3.1.85 \(\int \frac {\text {csch}^2(x)}{(a+b \sinh (x))^2} \, dx\) [85]

Optimal. Leaf size=115 \[ \frac {2 b \tanh ^{-1}(\cosh (x))}{a^3}-\frac {2 b^2 \left (3 a^2+2 b^2\right ) \tanh ^{-1}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right )^{3/2}}-\frac {\left (a^2+2 b^2\right ) \coth (x)}{a^2 \left (a^2+b^2\right )}+\frac {b^2 \coth (x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))} \]

[Out]

2*b*arctanh(cosh(x))/a^3-2*b^2*(3*a^2+2*b^2)*arctanh((b-a*tanh(1/2*x))/(a^2+b^2)^(1/2))/a^3/(a^2+b^2)^(3/2)-(a
^2+2*b^2)*coth(x)/a^2/(a^2+b^2)+b^2*coth(x)/a/(a^2+b^2)/(a+b*sinh(x))

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Rubi [A]
time = 0.27, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {2881, 3134, 3080, 3855, 2739, 632, 212} \begin {gather*} \frac {2 b \tanh ^{-1}(\cosh (x))}{a^3}-\frac {\left (a^2+2 b^2\right ) \coth (x)}{a^2 \left (a^2+b^2\right )}+\frac {b^2 \coth (x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))}-\frac {2 b^2 \left (3 a^2+2 b^2\right ) \tanh ^{-1}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Csch[x]^2/(a + b*Sinh[x])^2,x]

[Out]

(2*b*ArcTanh[Cosh[x]])/a^3 - (2*b^2*(3*a^2 + 2*b^2)*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a^3*(a^2 + b^
2)^(3/2)) - ((a^2 + 2*b^2)*Coth[x])/(a^2*(a^2 + b^2)) + (b^2*Coth[x])/(a*(a^2 + b^2)*(a + b*Sinh[x]))

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 632

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 2739

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 2881

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[(-b^2)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*((c + d*Sin[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2
- b^2))), x] + Dist[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])
^n*Simp[a*(b*c - a*d)*(m + 1) + b^2*d*(m + n + 2) - (b^2*c + b*(b*c - a*d)*(m + 1))*Sin[e + f*x] - b^2*d*(m +
n + 3)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]
&& NeQ[c^2 - d^2, 0] && LtQ[m, -1] && IntegersQ[2*m, 2*n] && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ[n]) ||
!(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] &&  !IntegerQ[m]) || EqQ[a, 0])))

Rule 3080

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.)
+ (f_.)*(x_)])), x_Symbol] :> Dist[(A*b - a*B)/(b*c - a*d), Int[1/(a + b*Sin[e + f*x]), x], x] + Dist[(B*c - A
*d)/(b*c - a*d), Int[1/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0]
&& NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]

Rule 3134

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*s
in[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b^2 - a*b*B + a^2*C))*Cos[e
+ f*x]*(a + b*Sin[e + f*x])^(m + 1)*((c + d*Sin[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 - b^2))), x] + D
ist[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[(m + 1)*
(b*c - a*d)*(a*A - b*B + a*C) + d*(A*b^2 - a*b*B + a^2*C)*(m + n + 2) - (c*(A*b^2 - a*b*B + a^2*C) + (m + 1)*(
b*c - a*d)*(A*b - a*B + b*C))*Sin[e + f*x] - d*(A*b^2 - a*b*B + a^2*C)*(m + n + 3)*Sin[e + f*x]^2, x], x], x]
/; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] &&
LtQ[m, -1] && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ[n]) ||  !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n]
&&  !IntegerQ[m]) || EqQ[a, 0])))

Rule 3855

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

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

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Mathematica [A]
time = 0.48, size = 118, normalized size = 1.03 \begin {gather*} -\frac {\frac {4 b^2 \left (3 a^2+2 b^2\right ) \text {ArcTan}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2-b^2}}\right )}{\left (-a^2-b^2\right )^{3/2}}+a \coth \left (\frac {x}{2}\right )+4 b \log \left (\tanh \left (\frac {x}{2}\right )\right )+\frac {2 a b^3 \cosh (x)}{\left (a^2+b^2\right ) (a+b \sinh (x))}+a \tanh \left (\frac {x}{2}\right )}{2 a^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*((4*b^2*(3*a^2 + 2*b^2)*ArcTan[(b - a*Tanh[x/2])/Sqrt[-a^2 - b^2]])/(-a^2 - b^2)^(3/2) + a*Coth[x/2] + 4*
b*Log[Tanh[x/2]] + (2*a*b^3*Cosh[x])/((a^2 + b^2)*(a + b*Sinh[x])) + a*Tanh[x/2])/a^3

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Maple [A]
time = 0.72, size = 141, normalized size = 1.23

method	result	size
default	\(-\frac {\tanh \left (\frac {x}{2}\right )}{2 a^{2}}-\frac {1}{2 a^{2} \tanh \left (\frac {x}{2}\right )}-\frac {2 b \ln \left (\tanh \left (\frac {x}{2}\right )\right )}{a^{3}}-\frac {2 b^{2} \left (\frac {-\frac {b^{2} \tanh \left (\frac {x}{2}\right )}{a^{2}+b^{2}}-\frac {a b}{a^{2}+b^{2}}}{a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-2 b \tanh \left (\frac {x}{2}\right )-a}-\frac {\left (3 a^{2}+2 b^{2}\right ) \arctanh \left (\frac {2 a \tanh \left (\frac {x}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{a^{3}}\)	\(141\)
risch	\(-\frac {2 \left (-a \,b^{2} {\mathrm e}^{3 x}+a^{2} b \,{\mathrm e}^{2 x}+2 b^{3} {\mathrm e}^{2 x}+2 a^{3} {\mathrm e}^{x}+3 a \,b^{2} {\mathrm e}^{x}-a^{2} b -2 b^{3}\right )}{a^{2} \left ({\mathrm e}^{2 x}-1\right ) \left (a^{2}+b^{2}\right ) \left (b \,{\mathrm e}^{2 x}+2 a \,{\mathrm e}^{x}-b \right )}-\frac {2 b \ln \left ({\mathrm e}^{x}-1\right )}{a^{3}}+\frac {2 b \ln \left ({\mathrm e}^{x}+1\right )}{a^{3}}+\frac {3 b^{2} \ln \left ({\mathrm e}^{x}+\frac {a \left (a^{2}+b^{2}\right )^{\frac {3}{2}}-a^{4}-2 a^{2} b^{2}-b^{4}}{b \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}} a}+\frac {2 b^{4} \ln \left ({\mathrm e}^{x}+\frac {a \left (a^{2}+b^{2}\right )^{\frac {3}{2}}-a^{4}-2 a^{2} b^{2}-b^{4}}{b \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}} a^{3}}-\frac {3 b^{2} \ln \left ({\mathrm e}^{x}+\frac {a \left (a^{2}+b^{2}\right )^{\frac {3}{2}}+a^{4}+2 a^{2} b^{2}+b^{4}}{b \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}} a}-\frac {2 b^{4} \ln \left ({\mathrm e}^{x}+\frac {a \left (a^{2}+b^{2}\right )^{\frac {3}{2}}+a^{4}+2 a^{2} b^{2}+b^{4}}{b \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}} a^{3}}\)	\(366\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/a^2*tanh(1/2*x)-1/2/a^2/tanh(1/2*x)-2/a^3*b*ln(tanh(1/2*x))-2*b^2/a^3*((-b^2/(a^2+b^2)*tanh(1/2*x)-a*b/(a
^2+b^2))/(a*tanh(1/2*x)^2-2*b*tanh(1/2*x)-a)-(3*a^2+2*b^2)/(a^2+b^2)^(3/2)*arctanh(1/2*(2*a*tanh(1/2*x)-2*b)/(
a^2+b^2)^(1/2)))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 251 vs. \(2 (111) = 222\).
time = 0.50, size = 251, normalized size = 2.18 \begin {gather*} \frac {{\left (3 \, a^{2} b^{2} + 2 \, b^{4}\right )} \log \left (\frac {b e^{\left (-x\right )} - a - \sqrt {a^{2} + b^{2}}}{b e^{\left (-x\right )} - a + \sqrt {a^{2} + b^{2}}}\right )}{{\left (a^{5} + a^{3} b^{2}\right )} \sqrt {a^{2} + b^{2}}} + \frac {2 \, {\left (a b^{2} e^{\left (-3 \, x\right )} - a^{2} b - 2 \, b^{3} - {\left (2 \, a^{3} + 3 \, a b^{2}\right )} e^{\left (-x\right )} + {\left (a^{2} b + 2 \, b^{3}\right )} e^{\left (-2 \, x\right )}\right )}}{a^{4} b + a^{2} b^{3} + 2 \, {\left (a^{5} + a^{3} b^{2}\right )} e^{\left (-x\right )} - 2 \, {\left (a^{4} b + a^{2} b^{3}\right )} e^{\left (-2 \, x\right )} - 2 \, {\left (a^{5} + a^{3} b^{2}\right )} e^{\left (-3 \, x\right )} + {\left (a^{4} b + a^{2} b^{3}\right )} e^{\left (-4 \, x\right )}} + \frac {2 \, b \log \left (e^{\left (-x\right )} + 1\right )}{a^{3}} - \frac {2 \, b \log \left (e^{\left (-x\right )} - 1\right )}{a^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x)^2/(a+b*sinh(x))^2,x, algorithm="maxima")

[Out]

(3*a^2*b^2 + 2*b^4)*log((b*e^(-x) - a - sqrt(a^2 + b^2))/(b*e^(-x) - a + sqrt(a^2 + b^2)))/((a^5 + a^3*b^2)*sq
rt(a^2 + b^2)) + 2*(a*b^2*e^(-3*x) - a^2*b - 2*b^3 - (2*a^3 + 3*a*b^2)*e^(-x) + (a^2*b + 2*b^3)*e^(-2*x))/(a^4
*b + a^2*b^3 + 2*(a^5 + a^3*b^2)*e^(-x) - 2*(a^4*b + a^2*b^3)*e^(-2*x) - 2*(a^5 + a^3*b^2)*e^(-3*x) + (a^4*b +
a^2*b^3)*e^(-4*x)) + 2*b*log(e^(-x) + 1)/a^3 - 2*b*log(e^(-x) - 1)/a^3

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1740 vs. \(2 (111) = 222\).
time = 0.53, size = 1740, normalized size = 15.13 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*a^5*b + 6*a^3*b^3 + 4*a*b^5 + 2*(a^4*b^2 + a^2*b^4)*cosh(x)^3 + 2*(a^4*b^2 + a^2*b^4)*sinh(x)^3 - 2*(a^5*b
+ 3*a^3*b^3 + 2*a*b^5)*cosh(x)^2 - 2*(a^5*b + 3*a^3*b^3 + 2*a*b^5 - 3*(a^4*b^2 + a^2*b^4)*cosh(x))*sinh(x)^2 +
(3*a^2*b^3 + 2*b^5 + (3*a^2*b^3 + 2*b^5)*cosh(x)^4 + (3*a^2*b^3 + 2*b^5)*sinh(x)^4 + 2*(3*a^3*b^2 + 2*a*b^4)*
cosh(x)^3 + 2*(3*a^3*b^2 + 2*a*b^4 + 2*(3*a^2*b^3 + 2*b^5)*cosh(x))*sinh(x)^3 - 2*(3*a^2*b^3 + 2*b^5)*cosh(x)^
2 - 2*(3*a^2*b^3 + 2*b^5 - 3*(3*a^2*b^3 + 2*b^5)*cosh(x)^2 - 3*(3*a^3*b^2 + 2*a*b^4)*cosh(x))*sinh(x)^2 - 2*(3
*a^3*b^2 + 2*a*b^4)*cosh(x) - 2*(3*a^3*b^2 + 2*a*b^4 - 2*(3*a^2*b^3 + 2*b^5)*cosh(x)^3 - 3*(3*a^3*b^2 + 2*a*b^
4)*cosh(x)^2 + 2*(3*a^2*b^3 + 2*b^5)*cosh(x))*sinh(x))*sqrt(a^2 + b^2)*log((b^2*cosh(x)^2 + b^2*sinh(x)^2 + 2*
a*b*cosh(x) + 2*a^2 + b^2 + 2*(b^2*cosh(x) + a*b)*sinh(x) - 2*sqrt(a^2 + b^2)*(b*cosh(x) + b*sinh(x) + a))/(b*
cosh(x)^2 + b*sinh(x)^2 + 2*a*cosh(x) + 2*(b*cosh(x) + a)*sinh(x) - b)) - 2*(2*a^6 + 5*a^4*b^2 + 3*a^2*b^4)*co
sh(x) + 2*(a^4*b^2 + 2*a^2*b^4 + b^6 + (a^4*b^2 + 2*a^2*b^4 + b^6)*cosh(x)^4 + (a^4*b^2 + 2*a^2*b^4 + b^6)*sin
h(x)^4 + 2*(a^5*b + 2*a^3*b^3 + a*b^5)*cosh(x)^3 + 2*(a^5*b + 2*a^3*b^3 + a*b^5 + 2*(a^4*b^2 + 2*a^2*b^4 + b^6
)*cosh(x))*sinh(x)^3 - 2*(a^4*b^2 + 2*a^2*b^4 + b^6)*cosh(x)^2 - 2*(a^4*b^2 + 2*a^2*b^4 + b^6 - 3*(a^4*b^2 + 2
*a^2*b^4 + b^6)*cosh(x)^2 - 3*(a^5*b + 2*a^3*b^3 + a*b^5)*cosh(x))*sinh(x)^2 - 2*(a^5*b + 2*a^3*b^3 + a*b^5)*c
osh(x) - 2*(a^5*b + 2*a^3*b^3 + a*b^5 - 2*(a^4*b^2 + 2*a^2*b^4 + b^6)*cosh(x)^3 - 3*(a^5*b + 2*a^3*b^3 + a*b^5
)*cosh(x)^2 + 2*(a^4*b^2 + 2*a^2*b^4 + b^6)*cosh(x))*sinh(x))*log(cosh(x) + sinh(x) + 1) - 2*(a^4*b^2 + 2*a^2*
b^4 + b^6 + (a^4*b^2 + 2*a^2*b^4 + b^6)*cosh(x)^4 + (a^4*b^2 + 2*a^2*b^4 + b^6)*sinh(x)^4 + 2*(a^5*b + 2*a^3*b
^3 + a*b^5)*cosh(x)^3 + 2*(a^5*b + 2*a^3*b^3 + a*b^5 + 2*(a^4*b^2 + 2*a^2*b^4 + b^6)*cosh(x))*sinh(x)^3 - 2*(a
^4*b^2 + 2*a^2*b^4 + b^6)*cosh(x)^2 - 2*(a^4*b^2 + 2*a^2*b^4 + b^6 - 3*(a^4*b^2 + 2*a^2*b^4 + b^6)*cosh(x)^2 -
3*(a^5*b + 2*a^3*b^3 + a*b^5)*cosh(x))*sinh(x)^2 - 2*(a^5*b + 2*a^3*b^3 + a*b^5)*cosh(x) - 2*(a^5*b + 2*a^3*b
^3 + a*b^5 - 2*(a^4*b^2 + 2*a^2*b^4 + b^6)*cosh(x)^3 - 3*(a^5*b + 2*a^3*b^3 + a*b^5)*cosh(x)^2 + 2*(a^4*b^2 +
2*a^2*b^4 + b^6)*cosh(x))*sinh(x))*log(cosh(x) + sinh(x) - 1) - 2*(2*a^6 + 5*a^4*b^2 + 3*a^2*b^4 - 3*(a^4*b^2
+ a^2*b^4)*cosh(x)^2 + 2*(a^5*b + 3*a^3*b^3 + 2*a*b^5)*cosh(x))*sinh(x))/(a^7*b + 2*a^5*b^3 + a^3*b^5 + (a^7*b
+ 2*a^5*b^3 + a^3*b^5)*cosh(x)^4 + (a^7*b + 2*a^5*b^3 + a^3*b^5)*sinh(x)^4 + 2*(a^8 + 2*a^6*b^2 + a^4*b^4)*co
sh(x)^3 + 2*(a^8 + 2*a^6*b^2 + a^4*b^4 + 2*(a^7*b + 2*a^5*b^3 + a^3*b^5)*cosh(x))*sinh(x)^3 - 2*(a^7*b + 2*a^5
*b^3 + a^3*b^5)*cosh(x)^2 - 2*(a^7*b + 2*a^5*b^3 + a^3*b^5 - 3*(a^7*b + 2*a^5*b^3 + a^3*b^5)*cosh(x)^2 - 3*(a^
8 + 2*a^6*b^2 + a^4*b^4)*cosh(x))*sinh(x)^2 - 2*(a^8 + 2*a^6*b^2 + a^4*b^4)*cosh(x) - 2*(a^8 + 2*a^6*b^2 + a^4
*b^4 - 2*(a^7*b + 2*a^5*b^3 + a^3*b^5)*cosh(x)^3 - 3*(a^8 + 2*a^6*b^2 + a^4*b^4)*cosh(x)^2 + 2*(a^7*b + 2*a^5*
b^3 + a^3*b^5)*cosh(x))*sinh(x))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(csch(x)**2/(a + b*sinh(x))**2, x)

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Giac [A]
time = 0.43, size = 205, normalized size = 1.78 \begin {gather*} \frac {{\left (3 \, a^{2} b^{2} + 2 \, b^{4}\right )} \log \left (\frac {{\left | 2 \, b e^{x} + 2 \, a - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, b e^{x} + 2 \, a + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{{\left (a^{5} + a^{3} b^{2}\right )} \sqrt {a^{2} + b^{2}}} + \frac {2 \, {\left (a b^{2} e^{\left (3 \, x\right )} - a^{2} b e^{\left (2 \, x\right )} - 2 \, b^{3} e^{\left (2 \, x\right )} - 2 \, a^{3} e^{x} - 3 \, a b^{2} e^{x} + a^{2} b + 2 \, b^{3}\right )}}{{\left (a^{4} + a^{2} b^{2}\right )} {\left (b e^{\left (4 \, x\right )} + 2 \, a e^{\left (3 \, x\right )} - 2 \, b e^{\left (2 \, x\right )} - 2 \, a e^{x} + b\right )}} + \frac {2 \, b \log \left (e^{x} + 1\right )}{a^{3}} - \frac {2 \, b \log \left ({\left | e^{x} - 1 \right |}\right )}{a^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(3*a^2*b^2 + 2*b^4)*log(abs(2*b*e^x + 2*a - 2*sqrt(a^2 + b^2))/abs(2*b*e^x + 2*a + 2*sqrt(a^2 + b^2)))/((a^5 +
a^3*b^2)*sqrt(a^2 + b^2)) + 2*(a*b^2*e^(3*x) - a^2*b*e^(2*x) - 2*b^3*e^(2*x) - 2*a^3*e^x - 3*a*b^2*e^x + a^2*
b + 2*b^3)/((a^4 + a^2*b^2)*(b*e^(4*x) + 2*a*e^(3*x) - 2*b*e^(2*x) - 2*a*e^x + b)) + 2*b*log(e^x + 1)/a^3 - 2*
b*log(abs(e^x - 1))/a^3

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Mupad [B]
time = 3.39, size = 1017, normalized size = 8.84 \begin {gather*} \frac {\frac {2\,\left (25\,a^8\,b^6+90\,a^6\,b^8+96\,a^4\,b^{10}+32\,a^2\,b^{12}\right )}{a^4\,b^2\,\left (25\,a^6\,b^3+65\,a^4\,b^5+56\,a^2\,b^7+16\,b^9\right )}-\frac {2\,{\mathrm {e}}^x\,\left (50\,a^9\,b^6+155\,a^7\,b^8+152\,a^5\,b^{10}+48\,a^3\,b^{12}\right )}{a^4\,b^3\,\left (25\,a^6\,b^3+65\,a^4\,b^5+56\,a^2\,b^7+16\,b^9\right )}-\frac {2\,{\mathrm {e}}^{2\,x}\,\left (25\,a^8\,b^6+90\,a^6\,b^8+96\,a^4\,b^{10}+32\,a^2\,b^{12}\right )}{a^4\,b^2\,\left (25\,a^6\,b^3+65\,a^4\,b^5+56\,a^2\,b^7+16\,b^9\right )}+\frac {2\,{\mathrm {e}}^{3\,x}\,\left (25\,a^7\,b^8+40\,a^5\,b^{10}+16\,a^3\,b^{12}\right )}{a^4\,b^3\,\left (25\,a^6\,b^3+65\,a^4\,b^5+56\,a^2\,b^7+16\,b^9\right )}}{b-2\,a\,{\mathrm {e}}^x+2\,a\,{\mathrm {e}}^{3\,x}-2\,b\,{\mathrm {e}}^{2\,x}+b\,{\mathrm {e}}^{4\,x}}-\frac {2\,b\,\ln \left ({\mathrm {e}}^x-1\right )}{a^3}+\frac {2\,b\,\ln \left ({\mathrm {e}}^x+1\right )}{a^3}+\frac {b^2\,\ln \left (-\frac {64\,\left (3\,a^2+2\,b^2\right )\,\left (-8\,{\mathrm {e}}^x\,a^3+4\,a^2\,b-7\,{\mathrm {e}}^x\,a\,b^2+4\,b^3\right )}{a^6\,b\,{\left (a^2+b^2\right )}^2}-\frac {32\,\left (3\,a^2+2\,b^2\right )\,\left (8\,a^9\,b-8\,b^7\,\sqrt {{\left (a^2+b^2\right )}^3}+3\,a^3\,b^7+13\,a^5\,b^5+18\,a^7\,b^3-16\,a^{10}\,{\mathrm {e}}^x-24\,a^2\,b^5\,\sqrt {{\left (a^2+b^2\right )}^3}-18\,a^4\,b^3\,\sqrt {{\left (a^2+b^2\right )}^3}-9\,a^4\,b^6\,{\mathrm {e}}^x-33\,a^6\,b^4\,{\mathrm {e}}^x-40\,a^8\,b^2\,{\mathrm {e}}^x+41\,a^3\,b^4\,{\mathrm {e}}^x\,\sqrt {{\left (a^2+b^2\right )}^3}+30\,a^5\,b^2\,{\mathrm {e}}^x\,\sqrt {{\left (a^2+b^2\right )}^3}+14\,a\,b^6\,{\mathrm {e}}^x\,\sqrt {{\left (a^2+b^2\right )}^3}\right )}{a^6\,b\,\sqrt {{\left (a^2+b^2\right )}^3}\,{\left (a^2+b^2\right )}^4}\right )\,\sqrt {{\left (a^2+b^2\right )}^3}\,\left (3\,a^2+2\,b^2\right )}{a^9+3\,a^7\,b^2+3\,a^5\,b^4+a^3\,b^6}-\frac {b^2\,\ln \left (\frac {32\,\left (3\,a^2+2\,b^2\right )\,\left (8\,a^9\,b+8\,b^7\,\sqrt {{\left (a^2+b^2\right )}^3}+3\,a^3\,b^7+13\,a^5\,b^5+18\,a^7\,b^3-16\,a^{10}\,{\mathrm {e}}^x+24\,a^2\,b^5\,\sqrt {{\left (a^2+b^2\right )}^3}+18\,a^4\,b^3\,\sqrt {{\left (a^2+b^2\right )}^3}-9\,a^4\,b^6\,{\mathrm {e}}^x-33\,a^6\,b^4\,{\mathrm {e}}^x-40\,a^8\,b^2\,{\mathrm {e}}^x-41\,a^3\,b^4\,{\mathrm {e}}^x\,\sqrt {{\left (a^2+b^2\right )}^3}-30\,a^5\,b^2\,{\mathrm {e}}^x\,\sqrt {{\left (a^2+b^2\right )}^3}-14\,a\,b^6\,{\mathrm {e}}^x\,\sqrt {{\left (a^2+b^2\right )}^3}\right )}{a^6\,b\,\sqrt {{\left (a^2+b^2\right )}^3}\,{\left (a^2+b^2\right )}^4}-\frac {64\,\left (3\,a^2+2\,b^2\right )\,\left (-8\,{\mathrm {e}}^x\,a^3+4\,a^2\,b-7\,{\mathrm {e}}^x\,a\,b^2+4\,b^3\right )}{a^6\,b\,{\left (a^2+b^2\right )}^2}\right )\,\sqrt {{\left (a^2+b^2\right )}^3}\,\left (3\,a^2+2\,b^2\right )}{a^9+3\,a^7\,b^2+3\,a^5\,b^4+a^3\,b^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((2*(32*a^2*b^12 + 96*a^4*b^10 + 90*a^6*b^8 + 25*a^8*b^6))/(a^4*b^2*(16*b^9 + 56*a^2*b^7 + 65*a^4*b^5 + 25*a^6
*b^3)) - (2*exp(x)*(48*a^3*b^12 + 152*a^5*b^10 + 155*a^7*b^8 + 50*a^9*b^6))/(a^4*b^3*(16*b^9 + 56*a^2*b^7 + 65
*a^4*b^5 + 25*a^6*b^3)) - (2*exp(2*x)*(32*a^2*b^12 + 96*a^4*b^10 + 90*a^6*b^8 + 25*a^8*b^6))/(a^4*b^2*(16*b^9
+ 56*a^2*b^7 + 65*a^4*b^5 + 25*a^6*b^3)) + (2*exp(3*x)*(16*a^3*b^12 + 40*a^5*b^10 + 25*a^7*b^8))/(a^4*b^3*(16*
b^9 + 56*a^2*b^7 + 65*a^4*b^5 + 25*a^6*b^3)))/(b - 2*a*exp(x) + 2*a*exp(3*x) - 2*b*exp(2*x) + b*exp(4*x)) - (2
*b*log(exp(x) - 1))/a^3 + (2*b*log(exp(x) + 1))/a^3 + (b^2*log(- (64*(3*a^2 + 2*b^2)*(4*a^2*b + 4*b^3 - 8*a^3*
exp(x) - 7*a*b^2*exp(x)))/(a^6*b*(a^2 + b^2)^2) - (32*(3*a^2 + 2*b^2)*(8*a^9*b - 8*b^7*((a^2 + b^2)^3)^(1/2) +
3*a^3*b^7 + 13*a^5*b^5 + 18*a^7*b^3 - 16*a^10*exp(x) - 24*a^2*b^5*((a^2 + b^2)^3)^(1/2) - 18*a^4*b^3*((a^2 +
b^2)^3)^(1/2) - 9*a^4*b^6*exp(x) - 33*a^6*b^4*exp(x) - 40*a^8*b^2*exp(x) + 41*a^3*b^4*exp(x)*((a^2 + b^2)^3)^(
1/2) + 30*a^5*b^2*exp(x)*((a^2 + b^2)^3)^(1/2) + 14*a*b^6*exp(x)*((a^2 + b^2)^3)^(1/2)))/(a^6*b*((a^2 + b^2)^3
)^(1/2)*(a^2 + b^2)^4))*((a^2 + b^2)^3)^(1/2)*(3*a^2 + 2*b^2))/(a^9 + a^3*b^6 + 3*a^5*b^4 + 3*a^7*b^2) - (b^2*
log((32*(3*a^2 + 2*b^2)*(8*a^9*b + 8*b^7*((a^2 + b^2)^3)^(1/2) + 3*a^3*b^7 + 13*a^5*b^5 + 18*a^7*b^3 - 16*a^10
*exp(x) + 24*a^2*b^5*((a^2 + b^2)^3)^(1/2) + 18*a^4*b^3*((a^2 + b^2)^3)^(1/2) - 9*a^4*b^6*exp(x) - 33*a^6*b^4*
exp(x) - 40*a^8*b^2*exp(x) - 41*a^3*b^4*exp(x)*((a^2 + b^2)^3)^(1/2) - 30*a^5*b^2*exp(x)*((a^2 + b^2)^3)^(1/2)
- 14*a*b^6*exp(x)*((a^2 + b^2)^3)^(1/2)))/(a^6*b*((a^2 + b^2)^3)^(1/2)*(a^2 + b^2)^4) - (64*(3*a^2 + 2*b^2)*(
4*a^2*b + 4*b^3 - 8*a^3*exp(x) - 7*a*b^2*exp(x)))/(a^6*b*(a^2 + b^2)^2))*((a^2 + b^2)^3)^(1/2)*(3*a^2 + 2*b^2)
)/(a^9 + a^3*b^6 + 3*a^5*b^4 + 3*a^7*b^2)

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