Integrand size = 20, antiderivative size = 293 \[ \int e^{\frac {5}{3} (a+b x)} \text {csch}^5(d+b x) \, dx=-\frac {4 e^{\frac {5 (a-d)}{3}+\frac {14}{3} (d+b x)}}{b \left (1-e^{2 (d+b x)}\right )^4}+\frac {28 e^{\frac {5 (a-d)}{3}+\frac {8}{3} (d+b x)}}{9 b \left (1-e^{2 (d+b x)}\right )^3}-\frac {56 e^{\frac {5 (a-d)}{3}+\frac {2}{3} (d+b x)}}{27 b \left (1-e^{2 (d+b x)}\right )^2}+\frac {56 e^{\frac {5 (a-d)}{3}+\frac {2}{3} (d+b x)}}{81 b \left (1-e^{2 (d+b x)}\right )}+\frac {112 e^{\frac {5 (a-d)}{3}} \arctan \left (\frac {1+2 e^{\frac {2}{3} (d+b x)}}{\sqrt {3}}\right )}{81 \sqrt {3} b}-\frac {112 e^{\frac {5 (a-d)}{3}} \log \left (1-e^{\frac {2}{3} (d+b x)}\right )}{243 b}+\frac {56 e^{\frac {5 (a-d)}{3}} \log \left (1+e^{\frac {2}{3} (d+b x)}+e^{\frac {4}{3} (d+b x)}\right )}{243 b} \] Output:
-4*exp(5/3*a+3*d+14/3*b*x)/b/(1-exp(2*b*x+2*d))^4+28/9*exp(5/3*a+d+8/3*b*x )/b/(1-exp(2*b*x+2*d))^3-56/27*exp(5/3*a-d+2/3*b*x)/b/(1-exp(2*b*x+2*d))^2 +56/81*exp(5/3*a-d+2/3*b*x)/b/(1-exp(2*b*x+2*d))+112/243*3^(1/2)*exp(5/3*a -5/3*d)*arctan(1/3*(1+2*exp(2/3*b*x+2/3*d))*3^(1/2))/b-112/243*exp(5/3*a-5 /3*d)*ln(1-exp(2/3*b*x+2/3*d))/b+56/243*exp(5/3*a-5/3*d)*ln(1+exp(2/3*b*x+ 2/3*d)+exp(4/3*b*x+4/3*d))/b
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.51 (sec) , antiderivative size = 323, normalized size of antiderivative = 1.10 \[ \int e^{\frac {5}{3} (a+b x)} \text {csch}^5(d+b x) \, dx=\frac {4 e^{5 a/3} (\cosh (d)-\sinh (d))^2 \left (-28 \text {RootSum}\left [\cosh \left (\frac {d}{2}\right )-\sinh \left (\frac {d}{2}\right )+\cosh \left (\frac {d}{2}\right ) \text {$\#$1}^3+\sinh \left (\frac {d}{2}\right ) \text {$\#$1}^3\&,\frac {b x-3 \log \left (e^{\frac {b x}{3}}-\text {$\#$1}\right )}{\text {$\#$1}}\&\right ]+28 \text {RootSum}\left [-\cosh \left (\frac {d}{2}\right )+\sinh \left (\frac {d}{2}\right )+\cosh \left (\frac {d}{2}\right ) \text {$\#$1}^3+\sinh \left (\frac {d}{2}\right ) \text {$\#$1}^3\&,\frac {b x-3 \log \left (e^{\frac {b x}{3}}-\text {$\#$1}\right )}{\text {$\#$1}}\&\right ]-\frac {729 e^{\frac {2 b x}{3}} (\cosh (d)-\sinh (d))^3}{\left (\left (-1+e^{2 b x}\right ) \cosh (d)+\left (1+e^{2 b x}\right ) \sinh (d)\right )^4}-\frac {2025 e^{\frac {2 b x}{3}} (\cosh (d)-\sinh (d))^2}{\left (\left (-1+e^{2 b x}\right ) \cosh (d)+\left (1+e^{2 b x}\right ) \sinh (d)\right )^3}-\frac {1674 e^{\frac {2 b x}{3}} (\cosh (d)-\sinh (d))}{\left (\left (-1+e^{2 b x}\right ) \cosh (d)+\left (1+e^{2 b x}\right ) \sinh (d)\right )^2}-\frac {126 e^{\frac {2 b x}{3}}}{\left (-1+e^{2 b x}\right ) \cosh (d)+\left (1+e^{2 b x}\right ) \sinh (d)}\right )}{729 b} \] Input:
Integrate[E^((5*(a + b*x))/3)*Csch[d + b*x]^5,x]
Output:
(4*E^((5*a)/3)*(Cosh[d] - Sinh[d])^2*(-28*RootSum[Cosh[d/2] - Sinh[d/2] + Cosh[d/2]*#1^3 + Sinh[d/2]*#1^3 & , (b*x - 3*Log[E^((b*x)/3) - #1])/#1 & ] + 28*RootSum[-Cosh[d/2] + Sinh[d/2] + Cosh[d/2]*#1^3 + Sinh[d/2]*#1^3 & , (b*x - 3*Log[E^((b*x)/3) - #1])/#1 & ] - (729*E^((2*b*x)/3)*(Cosh[d] - Si nh[d])^3)/((-1 + E^(2*b*x))*Cosh[d] + (1 + E^(2*b*x))*Sinh[d])^4 - (2025*E ^((2*b*x)/3)*(Cosh[d] - Sinh[d])^2)/((-1 + E^(2*b*x))*Cosh[d] + (1 + E^(2* b*x))*Sinh[d])^3 - (1674*E^((2*b*x)/3)*(Cosh[d] - Sinh[d]))/((-1 + E^(2*b* x))*Cosh[d] + (1 + E^(2*b*x))*Sinh[d])^2 - (126*E^((2*b*x)/3))/((-1 + E^(2 *b*x))*Cosh[d] + (1 + E^(2*b*x))*Sinh[d])))/(729*b)
Time = 0.36 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.65, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.650, Rules used = {2720, 27, 807, 817, 817, 817, 749, 750, 16, 1142, 1083, 217, 1103}
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.
| \(\displaystyle \int e^{\frac {5}{3} (a+b x)} \text {csch}^5(b x+d) \, dx\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle \frac {3 \int -\frac {32 e^{\frac {5 a}{3}+\frac {19 b x}{3}}}{\left (1-e^{2 b x}\right )^5}de^{\frac {b x}{3}}}{b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {96 e^{5 a/3} \int \frac {e^{\frac {19 b x}{3}}}{\left (1-e^{2 b x}\right )^5}de^{\frac {b x}{3}}}{b}\) |
| \(\Big \downarrow \) 807 |
| \(\displaystyle -\frac {48 e^{5 a/3} \int \frac {e^{3 b x}}{\left (1-e^{b x}\right )^5}de^{\frac {2 b x}{3}}}{b}\) |
| \(\Big \downarrow \) 817 |
| \(\displaystyle -\frac {48 e^{5 a/3} \left (\frac {e^{\frac {7 b x}{3}}}{12 \left (1-e^{b x}\right )^4}-\frac {7}{12} \int \frac {e^{2 b x}}{\left (1-e^{b x}\right )^4}de^{\frac {2 b x}{3}}\right )}{b}\) |
| \(\Big \downarrow \) 817 |
| \(\displaystyle -\frac {48 e^{5 a/3} \left (\frac {e^{\frac {7 b x}{3}}}{12 \left (1-e^{b x}\right )^4}-\frac {7}{12} \left (\frac {e^{\frac {4 b x}{3}}}{9 \left (1-e^{b x}\right )^3}-\frac {4}{9} \int \frac {e^{b x}}{\left (1-e^{b x}\right )^3}de^{\frac {2 b x}{3}}\right )\right )}{b}\) |
| \(\Big \downarrow \) 817 |
| \(\displaystyle -\frac {48 e^{5 a/3} \left (\frac {e^{\frac {7 b x}{3}}}{12 \left (1-e^{b x}\right )^4}-\frac {7}{12} \left (\frac {e^{\frac {4 b x}{3}}}{9 \left (1-e^{b x}\right )^3}-\frac {4}{9} \left (\frac {e^{\frac {2 b x}{3}}}{6 \left (1-e^{b x}\right )^2}-\frac {1}{6} \int \frac {1}{\left (1-e^{b x}\right )^2}de^{\frac {2 b x}{3}}\right )\right )\right )}{b}\) |
| \(\Big \downarrow \) 749 |
| \(\displaystyle -\frac {48 e^{5 a/3} \left (\frac {e^{\frac {7 b x}{3}}}{12 \left (1-e^{b x}\right )^4}-\frac {7}{12} \left (\frac {e^{\frac {4 b x}{3}}}{9 \left (1-e^{b x}\right )^3}-\frac {4}{9} \left (\frac {1}{6} \left (-\frac {2}{3} \int \frac {1}{1-e^{b x}}de^{\frac {2 b x}{3}}-\frac {e^{\frac {2 b x}{3}}}{3 \left (1-e^{b x}\right )}\right )+\frac {e^{\frac {2 b x}{3}}}{6 \left (1-e^{b x}\right )^2}\right )\right )\right )}{b}\) |
| \(\Big \downarrow \) 750 |
| \(\displaystyle -\frac {48 e^{5 a/3} \left (\frac {e^{\frac {7 b x}{3}}}{12 \left (1-e^{b x}\right )^4}-\frac {7}{12} \left (\frac {e^{\frac {4 b x}{3}}}{9 \left (1-e^{b x}\right )^3}-\frac {4}{9} \left (\frac {1}{6} \left (-\frac {2}{3} \left (\frac {1}{3} \int \frac {1}{1-e^{\frac {2 b x}{3}}}de^{\frac {2 b x}{3}}+\frac {1}{3} \int \frac {2+e^{\frac {2 b x}{3}}}{1+2 e^{\frac {2 b x}{3}}}de^{\frac {2 b x}{3}}\right )-\frac {e^{\frac {2 b x}{3}}}{3 \left (1-e^{b x}\right )}\right )+\frac {e^{\frac {2 b x}{3}}}{6 \left (1-e^{b x}\right )^2}\right )\right )\right )}{b}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle -\frac {48 e^{5 a/3} \left (\frac {e^{\frac {7 b x}{3}}}{12 \left (1-e^{b x}\right )^4}-\frac {7}{12} \left (\frac {e^{\frac {4 b x}{3}}}{9 \left (1-e^{b x}\right )^3}-\frac {4}{9} \left (\frac {1}{6} \left (-\frac {2}{3} \left (\frac {1}{3} \int \frac {2+e^{\frac {2 b x}{3}}}{1+2 e^{\frac {2 b x}{3}}}de^{\frac {2 b x}{3}}-\frac {1}{3} \log \left (1-e^{\frac {2 b x}{3}}\right )\right )-\frac {e^{\frac {2 b x}{3}}}{3 \left (1-e^{b x}\right )}\right )+\frac {e^{\frac {2 b x}{3}}}{6 \left (1-e^{b x}\right )^2}\right )\right )\right )}{b}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle -\frac {48 e^{5 a/3} \left (\frac {e^{\frac {7 b x}{3}}}{12 \left (1-e^{b x}\right )^4}-\frac {7}{12} \left (\frac {e^{\frac {4 b x}{3}}}{9 \left (1-e^{b x}\right )^3}-\frac {4}{9} \left (\frac {1}{6} \left (-\frac {2}{3} \left (\frac {1}{3} \left (\frac {1}{2} \int 1de^{\frac {2 b x}{3}}+\frac {3}{2} \int \frac {1}{1+2 e^{\frac {2 b x}{3}}}de^{\frac {2 b x}{3}}\right )-\frac {1}{3} \log \left (1-e^{\frac {2 b x}{3}}\right )\right )-\frac {e^{\frac {2 b x}{3}}}{3 \left (1-e^{b x}\right )}\right )+\frac {e^{\frac {2 b x}{3}}}{6 \left (1-e^{b x}\right )^2}\right )\right )\right )}{b}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle -\frac {48 e^{5 a/3} \left (\frac {e^{\frac {7 b x}{3}}}{12 \left (1-e^{b x}\right )^4}-\frac {7}{12} \left (\frac {e^{\frac {4 b x}{3}}}{9 \left (1-e^{b x}\right )^3}-\frac {4}{9} \left (\frac {1}{6} \left (-\frac {2}{3} \left (\frac {1}{3} \left (\frac {1}{2} \int 1de^{\frac {2 b x}{3}}-3 \int \frac {1}{-4-2 e^{\frac {2 b x}{3}}}d\left (1+2 e^{\frac {2 b x}{3}}\right )\right )-\frac {1}{3} \log \left (1-e^{\frac {2 b x}{3}}\right )\right )-\frac {e^{\frac {2 b x}{3}}}{3 \left (1-e^{b x}\right )}\right )+\frac {e^{\frac {2 b x}{3}}}{6 \left (1-e^{b x}\right )^2}\right )\right )\right )}{b}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {48 e^{5 a/3} \left (\frac {e^{\frac {7 b x}{3}}}{12 \left (1-e^{b x}\right )^4}-\frac {7}{12} \left (\frac {e^{\frac {4 b x}{3}}}{9 \left (1-e^{b x}\right )^3}-\frac {4}{9} \left (\frac {1}{6} \left (-\frac {2}{3} \left (\frac {1}{3} \left (\frac {1}{2} \int 1de^{\frac {2 b x}{3}}+\sqrt {3} \arctan \left (\frac {2 e^{\frac {2 b x}{3}}+1}{\sqrt {3}}\right )\right )-\frac {1}{3} \log \left (1-e^{\frac {2 b x}{3}}\right )\right )-\frac {e^{\frac {2 b x}{3}}}{3 \left (1-e^{b x}\right )}\right )+\frac {e^{\frac {2 b x}{3}}}{6 \left (1-e^{b x}\right )^2}\right )\right )\right )}{b}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle -\frac {48 e^{5 a/3} \left (\frac {e^{\frac {7 b x}{3}}}{12 \left (1-e^{b x}\right )^4}-\frac {7}{12} \left (\frac {e^{\frac {4 b x}{3}}}{9 \left (1-e^{b x}\right )^3}-\frac {4}{9} \left (\frac {1}{6} \left (-\frac {2}{3} \left (\frac {1}{3} \left (\sqrt {3} \arctan \left (\frac {2 e^{\frac {2 b x}{3}}+1}{\sqrt {3}}\right )+\frac {1}{2} \log \left (2 e^{\frac {2 b x}{3}}+1\right )\right )-\frac {1}{3} \log \left (1-e^{\frac {2 b x}{3}}\right )\right )-\frac {e^{\frac {2 b x}{3}}}{3 \left (1-e^{b x}\right )}\right )+\frac {e^{\frac {2 b x}{3}}}{6 \left (1-e^{b x}\right )^2}\right )\right )\right )}{b}\) |
Input:
Int[E^((5*(a + b*x))/3)*Csch[d + b*x]^5,x]
Output:
(-48*E^((5*a)/3)*(E^((7*b*x)/3)/(12*(1 - E^(b*x))^4) - (7*(E^((4*b*x)/3)/( 9*(1 - E^(b*x))^3) - (4*(E^((2*b*x)/3)/(6*(1 - E^(b*x))^2) + (-1/3*E^((2*b *x)/3)/(1 - E^(b*x)) - (2*(-1/3*Log[1 - E^((2*b*x)/3)] + (Sqrt[3]*ArcTan[( 1 + 2*E^((2*b*x)/3))/Sqrt[3]] + Log[1 + 2*E^((2*b*x)/3)]/2)/3))/3)/6))/9)) /12))/b
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
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)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] + Simp[(n*(p + 1) + 1)/(a*n*(p + 1)) Int[(a + b*x^ n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (Inte gerQ[2*p] || Denominator[p + 1/n] < Denominator[p])
Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Simp[1/(3*Rt[a, 3]^2) Int[1/ (Rt[a, 3] + Rt[b, 3]*x), x], x] + Simp[1/(3*Rt[a, 3]^2) Int[(2*Rt[a, 3] - Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2*x^2), x], x] /; FreeQ[{a, b}, x]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Simp[1/k Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^( n - 1)*(c*x)^(m - n + 1)*((a + b*x^n)^(p + 1)/(b*n*(p + 1))), x] - Simp[c^n *((m - n + 1)/(b*n*(p + 1))) Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x], x ] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] && ! ILtQ[(m + n*(p + 1) + 1)/n, 0] && IntBinomialQ[a, b, c, n, m, p, x]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, 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]]]
Result contains complex when optimal does not.
Time = 1.96 (sec) , antiderivative size = 224, normalized size of antiderivative = 0.76
| method | result | size |
| risch | \(-\frac {4 \left (14 \,{\mathrm e}^{6 b x +6 d}+144 \,{\mathrm e}^{4 b x +4 d}-105 \,{\mathrm e}^{2 b x +2 d}+28\right ) {\mathrm e}^{\frac {5 a}{3}-d +\frac {2 b x}{3}}}{81 \left ({\mathrm e}^{2 b x +2 d}-1\right )^{4} b}+\frac {56 \ln \left ({\mathrm e}^{\frac {2 b x}{3}+\frac {2 d}{3}}+\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) {\mathrm e}^{\frac {5 a}{3}-\frac {5 d}{3}}}{243 b}+\frac {56 i \ln \left ({\mathrm e}^{\frac {2 b x}{3}+\frac {2 d}{3}}+\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) {\mathrm e}^{\frac {5 a}{3}-\frac {5 d}{3}} \sqrt {3}}{243 b}+\frac {56 \ln \left ({\mathrm e}^{\frac {2 b x}{3}+\frac {2 d}{3}}+\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) {\mathrm e}^{\frac {5 a}{3}-\frac {5 d}{3}}}{243 b}-\frac {56 i \ln \left ({\mathrm e}^{\frac {2 b x}{3}+\frac {2 d}{3}}+\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) {\mathrm e}^{\frac {5 a}{3}-\frac {5 d}{3}} \sqrt {3}}{243 b}-\frac {112 \ln \left ({\mathrm e}^{\frac {2 b x}{3}+\frac {2 d}{3}}-1\right ) {\mathrm e}^{\frac {5 a}{3}-\frac {5 d}{3}}}{243 b}\) | \(224\) |
Input:
int(exp(5/3*b*x+5/3*a)*csch(b*x+d)^5,x,method=_RETURNVERBOSE)
Output:
-4/81/(exp(2*b*x+2*d)-1)^4/b*(14*exp(6*b*x+6*d)+144*exp(4*b*x+4*d)-105*exp (2*b*x+2*d)+28)*exp(5/3*a-d+2/3*b*x)+56/243*ln(exp(2/3*b*x+2/3*d)+1/2+1/2* I*3^(1/2))/b*exp(5/3*a-5/3*d)+56/243*I*ln(exp(2/3*b*x+2/3*d)+1/2+1/2*I*3^( 1/2))/b*exp(5/3*a-5/3*d)*3^(1/2)+56/243*ln(exp(2/3*b*x+2/3*d)+1/2-1/2*I*3^ (1/2))/b*exp(5/3*a-5/3*d)-56/243*I*ln(exp(2/3*b*x+2/3*d)+1/2-1/2*I*3^(1/2) )/b*exp(5/3*a-5/3*d)*3^(1/2)-112/243*ln(exp(2/3*b*x+2/3*d)-1)/b*exp(5/3*a- 5/3*d)
Leaf count of result is larger than twice the leaf count of optimal. 11702 vs. \(2 (215) = 430\).
Time = 0.28 (sec) , antiderivative size = 11702, normalized size of antiderivative = 39.94 \[ \int e^{\frac {5}{3} (a+b x)} \text {csch}^5(d+b x) \, dx=\text {Too large to display} \] Input:
integrate(exp(5/3*b*x+5/3*a)*csch(b*x+d)^5,x, algorithm="fricas")
Output:
Too large to include
\[ \int e^{\frac {5}{3} (a+b x)} \text {csch}^5(d+b x) \, dx=e^{\frac {5 a}{3}} \int e^{\frac {5 b x}{3}} \operatorname {csch}^{5}{\left (b x + d \right )}\, dx \] Input:
integrate(exp(5/3*b*x+5/3*a)*csch(b*x+d)**5,x)
Output:
exp(5*a/3)*Integral(exp(5*b*x/3)*csch(b*x + d)**5, x)
Time = 0.13 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.01 \[ \int e^{\frac {5}{3} (a+b x)} \text {csch}^5(d+b x) \, dx=\frac {112 \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, e^{\left (-\frac {1}{3} \, b x - \frac {1}{3} \, d\right )} + 1\right )}\right ) e^{\left (\frac {5}{3} \, a - \frac {5}{3} \, d\right )}}{243 \, b} - \frac {112 \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, e^{\left (-\frac {1}{3} \, b x - \frac {1}{3} \, d\right )} - 1\right )}\right ) e^{\left (\frac {5}{3} \, a - \frac {5}{3} \, d\right )}}{243 \, b} + \frac {56 \, e^{\left (\frac {5}{3} \, a - \frac {5}{3} \, d\right )} \log \left (e^{\left (-\frac {1}{3} \, b x - \frac {1}{3} \, d\right )} + e^{\left (-\frac {2}{3} \, b x - \frac {2}{3} \, d\right )} + 1\right )}{243 \, b} - \frac {112 \, e^{\left (\frac {5}{3} \, a - \frac {5}{3} \, d\right )} \log \left (e^{\left (-\frac {1}{3} \, b x - \frac {1}{3} \, d\right )} + 1\right )}{243 \, b} - \frac {112 \, e^{\left (\frac {5}{3} \, a - \frac {5}{3} \, d\right )} \log \left (e^{\left (-\frac {1}{3} \, b x - \frac {1}{3} \, d\right )} - 1\right )}{243 \, b} + \frac {56 \, e^{\left (\frac {5}{3} \, a - \frac {5}{3} \, d\right )} \log \left (-e^{\left (-\frac {1}{3} \, b x - \frac {1}{3} \, d\right )} + e^{\left (-\frac {2}{3} \, b x - \frac {2}{3} \, d\right )} + 1\right )}{243 \, b} + \frac {4 \, {\left (14 \, e^{\left (-\frac {4}{3} \, b x - \frac {4}{3} \, d\right )} + 144 \, e^{\left (-\frac {10}{3} \, b x - \frac {10}{3} \, d\right )} - 105 \, e^{\left (-\frac {16}{3} \, b x - \frac {16}{3} \, d\right )} + 28 \, e^{\left (-\frac {22}{3} \, b x - \frac {22}{3} \, d\right )}\right )} e^{\left (\frac {5}{3} \, a - \frac {5}{3} \, d\right )}}{81 \, b {\left (4 \, e^{\left (-2 \, b x - 2 \, d\right )} - 6 \, e^{\left (-4 \, b x - 4 \, d\right )} + 4 \, e^{\left (-6 \, b x - 6 \, d\right )} - e^{\left (-8 \, b x - 8 \, d\right )} - 1\right )}} \] Input:
integrate(exp(5/3*b*x+5/3*a)*csch(b*x+d)^5,x, algorithm="maxima")
Output:
112/243*sqrt(3)*arctan(1/3*sqrt(3)*(2*e^(-1/3*b*x - 1/3*d) + 1))*e^(5/3*a - 5/3*d)/b - 112/243*sqrt(3)*arctan(1/3*sqrt(3)*(2*e^(-1/3*b*x - 1/3*d) - 1))*e^(5/3*a - 5/3*d)/b + 56/243*e^(5/3*a - 5/3*d)*log(e^(-1/3*b*x - 1/3*d ) + e^(-2/3*b*x - 2/3*d) + 1)/b - 112/243*e^(5/3*a - 5/3*d)*log(e^(-1/3*b* x - 1/3*d) + 1)/b - 112/243*e^(5/3*a - 5/3*d)*log(e^(-1/3*b*x - 1/3*d) - 1 )/b + 56/243*e^(5/3*a - 5/3*d)*log(-e^(-1/3*b*x - 1/3*d) + e^(-2/3*b*x - 2 /3*d) + 1)/b + 4/81*(14*e^(-4/3*b*x - 4/3*d) + 144*e^(-10/3*b*x - 10/3*d) - 105*e^(-16/3*b*x - 16/3*d) + 28*e^(-22/3*b*x - 22/3*d))*e^(5/3*a - 5/3*d )/(b*(4*e^(-2*b*x - 2*d) - 6*e^(-4*b*x - 4*d) + 4*e^(-6*b*x - 6*d) - e^(-8 *b*x - 8*d) - 1))
Time = 0.12 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.52 \[ \int e^{\frac {5}{3} (a+b x)} \text {csch}^5(d+b x) \, dx=\frac {4 \, {\left (28 \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, e^{\left (\frac {2}{3} \, b x\right )} + e^{\left (-\frac {2}{3} \, d\right )}\right )} e^{\left (\frac {2}{3} \, d\right )}\right ) e^{\left (-\frac {20}{3} \, d\right )} + 14 \, e^{\left (-\frac {20}{3} \, d\right )} \log \left (e^{\left (\frac {4}{3} \, b x\right )} + e^{\left (\frac {2}{3} \, b x - \frac {2}{3} \, d\right )} + e^{\left (-\frac {4}{3} \, d\right )}\right ) - 28 \, e^{\left (-\frac {20}{3} \, d\right )} \log \left ({\left | e^{\left (\frac {2}{3} \, b x\right )} - e^{\left (-\frac {2}{3} \, d\right )} \right |}\right ) - \frac {3 \, {\left (14 \, e^{\left (\frac {20}{3} \, b x + 6 \, d\right )} + 144 \, e^{\left (\frac {14}{3} \, b x + 4 \, d\right )} - 105 \, e^{\left (\frac {8}{3} \, b x + 2 \, d\right )} + 28 \, e^{\left (\frac {2}{3} \, b x\right )}\right )} e^{\left (-6 \, d\right )}}{{\left (e^{\left (2 \, b x + 2 \, d\right )} - 1\right )}^{4}}\right )} e^{\left (\frac {5}{3} \, a + 5 \, d\right )}}{243 \, b} \] Input:
integrate(exp(5/3*b*x+5/3*a)*csch(b*x+d)^5,x, algorithm="giac")
Output:
4/243*(28*sqrt(3)*arctan(1/3*sqrt(3)*(2*e^(2/3*b*x) + e^(-2/3*d))*e^(2/3*d ))*e^(-20/3*d) + 14*e^(-20/3*d)*log(e^(4/3*b*x) + e^(2/3*b*x - 2/3*d) + e^ (-4/3*d)) - 28*e^(-20/3*d)*log(abs(e^(2/3*b*x) - e^(-2/3*d))) - 3*(14*e^(2 0/3*b*x + 6*d) + 144*e^(14/3*b*x + 4*d) - 105*e^(8/3*b*x + 2*d) + 28*e^(2/ 3*b*x))*e^(-6*d)/(e^(2*b*x + 2*d) - 1)^4)*e^(5/3*a + 5*d)/b
Time = 6.04 (sec) , antiderivative size = 373, normalized size of antiderivative = 1.27 \[ \int e^{\frac {5}{3} (a+b x)} \text {csch}^5(d+b x) \, dx=\frac {112\,{\left (-{\mathrm {e}}^{5\,a-5\,d}\right )}^{1/3}\,\ln \left (\frac {224\,{\left (-{\mathrm {e}}^{5\,a}\,{\mathrm {e}}^{-5\,d}\right )}^{1/3}}{243}+\frac {224\,{\mathrm {e}}^{\frac {5\,a}{3}}\,{\mathrm {e}}^{\frac {2\,d}{3}}\,{\mathrm {e}}^{-\frac {5\,d}{3}}\,{\mathrm {e}}^{\frac {2\,b\,x}{3}}}{243}\right )}{243\,b}-\frac {56\,{\mathrm {e}}^{\frac {5\,a}{3}-d+\frac {2\,b\,x}{3}}}{81\,b\,\left ({\mathrm {e}}^{2\,d+2\,b\,x}-1\right )}-\frac {56\,{\mathrm {e}}^{\frac {5\,a}{3}-d+\frac {2\,b\,x}{3}}}{27\,b\,\left ({\mathrm {e}}^{4\,d+4\,b\,x}-2\,{\mathrm {e}}^{2\,d+2\,b\,x}+1\right )}-\frac {4\,{\mathrm {e}}^{\frac {5\,a}{3}+3\,d+\frac {14\,b\,x}{3}}}{b\,\left (6\,{\mathrm {e}}^{4\,d+4\,b\,x}-4\,{\mathrm {e}}^{2\,d+2\,b\,x}-4\,{\mathrm {e}}^{6\,d+6\,b\,x}+{\mathrm {e}}^{8\,d+8\,b\,x}+1\right )}-\frac {28\,{\mathrm {e}}^{\frac {5\,a}{3}+d+\frac {8\,b\,x}{3}}}{9\,b\,\left (3\,{\mathrm {e}}^{2\,d+2\,b\,x}-3\,{\mathrm {e}}^{4\,d+4\,b\,x}+{\mathrm {e}}^{6\,d+6\,b\,x}-1\right )}+\frac {112\,{\left (-{\mathrm {e}}^{5\,a-5\,d}\right )}^{1/3}\,\ln \left (\frac {224\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (-{\mathrm {e}}^{5\,a}\,{\mathrm {e}}^{-5\,d}\right )}^{1/3}}{243}+\frac {224\,{\mathrm {e}}^{\frac {5\,a}{3}}\,{\mathrm {e}}^{\frac {2\,d}{3}}\,{\mathrm {e}}^{-\frac {5\,d}{3}}\,{\mathrm {e}}^{\frac {2\,b\,x}{3}}}{243}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{243\,b}-\frac {112\,{\left (-{\mathrm {e}}^{5\,a-5\,d}\right )}^{1/3}\,\ln \left (\frac {224\,{\mathrm {e}}^{\frac {5\,a}{3}}\,{\mathrm {e}}^{\frac {2\,d}{3}}\,{\mathrm {e}}^{-\frac {5\,d}{3}}\,{\mathrm {e}}^{\frac {2\,b\,x}{3}}}{243}-\frac {224\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (-{\mathrm {e}}^{5\,a}\,{\mathrm {e}}^{-5\,d}\right )}^{1/3}}{243}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{243\,b} \] Input:
int(exp((5*a)/3 + (5*b*x)/3)/sinh(d + b*x)^5,x)
Output:
(112*(-exp(5*a - 5*d))^(1/3)*log((224*(-exp(5*a)*exp(-5*d))^(1/3))/243 + ( 224*exp((5*a)/3)*exp((2*d)/3)*exp(-(5*d)/3)*exp((2*b*x)/3))/243))/(243*b) - (56*exp((5*a)/3 - d + (2*b*x)/3))/(81*b*(exp(2*d + 2*b*x) - 1)) - (56*ex p((5*a)/3 - d + (2*b*x)/3))/(27*b*(exp(4*d + 4*b*x) - 2*exp(2*d + 2*b*x) + 1)) - (4*exp((5*a)/3 + 3*d + (14*b*x)/3))/(b*(6*exp(4*d + 4*b*x) - 4*exp( 2*d + 2*b*x) - 4*exp(6*d + 6*b*x) + exp(8*d + 8*b*x) + 1)) - (28*exp((5*a) /3 + d + (8*b*x)/3))/(9*b*(3*exp(2*d + 2*b*x) - 3*exp(4*d + 4*b*x) + exp(6 *d + 6*b*x) - 1)) + (112*(-exp(5*a - 5*d))^(1/3)*log((224*((3^(1/2)*1i)/2 - 1/2)*(-exp(5*a)*exp(-5*d))^(1/3))/243 + (224*exp((5*a)/3)*exp((2*d)/3)*e xp(-(5*d)/3)*exp((2*b*x)/3))/243)*((3^(1/2)*1i)/2 - 1/2))/(243*b) - (112*( -exp(5*a - 5*d))^(1/3)*log((224*exp((5*a)/3)*exp((2*d)/3)*exp(-(5*d)/3)*ex p((2*b*x)/3))/243 - (224*((3^(1/2)*1i)/2 + 1/2)*(-exp(5*a)*exp(-5*d))^(1/3 ))/243)*((3^(1/2)*1i)/2 + 1/2))/(243*b)
\[ \int e^{\frac {5}{3} (a+b x)} \text {csch}^5(d+b x) \, dx=\int e^{\frac {5 b x}{3}+\frac {5 a}{3}} \mathrm {csch}\left (b x +d \right )^{5}d x \] Input:
int(exp(5/3*b*x+5/3*a)*csch(b*x+d)^5,x)
Output:
int(e**((5*a + 5*b*x)/3)*csch(b*x + d)**5,x)