Integrand size = 20, antiderivative size = 204 \[ \int e^{\frac {5}{3} (a+b x)} \text {csch}^2(d+b x) \, dx=\frac {2 e^{\frac {5 (a-d)}{3}+\frac {5}{3} (d+b x)}}{b \left (1-e^{2 (d+b x)}\right )}-\frac {5 e^{\frac {5 (a-d)}{3}} \arctan \left (\frac {1-2 e^{\frac {1}{3} (d+b x)}}{\sqrt {3}}\right )}{\sqrt {3} b}+\frac {5 e^{\frac {5 (a-d)}{3}} \arctan \left (\frac {1+2 e^{\frac {1}{3} (d+b x)}}{\sqrt {3}}\right )}{\sqrt {3} b}-\frac {10 e^{\frac {5 (a-d)}{3}} \text {arctanh}\left (e^{\frac {1}{3} (d+b x)}\right )}{3 b}-\frac {5 e^{\frac {5 (a-d)}{3}} \text {arctanh}\left (\frac {e^{\frac {1}{3} (d+b x)}}{1+e^{\frac {2}{3} (d+b x)}}\right )}{3 b} \] Output:
2*exp(5/3*b*x+5/3*a)/b/(1-exp(2*b*x+2*d))-5/3*3^(1/2)*exp(5/3*a-5/3*d)*arc tan(1/3*(1-2*exp(1/3*b*x+1/3*d))*3^(1/2))/b+5/3*3^(1/2)*exp(5/3*a-5/3*d)*a rctan(1/3*(1+2*exp(1/3*b*x+1/3*d))*3^(1/2))/b-10/3*exp(5/3*a-5/3*d)*arctan h(exp(1/3*b*x+1/3*d))/b-5/3*exp(5/3*a-5/3*d)*arctanh(exp(1/3*b*x+1/3*d)/(1 +exp(2/3*b*x+2/3*d)))/b
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.25 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.56 \[ \int e^{\frac {5}{3} (a+b x)} \text {csch}^2(d+b x) \, dx=\frac {e^{5 a/3} (\cosh (d)-\sinh (d)) \left (-5 \text {RootSum}\left [-\cosh (d)+\sinh (d)+\cosh (d) \text {$\#$1}^6+\sinh (d) \text {$\#$1}^6\&,\frac {b x-3 \log \left (e^{\frac {b x}{3}}-\text {$\#$1}\right )}{\text {$\#$1}}\&\right ] (\cosh (d)-\sinh (d))-\frac {18 e^{\frac {5 b x}{3}}}{\left (-1+e^{2 b x}\right ) \cosh (d)+\left (1+e^{2 b x}\right ) \sinh (d)}\right )}{9 b} \] Input:
Integrate[E^((5*(a + b*x))/3)*Csch[d + b*x]^2,x]
Output:
(E^((5*a)/3)*(Cosh[d] - Sinh[d])*(-5*RootSum[-Cosh[d] + Sinh[d] + Cosh[d]* #1^6 + Sinh[d]*#1^6 & , (b*x - 3*Log[E^((b*x)/3) - #1])/#1 & ]*(Cosh[d] - Sinh[d]) - (18*E^((5*b*x)/3))/((-1 + E^(2*b*x))*Cosh[d] + (1 + E^(2*b*x))* Sinh[d])))/(9*b)
Time = 0.56 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.81, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.550, Rules used = {2720, 27, 817, 825, 27, 219, 1142, 25, 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.
| \(\displaystyle \int e^{\frac {5}{3} (a+b x)} \text {csch}^2(b x+d) \, dx\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle \frac {3 \int \frac {4 e^{\frac {5 a}{3}+\frac {10 b x}{3}}}{\left (1-e^{2 b x}\right )^2}de^{\frac {b x}{3}}}{b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {12 e^{5 a/3} \int \frac {e^{\frac {10 b x}{3}}}{\left (1-e^{2 b x}\right )^2}de^{\frac {b x}{3}}}{b}\) |
| \(\Big \downarrow \) 817 |
| \(\displaystyle \frac {12 e^{5 a/3} \left (\frac {e^{\frac {5 b x}{3}}}{6 \left (1-e^{2 b x}\right )}-\frac {5}{6} \int \frac {e^{\frac {4 b x}{3}}}{1-e^{2 b x}}de^{\frac {b x}{3}}\right )}{b}\) |
| \(\Big \downarrow \) 825 |
| \(\displaystyle \frac {12 e^{5 a/3} \left (\frac {e^{\frac {5 b x}{3}}}{6 \left (1-e^{2 b x}\right )}-\frac {5}{6} \left (\frac {1}{3} \int \frac {1}{1-e^{\frac {2 b x}{3}}}de^{\frac {b x}{3}}+\frac {1}{3} \int -\frac {1+e^{\frac {b x}{3}}}{2 \left (1-e^{\frac {b x}{3}}+e^{\frac {2 b x}{3}}\right )}de^{\frac {b x}{3}}+\frac {1}{3} \int -\frac {1-e^{\frac {b x}{3}}}{2 \left (1+e^{\frac {b x}{3}}+e^{\frac {2 b x}{3}}\right )}de^{\frac {b x}{3}}\right )\right )}{b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {12 e^{5 a/3} \left (\frac {e^{\frac {5 b x}{3}}}{6 \left (1-e^{2 b x}\right )}-\frac {5}{6} \left (\frac {1}{3} \int \frac {1}{1-e^{\frac {2 b x}{3}}}de^{\frac {b x}{3}}-\frac {1}{6} \int \frac {1+e^{\frac {b x}{3}}}{1-e^{\frac {b x}{3}}+e^{\frac {2 b x}{3}}}de^{\frac {b x}{3}}-\frac {1}{6} \int \frac {1-e^{\frac {b x}{3}}}{1+e^{\frac {b x}{3}}+e^{\frac {2 b x}{3}}}de^{\frac {b x}{3}}\right )\right )}{b}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {12 e^{5 a/3} \left (\frac {e^{\frac {5 b x}{3}}}{6 \left (1-e^{2 b x}\right )}-\frac {5}{6} \left (-\frac {1}{6} \int \frac {1+e^{\frac {b x}{3}}}{1-e^{\frac {b x}{3}}+e^{\frac {2 b x}{3}}}de^{\frac {b x}{3}}-\frac {1}{6} \int \frac {1-e^{\frac {b x}{3}}}{1+e^{\frac {b x}{3}}+e^{\frac {2 b x}{3}}}de^{\frac {b x}{3}}+\frac {1}{3} \text {arctanh}\left (e^{\frac {b x}{3}}\right )\right )\right )}{b}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {12 e^{5 a/3} \left (\frac {e^{\frac {5 b x}{3}}}{6 \left (1-e^{2 b x}\right )}-\frac {5}{6} \left (\frac {1}{6} \left (-\frac {3}{2} \int \frac {1}{1-e^{\frac {b x}{3}}+e^{\frac {2 b x}{3}}}de^{\frac {b x}{3}}-\frac {1}{2} \int -\frac {1-2 e^{\frac {b x}{3}}}{1-e^{\frac {b x}{3}}+e^{\frac {2 b x}{3}}}de^{\frac {b x}{3}}\right )+\frac {1}{6} \left (\frac {1}{2} \int \frac {1+2 e^{\frac {b x}{3}}}{1+e^{\frac {b x}{3}}+e^{\frac {2 b x}{3}}}de^{\frac {b x}{3}}-\frac {3}{2} \int \frac {1}{1+e^{\frac {b x}{3}}+e^{\frac {2 b x}{3}}}de^{\frac {b x}{3}}\right )+\frac {1}{3} \text {arctanh}\left (e^{\frac {b x}{3}}\right )\right )\right )}{b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {12 e^{5 a/3} \left (\frac {e^{\frac {5 b x}{3}}}{6 \left (1-e^{2 b x}\right )}-\frac {5}{6} \left (\frac {1}{6} \left (\frac {1}{2} \int \frac {1-2 e^{\frac {b x}{3}}}{1-e^{\frac {b x}{3}}+e^{\frac {2 b x}{3}}}de^{\frac {b x}{3}}-\frac {3}{2} \int \frac {1}{1-e^{\frac {b x}{3}}+e^{\frac {2 b x}{3}}}de^{\frac {b x}{3}}\right )+\frac {1}{6} \left (\frac {1}{2} \int \frac {1+2 e^{\frac {b x}{3}}}{1+e^{\frac {b x}{3}}+e^{\frac {2 b x}{3}}}de^{\frac {b x}{3}}-\frac {3}{2} \int \frac {1}{1+e^{\frac {b x}{3}}+e^{\frac {2 b x}{3}}}de^{\frac {b x}{3}}\right )+\frac {1}{3} \text {arctanh}\left (e^{\frac {b x}{3}}\right )\right )\right )}{b}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {12 e^{5 a/3} \left (\frac {e^{\frac {5 b x}{3}}}{6 \left (1-e^{2 b x}\right )}-\frac {5}{6} \left (\frac {1}{6} \left (3 \int \frac {1}{-3-e^{\frac {2 b x}{3}}}d\left (-1+2 e^{\frac {b x}{3}}\right )+\frac {1}{2} \int \frac {1-2 e^{\frac {b x}{3}}}{1-e^{\frac {b x}{3}}+e^{\frac {2 b x}{3}}}de^{\frac {b x}{3}}\right )+\frac {1}{6} \left (3 \int \frac {1}{-3-e^{\frac {2 b x}{3}}}d\left (1+2 e^{\frac {b x}{3}}\right )+\frac {1}{2} \int \frac {1+2 e^{\frac {b x}{3}}}{1+e^{\frac {b x}{3}}+e^{\frac {2 b x}{3}}}de^{\frac {b x}{3}}\right )+\frac {1}{3} \text {arctanh}\left (e^{\frac {b x}{3}}\right )\right )\right )}{b}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {12 e^{5 a/3} \left (\frac {e^{\frac {5 b x}{3}}}{6 \left (1-e^{2 b x}\right )}-\frac {5}{6} \left (\frac {1}{6} \left (\frac {1}{2} \int \frac {1-2 e^{\frac {b x}{3}}}{1-e^{\frac {b x}{3}}+e^{\frac {2 b x}{3}}}de^{\frac {b x}{3}}-\sqrt {3} \arctan \left (\frac {2 e^{\frac {b x}{3}}-1}{\sqrt {3}}\right )\right )+\frac {1}{6} \left (\frac {1}{2} \int \frac {1+2 e^{\frac {b x}{3}}}{1+e^{\frac {b x}{3}}+e^{\frac {2 b x}{3}}}de^{\frac {b x}{3}}-\sqrt {3} \arctan \left (\frac {2 e^{\frac {b x}{3}}+1}{\sqrt {3}}\right )\right )+\frac {1}{3} \text {arctanh}\left (e^{\frac {b x}{3}}\right )\right )\right )}{b}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {12 e^{5 a/3} \left (\frac {e^{\frac {5 b x}{3}}}{6 \left (1-e^{2 b x}\right )}-\frac {5}{6} \left (\frac {1}{6} \left (-\sqrt {3} \arctan \left (\frac {2 e^{\frac {b x}{3}}-1}{\sqrt {3}}\right )-\frac {1}{2} \log \left (-e^{\frac {b x}{3}}+e^{\frac {2 b x}{3}}+1\right )\right )+\frac {1}{6} \left (\frac {1}{2} \log \left (e^{\frac {b x}{3}}+e^{\frac {2 b x}{3}}+1\right )-\sqrt {3} \arctan \left (\frac {2 e^{\frac {b x}{3}}+1}{\sqrt {3}}\right )\right )+\frac {1}{3} \text {arctanh}\left (e^{\frac {b x}{3}}\right )\right )\right )}{b}\) |
Input:
Int[E^((5*(a + b*x))/3)*Csch[d + b*x]^2,x]
Output:
(12*E^((5*a)/3)*(E^((5*b*x)/3)/(6*(1 - E^(2*b*x))) - (5*(ArcTanh[E^((b*x)/ 3)]/3 + (-(Sqrt[3]*ArcTan[(-1 + 2*E^((b*x)/3))/Sqrt[3]]) - Log[1 - E^((b*x )/3) + E^((2*b*x)/3)]/2)/6 + (-(Sqrt[3]*ArcTan[(1 + 2*E^((b*x)/3))/Sqrt[3] ]) + Log[1 + E^((b*x)/3) + E^((2*b*x)/3)]/2)/6))/6))/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)^(-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_)^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_)^(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[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Module[{r = Numerator [Rt[-a/b, n]], s = Denominator[Rt[-a/b, n]], k, u}, Simp[u = Int[(r*Cos[2*k *m*(Pi/n)] - s*Cos[2*k*(m + 1)*(Pi/n)]*x)/(r^2 - 2*r*s*Cos[2*k*(Pi/n)]*x + s^2*x^2), x] + Int[(r*Cos[2*k*m*(Pi/n)] + s*Cos[2*k*(m + 1)*(Pi/n)]*x)/(r^2 + 2*r*s*Cos[2*k*(Pi/n)]*x + s^2*x^2), x]; 2*(r^(m + 2)/(a*n*s^m)) Int[1/ (r^2 - s^2*x^2), x] + 2*(r^(m + 1)/(a*n*s^m)) Sum[u, {k, 1, (n - 2)/4}], x]] /; FreeQ[{a, b}, x] && IGtQ[(n - 2)/4, 0] && IGtQ[m, 0] && LtQ[m, n - 1 ] && NegQ[a/b]
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 = 0.35 (sec) , antiderivative size = 343, normalized size of antiderivative = 1.68
| method | result | size |
| risch | \(-\frac {2 \,{\mathrm e}^{\frac {5 b x}{3}+\frac {5 a}{3}}}{\left ({\mathrm e}^{2 b x +2 d}-1\right ) b}+\frac {5 \ln \left ({\mathrm e}^{\frac {b x}{3}+\frac {d}{3}}-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) {\mathrm e}^{\frac {5 a}{3}-\frac {5 d}{3}}}{6 b}+\frac {5 i \ln \left ({\mathrm e}^{\frac {b x}{3}+\frac {d}{3}}-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) {\mathrm e}^{\frac {5 a}{3}-\frac {5 d}{3}} \sqrt {3}}{6 b}+\frac {5 \ln \left ({\mathrm e}^{\frac {b x}{3}+\frac {d}{3}}-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) {\mathrm e}^{\frac {5 a}{3}-\frac {5 d}{3}}}{6 b}-\frac {5 i \ln \left ({\mathrm e}^{\frac {b x}{3}+\frac {d}{3}}-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) {\mathrm e}^{\frac {5 a}{3}-\frac {5 d}{3}} \sqrt {3}}{6 b}+\frac {5 \ln \left ({\mathrm e}^{\frac {b x}{3}+\frac {d}{3}}-1\right ) {\mathrm e}^{\frac {5 a}{3}-\frac {5 d}{3}}}{3 b}-\frac {5 \ln \left ({\mathrm e}^{\frac {b x}{3}+\frac {d}{3}}+\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) {\mathrm e}^{\frac {5 a}{3}-\frac {5 d}{3}}}{6 b}+\frac {5 i \ln \left ({\mathrm e}^{\frac {b x}{3}+\frac {d}{3}}+\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) {\mathrm e}^{\frac {5 a}{3}-\frac {5 d}{3}} \sqrt {3}}{6 b}-\frac {5 \ln \left ({\mathrm e}^{\frac {b x}{3}+\frac {d}{3}}+\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) {\mathrm e}^{\frac {5 a}{3}-\frac {5 d}{3}}}{6 b}-\frac {5 i \ln \left ({\mathrm e}^{\frac {b x}{3}+\frac {d}{3}}+\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) {\mathrm e}^{\frac {5 a}{3}-\frac {5 d}{3}} \sqrt {3}}{6 b}-\frac {5 \ln \left (1+{\mathrm e}^{\frac {b x}{3}+\frac {d}{3}}\right ) {\mathrm e}^{\frac {5 a}{3}-\frac {5 d}{3}}}{3 b}\) | \(343\) |
Input:
int(exp(5/3*b*x+5/3*a)*csch(b*x+d)^2,x,method=_RETURNVERBOSE)
Output:
-2/(exp(2*b*x+2*d)-1)/b*exp(5/3*b*x+5/3*a)+5/6/b*ln(exp(1/3*b*x+1/3*d)-1/2 +1/2*I*3^(1/2))*exp(5/3*a-5/3*d)+5/6*I/b*ln(exp(1/3*b*x+1/3*d)-1/2+1/2*I*3 ^(1/2))*exp(5/3*a-5/3*d)*3^(1/2)+5/6/b*ln(exp(1/3*b*x+1/3*d)-1/2-1/2*I*3^( 1/2))*exp(5/3*a-5/3*d)-5/6*I/b*ln(exp(1/3*b*x+1/3*d)-1/2-1/2*I*3^(1/2))*ex p(5/3*a-5/3*d)*3^(1/2)+5/3/b*ln(exp(1/3*b*x+1/3*d)-1)*exp(5/3*a-5/3*d)-5/6 /b*ln(exp(1/3*b*x+1/3*d)+1/2+1/2*I*3^(1/2))*exp(5/3*a-5/3*d)+5/6*I/b*ln(ex p(1/3*b*x+1/3*d)+1/2+1/2*I*3^(1/2))*exp(5/3*a-5/3*d)*3^(1/2)-5/6/b*ln(exp( 1/3*b*x+1/3*d)+1/2-1/2*I*3^(1/2))*exp(5/3*a-5/3*d)-5/6*I/b*ln(exp(1/3*b*x+ 1/3*d)+1/2-1/2*I*3^(1/2))*exp(5/3*a-5/3*d)*3^(1/2)-5/3/b*ln(1+exp(1/3*b*x+ 1/3*d))*exp(5/3*a-5/3*d)
Leaf count of result is larger than twice the leaf count of optimal. 2870 vs. \(2 (158) = 316\).
Time = 0.13 (sec) , antiderivative size = 2870, normalized size of antiderivative = 14.07 \[ \int e^{\frac {5}{3} (a+b x)} \text {csch}^2(d+b x) \, dx=\text {Too large to display} \] Input:
integrate(exp(5/3*b*x+5/3*a)*csch(b*x+d)^2,x, algorithm="fricas")
Output:
-1/6*(12*cosh(1/3*b*x + 1/3*d)^5*cosh(-5/3*a + 5/3*d) + 12*(cosh(-5/3*a + 5/3*d) - sinh(-5/3*a + 5/3*d))*sinh(1/3*b*x + 1/3*d)^5 - 12*cosh(1/3*b*x + 1/3*d)^5*sinh(-5/3*a + 5/3*d) + 60*(cosh(1/3*b*x + 1/3*d)*cosh(-5/3*a + 5 /3*d) - cosh(1/3*b*x + 1/3*d)*sinh(-5/3*a + 5/3*d))*sinh(1/3*b*x + 1/3*d)^ 4 + 120*(cosh(1/3*b*x + 1/3*d)^2*cosh(-5/3*a + 5/3*d) - cosh(1/3*b*x + 1/3 *d)^2*sinh(-5/3*a + 5/3*d))*sinh(1/3*b*x + 1/3*d)^3 + 120*(cosh(1/3*b*x + 1/3*d)^3*cosh(-5/3*a + 5/3*d) - cosh(1/3*b*x + 1/3*d)^3*sinh(-5/3*a + 5/3* d))*sinh(1/3*b*x + 1/3*d)^2 - 10*(sqrt(3)*cosh(1/3*b*x + 1/3*d)^6*cosh(-5/ 3*a + 5/3*d) + (sqrt(3)*cosh(-5/3*a + 5/3*d) - sqrt(3)*sinh(-5/3*a + 5/3*d ))*sinh(1/3*b*x + 1/3*d)^6 + 6*(sqrt(3)*cosh(1/3*b*x + 1/3*d)*cosh(-5/3*a + 5/3*d) - sqrt(3)*cosh(1/3*b*x + 1/3*d)*sinh(-5/3*a + 5/3*d))*sinh(1/3*b* x + 1/3*d)^5 + 15*(sqrt(3)*cosh(1/3*b*x + 1/3*d)^2*cosh(-5/3*a + 5/3*d) - sqrt(3)*cosh(1/3*b*x + 1/3*d)^2*sinh(-5/3*a + 5/3*d))*sinh(1/3*b*x + 1/3*d )^4 + 20*(sqrt(3)*cosh(1/3*b*x + 1/3*d)^3*cosh(-5/3*a + 5/3*d) - sqrt(3)*c osh(1/3*b*x + 1/3*d)^3*sinh(-5/3*a + 5/3*d))*sinh(1/3*b*x + 1/3*d)^3 + 15* (sqrt(3)*cosh(1/3*b*x + 1/3*d)^4*cosh(-5/3*a + 5/3*d) - sqrt(3)*cosh(1/3*b *x + 1/3*d)^4*sinh(-5/3*a + 5/3*d))*sinh(1/3*b*x + 1/3*d)^2 + 6*(sqrt(3)*c osh(1/3*b*x + 1/3*d)^5*cosh(-5/3*a + 5/3*d) - sqrt(3)*cosh(1/3*b*x + 1/3*d )^5*sinh(-5/3*a + 5/3*d))*sinh(1/3*b*x + 1/3*d) - (sqrt(3)*cosh(1/3*b*x + 1/3*d)^6 - sqrt(3))*sinh(-5/3*a + 5/3*d) - sqrt(3)*cosh(-5/3*a + 5/3*d)...
\[ \int e^{\frac {5}{3} (a+b x)} \text {csch}^2(d+b x) \, dx=e^{\frac {5 a}{3}} \int e^{\frac {5 b x}{3}} \operatorname {csch}^{2}{\left (b x + d \right )}\, dx \] Input:
integrate(exp(5/3*b*x+5/3*a)*csch(b*x+d)**2,x)
Output:
exp(5*a/3)*Integral(exp(5*b*x/3)*csch(b*x + d)**2, x)
Time = 0.12 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.08 \[ \int e^{\frac {5}{3} (a+b x)} \text {csch}^2(d+b x) \, dx=-\frac {5 \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, e^{\left (-\frac {1}{3} \, b x - \frac {1}{3} \, d\right )} + 1\right )}\right ) e^{\left (\frac {5}{3} \, a - \frac {5}{3} \, d\right )}}{3 \, b} - \frac {5 \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, e^{\left (-\frac {1}{3} \, b x - \frac {1}{3} \, d\right )} - 1\right )}\right ) e^{\left (\frac {5}{3} \, a - \frac {5}{3} \, d\right )}}{3 \, b} - \frac {5 \, e^{\left (\frac {5}{3} \, a - \frac {5}{3} \, d\right )} \log \left (e^{\left (-\frac {1}{3} \, b x - \frac {1}{3} \, d\right )} + e^{\left (-\frac {2}{3} \, b x - \frac {2}{3} \, d\right )} + 1\right )}{6 \, b} - \frac {5 \, e^{\left (\frac {5}{3} \, a - \frac {5}{3} \, d\right )} \log \left (e^{\left (-\frac {1}{3} \, b x - \frac {1}{3} \, d\right )} + 1\right )}{3 \, b} + \frac {5 \, e^{\left (\frac {5}{3} \, a - \frac {5}{3} \, d\right )} \log \left (e^{\left (-\frac {1}{3} \, b x - \frac {1}{3} \, d\right )} - 1\right )}{3 \, b} + \frac {5 \, e^{\left (\frac {5}{3} \, a - \frac {5}{3} \, d\right )} \log \left (-e^{\left (-\frac {1}{3} \, b x - \frac {1}{3} \, d\right )} + e^{\left (-\frac {2}{3} \, b x - \frac {2}{3} \, d\right )} + 1\right )}{6 \, b} + \frac {2 \, e^{\left (-\frac {1}{3} \, b x + \frac {5}{3} \, a - 2 \, d\right )}}{b {\left (e^{\left (-2 \, b x - 2 \, d\right )} - 1\right )}} \] Input:
integrate(exp(5/3*b*x+5/3*a)*csch(b*x+d)^2,x, algorithm="maxima")
Output:
-5/3*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 - 5/3*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 - 5/6*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 - 5/3*e^(5/3*a - 5/3*d)*log(e^(-1/3*b*x - 1/3*d) + 1 )/b + 5/3*e^(5/3*a - 5/3*d)*log(e^(-1/3*b*x - 1/3*d) - 1)/b + 5/6*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 + 2*e^(- 1/3*b*x + 5/3*a - 2*d)/(b*(e^(-2*b*x - 2*d) - 1))
Time = 0.13 (sec) , antiderivative size = 193, normalized size of antiderivative = 0.95 \[ \int e^{\frac {5}{3} (a+b x)} \text {csch}^2(d+b x) \, dx=\frac {{\left (10 \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, e^{\left (\frac {1}{3} \, b x\right )} + e^{\left (-\frac {1}{3} \, d\right )}\right )} e^{\left (\frac {1}{3} \, d\right )}\right ) e^{\left (-\frac {11}{3} \, d\right )} + 10 \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, e^{\left (\frac {1}{3} \, b x\right )} - e^{\left (-\frac {1}{3} \, d\right )}\right )} e^{\left (\frac {1}{3} \, d\right )}\right ) e^{\left (-\frac {11}{3} \, d\right )} - 5 \, e^{\left (-\frac {11}{3} \, d\right )} \log \left (e^{\left (\frac {1}{3} \, b x - \frac {1}{3} \, d\right )} + e^{\left (\frac {2}{3} \, b x\right )} + e^{\left (-\frac {2}{3} \, d\right )}\right ) + 5 \, e^{\left (-\frac {11}{3} \, d\right )} \log \left (-e^{\left (\frac {1}{3} \, b x - \frac {1}{3} \, d\right )} + e^{\left (\frac {2}{3} \, b x\right )} + e^{\left (-\frac {2}{3} \, d\right )}\right ) - 10 \, e^{\left (-\frac {11}{3} \, d\right )} \log \left (e^{\left (\frac {1}{3} \, b x\right )} + e^{\left (-\frac {1}{3} \, d\right )}\right ) + 10 \, e^{\left (-\frac {11}{3} \, d\right )} \log \left ({\left | e^{\left (\frac {1}{3} \, b x\right )} - e^{\left (-\frac {1}{3} \, d\right )} \right |}\right ) - \frac {12 \, e^{\left (\frac {5}{3} \, b x - 2 \, d\right )}}{e^{\left (2 \, b x + 2 \, d\right )} - 1}\right )} e^{\left (\frac {5}{3} \, a + 2 \, d\right )}}{6 \, b} \] Input:
integrate(exp(5/3*b*x+5/3*a)*csch(b*x+d)^2,x, algorithm="giac")
Output:
1/6*(10*sqrt(3)*arctan(1/3*sqrt(3)*(2*e^(1/3*b*x) + e^(-1/3*d))*e^(1/3*d)) *e^(-11/3*d) + 10*sqrt(3)*arctan(1/3*sqrt(3)*(2*e^(1/3*b*x) - e^(-1/3*d))* e^(1/3*d))*e^(-11/3*d) - 5*e^(-11/3*d)*log(e^(1/3*b*x - 1/3*d) + e^(2/3*b* x) + e^(-2/3*d)) + 5*e^(-11/3*d)*log(-e^(1/3*b*x - 1/3*d) + e^(2/3*b*x) + e^(-2/3*d)) - 10*e^(-11/3*d)*log(e^(1/3*b*x) + e^(-1/3*d)) + 10*e^(-11/3*d )*log(abs(e^(1/3*b*x) - e^(-1/3*d))) - 12*e^(5/3*b*x - 2*d)/(e^(2*b*x + 2* d) - 1))*e^(5/3*a + 2*d)/b
Time = 6.29 (sec) , antiderivative size = 434, normalized size of antiderivative = 2.13 \[ \int e^{\frac {5}{3} (a+b x)} \text {csch}^2(d+b x) \, dx =\text {Too large to display} \] Input:
int(exp((5*a)/3 + (5*b*x)/3)/sinh(d + b*x)^2,x)
Output:
(5*exp(10*a - 10*d)^(1/6)*log((100*exp((5*a)/3)*exp(d/3)*exp(-(5*d)/3)*exp ((b*x)/3)*(exp(10*a)*exp(-10*d))^(1/6))/9 - (100*exp((10*a)/3)*exp(-(10*d) /3))/9))/(3*b) - (5*exp(10*a - 10*d)^(1/6)*log(- (100*exp((10*a)/3)*exp(-( 10*d)/3))/9 - (100*exp((5*a)/3)*exp(d/3)*exp(-(5*d)/3)*exp((b*x)/3)*(exp(1 0*a)*exp(-10*d))^(1/6))/9))/(3*b) - (2*exp((5*a)/3 + (5*b*x)/3))/(b*(exp(2 *d + 2*b*x) - 1)) - (5*log(- (100*exp((10*a)/3)*exp(-(10*d)/3))/9 - (100*e xp((5*a)/3)*exp(d/3)*exp(-(5*d)/3)*exp((b*x)/3)*((3^(1/2)*1i)/2 - 1/2)*(ex p(10*a)*exp(-10*d))^(1/6))/9)*exp(10*a - 10*d)^(1/6)*((3^(1/2)*1i)/2 - 1/2 ))/(3*b) + (5*log((100*exp((5*a)/3)*exp(d/3)*exp(-(5*d)/3)*exp((b*x)/3)*(( 3^(1/2)*1i)/2 - 1/2)*(exp(10*a)*exp(-10*d))^(1/6))/9 - (100*exp((10*a)/3)* exp(-(10*d)/3))/9)*exp(10*a - 10*d)^(1/6)*((3^(1/2)*1i)/2 - 1/2))/(3*b) - (5*log(- (100*exp((10*a)/3)*exp(-(10*d)/3))/9 - (100*exp((5*a)/3)*exp(d/3) *exp(-(5*d)/3)*exp((b*x)/3)*((3^(1/2)*1i)/2 + 1/2)*(exp(10*a)*exp(-10*d))^ (1/6))/9)*exp(10*a - 10*d)^(1/6)*((3^(1/2)*1i)/2 + 1/2))/(3*b) + (5*log((1 00*exp((5*a)/3)*exp(d/3)*exp(-(5*d)/3)*exp((b*x)/3)*((3^(1/2)*1i)/2 + 1/2) *(exp(10*a)*exp(-10*d))^(1/6))/9 - (100*exp((10*a)/3)*exp(-(10*d)/3))/9)*e xp(10*a - 10*d)^(1/6)*((3^(1/2)*1i)/2 + 1/2))/(3*b)
\[ \int e^{\frac {5}{3} (a+b x)} \text {csch}^2(d+b x) \, dx=\int e^{\frac {5 b x}{3}+\frac {5 a}{3}} \mathrm {csch}\left (b x +d \right )^{2}d x \] Input:
int(exp(5/3*b*x+5/3*a)*csch(b*x+d)^2,x)
Output:
int(e**((5*a + 5*b*x)/3)*csch(b*x + d)**2,x)