\(\int \frac {\coth ^4(c+d x)}{(a+b \tanh ^2(c+d x))^3} \, dx\) [200]

Optimal result

Integrand size = 23, antiderivative size = 228 \[ \int \frac {\coth ^4(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\frac {x}{(a+b)^3}+\frac {b^{5/2} \left (63 a^2+90 a b+35 b^2\right ) \arctan \left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{8 a^{9/2} (a+b)^3 d}-\frac {\left (8 a^3-8 a^2 b-55 a b^2-35 b^3\right ) \coth (c+d x)}{8 a^4 (a+b)^2 d}-\frac {\left (8 a^2+55 a b+35 b^2\right ) \coth ^3(c+d x)}{24 a^3 (a+b)^2 d}+\frac {b \coth ^3(c+d x)}{4 a (a+b) d \left (a+b \tanh ^2(c+d x)\right )^2}+\frac {b (11 a+7 b) \coth ^3(c+d x)}{8 a^2 (a+b)^2 d \left (a+b \tanh ^2(c+d x)\right )} \]

[Out]

x/(a+b)^3+1/8*b^(5/2)*(63*a^2+90*a*b+35*b^2)*arctan(b^(1/2)*tanh(d*x+c)/a^(1/2))/a^(9/2)/(a+b)^3/d-1/8*(8*a^3-
8*a^2*b-55*a*b^2-35*b^3)*coth(d*x+c)/a^4/(a+b)^2/d-1/24*(8*a^2+55*a*b+35*b^2)*coth(d*x+c)^3/a^3/(a+b)^2/d+1/4*
b*coth(d*x+c)^3/a/(a+b)/d/(a+b*tanh(d*x+c)^2)^2+1/8*b*(11*a+7*b)*coth(d*x+c)^3/a^2/(a+b)^2/d/(a+b*tanh(d*x+c)^
2)

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {3751, 483, 593, 597, 536, 212, 211} \[ \int \frac {\coth ^4(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\frac {b (11 a+7 b) \coth ^3(c+d x)}{8 a^2 d (a+b)^2 \left (a+b \tanh ^2(c+d x)\right )}+\frac {b^{5/2} \left (63 a^2+90 a b+35 b^2\right ) \arctan \left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{8 a^{9/2} d (a+b)^3}-\frac {\left (8 a^2+55 a b+35 b^2\right ) \coth ^3(c+d x)}{24 a^3 d (a+b)^2}-\frac {\left (8 a^3-8 a^2 b-55 a b^2-35 b^3\right ) \coth (c+d x)}{8 a^4 d (a+b)^2}+\frac {b \coth ^3(c+d x)}{4 a d (a+b) \left (a+b \tanh ^2(c+d x)\right )^2}+\frac {x}{(a+b)^3} \]

[In]

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

[Out]

x/(a + b)^3 + (b^(5/2)*(63*a^2 + 90*a*b + 35*b^2)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/(8*a^(9/2)*(a + b)^
3*d) - ((8*a^3 - 8*a^2*b - 55*a*b^2 - 35*b^3)*Coth[c + d*x])/(8*a^4*(a + b)^2*d) - ((8*a^2 + 55*a*b + 35*b^2)*
Coth[c + d*x]^3)/(24*a^3*(a + b)^2*d) + (b*Coth[c + d*x]^3)/(4*a*(a + b)*d*(a + b*Tanh[c + d*x]^2)^2) + (b*(11
*a + 7*b)*Coth[c + d*x]^3)/(8*a^2*(a + b)^2*d*(a + b*Tanh[c + d*x]^2))

Rule 211

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

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 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 3751

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]))

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{x^4 \left (1-x^2\right ) \left (a+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 \tanh ^2(c+d x)\right )^2}-\frac {\text {Subst}\left (\int \frac {-4 a-7 b+7 b x^2}{x^4 \left (1-x^2\right ) \left (a+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 \tanh ^2(c+d x)\right )^2}+\frac {b (11 a+7 b) \coth ^3(c+d x)}{8 a^2 (a+b)^2 d \left (a+b \tanh ^2(c+d x)\right )}+\frac {\text {Subst}\left (\int \frac {8 a^2+55 a b+35 b^2-5 b (11 a+7 b) x^2}{x^4 \left (1-x^2\right ) \left (a+b x^2\right )} \, dx,x,\tanh (c+d x)\right )}{8 a^2 (a+b)^2 d} \\ & = -\frac {\left (8 a^2+55 a b+35 b^2\right ) \coth ^3(c+d x)}{24 a^3 (a+b)^2 d}+\frac {b \coth ^3(c+d x)}{4 a (a+b) d \left (a+b \tanh ^2(c+d x)\right )^2}+\frac {b (11 a+7 b) \coth ^3(c+d x)}{8 a^2 (a+b)^2 d \left (a+b \tanh ^2(c+d x)\right )}-\frac {\text {Subst}\left (\int \frac {-3 \left (8 a^3-8 a^2 b-55 a b^2-35 b^3\right )-3 b \left (8 a^2+55 a b+35 b^2\right ) x^2}{x^2 \left (1-x^2\right ) \left (a+b x^2\right )} \, dx,x,\tanh (c+d x)\right )}{24 a^3 (a+b)^2 d} \\ & = -\frac {\left (8 a^3-8 a^2 b-55 a b^2-35 b^3\right ) \coth (c+d x)}{8 a^4 (a+b)^2 d}-\frac {\left (8 a^2+55 a b+35 b^2\right ) \coth ^3(c+d x)}{24 a^3 (a+b)^2 d}+\frac {b \coth ^3(c+d x)}{4 a (a+b) d \left (a+b \tanh ^2(c+d x)\right )^2}+\frac {b (11 a+7 b) \coth ^3(c+d x)}{8 a^2 (a+b)^2 d \left (a+b \tanh ^2(c+d x)\right )}+\frac {\text {Subst}\left (\int \frac {3 \left (8 a^4-8 a^3 b+8 a^2 b^2+55 a b^3+35 b^4\right )+3 b \left (8 a^3-8 a^2 b-55 a b^2-35 b^3\right ) x^2}{\left (1-x^2\right ) \left (a+b x^2\right )} \, dx,x,\tanh (c+d x)\right )}{24 a^4 (a+b)^2 d} \\ & = -\frac {\left (8 a^3-8 a^2 b-55 a b^2-35 b^3\right ) \coth (c+d x)}{8 a^4 (a+b)^2 d}-\frac {\left (8 a^2+55 a b+35 b^2\right ) \coth ^3(c+d x)}{24 a^3 (a+b)^2 d}+\frac {b \coth ^3(c+d x)}{4 a (a+b) d \left (a+b \tanh ^2(c+d x)\right )^2}+\frac {b (11 a+7 b) \coth ^3(c+d x)}{8 a^2 (a+b)^2 d \left (a+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+b)^3 d}+\frac {\left (b^3 \left (63 a^2+90 a b+35 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\tanh (c+d x)\right )}{8 a^4 (a+b)^3 d} \\ & = \frac {x}{(a+b)^3}+\frac {b^{5/2} \left (63 a^2+90 a b+35 b^2\right ) \arctan \left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{8 a^{9/2} (a+b)^3 d}-\frac {\left (8 a^3-8 a^2 b-55 a b^2-35 b^3\right ) \coth (c+d x)}{8 a^4 (a+b)^2 d}-\frac {\left (8 a^2+55 a b+35 b^2\right ) \coth ^3(c+d x)}{24 a^3 (a+b)^2 d}+\frac {b \coth ^3(c+d x)}{4 a (a+b) d \left (a+b \tanh ^2(c+d x)\right )^2}+\frac {b (11 a+7 b) \coth ^3(c+d x)}{8 a^2 (a+b)^2 d \left (a+b \tanh ^2(c+d x)\right )} \\ \end{align*}

Mathematica [A] (verified)

Time = 3.91 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.85 \[ \int \frac {\coth ^4(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\frac {\frac {24 (c+d x)}{(a+b)^3}+\frac {3 b^{5/2} \left (63 a^2+90 a b+35 b^2\right ) \arctan \left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{a^{9/2} (a+b)^3}+\frac {8 (-4 a+9 b) \coth (c+d x)}{a^4}-\frac {8 \coth (c+d x) \text {csch}^2(c+d x)}{a^3}+\frac {12 b^4 \sinh (2 (c+d x))}{a^3 (a+b)^2 (a-b+(a+b) \cosh (2 (c+d x)))^2}+\frac {3 b^3 (17 a+11 b) \sinh (2 (c+d x))}{a^4 (a+b)^2 (a-b+(a+b) \cosh (2 (c+d x)))}}{24 d} \]

[In]

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

[Out]

((24*(c + d*x))/(a + b)^3 + (3*b^(5/2)*(63*a^2 + 90*a*b + 35*b^2)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/(a^
(9/2)*(a + b)^3) + (8*(-4*a + 9*b)*Coth[c + d*x])/a^4 - (8*Coth[c + d*x]*Csch[c + d*x]^2)/a^3 + (12*b^4*Sinh[2
*(c + d*x)])/(a^3*(a + b)^2*(a - b + (a + b)*Cosh[2*(c + d*x)])^2) + (3*b^3*(17*a + 11*b)*Sinh[2*(c + d*x)])/(
a^4*(a + b)^2*(a - b + (a + b)*Cosh[2*(c + d*x)])))/(24*d)

Maple [A] (verified)

Time = 0.53 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.83

method	result	size
derivativedivides	\(-\frac {\frac {\ln \left (\tanh \left (d x +c \right )-1\right )}{2 \left (a +b \right )^{3}}-\frac {\ln \left (\tanh \left (d x +c \right )+1\right )}{2 \left (a +b \right )^{3}}-\frac {b^{3} \left (\frac {\left (\frac {15}{8} a^{2} b +\frac {13}{4} a \,b^{2}+\frac {11}{8} b^{3}\right ) \tanh \left (d x +c \right )^{3}+\frac {a \left (17 a^{2}+30 a b +13 b^{2}\right ) \tanh \left (d x +c \right )}{8}}{\left (a +b \tanh \left (d x +c \right )^{2}\right )^{2}}+\frac {\left (63 a^{2}+90 a b +35 b^{2}\right ) \arctan \left (\frac {b \tanh \left (d x +c \right )}{\sqrt {a b}}\right )}{8 \sqrt {a b}}\right )}{\left (a +b \right )^{3} a^{4}}-\frac {-a +3 b}{a^{4} \tanh \left (d x +c \right )}+\frac {1}{3 a^{3} \tanh \left (d x +c \right )^{3}}}{d}\)	\(189\)
default	\(-\frac {\frac {\ln \left (\tanh \left (d x +c \right )-1\right )}{2 \left (a +b \right )^{3}}-\frac {\ln \left (\tanh \left (d x +c \right )+1\right )}{2 \left (a +b \right )^{3}}-\frac {b^{3} \left (\frac {\left (\frac {15}{8} a^{2} b +\frac {13}{4} a \,b^{2}+\frac {11}{8} b^{3}\right ) \tanh \left (d x +c \right )^{3}+\frac {a \left (17 a^{2}+30 a b +13 b^{2}\right ) \tanh \left (d x +c \right )}{8}}{\left (a +b \tanh \left (d x +c \right )^{2}\right )^{2}}+\frac {\left (63 a^{2}+90 a b +35 b^{2}\right ) \arctan \left (\frac {b \tanh \left (d x +c \right )}{\sqrt {a b}}\right )}{8 \sqrt {a b}}\right )}{\left (a +b \right )^{3} a^{4}}-\frac {-a +3 b}{a^{4} \tanh \left (d x +c \right )}+\frac {1}{3 a^{3} \tanh \left (d x +c \right )^{3}}}{d}\)	\(189\)
risch	\(\text {Expression too large to display}\)	\(1099\)

[In]

int(coth(d*x+c)^4/(a+b*tanh(d*x+c)^2)^3,x,method=_RETURNVERBOSE)

[Out]

-1/d*(1/2/(a+b)^3*ln(tanh(d*x+c)-1)-1/2/(a+b)^3*ln(tanh(d*x+c)+1)-b^3/(a+b)^3/a^4*(((15/8*a^2*b+13/4*a*b^2+11/
8*b^3)*tanh(d*x+c)^3+1/8*a*(17*a^2+30*a*b+13*b^2)*tanh(d*x+c))/(a+b*tanh(d*x+c)^2)^2+1/8*(63*a^2+90*a*b+35*b^2
)/(a*b)^(1/2)*arctan(b*tanh(d*x+c)/(a*b)^(1/2)))-(-a+3*b)/a^4/tanh(d*x+c)+1/3/a^3/tanh(d*x+c)^3)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 10858 vs. \(2 (210) = 420\).

Time = 0.55 (sec) , antiderivative size = 22038, normalized size of antiderivative = 96.66 \[ \int \frac {\coth ^4(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [F]

\[ \int \frac {\coth ^4(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\int \frac {\coth ^{4}{\left (c + d x \right )}}{\left (a + b \tanh ^{2}{\left (c + d x \right )}\right )^{3}}\, dx \]

[In]

integrate(coth(d*x+c)**4/(a+b*tanh(d*x+c)**2)**3,x)

[Out]

Integral(coth(c + d*x)**4/(a + b*tanh(c + d*x)**2)**3, x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4285 vs. \(2 (210) = 420\).

Time = 1.08 (sec) , antiderivative size = 4285, normalized size of antiderivative = 18.79 \[ \int \frac {\coth ^4(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/8*(3*a^3*b - 3*a^2*b^2 - 7*a*b^3 - 3*b^4)*log((a + b)*e^(4*d*x + 4*c) + 2*(a - b)*e^(2*d*x + 2*c) + a + b)/
((a^7 + 3*a^6*b + 3*a^5*b^2 + a^4*b^3)*d) + 1/8*(3*a^3*b - 3*a^2*b^2 - 7*a*b^3 - 3*b^4)*log(2*(a - b)*e^(-2*d*
x - 2*c) + (a + b)*e^(-4*d*x - 4*c) + a + b)/((a^7 + 3*a^6*b + 3*a^5*b^2 + a^4*b^3)*d) + 1/128*(15*a^4*b - 200
*a^3*b^2 - 186*a^2*b^3 + 35*b^5)*arctan(1/2*((a + b)*e^(2*d*x + 2*c) + a - b)/sqrt(a*b))/((a^7 + 3*a^6*b + 3*a
^5*b^2 + a^4*b^3)*sqrt(a*b)*d) - 1/128*(15*a^4*b - 200*a^3*b^2 - 186*a^2*b^3 + 35*b^5)*arctan(1/2*((a + b)*e^(
-2*d*x - 2*c) + a - b)/sqrt(a*b))/((a^7 + 3*a^6*b + 3*a^5*b^2 + a^4*b^3)*sqrt(a*b)*d) + 1/192*(176*a^6 + 781*a
^5*b + 1571*a^4*b^2 + 1538*a^3*b^3 + 502*a^2*b^4 - 175*a*b^5 - 105*b^6 + 3*(96*a^6 + 465*a^5*b + 665*a^4*b^2 +
706*a^3*b^3 + 506*a^2*b^4 + 61*a*b^5 - 35*b^6)*e^(12*d*x + 12*c) + 6*(120*a^6 + 192*a^5*b - 315*a^4*b^2 - 728
*a^3*b^3 - 1070*a^2*b^4 - 240*a*b^5 + 105*b^6)*e^(10*d*x + 10*c) + (176*a^6 - 281*a^5*b + 3509*a^4*b^2 + 3950*
a^3*b^3 + 12226*a^2*b^4 + 3755*a*b^5 - 1575*b^6)*e^(8*d*x + 8*c) - 4*(184*a^6 + 48*a^5*b + 473*a^4*b^2 + 970*a
^3*b^3 + 3684*a^2*b^4 + 1070*a*b^5 - 525*b^6)*e^(6*d*x + 6*c) - (384*a^6 + 1127*a^5*b - 861*a^4*b^2 - 7146*a^3
*b^3 - 11386*a^2*b^4 - 1965*a*b^5 + 1575*b^6)*e^(4*d*x + 4*c) + 2*(136*a^6 - 96*a^5*b - 1309*a^4*b^2 - 2996*a^
3*b^3 - 2238*a^2*b^4 - 4*a*b^5 + 315*b^6)*e^(2*d*x + 2*c))/((a^9 + 5*a^8*b + 10*a^7*b^2 + 10*a^6*b^3 + 5*a^5*b
^4 + a^4*b^5 - (a^9 + 5*a^8*b + 10*a^7*b^2 + 10*a^6*b^3 + 5*a^5*b^4 + a^4*b^5)*e^(14*d*x + 14*c) - (a^9 - 3*a^
8*b - 22*a^7*b^2 - 38*a^6*b^3 - 27*a^5*b^4 - 7*a^4*b^5)*e^(12*d*x + 12*c) + (3*a^9 + 7*a^8*b - 18*a^7*b^2 - 66
*a^6*b^3 - 65*a^5*b^4 - 21*a^4*b^5)*e^(10*d*x + 10*c) + (3*a^9 - a^8*b + 14*a^7*b^2 + 78*a^6*b^3 + 95*a^5*b^4
+ 35*a^4*b^5)*e^(8*d*x + 8*c) - (3*a^9 - a^8*b + 14*a^7*b^2 + 78*a^6*b^3 + 95*a^5*b^4 + 35*a^4*b^5)*e^(6*d*x +
6*c) - (3*a^9 + 7*a^8*b - 18*a^7*b^2 - 66*a^6*b^3 - 65*a^5*b^4 - 21*a^4*b^5)*e^(4*d*x + 4*c) + (a^9 - 3*a^8*b
- 22*a^7*b^2 - 38*a^6*b^3 - 27*a^5*b^4 - 7*a^4*b^5)*e^(2*d*x + 2*c))*d) - 1/192*(176*a^6 + 781*a^5*b + 1571*a
^4*b^2 + 1538*a^3*b^3 + 502*a^2*b^4 - 175*a*b^5 - 105*b^6 + 2*(136*a^6 - 96*a^5*b - 1309*a^4*b^2 - 2996*a^3*b^
3 - 2238*a^2*b^4 - 4*a*b^5 + 315*b^6)*e^(-2*d*x - 2*c) - (384*a^6 + 1127*a^5*b - 861*a^4*b^2 - 7146*a^3*b^3 -
11386*a^2*b^4 - 1965*a*b^5 + 1575*b^6)*e^(-4*d*x - 4*c) - 4*(184*a^6 + 48*a^5*b + 473*a^4*b^2 + 970*a^3*b^3 +
3684*a^2*b^4 + 1070*a*b^5 - 525*b^6)*e^(-6*d*x - 6*c) + (176*a^6 - 281*a^5*b + 3509*a^4*b^2 + 3950*a^3*b^3 + 1
2226*a^2*b^4 + 3755*a*b^5 - 1575*b^6)*e^(-8*d*x - 8*c) + 6*(120*a^6 + 192*a^5*b - 315*a^4*b^2 - 728*a^3*b^3 -
1070*a^2*b^4 - 240*a*b^5 + 105*b^6)*e^(-10*d*x - 10*c) + 3*(96*a^6 + 465*a^5*b + 665*a^4*b^2 + 706*a^3*b^3 + 5
06*a^2*b^4 + 61*a*b^5 - 35*b^6)*e^(-12*d*x - 12*c))/((a^9 + 5*a^8*b + 10*a^7*b^2 + 10*a^6*b^3 + 5*a^5*b^4 + a^
4*b^5 + (a^9 - 3*a^8*b - 22*a^7*b^2 - 38*a^6*b^3 - 27*a^5*b^4 - 7*a^4*b^5)*e^(-2*d*x - 2*c) - (3*a^9 + 7*a^8*b
- 18*a^7*b^2 - 66*a^6*b^3 - 65*a^5*b^4 - 21*a^4*b^5)*e^(-4*d*x - 4*c) - (3*a^9 - a^8*b + 14*a^7*b^2 + 78*a^6*
b^3 + 95*a^5*b^4 + 35*a^4*b^5)*e^(-6*d*x - 6*c) + (3*a^9 - a^8*b + 14*a^7*b^2 + 78*a^6*b^3 + 95*a^5*b^4 + 35*a
^4*b^5)*e^(-8*d*x - 8*c) + (3*a^9 + 7*a^8*b - 18*a^7*b^2 - 66*a^6*b^3 - 65*a^5*b^4 - 21*a^4*b^5)*e^(-10*d*x -
10*c) - (a^9 - 3*a^8*b - 22*a^7*b^2 - 38*a^6*b^3 - 27*a^5*b^4 - 7*a^4*b^5)*e^(-12*d*x - 12*c) - (a^9 + 5*a^8*b
+ 10*a^7*b^2 + 10*a^6*b^3 + 5*a^5*b^4 + a^4*b^5)*e^(-14*d*x - 14*c))*d) + 1/48*(32*a^5 + 83*a^4*b - 60*a^3*b^
2 - 346*a^2*b^3 - 340*a*b^4 - 105*b^5 + 3*(32*a^5 + 95*a^4*b + 154*a^3*b^2 + 84*a^2*b^3 - 42*a*b^4 - 35*b^5)*e
^(12*d*x + 12*c) + 6*(48*a^5 + 40*a^4*b - 117*a^3*b^2 - 201*a^2*b^3 + 45*a*b^4 + 105*b^5)*e^(10*d*x + 10*c) +
(224*a^5 + 281*a^4*b + 384*a^3*b^2 + 2318*a^2*b^3 - 160*a*b^4 - 1575*b^5)*e^(8*d*x + 8*c) - 4*(16*a^5 - 136*a^
4*b - 9*a^3*b^2 + 697*a^2*b^3 - 115*a*b^4 - 525*b^5)*e^(6*d*x + 6*c) - (96*a^5 + 137*a^4*b - 1262*a^3*b^2 - 18
40*a^2*b^3 + 1230*a*b^4 + 1575*b^5)*e^(4*d*x + 4*c) + 2*(16*a^5 - 136*a^4*b - 435*a^3*b^2 - 35*a^2*b^3 + 563*a
*b^4 + 315*b^5)*e^(2*d*x + 2*c))/((a^8 + 4*a^7*b + 6*a^6*b^2 + 4*a^5*b^3 + a^4*b^4 - (a^8 + 4*a^7*b + 6*a^6*b^
2 + 4*a^5*b^3 + a^4*b^4)*e^(14*d*x + 14*c) - (a^8 - 4*a^7*b - 18*a^6*b^2 - 20*a^5*b^3 - 7*a^4*b^4)*e^(12*d*x +
12*c) + (3*a^8 + 4*a^7*b - 22*a^6*b^2 - 44*a^5*b^3 - 21*a^4*b^4)*e^(10*d*x + 10*c) + (3*a^8 - 4*a^7*b + 18*a^
6*b^2 + 60*a^5*b^3 + 35*a^4*b^4)*e^(8*d*x + 8*c) - (3*a^8 - 4*a^7*b + 18*a^6*b^2 + 60*a^5*b^3 + 35*a^4*b^4)*e^
(6*d*x + 6*c) - (3*a^8 + 4*a^7*b - 22*a^6*b^2 - 44*a^5*b^3 - 21*a^4*b^4)*e^(4*d*x + 4*c) + (a^8 - 4*a^7*b - 18
*a^6*b^2 - 20*a^5*b^3 - 7*a^4*b^4)*e^(2*d*x + 2*c))*d) - 1/48*(32*a^5 + 83*a^4*b - 60*a^3*b^2 - 346*a^2*b^3 -
340*a*b^4 - 105*b^5 + 2*(16*a^5 - 136*a^4*b - 435*a^3*b^2 - 35*a^2*b^3 + 563*a*b^4 + 315*b^5)*e^(-2*d*x - 2*c)
- (96*a^5 + 137*a^4*b - 1262*a^3*b^2 - 1840*a^2*b^3 + 1230*a*b^4 + 1575*b^5)*e^(-4*d*x - 4*c) - 4*(16*a^5 - 1
36*a^4*b - 9*a^3*b^2 + 697*a^2*b^3 - 115*a*b^4 - 525*b^5)*e^(-6*d*x - 6*c) + (224*a^5 + 281*a^4*b + 384*a^3*b^
2 + 2318*a^2*b^3 - 160*a*b^4 - 1575*b^5)*e^(-8*d*x - 8*c) + 6*(48*a^5 + 40*a^4*b - 117*a^3*b^2 - 201*a^2*b^3 +
45*a*b^4 + 105*b^5)*e^(-10*d*x - 10*c) + 3*(32*a^5 + 95*a^4*b + 154*a^3*b^2 + 84*a^2*b^3 - 42*a*b^4 - 35*b^5)
*e^(-12*d*x - 12*c))/((a^8 + 4*a^7*b + 6*a^6*b^2 + 4*a^5*b^3 + a^4*b^4 + (a^8 - 4*a^7*b - 18*a^6*b^2 - 20*a^5*
b^3 - 7*a^4*b^4)*e^(-2*d*x - 2*c) - (3*a^8 + 4*a^7*b - 22*a^6*b^2 - 44*a^5*b^3 - 21*a^4*b^4)*e^(-4*d*x - 4*c)
- (3*a^8 - 4*a^7*b + 18*a^6*b^2 + 60*a^5*b^3 + 35*a^4*b^4)*e^(-6*d*x - 6*c) + (3*a^8 - 4*a^7*b + 18*a^6*b^2 +
60*a^5*b^3 + 35*a^4*b^4)*e^(-8*d*x - 8*c) + (3*a^8 + 4*a^7*b - 22*a^6*b^2 - 44*a^5*b^3 - 21*a^4*b^4)*e^(-10*d*
x - 10*c) - (a^8 - 4*a^7*b - 18*a^6*b^2 - 20*a^5*b^3 - 7*a^4*b^4)*e^(-12*d*x - 12*c) - (a^8 + 4*a^7*b + 6*a^6*
b^2 + 4*a^5*b^3 + a^4*b^4)*e^(-14*d*x - 14*c))*d) + 1/32*(16*a^4 + 147*a^3*b + 351*a^2*b^2 + 325*a*b^3 + 105*b
^4 + 2*(8*a^4 + 32*a^3*b - 251*a^2*b^2 - 590*a*b^3 - 315*b^4)*e^(-2*d*x - 2*c) - (96*a^4 + 313*a^3*b + 19*a^2*
b^2 - 1725*a*b^3 - 1575*b^4)*e^(-4*d*x - 4*c) - 4*(56*a^4 + 80*a^3*b - 65*a^2*b^2 + 400*a*b^3 + 525*b^4)*e^(-6
*d*x - 6*c) - (176*a^4 + 135*a^3*b + 15*a^2*b^2 - 1375*a*b^3 - 1575*b^4)*e^(-8*d*x - 8*c) - 6*(8*a^4 + 45*a^2*
b^2 + 150*a*b^3 + 105*b^4)*e^(-10*d*x - 10*c) + 15*(3*a^3*b + 13*a^2*b^2 + 17*a*b^3 + 7*b^4)*e^(-12*d*x - 12*c
))/((a^7 + 3*a^6*b + 3*a^5*b^2 + a^4*b^3 + (a^7 - 5*a^6*b - 13*a^5*b^2 - 7*a^4*b^3)*e^(-2*d*x - 2*c) - (3*a^7
+ a^6*b - 23*a^5*b^2 - 21*a^4*b^3)*e^(-4*d*x - 4*c) - (3*a^7 - 7*a^6*b + 25*a^5*b^2 + 35*a^4*b^3)*e^(-6*d*x -
6*c) + (3*a^7 - 7*a^6*b + 25*a^5*b^2 + 35*a^4*b^3)*e^(-8*d*x - 8*c) + (3*a^7 + a^6*b - 23*a^5*b^2 - 21*a^4*b^3
)*e^(-10*d*x - 10*c) - (a^7 - 5*a^6*b - 13*a^5*b^2 - 7*a^4*b^3)*e^(-12*d*x - 12*c) - (a^7 + 3*a^6*b + 3*a^5*b^
2 + a^4*b^3)*e^(-14*d*x - 14*c))*d) + 3/4*b*log((a + b)*e^(4*d*x + 4*c) + 2*(a - b)*e^(2*d*x + 2*c) + a + b)/(
a^4*d) - 3/4*b*log(2*(a - b)*e^(-2*d*x - 2*c) + (a + b)*e^(-4*d*x - 4*c) + a + b)/(a^4*d) + 1/4*(2*a - 3*b)*lo
g(e^(2*d*x + 2*c) - 1)/(a^4*d) - 3/2*b*log(e^(2*d*x + 2*c) - 1)/(a^4*d) - 1/4*(2*a - 3*b)*log(e^(-2*d*x - 2*c)
- 1)/(a^4*d) + 3/2*b*log(e^(-2*d*x - 2*c) - 1)/(a^4*d) - 5/32*(3*a*b - 7*b^2)*arctan(1/2*((a + b)*e^(2*d*x +
2*c) + a - b)/sqrt(a*b))/(sqrt(a*b)*a^4*d) - 15/64*(3*a*b + 7*b^2)*arctan(1/2*((a + b)*e^(-2*d*x - 2*c) + a -
b)/sqrt(a*b))/(sqrt(a*b)*a^4*d) + 5/32*(3*a*b - 7*b^2)*arctan(1/2*((a + b)*e^(-2*d*x - 2*c) + a - b)/sqrt(a*b)
)/(sqrt(a*b)*a^4*d)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 493 vs. \(2 (210) = 420\).

Time = 0.56 (sec) , antiderivative size = 493, normalized size of antiderivative = 2.16 \[ \int \frac {\coth ^4(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\frac {\frac {3 \, {\left (63 \, a^{2} b^{3} + 90 \, a b^{4} + 35 \, b^{5}\right )} \arctan \left (\frac {a e^{\left (2 \, d x + 2 \, c\right )} + b e^{\left (2 \, d x + 2 \, c\right )} + a - b}{2 \, \sqrt {a b}}\right )}{{\left (a^{7} + 3 \, a^{6} b + 3 \, a^{5} b^{2} + a^{4} b^{3}\right )} \sqrt {a b}} + \frac {24 \, {\left (d x + c\right )}}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}} - \frac {6 \, {\left (17 \, a^{3} b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 7 \, a^{2} b^{4} e^{\left (6 \, d x + 6 \, c\right )} - 21 \, a b^{5} e^{\left (6 \, d x + 6 \, c\right )} - 11 \, b^{6} e^{\left (6 \, d x + 6 \, c\right )} + 51 \, a^{3} b^{3} e^{\left (4 \, d x + 4 \, c\right )} - a^{2} b^{4} e^{\left (4 \, d x + 4 \, c\right )} + 29 \, a b^{5} e^{\left (4 \, d x + 4 \, c\right )} + 33 \, b^{6} e^{\left (4 \, d x + 4 \, c\right )} + 51 \, a^{3} b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 37 \, a^{2} b^{4} e^{\left (2 \, d x + 2 \, c\right )} - 47 \, a b^{5} e^{\left (2 \, d x + 2 \, c\right )} - 33 \, b^{6} e^{\left (2 \, d x + 2 \, c\right )} + 17 \, a^{3} b^{3} + 45 \, a^{2} b^{4} + 39 \, a b^{5} + 11 \, b^{6}\right )}}{{\left (a^{7} + 3 \, a^{6} b + 3 \, a^{5} b^{2} + a^{4} b^{3}\right )} {\left (a e^{\left (4 \, d x + 4 \, c\right )} + b e^{\left (4 \, d x + 4 \, c\right )} + 2 \, a e^{\left (2 \, d x + 2 \, c\right )} - 2 \, b e^{\left (2 \, d x + 2 \, c\right )} + a + b\right )}^{2}} - \frac {16 \, {\left (6 \, a e^{\left (4 \, d x + 4 \, c\right )} - 9 \, b e^{\left (4 \, d x + 4 \, c\right )} - 6 \, a e^{\left (2 \, d x + 2 \, c\right )} + 18 \, b e^{\left (2 \, d x + 2 \, c\right )} + 4 \, a - 9 \, b\right )}}{a^{4} {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{3}}}{24 \, d} \]

[In]

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

[Out]

1/24*(3*(63*a^2*b^3 + 90*a*b^4 + 35*b^5)*arctan(1/2*(a*e^(2*d*x + 2*c) + b*e^(2*d*x + 2*c) + a - b)/sqrt(a*b))
/((a^7 + 3*a^6*b + 3*a^5*b^2 + a^4*b^3)*sqrt(a*b)) + 24*(d*x + c)/(a^3 + 3*a^2*b + 3*a*b^2 + b^3) - 6*(17*a^3*
b^3*e^(6*d*x + 6*c) + 7*a^2*b^4*e^(6*d*x + 6*c) - 21*a*b^5*e^(6*d*x + 6*c) - 11*b^6*e^(6*d*x + 6*c) + 51*a^3*b
^3*e^(4*d*x + 4*c) - a^2*b^4*e^(4*d*x + 4*c) + 29*a*b^5*e^(4*d*x + 4*c) + 33*b^6*e^(4*d*x + 4*c) + 51*a^3*b^3*
e^(2*d*x + 2*c) + 37*a^2*b^4*e^(2*d*x + 2*c) - 47*a*b^5*e^(2*d*x + 2*c) - 33*b^6*e^(2*d*x + 2*c) + 17*a^3*b^3
+ 45*a^2*b^4 + 39*a*b^5 + 11*b^6)/((a^7 + 3*a^6*b + 3*a^5*b^2 + a^4*b^3)*(a*e^(4*d*x + 4*c) + b*e^(4*d*x + 4*c
) + 2*a*e^(2*d*x + 2*c) - 2*b*e^(2*d*x + 2*c) + a + b)^2) - 16*(6*a*e^(4*d*x + 4*c) - 9*b*e^(4*d*x + 4*c) - 6*
a*e^(2*d*x + 2*c) + 18*b*e^(2*d*x + 2*c) + 4*a - 9*b)/(a^4*(e^(2*d*x + 2*c) - 1)^3))/d

Mupad [F(-1)]

Timed out. \[ \int \frac {\coth ^4(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\int \frac {{\mathrm {coth}\left (c+d\,x\right )}^4}{{\left (b\,{\mathrm {tanh}\left (c+d\,x\right )}^2+a\right )}^3} \,d x \]

[In]

int(coth(c + d*x)^4/(a + b*tanh(c + d*x)^2)^3,x)

[Out]

int(coth(c + d*x)^4/(a + b*tanh(c + d*x)^2)^3, x)