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

Optimal result

Integrand size = 16, antiderivative size = 31 \[ \int e^{a+b x} \text {csch}^3(a+b x) \, dx=-\frac {2 e^{4 a+4 b x}}{b \left (1-e^{2 a+2 b x}\right )^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {2320, 12, 270} \[ \int e^{a+b x} \text {csch}^3(a+b x) \, dx=-\frac {2 e^{4 a+4 b x}}{b \left (1-e^{2 a+2 b x}\right )^2} \]

[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*} \text {integral}& = \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*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.94 \[ \int e^{a+b x} \text {csch}^3(a+b x) \, dx=-\frac {2 e^{4 a+4 b x}}{b \left (-1+e^{2 a+2 b x}\right )^2} \]

[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)

Maple [A] (verified)

Time = 1.56 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.77

method	result	size
derivativedivides	\(\frac {-\coth \left (b x +a \right )-\frac {1}{2 \sinh \left (b x +a \right )^{2}}}{b}\)	\(24\)
default	\(\frac {-\coth \left (b x +a \right )-\frac {1}{2 \sinh \left (b x +a \right )^{2}}}{b}\)	\(24\)
risch	\(-\frac {2 \left (2 \,{\mathrm e}^{2 b x +2 a}-1\right )}{b \left ({\mathrm e}^{2 b x +2 a}-1\right )^{2}}\)	\(32\)
parallelrisch	\(\frac {{\mathrm e}^{b x +a} \left (\tanh \left (\frac {b x}{2}+\frac {a}{2}\right )-1\right ) \left (\tanh \left (\frac {b x}{2}+\frac {a}{2}\right )+1\right )^{3}}{8 b \tanh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}}\)	\(47\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 88 vs. \(2 (27) = 54\).

Time = 0.27 (sec) , antiderivative size = 88, normalized size of antiderivative = 2.84 \[ \int e^{a+b x} \text {csch}^3(a+b x) \, dx=-\frac {2 \, {\left (\cosh \left (b x + a\right ) + 3 \, \sinh \left (b x + a\right )\right )}}{b \cosh \left (b x + a\right )^{3} + 3 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + b \sinh \left (b x + a\right )^{3} - b \cosh \left (b x + a\right ) + 3 \, {\left (b \cosh \left (b x + a\right )^{2} - b\right )} \sinh \left (b x + a\right )} \]

[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))

Sympy [F]

\[ \int e^{a+b x} \text {csch}^3(a+b x) \, dx=e^{a} \int e^{b x} \operatorname {csch}^{3}{\left (a + b x \right )}\, dx \]

[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)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (27) = 54\).

Time = 0.21 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.19 \[ \int e^{a+b x} \text {csch}^3(a+b x) \, dx=-\frac {4 \, e^{\left (2 \, b x + 2 \, a\right )}}{b {\left (e^{\left (4 \, b x + 4 \, a\right )} - 2 \, e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}} + \frac {2}{b {\left (e^{\left (4 \, b x + 4 \, a\right )} - 2 \, e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}} \]

[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))

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int e^{a+b x} \text {csch}^3(a+b x) \, dx=-\frac {2 \, {\left (2 \, e^{\left (2 \, b x + 2 \, a\right )} - 1\right )}}{b {\left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right )}^{2}} \]

[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)

Mupad [B] (verification not implemented)

Time = 1.31 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int e^{a+b x} \text {csch}^3(a+b x) \, dx=-\frac {2\,\left (2\,{\mathrm {e}}^{2\,a+2\,b\,x}-1\right )}{b\,{\left ({\mathrm {e}}^{2\,a+2\,b\,x}-1\right )}^2} \]

[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)