3.1.57 \(\int \frac {\text {csch}^2(c+d x)}{(a+b \sinh ^2(c+d x))^3} \, dx\) [57]

Optimal. Leaf size=215 \[ -\frac {3 b \left (8 a^2-12 a b+5 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{8 a^{7/2} (a-b)^{5/2} d}-\frac {(4 a-5 b) (2 a-3 b) \coth (c+d x)}{8 a^3 (a-b)^2 d}-\frac {b \text {csch}(c+d x) \text {sech}^3(c+d x)}{4 a (a-b) d \left (a-(a-b) \tanh ^2(c+d x)\right )^2}-\frac {b \coth (c+d x) \left (4 a-5 b-(4 a-b) \tanh ^2(c+d x)\right )}{8 a^2 (a-b)^2 d \left (a-(a-b) \tanh ^2(c+d x)\right )} \]

[Out]

-3/8*b*(8*a^2-12*a*b+5*b^2)*arctanh((a-b)^(1/2)*tanh(d*x+c)/a^(1/2))/a^(7/2)/(a-b)^(5/2)/d-1/8*(4*a-5*b)*(2*a-
3*b)*coth(d*x+c)/a^3/(a-b)^2/d-1/4*b*csch(d*x+c)*sech(d*x+c)^3/a/(a-b)/d/(a-(a-b)*tanh(d*x+c)^2)^2-1/8*b*coth(
d*x+c)*(4*a-5*b-(4*a-b)*tanh(d*x+c)^2)/a^2/(a-b)^2/d/(a-(a-b)*tanh(d*x+c)^2)

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Rubi [A]
time = 0.22, antiderivative size = 215, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3266, 479, 591, 464, 214} \begin {gather*} -\frac {(4 a-5 b) (2 a-3 b) \coth (c+d x)}{8 a^3 d (a-b)^2}-\frac {b \coth (c+d x) \left (-\left ((4 a-b) \tanh ^2(c+d x)\right )+4 a-5 b\right )}{8 a^2 d (a-b)^2 \left (a-(a-b) \tanh ^2(c+d x)\right )}-\frac {3 b \left (8 a^2-12 a b+5 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{8 a^{7/2} d (a-b)^{5/2}}-\frac {b \text {csch}(c+d x) \text {sech}^3(c+d x)}{4 a d (a-b) \left (a-(a-b) \tanh ^2(c+d x)\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Csch[c + d*x]^2/(a + b*Sinh[c + d*x]^2)^3,x]

[Out]

(-3*b*(8*a^2 - 12*a*b + 5*b^2)*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]])/(8*a^(7/2)*(a - b)^(5/2)*d) - ((4
*a - 5*b)*(2*a - 3*b)*Coth[c + d*x])/(8*a^3*(a - b)^2*d) - (b*Csch[c + d*x]*Sech[c + d*x]^3)/(4*a*(a - b)*d*(a
- (a - b)*Tanh[c + d*x]^2)^2) - (b*Coth[c + d*x]*(4*a - 5*b - (4*a - b)*Tanh[c + d*x]^2))/(8*a^2*(a - b)^2*d*
(a - (a - b)*Tanh[c + d*x]^2))

Rule 214

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

Rule 464

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[c*(e*x)^(m +
1)*((a + b*x^n)^(p + 1)/(a*e*(m + 1))), x] + Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*e^n*(m + 1)), In
t[(e*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && (IntegerQ[n] ||
GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -1])) &&  !ILtQ[p, -1]

Rule 479

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-(c*b -
a*d))*(e*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q - 1)/(a*b*e*n*(p + 1))), x] + Dist[1/(a*b*n*(p + 1)),
Int[(e*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 2)*Simp[c*(c*b*n*(p + 1) + (c*b - a*d)*(m + 1)) + d*(c*b*n*(
p + 1) + (c*b - a*d)*(m + n*(q - 1) + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0]
&& IGtQ[n, 0] && LtQ[p, -1] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]

Rule 591

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/(a*b*g*n*(p + 1))), x] + Dis
t[1/(a*b*n*(p + 1)), Int[(g*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1)*Simp[c*(b*e*n*(p + 1) + (b*e - a*f)*(
m + 1)) + d*(b*e*n*(p + 1) + (b*e - a*f)*(m + n*q + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x]
&& IGtQ[n, 0] && LtQ[p, -1] && GtQ[q, 0] &&  !(EqQ[q, 1] && SimplerQ[b*c - a*d, b*e - a*f])

Rule 3266

Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.), x_Symbol] :> With[{ff = FreeF
actors[Tan[e + f*x], x]}, Dist[ff^(m + 1)/f, Subst[Int[x^m*((a + (a + b)*ff^2*x^2)^p/(1 + ff^2*x^2)^(m/2 + p +
1)), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[m/2] && IntegerQ[p]

Rubi steps

\begin {align*} \int \frac {\text {csch}^2(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^3} \, dx &=\frac {\text {Subst}\left (\int \frac {\left (1-x^2\right )^3}{x^2 \left (a-(a-b) x^2\right )^3} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=-\frac {b \text {csch}(c+d x) \text {sech}^3(c+d x)}{4 a (a-b) d \left (a-(a-b) \tanh ^2(c+d x)\right )^2}+\frac {\text {Subst}\left (\int \frac {\left (1-x^2\right ) \left (4 a-5 b+(-4 a+b) x^2\right )}{x^2 \left (a+(-a+b) x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{4 a (a-b) d}\\ &=-\frac {b \text {csch}(c+d x) \text {sech}^3(c+d x)}{4 a (a-b) d \left (a-(a-b) \tanh ^2(c+d x)\right )^2}-\frac {b \coth (c+d x) \left (4 a-5 b-(4 a-b) \tanh ^2(c+d x)\right )}{8 a^2 (a-b)^2 d \left (a-(a-b) \tanh ^2(c+d x)\right )}+\frac {\text {Subst}\left (\int \frac {(4 a-5 b) (2 a-3 b)-(2 a-b) (4 a-b) x^2}{x^2 \left (a+(-a+b) x^2\right )} \, dx,x,\tanh (c+d x)\right )}{8 a^2 (a-b)^2 d}\\ &=-\frac {(4 a-5 b) (2 a-3 b) \coth (c+d x)}{8 a^3 (a-b)^2 d}-\frac {b \text {csch}(c+d x) \text {sech}^3(c+d x)}{4 a (a-b) d \left (a-(a-b) \tanh ^2(c+d x)\right )^2}-\frac {b \coth (c+d x) \left (4 a-5 b-(4 a-b) \tanh ^2(c+d x)\right )}{8 a^2 (a-b)^2 d \left (a-(a-b) \tanh ^2(c+d x)\right )}-\frac {\left (3 b \left (8 a^2-12 a b+5 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{a+(-a+b) x^2} \, dx,x,\tanh (c+d x)\right )}{8 a^3 (a-b)^2 d}\\ &=-\frac {3 b \left (8 a^2-12 a b+5 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{8 a^{7/2} (a-b)^{5/2} d}-\frac {(4 a-5 b) (2 a-3 b) \coth (c+d x)}{8 a^3 (a-b)^2 d}-\frac {b \text {csch}(c+d x) \text {sech}^3(c+d x)}{4 a (a-b) d \left (a-(a-b) \tanh ^2(c+d x)\right )^2}-\frac {b \coth (c+d x) \left (4 a-5 b-(4 a-b) \tanh ^2(c+d x)\right )}{8 a^2 (a-b)^2 d \left (a-(a-b) \tanh ^2(c+d x)\right )}\\ \end {align*}

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Mathematica [A]
time = 1.17, size = 225, normalized size = 1.05 \begin {gather*} \frac {(2 a-b+b \cosh (2 (c+d x))) \text {csch}^6(c+d x) \left (-\frac {3 b \left (8 a^2-12 a b+5 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right ) (2 a-b+b \cosh (2 (c+d x)))^2}{(a-b)^{5/2}}-8 \sqrt {a} (2 a-b+b \cosh (2 (c+d x)))^2 \coth (c+d x)+\frac {4 a^{3/2} b^2 \sinh (2 (c+d x))}{a-b}+\frac {\sqrt {a} (10 a-7 b) b^2 (2 a-b+b \cosh (2 (c+d x))) \sinh (2 (c+d x))}{(a-b)^2}\right )}{64 a^{7/2} d \left (b+a \text {csch}^2(c+d x)\right )^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((2*a - b + b*Cosh[2*(c + d*x)])*Csch[c + d*x]^6*((-3*b*(8*a^2 - 12*a*b + 5*b^2)*ArcTanh[(Sqrt[a - b]*Tanh[c +
d*x])/Sqrt[a]]*(2*a - b + b*Cosh[2*(c + d*x)])^2)/(a - b)^(5/2) - 8*Sqrt[a]*(2*a - b + b*Cosh[2*(c + d*x)])^2
*Coth[c + d*x] + (4*a^(3/2)*b^2*Sinh[2*(c + d*x)])/(a - b) + (Sqrt[a]*(10*a - 7*b)*b^2*(2*a - b + b*Cosh[2*(c
+ d*x)])*Sinh[2*(c + d*x)])/(a - b)^2))/(64*a^(7/2)*d*(b + a*Csch[c + d*x]^2)^3)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(441\) vs. \(2(199)=398\).
time = 1.67, size = 442, normalized size = 2.06 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(d*x+c)^2/(a+b*sinh(d*x+c)^2)^3,x,method=_RETURNVERBOSE)

[Out]

1/d*(-1/2/a^3*tanh(1/2*d*x+1/2*c)-1/2/a^3/tanh(1/2*d*x+1/2*c)+2/a^3*b*((3/8*a*b*(4*a-3*b)/(a^2-2*a*b+b^2)*tanh
(1/2*d*x+1/2*c)^7-1/8*(12*a^2-49*a*b+28*b^2)*b/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^5-1/8*(12*a^2-49*a*b+28*b^2
)*b/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^3+3/8*a*b*(4*a-3*b)/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c))/(a*tanh(1/2*d
*x+1/2*c)^4-2*a*tanh(1/2*d*x+1/2*c)^2+4*b*tanh(1/2*d*x+1/2*c)^2+a)^2+3/8*(8*a^2-12*a*b+5*b^2)/(a^2-2*a*b+b^2)*
a*(1/2*((-b*(a-b))^(1/2)+b)/a/(-b*(a-b))^(1/2)/((2*(-b*(a-b))^(1/2)-a+2*b)*a)^(1/2)*arctan(a*tanh(1/2*d*x+1/2*
c)/((2*(-b*(a-b))^(1/2)-a+2*b)*a)^(1/2))-1/2*((-b*(a-b))^(1/2)-b)/a/(-b*(a-b))^(1/2)/((2*(-b*(a-b))^(1/2)+a-2*
b)*a)^(1/2)*arctanh(a*tanh(1/2*d*x+1/2*c)/((2*(-b*(a-b))^(1/2)+a-2*b)*a)^(1/2)))))

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 4423 vs. \(2 (200) = 400\).
time = 0.48, size = 9102, normalized size = 42.33 \begin {gather*} \text {too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/16*(12*(8*a^4*b^2 - 20*a^3*b^3 + 17*a^2*b^4 - 5*a*b^5)*cosh(d*x + c)^8 + 96*(8*a^4*b^2 - 20*a^3*b^3 + 17*a
^2*b^4 - 5*a*b^5)*cosh(d*x + c)*sinh(d*x + c)^7 + 12*(8*a^4*b^2 - 20*a^3*b^3 + 17*a^2*b^4 - 5*a*b^5)*sinh(d*x
+ c)^8 + 24*(24*a^5*b - 76*a^4*b^2 + 91*a^3*b^3 - 49*a^2*b^4 + 10*a*b^5)*cosh(d*x + c)^6 + 24*(24*a^5*b - 76*a
^4*b^2 + 91*a^3*b^3 - 49*a^2*b^4 + 10*a*b^5 + 14*(8*a^4*b^2 - 20*a^3*b^3 + 17*a^2*b^4 - 5*a*b^5)*cosh(d*x + c)
^2)*sinh(d*x + c)^6 + 32*a^4*b^2 - 136*a^3*b^3 + 164*a^2*b^4 - 60*a*b^5 + 48*(14*(8*a^4*b^2 - 20*a^3*b^3 + 17*
a^2*b^4 - 5*a*b^5)*cosh(d*x + c)^3 + 3*(24*a^5*b - 76*a^4*b^2 + 91*a^3*b^3 - 49*a^2*b^4 + 10*a*b^5)*cosh(d*x +
c))*sinh(d*x + c)^5 + 8*(64*a^6 - 296*a^5*b + 548*a^4*b^2 - 509*a^3*b^3 + 238*a^2*b^4 - 45*a*b^5)*cosh(d*x +
c)^4 + 8*(64*a^6 - 296*a^5*b + 548*a^4*b^2 - 509*a^3*b^3 + 238*a^2*b^4 - 45*a*b^5 + 105*(8*a^4*b^2 - 20*a^3*b^
3 + 17*a^2*b^4 - 5*a*b^5)*cosh(d*x + c)^4 + 45*(24*a^5*b - 76*a^4*b^2 + 91*a^3*b^3 - 49*a^2*b^4 + 10*a*b^5)*co
sh(d*x + c)^2)*sinh(d*x + c)^4 + 32*(21*(8*a^4*b^2 - 20*a^3*b^3 + 17*a^2*b^4 - 5*a*b^5)*cosh(d*x + c)^5 + 15*(
24*a^5*b - 76*a^4*b^2 + 91*a^3*b^3 - 49*a^2*b^4 + 10*a*b^5)*cosh(d*x + c)^3 + (64*a^6 - 296*a^5*b + 548*a^4*b^
2 - 509*a^3*b^3 + 238*a^2*b^4 - 45*a*b^5)*cosh(d*x + c))*sinh(d*x + c)^3 + 8*(32*a^5*b - 144*a^4*b^2 + 219*a^3
*b^3 - 137*a^2*b^4 + 30*a*b^5)*cosh(d*x + c)^2 + 8*(42*(8*a^4*b^2 - 20*a^3*b^3 + 17*a^2*b^4 - 5*a*b^5)*cosh(d*
x + c)^6 + 32*a^5*b - 144*a^4*b^2 + 219*a^3*b^3 - 137*a^2*b^4 + 30*a*b^5 + 45*(24*a^5*b - 76*a^4*b^2 + 91*a^3*
b^3 - 49*a^2*b^4 + 10*a*b^5)*cosh(d*x + c)^4 + 6*(64*a^6 - 296*a^5*b + 548*a^4*b^2 - 509*a^3*b^3 + 238*a^2*b^4
- 45*a*b^5)*cosh(d*x + c)^2)*sinh(d*x + c)^2 - 3*((8*a^2*b^3 - 12*a*b^4 + 5*b^5)*cosh(d*x + c)^10 + 10*(8*a^2
*b^3 - 12*a*b^4 + 5*b^5)*cosh(d*x + c)*sinh(d*x + c)^9 + (8*a^2*b^3 - 12*a*b^4 + 5*b^5)*sinh(d*x + c)^10 + (64
*a^3*b^2 - 136*a^2*b^3 + 100*a*b^4 - 25*b^5)*cosh(d*x + c)^8 + (64*a^3*b^2 - 136*a^2*b^3 + 100*a*b^4 - 25*b^5
+ 45*(8*a^2*b^3 - 12*a*b^4 + 5*b^5)*cosh(d*x + c)^2)*sinh(d*x + c)^8 + 8*(15*(8*a^2*b^3 - 12*a*b^4 + 5*b^5)*co
sh(d*x + c)^3 + (64*a^3*b^2 - 136*a^2*b^3 + 100*a*b^4 - 25*b^5)*cosh(d*x + c))*sinh(d*x + c)^7 + 2*(64*a^4*b -
192*a^3*b^2 + 224*a^2*b^3 - 120*a*b^4 + 25*b^5)*cosh(d*x + c)^6 + 2*(64*a^4*b - 192*a^3*b^2 + 224*a^2*b^3 - 1
20*a*b^4 + 25*b^5 + 105*(8*a^2*b^3 - 12*a*b^4 + 5*b^5)*cosh(d*x + c)^4 + 14*(64*a^3*b^2 - 136*a^2*b^3 + 100*a*
b^4 - 25*b^5)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 4*(63*(8*a^2*b^3 - 12*a*b^4 + 5*b^5)*cosh(d*x + c)^5 + 14*(64
*a^3*b^2 - 136*a^2*b^3 + 100*a*b^4 - 25*b^5)*cosh(d*x + c)^3 + 3*(64*a^4*b - 192*a^3*b^2 + 224*a^2*b^3 - 120*a
*b^4 + 25*b^5)*cosh(d*x + c))*sinh(d*x + c)^5 - 8*a^2*b^3 + 12*a*b^4 - 5*b^5 - 2*(64*a^4*b - 192*a^3*b^2 + 224
*a^2*b^3 - 120*a*b^4 + 25*b^5)*cosh(d*x + c)^4 + 2*(105*(8*a^2*b^3 - 12*a*b^4 + 5*b^5)*cosh(d*x + c)^6 - 64*a^
4*b + 192*a^3*b^2 - 224*a^2*b^3 + 120*a*b^4 - 25*b^5 + 35*(64*a^3*b^2 - 136*a^2*b^3 + 100*a*b^4 - 25*b^5)*cosh
(d*x + c)^4 + 15*(64*a^4*b - 192*a^3*b^2 + 224*a^2*b^3 - 120*a*b^4 + 25*b^5)*cosh(d*x + c)^2)*sinh(d*x + c)^4
+ 8*(15*(8*a^2*b^3 - 12*a*b^4 + 5*b^5)*cosh(d*x + c)^7 + 7*(64*a^3*b^2 - 136*a^2*b^3 + 100*a*b^4 - 25*b^5)*cos
h(d*x + c)^5 + 5*(64*a^4*b - 192*a^3*b^2 + 224*a^2*b^3 - 120*a*b^4 + 25*b^5)*cosh(d*x + c)^3 - (64*a^4*b - 192
*a^3*b^2 + 224*a^2*b^3 - 120*a*b^4 + 25*b^5)*cosh(d*x + c))*sinh(d*x + c)^3 - (64*a^3*b^2 - 136*a^2*b^3 + 100*
a*b^4 - 25*b^5)*cosh(d*x + c)^2 + (45*(8*a^2*b^3 - 12*a*b^4 + 5*b^5)*cosh(d*x + c)^8 + 28*(64*a^3*b^2 - 136*a^
2*b^3 + 100*a*b^4 - 25*b^5)*cosh(d*x + c)^6 - 64*a^3*b^2 + 136*a^2*b^3 - 100*a*b^4 + 25*b^5 + 30*(64*a^4*b - 1
92*a^3*b^2 + 224*a^2*b^3 - 120*a*b^4 + 25*b^5)*cosh(d*x + c)^4 - 12*(64*a^4*b - 192*a^3*b^2 + 224*a^2*b^3 - 12
0*a*b^4 + 25*b^5)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 2*(5*(8*a^2*b^3 - 12*a*b^4 + 5*b^5)*cosh(d*x + c)^9 + 4*(
64*a^3*b^2 - 136*a^2*b^3 + 100*a*b^4 - 25*b^5)*cosh(d*x + c)^7 + 6*(64*a^4*b - 192*a^3*b^2 + 224*a^2*b^3 - 120
*a*b^4 + 25*b^5)*cosh(d*x + c)^5 - 4*(64*a^4*b - 192*a^3*b^2 + 224*a^2*b^3 - 120*a*b^4 + 25*b^5)*cosh(d*x + c)
^3 - (64*a^3*b^2 - 136*a^2*b^3 + 100*a*b^4 - 25*b^5)*cosh(d*x + c))*sinh(d*x + c))*sqrt(a^2 - a*b)*log((b^2*co
sh(d*x + c)^4 + 4*b^2*cosh(d*x + c)*sinh(d*x + c)^3 + b^2*sinh(d*x + c)^4 + 2*(2*a*b - b^2)*cosh(d*x + c)^2 +
2*(3*b^2*cosh(d*x + c)^2 + 2*a*b - b^2)*sinh(d*x + c)^2 + 8*a^2 - 8*a*b + b^2 + 4*(b^2*cosh(d*x + c)^3 + (2*a*
b - b^2)*cosh(d*x + c))*sinh(d*x + c) + 4*(b*cosh(d*x + c)^2 + 2*b*cosh(d*x + c)*sinh(d*x + c) + b*sinh(d*x +
c)^2 + 2*a - b)*sqrt(a^2 - a*b))/(b*cosh(d*x + c)^4 + 4*b*cosh(d*x + c)*sinh(d*x + c)^3 + b*sinh(d*x + c)^4 +
2*(2*a - b)*cosh(d*x + c)^2 + 2*(3*b*cosh(d*x + c)^2 + 2*a - b)*sinh(d*x + c)^2 + 4*(b*cosh(d*x + c)^3 + (2*a
- b)*cosh(d*x + c))*sinh(d*x + c) + b)) + 16*(6*(8*a^4*b^2 - 20*a^3*b^3 + 17*a^2*b^4 - 5*a*b^5)*cosh(d*x + c)^
7 + 9*(24*a^5*b - 76*a^4*b^2 + 91*a^3*b^3 - 49*a^2*b^4 + 10*a*b^5)*cosh(d*x + c)^5 + 2*(64*a^6 - 296*a^5*b + 5
48*a^4*b^2 - 509*a^3*b^3 + 238*a^2*b^4 - 45*a*b...

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]
time = 0.81, size = 331, normalized size = 1.54 \begin {gather*} -\frac {\frac {3 \, {\left (8 \, a^{2} b - 12 \, a b^{2} + 5 \, b^{3}\right )} \arctan \left (\frac {b e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a - b}{2 \, \sqrt {-a^{2} + a b}}\right )}{{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} \sqrt {-a^{2} + a b}} + \frac {2 \, {\left (16 \, a^{2} b^{2} e^{\left (6 \, d x + 6 \, c\right )} - 20 \, a b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 7 \, b^{4} e^{\left (6 \, d x + 6 \, c\right )} + 80 \, a^{3} b e^{\left (4 \, d x + 4 \, c\right )} - 136 \, a^{2} b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 86 \, a b^{3} e^{\left (4 \, d x + 4 \, c\right )} - 21 \, b^{4} e^{\left (4 \, d x + 4 \, c\right )} + 64 \, a^{2} b^{2} e^{\left (2 \, d x + 2 \, c\right )} - 76 \, a b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 21 \, b^{4} e^{\left (2 \, d x + 2 \, c\right )} + 10 \, a b^{3} - 7 \, b^{4}\right )}}{{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} {\left (b e^{\left (4 \, d x + 4 \, c\right )} + 4 \, a e^{\left (2 \, d x + 2 \, c\right )} - 2 \, b e^{\left (2 \, d x + 2 \, c\right )} + b\right )}^{2}} + \frac {16}{a^{3} {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}}}{8 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(3*(8*a^2*b - 12*a*b^2 + 5*b^3)*arctan(1/2*(b*e^(2*d*x + 2*c) + 2*a - b)/sqrt(-a^2 + a*b))/((a^5 - 2*a^4*
b + a^3*b^2)*sqrt(-a^2 + a*b)) + 2*(16*a^2*b^2*e^(6*d*x + 6*c) - 20*a*b^3*e^(6*d*x + 6*c) + 7*b^4*e^(6*d*x + 6
*c) + 80*a^3*b*e^(4*d*x + 4*c) - 136*a^2*b^2*e^(4*d*x + 4*c) + 86*a*b^3*e^(4*d*x + 4*c) - 21*b^4*e^(4*d*x + 4*
c) + 64*a^2*b^2*e^(2*d*x + 2*c) - 76*a*b^3*e^(2*d*x + 2*c) + 21*b^4*e^(2*d*x + 2*c) + 10*a*b^3 - 7*b^4)/((a^5
- 2*a^4*b + a^3*b^2)*(b*e^(4*d*x + 4*c) + 4*a*e^(2*d*x + 2*c) - 2*b*e^(2*d*x + 2*c) + b)^2) + 16/(a^3*(e^(2*d*
x + 2*c) - 1)))/d

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{{\mathrm {sinh}\left (c+d\,x\right )}^2\,{\left (b\,{\mathrm {sinh}\left (c+d\,x\right )}^2+a\right )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sinh(c + d*x)^2*(a + b*sinh(c + d*x)^2)^3),x)

[Out]

int(1/(sinh(c + d*x)^2*(a + b*sinh(c + d*x)^2)^3), x)

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