3.1.6 \(\int \frac {1}{(a+b \text {csch}^2(c+d x))^2} \, dx\) [6]

Optimal. Leaf size=100 \[ \frac {x}{a^2}-\frac {(3 a-2 b) \sqrt {b} \text {ArcTan}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {b}}\right )}{2 a^2 (a-b)^{3/2} d}+\frac {b \coth (c+d x)}{2 a (a-b) d \left (a-b+b \coth ^2(c+d x)\right )} \]

[Out]

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

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Rubi [A]
time = 0.10, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {4213, 425, 536, 212, 211} \begin {gather*} -\frac {\sqrt {b} (3 a-2 b) \text {ArcTan}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {b}}\right )}{2 a^2 d (a-b)^{3/2}}+\frac {x}{a^2}+\frac {b \coth (c+d x)}{2 a d (a-b) \left (a+b \coth ^2(c+d x)-b\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*Csch[c + d*x]^2)^(-2),x]

[Out]

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

Rule 211

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

Rule 212

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

Rule 425

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]) && IntBinomi
alQ[a, b, c, d, n, p, q, x]

Rule 536

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 4213

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]

Rubi steps

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

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Mathematica [A]
time = 0.47, size = 199, normalized size = 1.99 \begin {gather*} \frac {(-a+2 b+a \cosh (2 (c+d x))) \text {csch}^4(c+d x) \left ((a-2 b) \left (-2 (a-b)^{3/2} (c+d x)+(3 a-2 b) \sqrt {b} \text {ArcTan}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {b}}\right )\right )+a \left (2 (a-b)^{3/2} (c+d x)+\sqrt {b} (-3 a+2 b) \text {ArcTan}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {b}}\right )\right ) \cosh (2 (c+d x))+a \sqrt {a-b} b \sinh (2 (c+d x))\right )}{8 a^2 (a-b)^{3/2} d \left (a+b \text {csch}^2(c+d x)\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-a + 2*b + a*Cosh[2*(c + d*x)])*Csch[c + d*x]^4*((a - 2*b)*(-2*(a - b)^(3/2)*(c + d*x) + (3*a - 2*b)*Sqrt[b]
*ArcTan[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[b]]) + a*(2*(a - b)^(3/2)*(c + d*x) + Sqrt[b]*(-3*a + 2*b)*ArcTan[(Sq
rt[a - b]*Tanh[c + d*x])/Sqrt[b]])*Cosh[2*(c + d*x)] + a*Sqrt[a - b]*b*Sinh[2*(c + d*x)]))/(8*a^2*(a - b)^(3/2
)*d*(a + b*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. \(314\) vs. \(2(88)=176\).
time = 1.98, size = 315, normalized size = 3.15

method	result	size
derivativedivides	\(\frac {\frac {2 b \left (\frac {\frac {a \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a -8 b}+\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a -8 b}}{\frac {b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}+a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {b}{4}}+\frac {2 \left (3 a -2 b \right ) b \left (-\frac {\left (\sqrt {a \left (a -b \right )}-a \right ) \arctanh \left (\frac {b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {a \left (a -b \right )}-2 a +b \right ) b}}\right )}{2 \sqrt {a \left (a -b \right )}\, b \sqrt {\left (2 \sqrt {a \left (a -b \right )}-2 a +b \right ) b}}+\frac {\left (\sqrt {a \left (a -b \right )}+a \right ) \arctan \left (\frac {b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {a \left (a -b \right )}+2 a -b \right ) b}}\right )}{2 \sqrt {a \left (a -b \right )}\, b \sqrt {\left (2 \sqrt {a \left (a -b \right )}+2 a -b \right ) b}}\right )}{4 a -4 b}\right )}{a^{2}}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a^{2}}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a^{2}}}{d}\)	\(315\)
default	\(\frac {\frac {2 b \left (\frac {\frac {a \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a -8 b}+\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a -8 b}}{\frac {b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}+a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {b}{4}}+\frac {2 \left (3 a -2 b \right ) b \left (-\frac {\left (\sqrt {a \left (a -b \right )}-a \right ) \arctanh \left (\frac {b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {a \left (a -b \right )}-2 a +b \right ) b}}\right )}{2 \sqrt {a \left (a -b \right )}\, b \sqrt {\left (2 \sqrt {a \left (a -b \right )}-2 a +b \right ) b}}+\frac {\left (\sqrt {a \left (a -b \right )}+a \right ) \arctan \left (\frac {b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {a \left (a -b \right )}+2 a -b \right ) b}}\right )}{2 \sqrt {a \left (a -b \right )}\, b \sqrt {\left (2 \sqrt {a \left (a -b \right )}+2 a -b \right ) b}}\right )}{4 a -4 b}\right )}{a^{2}}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a^{2}}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a^{2}}}{d}\)	\(315\)
risch	\(\frac {x}{a^{2}}+\frac {b \left (a \,{\mathrm e}^{2 d x +2 c}-2 b \,{\mathrm e}^{2 d x +2 c}-a \right )}{a^{2} \left (a -b \right ) d \left (a \,{\mathrm e}^{4 d x +4 c}-2 a \,{\mathrm e}^{2 d x +2 c}+4 b \,{\mathrm e}^{2 d x +2 c}+a \right )}+\frac {3 \sqrt {-b \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-b \left (a -b \right )}+a -2 b}{a}\right )}{4 \left (a -b \right )^{2} d a}-\frac {\sqrt {-b \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-b \left (a -b \right )}+a -2 b}{a}\right ) b}{2 \left (a -b \right )^{2} d \,a^{2}}-\frac {3 \sqrt {-b \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-b \left (a -b \right )}-a +2 b}{a}\right )}{4 \left (a -b \right )^{2} d a}+\frac {\sqrt {-b \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-b \left (a -b \right )}-a +2 b}{a}\right ) b}{2 \left (a -b \right )^{2} d \,a^{2}}\)	\(324\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(2*b/a^2*((1/8*a/(a-b)*tanh(1/2*d*x+1/2*c)^3+1/8*a/(a-b)*tanh(1/2*d*x+1/2*c))/(1/4*b*tanh(1/2*d*x+1/2*c)^4
+a*tanh(1/2*d*x+1/2*c)^2-1/2*b*tanh(1/2*d*x+1/2*c)^2+1/4*b)+2*(3*a-2*b)/(4*a-4*b)*b*(-1/2*((a*(a-b))^(1/2)-a)/
(a*(a-b))^(1/2)/b/((2*(a*(a-b))^(1/2)-2*a+b)*b)^(1/2)*arctanh(b*tanh(1/2*d*x+1/2*c)/((2*(a*(a-b))^(1/2)-2*a+b)
*b)^(1/2))+1/2*((a*(a-b))^(1/2)+a)/(a*(a-b))^(1/2)/b/((2*(a*(a-b))^(1/2)+2*a-b)*b)^(1/2)*arctan(b*tanh(1/2*d*x
+1/2*c)/((2*(a*(a-b))^(1/2)+2*a-b)*b)^(1/2))))+1/a^2*ln(tanh(1/2*d*x+1/2*c)+1)-1/a^2*ln(tanh(1/2*d*x+1/2*c)-1)
)

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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(1/(a+b*csch(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. 729 vs. \(2 (88) = 176\).
time = 0.43, size = 1747, normalized size = 17.47 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*(4*(a^2 - a*b)*d*x*cosh(d*x + c)^4 + 16*(a^2 - a*b)*d*x*cosh(d*x + c)*sinh(d*x + c)^3 + 4*(a^2 - a*b)*d*x
*sinh(d*x + c)^4 + 4*(a^2 - a*b)*d*x - 4*(2*(a^2 - 3*a*b + 2*b^2)*d*x - a*b + 2*b^2)*cosh(d*x + c)^2 + 4*(6*(a
^2 - a*b)*d*x*cosh(d*x + c)^2 - 2*(a^2 - 3*a*b + 2*b^2)*d*x + a*b - 2*b^2)*sinh(d*x + c)^2 + ((3*a^2 - 2*a*b)*
cosh(d*x + c)^4 + 4*(3*a^2 - 2*a*b)*cosh(d*x + c)*sinh(d*x + c)^3 + (3*a^2 - 2*a*b)*sinh(d*x + c)^4 - 2*(3*a^2
- 8*a*b + 4*b^2)*cosh(d*x + c)^2 + 2*(3*(3*a^2 - 2*a*b)*cosh(d*x + c)^2 - 3*a^2 + 8*a*b - 4*b^2)*sinh(d*x + c
)^2 + 3*a^2 - 2*a*b + 4*((3*a^2 - 2*a*b)*cosh(d*x + c)^3 - (3*a^2 - 8*a*b + 4*b^2)*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)) - 4*a*b + 8*(2*(a^2 - a*b)*d*x*cosh(d*x + c)^3 - (2*(a^2 - 3*a*b + 2*b^2)*d*x - a*b + 2*b^2)*cosh(d*x + c)
)*sinh(d*x + c))/((a^4 - a^3*b)*d*cosh(d*x + c)^4 + 4*(a^4 - a^3*b)*d*cosh(d*x + c)*sinh(d*x + c)^3 + (a^4 - a
^3*b)*d*sinh(d*x + c)^4 - 2*(a^4 - 3*a^3*b + 2*a^2*b^2)*d*cosh(d*x + c)^2 + 2*(3*(a^4 - a^3*b)*d*cosh(d*x + c)
^2 - (a^4 - 3*a^3*b + 2*a^2*b^2)*d)*sinh(d*x + c)^2 + (a^4 - a^3*b)*d + 4*((a^4 - a^3*b)*d*cosh(d*x + c)^3 - (
a^4 - 3*a^3*b + 2*a^2*b^2)*d*cosh(d*x + c))*sinh(d*x + c)), 1/2*(2*(a^2 - a*b)*d*x*cosh(d*x + c)^4 + 8*(a^2 -
a*b)*d*x*cosh(d*x + c)*sinh(d*x + c)^3 + 2*(a^2 - a*b)*d*x*sinh(d*x + c)^4 + 2*(a^2 - a*b)*d*x - 2*(2*(a^2 - 3
*a*b + 2*b^2)*d*x - a*b + 2*b^2)*cosh(d*x + c)^2 + 2*(6*(a^2 - a*b)*d*x*cosh(d*x + c)^2 - 2*(a^2 - 3*a*b + 2*b
^2)*d*x + a*b - 2*b^2)*sinh(d*x + c)^2 - ((3*a^2 - 2*a*b)*cosh(d*x + c)^4 + 4*(3*a^2 - 2*a*b)*cosh(d*x + c)*si
nh(d*x + c)^3 + (3*a^2 - 2*a*b)*sinh(d*x + c)^4 - 2*(3*a^2 - 8*a*b + 4*b^2)*cosh(d*x + c)^2 + 2*(3*(3*a^2 - 2*
a*b)*cosh(d*x + c)^2 - 3*a^2 + 8*a*b - 4*b^2)*sinh(d*x + c)^2 + 3*a^2 - 2*a*b + 4*((3*a^2 - 2*a*b)*cosh(d*x +
c)^3 - (3*a^2 - 8*a*b + 4*b^2)*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) - 2*a*b + 4*(2*(a^2 - a*b)*d*
x*cosh(d*x + c)^3 - (2*(a^2 - 3*a*b + 2*b^2)*d*x - a*b + 2*b^2)*cosh(d*x + c))*sinh(d*x + c))/((a^4 - a^3*b)*d
*cosh(d*x + c)^4 + 4*(a^4 - a^3*b)*d*cosh(d*x + c)*sinh(d*x + c)^3 + (a^4 - a^3*b)*d*sinh(d*x + c)^4 - 2*(a^4
- 3*a^3*b + 2*a^2*b^2)*d*cosh(d*x + c)^2 + 2*(3*(a^4 - a^3*b)*d*cosh(d*x + c)^2 - (a^4 - 3*a^3*b + 2*a^2*b^2)*
d)*sinh(d*x + c)^2 + (a^4 - a^3*b)*d + 4*((a^4 - a^3*b)*d*cosh(d*x + c)^3 - (a^4 - 3*a^3*b + 2*a^2*b^2)*d*cosh
(d*x + c))*sinh(d*x + c))]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a + b \operatorname {csch}^{2}{\left (c + d x \right )}\right )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.49, size = 166, normalized size = 1.66 \begin {gather*} -\frac {\frac {{\left (3 \, a b - 2 \, b^{2}\right )} \arctan \left (\frac {a e^{\left (2 \, d x + 2 \, c\right )} - a + 2 \, b}{2 \, \sqrt {a b - b^{2}}}\right )}{{\left (a^{3} - a^{2} b\right )} \sqrt {a b - b^{2}}} - \frac {2 \, {\left (a b e^{\left (2 \, d x + 2 \, c\right )} - 2 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} - a b\right )}}{{\left (a^{3} - a^{2} b\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 )}} - \frac {2 \, {\left (d x + c\right )}}{a^{2}}}{2 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{{\left (a+\frac {b}{{\mathrm {sinh}\left (c+d\,x\right )}^2}\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b/sinh(c + d*x)^2)^2,x)

[Out]

int(1/(a + b/sinh(c + d*x)^2)^2, x)

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