3.1.57 \(\int \frac {\text {csch}^2(x)}{a+a \text {sech}(x)} \, dx\) [57]

Optimal. Leaf size=23 \[ -\frac {\coth ^3(x)}{3 a}+\frac {\text {csch}^3(x)}{3 a} \]

[Out]

-1/3*coth(x)^3/a+1/3*csch(x)^3/a

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Rubi [A]
time = 0.09, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {3957, 2918, 2686, 30, 2687} \begin {gather*} \frac {\text {csch}^3(x)}{3 a}-\frac {\coth ^3(x)}{3 a} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Csch[x]^2/(a + a*Sech[x]),x]

[Out]

-1/3*Coth[x]^3/a + Csch[x]^3/(3*a)

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2686

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a/f, Subst[
Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n -
1)/2] &&  !(IntegerQ[m/2] && LtQ[0, m, n + 1])

Rule 2687

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[1/f, Subst[Int[(b*x)
^n*(1 + x^2)^(m/2 - 1), x], x, Tan[e + f*x]], x] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2] &&  !(IntegerQ[(n
- 1)/2] && LtQ[0, n, m - 1])

Rule 2918

Int[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.))/((a_) + (b_.)*sin[(e_.) + (f_
.)*(x_)]), x_Symbol] :> Dist[g^2/a, Int[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^n, x], x] - Dist[g^2/(b*d),
Int[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2
- b^2, 0]

Rule 3957

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[(g*Co
s[e + f*x])^p*((b + a*Sin[e + f*x])^m/Sin[e + f*x]^m), x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]

Rubi steps

\begin {align*} \int \frac {\text {csch}^2(x)}{a+a \text {sech}(x)} \, dx &=-\int \frac {\coth (x) \text {csch}(x)}{-a-a \cosh (x)} \, dx\\ &=\frac {\int \coth ^2(x) \text {csch}^2(x) \, dx}{a}-\frac {\int \coth (x) \text {csch}^3(x) \, dx}{a}\\ &=-\frac {i \text {Subst}\left (\int x^2 \, dx,x,i \coth (x)\right )}{a}-\frac {i \text {Subst}\left (\int x^2 \, dx,x,-i \text {csch}(x)\right )}{a}\\ &=-\frac {\coth ^3(x)}{3 a}+\frac {\text {csch}^3(x)}{3 a}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 25, normalized size = 1.09 \begin {gather*} -\frac {(3+2 \cosh (x)+\cosh (2 x)) \text {csch}(x)}{6 a (1+\cosh (x))} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/6*((3 + 2*Cosh[x] + Cosh[2*x])*Csch[x])/(a*(1 + Cosh[x]))

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Maple [A]
time = 0.77, size = 23, normalized size = 1.00

method	result	size
default	\(\frac {-\frac {\left (\tanh ^{3}\left (\frac {x}{2}\right )\right )}{3}-\frac {1}{\tanh \left (\frac {x}{2}\right )}}{4 a}\)	\(23\)
risch	\(-\frac {2 \left (3 \,{\mathrm e}^{2 x}+2 \,{\mathrm e}^{x}+1\right )}{3 \left ({\mathrm e}^{x}+1\right )^{3} a \left ({\mathrm e}^{x}-1\right )}\)	\(30\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4/a*(-1/3*tanh(1/2*x)^3-1/tanh(1/2*x))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 90 vs. \(2 (19) = 38\).
time = 0.30, size = 90, normalized size = 3.91 \begin {gather*} -\frac {4 \, e^{\left (-x\right )}}{3 \, {\left (2 \, a e^{\left (-x\right )} - 2 \, a e^{\left (-3 \, x\right )} - a e^{\left (-4 \, x\right )} + a\right )}} - \frac {2 \, e^{\left (-2 \, x\right )}}{2 \, a e^{\left (-x\right )} - 2 \, a e^{\left (-3 \, x\right )} - a e^{\left (-4 \, x\right )} + a} - \frac {2}{3 \, {\left (2 \, a e^{\left (-x\right )} - 2 \, a e^{\left (-3 \, x\right )} - a e^{\left (-4 \, x\right )} + a\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/3*e^(-x)/(2*a*e^(-x) - 2*a*e^(-3*x) - a*e^(-4*x) + a) - 2*e^(-2*x)/(2*a*e^(-x) - 2*a*e^(-3*x) - a*e^(-4*x)
+ a) - 2/3/(2*a*e^(-x) - 2*a*e^(-3*x) - a*e^(-4*x) + a)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (19) = 38\).
time = 0.35, size = 71, normalized size = 3.09 \begin {gather*} -\frac {4 \, {\left (2 \, \cosh \left (x\right ) + \sinh \left (x\right ) + 1\right )}}{3 \, {\left (a \cosh \left (x\right )^{3} + a \sinh \left (x\right )^{3} + 2 \, a \cosh \left (x\right )^{2} + {\left (3 \, a \cosh \left (x\right ) + 2 \, a\right )} \sinh \left (x\right )^{2} - a \cosh \left (x\right ) + {\left (3 \, a \cosh \left (x\right )^{2} + 4 \, a \cosh \left (x\right ) + a\right )} \sinh \left (x\right ) - 2 \, a\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/3*(2*cosh(x) + sinh(x) + 1)/(a*cosh(x)^3 + a*sinh(x)^3 + 2*a*cosh(x)^2 + (3*a*cosh(x) + 2*a)*sinh(x)^2 - a*
cosh(x) + (3*a*cosh(x)^2 + 4*a*cosh(x) + a)*sinh(x) - 2*a)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x)**2/(a+a*sech(x)),x)

[Out]

Integral(csch(x)**2/(sech(x) + 1), x)/a

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Giac [A]
time = 0.39, size = 31, normalized size = 1.35 \begin {gather*} -\frac {1}{2 \, a {\left (e^{x} - 1\right )}} + \frac {3 \, e^{\left (2 \, x\right )} + 1}{6 \, a {\left (e^{x} + 1\right )}^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/(a*(e^x - 1)) + 1/6*(3*e^(2*x) + 1)/(a*(e^x + 1)^3)

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Mupad [B]
time = 1.35, size = 91, normalized size = 3.96 \begin {gather*} \frac {\frac {{\mathrm {e}}^{2\,x}}{6\,a}+\frac {1}{6\,a}-\frac {{\mathrm {e}}^x}{3\,a}}{3\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{3\,x}+3\,{\mathrm {e}}^x+1}-\frac {\frac {1}{6\,a}-\frac {{\mathrm {e}}^x}{6\,a}}{{\mathrm {e}}^{2\,x}+2\,{\mathrm {e}}^x+1}-\frac {1}{2\,a\,\left ({\mathrm {e}}^x-1\right )}+\frac {1}{6\,a\,\left ({\mathrm {e}}^x+1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sinh(x)^2*(a + a/cosh(x))),x)

[Out]

(exp(2*x)/(6*a) + 1/(6*a) - exp(x)/(3*a))/(3*exp(2*x) + exp(3*x) + 3*exp(x) + 1) - (1/(6*a) - exp(x)/(6*a))/(e
xp(2*x) + 2*exp(x) + 1) - 1/(2*a*(exp(x) - 1)) + 1/(6*a*(exp(x) + 1))

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