3.7.49 \(\int \frac {1}{a \coth (x)+b \text {csch}(x)} \, dx\) [649]

Optimal. Leaf size=11 \[ \frac {\log (b+a \cosh (x))}{a} \]

[Out]

ln(b+a*cosh(x))/a

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Rubi [A]
time = 0.03, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {3239, 2747, 31} \begin {gather*} \frac {\log (a \cosh (x)+b)}{a} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a*Coth[x] + b*Csch[x])^(-1),x]

[Out]

Log[b + a*Cosh[x]]/a

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 2747

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^m*(b^2 - x^2)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && Integer
Q[(p - 1)/2] && NeQ[a^2 - b^2, 0]

Rule 3239

Int[((a_.) + csc[(d_.) + (e_.)*(x_)]*(b_.) + cot[(d_.) + (e_.)*(x_)]*(c_.))^(-1), x_Symbol] :> Int[Sin[d + e*x
]/(b + a*Sin[d + e*x] + c*Cos[d + e*x]), x] /; FreeQ[{a, b, c, d, e}, x]

Rubi steps

\begin {align*} \int \frac {1}{a \coth (x)+b \text {csch}(x)} \, dx &=i \int \frac {\sinh (x)}{i b+i a \cosh (x)} \, dx\\ &=\frac {\text {Subst}\left (\int \frac {1}{i b+x} \, dx,x,i a \cosh (x)\right )}{a}\\ &=\frac {\log (b+a \cosh (x))}{a}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 11, normalized size = 1.00 \begin {gather*} \frac {\log (b+a \cosh (x))}{a} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a*Coth[x] + b*Csch[x])^(-1),x]

[Out]

Log[b + a*Cosh[x]]/a

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(50\) vs. \(2(11)=22\).
time = 1.13, size = 51, normalized size = 4.64

method	result	size
risch	\(-\frac {x}{a}+\frac {\ln \left ({\mathrm e}^{2 x}+\frac {2 b \,{\mathrm e}^{x}}{a}+1\right )}{a}\)	\(27\)
default	\(-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{a}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{a}+\frac {\ln \left (a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-b \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )+a +b \right )}{a}\)	\(51\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*coth(x)+b*csch(x)),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 26 vs. \(2 (11) = 22\).
time = 0.26, size = 26, normalized size = 2.36 \begin {gather*} \frac {x}{a} + \frac {\log \left (2 \, b e^{\left (-x\right )} + a e^{\left (-2 \, x\right )} + a\right )}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 27 vs. \(2 (11) = 22\).
time = 0.37, size = 27, normalized size = 2.45 \begin {gather*} -\frac {x - \log \left (\frac {2 \, {\left (a \cosh \left (x\right ) + b\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right )}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(x - log(2*(a*cosh(x) + b)/(cosh(x) - sinh(x))))/a

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.43, size = 19, normalized size = 1.73 \begin {gather*} \frac {\log \left ({\left | a {\left (e^{\left (-x\right )} + e^{x}\right )} + 2 \, b \right |}\right )}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(abs(a*(e^(-x) + e^x) + 2*b))/a

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Mupad [B]
time = 0.09, size = 23, normalized size = 2.09 \begin {gather*} -\frac {x-\ln \left (a+2\,b\,{\mathrm {e}}^x+a\,{\mathrm {e}}^{2\,x}\right )}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b/sinh(x) + a*coth(x)),x)

[Out]

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

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