x + 2*c))*d) - 1/16*(8*a^5 + 9*a^4*b + 28*a^3*b^2 + 12*a^2*b^3 + 2*(16*a^5 + 41*a^4*b + 77*a^3*b^2 + 130*a^2*
b^3 + 48*a*b^4)*e^(-2*d*x - 2*c) + 2*(24*a^5 + 64*a^4*b + 99*a^3*b^2 + 190*a^2*b^3 + 272*a*b^4 + 96*b^5)*e^(-4
*d*x - 4*c) + 2*(16*a^5 + 23*a^4*b - 77*a^3*b^2 - 246*a^2*b^3 - 288*a*b^4 - 96*b^5)*e^(-6*d*x - 6*c) + (8*a^5
- 9*a^4*b - 98*a^3*b^2 - 160*a^2*b^3 - 64*a*b^4)*e^(-8*d*x - 8*c))/((a^8 + 3*a^7*b + 3*a^6*b^2 + a^5*b^3 + (3*
a^8 + 17*a^7*b + 33*a^6*b^2 + 27*a^5*b^3 + 8*a^4*b^4)*e^(-2*d*x - 2*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)*e^(-4*d*x - 4*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)*e^(-6*d*x - 6*c) - (3*a^8 + 17*a^7*b + 33*a^6*b^2 + 27*a^5*b^3 + 8*a^4*b^4)*e^(-8*d*x - 8*c) - (a^8
+ 3*a^7*b + 3*a^6*b^2 + a^5*b^3)*e^(-10*d*x - 10*c))*d) - 1/8*(8*a^4 - 9*a^3*b - 2*a^2*b^2 + 2*(16*a^4 + 23*a^
3*b - 27*a^2*b^2 - 4*a*b^3)*e^(-2*d*x - 2*c) + 2*(24*a^4 + 64*a^3*b + 53*a^2*b^2 - 40*a*b^3 - 8*b^4)*e^(-4*d*x
- 4*c) + 2*(16*a^4 + 41*a^3*b + 27*a^2*b^2 + 40*a*b^3 + 8*b^4)*e^(-6*d*x - 6*c) + (8*a^4 + 9*a^3*b + 24*a^2*b
^2 + 8*a*b^3)*e^(-8*d*x - 8*c))/((a^7 + 3*a^6*b + 3*a^5*b^2 + a^4*b^3 + (3*a^7 + 17*a^6*b + 33*a^5*b^2 + 27*a^
4*b^3 + 8*a^3*b^4)*e^(-2*d*x - 2*c) + 2*(a^7 + 7*a^6*b + 23*a^5*b^2 + 37*a^4*b^3 + 28*a^3*b^4 + 8*a^2*b^5)*e^(
-4*d*x - 4*c) - 2*(a^7 + 7*a^6*b + 23*a^5*b^2 + 37*a^4*b^3 + 28*a^3*b^4 + 8*a^2*b^5)*e^(-6*d*x - 6*c) - (3*a^7
+ 17*a^6*b + 33*a^5*b^2 + 27*a^4*b^3 + 8*a^3*b^4)*e^(-8*d*x - 8*c) - (a^7 + 3*a^6*b + 3*a^5*b^2 + a^4*b^3)*e^
(-10*d*x - 10*c))*d) + 1/2*log(e^(2*d*x + 2*c) - 1)/((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d) - 1/2*log(e^(-2*d*x -
2*c) - 1)/((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 5665 vs. \(2 (172) = 344\).
time = 0.52, size = 11606, normalized size = 63.77 \begin {gather*} \text {Too large to display} \end {gather*}

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...

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\coth ^{2}{\left (c + d x \right )}}{\left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right )^{3}}\, dx \end {gather*}

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]

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

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 389 vs. \(2 (172) = 344\).
time = 1.76, size = 389, normalized size = 2.14 \begin {gather*} -\frac {\frac {{\left (35 \, a^{2} b^{2} + 28 \, a b^{3} + 8 \, b^{4}\right )} \arctan \left (\frac {a e^{\left (2 \, d x + 2 \, c\right )} + a + 2 \, b}{2 \, \sqrt {-a b - b^{2}}}\right )}{{\left (a^{6} + 3 \, a^{5} b + 3 \, a^{4} b^{2} + a^{3} b^{3}\right )} \sqrt {-a b - b^{2}}} - \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 {8 \, {\left (d x + c\right )}}{a^{3}} + \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 {gather*}

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 + 28*a*b^3 + 8*b^4)*arctan(1/2*(a*e^(2*d*x + 2*c) + a + 2*b)/sqrt(-a*b - b^2))/((a^6 + 3*a^5
*b + 3*a^4*b^2 + a^3*b^3)*sqrt(-a*b - b^2)) - 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) - 8*(d*x + c)/a^3 + 16/((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*(e^(2*d*x + 2*c) - 1)))/d

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^6\,{\mathrm {coth}\left (c+d\,x\right )}^2}{{\left (a\,{\mathrm {cosh}\left (c+d\,x\right )}^2+b\right )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(c + d*x)^2/(a + b/cosh(c + d*x)^2)^3,x)

[Out]

int((cosh(c + d*x)^6*coth(c + d*x)^2)/(b + a*cosh(c + d*x)^2)^3, x)

________________________________________________________________________________________