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

Optimal. Leaf size=220 \[ \frac{b \left (19 a^2-22 a b+8 b^2\right ) \coth (c+d x)}{16 a^3 d (a-b)^3 \left (a+b \coth ^2(c+d x)-b\right )}-\frac{\sqrt{b} \left (-70 a^2 b+35 a^3+56 a b^2-16 b^3\right ) \tan ^{-1}\left (\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{b}}\right )}{16 a^4 d (a-b)^{7/2}}+\frac{b (11 a-6 b) \coth (c+d x)}{24 a^2 d (a-b)^2 \left (a+b \coth ^2(c+d x)-b\right )^2}+\frac{x}{a^4}+\frac{b \coth (c+d x)}{6 a d (a-b) \left (a+b \coth ^2(c+d x)-b\right )^3} \]

[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))

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Rubi [A]  time = 0.344481, antiderivative size = 220, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {4128, 414, 527, 522, 206, 205} \[ \frac{b \left (19 a^2-22 a b+8 b^2\right ) \coth (c+d x)}{16 a^3 d (a-b)^3 \left (a+b \coth ^2(c+d x)-b\right )}-\frac{\sqrt{b} \left (-70 a^2 b+35 a^3+56 a b^2-16 b^3\right ) \tan ^{-1}\left (\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{b}}\right )}{16 a^4 d (a-b)^{7/2}}+\frac{b (11 a-6 b) \coth (c+d x)}{24 a^2 d (a-b)^2 \left (a+b \coth ^2(c+d x)-b\right )^2}+\frac{x}{a^4}+\frac{b \coth (c+d x)}{6 a d (a-b) \left (a+b \coth ^2(c+d x)-b\right )^3} \]

Antiderivative was successfully verified.

[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 4128

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]

Rule 414

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]) && IntBinomial
Q[a, b, c, d, n, p, q, x]

Rule 527

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 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 205

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

Rubi steps

\begin{align*} \int \frac{1}{\left (a+b \text{csch}^2(c+d x)\right )^4} \, dx &=\frac{\operatorname{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{\operatorname{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{\operatorname{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{\operatorname{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{\operatorname{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 ) \operatorname{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 ) \tan ^{-1}\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]  time = 3.66612, size = 273, normalized size = 1.24 \[ \frac{\text{csch}^8(c+d x) (a \cosh (2 (c+d x))-a+2 b) \left (\frac{a b \left (87 a^2-116 a b+44 b^2\right ) \sinh (2 (c+d x)) (a (-\cosh (2 (c+d x)))+a-2 b)^2}{(a-b)^3}+\frac{3 \sqrt{b} \left (70 a^2 b-35 a^3-56 a b^2+16 b^3\right ) (a \cosh (2 (c+d x))-a+2 b)^3 \tan ^{-1}\left (\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{b}}\right )}{(a-b)^{7/2}}+\frac{32 a b^3 \sinh (2 (c+d x))}{a-b}-\frac{4 a b^2 (19 a-14 b) \sinh (2 (c+d x)) (a \cosh (2 (c+d x))-a+2 b)}{(a-b)^2}+48 (c+d x) (a \cosh (2 (c+d x))-a+2 b)^3\right )}{768 a^4 d \left (a+b \text{csch}^2(c+d x)\right )^4} \]

Antiderivative was successfully verified.

[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]  time = 0.079, size = 3638, normalized size = 16.5 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/d*b^4/a^3/(a^3-3*a^2*b+3*a*b^2-b^3)/(a*(a-b))^(1/2)/((2*(a*(a-b))^(1/2)+2*a-b)*b)^(1/2)*arctan(tanh(1/2*d*x
+1/2*c)*b/((2*(a*(a-b))^(1/2)+2*a-b)*b)^(1/2))-35/8/d*b^2/a/(a^3-3*a^2*b+3*a*b^2-b^3)/(a*(a-b))^(1/2)/((2*(a*(
a-b))^(1/2)+2*a-b)*b)^(1/2)*arctan(tanh(1/2*d*x+1/2*c)*b/((2*(a*(a-b))^(1/2)+2*a-b)*b)^(1/2))-35/8/d*b^2/a/(a^
3-3*a^2*b+3*a*b^2-b^3)/(a*(a-b))^(1/2)/((2*(a*(a-b))^(1/2)-2*a+b)*b)^(1/2)*arctanh(tanh(1/2*d*x+1/2*c)*b/((2*(
a*(a-b))^(1/2)-2*a+b)*b)^(1/2))+7/2/d*b^3/a^2/(a^3-3*a^2*b+3*a*b^2-b^3)/(a*(a-b))^(1/2)/((2*(a*(a-b))^(1/2)-2*
a+b)*b)^(1/2)*arctanh(tanh(1/2*d*x+1/2*c)*b/((2*(a*(a-b))^(1/2)-2*a+b)*b)^(1/2))+7/2/d*b^3/a^2/(a^3-3*a^2*b+3*
a*b^2-b^3)/(a*(a-b))^(1/2)/((2*(a*(a-b))^(1/2)+2*a-b)*b)^(1/2)*arctan(tanh(1/2*d*x+1/2*c)*b/((2*(a*(a-b))^(1/2
)+2*a-b)*b)^(1/2))-11/4/d*b^4/a^2/(b*tanh(1/2*d*x+1/2*c)^4+4*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b
+b)^3/(a^3-3*a^2*b+3*a*b^2-b^3)*tanh(1/2*d*x+1/2*c)^11-31/2/d*b^4/a^2/(b*tanh(1/2*d*x+1/2*c)^4+4*tanh(1/2*d*x+
1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+b)^3/(a^3-3*a^2*b+3*a*b^2-b^3)*tanh(1/2*d*x+1/2*c)^7+35/16/d*b/(a^3-3*a^2
*b+3*a*b^2-b^3)/(a*(a-b))^(1/2)/((2*(a*(a-b))^(1/2)+2*a-b)*b)^(1/2)*arctan(tanh(1/2*d*x+1/2*c)*b/((2*(a*(a-b))
^(1/2)+2*a-b)*b)^(1/2))-11/4/d*b^4/a^2/(b*tanh(1/2*d*x+1/2*c)^4+4*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c
)^2*b+b)^3/(a^3-3*a^2*b+3*a*b^2-b^3)*tanh(1/2*d*x+1/2*c)+1/d*b^5/a^3/(b*tanh(1/2*d*x+1/2*c)^4+4*tanh(1/2*d*x+1
/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+b)^3/(a^3-3*a^2*b+3*a*b^2-b^3)*tanh(1/2*d*x+1/2*c)+35/16/d*b/a/(a^3-3*a^2*
b+3*a*b^2-b^3)/((2*(a*(a-b))^(1/2)+2*a-b)*b)^(1/2)*arctan(tanh(1/2*d*x+1/2*c)*b/((2*(a*(a-b))^(1/2)+2*a-b)*b)^
(1/2))-1/d*b^4/a^4/(a^3-3*a^2*b+3*a*b^2-b^3)/((2*(a*(a-b))^(1/2)+2*a-b)*b)^(1/2)*arctan(tanh(1/2*d*x+1/2*c)*b/
((2*(a*(a-b))^(1/2)+2*a-b)*b)^(1/2))+1/d*b^4/a^4/(a^3-3*a^2*b+3*a*b^2-b^3)/((2*(a*(a-b))^(1/2)-2*a+b)*b)^(1/2)
*arctanh(tanh(1/2*d*x+1/2*c)*b/((2*(a*(a-b))^(1/2)-2*a+b)*b)^(1/2))-100/d*b^2/(b*tanh(1/2*d*x+1/2*c)^4+4*tanh(
1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+b)^3/(a^3-3*a^2*b+3*a*b^2-b^3)*tanh(1/2*d*x+1/2*c)^5+68/3/d*b^2/(
b*tanh(1/2*d*x+1/2*c)^4+4*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+b)^3/(a^3-3*a^2*b+3*a*b^2-b^3)*tan
h(1/2*d*x+1/2*c)^3+68/3/d*b^2/(b*tanh(1/2*d*x+1/2*c)^4+4*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+b)^
3/(a^3-3*a^2*b+3*a*b^2-b^3)*tanh(1/2*d*x+1/2*c)^9+1/d/a^4*ln(tanh(1/2*d*x+1/2*c)+1)-1/d/a^4*ln(tanh(1/2*d*x+1/
2*c)-1)-100/d*b^2/(b*tanh(1/2*d*x+1/2*c)^4+4*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+b)^3/(a^3-3*a^2
*b+3*a*b^2-b^3)*tanh(1/2*d*x+1/2*c)^7+2/d*b^5/a^3/(b*tanh(1/2*d*x+1/2*c)^4+4*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/
2*d*x+1/2*c)^2*b+b)^3/(a^3-3*a^2*b+3*a*b^2-b^3)*tanh(1/2*d*x+1/2*c)^7+247/4/d*b^3/a/(b*tanh(1/2*d*x+1/2*c)^4+4
*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+b)^3/(a^3-3*a^2*b+3*a*b^2-b^3)*tanh(1/2*d*x+1/2*c)^5-31/2/d
*b^4/a^2/(b*tanh(1/2*d*x+1/2*c)^4+4*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+b)^3/(a^3-3*a^2*b+3*a*b^
2-b^3)*tanh(1/2*d*x+1/2*c)^5-35/8/d*b^2/a^2/(a^3-3*a^2*b+3*a*b^2-b^3)/((2*(a*(a-b))^(1/2)+2*a-b)*b)^(1/2)*arct
an(tanh(1/2*d*x+1/2*c)*b/((2*(a*(a-b))^(1/2)+2*a-b)*b)^(1/2))+2/d*b^5/a^3/(b*tanh(1/2*d*x+1/2*c)^4+4*tanh(1/2*
d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+b)^3/(a^3-3*a^2*b+3*a*b^2-b^3)*tanh(1/2*d*x+1/2*c)^5-835/24/d*b^3/a/(
b*tanh(1/2*d*x+1/2*c)^4+4*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+b)^3/(a^3-3*a^2*b+3*a*b^2-b^3)*tan
h(1/2*d*x+1/2*c)^3+73/4/d*b^4/a^2/(b*tanh(1/2*d*x+1/2*c)^4+4*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b
+b)^3/(a^3-3*a^2*b+3*a*b^2-b^3)*tanh(1/2*d*x+1/2*c)^3-3/d*b^5/a^3/(b*tanh(1/2*d*x+1/2*c)^4+4*tanh(1/2*d*x+1/2*
c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+b)^3/(a^3-3*a^2*b+3*a*b^2-b^3)*tanh(1/2*d*x+1/2*c)^3+19/8/d*b^3/a/(b*tanh(1/2
*d*x+1/2*c)^4+4*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+b)^3/(a^3-3*a^2*b+3*a*b^2-b^3)*tanh(1/2*d*x+
1/2*c)+35/16/d*b/(a^3-3*a^2*b+3*a*b^2-b^3)/(a*(a-b))^(1/2)/((2*(a*(a-b))^(1/2)-2*a+b)*b)^(1/2)*arctanh(tanh(1/
2*d*x+1/2*c)*b/((2*(a*(a-b))^(1/2)-2*a+b)*b)^(1/2))+7/2/d*b^3/a^3/(a^3-3*a^2*b+3*a*b^2-b^3)/((2*(a*(a-b))^(1/2
)+2*a-b)*b)^(1/2)*arctan(tanh(1/2*d*x+1/2*c)*b/((2*(a*(a-b))^(1/2)+2*a-b)*b)^(1/2))-35/16/d*b/a/(a^3-3*a^2*b+3
*a*b^2-b^3)/((2*(a*(a-b))^(1/2)-2*a+b)*b)^(1/2)*arctanh(tanh(1/2*d*x+1/2*c)*b/((2*(a*(a-b))^(1/2)-2*a+b)*b)^(1
/2))+35/8/d*b^2/a^2/(a^3-3*a^2*b+3*a*b^2-b^3)/((2*(a*(a-b))^(1/2)-2*a+b)*b)^(1/2)*arctanh(tanh(1/2*d*x+1/2*c)*
b/((2*(a*(a-b))^(1/2)-2*a+b)*b)^(1/2))-7/2/d*b^3/a^3/(a^3-3*a^2*b+3*a*b^2-b^3)/((2*(a*(a-b))^(1/2)-2*a+b)*b)^(
1/2)*arctanh(tanh(1/2*d*x+1/2*c)*b/((2*(a*(a-b))^(1/2)-2*a+b)*b)^(1/2))+1/d*b^5/a^3/(b*tanh(1/2*d*x+1/2*c)^4+4
*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+b)^3/(a^3-3*a^2*b+3*a*b^2-b^3)*tanh(1/2*d*x+1/2*c)^11-835/2
4/d*b^3/a/(b*tanh(1/2*d*x+1/2*c)^4+4*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+b)^3/(a^3-3*a^2*b+3*a*b
^2-b^3)*tanh(1/2*d*x+1/2*c)^9+73/4/d*b^4/a^2/(b*tanh(1/2*d*x+1/2*c)^4+4*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x
+1/2*c)^2*b+b)^3/(a^3-3*a^2*b+3*a*b^2-b^3)*tanh(1/2*d*x+1/2*c)^9-3/d*b^5/a^3/(b*tanh(1/2*d*x+1/2*c)^4+4*tanh(1
/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+b)^3/(a^3-3*a^2*b+3*a*b^2-b^3)*tanh(1/2*d*x+1/2*c)^9+247/4/d*b^3/a
/(b*tanh(1/2*d*x+1/2*c)^4+4*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+b)^3/(a^3-3*a^2*b+3*a*b^2-b^3)*t
anh(1/2*d*x+1/2*c)^7-1/d*b^4/a^3/(a^3-3*a^2*b+3*a*b^2-b^3)/(a*(a-b))^(1/2)/((2*(a*(a-b))^(1/2)-2*a+b)*b)^(1/2)
*arctanh(tanh(1/2*d*x+1/2*c)*b/((2*(a*(a-b))^(1/2)-2*a+b)*b)^(1/2))+19/8/d*b^3/a/(b*tanh(1/2*d*x+1/2*c)^4+4*ta
nh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+b)^3/(a^3-3*a^2*b+3*a*b^2-b^3)*tanh(1/2*d*x+1/2*c)^11+58/d*b*a
/(b*tanh(1/2*d*x+1/2*c)^4+4*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+b)^3/(a^3-3*a^2*b+3*a*b^2-b^3)*t
anh(1/2*d*x+1/2*c)^7+58/d*b*a/(b*tanh(1/2*d*x+1/2*c)^4+4*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+b)^
3/(a^3-3*a^2*b+3*a*b^2-b^3)*tanh(1/2*d*x+1/2*c)^5

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

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a + b \operatorname{csch}^{2}{\left (c + d x \right )}\right )^{4}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.33571, size = 814, normalized size = 3.7 \begin{align*} -\frac{{\left (35 \, a^{3} b - 70 \, a^{2} b^{2} + 56 \, a b^{3} - 16 \, 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 )}{16 \,{\left (a^{7} d - 3 \, a^{6} b d + 3 \, a^{5} b^{2} d - a^{4} b^{3} d\right )} \sqrt{a b - b^{2}}} + \frac{87 \, a^{5} b e^{\left (10 \, d x + 10 \, c\right )} - 366 \, a^{4} b^{2} e^{\left (10 \, d x + 10 \, c\right )} + 408 \, a^{3} b^{3} e^{\left (10 \, d x + 10 \, c\right )} - 144 \, a^{2} b^{4} e^{\left (10 \, d x + 10 \, c\right )} - 435 \, a^{5} b e^{\left (8 \, d x + 8 \, c\right )} + 2124 \, a^{4} b^{2} e^{\left (8 \, d x + 8 \, c\right )} - 3972 \, a^{3} b^{3} e^{\left (8 \, d x + 8 \, c\right )} + 3072 \, a^{2} b^{4} e^{\left (8 \, d x + 8 \, c\right )} - 864 \, a b^{5} e^{\left (8 \, d x + 8 \, c\right )} + 870 \, a^{5} b e^{\left (6 \, d x + 6 \, c\right )} - 4292 \, a^{4} b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 8792 \, a^{3} b^{3} e^{\left (6 \, d x + 6 \, c\right )} - 9936 \, a^{2} b^{4} e^{\left (6 \, d x + 6 \, c\right )} + 5824 \, a b^{5} e^{\left (6 \, d x + 6 \, c\right )} - 1408 \, b^{6} e^{\left (6 \, d x + 6 \, c\right )} - 870 \, a^{5} b e^{\left (4 \, d x + 4 \, c\right )} + 3792 \, a^{4} b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 6432 \, a^{3} b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 4608 \, a^{2} b^{4} e^{\left (4 \, d x + 4 \, c\right )} - 1248 \, a b^{5} e^{\left (4 \, d x + 4 \, c\right )} + 435 \, a^{5} b e^{\left (2 \, d x + 2 \, c\right )} - 1374 \, a^{4} b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 1248 \, a^{3} b^{3} e^{\left (2 \, d x + 2 \, c\right )} - 384 \, a^{2} b^{4} e^{\left (2 \, d x + 2 \, c\right )} - 87 \, a^{5} b + 116 \, a^{4} b^{2} - 44 \, a^{3} b^{3}}{24 \,{\left (a^{7} d - 3 \, a^{6} b d + 3 \, a^{5} b^{2} d - a^{4} b^{3} d\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 )}^{3}} + \frac{d x + c}{a^{4} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16*(35*a^3*b - 70*a^2*b^2 + 56*a*b^3 - 16*b^4)*arctan(1/2*(a*e^(2*d*x + 2*c) - a + 2*b)/sqrt(a*b - b^2))/((
a^7*d - 3*a^6*b*d + 3*a^5*b^2*d - a^4*b^3*d)*sqrt(a*b - b^2)) + 1/24*(87*a^5*b*e^(10*d*x + 10*c) - 366*a^4*b^2
*e^(10*d*x + 10*c) + 408*a^3*b^3*e^(10*d*x + 10*c) - 144*a^2*b^4*e^(10*d*x + 10*c) - 435*a^5*b*e^(8*d*x + 8*c)
+ 2124*a^4*b^2*e^(8*d*x + 8*c) - 3972*a^3*b^3*e^(8*d*x + 8*c) + 3072*a^2*b^4*e^(8*d*x + 8*c) - 864*a*b^5*e^(8
*d*x + 8*c) + 870*a^5*b*e^(6*d*x + 6*c) - 4292*a^4*b^2*e^(6*d*x + 6*c) + 8792*a^3*b^3*e^(6*d*x + 6*c) - 9936*a
^2*b^4*e^(6*d*x + 6*c) + 5824*a*b^5*e^(6*d*x + 6*c) - 1408*b^6*e^(6*d*x + 6*c) - 870*a^5*b*e^(4*d*x + 4*c) + 3
792*a^4*b^2*e^(4*d*x + 4*c) - 6432*a^3*b^3*e^(4*d*x + 4*c) + 4608*a^2*b^4*e^(4*d*x + 4*c) - 1248*a*b^5*e^(4*d*
x + 4*c) + 435*a^5*b*e^(2*d*x + 2*c) - 1374*a^4*b^2*e^(2*d*x + 2*c) + 1248*a^3*b^3*e^(2*d*x + 2*c) - 384*a^2*b
^4*e^(2*d*x + 2*c) - 87*a^5*b + 116*a^4*b^2 - 44*a^3*b^3)/((a^7*d - 3*a^6*b*d + 3*a^5*b^2*d - a^4*b^3*d)*(a*e^
(4*d*x + 4*c) - 2*a*e^(2*d*x + 2*c) + 4*b*e^(2*d*x + 2*c) + a)^3) + (d*x + c)/(a^4*d)