3.4.9 \(\int e^{a+b x} \text {csch}^5(a+b x) \, dx\) [309]

Optimal. Leaf size=66 \[ -\frac {4}{b \left (1-e^{2 a+2 b x}\right )^4}+\frac {32}{3 b \left (1-e^{2 a+2 b x}\right )^3}-\frac {8}{b \left (1-e^{2 a+2 b x}\right )^2} \]

[Out]

-4/b/(1-exp(2*b*x+2*a))^4+32/3/b/(1-exp(2*b*x+2*a))^3-8/b/(1-exp(2*b*x+2*a))^2

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Rubi [A]
time = 0.04, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2320, 12, 272, 45} \begin {gather*} -\frac {8}{b \left (1-e^{2 a+2 b x}\right )^2}+\frac {32}{3 b \left (1-e^{2 a+2 b x}\right )^3}-\frac {4}{b \left (1-e^{2 a+2 b x}\right )^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(a + b*x)*Csch[a + b*x]^5,x]

[Out]

-4/(b*(1 - E^(2*a + 2*b*x))^4) + 32/(3*b*(1 - E^(2*a + 2*b*x))^3) - 8/(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 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

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}^5(a+b x) \, dx &=\frac {\text {Subst}\left (\int \frac {32 x^5}{\left (-1+x^2\right )^5} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac {32 \text {Subst}\left (\int \frac {x^5}{\left (-1+x^2\right )^5} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac {16 \text {Subst}\left (\int \frac {x^2}{(-1+x)^5} \, dx,x,e^{2 a+2 b x}\right )}{b}\\ &=\frac {16 \text {Subst}\left (\int \left (\frac {1}{(-1+x)^5}+\frac {2}{(-1+x)^4}+\frac {1}{(-1+x)^3}\right ) \, dx,x,e^{2 a+2 b x}\right )}{b}\\ &=-\frac {4}{b \left (1-e^{2 a+2 b x}\right )^4}+\frac {32}{3 b \left (1-e^{2 a+2 b x}\right )^3}-\frac {8}{b \left (1-e^{2 a+2 b x}\right )^2}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 44, normalized size = 0.67 \begin {gather*} -\frac {4 \left (1-4 e^{2 (a+b x)}+6 e^{4 (a+b x)}\right )}{3 b \left (-1+e^{2 (a+b x)}\right )^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(a + b*x)*Csch[a + b*x]^5,x]

[Out]

(-4*(1 - 4*E^(2*(a + b*x)) + 6*E^(4*(a + b*x))))/(3*b*(-1 + E^(2*(a + b*x)))^4)

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Maple [A]
time = 0.99, size = 43, normalized size = 0.65

method	result	size
risch	\(-\frac {4 \left (6 \,{\mathrm e}^{4 b x +4 a}-4 \,{\mathrm e}^{2 b x +2 a}+1\right )}{3 b \left ({\mathrm e}^{2 b x +2 a}-1\right )^{4}}\)	\(43\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(b*x+a)*csch(b*x+a)^5,x,method=_RETURNVERBOSE)

[Out]

-4/3*(6*exp(4*b*x+4*a)-4*exp(2*b*x+2*a)+1)/b/(exp(2*b*x+2*a)-1)^4

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 172 vs. \(2 (55) = 110\).
time = 0.26, size = 172, normalized size = 2.61 \begin {gather*} -\frac {8 \, e^{\left (4 \, b x + 4 \, a\right )}}{b {\left (e^{\left (8 \, b x + 8 \, a\right )} - 4 \, e^{\left (6 \, b x + 6 \, a\right )} + 6 \, e^{\left (4 \, b x + 4 \, a\right )} - 4 \, e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}} + \frac {16 \, e^{\left (2 \, b x + 2 \, a\right )}}{3 \, b {\left (e^{\left (8 \, b x + 8 \, a\right )} - 4 \, e^{\left (6 \, b x + 6 \, a\right )} + 6 \, e^{\left (4 \, b x + 4 \, a\right )} - 4 \, e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}} - \frac {4}{3 \, b {\left (e^{\left (8 \, b x + 8 \, a\right )} - 4 \, e^{\left (6 \, b x + 6 \, a\right )} + 6 \, e^{\left (4 \, b x + 4 \, a\right )} - 4 \, 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)^5,x, algorithm="maxima")

[Out]

-8*e^(4*b*x + 4*a)/(b*(e^(8*b*x + 8*a) - 4*e^(6*b*x + 6*a) + 6*e^(4*b*x + 4*a) - 4*e^(2*b*x + 2*a) + 1)) + 16/
3*e^(2*b*x + 2*a)/(b*(e^(8*b*x + 8*a) - 4*e^(6*b*x + 6*a) + 6*e^(4*b*x + 4*a) - 4*e^(2*b*x + 2*a) + 1)) - 4/3/
(b*(e^(8*b*x + 8*a) - 4*e^(6*b*x + 6*a) + 6*e^(4*b*x + 4*a) - 4*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. 233 vs. \(2 (55) = 110\).
time = 0.39, size = 233, normalized size = 3.53 \begin {gather*} -\frac {4 \, {\left (7 \, \cosh \left (b x + a\right )^{2} + 10 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + 7 \, \sinh \left (b x + a\right )^{2} - 4\right )}}{3 \, {\left (b \cosh \left (b x + a\right )^{6} + 6 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{5} + b \sinh \left (b x + a\right )^{6} - 4 \, b \cosh \left (b x + a\right )^{4} + {\left (15 \, b \cosh \left (b x + a\right )^{2} - 4 \, b\right )} \sinh \left (b x + a\right )^{4} + 4 \, {\left (5 \, b \cosh \left (b x + a\right )^{3} - 4 \, b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{3} + 7 \, b \cosh \left (b x + a\right )^{2} + {\left (15 \, b \cosh \left (b x + a\right )^{4} - 24 \, b \cosh \left (b x + a\right )^{2} + 7 \, b\right )} \sinh \left (b x + a\right )^{2} + 2 \, {\left (3 \, b \cosh \left (b x + a\right )^{5} - 8 \, b \cosh \left (b x + a\right )^{3} + 5 \, b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) - 4 \, b\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(b*x+a)*csch(b*x+a)^5,x, algorithm="fricas")

[Out]

-4/3*(7*cosh(b*x + a)^2 + 10*cosh(b*x + a)*sinh(b*x + a) + 7*sinh(b*x + a)^2 - 4)/(b*cosh(b*x + a)^6 + 6*b*cos
h(b*x + a)*sinh(b*x + a)^5 + b*sinh(b*x + a)^6 - 4*b*cosh(b*x + a)^4 + (15*b*cosh(b*x + a)^2 - 4*b)*sinh(b*x +
a)^4 + 4*(5*b*cosh(b*x + a)^3 - 4*b*cosh(b*x + a))*sinh(b*x + a)^3 + 7*b*cosh(b*x + a)^2 + (15*b*cosh(b*x + a
)^4 - 24*b*cosh(b*x + a)^2 + 7*b)*sinh(b*x + a)^2 + 2*(3*b*cosh(b*x + a)^5 - 8*b*cosh(b*x + a)^3 + 5*b*cosh(b*
x + a))*sinh(b*x + a) - 4*b)

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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}^{5}{\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)**5,x)

[Out]

exp(a)*Integral(exp(b*x)*csch(a + b*x)**5, x)

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Giac [A]
time = 0.41, size = 42, normalized size = 0.64 \begin {gather*} -\frac {4 \, {\left (6 \, e^{\left (4 \, b x + 4 \, a\right )} - 4 \, e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}}{3 \, b {\left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right )}^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(b*x+a)*csch(b*x+a)^5,x, algorithm="giac")

[Out]

-4/3*(6*e^(4*b*x + 4*a) - 4*e^(2*b*x + 2*a) + 1)/(b*(e^(2*b*x + 2*a) - 1)^4)

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Mupad [B]
time = 0.60, size = 42, normalized size = 0.64 \begin {gather*} -\frac {4\,\left (6\,{\mathrm {e}}^{4\,a+4\,b\,x}-4\,{\mathrm {e}}^{2\,a+2\,b\,x}+1\right )}{3\,b\,{\left ({\mathrm {e}}^{2\,a+2\,b\,x}-1\right )}^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(a + b*x)/sinh(a + b*x)^5,x)

[Out]

-(4*(6*exp(4*a + 4*b*x) - 4*exp(2*a + 2*b*x) + 1))/(3*b*(exp(2*a + 2*b*x) - 1)^4)

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