Integrand size = 16, antiderivative size = 66 \[ \int e^{a+b x} \text {csch}^5(a+b x) \, dx=-\frac {4}{b \left (1-e^{2 a+2 b x}\right )^4}+\frac {32}{3 b \left (1-e^{2 a+2 b x}\right )^3}-\frac {8}{b \left (1-e^{2 a+2 b x}\right )^2} \] Output:
-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
Time = 0.03 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.67 \[ \int e^{a+b x} \text {csch}^5(a+b x) \, dx=-\frac {4 \left (1-4 e^{2 (a+b x)}+6 e^{4 (a+b x)}\right )}{3 b \left (-1+e^{2 (a+b x)}\right )^4} \] Input:
Integrate[E^(a + b*x)*Csch[a + b*x]^5,x]
Output:
(-4*(1 - 4*E^(2*(a + b*x)) + 6*E^(4*(a + b*x))))/(3*b*(-1 + E^(2*(a + b*x) ))^4)
Time = 0.38 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {2720, 27, 243, 53, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int e^{a+b x} \text {csch}^5(a+b x) \, dx\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle \frac {\int -\frac {32 e^{5 a+5 b x}}{\left (1-e^{2 a+2 b x}\right )^5}de^{a+b x}}{b}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {32 \int \frac {e^{5 a+5 b x}}{\left (1-e^{2 a+2 b x}\right )^5}de^{a+b x}}{b}\) |
\(\Big \downarrow \) 243 |
\(\displaystyle -\frac {16 \int \frac {e^{2 a+2 b x}}{\left (1-e^{2 a+2 b x}\right )^5}de^{2 a+2 b x}}{b}\) |
\(\Big \downarrow \) 53 |
\(\displaystyle -\frac {16 \int \left (-\frac {1}{\left (-1+e^{2 a+2 b x}\right )^3}-\frac {2}{\left (-1+e^{2 a+2 b x}\right )^4}-\frac {1}{\left (-1+e^{2 a+2 b x}\right )^5}\right )de^{2 a+2 b x}}{b}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {16 \left (\frac {1}{2 \left (1-e^{2 a+2 b x}\right )^2}-\frac {2}{3 \left (1-e^{2 a+2 b x}\right )^3}+\frac {1}{4 \left (1-e^{2 a+2 b x}\right )^4}\right )}{b}\) |
Input:
Int[E^(a + b*x)*Csch[a + b*x]^5,x]
Output:
(-16*(1/(4*(1 - E^(2*a + 2*b*x))^4) - 2/(3*(1 - E^(2*a + 2*b*x))^3) + 1/(2 *(1 - E^(2*a + 2*b*x))^2)))/b
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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] && 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])
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct ionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ [{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) *(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
Time = 2.34 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.53
method | result | size |
derivativedivides | \(\frac {\left (\frac {2}{3}-\frac {\operatorname {csch}\left (b x +a \right )^{2}}{3}\right ) \coth \left (b x +a \right )-\frac {1}{4 \sinh \left (b x +a \right )^{4}}}{b}\) | \(35\) |
default | \(\frac {\left (\frac {2}{3}-\frac {\operatorname {csch}\left (b x +a \right )^{2}}{3}\right ) \coth \left (b x +a \right )-\frac {1}{4 \sinh \left (b x +a \right )^{4}}}{b}\) | \(35\) |
risch | \(-\frac {4 \left (6 \,{\mathrm e}^{4 b x +4 a}-4 \,{\mathrm e}^{2 b x +2 a}+1\right )}{3 b \left ({\mathrm e}^{2 b x +2 a}-1\right )^{4}}\) | \(43\) |
parallelrisch | \(\frac {{\mathrm e}^{b x +a} \left (3 \tanh \left (\frac {b x}{2}+\frac {a}{2}\right )^{4}-3 \coth \left (\frac {b x}{2}+\frac {a}{2}\right )^{4}+2 \tanh \left (\frac {b x}{2}+\frac {a}{2}\right )^{3}-2 \coth \left (\frac {b x}{2}+\frac {a}{2}\right )^{3}-22 \tanh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}+22 \coth \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}-38 \tanh \left (\frac {b x}{2}+\frac {a}{2}\right )+38 \coth \left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{192 b}\) | \(113\) |
Input:
int(exp(b*x+a)*csch(b*x+a)^5,x,method=_RETURNVERBOSE)
Output:
1/b*((2/3-1/3*csch(b*x+a)^2)*coth(b*x+a)-1/4/sinh(b*x+a)^4)
Leaf count of result is larger than twice the leaf count of optimal. 233 vs. \(2 (55) = 110\).
Time = 0.08 (sec) , antiderivative size = 233, normalized size of antiderivative = 3.53 \[ \int e^{a+b x} \text {csch}^5(a+b x) \, dx=-\frac {4 \, {\left (7 \, \cosh \left (b x + a\right )^{2} + 10 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + 7 \, \sinh \left (b x + a\right )^{2} - 4\right )}}{3 \, {\left (b \cosh \left (b x + a\right )^{6} + 6 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{5} + b \sinh \left (b x + a\right )^{6} - 4 \, b \cosh \left (b x + a\right )^{4} + {\left (15 \, b \cosh \left (b x + a\right )^{2} - 4 \, b\right )} \sinh \left (b x + a\right )^{4} + 4 \, {\left (5 \, b \cosh \left (b x + a\right )^{3} - 4 \, b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{3} + 7 \, b \cosh \left (b x + a\right )^{2} + {\left (15 \, b \cosh \left (b x + a\right )^{4} - 24 \, b \cosh \left (b x + a\right )^{2} + 7 \, b\right )} \sinh \left (b x + a\right )^{2} + 2 \, {\left (3 \, b \cosh \left (b x + a\right )^{5} - 8 \, b \cosh \left (b x + a\right )^{3} + 5 \, b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) - 4 \, b\right )}} \] Input:
integrate(exp(b*x+a)*csch(b*x+a)^5,x, algorithm="fricas")
Output:
-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*cosh(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*co sh(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)
\[ \int e^{a+b x} \text {csch}^5(a+b x) \, dx=e^{a} \int e^{b x} \operatorname {csch}^{5}{\left (a + b x \right )}\, dx \] Input:
integrate(exp(b*x+a)*csch(b*x+a)**5,x)
Output:
exp(a)*Integral(exp(b*x)*csch(a + b*x)**5, x)
Leaf count of result is larger than twice the leaf count of optimal. 172 vs. \(2 (55) = 110\).
Time = 0.04 (sec) , antiderivative size = 172, normalized size of antiderivative = 2.61 \[ \int e^{a+b x} \text {csch}^5(a+b x) \, dx=-\frac {8 \, e^{\left (4 \, b x + 4 \, a\right )}}{b {\left (e^{\left (8 \, b x + 8 \, a\right )} - 4 \, e^{\left (6 \, b x + 6 \, a\right )} + 6 \, e^{\left (4 \, b x + 4 \, a\right )} - 4 \, e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}} + \frac {16 \, e^{\left (2 \, b x + 2 \, a\right )}}{3 \, b {\left (e^{\left (8 \, b x + 8 \, a\right )} - 4 \, e^{\left (6 \, b x + 6 \, a\right )} + 6 \, e^{\left (4 \, b x + 4 \, a\right )} - 4 \, e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}} - \frac {4}{3 \, b {\left (e^{\left (8 \, b x + 8 \, a\right )} - 4 \, e^{\left (6 \, b x + 6 \, a\right )} + 6 \, e^{\left (4 \, b x + 4 \, a\right )} - 4 \, e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}} \] Input:
integrate(exp(b*x+a)*csch(b*x+a)^5,x, algorithm="maxima")
Output:
-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))
Time = 0.14 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.64 \[ \int e^{a+b x} \text {csch}^5(a+b x) \, dx=-\frac {4 \, {\left (6 \, e^{\left (4 \, b x + 4 \, a\right )} - 4 \, e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}}{3 \, b {\left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right )}^{4}} \] Input:
integrate(exp(b*x+a)*csch(b*x+a)^5,x, algorithm="giac")
Output:
-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)
Time = 1.62 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.64 \[ \int e^{a+b x} \text {csch}^5(a+b x) \, dx=-\frac {4\,\left (6\,{\mathrm {e}}^{4\,a+4\,b\,x}-4\,{\mathrm {e}}^{2\,a+2\,b\,x}+1\right )}{3\,b\,{\left ({\mathrm {e}}^{2\,a+2\,b\,x}-1\right )}^4} \] Input:
int(exp(a + b*x)/sinh(a + b*x)^5,x)
Output:
-(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)
Time = 0.15 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.23 \[ \int e^{a+b x} \text {csch}^5(a+b x) \, dx=\frac {-8 e^{4 b x +4 a}+\frac {16 e^{2 b x +2 a}}{3}-\frac {4}{3}}{b \left (e^{8 b x +8 a}-4 e^{6 b x +6 a}+6 e^{4 b x +4 a}-4 e^{2 b x +2 a}+1\right )} \] Input:
int(exp(b*x+a)*csch(b*x+a)^5,x)
Output:
(4*( - 6*e**(4*a + 4*b*x) + 4*e**(2*a + 2*b*x) - 1))/(3*b*(e**(8*a + 8*b*x ) - 4*e**(6*a + 6*b*x) + 6*e**(4*a + 4*b*x) - 4*e**(2*a + 2*b*x) + 1))