3.1.48 \(\int \frac {\text {csch}^2(c+d x)}{(a+b \sinh ^2(c+d x))^2} \, dx\) [48]

Optimal. Leaf size=142 \[ -\frac {(4 a-3 b) b \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{2 a^{5/2} (a-b)^{3/2} d}-\frac {\coth (c+d x)}{a d \left (a-(a-b) \tanh ^2(c+d x)\right )}+\frac {\left (2 a^2-4 a b+3 b^2\right ) \tanh (c+d x)}{2 a^2 (a-b) d \left (a-(a-b) \tanh ^2(c+d x)\right )} \]

[Out]

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

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Rubi [A]
time = 0.11, antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3266, 473, 393, 214} \begin {gather*} -\frac {b (4 a-3 b) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{2 a^{5/2} d (a-b)^{3/2}}+\frac {\left (2 a^2-4 a b+3 b^2\right ) \tanh (c+d x)}{2 a^2 d (a-b) \left (a-(a-b) \tanh ^2(c+d x)\right )}-\frac {\coth (c+d x)}{a d \left (a-(a-b) \tanh ^2(c+d x)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*((4*a - 3*b)*b*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]])/(a^(5/2)*(a - b)^(3/2)*d) - Coth[c + d*x]/(a
*d*(a - (a - b)*Tanh[c + d*x]^2)) + ((2*a^2 - 4*a*b + 3*b^2)*Tanh[c + d*x])/(2*a^2*(a - b)*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 393

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

Rule 473

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

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

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Mathematica [A]
time = 0.52, size = 170, normalized size = 1.20 \begin {gather*} -\frac {(2 a-b+b \cosh (2 (c+d x))) \text {csch}^5(c+d x) \left (2 \sqrt {a} \sqrt {a-b} \cosh (c+d x) \left (4 a^2-6 a b+3 b^2+(2 a-3 b) b \cosh (2 (c+d x))\right )-2 b (-4 a+3 b) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right ) (2 a-b+b \cosh (2 (c+d x))) \sinh (c+d x)\right )}{16 a^{5/2} (a-b)^{3/2} d \left (b+a \text {csch}^2(c+d x)\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/16*((2*a - b + b*Cosh[2*(c + d*x)])*Csch[c + d*x]^5*(2*Sqrt[a]*Sqrt[a - b]*Cosh[c + d*x]*(4*a^2 - 6*a*b + 3
*b^2 + (2*a - 3*b)*b*Cosh[2*(c + d*x)]) - 2*b*(-4*a + 3*b)*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]]*(2*a -
b + b*Cosh[2*(c + d*x)])*Sinh[c + d*x]))/(a^(5/2)*(a - b)^(3/2)*d*(b + a*Csch[c + d*x]^2)^2)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(315\) vs. \(2(130)=260\).
time = 1.51, size = 316, normalized size = 2.23

method	result	size
derivativedivides	\(\frac {-\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a^{2}}+\frac {4 b \left (\frac {\frac {b \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a -4 b}+\frac {b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a -4 b}}{a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a}+\frac {\left (4 a -3 b \right ) a \left (\frac {\left (\sqrt {-b \left (a -b \right )}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}-\frac {\left (\sqrt {-b \left (a -b \right )}-b \right ) \arctanh \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{4 a -4 b}\right )}{a^{2}}-\frac {1}{2 a^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}}{d}\)	\(316\)
default	\(\frac {-\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a^{2}}+\frac {4 b \left (\frac {\frac {b \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a -4 b}+\frac {b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a -4 b}}{a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a}+\frac {\left (4 a -3 b \right ) a \left (\frac {\left (\sqrt {-b \left (a -b \right )}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}-\frac {\left (\sqrt {-b \left (a -b \right )}-b \right ) \arctanh \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{4 a -4 b}\right )}{a^{2}}-\frac {1}{2 a^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}}{d}\)	\(316\)
risch	\(-\frac {4 a b \,{\mathrm e}^{4 d x +4 c}-3 b^{2} {\mathrm e}^{4 d x +4 c}+8 a^{2} {\mathrm e}^{2 d x +2 c}-14 a b \,{\mathrm e}^{2 d x +2 c}+6 b^{2} {\mathrm e}^{2 d x +2 c}+2 a b -3 b^{2}}{a^{2} \left (a -b \right ) d \left (b \,{\mathrm e}^{4 d x +4 c}+4 a \,{\mathrm e}^{2 d x +2 c}-2 b \,{\mathrm e}^{2 d x +2 c}+b \right ) \left ({\mathrm e}^{2 d x +2 c}-1\right )}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \sqrt {a^{2}-a b}-b \sqrt {a^{2}-a b}+2 a^{2}-2 a b}{b \sqrt {a^{2}-a b}}\right ) b}{\sqrt {a^{2}-a b}\, \left (a -b \right ) d a}-\frac {3 b^{2} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \sqrt {a^{2}-a b}-b \sqrt {a^{2}-a b}+2 a^{2}-2 a b}{b \sqrt {a^{2}-a b}}\right )}{4 \sqrt {a^{2}-a b}\, \left (a -b \right ) d \,a^{2}}-\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \sqrt {a^{2}-a b}-b \sqrt {a^{2}-a b}-2 a^{2}+2 a b}{b \sqrt {a^{2}-a b}}\right ) b}{\sqrt {a^{2}-a b}\, \left (a -b \right ) d a}+\frac {3 b^{2} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \sqrt {a^{2}-a b}-b \sqrt {a^{2}-a b}-2 a^{2}+2 a b}{b \sqrt {a^{2}-a b}}\right )}{4 \sqrt {a^{2}-a b}\, \left (a -b \right ) d \,a^{2}}\)	\(498\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(-1/2/a^2*tanh(1/2*d*x+1/2*c)+4*b/a^2*((1/4*b/(a-b)*tanh(1/2*d*x+1/2*c)^3+1/4*b/(a-b)*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)+1/4*(4*a-3*b)/(a-b)*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))))-1/2/a^2/tanh(1/2*d*x+1/2*c))

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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)^2,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. 1366 vs. \(2 (131) = 262\).
time = 0.43, size = 2988, normalized size = 21.04 \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)^2,x, algorithm="fricas")

[Out]

[-1/4*(4*(4*a^3*b - 7*a^2*b^2 + 3*a*b^3)*cosh(d*x + c)^4 + 16*(4*a^3*b - 7*a^2*b^2 + 3*a*b^3)*cosh(d*x + c)*si
nh(d*x + c)^3 + 4*(4*a^3*b - 7*a^2*b^2 + 3*a*b^3)*sinh(d*x + c)^4 + 8*a^3*b - 20*a^2*b^2 + 12*a*b^3 + 8*(4*a^4
- 11*a^3*b + 10*a^2*b^2 - 3*a*b^3)*cosh(d*x + c)^2 + 8*(4*a^4 - 11*a^3*b + 10*a^2*b^2 - 3*a*b^3 + 3*(4*a^3*b
- 7*a^2*b^2 + 3*a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 - ((4*a*b^2 - 3*b^3)*cosh(d*x + c)^6 + 6*(4*a*b^2 - 3*
b^3)*cosh(d*x + c)*sinh(d*x + c)^5 + (4*a*b^2 - 3*b^3)*sinh(d*x + c)^6 + (16*a^2*b - 24*a*b^2 + 9*b^3)*cosh(d*
x + c)^4 + (16*a^2*b - 24*a*b^2 + 9*b^3 + 15*(4*a*b^2 - 3*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 4*(5*(4*a*b^
2 - 3*b^3)*cosh(d*x + c)^3 + (16*a^2*b - 24*a*b^2 + 9*b^3)*cosh(d*x + c))*sinh(d*x + c)^3 - 4*a*b^2 + 3*b^3 -
(16*a^2*b - 24*a*b^2 + 9*b^3)*cosh(d*x + c)^2 + (15*(4*a*b^2 - 3*b^3)*cosh(d*x + c)^4 - 16*a^2*b + 24*a*b^2 -
9*b^3 + 6*(16*a^2*b - 24*a*b^2 + 9*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 2*(3*(4*a*b^2 - 3*b^3)*cosh(d*x + c
)^5 + 2*(16*a^2*b - 24*a*b^2 + 9*b^3)*cosh(d*x + c)^3 - (16*a^2*b - 24*a*b^2 + 9*b^3)*cosh(d*x + c))*sinh(d*x
+ c))*sqrt(a^2 - a*b)*log((b^2*cosh(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*((4*a^3*b - 7*a^2*b^2 + 3*a*b^3)
*cosh(d*x + c)^3 + (4*a^4 - 11*a^3*b + 10*a^2*b^2 - 3*a*b^3)*cosh(d*x + c))*sinh(d*x + c))/((a^5*b - 2*a^4*b^2
+ a^3*b^3)*d*cosh(d*x + c)^6 + 6*(a^5*b - 2*a^4*b^2 + a^3*b^3)*d*cosh(d*x + c)*sinh(d*x + c)^5 + (a^5*b - 2*a
^4*b^2 + a^3*b^3)*d*sinh(d*x + c)^6 + (4*a^6 - 11*a^5*b + 10*a^4*b^2 - 3*a^3*b^3)*d*cosh(d*x + c)^4 + (15*(a^5
*b - 2*a^4*b^2 + a^3*b^3)*d*cosh(d*x + c)^2 + (4*a^6 - 11*a^5*b + 10*a^4*b^2 - 3*a^3*b^3)*d)*sinh(d*x + c)^4 -
(4*a^6 - 11*a^5*b + 10*a^4*b^2 - 3*a^3*b^3)*d*cosh(d*x + c)^2 + 4*(5*(a^5*b - 2*a^4*b^2 + a^3*b^3)*d*cosh(d*x
+ c)^3 + (4*a^6 - 11*a^5*b + 10*a^4*b^2 - 3*a^3*b^3)*d*cosh(d*x + c))*sinh(d*x + c)^3 + (15*(a^5*b - 2*a^4*b^
2 + a^3*b^3)*d*cosh(d*x + c)^4 + 6*(4*a^6 - 11*a^5*b + 10*a^4*b^2 - 3*a^3*b^3)*d*cosh(d*x + c)^2 - (4*a^6 - 11
*a^5*b + 10*a^4*b^2 - 3*a^3*b^3)*d)*sinh(d*x + c)^2 - (a^5*b - 2*a^4*b^2 + a^3*b^3)*d + 2*(3*(a^5*b - 2*a^4*b^
2 + a^3*b^3)*d*cosh(d*x + c)^5 + 2*(4*a^6 - 11*a^5*b + 10*a^4*b^2 - 3*a^3*b^3)*d*cosh(d*x + c)^3 - (4*a^6 - 11
*a^5*b + 10*a^4*b^2 - 3*a^3*b^3)*d*cosh(d*x + c))*sinh(d*x + c)), -1/2*(2*(4*a^3*b - 7*a^2*b^2 + 3*a*b^3)*cosh
(d*x + c)^4 + 8*(4*a^3*b - 7*a^2*b^2 + 3*a*b^3)*cosh(d*x + c)*sinh(d*x + c)^3 + 2*(4*a^3*b - 7*a^2*b^2 + 3*a*b
^3)*sinh(d*x + c)^4 + 4*a^3*b - 10*a^2*b^2 + 6*a*b^3 + 4*(4*a^4 - 11*a^3*b + 10*a^2*b^2 - 3*a*b^3)*cosh(d*x +
c)^2 + 4*(4*a^4 - 11*a^3*b + 10*a^2*b^2 - 3*a*b^3 + 3*(4*a^3*b - 7*a^2*b^2 + 3*a*b^3)*cosh(d*x + c)^2)*sinh(d*
x + c)^2 - ((4*a*b^2 - 3*b^3)*cosh(d*x + c)^6 + 6*(4*a*b^2 - 3*b^3)*cosh(d*x + c)*sinh(d*x + c)^5 + (4*a*b^2 -
3*b^3)*sinh(d*x + c)^6 + (16*a^2*b - 24*a*b^2 + 9*b^3)*cosh(d*x + c)^4 + (16*a^2*b - 24*a*b^2 + 9*b^3 + 15*(4
*a*b^2 - 3*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 4*(5*(4*a*b^2 - 3*b^3)*cosh(d*x + c)^3 + (16*a^2*b - 24*a*b
^2 + 9*b^3)*cosh(d*x + c))*sinh(d*x + c)^3 - 4*a*b^2 + 3*b^3 - (16*a^2*b - 24*a*b^2 + 9*b^3)*cosh(d*x + c)^2 +
(15*(4*a*b^2 - 3*b^3)*cosh(d*x + c)^4 - 16*a^2*b + 24*a*b^2 - 9*b^3 + 6*(16*a^2*b - 24*a*b^2 + 9*b^3)*cosh(d*
x + c)^2)*sinh(d*x + c)^2 + 2*(3*(4*a*b^2 - 3*b^3)*cosh(d*x + c)^5 + 2*(16*a^2*b - 24*a*b^2 + 9*b^3)*cosh(d*x
+ c)^3 - (16*a^2*b - 24*a*b^2 + 9*b^3)*cosh(d*x + c))*sinh(d*x + c))*sqrt(-a^2 + a*b)*arctan(-1/2*(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)/(a^2 - a*b)) + 8*((4*
a^3*b - 7*a^2*b^2 + 3*a*b^3)*cosh(d*x + c)^3 + (4*a^4 - 11*a^3*b + 10*a^2*b^2 - 3*a*b^3)*cosh(d*x + c))*sinh(d
*x + c))/((a^5*b - 2*a^4*b^2 + a^3*b^3)*d*cosh(d*x + c)^6 + 6*(a^5*b - 2*a^4*b^2 + a^3*b^3)*d*cosh(d*x + c)*si
nh(d*x + c)^5 + (a^5*b - 2*a^4*b^2 + a^3*b^3)*d*sinh(d*x + c)^6 + (4*a^6 - 11*a^5*b + 10*a^4*b^2 - 3*a^3*b^3)*
d*cosh(d*x + c)^4 + (15*(a^5*b - 2*a^4*b^2 + a^3*b^3)*d*cosh(d*x + c)^2 + (4*a^6 - 11*a^5*b + 10*a^4*b^2 - 3*a
^3*b^3)*d)*sinh(d*x + c)^4 - (4*a^6 - 11*a^5*b + 10*a^4*b^2 - 3*a^3*b^3)*d*cosh(d*x + c)^2 + 4*(5*(a^5*b - 2*a
^4*b^2 + a^3*b^3)*d*cosh(d*x + c)^3 + (4*a^6 - 11*a^5*b + 10*a^4*b^2 - 3*a^3*b^3)*d*cosh(d*x + c))*sinh(d*x +
c)^3 + (15*(a^5*b - 2*a^4*b^2 + a^3*b^3)*d*cosh(d*x + c)^4 + 6*(4*a^6 - 11*a^5*b + 10*a^4*b^2 - 3*a^3*b^3)*d*c
osh(d*x + c)^2 - (4*a^6 - 11*a^5*b + 10*a^4*b^2 - 3*a^3*b^3)*d)*sinh(d*x + c)^2 - (a^5*b - 2*a^4*b^2 + a^3*b^3
)*d + 2*(3*(a^5*b - 2*a^4*b^2 + a^3*b^3)*d*cosh...

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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)**2,x)

[Out]

Timed out

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Giac [A]
time = 0.68, size = 229, normalized size = 1.61 \begin {gather*} -\frac {\frac {{\left (4 \, a b - 3 \, b^{2}\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^{3} - a^{2} b\right )} \sqrt {-a^{2} + a b}} + \frac {2 \, {\left (4 \, a b e^{\left (4 \, d x + 4 \, c\right )} - 3 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 8 \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} - 14 \, a b e^{\left (2 \, d x + 2 \, c\right )} + 6 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a b - 3 \, b^{2}\right )}}{{\left (a^{3} - a^{2} b\right )} {\left (b e^{\left (6 \, d x + 6 \, c\right )} + 4 \, a e^{\left (4 \, d x + 4 \, c\right )} - 3 \, b e^{\left (4 \, d x + 4 \, c\right )} - 4 \, a e^{\left (2 \, d x + 2 \, c\right )} + 3 \, b e^{\left (2 \, d x + 2 \, c\right )} - b\right )}}}{2 \, 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)^2,x, algorithm="giac")

[Out]

-1/2*((4*a*b - 3*b^2)*arctan(1/2*(b*e^(2*d*x + 2*c) + 2*a - b)/sqrt(-a^2 + a*b))/((a^3 - a^2*b)*sqrt(-a^2 + a*
b)) + 2*(4*a*b*e^(4*d*x + 4*c) - 3*b^2*e^(4*d*x + 4*c) + 8*a^2*e^(2*d*x + 2*c) - 14*a*b*e^(2*d*x + 2*c) + 6*b^
2*e^(2*d*x + 2*c) + 2*a*b - 3*b^2)/((a^3 - a^2*b)*(b*e^(6*d*x + 6*c) + 4*a*e^(4*d*x + 4*c) - 3*b*e^(4*d*x + 4*
c) - 4*a*e^(2*d*x + 2*c) + 3*b*e^(2*d*x + 2*c) - b)))/d

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \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 )}^2} \,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)^2),x)

[Out]

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

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