3.168 \(\int \frac{\coth ^4(c+d x)}{(a+b \text{sech}^2(c+d x))^3} \, dx\)

Optimal. Leaf size=232 \[ -\frac{b^{5/2} \left (63 a^2+36 a b+8 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right )}{8 a^3 d (a+b)^{9/2}}-\frac{\left (8 a^2-39 a b-12 b^2\right ) \coth ^3(c+d x)}{24 a^2 d (a+b)^3}-\frac{\left (32 a^2 b+8 a^3-15 a b^2-4 b^3\right ) \coth (c+d x)}{8 a^2 d (a+b)^4}-\frac{b (11 a+4 b) \coth ^3(c+d x)}{8 a^2 d (a+b)^2 \left (a-b \tanh ^2(c+d x)+b\right )}+\frac{x}{a^3}-\frac{b \coth ^3(c+d x)}{4 a d (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^2} \]

[Out]

x/a^3 - (b^(5/2)*(63*a^2 + 36*a*b + 8*b^2)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(8*a^3*(a + b)^(9/2)*
d) - ((8*a^3 + 32*a^2*b - 15*a*b^2 - 4*b^3)*Coth[c + d*x])/(8*a^2*(a + b)^4*d) - ((8*a^2 - 39*a*b - 12*b^2)*Co
th[c + d*x]^3)/(24*a^2*(a + b)^3*d) - (b*Coth[c + d*x]^3)/(4*a*(a + b)*d*(a + b - b*Tanh[c + d*x]^2)^2) - (b*(
11*a + 4*b)*Coth[c + d*x]^3)/(8*a^2*(a + b)^2*d*(a + b - b*Tanh[c + d*x]^2))

________________________________________________________________________________________

Rubi [A]  time = 0.513631, antiderivative size = 232, normalized size of antiderivative = 1., number of steps used = 9, 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} \[ -\frac{b^{5/2} \left (63 a^2+36 a b+8 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right )}{8 a^3 d (a+b)^{9/2}}-\frac{\left (8 a^2-39 a b-12 b^2\right ) \coth ^3(c+d x)}{24 a^2 d (a+b)^3}-\frac{\left (32 a^2 b+8 a^3-15 a b^2-4 b^3\right ) \coth (c+d x)}{8 a^2 d (a+b)^4}-\frac{b (11 a+4 b) \coth ^3(c+d x)}{8 a^2 d (a+b)^2 \left (a-b \tanh ^2(c+d x)+b\right )}+\frac{x}{a^3}-\frac{b \coth ^3(c+d x)}{4 a d (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[Coth[c + d*x]^4/(a + b*Sech[c + d*x]^2)^3,x]

[Out]

x/a^3 - (b^(5/2)*(63*a^2 + 36*a*b + 8*b^2)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(8*a^3*(a + b)^(9/2)*
d) - ((8*a^3 + 32*a^2*b - 15*a*b^2 - 4*b^3)*Coth[c + d*x])/(8*a^2*(a + b)^4*d) - ((8*a^2 - 39*a*b - 12*b^2)*Co
th[c + d*x]^3)/(24*a^2*(a + b)^3*d) - (b*Coth[c + d*x]^3)/(4*a*(a + b)*d*(a + b - b*Tanh[c + d*x]^2)^2) - (b*(
11*a + 4*b)*Coth[c + d*x]^3)/(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 ^4(c+d x)}{\left (a+b \text{sech}^2(c+d x)\right )^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x^4 \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^4 \left (1-x^2\right ) \left (a+b-b x^2\right )^3} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=-\frac{b \coth ^3(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+3 b-7 b x^2}{x^4 \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 ^3(c+d x)}{4 a (a+b) d \left (a+b-b \tanh ^2(c+d x)\right )^2}-\frac{b (11 a+4 b) \coth ^3(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^2-39 a b-12 b^2+5 b (11 a+4 b) x^2}{x^4 \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^2-39 a b-12 b^2\right ) \coth ^3(c+d x)}{24 a^2 (a+b)^3 d}-\frac{b \coth ^3(c+d x)}{4 a (a+b) d \left (a+b-b \tanh ^2(c+d x)\right )^2}-\frac{b (11 a+4 b) \coth ^3(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{-3 \left (8 a^3+32 a^2 b-15 a b^2-4 b^3\right )+3 b \left (8 a^2-39 a b-12 b^2\right ) x^2}{x^2 \left (1-x^2\right ) \left (a+b-b x^2\right )} \, dx,x,\tanh (c+d x)\right )}{24 a^2 (a+b)^3 d}\\ &=-\frac{\left (8 a^3+32 a^2 b-15 a b^2-4 b^3\right ) \coth (c+d x)}{8 a^2 (a+b)^4 d}-\frac{\left (8 a^2-39 a b-12 b^2\right ) \coth ^3(c+d x)}{24 a^2 (a+b)^3 d}-\frac{b \coth ^3(c+d x)}{4 a (a+b) d \left (a+b-b \tanh ^2(c+d x)\right )^2}-\frac{b (11 a+4 b) \coth ^3(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{3 \left (8 a^4+40 a^3 b+80 a^2 b^2+17 a b^3+4 b^4\right )-3 b \left (8 a^3+32 a^2 b-15 a b^2-4 b^3\right ) x^2}{\left (1-x^2\right ) \left (a+b-b x^2\right )} \, dx,x,\tanh (c+d x)\right )}{24 a^2 (a+b)^4 d}\\ &=-\frac{\left (8 a^3+32 a^2 b-15 a b^2-4 b^3\right ) \coth (c+d x)}{8 a^2 (a+b)^4 d}-\frac{\left (8 a^2-39 a b-12 b^2\right ) \coth ^3(c+d x)}{24 a^2 (a+b)^3 d}-\frac{b \coth ^3(c+d x)}{4 a (a+b) d \left (a+b-b \tanh ^2(c+d x)\right )^2}-\frac{b (11 a+4 b) \coth ^3(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^3 \left (63 a^2+36 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)^4 d}\\ &=\frac{x}{a^3}-\frac{b^{5/2} \left (63 a^2+36 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)^{9/2} d}-\frac{\left (8 a^3+32 a^2 b-15 a b^2-4 b^3\right ) \coth (c+d x)}{8 a^2 (a+b)^4 d}-\frac{\left (8 a^2-39 a b-12 b^2\right ) \coth ^3(c+d x)}{24 a^2 (a+b)^3 d}-\frac{b \coth ^3(c+d x)}{4 a (a+b) d \left (a+b-b \tanh ^2(c+d x)\right )^2}-\frac{b (11 a+4 b) \coth ^3(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 = 7.52759, size = 3334, normalized size = 14.37 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Coth[c + d*x]^4/(a + b*Sech[c + d*x]^2)^3,x]

[Out]

((63*a^2 + 36*a*b + 8*b^2)*(a + 2*b + a*Cosh[2*c + 2*d*x])^3*Sech[c + d*x]^6*(((I/64)*b^3*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^3*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)^4*(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]^3*Sech[2*c]*Sech[c + d*x]^6*(-36*a^
6*d*x*Cosh[d*x] - 336*a^5*b*d*x*Cosh[d*x] - 1560*a^4*b^2*d*x*Cosh[d*x] - 3600*a^3*b^3*d*x*Cosh[d*x] - 4260*a^2
*b^4*d*x*Cosh[d*x] - 2496*a*b^5*d*x*Cosh[d*x] - 576*b^6*d*x*Cosh[d*x] + 36*a^6*d*x*Cosh[3*d*x] + 240*a^5*b*d*x
*Cosh[3*d*x] + 408*a^4*b^2*d*x*Cosh[3*d*x] - 48*a^3*b^3*d*x*Cosh[3*d*x] - 732*a^2*b^4*d*x*Cosh[3*d*x] - 672*a*
b^5*d*x*Cosh[3*d*x] - 192*b^6*d*x*Cosh[3*d*x] + 36*a^6*d*x*Cosh[2*c - d*x] + 336*a^5*b*d*x*Cosh[2*c - d*x] + 1
560*a^4*b^2*d*x*Cosh[2*c - d*x] + 3600*a^3*b^3*d*x*Cosh[2*c - d*x] + 4260*a^2*b^4*d*x*Cosh[2*c - d*x] + 2496*a
*b^5*d*x*Cosh[2*c - d*x] + 576*b^6*d*x*Cosh[2*c - d*x] + 36*a^6*d*x*Cosh[2*c + d*x] + 336*a^5*b*d*x*Cosh[2*c +
d*x] + 1560*a^4*b^2*d*x*Cosh[2*c + d*x] + 3600*a^3*b^3*d*x*Cosh[2*c + d*x] + 4260*a^2*b^4*d*x*Cosh[2*c + d*x]
+ 2496*a*b^5*d*x*Cosh[2*c + d*x] + 576*b^6*d*x*Cosh[2*c + d*x] - 36*a^6*d*x*Cosh[4*c + d*x] - 336*a^5*b*d*x*C
osh[4*c + d*x] - 1560*a^4*b^2*d*x*Cosh[4*c + d*x] - 3600*a^3*b^3*d*x*Cosh[4*c + d*x] - 4260*a^2*b^4*d*x*Cosh[4
*c + d*x] - 2496*a*b^5*d*x*Cosh[4*c + d*x] - 576*b^6*d*x*Cosh[4*c + d*x] - 36*a^6*d*x*Cosh[2*c + 3*d*x] - 240*
a^5*b*d*x*Cosh[2*c + 3*d*x] - 408*a^4*b^2*d*x*Cosh[2*c + 3*d*x] + 48*a^3*b^3*d*x*Cosh[2*c + 3*d*x] + 732*a^2*b
^4*d*x*Cosh[2*c + 3*d*x] + 672*a*b^5*d*x*Cosh[2*c + 3*d*x] + 192*b^6*d*x*Cosh[2*c + 3*d*x] + 36*a^6*d*x*Cosh[4
*c + 3*d*x] + 240*a^5*b*d*x*Cosh[4*c + 3*d*x] + 408*a^4*b^2*d*x*Cosh[4*c + 3*d*x] - 48*a^3*b^3*d*x*Cosh[4*c +
3*d*x] - 732*a^2*b^4*d*x*Cosh[4*c + 3*d*x] - 672*a*b^5*d*x*Cosh[4*c + 3*d*x] - 192*b^6*d*x*Cosh[4*c + 3*d*x] -
36*a^6*d*x*Cosh[6*c + 3*d*x] - 240*a^5*b*d*x*Cosh[6*c + 3*d*x] - 408*a^4*b^2*d*x*Cosh[6*c + 3*d*x] + 48*a^3*b
^3*d*x*Cosh[6*c + 3*d*x] + 732*a^2*b^4*d*x*Cosh[6*c + 3*d*x] + 672*a*b^5*d*x*Cosh[6*c + 3*d*x] + 192*b^6*d*x*C
osh[6*c + 3*d*x] - 12*a^6*d*x*Cosh[2*c + 5*d*x] - 144*a^5*b*d*x*Cosh[2*c + 5*d*x] - 456*a^4*b^2*d*x*Cosh[2*c +
5*d*x] - 624*a^3*b^3*d*x*Cosh[2*c + 5*d*x] - 396*a^2*b^4*d*x*Cosh[2*c + 5*d*x] - 96*a*b^5*d*x*Cosh[2*c + 5*d*
x] + 12*a^6*d*x*Cosh[4*c + 5*d*x] + 144*a^5*b*d*x*Cosh[4*c + 5*d*x] + 456*a^4*b^2*d*x*Cosh[4*c + 5*d*x] + 624*
a^3*b^3*d*x*Cosh[4*c + 5*d*x] + 396*a^2*b^4*d*x*Cosh[4*c + 5*d*x] + 96*a*b^5*d*x*Cosh[4*c + 5*d*x] - 12*a^6*d*
x*Cosh[6*c + 5*d*x] - 144*a^5*b*d*x*Cosh[6*c + 5*d*x] - 456*a^4*b^2*d*x*Cosh[6*c + 5*d*x] - 624*a^3*b^3*d*x*Co
sh[6*c + 5*d*x] - 396*a^2*b^4*d*x*Cosh[6*c + 5*d*x] - 96*a*b^5*d*x*Cosh[6*c + 5*d*x] + 12*a^6*d*x*Cosh[8*c + 5
*d*x] + 144*a^5*b*d*x*Cosh[8*c + 5*d*x] + 456*a^4*b^2*d*x*Cosh[8*c + 5*d*x] + 624*a^3*b^3*d*x*Cosh[8*c + 5*d*x
] + 396*a^2*b^4*d*x*Cosh[8*c + 5*d*x] + 96*a*b^5*d*x*Cosh[8*c + 5*d*x] - 12*a^6*d*x*Cosh[4*c + 7*d*x] - 48*a^5
*b*d*x*Cosh[4*c + 7*d*x] - 72*a^4*b^2*d*x*Cosh[4*c + 7*d*x] - 48*a^3*b^3*d*x*Cosh[4*c + 7*d*x] - 12*a^2*b^4*d*
x*Cosh[4*c + 7*d*x] + 12*a^6*d*x*Cosh[6*c + 7*d*x] + 48*a^5*b*d*x*Cosh[6*c + 7*d*x] + 72*a^4*b^2*d*x*Cosh[6*c
+ 7*d*x] + 48*a^3*b^3*d*x*Cosh[6*c + 7*d*x] + 12*a^2*b^4*d*x*Cosh[6*c + 7*d*x] - 12*a^6*d*x*Cosh[8*c + 7*d*x]
- 48*a^5*b*d*x*Cosh[8*c + 7*d*x] - 72*a^4*b^2*d*x*Cosh[8*c + 7*d*x] - 48*a^3*b^3*d*x*Cosh[8*c + 7*d*x] - 12*a^
2*b^4*d*x*Cosh[8*c + 7*d*x] + 12*a^6*d*x*Cosh[10*c + 7*d*x] + 48*a^5*b*d*x*Cosh[10*c + 7*d*x] + 72*a^4*b^2*d*x
*Cosh[10*c + 7*d*x] + 48*a^3*b^3*d*x*Cosh[10*c + 7*d*x] + 12*a^2*b^4*d*x*Cosh[10*c + 7*d*x] - 128*a^6*Sinh[d*x
] - 440*a^5*b*Sinh[d*x] - 1152*a^4*b^2*Sinh[d*x] - 1920*a^3*b^3*Sinh[d*x] + 228*a^2*b^4*Sinh[d*x] + 1320*a*b^5
*Sinh[d*x] + 432*b^6*Sinh[d*x] + 48*a^6*Sinh[3*d*x] + 104*a^5*b*Sinh[3*d*x] + 640*a^4*b^2*Sinh[3*d*x] + 1511*a
^3*b^3*Sinh[3*d*x] - 528*a^2*b^4*Sinh[3*d*x] + 264*a*b^5*Sinh[3*d*x] + 144*b^6*Sinh[3*d*x] - 32*a^6*Sinh[2*c -
d*x] + 384*a^5*b*Sinh[2*c - d*x] + 2048*a^4*b^2*Sinh[2*c - d*x] + 3072*a^3*b^3*Sinh[2*c - d*x] + 228*a^2*b^4*
Sinh[2*c - d*x] + 1320*a*b^5*Sinh[2*c - d*x] + 432*b^6*Sinh[2*c - d*x] + 32*a^6*Sinh[2*c + d*x] - 384*a^5*b*Si
nh[2*c + d*x] - 2048*a^4*b^2*Sinh[2*c + d*x] - 2919*a^3*b^3*Sinh[2*c + d*x] + 642*a^2*b^4*Sinh[2*c + d*x] + 14
16*a*b^5*Sinh[2*c + d*x] + 432*b^6*Sinh[2*c + d*x] - 128*a^6*Sinh[4*c + d*x] - 440*a^5*b*Sinh[4*c + d*x] - 115
2*a^4*b^2*Sinh[4*c + d*x] - 2073*a^3*b^3*Sinh[4*c + d*x] - 642*a^2*b^4*Sinh[4*c + d*x] - 1416*a*b^5*Sinh[4*c +
d*x] - 432*b^6*Sinh[4*c + d*x] - 144*a^6*Sinh[2*c + 3*d*x] - 672*a^5*b*Sinh[2*c + 3*d*x] - 960*a^4*b^2*Sinh[2
*c + 3*d*x] + 153*a^3*b^3*Sinh[2*c + 3*d*x] + 528*a^2*b^4*Sinh[2*c + 3*d*x] - 264*a*b^5*Sinh[2*c + 3*d*x] - 14
4*b^6*Sinh[2*c + 3*d*x] + 48*a^6*Sinh[4*c + 3*d*x] + 104*a^5*b*Sinh[4*c + 3*d*x] + 640*a^4*b^2*Sinh[4*c + 3*d*
x] + 1664*a^3*b^3*Sinh[4*c + 3*d*x] - 66*a^2*b^4*Sinh[4*c + 3*d*x] - 408*a*b^5*Sinh[4*c + 3*d*x] - 144*b^6*Sin
h[4*c + 3*d*x] - 144*a^6*Sinh[6*c + 3*d*x] - 672*a^5*b*Sinh[6*c + 3*d*x] - 960*a^4*b^2*Sinh[6*c + 3*d*x] + 66*
a^2*b^4*Sinh[6*c + 3*d*x] + 408*a*b^5*Sinh[6*c + 3*d*x] + 144*b^6*Sinh[6*c + 3*d*x] + 80*a^6*Sinh[2*c + 5*d*x]
+ 480*a^5*b*Sinh[2*c + 5*d*x] + 832*a^4*b^2*Sinh[2*c + 5*d*x] + 294*a^2*b^4*Sinh[2*c + 5*d*x] + 96*a*b^5*Sinh
[2*c + 5*d*x] - 48*a^6*Sinh[4*c + 5*d*x] - 120*a^5*b*Sinh[4*c + 5*d*x] - 294*a^2*b^4*Sinh[4*c + 5*d*x] - 96*a*
b^5*Sinh[4*c + 5*d*x] + 80*a^6*Sinh[6*c + 5*d*x] + 480*a^5*b*Sinh[6*c + 5*d*x] + 832*a^4*b^2*Sinh[6*c + 5*d*x]
- 51*a^3*b^3*Sinh[6*c + 5*d*x] - 132*a^2*b^4*Sinh[6*c + 5*d*x] - 48*a*b^5*Sinh[6*c + 5*d*x] - 48*a^6*Sinh[8*c
+ 5*d*x] - 120*a^5*b*Sinh[8*c + 5*d*x] + 51*a^3*b^3*Sinh[8*c + 5*d*x] + 132*a^2*b^4*Sinh[8*c + 5*d*x] + 48*a*
b^5*Sinh[8*c + 5*d*x] + 32*a^6*Sinh[4*c + 7*d*x] + 104*a^5*b*Sinh[4*c + 7*d*x] + 51*a^3*b^3*Sinh[4*c + 7*d*x]
+ 18*a^2*b^4*Sinh[4*c + 7*d*x] - 51*a^3*b^3*Sinh[6*c + 7*d*x] - 18*a^2*b^4*Sinh[6*c + 7*d*x] + 32*a^6*Sinh[8*c
+ 7*d*x] + 104*a^5*b*Sinh[8*c + 7*d*x]))/(6144*a^3*(a + b)^4*d*(a + b*Sech[c + d*x]^2)^3)

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Maple [B]  time = 0.134, size = 1610, normalized size = 6.9 \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)^4/(a+b*sech(d*x+c)^2)^3,x)

[Out]

1/2/d*b^(9/2)/(a+b)^(9/2)/a^3*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^(9/2)/(a+b)^(9/2)/a^3*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))-
63/16/d*b^(5/2)/(a+b)^(9/2)/a*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))+
9/4/d*b^(7/2)/(a+b)^(9/2)/a^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))+
63/16/d*b^(5/2)/(a+b)^(9/2)/a*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))-
5/8/d/(a+b)^4/tanh(1/2*d*x+1/2*c)*a-17/8/d/(a+b)^4/tanh(1/2*d*x+1/2*c)*b-9/4/d*b^(7/2)/(a+b)^(9/2)/a^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))-1/24/d/(a+b)^3/tanh(1/2*d*x+1/2*c)^3-
1/24/d/(a+b)/(a^3+3*a^2*b+3*a*b^2+b^3)*a*tanh(1/2*d*x+1/2*c)^3-1/d/a^3*ln(tanh(1/2*d*x+1/2*c)-1)+1/d/a^3*ln(ta
nh(1/2*d*x+1/2*c)+1)-1/24/d/(a+b)/(a^3+3*a^2*b+3*a*b^2+b^3)*b*tanh(1/2*d*x+1/2*c)^3-5/8/d/(a+b)/(a^3+3*a^2*b+3
*a*b^2+b^3)*a*tanh(1/2*d*x+1/2*c)-17/8/d/(a+b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tanh(1/2*d*x+1/2*c)*b-17/4/d*b^3/(a+b
)^4/(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)^7-51/4/d*b^3/(a+b)^4/(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)^5-51/4/d*b^3/(a+b)^4/(tanh(1/2*d*x+1/2*c)^4*a+b*t
anh(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)^3-17/4/d*b
^3/(a+b)^4/(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)-21/4/d*b^4/(a+b)^4/a/(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^5/(a+b)^4/a^2/(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-
3/4/d*b^4/(a+b)^4/a/(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^5/(a+b)^4/a^2/(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)^5-3/4/d*b^4/(a+b)^4/a/(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^5/(a+b)^4/a^2/(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*tan
h(1/2*d*x+1/2*c)^2*b+a+b)^2*tanh(1/2*d*x+1/2*c)^3-21/4/d*b^4/(a+b)^4/a/(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^5/(a+b)^4/a^2/(
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*tan
h(1/2*d*x+1/2*c)

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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)^4/(a+b*sech(d*x+c)^2)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [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)^4/(a+b*sech(d*x+c)^2)^3,x, algorithm="fricas")

[Out]

Timed out

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

[Out]

Timed out

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Giac [B]  time = 3.9256, size = 651, normalized size = 2.81 \begin{align*} -\frac{\frac{3 \,{\left (63 \, a^{2} b^{3} e^{\left (2 \, c\right )} + 36 \, a b^{4} e^{\left (2 \, c\right )} + 8 \, b^{5} 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^{7} + 4 \, a^{6} b + 6 \, a^{5} b^{2} + 4 \, a^{4} b^{3} + a^{3} b^{4}\right )} \sqrt{-a b - b^{2}}} - \frac{24 \, d x}{a^{3}} - \frac{6 \,{\left (17 \, a^{3} b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 44 \, a^{2} b^{4} e^{\left (6 \, d x + 6 \, c\right )} + 16 \, a b^{5} e^{\left (6 \, d x + 6 \, c\right )} + 51 \, a^{3} b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 154 \, a^{2} b^{4} e^{\left (4 \, d x + 4 \, c\right )} + 184 \, a b^{5} e^{\left (4 \, d x + 4 \, c\right )} + 48 \, b^{6} e^{\left (4 \, d x + 4 \, c\right )} + 51 \, a^{3} b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 116 \, a^{2} b^{4} e^{\left (2 \, d x + 2 \, c\right )} + 32 \, a b^{5} e^{\left (2 \, d x + 2 \, c\right )} + 17 \, a^{3} b^{3} + 6 \, a^{2} b^{4}\right )}}{{\left (a^{7} + 4 \, a^{6} b + 6 \, a^{5} b^{2} + 4 \, a^{4} b^{3} + a^{3} b^{4}\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 (6 \, a e^{\left (4 \, d x + 4 \, c\right )} + 15 \, b e^{\left (4 \, d x + 4 \, c\right )} - 6 \, a e^{\left (2 \, d x + 2 \, c\right )} - 24 \, b e^{\left (2 \, d x + 2 \, c\right )} + 4 \, a + 13 \, b\right )}}{{\left (a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )}{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{3}}}{24 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(d*x+c)^4/(a+b*sech(d*x+c)^2)^3,x, algorithm="giac")

[Out]

-1/24*(3*(63*a^2*b^3*e^(2*c) + 36*a*b^4*e^(2*c) + 8*b^5*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^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*sqrt(-a*b - b^2)) - 24*d*x/a^3 - 6*(
17*a^3*b^3*e^(6*d*x + 6*c) + 44*a^2*b^4*e^(6*d*x + 6*c) + 16*a*b^5*e^(6*d*x + 6*c) + 51*a^3*b^3*e^(4*d*x + 4*c
) + 154*a^2*b^4*e^(4*d*x + 4*c) + 184*a*b^5*e^(4*d*x + 4*c) + 48*b^6*e^(4*d*x + 4*c) + 51*a^3*b^3*e^(2*d*x + 2
*c) + 116*a^2*b^4*e^(2*d*x + 2*c) + 32*a*b^5*e^(2*d*x + 2*c) + 17*a^3*b^3 + 6*a^2*b^4)/((a^7 + 4*a^6*b + 6*a^5
*b^2 + 4*a^4*b^3 + a^3*b^4)*(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*(6*a*e
^(4*d*x + 4*c) + 15*b*e^(4*d*x + 4*c) - 6*a*e^(2*d*x + 2*c) - 24*b*e^(2*d*x + 2*c) + 4*a + 13*b)/((a^4 + 4*a^3
*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*(e^(2*d*x + 2*c) - 1)^3))/d