Integrand size = 16, antiderivative size = 101 \[ \int e^{a+b x} \text {csch}^4(a+b x) \, dx=\frac {8 e^{3 a+3 b x}}{3 b \left (1-e^{2 a+2 b x}\right )^3}-\frac {2 e^{a+b x}}{b \left (1-e^{2 a+2 b x}\right )^2}+\frac {e^{a+b x}}{b \left (1-e^{2 a+2 b x}\right )}+\frac {\text {arctanh}\left (e^{a+b x}\right )}{b} \] Output:
8/3*exp(3*b*x+3*a)/b/(1-exp(2*b*x+2*a))^3-2*exp(b*x+a)/b/(1-exp(2*b*x+2*a) )^2+exp(b*x+a)/b/(1-exp(2*b*x+2*a))+arctanh(exp(b*x+a))/b
Time = 0.05 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.74 \[ \int e^{a+b x} \text {csch}^4(a+b x) \, dx=\frac {3 e^{a+b x}-8 e^{3 (a+b x)}-3 e^{5 (a+b x)}+3 \left (-1+e^{2 (a+b x)}\right )^3 \text {arctanh}\left (e^{a+b x}\right )}{3 b \left (-1+e^{2 (a+b x)}\right )^3} \] Input:
Integrate[E^(a + b*x)*Csch[a + b*x]^4,x]
Output:
(3*E^(a + b*x) - 8*E^(3*(a + b*x)) - 3*E^(5*(a + b*x)) + 3*(-1 + E^(2*(a + b*x)))^3*ArcTanh[E^(a + b*x)])/(3*b*(-1 + E^(2*(a + b*x)))^3)
Time = 0.37 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.11, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {2720, 27, 252, 252, 215, 219}
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.
\(\displaystyle \int e^{a+b x} \text {csch}^4(a+b x) \, dx\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle \frac {\int \frac {16 e^{4 a+4 b x}}{\left (1-e^{2 a+2 b x}\right )^4}de^{a+b x}}{b}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {16 \int \frac {e^{4 a+4 b x}}{\left (1-e^{2 a+2 b x}\right )^4}de^{a+b x}}{b}\) |
\(\Big \downarrow \) 252 |
\(\displaystyle \frac {16 \left (\frac {e^{3 a+3 b x}}{6 \left (1-e^{2 a+2 b x}\right )^3}-\frac {1}{2} \int \frac {e^{2 a+2 b x}}{\left (1-e^{2 a+2 b x}\right )^3}de^{a+b x}\right )}{b}\) |
\(\Big \downarrow \) 252 |
\(\displaystyle \frac {16 \left (\frac {1}{2} \left (\frac {1}{4} \int \frac {1}{\left (1-e^{2 a+2 b x}\right )^2}de^{a+b x}-\frac {e^{a+b x}}{4 \left (1-e^{2 a+2 b x}\right )^2}\right )+\frac {e^{3 a+3 b x}}{6 \left (1-e^{2 a+2 b x}\right )^3}\right )}{b}\) |
\(\Big \downarrow \) 215 |
\(\displaystyle \frac {16 \left (\frac {1}{2} \left (\frac {1}{4} \left (\frac {1}{2} \int \frac {1}{1-e^{2 a+2 b x}}de^{a+b x}+\frac {e^{a+b x}}{2 \left (1-e^{2 a+2 b x}\right )}\right )-\frac {e^{a+b x}}{4 \left (1-e^{2 a+2 b x}\right )^2}\right )+\frac {e^{3 a+3 b x}}{6 \left (1-e^{2 a+2 b x}\right )^3}\right )}{b}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {16 \left (\frac {1}{2} \left (\frac {1}{4} \left (\frac {1}{2} \text {arctanh}\left (e^{a+b x}\right )+\frac {e^{a+b x}}{2 \left (1-e^{2 a+2 b x}\right )}\right )-\frac {e^{a+b x}}{4 \left (1-e^{2 a+2 b x}\right )^2}\right )+\frac {e^{3 a+3 b x}}{6 \left (1-e^{2 a+2 b x}\right )^3}\right )}{b}\) |
Input:
Int[E^(a + b*x)*Csch[a + b*x]^4,x]
Output:
(16*(E^(3*a + 3*b*x)/(6*(1 - E^(2*a + 2*b*x))^3) + (-1/4*E^(a + b*x)/(1 - E^(2*a + 2*b*x))^2 + (E^(a + b*x)/(2*(1 - E^(2*a + 2*b*x))) + ArcTanh[E^(a + b*x)]/2)/4)/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_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) /(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1)) Int[(a + b*x^2)^(p + 1 ), x], x] /; FreeQ[{a, b}, x] && LtQ[p, -1] && (IntegerQ[4*p] || IntegerQ[6 *p])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x )^(m - 1)*((a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] - Simp[c^2*((m - 1)/(2*b* (p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c }, x] && LtQ[p, -1] && GtQ[m, 1] && !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomi alQ[a, b, c, 2, m, p, x]
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 = 1.75 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.37
method | result | size |
derivativedivides | \(\frac {-\frac {\coth \left (b x +a \right ) \operatorname {csch}\left (b x +a \right )}{2}+\operatorname {arctanh}\left ({\mathrm e}^{b x +a}\right )-\frac {1}{3 \sinh \left (b x +a \right )^{3}}}{b}\) | \(37\) |
default | \(\frac {-\frac {\coth \left (b x +a \right ) \operatorname {csch}\left (b x +a \right )}{2}+\operatorname {arctanh}\left ({\mathrm e}^{b x +a}\right )-\frac {1}{3 \sinh \left (b x +a \right )^{3}}}{b}\) | \(37\) |
risch | \(-\frac {{\mathrm e}^{b x +a} \left (3 \,{\mathrm e}^{4 b x +4 a}+8 \,{\mathrm e}^{2 b x +2 a}-3\right )}{3 b \left ({\mathrm e}^{2 b x +2 a}-1\right )^{3}}+\frac {\ln \left ({\mathrm e}^{b x +a}+1\right )}{2 b}-\frac {\ln \left ({\mathrm e}^{b x +a}-1\right )}{2 b}\) | \(78\) |
Input:
int(exp(b*x+a)*csch(b*x+a)^4,x,method=_RETURNVERBOSE)
Output:
1/b*(-1/2*coth(b*x+a)*csch(b*x+a)+arctanh(exp(b*x+a))-1/3/sinh(b*x+a)^3)
Leaf count of result is larger than twice the leaf count of optimal. 705 vs. \(2 (87) = 174\).
Time = 0.09 (sec) , antiderivative size = 705, normalized size of antiderivative = 6.98 \[ \int e^{a+b x} \text {csch}^4(a+b x) \, dx =\text {Too large to display} \] Input:
integrate(exp(b*x+a)*csch(b*x+a)^4,x, algorithm="fricas")
Output:
-1/6*(6*cosh(b*x + a)^5 + 30*cosh(b*x + a)*sinh(b*x + a)^4 + 6*sinh(b*x + a)^5 + 4*(15*cosh(b*x + a)^2 + 4)*sinh(b*x + a)^3 + 16*cosh(b*x + a)^3 + 1 2*(5*cosh(b*x + a)^3 + 4*cosh(b*x + a))*sinh(b*x + a)^2 - 3*(cosh(b*x + a) ^6 + 6*cosh(b*x + a)*sinh(b*x + a)^5 + sinh(b*x + a)^6 + 3*(5*cosh(b*x + a )^2 - 1)*sinh(b*x + a)^4 - 3*cosh(b*x + a)^4 + 4*(5*cosh(b*x + a)^3 - 3*co sh(b*x + a))*sinh(b*x + a)^3 + 3*(5*cosh(b*x + a)^4 - 6*cosh(b*x + a)^2 + 1)*sinh(b*x + a)^2 + 3*cosh(b*x + a)^2 + 6*(cosh(b*x + a)^5 - 2*cosh(b*x + a)^3 + cosh(b*x + a))*sinh(b*x + a) - 1)*log(cosh(b*x + a) + sinh(b*x + a ) + 1) + 3*(cosh(b*x + a)^6 + 6*cosh(b*x + a)*sinh(b*x + a)^5 + sinh(b*x + a)^6 + 3*(5*cosh(b*x + a)^2 - 1)*sinh(b*x + a)^4 - 3*cosh(b*x + a)^4 + 4* (5*cosh(b*x + a)^3 - 3*cosh(b*x + a))*sinh(b*x + a)^3 + 3*(5*cosh(b*x + a) ^4 - 6*cosh(b*x + a)^2 + 1)*sinh(b*x + a)^2 + 3*cosh(b*x + a)^2 + 6*(cosh( b*x + a)^5 - 2*cosh(b*x + a)^3 + cosh(b*x + a))*sinh(b*x + a) - 1)*log(cos h(b*x + a) + sinh(b*x + a) - 1) + 6*(5*cosh(b*x + a)^4 + 8*cosh(b*x + a)^2 - 1)*sinh(b*x + a) - 6*cosh(b*x + a))/(b*cosh(b*x + a)^6 + 6*b*cosh(b*x + a)*sinh(b*x + a)^5 + b*sinh(b*x + a)^6 - 3*b*cosh(b*x + a)^4 + 3*(5*b*cos h(b*x + a)^2 - b)*sinh(b*x + a)^4 + 4*(5*b*cosh(b*x + a)^3 - 3*b*cosh(b*x + a))*sinh(b*x + a)^3 + 3*b*cosh(b*x + a)^2 + 3*(5*b*cosh(b*x + a)^4 - 6*b *cosh(b*x + a)^2 + b)*sinh(b*x + a)^2 + 6*(b*cosh(b*x + a)^5 - 2*b*cosh(b* x + a)^3 + b*cosh(b*x + a))*sinh(b*x + a) - b)
\[ \int e^{a+b x} \text {csch}^4(a+b x) \, dx=e^{a} \int e^{b x} \operatorname {csch}^{4}{\left (a + b x \right )}\, dx \] Input:
integrate(exp(b*x+a)*csch(b*x+a)**4,x)
Output:
exp(a)*Integral(exp(b*x)*csch(a + b*x)**4, x)
Time = 0.04 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.99 \[ \int e^{a+b x} \text {csch}^4(a+b x) \, dx=\frac {\log \left (e^{\left (b x + a\right )} + 1\right )}{2 \, b} - \frac {\log \left (e^{\left (b x + a\right )} - 1\right )}{2 \, b} - \frac {3 \, e^{\left (5 \, b x + 5 \, a\right )} + 8 \, e^{\left (3 \, b x + 3 \, a\right )} - 3 \, e^{\left (b x + a\right )}}{3 \, b {\left (e^{\left (6 \, b x + 6 \, a\right )} - 3 \, e^{\left (4 \, b x + 4 \, a\right )} + 3 \, e^{\left (2 \, b x + 2 \, a\right )} - 1\right )}} \] Input:
integrate(exp(b*x+a)*csch(b*x+a)^4,x, algorithm="maxima")
Output:
1/2*log(e^(b*x + a) + 1)/b - 1/2*log(e^(b*x + a) - 1)/b - 1/3*(3*e^(5*b*x + 5*a) + 8*e^(3*b*x + 3*a) - 3*e^(b*x + a))/(b*(e^(6*b*x + 6*a) - 3*e^(4*b *x + 4*a) + 3*e^(2*b*x + 2*a) - 1))
Time = 0.12 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.74 \[ \int e^{a+b x} \text {csch}^4(a+b x) \, dx=-\frac {\frac {2 \, {\left (3 \, e^{\left (5 \, b x + 5 \, a\right )} + 8 \, e^{\left (3 \, b x + 3 \, a\right )} - 3 \, e^{\left (b x + a\right )}\right )}}{{\left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right )}^{3}} - 3 \, \log \left (e^{\left (b x + a\right )} + 1\right ) + 3 \, \log \left ({\left | e^{\left (b x + a\right )} - 1 \right |}\right )}{6 \, b} \] Input:
integrate(exp(b*x+a)*csch(b*x+a)^4,x, algorithm="giac")
Output:
-1/6*(2*(3*e^(5*b*x + 5*a) + 8*e^(3*b*x + 3*a) - 3*e^(b*x + a))/(e^(2*b*x + 2*a) - 1)^3 - 3*log(e^(b*x + a) + 1) + 3*log(abs(e^(b*x + a) - 1)))/b
Time = 1.60 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.34 \[ \int e^{a+b x} \text {csch}^4(a+b x) \, dx=\frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{b\,x}\,{\mathrm {e}}^a\,\sqrt {-b^2}}{b}\right )}{\sqrt {-b^2}}-\frac {2\,{\mathrm {e}}^{a+b\,x}}{b\,\left ({\mathrm {e}}^{4\,a+4\,b\,x}-2\,{\mathrm {e}}^{2\,a+2\,b\,x}+1\right )}-\frac {8\,{\mathrm {e}}^{3\,a+3\,b\,x}}{3\,b\,\left (3\,{\mathrm {e}}^{2\,a+2\,b\,x}-3\,{\mathrm {e}}^{4\,a+4\,b\,x}+{\mathrm {e}}^{6\,a+6\,b\,x}-1\right )}-\frac {{\mathrm {e}}^{a+b\,x}}{b\,\left ({\mathrm {e}}^{2\,a+2\,b\,x}-1\right )} \] Input:
int(exp(a + b*x)/sinh(a + b*x)^4,x)
Output:
atan((exp(b*x)*exp(a)*(-b^2)^(1/2))/b)/(-b^2)^(1/2) - (2*exp(a + b*x))/(b* (exp(4*a + 4*b*x) - 2*exp(2*a + 2*b*x) + 1)) - (8*exp(3*a + 3*b*x))/(3*b*( 3*exp(2*a + 2*b*x) - 3*exp(4*a + 4*b*x) + exp(6*a + 6*b*x) - 1)) - exp(a + b*x)/(b*(exp(2*a + 2*b*x) - 1))
Time = 0.15 (sec) , antiderivative size = 233, normalized size of antiderivative = 2.31 \[ \int e^{a+b x} \text {csch}^4(a+b x) \, dx=\frac {-3 e^{6 b x +6 a} \mathrm {log}\left (e^{b x +a}-1\right )+3 e^{6 b x +6 a} \mathrm {log}\left (e^{b x +a}+1\right )-6 e^{5 b x +5 a}+9 e^{4 b x +4 a} \mathrm {log}\left (e^{b x +a}-1\right )-9 e^{4 b x +4 a} \mathrm {log}\left (e^{b x +a}+1\right )-16 e^{3 b x +3 a}-9 e^{2 b x +2 a} \mathrm {log}\left (e^{b x +a}-1\right )+9 e^{2 b x +2 a} \mathrm {log}\left (e^{b x +a}+1\right )+6 e^{b x +a}+3 \,\mathrm {log}\left (e^{b x +a}-1\right )-3 \,\mathrm {log}\left (e^{b x +a}+1\right )}{6 b \left (e^{6 b x +6 a}-3 e^{4 b x +4 a}+3 e^{2 b x +2 a}-1\right )} \] Input:
int(exp(b*x+a)*csch(b*x+a)^4,x)
Output:
( - 3*e**(6*a + 6*b*x)*log(e**(a + b*x) - 1) + 3*e**(6*a + 6*b*x)*log(e**( a + b*x) + 1) - 6*e**(5*a + 5*b*x) + 9*e**(4*a + 4*b*x)*log(e**(a + b*x) - 1) - 9*e**(4*a + 4*b*x)*log(e**(a + b*x) + 1) - 16*e**(3*a + 3*b*x) - 9*e **(2*a + 2*b*x)*log(e**(a + b*x) - 1) + 9*e**(2*a + 2*b*x)*log(e**(a + b*x ) + 1) + 6*e**(a + b*x) + 3*log(e**(a + b*x) - 1) - 3*log(e**(a + b*x) + 1 ))/(6*b*(e**(6*a + 6*b*x) - 3*e**(4*a + 4*b*x) + 3*e**(2*a + 2*b*x) - 1))