3.650 \(\int \frac {1}{(a \coth (x)+b \text {csch}(x))^2} \, dx\)

Optimal. Leaf size=67 \[ -\frac {2 b \tan ^{-1}\left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{a^2 \sqrt {a-b} \sqrt {a+b}}+\frac {x}{a^2}-\frac {\sinh (x)}{a (a \cosh (x)+b)} \]

[Out]

x/a^2-sinh(x)/a/(b+a*cosh(x))-2*b*arctan((a-b)^(1/2)*tanh(1/2*x)/(a+b)^(1/2))/a^2/(a-b)^(1/2)/(a+b)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.13, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.454, Rules used = {4392, 2693, 2735, 2659, 205} \[ -\frac {2 b \tan ^{-1}\left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{a^2 \sqrt {a-b} \sqrt {a+b}}+\frac {x}{a^2}-\frac {\sinh (x)}{a (a \cosh (x)+b)} \]

Antiderivative was successfully verified.

[In]

Int[(a*Coth[x] + b*Csch[x])^(-2),x]

[Out]

x/a^2 - (2*b*ArcTan[(Sqrt[a - b]*Tanh[x/2])/Sqrt[a + b]])/(a^2*Sqrt[a - b]*Sqrt[a + b]) - Sinh[x]/(a*(b + a*Co
sh[x]))

Rule 205

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

Rule 2659

Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x
]}, Dist[(2*e)/d, Subst[Int[1/(a + b + (a - b)*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 2693

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(g*(g*
Cos[e + f*x])^(p - 1)*(a + b*Sin[e + f*x])^(m + 1))/(b*f*(m + 1)), x] + Dist[(g^2*(p - 1))/(b*(m + 1)), Int[(g
*Cos[e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^(m + 1)*Sin[e + f*x], x], x] /; FreeQ[{a, b, e, f, g}, x] && NeQ[a
^2 - b^2, 0] && LtQ[m, -1] && GtQ[p, 1] && IntegersQ[2*m, 2*p]

Rule 2735

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

Rule 4392

Int[(cot[(c_.) + (d_.)*(x_)]^(n_.)*(a_.) + csc[(c_.) + (d_.)*(x_)]^(n_.)*(b_.))^(p_)*(u_.), x_Symbol] :> Int[A
ctivateTrig[u]*Csc[c + d*x]^(n*p)*(b + a*Cos[c + d*x]^n)^p, x] /; FreeQ[{a, b, c, d}, x] && IntegersQ[n, p]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.35, size = 61, normalized size = 0.91 \[ \frac {\frac {2 b \tan ^{-1}\left (\frac {(b-a) \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}-\frac {a \sinh (x)}{a \cosh (x)+b}+x}{a^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [B]  time = 0.45, size = 682, normalized size = 10.18 \[ \left [\frac {{\left (a^{3} - a b^{2}\right )} x \cosh \relax (x)^{2} + {\left (a^{3} - a b^{2}\right )} x \sinh \relax (x)^{2} + 2 \, a^{3} - 2 \, a b^{2} - {\left (a b \cosh \relax (x)^{2} + a b \sinh \relax (x)^{2} + 2 \, b^{2} \cosh \relax (x) + a b + 2 \, {\left (a b \cosh \relax (x) + b^{2}\right )} \sinh \relax (x)\right )} \sqrt {-a^{2} + b^{2}} \log \left (\frac {a^{2} \cosh \relax (x)^{2} + a^{2} \sinh \relax (x)^{2} + 2 \, a b \cosh \relax (x) - a^{2} + 2 \, b^{2} + 2 \, {\left (a^{2} \cosh \relax (x) + a b\right )} \sinh \relax (x) + 2 \, \sqrt {-a^{2} + b^{2}} {\left (a \cosh \relax (x) + a \sinh \relax (x) + b\right )}}{a \cosh \relax (x)^{2} + a \sinh \relax (x)^{2} + 2 \, b \cosh \relax (x) + 2 \, {\left (a \cosh \relax (x) + b\right )} \sinh \relax (x) + a}\right ) + {\left (a^{3} - a b^{2}\right )} x + 2 \, {\left (a^{2} b - b^{3} + {\left (a^{2} b - b^{3}\right )} x\right )} \cosh \relax (x) + 2 \, {\left (a^{2} b - b^{3} + {\left (a^{3} - a b^{2}\right )} x \cosh \relax (x) + {\left (a^{2} b - b^{3}\right )} x\right )} \sinh \relax (x)}{a^{5} - a^{3} b^{2} + {\left (a^{5} - a^{3} b^{2}\right )} \cosh \relax (x)^{2} + {\left (a^{5} - a^{3} b^{2}\right )} \sinh \relax (x)^{2} + 2 \, {\left (a^{4} b - a^{2} b^{3}\right )} \cosh \relax (x) + 2 \, {\left (a^{4} b - a^{2} b^{3} + {\left (a^{5} - a^{3} b^{2}\right )} \cosh \relax (x)\right )} \sinh \relax (x)}, \frac {{\left (a^{3} - a b^{2}\right )} x \cosh \relax (x)^{2} + {\left (a^{3} - a b^{2}\right )} x \sinh \relax (x)^{2} + 2 \, a^{3} - 2 \, a b^{2} + 2 \, {\left (a b \cosh \relax (x)^{2} + a b \sinh \relax (x)^{2} + 2 \, b^{2} \cosh \relax (x) + a b + 2 \, {\left (a b \cosh \relax (x) + b^{2}\right )} \sinh \relax (x)\right )} \sqrt {a^{2} - b^{2}} \arctan \left (-\frac {a \cosh \relax (x) + a \sinh \relax (x) + b}{\sqrt {a^{2} - b^{2}}}\right ) + {\left (a^{3} - a b^{2}\right )} x + 2 \, {\left (a^{2} b - b^{3} + {\left (a^{2} b - b^{3}\right )} x\right )} \cosh \relax (x) + 2 \, {\left (a^{2} b - b^{3} + {\left (a^{3} - a b^{2}\right )} x \cosh \relax (x) + {\left (a^{2} b - b^{3}\right )} x\right )} \sinh \relax (x)}{a^{5} - a^{3} b^{2} + {\left (a^{5} - a^{3} b^{2}\right )} \cosh \relax (x)^{2} + {\left (a^{5} - a^{3} b^{2}\right )} \sinh \relax (x)^{2} + 2 \, {\left (a^{4} b - a^{2} b^{3}\right )} \cosh \relax (x) + 2 \, {\left (a^{4} b - a^{2} b^{3} + {\left (a^{5} - a^{3} b^{2}\right )} \cosh \relax (x)\right )} \sinh \relax (x)}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[((a^3 - a*b^2)*x*cosh(x)^2 + (a^3 - a*b^2)*x*sinh(x)^2 + 2*a^3 - 2*a*b^2 - (a*b*cosh(x)^2 + a*b*sinh(x)^2 + 2
*b^2*cosh(x) + a*b + 2*(a*b*cosh(x) + b^2)*sinh(x))*sqrt(-a^2 + b^2)*log((a^2*cosh(x)^2 + a^2*sinh(x)^2 + 2*a*
b*cosh(x) - a^2 + 2*b^2 + 2*(a^2*cosh(x) + a*b)*sinh(x) + 2*sqrt(-a^2 + b^2)*(a*cosh(x) + a*sinh(x) + b))/(a*c
osh(x)^2 + a*sinh(x)^2 + 2*b*cosh(x) + 2*(a*cosh(x) + b)*sinh(x) + a)) + (a^3 - a*b^2)*x + 2*(a^2*b - b^3 + (a
^2*b - b^3)*x)*cosh(x) + 2*(a^2*b - b^3 + (a^3 - a*b^2)*x*cosh(x) + (a^2*b - b^3)*x)*sinh(x))/(a^5 - a^3*b^2 +
(a^5 - a^3*b^2)*cosh(x)^2 + (a^5 - a^3*b^2)*sinh(x)^2 + 2*(a^4*b - a^2*b^3)*cosh(x) + 2*(a^4*b - a^2*b^3 + (a
^5 - a^3*b^2)*cosh(x))*sinh(x)), ((a^3 - a*b^2)*x*cosh(x)^2 + (a^3 - a*b^2)*x*sinh(x)^2 + 2*a^3 - 2*a*b^2 + 2*
(a*b*cosh(x)^2 + a*b*sinh(x)^2 + 2*b^2*cosh(x) + a*b + 2*(a*b*cosh(x) + b^2)*sinh(x))*sqrt(a^2 - b^2)*arctan(-
(a*cosh(x) + a*sinh(x) + b)/sqrt(a^2 - b^2)) + (a^3 - a*b^2)*x + 2*(a^2*b - b^3 + (a^2*b - b^3)*x)*cosh(x) + 2
*(a^2*b - b^3 + (a^3 - a*b^2)*x*cosh(x) + (a^2*b - b^3)*x)*sinh(x))/(a^5 - a^3*b^2 + (a^5 - a^3*b^2)*cosh(x)^2
+ (a^5 - a^3*b^2)*sinh(x)^2 + 2*(a^4*b - a^2*b^3)*cosh(x) + 2*(a^4*b - a^2*b^3 + (a^5 - a^3*b^2)*cosh(x))*sin
h(x))]

________________________________________________________________________________________

giac [A]  time = 0.12, size = 68, normalized size = 1.01 \[ -\frac {2 \, b \arctan \left (\frac {a e^{x} + b}{\sqrt {a^{2} - b^{2}}}\right )}{\sqrt {a^{2} - b^{2}} a^{2}} + \frac {x}{a^{2}} + \frac {2 \, {\left (b e^{x} + a\right )}}{{\left (a e^{\left (2 \, x\right )} + 2 \, b e^{x} + a\right )} a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*b*arctan((a*e^x + b)/sqrt(a^2 - b^2))/(sqrt(a^2 - b^2)*a^2) + x/a^2 + 2*(b*e^x + a)/((a*e^(2*x) + 2*b*e^x +
a)*a^2)

________________________________________________________________________________________

maple [A]  time = 0.22, size = 95, normalized size = 1.42 \[ -\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{a^{2}}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{a^{2}}-\frac {2 \tanh \left (\frac {x}{2}\right )}{a \left (a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-\left (\tanh ^{2}\left (\frac {x}{2}\right )\right ) b +a +b \right )}-\frac {2 b \arctan \left (\frac {\left (a -b \right ) \tanh \left (\frac {x}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{a^{2} \sqrt {\left (a +b \right ) \left (a -b \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*coth(x)+b*csch(x))^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(4*b^2-4*a^2>0)', see `assume?`
for more details)Is 4*b^2-4*a^2 positive or negative?

________________________________________________________________________________________

mupad [B]  time = 1.79, size = 139, normalized size = 2.07 \[ \frac {x}{a^2}+\frac {\frac {2}{a}+\frac {2\,b\,{\mathrm {e}}^x}{a^2}}{a+2\,b\,{\mathrm {e}}^x+a\,{\mathrm {e}}^{2\,x}}+\frac {b\,\ln \left (\frac {2\,b\,{\mathrm {e}}^x}{a^3}-\frac {2\,b\,\left (a+b\,{\mathrm {e}}^x\right )}{a^3\,\sqrt {a+b}\,\sqrt {b-a}}\right )}{a^2\,\sqrt {a+b}\,\sqrt {b-a}}-\frac {b\,\ln \left (\frac {2\,b\,{\mathrm {e}}^x}{a^3}+\frac {2\,b\,\left (a+b\,{\mathrm {e}}^x\right )}{a^3\,\sqrt {a+b}\,\sqrt {b-a}}\right )}{a^2\,\sqrt {a+b}\,\sqrt {b-a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b/sinh(x) + a*coth(x))^2,x)

[Out]

x/a^2 + (2/a + (2*b*exp(x))/a^2)/(a + 2*b*exp(x) + a*exp(2*x)) + (b*log((2*b*exp(x))/a^3 - (2*b*(a + b*exp(x))
)/(a^3*(a + b)^(1/2)*(b - a)^(1/2))))/(a^2*(a + b)^(1/2)*(b - a)^(1/2)) - (b*log((2*b*exp(x))/a^3 + (2*b*(a +
b*exp(x)))/(a^3*(a + b)^(1/2)*(b - a)^(1/2))))/(a^2*(a + b)^(1/2)*(b - a)^(1/2))

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a \coth {\relax (x )} + b \operatorname {csch}{\relax (x )}\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________