\(\int \frac {1}{(a+b \text {csch}^2(c+d x))^4} \, dx\) [8]

Optimal result

Integrand size = 14, antiderivative size = 220 \[ \int \frac {1}{\left (a+b \text {csch}^2(c+d x)\right )^4} \, dx=\frac {x}{a^4}-\frac {\sqrt {b} \left (35 a^3-70 a^2 b+56 a b^2-16 b^3\right ) \arctan \left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {b}}\right )}{16 a^4 (a-b)^{7/2} d}+\frac {b \coth (c+d x)}{6 a (a-b) d \left (a-b+b \coth ^2(c+d x)\right )^3}+\frac {(11 a-6 b) b \coth (c+d x)}{24 a^2 (a-b)^2 d \left (a-b+b \coth ^2(c+d x)\right )^2}+\frac {b \left (19 a^2-22 a b+8 b^2\right ) \coth (c+d x)}{16 a^3 (a-b)^3 d \left (a-b+b \coth ^2(c+d x)\right )} \]

[Out]

x/a^4+1/6*b*coth(d*x+c)/a/(a-b)/d/(a-b+b*coth(d*x+c)^2)^3+1/24*(11*a-6*b)*b*coth(d*x+c)/a^2/(a-b)^2/d/(a-b+b*c
oth(d*x+c)^2)^2+1/16*b*(19*a^2-22*a*b+8*b^2)*coth(d*x+c)/a^3/(a-b)^3/d/(a-b+b*coth(d*x+c)^2)-1/16*(35*a^3-70*a
^2*b+56*a*b^2-16*b^3)*arctan((a-b)^(1/2)*tanh(d*x+c)/b^(1/2))*b^(1/2)/a^4/(a-b)^(7/2)/d

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {4213, 425, 541, 536, 212, 211} \[ \int \frac {1}{\left (a+b \text {csch}^2(c+d x)\right )^4} \, dx=\frac {x}{a^4}+\frac {b (11 a-6 b) \coth (c+d x)}{24 a^2 d (a-b)^2 \left (a+b \coth ^2(c+d x)-b\right )^2}+\frac {b \left (19 a^2-22 a b+8 b^2\right ) \coth (c+d x)}{16 a^3 d (a-b)^3 \left (a+b \coth ^2(c+d x)-b\right )}-\frac {\sqrt {b} \left (35 a^3-70 a^2 b+56 a b^2-16 b^3\right ) \arctan \left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {b}}\right )}{16 a^4 d (a-b)^{7/2}}+\frac {b \coth (c+d x)}{6 a d (a-b) \left (a+b \coth ^2(c+d x)-b\right )^3} \]

[In]

Int[(a + b*Csch[c + d*x]^2)^(-4),x]

[Out]

x/a^4 - (Sqrt[b]*(35*a^3 - 70*a^2*b + 56*a*b^2 - 16*b^3)*ArcTan[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[b]])/(16*a^4*
(a - b)^(7/2)*d) + (b*Coth[c + d*x])/(6*a*(a - b)*d*(a - b + b*Coth[c + d*x]^2)^3) + ((11*a - 6*b)*b*Coth[c +
d*x])/(24*a^2*(a - b)^2*d*(a - b + b*Coth[c + d*x]^2)^2) + (b*(19*a^2 - 22*a*b + 8*b^2)*Coth[c + d*x])/(16*a^3
*(a - b)^3*d*(a - b + b*Coth[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 425

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-b)*x*(a + b*x^n)^(p + 1)*
((c + d*x^n)^(q + 1)/(a*n*(p + 1)*(b*c - a*d))), x] + Dist[1/(a*n*(p + 1)*(b*c - a*d)), Int[(a + b*x^n)^(p + 1
)*(c + d*x^n)^q*Simp[b*c + n*(p + 1)*(b*c - a*d) + d*b*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c,
d, n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &&  !( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomi
alQ[a, b, c, d, 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 541

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

Rule 4213

Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist
[ff/f, Subst[Int[(a + b + b*ff^2*x^2)^p/(1 + ff^2*x^2), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p},
x] && NeQ[a + b, 0] && NeQ[p, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \left (a-b+b x^2\right )^4} \, dx,x,\coth (c+d x)\right )}{d} \\ & = \frac {b \coth (c+d x)}{6 a (a-b) d \left (a-b+b \coth ^2(c+d x)\right )^3}-\frac {\text {Subst}\left (\int \frac {-6 a+b+5 b x^2}{\left (1-x^2\right ) \left (a-b+b x^2\right )^3} \, dx,x,\coth (c+d x)\right )}{6 a (a-b) d} \\ & = \frac {b \coth (c+d x)}{6 a (a-b) d \left (a-b+b \coth ^2(c+d x)\right )^3}+\frac {(11 a-6 b) b \coth (c+d x)}{24 a^2 (a-b)^2 d \left (a-b+b \coth ^2(c+d x)\right )^2}+\frac {\text {Subst}\left (\int \frac {3 \left (8 a^2-5 a b+2 b^2\right )-3 (11 a-6 b) b x^2}{\left (1-x^2\right ) \left (a-b+b x^2\right )^2} \, dx,x,\coth (c+d x)\right )}{24 a^2 (a-b)^2 d} \\ & = \frac {b \coth (c+d x)}{6 a (a-b) d \left (a-b+b \coth ^2(c+d x)\right )^3}+\frac {(11 a-6 b) b \coth (c+d x)}{24 a^2 (a-b)^2 d \left (a-b+b \coth ^2(c+d x)\right )^2}+\frac {b \left (19 a^2-22 a b+8 b^2\right ) \coth (c+d x)}{16 a^3 (a-b)^3 d \left (a-b+b \coth ^2(c+d x)\right )}-\frac {\text {Subst}\left (\int \frac {-3 \left (16 a^3-29 a^2 b+26 a b^2-8 b^3\right )+3 b \left (19 a^2-22 a b+8 b^2\right ) x^2}{\left (1-x^2\right ) \left (a-b+b x^2\right )} \, dx,x,\coth (c+d x)\right )}{48 a^3 (a-b)^3 d} \\ & = \frac {b \coth (c+d x)}{6 a (a-b) d \left (a-b+b \coth ^2(c+d x)\right )^3}+\frac {(11 a-6 b) b \coth (c+d x)}{24 a^2 (a-b)^2 d \left (a-b+b \coth ^2(c+d x)\right )^2}+\frac {b \left (19 a^2-22 a b+8 b^2\right ) \coth (c+d x)}{16 a^3 (a-b)^3 d \left (a-b+b \coth ^2(c+d x)\right )}+\frac {\text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\coth (c+d x)\right )}{a^4 d}+\frac {\left (b \left (35 a^3-70 a^2 b+56 a b^2-16 b^3\right )\right ) \text {Subst}\left (\int \frac {1}{a-b+b x^2} \, dx,x,\coth (c+d x)\right )}{16 a^4 (a-b)^3 d} \\ & = \frac {x}{a^4}-\frac {\sqrt {b} \left (35 a^3-70 a^2 b+56 a b^2-16 b^3\right ) \arctan \left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {b}}\right )}{16 a^4 (a-b)^{7/2} d}+\frac {b \coth (c+d x)}{6 a (a-b) d \left (a-b+b \coth ^2(c+d x)\right )^3}+\frac {(11 a-6 b) b \coth (c+d x)}{24 a^2 (a-b)^2 d \left (a-b+b \coth ^2(c+d x)\right )^2}+\frac {b \left (19 a^2-22 a b+8 b^2\right ) \coth (c+d x)}{16 a^3 (a-b)^3 d \left (a-b+b \coth ^2(c+d x)\right )} \\ \end{align*}

Mathematica [A] (verified)

Time = 8.44 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.24 \[ \int \frac {1}{\left (a+b \text {csch}^2(c+d x)\right )^4} \, dx=\frac {(-a+2 b+a \cosh (2 (c+d x))) \text {csch}^8(c+d x) \left (48 (c+d x) (-a+2 b+a \cosh (2 (c+d x)))^3+\frac {3 \sqrt {b} \left (-35 a^3+70 a^2 b-56 a b^2+16 b^3\right ) \arctan \left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {b}}\right ) (-a+2 b+a \cosh (2 (c+d x)))^3}{(a-b)^{7/2}}+\frac {32 a b^3 \sinh (2 (c+d x))}{a-b}+\frac {a b \left (87 a^2-116 a b+44 b^2\right ) (a-2 b-a \cosh (2 (c+d x)))^2 \sinh (2 (c+d x))}{(a-b)^3}-\frac {4 a (19 a-14 b) b^2 (-a+2 b+a \cosh (2 (c+d x))) \sinh (2 (c+d x))}{(a-b)^2}\right )}{768 a^4 d \left (a+b \text {csch}^2(c+d x)\right )^4} \]

[In]

Integrate[(a + b*Csch[c + d*x]^2)^(-4),x]

[Out]

((-a + 2*b + a*Cosh[2*(c + d*x)])*Csch[c + d*x]^8*(48*(c + d*x)*(-a + 2*b + a*Cosh[2*(c + d*x)])^3 + (3*Sqrt[b
]*(-35*a^3 + 70*a^2*b - 56*a*b^2 + 16*b^3)*ArcTan[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[b]]*(-a + 2*b + a*Cosh[2*(c
+ d*x)])^3)/(a - b)^(7/2) + (32*a*b^3*Sinh[2*(c + d*x)])/(a - b) + (a*b*(87*a^2 - 116*a*b + 44*b^2)*(a - 2*b
- a*Cosh[2*(c + d*x)])^2*Sinh[2*(c + d*x)])/(a - b)^3 - (4*a*(19*a - 14*b)*b^2*(-a + 2*b + a*Cosh[2*(c + d*x)]
)*Sinh[2*(c + d*x)])/(a - b)^2))/(768*a^4*d*(a + b*Csch[c + d*x]^2)^4)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(671\) vs. \(2(204)=408\).

Time = 2.31 (sec) , antiderivative size = 672, normalized size of antiderivative = 3.05

method	result	size
derivativedivides	\(\frac {\frac {2 b \left (\frac {\frac {64 b^{2} a \left (19 a^{2}-22 a b +8 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{1024 a^{3}-3072 a^{2} b +3072 a \,b^{2}-1024 b^{3}}+\frac {64 \left (544 a^{3}-835 a^{2} b +438 a \,b^{2}-72 b^{3}\right ) a b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{3072 a^{3}-9216 a^{2} b +9216 a \,b^{2}-3072 b^{3}}+\frac {64 a \left (232 a^{4}-400 a^{3} b +247 a^{2} b^{2}-62 a \,b^{3}+8 b^{4}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{512 a^{3}-1536 a^{2} b +1536 a \,b^{2}-512 b^{3}}+\frac {64 a \left (232 a^{4}-400 a^{3} b +247 a^{2} b^{2}-62 a \,b^{3}+8 b^{4}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{512 a^{3}-1536 a^{2} b +1536 a \,b^{2}-512 b^{3}}+\frac {64 \left (544 a^{3}-835 a^{2} b +438 a \,b^{2}-72 b^{3}\right ) a b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3072 a^{3}-9216 a^{2} b +9216 a \,b^{2}-3072 b^{3}}+\frac {64 b^{2} a \left (19 a^{2}-22 a b +8 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{1024 a^{3}-3072 a^{2} b +3072 a \,b^{2}-1024 b^{3}}}{\left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} b +4 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +b \right )^{3}}+\frac {4 \left (35 a^{3}-70 a^{2} b +56 a \,b^{2}-16 b^{3}\right ) b \left (\frac {\left (\sqrt {a \left (a -b \right )}+a \right ) \arctan \left (\frac {b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {a \left (a -b \right )}+2 a -b \right ) b}}\right )}{2 b \sqrt {a \left (a -b \right )}\, \sqrt {\left (2 \sqrt {a \left (a -b \right )}+2 a -b \right ) b}}-\frac {\left (\sqrt {a \left (a -b \right )}-a \right ) \operatorname {arctanh}\left (\frac {b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {a \left (a -b \right )}-2 a +b \right ) b}}\right )}{2 b \sqrt {a \left (a -b \right )}\, \sqrt {\left (2 \sqrt {a \left (a -b \right )}-2 a +b \right ) b}}\right )}{64 a^{3}-192 a^{2} b +192 a \,b^{2}-64 b^{3}}\right )}{a^{4}}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a^{4}}+\frac {\ln \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{4}}}{d}\)	\(672\)
default	\(\frac {\frac {2 b \left (\frac {\frac {64 b^{2} a \left (19 a^{2}-22 a b +8 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{1024 a^{3}-3072 a^{2} b +3072 a \,b^{2}-1024 b^{3}}+\frac {64 \left (544 a^{3}-835 a^{2} b +438 a \,b^{2}-72 b^{3}\right ) a b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{3072 a^{3}-9216 a^{2} b +9216 a \,b^{2}-3072 b^{3}}+\frac {64 a \left (232 a^{4}-400 a^{3} b +247 a^{2} b^{2}-62 a \,b^{3}+8 b^{4}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{512 a^{3}-1536 a^{2} b +1536 a \,b^{2}-512 b^{3}}+\frac {64 a \left (232 a^{4}-400 a^{3} b +247 a^{2} b^{2}-62 a \,b^{3}+8 b^{4}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{512 a^{3}-1536 a^{2} b +1536 a \,b^{2}-512 b^{3}}+\frac {64 \left (544 a^{3}-835 a^{2} b +438 a \,b^{2}-72 b^{3}\right ) a b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3072 a^{3}-9216 a^{2} b +9216 a \,b^{2}-3072 b^{3}}+\frac {64 b^{2} a \left (19 a^{2}-22 a b +8 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{1024 a^{3}-3072 a^{2} b +3072 a \,b^{2}-1024 b^{3}}}{\left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} b +4 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +b \right )^{3}}+\frac {4 \left (35 a^{3}-70 a^{2} b +56 a \,b^{2}-16 b^{3}\right ) b \left (\frac {\left (\sqrt {a \left (a -b \right )}+a \right ) \arctan \left (\frac {b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {a \left (a -b \right )}+2 a -b \right ) b}}\right )}{2 b \sqrt {a \left (a -b \right )}\, \sqrt {\left (2 \sqrt {a \left (a -b \right )}+2 a -b \right ) b}}-\frac {\left (\sqrt {a \left (a -b \right )}-a \right ) \operatorname {arctanh}\left (\frac {b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {a \left (a -b \right )}-2 a +b \right ) b}}\right )}{2 b \sqrt {a \left (a -b \right )}\, \sqrt {\left (2 \sqrt {a \left (a -b \right )}-2 a +b \right ) b}}\right )}{64 a^{3}-192 a^{2} b +192 a \,b^{2}-64 b^{3}}\right )}{a^{4}}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a^{4}}+\frac {\ln \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{4}}}{d}\)	\(672\)
risch	\(\frac {x}{a^{4}}+\frac {b \left (-435 a^{5} {\mathrm e}^{8 d x +8 c}-1408 b^{5} {\mathrm e}^{6 d x +6 c}-870 a^{5} {\mathrm e}^{4 d x +4 c}-87 a^{5}-4292 a^{4} b \,{\mathrm e}^{6 d x +6 c}+8792 a^{3} b^{2} {\mathrm e}^{6 d x +6 c}-9936 a^{2} b^{3} {\mathrm e}^{6 d x +6 c}+5824 a \,b^{4} {\mathrm e}^{6 d x +6 c}+3792 a^{4} b \,{\mathrm e}^{4 d x +4 c}-6432 a^{3} b^{2} {\mathrm e}^{4 d x +4 c}+4608 a^{2} b^{3} {\mathrm e}^{4 d x +4 c}-366 a^{4} b \,{\mathrm e}^{10 d x +10 c}+408 a^{3} b^{2} {\mathrm e}^{10 d x +10 c}-144 a^{2} b^{3} {\mathrm e}^{10 d x +10 c}+116 a^{4} b -44 a^{3} b^{2}+2124 a^{4} b \,{\mathrm e}^{8 d x +8 c}-1248 a \,b^{4} {\mathrm e}^{4 d x +4 c}-1374 a^{4} b \,{\mathrm e}^{2 d x +2 c}+1248 a^{3} b^{2} {\mathrm e}^{2 d x +2 c}-384 a^{2} b^{3} {\mathrm e}^{2 d x +2 c}-3972 a^{3} b^{2} {\mathrm e}^{8 d x +8 c}+3072 a^{2} b^{3} {\mathrm e}^{8 d x +8 c}-864 a \,b^{4} {\mathrm e}^{8 d x +8 c}+435 a^{5} {\mathrm e}^{2 d x +2 c}+87 a^{5} {\mathrm e}^{10 d x +10 c}+870 a^{5} {\mathrm e}^{6 d x +6 c}\right )}{24 a^{4} \left (a -b \right )^{3} d \left (a \,{\mathrm e}^{4 d x +4 c}-2 \,{\mathrm e}^{2 d x +2 c} a +4 b \,{\mathrm e}^{2 d x +2 c}+a \right )^{3}}+\frac {35 \sqrt {-b \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {a +2 \sqrt {-b \left (a -b \right )}-2 b}{a}\right )}{32 \left (a -b \right )^{4} d a}-\frac {35 \sqrt {-b \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {a +2 \sqrt {-b \left (a -b \right )}-2 b}{a}\right ) b}{16 \left (a -b \right )^{4} d \,a^{2}}+\frac {7 \sqrt {-b \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {a +2 \sqrt {-b \left (a -b \right )}-2 b}{a}\right ) b^{2}}{4 \left (a -b \right )^{4} d \,a^{3}}-\frac {\sqrt {-b \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {a +2 \sqrt {-b \left (a -b \right )}-2 b}{a}\right ) b^{3}}{2 \left (a -b \right )^{4} d \,a^{4}}-\frac {35 \sqrt {-b \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {-a +2 \sqrt {-b \left (a -b \right )}+2 b}{a}\right )}{32 \left (a -b \right )^{4} d a}+\frac {35 \sqrt {-b \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {-a +2 \sqrt {-b \left (a -b \right )}+2 b}{a}\right ) b}{16 \left (a -b \right )^{4} d \,a^{2}}-\frac {7 \sqrt {-b \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {-a +2 \sqrt {-b \left (a -b \right )}+2 b}{a}\right ) b^{2}}{4 \left (a -b \right )^{4} d \,a^{3}}+\frac {\sqrt {-b \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {-a +2 \sqrt {-b \left (a -b \right )}+2 b}{a}\right ) b^{3}}{2 \left (a -b \right )^{4} d \,a^{4}}\)	\(938\)

[In]

int(1/(a+b*csch(d*x+c)^2)^4,x,method=_RETURNVERBOSE)

[Out]

1/d*(2*b/a^4*(64*(1/1024*b^2*a*(19*a^2-22*a*b+8*b^2)/(a^3-3*a^2*b+3*a*b^2-b^3)*tanh(1/2*d*x+1/2*c)^11+1/3072*(
544*a^3-835*a^2*b+438*a*b^2-72*b^3)*a*b/(a^3-3*a^2*b+3*a*b^2-b^3)*tanh(1/2*d*x+1/2*c)^9+1/512*a*(232*a^4-400*a
^3*b+247*a^2*b^2-62*a*b^3+8*b^4)/(a^3-3*a^2*b+3*a*b^2-b^3)*tanh(1/2*d*x+1/2*c)^7+1/512*a*(232*a^4-400*a^3*b+24
7*a^2*b^2-62*a*b^3+8*b^4)/(a^3-3*a^2*b+3*a*b^2-b^3)*tanh(1/2*d*x+1/2*c)^5+1/3072*(544*a^3-835*a^2*b+438*a*b^2-
72*b^3)*a*b/(a^3-3*a^2*b+3*a*b^2-b^3)*tanh(1/2*d*x+1/2*c)^3+1/1024*b^2*a*(19*a^2-22*a*b+8*b^2)/(a^3-3*a^2*b+3*
a*b^2-b^3)*tanh(1/2*d*x+1/2*c))/(tanh(1/2*d*x+1/2*c)^4*b+4*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+b
)^3+4*(35*a^3-70*a^2*b+56*a*b^2-16*b^3)/(64*a^3-192*a^2*b+192*a*b^2-64*b^3)*b*(1/2*((a*(a-b))^(1/2)+a)/b/(a*(a
-b))^(1/2)/((2*(a*(a-b))^(1/2)+2*a-b)*b)^(1/2)*arctan(b*tanh(1/2*d*x+1/2*c)/((2*(a*(a-b))^(1/2)+2*a-b)*b)^(1/2
))-1/2*((a*(a-b))^(1/2)-a)/b/(a*(a-b))^(1/2)/((2*(a*(a-b))^(1/2)-2*a+b)*b)^(1/2)*arctanh(b*tanh(1/2*d*x+1/2*c)
/((2*(a*(a-b))^(1/2)-2*a+b)*b)^(1/2))))-1/a^4*ln(tanh(1/2*d*x+1/2*c)-1)+1/a^4*ln(1+tanh(1/2*d*x+1/2*c)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 8543 vs. \(2 (204) = 408\).

Time = 0.44 (sec) , antiderivative size = 17376, normalized size of antiderivative = 78.98 \[ \int \frac {1}{\left (a+b \text {csch}^2(c+d x)\right )^4} \, dx=\text {Too large to display} \]

[In]

integrate(1/(a+b*csch(d*x+c)^2)^4,x, algorithm="fricas")

[Out]

Too large to include

Sympy [F]

\[ \int \frac {1}{\left (a+b \text {csch}^2(c+d x)\right )^4} \, dx=\int \frac {1}{\left (a + b \operatorname {csch}^{2}{\left (c + d x \right )}\right )^{4}}\, dx \]

[In]

integrate(1/(a+b*csch(d*x+c)**2)**4,x)

[Out]

Integral((a + b*csch(c + d*x)**2)**(-4), x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{\left (a+b \text {csch}^2(c+d x)\right )^4} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate(1/(a+b*csch(d*x+c)^2)^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(b-a>0)', see `assume?` for mor
e details)Is

Giac [F]

\[ \int \frac {1}{\left (a+b \text {csch}^2(c+d x)\right )^4} \, dx=\int { \frac {1}{{\left (b \operatorname {csch}\left (d x + c\right )^{2} + a\right )}^{4}} \,d x } \]

[In]

integrate(1/(a+b*csch(d*x+c)^2)^4,x, algorithm="giac")

[Out]

sage0*x

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (a+b \text {csch}^2(c+d x)\right )^4} \, dx=\int \frac {1}{{\left (a+\frac {b}{{\mathrm {sinh}\left (c+d\,x\right )}^2}\right )}^4} \,d x \]

[In]

int(1/(a + b/sinh(c + d*x)^2)^4,x)

[Out]

int(1/(a + b/sinh(c + d*x)^2)^4, x)