3.7 \(\int \frac{1}{(a+b \text{csch}^2(c+d x))^3} \, dx\)

Optimal. Leaf size=156 \[ -\frac{\sqrt{b} \left (15 a^2-20 a b+8 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{b}}\right )}{8 a^3 d (a-b)^{5/2}}+\frac{b (7 a-4 b) \coth (c+d x)}{8 a^2 d (a-b)^2 \left (a+b \coth ^2(c+d x)-b\right )}+\frac{x}{a^3}+\frac{b \coth (c+d x)}{4 a d (a-b) \left (a+b \coth ^2(c+d x)-b\right )^2} \]

[Out]

x/a^3 - (Sqrt[b]*(15*a^2 - 20*a*b + 8*b^2)*ArcTan[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[b]])/(8*a^3*(a - b)^(5/2)*d
) + (b*Coth[c + d*x])/(4*a*(a - b)*d*(a - b + b*Coth[c + d*x]^2)^2) + ((7*a - 4*b)*b*Coth[c + d*x])/(8*a^2*(a
- b)^2*d*(a - b + b*Coth[c + d*x]^2))

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Rubi [A]  time = 0.215668, antiderivative size = 156, normalized size of antiderivative = 1., number of steps used = 6, 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} \[ -\frac{\sqrt{b} \left (15 a^2-20 a b+8 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{b}}\right )}{8 a^3 d (a-b)^{5/2}}+\frac{b (7 a-4 b) \coth (c+d x)}{8 a^2 d (a-b)^2 \left (a+b \coth ^2(c+d x)-b\right )}+\frac{x}{a^3}+\frac{b \coth (c+d x)}{4 a d (a-b) \left (a+b \coth ^2(c+d x)-b\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Csch[c + d*x]^2)^(-3),x]

[Out]

x/a^3 - (Sqrt[b]*(15*a^2 - 20*a*b + 8*b^2)*ArcTan[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[b]])/(8*a^3*(a - b)^(5/2)*d
) + (b*Coth[c + d*x])/(4*a*(a - b)*d*(a - b + b*Coth[c + d*x]^2)^2) + ((7*a - 4*b)*b*Coth[c + d*x])/(8*a^2*(a
- b)^2*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 )^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\left (1-x^2\right ) \left (a-b+b x^2\right )^3} \, dx,x,\coth (c+d x)\right )}{d}\\ &=\frac{b \coth (c+d x)}{4 a (a-b) d \left (a-b+b \coth ^2(c+d x)\right )^2}-\frac{\operatorname{Subst}\left (\int \frac{-4 a+b+3 b x^2}{\left (1-x^2\right ) \left (a-b+b x^2\right )^2} \, dx,x,\coth (c+d x)\right )}{4 a (a-b) d}\\ &=\frac{b \coth (c+d x)}{4 a (a-b) d \left (a-b+b \coth ^2(c+d x)\right )^2}+\frac{(7 a-4 b) b \coth (c+d x)}{8 a^2 (a-b)^2 d \left (a-b+b \coth ^2(c+d x)\right )}+\frac{\operatorname{Subst}\left (\int \frac{8 a^2-9 a b+4 b^2-(7 a-4 b) b x^2}{\left (1-x^2\right ) \left (a-b+b x^2\right )} \, dx,x,\coth (c+d x)\right )}{8 a^2 (a-b)^2 d}\\ &=\frac{b \coth (c+d x)}{4 a (a-b) d \left (a-b+b \coth ^2(c+d x)\right )^2}+\frac{(7 a-4 b) b \coth (c+d x)}{8 a^2 (a-b)^2 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^3 d}+\frac{\left (b \left (15 a^2-20 a b+8 b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a-b+b x^2} \, dx,x,\coth (c+d x)\right )}{8 a^3 (a-b)^2 d}\\ &=\frac{x}{a^3}-\frac{\sqrt{b} \left (15 a^2-20 a b+8 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{b}}\right )}{8 a^3 (a-b)^{5/2} d}+\frac{b \coth (c+d x)}{4 a (a-b) d \left (a-b+b \coth ^2(c+d x)\right )^2}+\frac{(7 a-4 b) b \coth (c+d x)}{8 a^2 (a-b)^2 d \left (a-b+b \coth ^2(c+d x)\right )}\\ \end{align*}

Mathematica [A]  time = 1.53437, size = 210, normalized size = 1.35 \[ \frac{\text{csch}^6(c+d x) (a \cosh (2 (c+d x))-a+2 b) \left (-\frac{\sqrt{b} \left (15 a^2-20 a b+8 b^2\right ) (a (-\cosh (2 (c+d x)))+a-2 b)^2 \tan ^{-1}\left (\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{b}}\right )}{(a-b)^{5/2}}-\frac{4 a b^2 \sinh (2 (c+d x))}{a-b}+8 (c+d x) (a (-\cosh (2 (c+d x)))+a-2 b)^2+\frac{3 a b (3 a-2 b) \sinh (2 (c+d x)) (a \cosh (2 (c+d x))-a+2 b)}{(a-b)^2}\right )}{64 a^3 d \left (a+b \text{csch}^2(c+d x)\right )^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Csch[c + d*x]^2)^(-3),x]

[Out]

((-a + 2*b + a*Cosh[2*(c + d*x)])*Csch[c + d*x]^6*(8*(c + d*x)*(a - 2*b - a*Cosh[2*(c + d*x)])^2 - (Sqrt[b]*(1
5*a^2 - 20*a*b + 8*b^2)*ArcTan[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[b]]*(a - 2*b - a*Cosh[2*(c + d*x)])^2)/(a - b)
^(5/2) - (4*a*b^2*Sinh[2*(c + d*x)])/(a - b) + (3*a*(3*a - 2*b)*b*(-a + 2*b + a*Cosh[2*(c + d*x)])*Sinh[2*(c +
d*x)])/(a - b)^2))/(64*a^3*d*(a + b*Csch[c + d*x]^2)^3)

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Maple [B]  time = 0.069, size = 1798, normalized size = 11.5 \begin{align*} \text{result 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)^3,x)

[Out]

1/d/a^3*ln(tanh(1/2*d*x+1/2*c)+1)+7/4/d*b^2/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)^2/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^7-1/d*b^3/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)^2/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^7+9/d*b/(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)^2/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^5-31/4/d*b^2/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)^2/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/
2*c)^5+1/d*b^3/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)^2/(a^2-2*a*
b+b^2)*tanh(1/2*d*x+1/2*c)^5+9/d*b/(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)^2/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^3-31/4/d*b^2/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)^2/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^3+1/d*b^3/a^2/(b*tanh(1/2*d*x+1/2*c)^4+4*tan
h(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+b)^2/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^3+7/4/d*b^2/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)^2/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)-
1/d*b^3/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)^2/(a^2-2*a*b+b^2)*
tanh(1/2*d*x+1/2*c)+15/8/d*b/a/(a^2-2*a*b+b^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))-5/2/d*b^2/a^2/(a^2-2*a*b+b^2)/((2*(a*(a-b))^(1/2)+2*a-b)*b)^(1/2)*arcta
n(tanh(1/2*d*x+1/2*c)*b/((2*(a*(a-b))^(1/2)+2*a-b)*b)^(1/2))+1/d*b^3/a^3/(a^2-2*a*b+b^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))+15/8/d*b/(a^2-2*a*b+b^2)/(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))-5/2/d*b^2/a/(a^2-2*a*b+b^2)/(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))+1/d*b^3/a^2/(a^2-2*a*b+b^2)/(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))-15/8/d*b/a/(a^2-2*a*b+b^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))+5/2/d*b^2/a^2/
(a^2-2*a*b+b^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))-1/d*b^3/a^3/(a^2-2*a*b+b^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))+15/8/d*b/(a^2-2*a*b+b^2)/(a*(a-b))^(1/2)/((2*(a*(a-b))^(1/2)-2*a+b)*b)^(1/2)*arc
tanh(tanh(1/2*d*x+1/2*c)*b/((2*(a*(a-b))^(1/2)-2*a+b)*b)^(1/2))-5/2/d*b^2/a/(a^2-2*a*b+b^2)/(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))+1/d*b^3/a
^2/(a^2-2*a*b+b^2)/(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))-1/d/a^3*ln(tanh(1/2*d*x+1/2*c)-1)

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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)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.50347, size = 15505, normalized size = 99.39 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*csch(d*x+c)^2)^3,x, algorithm="fricas")

[Out]

[1/16*(16*(a^4 - 2*a^3*b + a^2*b^2)*d*x*cosh(d*x + c)^8 + 128*(a^4 - 2*a^3*b + a^2*b^2)*d*x*cosh(d*x + c)*sinh
(d*x + c)^7 + 16*(a^4 - 2*a^3*b + a^2*b^2)*d*x*sinh(d*x + c)^8 + 4*(9*a^3*b - 28*a^2*b^2 + 16*a*b^3 - 16*(a^4
- 4*a^3*b + 5*a^2*b^2 - 2*a*b^3)*d*x)*cosh(d*x + c)^6 + 4*(112*(a^4 - 2*a^3*b + a^2*b^2)*d*x*cosh(d*x + c)^2 +
9*a^3*b - 28*a^2*b^2 + 16*a*b^3 - 16*(a^4 - 4*a^3*b + 5*a^2*b^2 - 2*a*b^3)*d*x)*sinh(d*x + c)^6 + 8*(112*(a^4
- 2*a^3*b + a^2*b^2)*d*x*cosh(d*x + c)^3 + 3*(9*a^3*b - 28*a^2*b^2 + 16*a*b^3 - 16*(a^4 - 4*a^3*b + 5*a^2*b^2
- 2*a*b^3)*d*x)*cosh(d*x + c))*sinh(d*x + c)^5 - 4*(27*a^3*b - 90*a^2*b^2 + 120*a*b^3 - 48*b^4 - 8*(3*a^4 - 1
4*a^3*b + 27*a^2*b^2 - 24*a*b^3 + 8*b^4)*d*x)*cosh(d*x + c)^4 + 4*(280*(a^4 - 2*a^3*b + a^2*b^2)*d*x*cosh(d*x
+ c)^4 - 27*a^3*b + 90*a^2*b^2 - 120*a*b^3 + 48*b^4 + 8*(3*a^4 - 14*a^3*b + 27*a^2*b^2 - 24*a*b^3 + 8*b^4)*d*x
+ 15*(9*a^3*b - 28*a^2*b^2 + 16*a*b^3 - 16*(a^4 - 4*a^3*b + 5*a^2*b^2 - 2*a*b^3)*d*x)*cosh(d*x + c)^2)*sinh(d
*x + c)^4 - 36*a^3*b + 24*a^2*b^2 + 16*(56*(a^4 - 2*a^3*b + a^2*b^2)*d*x*cosh(d*x + c)^5 + 5*(9*a^3*b - 28*a^2
*b^2 + 16*a*b^3 - 16*(a^4 - 4*a^3*b + 5*a^2*b^2 - 2*a*b^3)*d*x)*cosh(d*x + c)^3 - (27*a^3*b - 90*a^2*b^2 + 120
*a*b^3 - 48*b^4 - 8*(3*a^4 - 14*a^3*b + 27*a^2*b^2 - 24*a*b^3 + 8*b^4)*d*x)*cosh(d*x + c))*sinh(d*x + c)^3 + 1
6*(a^4 - 2*a^3*b + a^2*b^2)*d*x + 4*(27*a^3*b - 68*a^2*b^2 + 32*a*b^3 - 16*(a^4 - 4*a^3*b + 5*a^2*b^2 - 2*a*b^
3)*d*x)*cosh(d*x + c)^2 + 4*(112*(a^4 - 2*a^3*b + a^2*b^2)*d*x*cosh(d*x + c)^6 + 15*(9*a^3*b - 28*a^2*b^2 + 16
*a*b^3 - 16*(a^4 - 4*a^3*b + 5*a^2*b^2 - 2*a*b^3)*d*x)*cosh(d*x + c)^4 + 27*a^3*b - 68*a^2*b^2 + 32*a*b^3 - 16
*(a^4 - 4*a^3*b + 5*a^2*b^2 - 2*a*b^3)*d*x - 6*(27*a^3*b - 90*a^2*b^2 + 120*a*b^3 - 48*b^4 - 8*(3*a^4 - 14*a^3
*b + 27*a^2*b^2 - 24*a*b^3 + 8*b^4)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + ((15*a^4 - 20*a^3*b + 8*a^2*b^2)*c
osh(d*x + c)^8 + 8*(15*a^4 - 20*a^3*b + 8*a^2*b^2)*cosh(d*x + c)*sinh(d*x + c)^7 + (15*a^4 - 20*a^3*b + 8*a^2*
b^2)*sinh(d*x + c)^8 - 4*(15*a^4 - 50*a^3*b + 48*a^2*b^2 - 16*a*b^3)*cosh(d*x + c)^6 - 4*(15*a^4 - 50*a^3*b +
48*a^2*b^2 - 16*a*b^3 - 7*(15*a^4 - 20*a^3*b + 8*a^2*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 8*(7*(15*a^4 - 20
*a^3*b + 8*a^2*b^2)*cosh(d*x + c)^3 - 3*(15*a^4 - 50*a^3*b + 48*a^2*b^2 - 16*a*b^3)*cosh(d*x + c))*sinh(d*x +
c)^5 + 2*(45*a^4 - 180*a^3*b + 304*a^2*b^2 - 224*a*b^3 + 64*b^4)*cosh(d*x + c)^4 + 2*(35*(15*a^4 - 20*a^3*b +
8*a^2*b^2)*cosh(d*x + c)^4 + 45*a^4 - 180*a^3*b + 304*a^2*b^2 - 224*a*b^3 + 64*b^4 - 30*(15*a^4 - 50*a^3*b + 4
8*a^2*b^2 - 16*a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 15*a^4 - 20*a^3*b + 8*a^2*b^2 + 8*(7*(15*a^4 - 20*a^3
*b + 8*a^2*b^2)*cosh(d*x + c)^5 - 10*(15*a^4 - 50*a^3*b + 48*a^2*b^2 - 16*a*b^3)*cosh(d*x + c)^3 + (45*a^4 - 1
80*a^3*b + 304*a^2*b^2 - 224*a*b^3 + 64*b^4)*cosh(d*x + c))*sinh(d*x + c)^3 - 4*(15*a^4 - 50*a^3*b + 48*a^2*b^
2 - 16*a*b^3)*cosh(d*x + c)^2 + 4*(7*(15*a^4 - 20*a^3*b + 8*a^2*b^2)*cosh(d*x + c)^6 - 15*(15*a^4 - 50*a^3*b +
48*a^2*b^2 - 16*a*b^3)*cosh(d*x + c)^4 - 15*a^4 + 50*a^3*b - 48*a^2*b^2 + 16*a*b^3 + 3*(45*a^4 - 180*a^3*b +
304*a^2*b^2 - 224*a*b^3 + 64*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*((15*a^4 - 20*a^3*b + 8*a^2*b^2)*cosh(d
*x + c)^7 - 3*(15*a^4 - 50*a^3*b + 48*a^2*b^2 - 16*a*b^3)*cosh(d*x + c)^5 + (45*a^4 - 180*a^3*b + 304*a^2*b^2
- 224*a*b^3 + 64*b^4)*cosh(d*x + c)^3 - (15*a^4 - 50*a^3*b + 48*a^2*b^2 - 16*a*b^3)*cosh(d*x + c))*sinh(d*x +
c))*sqrt(-b/(a - b))*log((a^2*cosh(d*x + c)^4 + 4*a^2*cosh(d*x + c)*sinh(d*x + c)^3 + a^2*sinh(d*x + c)^4 - 2*
(a^2 - 2*a*b)*cosh(d*x + c)^2 + 2*(3*a^2*cosh(d*x + c)^2 - a^2 + 2*a*b)*sinh(d*x + c)^2 + a^2 - 8*a*b + 8*b^2
+ 4*(a^2*cosh(d*x + c)^3 - (a^2 - 2*a*b)*cosh(d*x + c))*sinh(d*x + c) - 4*((a^2 - a*b)*cosh(d*x + c)^2 + 2*(a^
2 - a*b)*cosh(d*x + c)*sinh(d*x + c) + (a^2 - a*b)*sinh(d*x + c)^2 - a^2 + 3*a*b - 2*b^2)*sqrt(-b/(a - b)))/(a
*cosh(d*x + c)^4 + 4*a*cosh(d*x + c)*sinh(d*x + c)^3 + a*sinh(d*x + c)^4 - 2*(a - 2*b)*cosh(d*x + c)^2 + 2*(3*
a*cosh(d*x + c)^2 - a + 2*b)*sinh(d*x + c)^2 + 4*(a*cosh(d*x + c)^3 - (a - 2*b)*cosh(d*x + c))*sinh(d*x + c) +
a)) + 8*(16*(a^4 - 2*a^3*b + a^2*b^2)*d*x*cosh(d*x + c)^7 + 3*(9*a^3*b - 28*a^2*b^2 + 16*a*b^3 - 16*(a^4 - 4*
a^3*b + 5*a^2*b^2 - 2*a*b^3)*d*x)*cosh(d*x + c)^5 - 2*(27*a^3*b - 90*a^2*b^2 + 120*a*b^3 - 48*b^4 - 8*(3*a^4 -
14*a^3*b + 27*a^2*b^2 - 24*a*b^3 + 8*b^4)*d*x)*cosh(d*x + c)^3 + (27*a^3*b - 68*a^2*b^2 + 32*a*b^3 - 16*(a^4
- 4*a^3*b + 5*a^2*b^2 - 2*a*b^3)*d*x)*cosh(d*x + c))*sinh(d*x + c))/((a^7 - 2*a^6*b + a^5*b^2)*d*cosh(d*x + c)
^8 + 8*(a^7 - 2*a^6*b + a^5*b^2)*d*cosh(d*x + c)*sinh(d*x + c)^7 + (a^7 - 2*a^6*b + a^5*b^2)*d*sinh(d*x + c)^8
- 4*(a^7 - 4*a^6*b + 5*a^5*b^2 - 2*a^4*b^3)*d*cosh(d*x + c)^6 + 4*(7*(a^7 - 2*a^6*b + a^5*b^2)*d*cosh(d*x + c
)^2 - (a^7 - 4*a^6*b + 5*a^5*b^2 - 2*a^4*b^3)*d)*sinh(d*x + c)^6 + 2*(3*a^7 - 14*a^6*b + 27*a^5*b^2 - 24*a^4*b
^3 + 8*a^3*b^4)*d*cosh(d*x + c)^4 + 8*(7*(a^7 - 2*a^6*b + a^5*b^2)*d*cosh(d*x + c)^3 - 3*(a^7 - 4*a^6*b + 5*a^
5*b^2 - 2*a^4*b^3)*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(35*(a^7 - 2*a^6*b + a^5*b^2)*d*cosh(d*x + c)^4 - 30*(
a^7 - 4*a^6*b + 5*a^5*b^2 - 2*a^4*b^3)*d*cosh(d*x + c)^2 + (3*a^7 - 14*a^6*b + 27*a^5*b^2 - 24*a^4*b^3 + 8*a^3
*b^4)*d)*sinh(d*x + c)^4 - 4*(a^7 - 4*a^6*b + 5*a^5*b^2 - 2*a^4*b^3)*d*cosh(d*x + c)^2 + 8*(7*(a^7 - 2*a^6*b +
a^5*b^2)*d*cosh(d*x + c)^5 - 10*(a^7 - 4*a^6*b + 5*a^5*b^2 - 2*a^4*b^3)*d*cosh(d*x + c)^3 + (3*a^7 - 14*a^6*b
+ 27*a^5*b^2 - 24*a^4*b^3 + 8*a^3*b^4)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*(a^7 - 2*a^6*b + a^5*b^2)*d*co
sh(d*x + c)^6 - 15*(a^7 - 4*a^6*b + 5*a^5*b^2 - 2*a^4*b^3)*d*cosh(d*x + c)^4 + 3*(3*a^7 - 14*a^6*b + 27*a^5*b^
2 - 24*a^4*b^3 + 8*a^3*b^4)*d*cosh(d*x + c)^2 - (a^7 - 4*a^6*b + 5*a^5*b^2 - 2*a^4*b^3)*d)*sinh(d*x + c)^2 + (
a^7 - 2*a^6*b + a^5*b^2)*d + 8*((a^7 - 2*a^6*b + a^5*b^2)*d*cosh(d*x + c)^7 - 3*(a^7 - 4*a^6*b + 5*a^5*b^2 - 2
*a^4*b^3)*d*cosh(d*x + c)^5 + (3*a^7 - 14*a^6*b + 27*a^5*b^2 - 24*a^4*b^3 + 8*a^3*b^4)*d*cosh(d*x + c)^3 - (a^
7 - 4*a^6*b + 5*a^5*b^2 - 2*a^4*b^3)*d*cosh(d*x + c))*sinh(d*x + c)), 1/8*(8*(a^4 - 2*a^3*b + a^2*b^2)*d*x*cos
h(d*x + c)^8 + 64*(a^4 - 2*a^3*b + a^2*b^2)*d*x*cosh(d*x + c)*sinh(d*x + c)^7 + 8*(a^4 - 2*a^3*b + a^2*b^2)*d*
x*sinh(d*x + c)^8 + 2*(9*a^3*b - 28*a^2*b^2 + 16*a*b^3 - 16*(a^4 - 4*a^3*b + 5*a^2*b^2 - 2*a*b^3)*d*x)*cosh(d*
x + c)^6 + 2*(112*(a^4 - 2*a^3*b + a^2*b^2)*d*x*cosh(d*x + c)^2 + 9*a^3*b - 28*a^2*b^2 + 16*a*b^3 - 16*(a^4 -
4*a^3*b + 5*a^2*b^2 - 2*a*b^3)*d*x)*sinh(d*x + c)^6 + 4*(112*(a^4 - 2*a^3*b + a^2*b^2)*d*x*cosh(d*x + c)^3 + 3
*(9*a^3*b - 28*a^2*b^2 + 16*a*b^3 - 16*(a^4 - 4*a^3*b + 5*a^2*b^2 - 2*a*b^3)*d*x)*cosh(d*x + c))*sinh(d*x + c)
^5 - 2*(27*a^3*b - 90*a^2*b^2 + 120*a*b^3 - 48*b^4 - 8*(3*a^4 - 14*a^3*b + 27*a^2*b^2 - 24*a*b^3 + 8*b^4)*d*x)
*cosh(d*x + c)^4 + 2*(280*(a^4 - 2*a^3*b + a^2*b^2)*d*x*cosh(d*x + c)^4 - 27*a^3*b + 90*a^2*b^2 - 120*a*b^3 +
48*b^4 + 8*(3*a^4 - 14*a^3*b + 27*a^2*b^2 - 24*a*b^3 + 8*b^4)*d*x + 15*(9*a^3*b - 28*a^2*b^2 + 16*a*b^3 - 16*(
a^4 - 4*a^3*b + 5*a^2*b^2 - 2*a*b^3)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c)^4 - 18*a^3*b + 12*a^2*b^2 + 8*(56*(a^
4 - 2*a^3*b + a^2*b^2)*d*x*cosh(d*x + c)^5 + 5*(9*a^3*b - 28*a^2*b^2 + 16*a*b^3 - 16*(a^4 - 4*a^3*b + 5*a^2*b^
2 - 2*a*b^3)*d*x)*cosh(d*x + c)^3 - (27*a^3*b - 90*a^2*b^2 + 120*a*b^3 - 48*b^4 - 8*(3*a^4 - 14*a^3*b + 27*a^2
*b^2 - 24*a*b^3 + 8*b^4)*d*x)*cosh(d*x + c))*sinh(d*x + c)^3 + 8*(a^4 - 2*a^3*b + a^2*b^2)*d*x + 2*(27*a^3*b -
68*a^2*b^2 + 32*a*b^3 - 16*(a^4 - 4*a^3*b + 5*a^2*b^2 - 2*a*b^3)*d*x)*cosh(d*x + c)^2 + 2*(112*(a^4 - 2*a^3*b
+ a^2*b^2)*d*x*cosh(d*x + c)^6 + 15*(9*a^3*b - 28*a^2*b^2 + 16*a*b^3 - 16*(a^4 - 4*a^3*b + 5*a^2*b^2 - 2*a*b^
3)*d*x)*cosh(d*x + c)^4 + 27*a^3*b - 68*a^2*b^2 + 32*a*b^3 - 16*(a^4 - 4*a^3*b + 5*a^2*b^2 - 2*a*b^3)*d*x - 6*
(27*a^3*b - 90*a^2*b^2 + 120*a*b^3 - 48*b^4 - 8*(3*a^4 - 14*a^3*b + 27*a^2*b^2 - 24*a*b^3 + 8*b^4)*d*x)*cosh(d
*x + c)^2)*sinh(d*x + c)^2 - ((15*a^4 - 20*a^3*b + 8*a^2*b^2)*cosh(d*x + c)^8 + 8*(15*a^4 - 20*a^3*b + 8*a^2*b
^2)*cosh(d*x + c)*sinh(d*x + c)^7 + (15*a^4 - 20*a^3*b + 8*a^2*b^2)*sinh(d*x + c)^8 - 4*(15*a^4 - 50*a^3*b + 4
8*a^2*b^2 - 16*a*b^3)*cosh(d*x + c)^6 - 4*(15*a^4 - 50*a^3*b + 48*a^2*b^2 - 16*a*b^3 - 7*(15*a^4 - 20*a^3*b +
8*a^2*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 8*(7*(15*a^4 - 20*a^3*b + 8*a^2*b^2)*cosh(d*x + c)^3 - 3*(15*a^4
- 50*a^3*b + 48*a^2*b^2 - 16*a*b^3)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(45*a^4 - 180*a^3*b + 304*a^2*b^2 - 22
4*a*b^3 + 64*b^4)*cosh(d*x + c)^4 + 2*(35*(15*a^4 - 20*a^3*b + 8*a^2*b^2)*cosh(d*x + c)^4 + 45*a^4 - 180*a^3*b
+ 304*a^2*b^2 - 224*a*b^3 + 64*b^4 - 30*(15*a^4 - 50*a^3*b + 48*a^2*b^2 - 16*a*b^3)*cosh(d*x + c)^2)*sinh(d*x
+ c)^4 + 15*a^4 - 20*a^3*b + 8*a^2*b^2 + 8*(7*(15*a^4 - 20*a^3*b + 8*a^2*b^2)*cosh(d*x + c)^5 - 10*(15*a^4 -
50*a^3*b + 48*a^2*b^2 - 16*a*b^3)*cosh(d*x + c)^3 + (45*a^4 - 180*a^3*b + 304*a^2*b^2 - 224*a*b^3 + 64*b^4)*co
sh(d*x + c))*sinh(d*x + c)^3 - 4*(15*a^4 - 50*a^3*b + 48*a^2*b^2 - 16*a*b^3)*cosh(d*x + c)^2 + 4*(7*(15*a^4 -
20*a^3*b + 8*a^2*b^2)*cosh(d*x + c)^6 - 15*(15*a^4 - 50*a^3*b + 48*a^2*b^2 - 16*a*b^3)*cosh(d*x + c)^4 - 15*a^
4 + 50*a^3*b - 48*a^2*b^2 + 16*a*b^3 + 3*(45*a^4 - 180*a^3*b + 304*a^2*b^2 - 224*a*b^3 + 64*b^4)*cosh(d*x + c)
^2)*sinh(d*x + c)^2 + 8*((15*a^4 - 20*a^3*b + 8*a^2*b^2)*cosh(d*x + c)^7 - 3*(15*a^4 - 50*a^3*b + 48*a^2*b^2 -
16*a*b^3)*cosh(d*x + c)^5 + (45*a^4 - 180*a^3*b + 304*a^2*b^2 - 224*a*b^3 + 64*b^4)*cosh(d*x + c)^3 - (15*a^4
- 50*a^3*b + 48*a^2*b^2 - 16*a*b^3)*cosh(d*x + c))*sinh(d*x + c))*sqrt(b/(a - b))*arctan(1/2*(a*cosh(d*x + c)
^2 + 2*a*cosh(d*x + c)*sinh(d*x + c) + a*sinh(d*x + c)^2 - a + 2*b)*sqrt(b/(a - b))/b) + 4*(16*(a^4 - 2*a^3*b
+ a^2*b^2)*d*x*cosh(d*x + c)^7 + 3*(9*a^3*b - 28*a^2*b^2 + 16*a*b^3 - 16*(a^4 - 4*a^3*b + 5*a^2*b^2 - 2*a*b^3)
*d*x)*cosh(d*x + c)^5 - 2*(27*a^3*b - 90*a^2*b^2 + 120*a*b^3 - 48*b^4 - 8*(3*a^4 - 14*a^3*b + 27*a^2*b^2 - 24*
a*b^3 + 8*b^4)*d*x)*cosh(d*x + c)^3 + (27*a^3*b - 68*a^2*b^2 + 32*a*b^3 - 16*(a^4 - 4*a^3*b + 5*a^2*b^2 - 2*a*
b^3)*d*x)*cosh(d*x + c))*sinh(d*x + c))/((a^7 - 2*a^6*b + a^5*b^2)*d*cosh(d*x + c)^8 + 8*(a^7 - 2*a^6*b + a^5*
b^2)*d*cosh(d*x + c)*sinh(d*x + c)^7 + (a^7 - 2*a^6*b + a^5*b^2)*d*sinh(d*x + c)^8 - 4*(a^7 - 4*a^6*b + 5*a^5*
b^2 - 2*a^4*b^3)*d*cosh(d*x + c)^6 + 4*(7*(a^7 - 2*a^6*b + a^5*b^2)*d*cosh(d*x + c)^2 - (a^7 - 4*a^6*b + 5*a^5
*b^2 - 2*a^4*b^3)*d)*sinh(d*x + c)^6 + 2*(3*a^7 - 14*a^6*b + 27*a^5*b^2 - 24*a^4*b^3 + 8*a^3*b^4)*d*cosh(d*x +
c)^4 + 8*(7*(a^7 - 2*a^6*b + a^5*b^2)*d*cosh(d*x + c)^3 - 3*(a^7 - 4*a^6*b + 5*a^5*b^2 - 2*a^4*b^3)*d*cosh(d*
x + c))*sinh(d*x + c)^5 + 2*(35*(a^7 - 2*a^6*b + a^5*b^2)*d*cosh(d*x + c)^4 - 30*(a^7 - 4*a^6*b + 5*a^5*b^2 -
2*a^4*b^3)*d*cosh(d*x + c)^2 + (3*a^7 - 14*a^6*b + 27*a^5*b^2 - 24*a^4*b^3 + 8*a^3*b^4)*d)*sinh(d*x + c)^4 - 4
*(a^7 - 4*a^6*b + 5*a^5*b^2 - 2*a^4*b^3)*d*cosh(d*x + c)^2 + 8*(7*(a^7 - 2*a^6*b + a^5*b^2)*d*cosh(d*x + c)^5
- 10*(a^7 - 4*a^6*b + 5*a^5*b^2 - 2*a^4*b^3)*d*cosh(d*x + c)^3 + (3*a^7 - 14*a^6*b + 27*a^5*b^2 - 24*a^4*b^3 +
8*a^3*b^4)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*(a^7 - 2*a^6*b + a^5*b^2)*d*cosh(d*x + c)^6 - 15*(a^7 - 4*
a^6*b + 5*a^5*b^2 - 2*a^4*b^3)*d*cosh(d*x + c)^4 + 3*(3*a^7 - 14*a^6*b + 27*a^5*b^2 - 24*a^4*b^3 + 8*a^3*b^4)*
d*cosh(d*x + c)^2 - (a^7 - 4*a^6*b + 5*a^5*b^2 - 2*a^4*b^3)*d)*sinh(d*x + c)^2 + (a^7 - 2*a^6*b + a^5*b^2)*d +
8*((a^7 - 2*a^6*b + a^5*b^2)*d*cosh(d*x + c)^7 - 3*(a^7 - 4*a^6*b + 5*a^5*b^2 - 2*a^4*b^3)*d*cosh(d*x + c)^5
+ (3*a^7 - 14*a^6*b + 27*a^5*b^2 - 24*a^4*b^3 + 8*a^3*b^4)*d*cosh(d*x + c)^3 - (a^7 - 4*a^6*b + 5*a^5*b^2 - 2*
a^4*b^3)*d*cosh(d*x + c))*sinh(d*x + c))]

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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 )^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*csch(d*x+c)**2)**3,x)

[Out]

Integral((a + b*csch(c + d*x)**2)**(-3), x)

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Giac [B]  time = 1.27022, size = 450, normalized size = 2.88 \begin{align*} -\frac{{\left (15 \, a^{2} b - 20 \, a b^{2} + 8 \, b^{3}\right )} \arctan \left (\frac{a e^{\left (2 \, d x + 2 \, c\right )} - a + 2 \, b}{2 \, \sqrt{a b - b^{2}}}\right )}{8 \,{\left (a^{5} d - 2 \, a^{4} b d + a^{3} b^{2} d\right )} \sqrt{a b - b^{2}}} + \frac{9 \, a^{3} b e^{\left (6 \, d x + 6 \, c\right )} - 28 \, a^{2} b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 16 \, a b^{3} e^{\left (6 \, d x + 6 \, c\right )} - 27 \, a^{3} b e^{\left (4 \, d x + 4 \, c\right )} + 90 \, a^{2} b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 120 \, a b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 48 \, b^{4} e^{\left (4 \, d x + 4 \, c\right )} + 27 \, a^{3} b e^{\left (2 \, d x + 2 \, c\right )} - 68 \, a^{2} b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 32 \, a b^{3} e^{\left (2 \, d x + 2 \, c\right )} - 9 \, a^{3} b + 6 \, a^{2} b^{2}}{4 \,{\left (a^{5} d - 2 \, a^{4} b d + a^{3} b^{2} 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 )}^{2}} + \frac{d x + c}{a^{3} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*csch(d*x+c)^2)^3,x, algorithm="giac")

[Out]

-1/8*(15*a^2*b - 20*a*b^2 + 8*b^3)*arctan(1/2*(a*e^(2*d*x + 2*c) - a + 2*b)/sqrt(a*b - b^2))/((a^5*d - 2*a^4*b
*d + a^3*b^2*d)*sqrt(a*b - b^2)) + 1/4*(9*a^3*b*e^(6*d*x + 6*c) - 28*a^2*b^2*e^(6*d*x + 6*c) + 16*a*b^3*e^(6*d
*x + 6*c) - 27*a^3*b*e^(4*d*x + 4*c) + 90*a^2*b^2*e^(4*d*x + 4*c) - 120*a*b^3*e^(4*d*x + 4*c) + 48*b^4*e^(4*d*
x + 4*c) + 27*a^3*b*e^(2*d*x + 2*c) - 68*a^2*b^2*e^(2*d*x + 2*c) + 32*a*b^3*e^(2*d*x + 2*c) - 9*a^3*b + 6*a^2*
b^2)/((a^5*d - 2*a^4*b*d + a^3*b^2*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)^2) +
(d*x + c)/(a^3*d)