3.3.41 \(\int \frac {\coth ^2(x)}{(a+b \sinh (x))^2} \, dx\) [241]

Optimal. Leaf size=80 \[ \frac {2 b \tanh ^{-1}(\cosh (x))}{a^3}-\frac {2 \left (a^2+2 b^2\right ) \tanh ^{-1}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{a^3 \sqrt {a^2+b^2}}-\frac {2 \coth (x)}{a^2}+\frac {\coth (x)}{a (a+b \sinh (x))} \]

[Out]

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

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Rubi [A]
time = 0.29, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {2802, 3135, 3080, 3855, 2739, 632, 212} \begin {gather*} \frac {2 b \tanh ^{-1}(\cosh (x))}{a^3}-\frac {2 \coth (x)}{a^2}-\frac {2 \left (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 \sqrt {a^2+b^2}}+\frac {\coth (x)}{a (a+b \sinh (x))} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Coth[x]^2/(a + b*Sinh[x])^2,x]

[Out]

(2*b*ArcTanh[Cosh[x]])/a^3 - (2*(a^2 + 2*b^2)*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a^3*Sqrt[a^2 + b^2]
) - (2*Coth[x])/a^2 + Coth[x]/(a*(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 2802

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)/tan[(e_.) + (f_.)*(x_)]^2, x_Symbol] :> Int[(a + b*Sin[e + f*x
])^m*((1 - Sin[e + f*x]^2)/Sin[e + f*x]^2), x] /; FreeQ[{a, b, e, f, m}, x] && NeQ[a^2 - b^2, 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 3135

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (C_.)*s
in[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b^2 + 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] + 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*(m + 1)*(b*c - a*d)*(A + C) + d*(A*b^2 + a^2*C
)*(m + n + 2) - (c*(A*b^2 + a^2*C) + b*(m + 1)*(b*c - a*d)*(A + C))*Sin[e + f*x] - d*(A*b^2 + a^2*C)*(m + n +
3)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, 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] && L
tQ[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 {\coth ^2(x)}{(a+b \sinh (x))^2} \, dx &=\int \frac {\text {csch}^2(x) \left (1+\sinh ^2(x)\right )}{(a+b \sinh (x))^2} \, dx\\ &=\frac {\coth (x)}{a (a+b \sinh (x))}+\frac {\int \frac {\text {csch}^2(x) \left (2 \left (a^2+b^2\right )+\left (a^2+b^2\right ) \sinh ^2(x)\right )}{a+b \sinh (x)} \, dx}{a \left (a^2+b^2\right )}\\ &=-\frac {2 \coth (x)}{a^2}+\frac {\coth (x)}{a (a+b \sinh (x))}+\frac {i \int \frac {\text {csch}(x) \left (2 i b \left (a^2+b^2\right )-i a \left (a^2+b^2\right ) \sinh (x)\right )}{a+b \sinh (x)} \, dx}{a^2 \left (a^2+b^2\right )}\\ &=-\frac {2 \coth (x)}{a^2}+\frac {\coth (x)}{a (a+b \sinh (x))}-\frac {(2 b) \int \text {csch}(x) \, dx}{a^3}+\frac {\left (a^2+2 b^2\right ) \int \frac {1}{a+b \sinh (x)} \, dx}{a^3}\\ &=\frac {2 b \tanh ^{-1}(\cosh (x))}{a^3}-\frac {2 \coth (x)}{a^2}+\frac {\coth (x)}{a (a+b \sinh (x))}+\frac {\left (2 \left (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}\\ &=\frac {2 b \tanh ^{-1}(\cosh (x))}{a^3}-\frac {2 \coth (x)}{a^2}+\frac {\coth (x)}{a (a+b \sinh (x))}-\frac {\left (4 \left (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}\\ &=\frac {2 b \tanh ^{-1}(\cosh (x))}{a^3}-\frac {2 \left (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 \sqrt {a^2+b^2}}-\frac {2 \coth (x)}{a^2}+\frac {\coth (x)}{a (a+b \sinh (x))}\\ \end {align*}

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Mathematica [A]
time = 0.38, size = 102, normalized size = 1.28 \begin {gather*} -\frac {-\frac {4 \left (a^2+2 b^2\right ) \text {ArcTan}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-a^2-b^2}}+a \coth \left (\frac {x}{2}\right )+4 b \log \left (\tanh \left (\frac {x}{2}\right )\right )+\frac {2 a b \cosh (x)}{a+b \sinh (x)}+a \tanh \left (\frac {x}{2}\right )}{2 a^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.65, size = 118, normalized size = 1.48

method	result	size
default	\(-\frac {\tanh \left (\frac {x}{2}\right )}{2 a^{2}}-\frac {2 \left (\frac {-b^{2} \tanh \left (\frac {x}{2}\right )-a b}{a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-2 b \tanh \left (\frac {x}{2}\right )-a}-\frac {\left (a^{2}+2 b^{2}\right ) \arctanh \left (\frac {2 a \tanh \left (\frac {x}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{\sqrt {a^{2}+b^{2}}}\right )}{a^{3}}-\frac {1}{2 a^{2} \tanh \left (\frac {x}{2}\right )}-\frac {2 b \ln \left (\tanh \left (\frac {x}{2}\right )\right )}{a^{3}}\)	\(118\)
risch	\(\frac {2 a \,{\mathrm e}^{3 x}-4 b \,{\mathrm e}^{2 x}-6 a \,{\mathrm e}^{x}+4 b}{a^{2} \left ({\mathrm e}^{2 x}-1\right ) \left (b \,{\mathrm e}^{2 x}+2 a \,{\mathrm e}^{x}-b \right )}+\frac {2 b \ln \left ({\mathrm e}^{x}+1\right )}{a^{3}}-\frac {2 b \ln \left ({\mathrm e}^{x}-1\right )}{a^{3}}+\frac {\ln \left ({\mathrm e}^{x}+\frac {a \sqrt {a^{2}+b^{2}}-a^{2}-b^{2}}{\sqrt {a^{2}+b^{2}}\, b}\right )}{\sqrt {a^{2}+b^{2}}\, a}+\frac {2 \ln \left ({\mathrm e}^{x}+\frac {a \sqrt {a^{2}+b^{2}}-a^{2}-b^{2}}{\sqrt {a^{2}+b^{2}}\, b}\right ) b^{2}}{\sqrt {a^{2}+b^{2}}\, a^{3}}-\frac {\ln \left ({\mathrm e}^{x}+\frac {a \sqrt {a^{2}+b^{2}}+a^{2}+b^{2}}{\sqrt {a^{2}+b^{2}}\, b}\right )}{\sqrt {a^{2}+b^{2}}\, a}-\frac {2 \ln \left ({\mathrm e}^{x}+\frac {a \sqrt {a^{2}+b^{2}}+a^{2}+b^{2}}{\sqrt {a^{2}+b^{2}}\, b}\right ) b^{2}}{\sqrt {a^{2}+b^{2}}\, a^{3}}\)	\(285\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 165 vs. \(2 (76) = 152\).
time = 0.49, size = 165, normalized size = 2.06 \begin {gather*} -\frac {2 \, {\left (3 \, a e^{\left (-x\right )} - 2 \, b e^{\left (-2 \, x\right )} - a e^{\left (-3 \, x\right )} + 2 \, b\right )}}{2 \, a^{3} e^{\left (-x\right )} - 2 \, a^{2} b e^{\left (-2 \, x\right )} - 2 \, a^{3} e^{\left (-3 \, x\right )} + a^{2} b e^{\left (-4 \, x\right )} + a^{2} b} + \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}} + \frac {{\left (a^{2} + 2 \, b^{2}\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 )}{\sqrt {a^{2} + b^{2}} a^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1257 vs. \(2 (76) = 152\).
time = 0.48, size = 1257, normalized size = 15.71 \begin {gather*} \frac {4 \, a^{3} b + 4 \, a b^{3} + 2 \, {\left (a^{4} + a^{2} b^{2}\right )} \cosh \left (x\right )^{3} + 2 \, {\left (a^{4} + a^{2} b^{2}\right )} \sinh \left (x\right )^{3} - 4 \, {\left (a^{3} b + a b^{3}\right )} \cosh \left (x\right )^{2} - 2 \, {\left (2 \, a^{3} b + 2 \, a b^{3} - 3 \, {\left (a^{4} + a^{2} b^{2}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} + {\left ({\left (a^{2} b + 2 \, b^{3}\right )} \cosh \left (x\right )^{4} + {\left (a^{2} b + 2 \, b^{3}\right )} \sinh \left (x\right )^{4} + 2 \, {\left (a^{3} + 2 \, a b^{2}\right )} \cosh \left (x\right )^{3} + 2 \, {\left (a^{3} + 2 \, a b^{2} + 2 \, {\left (a^{2} b + 2 \, b^{3}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + a^{2} b + 2 \, b^{3} - 2 \, {\left (a^{2} b + 2 \, b^{3}\right )} \cosh \left (x\right )^{2} - 2 \, {\left (a^{2} b + 2 \, b^{3} - 3 \, {\left (a^{2} b + 2 \, b^{3}\right )} \cosh \left (x\right )^{2} - 3 \, {\left (a^{3} + 2 \, a b^{2}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} - 2 \, {\left (a^{3} + 2 \, a b^{2}\right )} \cosh \left (x\right ) + 2 \, {\left (2 \, {\left (a^{2} b + 2 \, b^{3}\right )} \cosh \left (x\right )^{3} - a^{3} - 2 \, a b^{2} + 3 \, {\left (a^{3} + 2 \, a b^{2}\right )} \cosh \left (x\right )^{2} - 2 \, {\left (a^{2} b + 2 \, b^{3}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )\right )} \sqrt {a^{2} + b^{2}} \log \left (\frac {b^{2} \cosh \left (x\right )^{2} + b^{2} \sinh \left (x\right )^{2} + 2 \, a b \cosh \left (x\right ) + 2 \, a^{2} + b^{2} + 2 \, {\left (b^{2} \cosh \left (x\right ) + a b\right )} \sinh \left (x\right ) - 2 \, \sqrt {a^{2} + b^{2}} {\left (b \cosh \left (x\right ) + b \sinh \left (x\right ) + a\right )}}{b \cosh \left (x\right )^{2} + b \sinh \left (x\right )^{2} + 2 \, a \cosh \left (x\right ) + 2 \, {\left (b \cosh \left (x\right ) + a\right )} \sinh \left (x\right ) - b}\right ) - 6 \, {\left (a^{4} + a^{2} b^{2}\right )} \cosh \left (x\right ) + 2 \, {\left ({\left (a^{2} b^{2} + b^{4}\right )} \cosh \left (x\right )^{4} + {\left (a^{2} b^{2} + b^{4}\right )} \sinh \left (x\right )^{4} + a^{2} b^{2} + b^{4} + 2 \, {\left (a^{3} b + a b^{3}\right )} \cosh \left (x\right )^{3} + 2 \, {\left (a^{3} b + a b^{3} + 2 \, {\left (a^{2} b^{2} + b^{4}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} - 2 \, {\left (a^{2} b^{2} + b^{4}\right )} \cosh \left (x\right )^{2} - 2 \, {\left (a^{2} b^{2} + b^{4} - 3 \, {\left (a^{2} b^{2} + b^{4}\right )} \cosh \left (x\right )^{2} - 3 \, {\left (a^{3} b + a b^{3}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} - 2 \, {\left (a^{3} b + a b^{3}\right )} \cosh \left (x\right ) - 2 \, {\left (a^{3} b + a b^{3} - 2 \, {\left (a^{2} b^{2} + b^{4}\right )} \cosh \left (x\right )^{3} - 3 \, {\left (a^{3} b + a b^{3}\right )} \cosh \left (x\right )^{2} + 2 \, {\left (a^{2} b^{2} + b^{4}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) - 2 \, {\left ({\left (a^{2} b^{2} + b^{4}\right )} \cosh \left (x\right )^{4} + {\left (a^{2} b^{2} + b^{4}\right )} \sinh \left (x\right )^{4} + a^{2} b^{2} + b^{4} + 2 \, {\left (a^{3} b + a b^{3}\right )} \cosh \left (x\right )^{3} + 2 \, {\left (a^{3} b + a b^{3} + 2 \, {\left (a^{2} b^{2} + b^{4}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} - 2 \, {\left (a^{2} b^{2} + b^{4}\right )} \cosh \left (x\right )^{2} - 2 \, {\left (a^{2} b^{2} + b^{4} - 3 \, {\left (a^{2} b^{2} + b^{4}\right )} \cosh \left (x\right )^{2} - 3 \, {\left (a^{3} b + a b^{3}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} - 2 \, {\left (a^{3} b + a b^{3}\right )} \cosh \left (x\right ) - 2 \, {\left (a^{3} b + a b^{3} - 2 \, {\left (a^{2} b^{2} + b^{4}\right )} \cosh \left (x\right )^{3} - 3 \, {\left (a^{3} b + a b^{3}\right )} \cosh \left (x\right )^{2} + 2 \, {\left (a^{2} b^{2} + b^{4}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\right ) - 2 \, {\left (3 \, a^{4} + 3 \, a^{2} b^{2} - 3 \, {\left (a^{4} + a^{2} b^{2}\right )} \cosh \left (x\right )^{2} + 4 \, {\left (a^{3} b + a b^{3}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )}{a^{5} b + a^{3} b^{3} + {\left (a^{5} b + a^{3} b^{3}\right )} \cosh \left (x\right )^{4} + {\left (a^{5} b + a^{3} b^{3}\right )} \sinh \left (x\right )^{4} + 2 \, {\left (a^{6} + a^{4} b^{2}\right )} \cosh \left (x\right )^{3} + 2 \, {\left (a^{6} + a^{4} b^{2} + 2 \, {\left (a^{5} b + a^{3} b^{3}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} - 2 \, {\left (a^{5} b + a^{3} b^{3}\right )} \cosh \left (x\right )^{2} - 2 \, {\left (a^{5} b + a^{3} b^{3} - 3 \, {\left (a^{5} b + a^{3} b^{3}\right )} \cosh \left (x\right )^{2} - 3 \, {\left (a^{6} + a^{4} b^{2}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} - 2 \, {\left (a^{6} + a^{4} b^{2}\right )} \cosh \left (x\right ) - 2 \, {\left (a^{6} + a^{4} b^{2} - 2 \, {\left (a^{5} b + a^{3} b^{3}\right )} \cosh \left (x\right )^{3} - 3 \, {\left (a^{6} + a^{4} b^{2}\right )} \cosh \left (x\right )^{2} + 2 \, {\left (a^{5} b + a^{3} b^{3}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(4*a^3*b + 4*a*b^3 + 2*(a^4 + a^2*b^2)*cosh(x)^3 + 2*(a^4 + a^2*b^2)*sinh(x)^3 - 4*(a^3*b + a*b^3)*cosh(x)^2 -
2*(2*a^3*b + 2*a*b^3 - 3*(a^4 + a^2*b^2)*cosh(x))*sinh(x)^2 + ((a^2*b + 2*b^3)*cosh(x)^4 + (a^2*b + 2*b^3)*si
nh(x)^4 + 2*(a^3 + 2*a*b^2)*cosh(x)^3 + 2*(a^3 + 2*a*b^2 + 2*(a^2*b + 2*b^3)*cosh(x))*sinh(x)^3 + a^2*b + 2*b^
3 - 2*(a^2*b + 2*b^3)*cosh(x)^2 - 2*(a^2*b + 2*b^3 - 3*(a^2*b + 2*b^3)*cosh(x)^2 - 3*(a^3 + 2*a*b^2)*cosh(x))*
sinh(x)^2 - 2*(a^3 + 2*a*b^2)*cosh(x) + 2*(2*(a^2*b + 2*b^3)*cosh(x)^3 - a^3 - 2*a*b^2 + 3*(a^3 + 2*a*b^2)*cos
h(x)^2 - 2*(a^2*b + 2*b^3)*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)) - 6*(a^4 + a^2*b^2)*cosh(x) + 2*((a^2*b^2 + b^4)
*cosh(x)^4 + (a^2*b^2 + b^4)*sinh(x)^4 + a^2*b^2 + b^4 + 2*(a^3*b + a*b^3)*cosh(x)^3 + 2*(a^3*b + a*b^3 + 2*(a
^2*b^2 + b^4)*cosh(x))*sinh(x)^3 - 2*(a^2*b^2 + b^4)*cosh(x)^2 - 2*(a^2*b^2 + b^4 - 3*(a^2*b^2 + b^4)*cosh(x)^
2 - 3*(a^3*b + a*b^3)*cosh(x))*sinh(x)^2 - 2*(a^3*b + a*b^3)*cosh(x) - 2*(a^3*b + a*b^3 - 2*(a^2*b^2 + b^4)*co
sh(x)^3 - 3*(a^3*b + a*b^3)*cosh(x)^2 + 2*(a^2*b^2 + b^4)*cosh(x))*sinh(x))*log(cosh(x) + sinh(x) + 1) - 2*((a
^2*b^2 + b^4)*cosh(x)^4 + (a^2*b^2 + b^4)*sinh(x)^4 + a^2*b^2 + b^4 + 2*(a^3*b + a*b^3)*cosh(x)^3 + 2*(a^3*b +
a*b^3 + 2*(a^2*b^2 + b^4)*cosh(x))*sinh(x)^3 - 2*(a^2*b^2 + b^4)*cosh(x)^2 - 2*(a^2*b^2 + b^4 - 3*(a^2*b^2 +
b^4)*cosh(x)^2 - 3*(a^3*b + a*b^3)*cosh(x))*sinh(x)^2 - 2*(a^3*b + a*b^3)*cosh(x) - 2*(a^3*b + a*b^3 - 2*(a^2*
b^2 + b^4)*cosh(x)^3 - 3*(a^3*b + a*b^3)*cosh(x)^2 + 2*(a^2*b^2 + b^4)*cosh(x))*sinh(x))*log(cosh(x) + sinh(x)
- 1) - 2*(3*a^4 + 3*a^2*b^2 - 3*(a^4 + a^2*b^2)*cosh(x)^2 + 4*(a^3*b + a*b^3)*cosh(x))*sinh(x))/(a^5*b + a^3*
b^3 + (a^5*b + a^3*b^3)*cosh(x)^4 + (a^5*b + a^3*b^3)*sinh(x)^4 + 2*(a^6 + a^4*b^2)*cosh(x)^3 + 2*(a^6 + a^4*b
^2 + 2*(a^5*b + a^3*b^3)*cosh(x))*sinh(x)^3 - 2*(a^5*b + a^3*b^3)*cosh(x)^2 - 2*(a^5*b + a^3*b^3 - 3*(a^5*b +
a^3*b^3)*cosh(x)^2 - 3*(a^6 + a^4*b^2)*cosh(x))*sinh(x)^2 - 2*(a^6 + a^4*b^2)*cosh(x) - 2*(a^6 + a^4*b^2 - 2*(
a^5*b + a^3*b^3)*cosh(x)^3 - 3*(a^6 + a^4*b^2)*cosh(x)^2 + 2*(a^5*b + a^3*b^3)*cosh(x))*sinh(x))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\coth ^{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(coth(x)**2/(a+b*sinh(x))**2,x)

[Out]

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

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Giac [A]
time = 0.43, size = 148, normalized size = 1.85 \begin {gather*} \frac {2 \, b \log \left (e^{x} + 1\right )}{a^{3}} - \frac {2 \, b \log \left ({\left | e^{x} - 1 \right |}\right )}{a^{3}} + \frac {{\left (a^{2} + 2 \, b^{2}\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 )}{\sqrt {a^{2} + b^{2}} a^{3}} + \frac {2 \, {\left (a e^{\left (3 \, x\right )} - 2 \, b e^{\left (2 \, x\right )} - 3 \, a e^{x} + 2 \, b\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 )} a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 1.78, size = 897, normalized size = 11.21 \begin {gather*} \frac {\frac {4\,\left (25\,a^8\,b^8+65\,a^6\,b^{10}+56\,a^4\,b^{12}+16\,a^2\,b^{14}\right )}{a^4\,b^4\,\left (25\,a^6\,b^3+65\,a^4\,b^5+56\,a^2\,b^7+16\,b^9\right )}-\frac {6\,{\mathrm {e}}^x\,\left (25\,a^9\,b^8+65\,a^7\,b^{10}+56\,a^5\,b^{12}+16\,a^3\,b^{14}\right )}{a^4\,b^5\,\left (25\,a^6\,b^3+65\,a^4\,b^5+56\,a^2\,b^7+16\,b^9\right )}-\frac {4\,{\mathrm {e}}^{2\,x}\,\left (25\,a^8\,b^8+65\,a^6\,b^{10}+56\,a^4\,b^{12}+16\,a^2\,b^{14}\right )}{a^4\,b^4\,\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^9\,b^8+65\,a^7\,b^{10}+56\,a^5\,b^{12}+16\,a^3\,b^{14}\right )}{a^4\,b^5\,\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 (64\,{\mathrm {e}}^x-64\right )}{a^3}+\frac {2\,b\,\ln \left (64\,{\mathrm {e}}^x+64\right )}{a^3}-\frac {\ln \left (\frac {\left (a^2+2\,b^2\right )\,\left (\frac {32\,\left (a^4-16\,{\mathrm {e}}^x\,a^3\,b+12\,a^2\,b^2-12\,{\mathrm {e}}^x\,a\,b^3+8\,b^4\right )}{a^4\,b^4}+\frac {\left (a^2+2\,b^2\right )\,\left (\frac {32\,\left (-4\,{\mathrm {e}}^x\,a^3+2\,a^2\,b-7\,{\mathrm {e}}^x\,a\,b^2+4\,b^3\right )}{b^5}-\frac {32\,\left (a^2+2\,b^2\right )\,\sqrt {a^2+b^2}\,\left (-4\,{\mathrm {e}}^x\,a^5+3\,a^4\,b-3\,{\mathrm {e}}^x\,a^3\,b^2+2\,a^2\,b^3\right )}{b^5\,\left (a^5+a^3\,b^2\right )}\right )\,\sqrt {a^2+b^2}}{a^5+a^3\,b^2}\right )\,\sqrt {a^2+b^2}}{a^5+a^3\,b^2}-\frac {64\,\left (a^2+2\,b^2\right )\,\left (4\,b-7\,a\,{\mathrm {e}}^x\right )}{a^6\,b^3}\right )\,\left (a^2+2\,b^2\right )\,\sqrt {a^2+b^2}}{a^5+a^3\,b^2}+\frac {\ln \left (-\frac {\left (a^2+2\,b^2\right )\,\left (\frac {32\,\left (a^4-16\,{\mathrm {e}}^x\,a^3\,b+12\,a^2\,b^2-12\,{\mathrm {e}}^x\,a\,b^3+8\,b^4\right )}{a^4\,b^4}-\frac {\left (a^2+2\,b^2\right )\,\left (\frac {32\,\left (-4\,{\mathrm {e}}^x\,a^3+2\,a^2\,b-7\,{\mathrm {e}}^x\,a\,b^2+4\,b^3\right )}{b^5}+\frac {32\,\left (a^2+2\,b^2\right )\,\sqrt {a^2+b^2}\,\left (-4\,{\mathrm {e}}^x\,a^5+3\,a^4\,b-3\,{\mathrm {e}}^x\,a^3\,b^2+2\,a^2\,b^3\right )}{b^5\,\left (a^5+a^3\,b^2\right )}\right )\,\sqrt {a^2+b^2}}{a^5+a^3\,b^2}\right )\,\sqrt {a^2+b^2}}{a^5+a^3\,b^2}-\frac {64\,\left (a^2+2\,b^2\right )\,\left (4\,b-7\,a\,{\mathrm {e}}^x\right )}{a^6\,b^3}\right )\,\left (a^2+2\,b^2\right )\,\sqrt {a^2+b^2}}{a^5+a^3\,b^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(x)^2/(a + b*sinh(x))^2,x)

[Out]

((4*(16*a^2*b^14 + 56*a^4*b^12 + 65*a^6*b^10 + 25*a^8*b^8))/(a^4*b^4*(16*b^9 + 56*a^2*b^7 + 65*a^4*b^5 + 25*a^
6*b^3)) - (6*exp(x)*(16*a^3*b^14 + 56*a^5*b^12 + 65*a^7*b^10 + 25*a^9*b^8))/(a^4*b^5*(16*b^9 + 56*a^2*b^7 + 65
*a^4*b^5 + 25*a^6*b^3)) - (4*exp(2*x)*(16*a^2*b^14 + 56*a^4*b^12 + 65*a^6*b^10 + 25*a^8*b^8))/(a^4*b^4*(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^14 + 56*a^5*b^12 + 65*a^7*b^10 + 25*a^9*b^8)
)/(a^4*b^5*(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(64*exp(x) - 64))/a^3 + (2*b*log(64*exp(x) + 64))/a^3 - (log(((a^2 + 2*b^2)*((32*(a^4 + 8
*b^4 + 12*a^2*b^2 - 12*a*b^3*exp(x) - 16*a^3*b*exp(x)))/(a^4*b^4) + ((a^2 + 2*b^2)*((32*(2*a^2*b + 4*b^3 - 4*a
^3*exp(x) - 7*a*b^2*exp(x)))/b^5 - (32*(a^2 + 2*b^2)*(a^2 + b^2)^(1/2)*(3*a^4*b + 2*a^2*b^3 - 4*a^5*exp(x) - 3
*a^3*b^2*exp(x)))/(b^5*(a^5 + a^3*b^2)))*(a^2 + b^2)^(1/2))/(a^5 + a^3*b^2))*(a^2 + b^2)^(1/2))/(a^5 + a^3*b^2
) - (64*(a^2 + 2*b^2)*(4*b - 7*a*exp(x)))/(a^6*b^3))*(a^2 + 2*b^2)*(a^2 + b^2)^(1/2))/(a^5 + a^3*b^2) + (log(-
((a^2 + 2*b^2)*((32*(a^4 + 8*b^4 + 12*a^2*b^2 - 12*a*b^3*exp(x) - 16*a^3*b*exp(x)))/(a^4*b^4) - ((a^2 + 2*b^2
)*((32*(2*a^2*b + 4*b^3 - 4*a^3*exp(x) - 7*a*b^2*exp(x)))/b^5 + (32*(a^2 + 2*b^2)*(a^2 + b^2)^(1/2)*(3*a^4*b +
2*a^2*b^3 - 4*a^5*exp(x) - 3*a^3*b^2*exp(x)))/(b^5*(a^5 + a^3*b^2)))*(a^2 + b^2)^(1/2))/(a^5 + a^3*b^2))*(a^2
+ b^2)^(1/2))/(a^5 + a^3*b^2) - (64*(a^2 + 2*b^2)*(4*b - 7*a*exp(x)))/(a^6*b^3))*(a^2 + 2*b^2)*(a^2 + b^2)^(1
/2))/(a^5 + a^3*b^2)

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