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

Optimal. Leaf size=163 \[ \frac {x}{a^3}-\frac {b^2 \left (5 a^2+2 b^2\right ) \coth (c+d x)}{2 a^2 d \left (a^2+b^2\right )^2 (a+b \text {csch}(c+d x))}-\frac {b^2 \coth (c+d x)}{2 a d \left (a^2+b^2\right ) (a+b \text {csch}(c+d x))^2}+\frac {b \left (6 a^4+5 a^2 b^2+2 b^4\right ) \tanh ^{-1}\left (\frac {a-b \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )}{a^3 d \left (a^2+b^2\right )^{5/2}} \]

[Out]

x/a^3+b*(6*a^4+5*a^2*b^2+2*b^4)*arctanh((a-b*tanh(1/2*d*x+1/2*c))/(a^2+b^2)^(1/2))/a^3/(a^2+b^2)^(5/2)/d-1/2*b
^2*coth(d*x+c)/a/(a^2+b^2)/d/(a+b*csch(d*x+c))^2-1/2*b^2*(5*a^2+2*b^2)*coth(d*x+c)/a^2/(a^2+b^2)^2/d/(a+b*csch
(d*x+c))

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Rubi [A]  time = 0.32, antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {3785, 4060, 3919, 3831, 2660, 618, 204} \[ \frac {b \left (5 a^2 b^2+6 a^4+2 b^4\right ) \tanh ^{-1}\left (\frac {a-b \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )}{a^3 d \left (a^2+b^2\right )^{5/2}}-\frac {b^2 \left (5 a^2+2 b^2\right ) \coth (c+d x)}{2 a^2 d \left (a^2+b^2\right )^2 (a+b \text {csch}(c+d x))}-\frac {b^2 \coth (c+d x)}{2 a d \left (a^2+b^2\right ) (a+b \text {csch}(c+d x))^2}+\frac {x}{a^3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Csch[c + d*x])^(-3),x]

[Out]

x/a^3 + (b*(6*a^4 + 5*a^2*b^2 + 2*b^4)*ArcTanh[(a - b*Tanh[(c + d*x)/2])/Sqrt[a^2 + b^2]])/(a^3*(a^2 + b^2)^(5
/2)*d) - (b^2*Coth[c + d*x])/(2*a*(a^2 + b^2)*d*(a + b*Csch[c + d*x])^2) - (b^2*(5*a^2 + 2*b^2)*Coth[c + d*x])
/(2*a^2*(a^2 + b^2)^2*d*(a + b*Csch[c + d*x]))

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 2660

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x]}, Dis
t[(2*e)/d, Subst[Int[1/(a + 2*b*e*x + a*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] &&
NeQ[a^2 - b^2, 0]

Rule 3785

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> Simp[(b^2*Cot[c + d*x]*(a + b*Csc[c + d*x])^(n +
1))/(a*d*(n + 1)*(a^2 - b^2)), x] + Dist[1/(a*(n + 1)*(a^2 - b^2)), Int[(a + b*Csc[c + d*x])^(n + 1)*Simp[(a^
2 - b^2)*(n + 1) - a*b*(n + 1)*Csc[c + d*x] + b^2*(n + 2)*Csc[c + d*x]^2, x], x], x] /; FreeQ[{a, b, c, d}, x]
&& NeQ[a^2 - b^2, 0] && LtQ[n, -1] && IntegerQ[2*n]

Rule 3831

Int[csc[(e_.) + (f_.)*(x_)]/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Dist[1/b, Int[1/(1 + (a*Sin[e
+ f*x])/b), x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]

Rule 3919

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[(c*x)/a,
x] - Dist[(b*c - a*d)/a, Int[Csc[e + f*x]/(a + b*Csc[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[
b*c - a*d, 0]

Rule 4060

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(b_.) +
(a_))^(m_), x_Symbol] :> Simp[((A*b^2 - a*b*B + a^2*C)*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1))/(a*f*(m + 1
)*(a^2 - b^2)), x] + Dist[1/(a*(m + 1)*(a^2 - b^2)), Int[(a + b*Csc[e + f*x])^(m + 1)*Simp[A*(a^2 - b^2)*(m +
1) - a*(A*b - a*B + b*C)*(m + 1)*Csc[e + f*x] + (A*b^2 - a*b*B + a^2*C)*(m + 2)*Csc[e + f*x]^2, x], x], x] /;
FreeQ[{a, b, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1]

Rubi steps

\begin {align*} \int \frac {1}{(a+b \text {csch}(c+d x))^3} \, dx &=-\frac {b^2 \coth (c+d x)}{2 a \left (a^2+b^2\right ) d (a+b \text {csch}(c+d x))^2}-\frac {\int \frac {-2 \left (a^2+b^2\right )+2 a b \text {csch}(c+d x)-b^2 \text {csch}^2(c+d x)}{(a+b \text {csch}(c+d x))^2} \, dx}{2 a \left (a^2+b^2\right )}\\ &=-\frac {b^2 \coth (c+d x)}{2 a \left (a^2+b^2\right ) d (a+b \text {csch}(c+d x))^2}-\frac {b^2 \left (5 a^2+2 b^2\right ) \coth (c+d x)}{2 a^2 \left (a^2+b^2\right )^2 d (a+b \text {csch}(c+d x))}+\frac {\int \frac {2 \left (a^2+b^2\right )^2-a b \left (4 a^2+b^2\right ) \text {csch}(c+d x)}{a+b \text {csch}(c+d x)} \, dx}{2 a^2 \left (a^2+b^2\right )^2}\\ &=\frac {x}{a^3}-\frac {b^2 \coth (c+d x)}{2 a \left (a^2+b^2\right ) d (a+b \text {csch}(c+d x))^2}-\frac {b^2 \left (5 a^2+2 b^2\right ) \coth (c+d x)}{2 a^2 \left (a^2+b^2\right )^2 d (a+b \text {csch}(c+d x))}-\frac {\left (b \left (6 a^4+5 a^2 b^2+2 b^4\right )\right ) \int \frac {\text {csch}(c+d x)}{a+b \text {csch}(c+d x)} \, dx}{2 a^3 \left (a^2+b^2\right )^2}\\ &=\frac {x}{a^3}-\frac {b^2 \coth (c+d x)}{2 a \left (a^2+b^2\right ) d (a+b \text {csch}(c+d x))^2}-\frac {b^2 \left (5 a^2+2 b^2\right ) \coth (c+d x)}{2 a^2 \left (a^2+b^2\right )^2 d (a+b \text {csch}(c+d x))}-\frac {\left (6 a^4+5 a^2 b^2+2 b^4\right ) \int \frac {1}{1+\frac {a \sinh (c+d x)}{b}} \, dx}{2 a^3 \left (a^2+b^2\right )^2}\\ &=\frac {x}{a^3}-\frac {b^2 \coth (c+d x)}{2 a \left (a^2+b^2\right ) d (a+b \text {csch}(c+d x))^2}-\frac {b^2 \left (5 a^2+2 b^2\right ) \coth (c+d x)}{2 a^2 \left (a^2+b^2\right )^2 d (a+b \text {csch}(c+d x))}+\frac {\left (i \left (6 a^4+5 a^2 b^2+2 b^4\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1-\frac {2 i a x}{b}+x^2} \, dx,x,\tan \left (\frac {1}{2} (i c+i d x)\right )\right )}{a^3 \left (a^2+b^2\right )^2 d}\\ &=\frac {x}{a^3}-\frac {b^2 \coth (c+d x)}{2 a \left (a^2+b^2\right ) d (a+b \text {csch}(c+d x))^2}-\frac {b^2 \left (5 a^2+2 b^2\right ) \coth (c+d x)}{2 a^2 \left (a^2+b^2\right )^2 d (a+b \text {csch}(c+d x))}-\frac {\left (2 i \left (6 a^4+5 a^2 b^2+2 b^4\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-4 \left (1+\frac {a^2}{b^2}\right )-x^2} \, dx,x,-\frac {2 i a}{b}+2 \tan \left (\frac {1}{2} (i c+i d x)\right )\right )}{a^3 \left (a^2+b^2\right )^2 d}\\ &=\frac {x}{a^3}+\frac {b \left (6 a^4+5 a^2 b^2+2 b^4\right ) \tanh ^{-1}\left (\frac {b \left (\frac {a}{b}-\tanh \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right )^{5/2} d}-\frac {b^2 \coth (c+d x)}{2 a \left (a^2+b^2\right ) d (a+b \text {csch}(c+d x))^2}-\frac {b^2 \left (5 a^2+2 b^2\right ) \coth (c+d x)}{2 a^2 \left (a^2+b^2\right )^2 d (a+b \text {csch}(c+d x))}\\ \end {align*}

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Mathematica [A]  time = 0.93, size = 213, normalized size = 1.31 \[ \frac {\text {csch}^2(c+d x) (a \sinh (c+d x)+b) \left (-\frac {3 a b^2 \left (2 a^2+b^2\right ) \coth (c+d x) (a \sinh (c+d x)+b)}{\left (a^2+b^2\right )^2}+\frac {a b^3 \coth (c+d x)}{a^2+b^2}-\frac {2 b \left (6 a^4+5 a^2 b^2+2 b^4\right ) \text {csch}(c+d x) (a \sinh (c+d x)+b)^2 \tan ^{-1}\left (\frac {a-b \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2-b^2}}\right )}{\left (-a^2-b^2\right )^{5/2}}+2 (c+d x) \text {csch}(c+d x) (a \sinh (c+d x)+b)^2\right )}{2 a^3 d (a+b \text {csch}(c+d x))^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.46, size = 2094, normalized size = 12.85 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(12*a^6*b^2 + 18*a^4*b^4 + 6*a^2*b^6 + 2*(a^8 + 3*a^6*b^2 + 3*a^4*b^4 + a^2*b^6)*d*x*cosh(d*x + c)^4 + 2*(
a^8 + 3*a^6*b^2 + 3*a^4*b^4 + a^2*b^6)*d*x*sinh(d*x + c)^4 + 2*(7*a^5*b^3 + 11*a^3*b^5 + 4*a*b^7 + 4*(a^7*b +
3*a^5*b^3 + 3*a^3*b^5 + a*b^7)*d*x)*cosh(d*x + c)^3 + 2*(7*a^5*b^3 + 11*a^3*b^5 + 4*a*b^7 + 4*(a^8 + 3*a^6*b^2
+ 3*a^4*b^4 + a^2*b^6)*d*x*cosh(d*x + c) + 4*(a^7*b + 3*a^5*b^3 + 3*a^3*b^5 + a*b^7)*d*x)*sinh(d*x + c)^3 + 2
*(a^8 + 3*a^6*b^2 + 3*a^4*b^4 + a^2*b^6)*d*x - 2*(6*a^6*b^2 - 3*a^4*b^4 - 15*a^2*b^6 - 6*b^8 + 2*(a^8 + a^6*b^
2 - 3*a^4*b^4 - 5*a^2*b^6 - 2*b^8)*d*x)*cosh(d*x + c)^2 - 2*(6*a^6*b^2 - 3*a^4*b^4 - 15*a^2*b^6 - 6*b^8 - 6*(a
^8 + 3*a^6*b^2 + 3*a^4*b^4 + a^2*b^6)*d*x*cosh(d*x + c)^2 + 2*(a^8 + a^6*b^2 - 3*a^4*b^4 - 5*a^2*b^6 - 2*b^8)*
d*x - 3*(7*a^5*b^3 + 11*a^3*b^5 + 4*a*b^7 + 4*(a^7*b + 3*a^5*b^3 + 3*a^3*b^5 + a*b^7)*d*x)*cosh(d*x + c))*sinh
(d*x + c)^2 + (6*a^6*b + 5*a^4*b^3 + 2*a^2*b^5 + (6*a^6*b + 5*a^4*b^3 + 2*a^2*b^5)*cosh(d*x + c)^4 + (6*a^6*b
+ 5*a^4*b^3 + 2*a^2*b^5)*sinh(d*x + c)^4 + 4*(6*a^5*b^2 + 5*a^3*b^4 + 2*a*b^6)*cosh(d*x + c)^3 + 4*(6*a^5*b^2
+ 5*a^3*b^4 + 2*a*b^6 + (6*a^6*b + 5*a^4*b^3 + 2*a^2*b^5)*cosh(d*x + c))*sinh(d*x + c)^3 - 2*(6*a^6*b - 7*a^4*
b^3 - 8*a^2*b^5 - 4*b^7)*cosh(d*x + c)^2 - 2*(6*a^6*b - 7*a^4*b^3 - 8*a^2*b^5 - 4*b^7 - 3*(6*a^6*b + 5*a^4*b^3
+ 2*a^2*b^5)*cosh(d*x + c)^2 - 6*(6*a^5*b^2 + 5*a^3*b^4 + 2*a*b^6)*cosh(d*x + c))*sinh(d*x + c)^2 - 4*(6*a^5*
b^2 + 5*a^3*b^4 + 2*a*b^6)*cosh(d*x + c) - 4*(6*a^5*b^2 + 5*a^3*b^4 + 2*a*b^6 - (6*a^6*b + 5*a^4*b^3 + 2*a^2*b
^5)*cosh(d*x + c)^3 - 3*(6*a^5*b^2 + 5*a^3*b^4 + 2*a*b^6)*cosh(d*x + c)^2 + (6*a^6*b - 7*a^4*b^3 - 8*a^2*b^5 -
4*b^7)*cosh(d*x + c))*sinh(d*x + c))*sqrt(a^2 + b^2)*log((a^2*cosh(d*x + c)^2 + a^2*sinh(d*x + c)^2 + 2*a*b*c
osh(d*x + c) + a^2 + 2*b^2 + 2*(a^2*cosh(d*x + c) + a*b)*sinh(d*x + c) + 2*sqrt(a^2 + b^2)*(a*cosh(d*x + c) +
a*sinh(d*x + c) + b))/(a*cosh(d*x + c)^2 + a*sinh(d*x + c)^2 + 2*b*cosh(d*x + c) + 2*(a*cosh(d*x + c) + b)*sin
h(d*x + c) - a)) - 2*(17*a^5*b^3 + 25*a^3*b^5 + 8*a*b^7 + 4*(a^7*b + 3*a^5*b^3 + 3*a^3*b^5 + a*b^7)*d*x)*cosh(
d*x + c) - 2*(17*a^5*b^3 + 25*a^3*b^5 + 8*a*b^7 - 4*(a^8 + 3*a^6*b^2 + 3*a^4*b^4 + a^2*b^6)*d*x*cosh(d*x + c)^
3 + 4*(a^7*b + 3*a^5*b^3 + 3*a^3*b^5 + a*b^7)*d*x - 3*(7*a^5*b^3 + 11*a^3*b^5 + 4*a*b^7 + 4*(a^7*b + 3*a^5*b^3
+ 3*a^3*b^5 + a*b^7)*d*x)*cosh(d*x + c)^2 + 2*(6*a^6*b^2 - 3*a^4*b^4 - 15*a^2*b^6 - 6*b^8 + 2*(a^8 + a^6*b^2
- 3*a^4*b^4 - 5*a^2*b^6 - 2*b^8)*d*x)*cosh(d*x + c))*sinh(d*x + c))/((a^11 + 3*a^9*b^2 + 3*a^7*b^4 + a^5*b^6)*
d*cosh(d*x + c)^4 + (a^11 + 3*a^9*b^2 + 3*a^7*b^4 + a^5*b^6)*d*sinh(d*x + c)^4 + 4*(a^10*b + 3*a^8*b^3 + 3*a^6
*b^5 + a^4*b^7)*d*cosh(d*x + c)^3 - 2*(a^11 + a^9*b^2 - 3*a^7*b^4 - 5*a^5*b^6 - 2*a^3*b^8)*d*cosh(d*x + c)^2 +
4*((a^11 + 3*a^9*b^2 + 3*a^7*b^4 + a^5*b^6)*d*cosh(d*x + c) + (a^10*b + 3*a^8*b^3 + 3*a^6*b^5 + a^4*b^7)*d)*s
inh(d*x + c)^3 - 4*(a^10*b + 3*a^8*b^3 + 3*a^6*b^5 + a^4*b^7)*d*cosh(d*x + c) + 2*(3*(a^11 + 3*a^9*b^2 + 3*a^7
*b^4 + a^5*b^6)*d*cosh(d*x + c)^2 + 6*(a^10*b + 3*a^8*b^3 + 3*a^6*b^5 + a^4*b^7)*d*cosh(d*x + c) - (a^11 + a^9
*b^2 - 3*a^7*b^4 - 5*a^5*b^6 - 2*a^3*b^8)*d)*sinh(d*x + c)^2 + (a^11 + 3*a^9*b^2 + 3*a^7*b^4 + a^5*b^6)*d + 4*
((a^11 + 3*a^9*b^2 + 3*a^7*b^4 + a^5*b^6)*d*cosh(d*x + c)^3 + 3*(a^10*b + 3*a^8*b^3 + 3*a^6*b^5 + a^4*b^7)*d*c
osh(d*x + c)^2 - (a^11 + a^9*b^2 - 3*a^7*b^4 - 5*a^5*b^6 - 2*a^3*b^8)*d*cosh(d*x + c) - (a^10*b + 3*a^8*b^3 +
3*a^6*b^5 + a^4*b^7)*d)*sinh(d*x + c))

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giac [A]  time = 0.16, size = 293, normalized size = 1.80 \[ -\frac {\frac {{\left (6 \, a^{4} b + 5 \, a^{2} b^{3} + 2 \, b^{5}\right )} \log \left (\frac {{\left | 2 \, a e^{\left (d x + c\right )} + 2 \, b - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, a e^{\left (d x + c\right )} + 2 \, b + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{{\left (a^{7} + 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} \sqrt {a^{2} + b^{2}}} - \frac {2 \, {\left (7 \, a^{3} b^{3} e^{\left (3 \, d x + 3 \, c\right )} + 4 \, a b^{5} e^{\left (3 \, d x + 3 \, c\right )} - 6 \, a^{4} b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 9 \, a^{2} b^{4} e^{\left (2 \, d x + 2 \, c\right )} + 6 \, b^{6} e^{\left (2 \, d x + 2 \, c\right )} - 17 \, a^{3} b^{3} e^{\left (d x + c\right )} - 8 \, a b^{5} e^{\left (d x + c\right )} + 6 \, a^{4} b^{2} + 3 \, a^{2} b^{4}\right )}}{{\left (a^{7} + 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} {\left (a e^{\left (2 \, d x + 2 \, c\right )} + 2 \, b e^{\left (d x + c\right )} - a\right )}^{2}} - \frac {2 \, {\left (d x + c\right )}}{a^{3}}}{2 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*((6*a^4*b + 5*a^2*b^3 + 2*b^5)*log(abs(2*a*e^(d*x + c) + 2*b - 2*sqrt(a^2 + b^2))/abs(2*a*e^(d*x + c) + 2
*b + 2*sqrt(a^2 + b^2)))/((a^7 + 2*a^5*b^2 + a^3*b^4)*sqrt(a^2 + b^2)) - 2*(7*a^3*b^3*e^(3*d*x + 3*c) + 4*a*b^
5*e^(3*d*x + 3*c) - 6*a^4*b^2*e^(2*d*x + 2*c) + 9*a^2*b^4*e^(2*d*x + 2*c) + 6*b^6*e^(2*d*x + 2*c) - 17*a^3*b^3
*e^(d*x + c) - 8*a*b^5*e^(d*x + c) + 6*a^4*b^2 + 3*a^2*b^4)/((a^7 + 2*a^5*b^2 + a^3*b^4)*(a*e^(2*d*x + 2*c) +
2*b*e^(d*x + c) - a)^2) - 2*(d*x + c)/a^3)/d

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maple [B]  time = 0.34, size = 822, normalized size = 5.04 \[ -\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d \,a^{3}}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d \,a^{3}}+\frac {4 a \,b^{2} \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \left (\left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b -2 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-b \right )^{2} \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {b^{4} \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a \left (\left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b -2 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-b \right )^{2} \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {10 a^{2} b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \left (\left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b -2 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-b \right )^{2} \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {b^{3} \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \left (\left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b -2 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-b \right )^{2} \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {2 b^{5} \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{2} \left (\left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b -2 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-b \right )^{2} \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {16 a \,b^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{d \left (\left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b -2 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-b \right )^{2} \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {7 b^{4} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{d a \left (\left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b -2 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-b \right )^{2} \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {5 b^{3}}{d \left (\left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b -2 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-b \right )^{2} \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {2 b^{5}}{d \,a^{2} \left (\left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b -2 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-b \right )^{2} \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {6 a b \arctanh \left (\frac {2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) b -2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \sqrt {a^{2}+b^{2}}}-\frac {5 b^{3} \arctanh \left (\frac {2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) b -2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{d a \left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \sqrt {a^{2}+b^{2}}}-\frac {2 b^{5} \arctanh \left (\frac {2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) b -2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{d \,a^{3} \left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \sqrt {a^{2}+b^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*csch(d*x+c))^3,x)

[Out]

-1/d/a^3*ln(tanh(1/2*d*x+1/2*c)-1)+1/d/a^3*ln(tanh(1/2*d*x+1/2*c)+1)+4/d*a*b^2/(tanh(1/2*d*x+1/2*c)^2*b-2*a*ta
nh(1/2*d*x+1/2*c)-b)^2/(a^4+2*a^2*b^2+b^4)*tanh(1/2*d*x+1/2*c)^3+1/d/a*b^4/(tanh(1/2*d*x+1/2*c)^2*b-2*a*tanh(1
/2*d*x+1/2*c)-b)^2/(a^4+2*a^2*b^2+b^4)*tanh(1/2*d*x+1/2*c)^3-10/d*a^2*b/(tanh(1/2*d*x+1/2*c)^2*b-2*a*tanh(1/2*
d*x+1/2*c)-b)^2/(a^4+2*a^2*b^2+b^4)*tanh(1/2*d*x+1/2*c)^2+1/d*b^3/(tanh(1/2*d*x+1/2*c)^2*b-2*a*tanh(1/2*d*x+1/
2*c)-b)^2/(a^4+2*a^2*b^2+b^4)*tanh(1/2*d*x+1/2*c)^2+2/d/a^2*b^5/(tanh(1/2*d*x+1/2*c)^2*b-2*a*tanh(1/2*d*x+1/2*
c)-b)^2/(a^4+2*a^2*b^2+b^4)*tanh(1/2*d*x+1/2*c)^2-16/d*a*b^2/(tanh(1/2*d*x+1/2*c)^2*b-2*a*tanh(1/2*d*x+1/2*c)-
b)^2/(a^4+2*a^2*b^2+b^4)*tanh(1/2*d*x+1/2*c)-7/d/a*b^4/(tanh(1/2*d*x+1/2*c)^2*b-2*a*tanh(1/2*d*x+1/2*c)-b)^2/(
a^4+2*a^2*b^2+b^4)*tanh(1/2*d*x+1/2*c)-5/d*b^3/(tanh(1/2*d*x+1/2*c)^2*b-2*a*tanh(1/2*d*x+1/2*c)-b)^2/(a^4+2*a^
2*b^2+b^4)-2/d/a^2*b^5/(tanh(1/2*d*x+1/2*c)^2*b-2*a*tanh(1/2*d*x+1/2*c)-b)^2/(a^4+2*a^2*b^2+b^4)-6/d*a*b/(a^4+
2*a^2*b^2+b^4)/(a^2+b^2)^(1/2)*arctanh(1/2*(2*tanh(1/2*d*x+1/2*c)*b-2*a)/(a^2+b^2)^(1/2))-5/d/a*b^3/(a^4+2*a^2
*b^2+b^4)/(a^2+b^2)^(1/2)*arctanh(1/2*(2*tanh(1/2*d*x+1/2*c)*b-2*a)/(a^2+b^2)^(1/2))-2/d/a^3*b^5/(a^4+2*a^2*b^
2+b^4)/(a^2+b^2)^(1/2)*arctanh(1/2*(2*tanh(1/2*d*x+1/2*c)*b-2*a)/(a^2+b^2)^(1/2))

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maxima [B]  time = 0.42, size = 373, normalized size = 2.29 \[ -\frac {{\left (6 \, a^{4} b + 5 \, a^{2} b^{3} + 2 \, b^{5}\right )} \log \left (\frac {a e^{\left (-d x - c\right )} - b - \sqrt {a^{2} + b^{2}}}{a e^{\left (-d x - c\right )} - b + \sqrt {a^{2} + b^{2}}}\right )}{2 \, {\left (a^{7} + 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} \sqrt {a^{2} + b^{2}} d} - \frac {6 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + {\left (17 \, a^{3} b^{3} + 8 \, a b^{5}\right )} e^{\left (-d x - c\right )} - 3 \, {\left (2 \, a^{4} b^{2} - 3 \, a^{2} b^{4} - 2 \, b^{6}\right )} e^{\left (-2 \, d x - 2 \, c\right )} - {\left (7 \, a^{3} b^{3} + 4 \, a b^{5}\right )} e^{\left (-3 \, d x - 3 \, c\right )}}{{\left (a^{9} + 2 \, a^{7} b^{2} + a^{5} b^{4} + 4 \, {\left (a^{8} b + 2 \, a^{6} b^{3} + a^{4} b^{5}\right )} e^{\left (-d x - c\right )} - 2 \, {\left (a^{9} - 3 \, a^{5} b^{4} - 2 \, a^{3} b^{6}\right )} e^{\left (-2 \, d x - 2 \, c\right )} - 4 \, {\left (a^{8} b + 2 \, a^{6} b^{3} + a^{4} b^{5}\right )} e^{\left (-3 \, d x - 3 \, c\right )} + {\left (a^{9} + 2 \, a^{7} b^{2} + a^{5} b^{4}\right )} e^{\left (-4 \, d x - 4 \, c\right )}\right )} d} + \frac {d x + c}{a^{3} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(6*a^4*b + 5*a^2*b^3 + 2*b^5)*log((a*e^(-d*x - c) - b - sqrt(a^2 + b^2))/(a*e^(-d*x - c) - b + sqrt(a^2 +
b^2)))/((a^7 + 2*a^5*b^2 + a^3*b^4)*sqrt(a^2 + b^2)*d) - (6*a^4*b^2 + 3*a^2*b^4 + (17*a^3*b^3 + 8*a*b^5)*e^(-
d*x - c) - 3*(2*a^4*b^2 - 3*a^2*b^4 - 2*b^6)*e^(-2*d*x - 2*c) - (7*a^3*b^3 + 4*a*b^5)*e^(-3*d*x - 3*c))/((a^9
+ 2*a^7*b^2 + a^5*b^4 + 4*(a^8*b + 2*a^6*b^3 + a^4*b^5)*e^(-d*x - c) - 2*(a^9 - 3*a^5*b^4 - 2*a^3*b^6)*e^(-2*d
*x - 2*c) - 4*(a^8*b + 2*a^6*b^3 + a^4*b^5)*e^(-3*d*x - 3*c) + (a^9 + 2*a^7*b^2 + a^5*b^4)*e^(-4*d*x - 4*c))*d
) + (d*x + c)/(a^3*d)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (a+\frac {b}{\mathrm {sinh}\left (c+d\,x\right )}\right )}^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b/sinh(c + d*x))^3,x)

[Out]

int(1/(a + b/sinh(c + d*x))^3, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a + b \operatorname {csch}{\left (c + d x \right )}\right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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