3.1.59 \(\int \frac {\text {csch}^4(x)}{a+a \text {sech}(x)} \, dx\) [59]

Optimal. Leaf size=34 \[ \frac {\coth ^3(x)}{3 a}-\frac {\coth ^5(x)}{5 a}+\frac {\text {csch}^5(x)}{5 a} \]

[Out]

1/3*coth(x)^3/a-1/5*coth(x)^5/a+1/5*csch(x)^5/a

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Rubi [A]
time = 0.10, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {3957, 2918, 2686, 30, 2687, 14} \begin {gather*} -\frac {\coth ^5(x)}{5 a}+\frac {\coth ^3(x)}{3 a}+\frac {\text {csch}^5(x)}{5 a} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Csch[x]^4/(a + a*Sech[x]),x]

[Out]

Coth[x]^3/(3*a) - Coth[x]^5/(5*a) + Csch[x]^5/(5*a)

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

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}^4(x)}{a+a \text {sech}(x)} \, dx &=-\int \frac {\coth (x) \text {csch}^3(x)}{-a-a \cosh (x)} \, dx\\ &=\frac {\int \coth ^2(x) \text {csch}^4(x) \, dx}{a}-\frac {\int \coth (x) \text {csch}^5(x) \, dx}{a}\\ &=\frac {i \text {Subst}\left (\int x^4 \, dx,x,-i \text {csch}(x)\right )}{a}+\frac {i \text {Subst}\left (\int x^2 \left (1+x^2\right ) \, dx,x,i \coth (x)\right )}{a}\\ &=\frac {\text {csch}^5(x)}{5 a}+\frac {i \text {Subst}\left (\int \left (x^2+x^4\right ) \, dx,x,i \coth (x)\right )}{a}\\ &=\frac {\coth ^3(x)}{3 a}-\frac {\coth ^5(x)}{5 a}+\frac {\text {csch}^5(x)}{5 a}\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 39, normalized size = 1.15 \begin {gather*} \frac {(-15-6 \cosh (x)-2 \cosh (2 x)+2 \cosh (3 x)+\cosh (4 x)) \text {csch}^3(x)}{60 a (1+\cosh (x))} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Csch[x]^4/(a + a*Sech[x]),x]

[Out]

((-15 - 6*Cosh[x] - 2*Cosh[2*x] + 2*Cosh[3*x] + Cosh[4*x])*Csch[x]^3)/(60*a*(1 + Cosh[x]))

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Maple [A]
time = 0.81, size = 39, normalized size = 1.15

method	result	size
default	\(\frac {-\frac {\left (\tanh ^{5}\left (\frac {x}{2}\right )\right )}{5}+\frac {2 \left (\tanh ^{3}\left (\frac {x}{2}\right )\right )}{3}-\frac {1}{3 \tanh \left (\frac {x}{2}\right )^{3}}+\frac {2}{\tanh \left (\frac {x}{2}\right )}}{16 a}\)	\(39\)
risch	\(-\frac {4 \left (15 \,{\mathrm e}^{4 x}+6 \,{\mathrm e}^{3 x}+2 \,{\mathrm e}^{2 x}-2 \,{\mathrm e}^{x}-1\right )}{15 \left ({\mathrm e}^{x}-1\right )^{3} a \left ({\mathrm e}^{x}+1\right )^{5}}\)	\(42\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(x)^4/(a+a*sech(x)),x,method=_RETURNVERBOSE)

[Out]

1/16/a*(-1/5*tanh(1/2*x)^5+2/3*tanh(1/2*x)^3-1/3/tanh(1/2*x)^3+2/tanh(1/2*x))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 292 vs. \(2 (28) = 56\).
time = 0.26, size = 292, normalized size = 8.59 \begin {gather*} \frac {8 \, e^{\left (-x\right )}}{15 \, {\left (2 \, a e^{\left (-x\right )} - 2 \, a e^{\left (-2 \, x\right )} - 6 \, a e^{\left (-3 \, x\right )} + 6 \, a e^{\left (-5 \, x\right )} + 2 \, a e^{\left (-6 \, x\right )} - 2 \, a e^{\left (-7 \, x\right )} - a e^{\left (-8 \, x\right )} + a\right )}} - \frac {8 \, e^{\left (-2 \, x\right )}}{15 \, {\left (2 \, a e^{\left (-x\right )} - 2 \, a e^{\left (-2 \, x\right )} - 6 \, a e^{\left (-3 \, x\right )} + 6 \, a e^{\left (-5 \, x\right )} + 2 \, a e^{\left (-6 \, x\right )} - 2 \, a e^{\left (-7 \, x\right )} - a e^{\left (-8 \, x\right )} + a\right )}} - \frac {8 \, e^{\left (-3 \, x\right )}}{5 \, {\left (2 \, a e^{\left (-x\right )} - 2 \, a e^{\left (-2 \, x\right )} - 6 \, a e^{\left (-3 \, x\right )} + 6 \, a e^{\left (-5 \, x\right )} + 2 \, a e^{\left (-6 \, x\right )} - 2 \, a e^{\left (-7 \, x\right )} - a e^{\left (-8 \, x\right )} + a\right )}} - \frac {4 \, e^{\left (-4 \, x\right )}}{2 \, a e^{\left (-x\right )} - 2 \, a e^{\left (-2 \, x\right )} - 6 \, a e^{\left (-3 \, x\right )} + 6 \, a e^{\left (-5 \, x\right )} + 2 \, a e^{\left (-6 \, x\right )} - 2 \, a e^{\left (-7 \, x\right )} - a e^{\left (-8 \, x\right )} + a} + \frac {4}{15 \, {\left (2 \, a e^{\left (-x\right )} - 2 \, a e^{\left (-2 \, x\right )} - 6 \, a e^{\left (-3 \, x\right )} + 6 \, a e^{\left (-5 \, x\right )} + 2 \, a e^{\left (-6 \, x\right )} - 2 \, a e^{\left (-7 \, x\right )} - a e^{\left (-8 \, x\right )} + a\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x)^4/(a+a*sech(x)),x, algorithm="maxima")

[Out]

8/15*e^(-x)/(2*a*e^(-x) - 2*a*e^(-2*x) - 6*a*e^(-3*x) + 6*a*e^(-5*x) + 2*a*e^(-6*x) - 2*a*e^(-7*x) - a*e^(-8*x
) + a) - 8/15*e^(-2*x)/(2*a*e^(-x) - 2*a*e^(-2*x) - 6*a*e^(-3*x) + 6*a*e^(-5*x) + 2*a*e^(-6*x) - 2*a*e^(-7*x)
- a*e^(-8*x) + a) - 8/5*e^(-3*x)/(2*a*e^(-x) - 2*a*e^(-2*x) - 6*a*e^(-3*x) + 6*a*e^(-5*x) + 2*a*e^(-6*x) - 2*a
*e^(-7*x) - a*e^(-8*x) + a) - 4*e^(-4*x)/(2*a*e^(-x) - 2*a*e^(-2*x) - 6*a*e^(-3*x) + 6*a*e^(-5*x) + 2*a*e^(-6*
x) - 2*a*e^(-7*x) - a*e^(-8*x) + a) + 4/15/(2*a*e^(-x) - 2*a*e^(-2*x) - 6*a*e^(-3*x) + 6*a*e^(-5*x) + 2*a*e^(-
6*x) - 2*a*e^(-7*x) - a*e^(-8*x) + a)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 219 vs. \(2 (28) = 56\).
time = 0.35, size = 219, normalized size = 6.44 \begin {gather*} -\frac {8 \, {\left (7 \, \cosh \left (x\right )^{2} + 4 \, {\left (4 \, \cosh \left (x\right ) + 1\right )} \sinh \left (x\right ) + 7 \, \sinh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) + 1\right )}}{15 \, {\left (a \cosh \left (x\right )^{6} + a \sinh \left (x\right )^{6} + 2 \, a \cosh \left (x\right )^{5} + 2 \, {\left (3 \, a \cosh \left (x\right ) + a\right )} \sinh \left (x\right )^{5} - 2 \, a \cosh \left (x\right )^{4} + {\left (15 \, a \cosh \left (x\right )^{2} + 10 \, a \cosh \left (x\right ) - 2 \, a\right )} \sinh \left (x\right )^{4} - 6 \, a \cosh \left (x\right )^{3} + 2 \, {\left (10 \, a \cosh \left (x\right )^{3} + 10 \, a \cosh \left (x\right )^{2} - 4 \, a \cosh \left (x\right ) - 3 \, a\right )} \sinh \left (x\right )^{3} - a \cosh \left (x\right )^{2} + {\left (15 \, a \cosh \left (x\right )^{4} + 20 \, a \cosh \left (x\right )^{3} - 12 \, a \cosh \left (x\right )^{2} - 18 \, a \cosh \left (x\right ) - a\right )} \sinh \left (x\right )^{2} + 4 \, a \cosh \left (x\right ) + 2 \, {\left (3 \, a \cosh \left (x\right )^{5} + 5 \, a \cosh \left (x\right )^{4} - 4 \, a \cosh \left (x\right )^{3} - 9 \, a \cosh \left (x\right )^{2} + a \cosh \left (x\right ) + 4 \, 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)^4/(a+a*sech(x)),x, algorithm="fricas")

[Out]

-8/15*(7*cosh(x)^2 + 4*(4*cosh(x) + 1)*sinh(x) + 7*sinh(x)^2 + 2*cosh(x) + 1)/(a*cosh(x)^6 + a*sinh(x)^6 + 2*a
*cosh(x)^5 + 2*(3*a*cosh(x) + a)*sinh(x)^5 - 2*a*cosh(x)^4 + (15*a*cosh(x)^2 + 10*a*cosh(x) - 2*a)*sinh(x)^4 -
6*a*cosh(x)^3 + 2*(10*a*cosh(x)^3 + 10*a*cosh(x)^2 - 4*a*cosh(x) - 3*a)*sinh(x)^3 - a*cosh(x)^2 + (15*a*cosh(
x)^4 + 20*a*cosh(x)^3 - 12*a*cosh(x)^2 - 18*a*cosh(x) - a)*sinh(x)^2 + 4*a*cosh(x) + 2*(3*a*cosh(x)^5 + 5*a*co
sh(x)^4 - 4*a*cosh(x)^3 - 9*a*cosh(x)^2 + a*cosh(x) + 4*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}^{4}{\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)**4/(a+a*sech(x)),x)

[Out]

Integral(csch(x)**4/(sech(x) + 1), x)/a

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 59 vs. \(2 (28) = 56\).
time = 0.39, size = 59, normalized size = 1.74 \begin {gather*} \frac {3 \, e^{\left (2 \, x\right )} - 12 \, e^{x} + 5}{24 \, a {\left (e^{x} - 1\right )}^{3}} - \frac {15 \, e^{\left (4 \, x\right )} + 60 \, e^{\left (3 \, x\right )} + 10 \, e^{\left (2 \, x\right )} + 20 \, e^{x} + 7}{120 \, a {\left (e^{x} + 1\right )}^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x)^4/(a+a*sech(x)),x, algorithm="giac")

[Out]

1/24*(3*e^(2*x) - 12*e^x + 5)/(a*(e^x - 1)^3) - 1/120*(15*e^(4*x) + 60*e^(3*x) + 10*e^(2*x) + 20*e^x + 7)/(a*(
e^x + 1)^5)

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Mupad [B]
time = 1.38, size = 236, normalized size = 6.94 \begin {gather*} \frac {1}{6\,a\,\left (3\,{\mathrm {e}}^{2\,x}-{\mathrm {e}}^{3\,x}-3\,{\mathrm {e}}^x+1\right )}-\frac {\frac {3\,{\mathrm {e}}^{2\,x}}{40\,a}+\frac {{\mathrm {e}}^{3\,x}}{40\,a}+\frac {1}{40\,a}-\frac {{\mathrm {e}}^x}{8\,a}}{6\,{\mathrm {e}}^{2\,x}+4\,{\mathrm {e}}^{3\,x}+{\mathrm {e}}^{4\,x}+4\,{\mathrm {e}}^x+1}-\frac {\frac {{\mathrm {e}}^{2\,x}}{40\,a}-\frac {1}{24\,a}+\frac {{\mathrm {e}}^x}{20\,a}}{3\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{3\,x}+3\,{\mathrm {e}}^x+1}-\frac {\frac {{\mathrm {e}}^{3\,x}}{10\,a}-\frac {{\mathrm {e}}^{2\,x}}{4\,a}+\frac {{\mathrm {e}}^{4\,x}}{40\,a}+\frac {1}{40\,a}+\frac {{\mathrm {e}}^x}{10\,a}}{10\,{\mathrm {e}}^{2\,x}+10\,{\mathrm {e}}^{3\,x}+5\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{5\,x}+5\,{\mathrm {e}}^x+1}-\frac {1}{4\,a\,\left ({\mathrm {e}}^{2\,x}-2\,{\mathrm {e}}^x+1\right )}+\frac {1}{8\,a\,\left ({\mathrm {e}}^x-1\right )}-\frac {1}{20\,a\,\left ({\mathrm {e}}^x+1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sinh(x)^4*(a + a/cosh(x))),x)

[Out]

1/(6*a*(3*exp(2*x) - exp(3*x) - 3*exp(x) + 1)) - ((3*exp(2*x))/(40*a) + exp(3*x)/(40*a) + 1/(40*a) - exp(x)/(8
*a))/(6*exp(2*x) + 4*exp(3*x) + exp(4*x) + 4*exp(x) + 1) - (exp(2*x)/(40*a) - 1/(24*a) + exp(x)/(20*a))/(3*exp
(2*x) + exp(3*x) + 3*exp(x) + 1) - (exp(3*x)/(10*a) - exp(2*x)/(4*a) + exp(4*x)/(40*a) + 1/(40*a) + exp(x)/(10
*a))/(10*exp(2*x) + 10*exp(3*x) + 5*exp(4*x) + exp(5*x) + 5*exp(x) + 1) - 1/(4*a*(exp(2*x) - 2*exp(x) + 1)) +
1/(8*a*(exp(x) - 1)) - 1/(20*a*(exp(x) + 1))

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