3.4.7 \(\int e^{a+b x} \text {csch}^3(a+b x) \, dx\) [307]

Optimal. Leaf size=31 \[ -\frac {2 e^{4 a+4 b x}}{b \left (1-e^{2 a+2 b x}\right )^2} \]

[Out]

-2*exp(4*b*x+4*a)/b/(1-exp(2*b*x+2*a))^2

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Rubi [A]
time = 0.02, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {2320, 12, 270} \begin {gather*} -\frac {2 e^{4 a+4 b x}}{b \left (1-e^{2 a+2 b x}\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(a + b*x)*Csch[a + b*x]^3,x]

[Out]

(-2*E^(4*a + 4*b*x))/(b*(1 - E^(2*a + 2*b*x))^2)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*
c*(m + 1))), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rule 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rubi steps

\begin {align*} \int e^{a+b x} \text {csch}^3(a+b x) \, dx &=\frac {\text {Subst}\left (\int \frac {8 x^3}{\left (-1+x^2\right )^3} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac {8 \text {Subst}\left (\int \frac {x^3}{\left (-1+x^2\right )^3} \, dx,x,e^{a+b x}\right )}{b}\\ &=-\frac {2 e^{4 a+4 b x}}{b \left (1-e^{2 a+2 b x}\right )^2}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 29, normalized size = 0.94 \begin {gather*} -\frac {2 e^{4 a+4 b x}}{b \left (-1+e^{2 a+2 b x}\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(a + b*x)*Csch[a + b*x]^3,x]

[Out]

(-2*E^(4*a + 4*b*x))/(b*(-1 + E^(2*a + 2*b*x))^2)

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Maple [A]
time = 1.13, size = 32, normalized size = 1.03

method	result	size
risch	\(-\frac {2 \left (2 \,{\mathrm e}^{2 b x +2 a}-1\right )}{b \left ({\mathrm e}^{2 b x +2 a}-1\right )^{2}}\)	\(32\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(b*x+a)*csch(b*x+a)^3,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (27) = 54\).
time = 0.27, size = 68, normalized size = 2.19 \begin {gather*} -\frac {4 \, e^{\left (2 \, b x + 2 \, a\right )}}{b {\left (e^{\left (4 \, b x + 4 \, a\right )} - 2 \, e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}} + \frac {2}{b {\left (e^{\left (4 \, b x + 4 \, a\right )} - 2 \, e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(b*x+a)*csch(b*x+a)^3,x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 88 vs. \(2 (27) = 54\).
time = 0.45, size = 88, normalized size = 2.84 \begin {gather*} -\frac {2 \, {\left (\cosh \left (b x + a\right ) + 3 \, \sinh \left (b x + a\right )\right )}}{b \cosh \left (b x + a\right )^{3} + 3 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + b \sinh \left (b x + a\right )^{3} - b \cosh \left (b x + a\right ) + 3 \, {\left (b \cosh \left (b x + a\right )^{2} - b\right )} \sinh \left (b x + a\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(b*x+a)*csch(b*x+a)^3,x, algorithm="fricas")

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} e^{a} \int e^{b x} \operatorname {csch}^{3}{\left (a + b x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(b*x+a)*csch(b*x+a)**3,x)

[Out]

exp(a)*Integral(exp(b*x)*csch(a + b*x)**3, x)

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Giac [A]
time = 0.43, size = 31, normalized size = 1.00 \begin {gather*} -\frac {2 \, {\left (2 \, e^{\left (2 \, b x + 2 \, a\right )} - 1\right )}}{b {\left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(b*x+a)*csch(b*x+a)^3,x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 0.58, size = 31, normalized size = 1.00 \begin {gather*} -\frac {2\,\left (2\,{\mathrm {e}}^{2\,a+2\,b\,x}-1\right )}{b\,{\left ({\mathrm {e}}^{2\,a+2\,b\,x}-1\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(a + b*x)/sinh(a + b*x)^3,x)

[Out]

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

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