Integrand size = 31, antiderivative size = 341 \[ \int \frac {e^{\frac {5}{3} (a+b x)}}{(g \cosh (d+b x)+f \sinh (d+b x))^3} \, dx=-\frac {2 e^{\frac {5 (a-d)}{3}+\frac {8}{3} (d+b x)}}{b (f+g) \left (f-g-e^{2 (d+b x)} (f+g)\right )^2}+\frac {8 e^{\frac {5 (a-d)}{3}+\frac {2}{3} (d+b x)}}{3 b (f+g)^2 \left (f-g-e^{2 (d+b x)} (f+g)\right )}-\frac {8 e^{\frac {5 (a-d)}{3}} \arctan \left (\frac {1+\frac {2 e^{\frac {2}{3} (d+b x)} \sqrt [3]{f+g}}{\sqrt [3]{f-g}}}{\sqrt {3}}\right )}{3 \sqrt {3} b (f-g)^{2/3} (f+g)^{7/3}}+\frac {8 e^{\frac {5 (a-d)}{3}} \log \left (\sqrt [3]{f-g}-e^{\frac {2}{3} (d+b x)} \sqrt [3]{f+g}\right )}{9 b (f-g)^{2/3} (f+g)^{7/3}}-\frac {4 e^{\frac {5 (a-d)}{3}} \log \left ((f-g)^{2/3}+e^{\frac {2}{3} (d+b x)} \sqrt [3]{f-g} \sqrt [3]{f+g}+e^{\frac {4}{3} (d+b x)} (f+g)^{2/3}\right )}{9 b (f-g)^{2/3} (f+g)^{7/3}} \] Output:
-2*exp(5/3*a+d+8/3*b*x)/b/(f+g)/(f-g-exp(2*b*x+2*d)*(f+g))^2+8/3*exp(5/3*a -d+2/3*b*x)/b/(f+g)^2/(f-g-exp(2*b*x+2*d)*(f+g))-8/9*exp(5/3*a-5/3*d)*arct an(1/3*(1+2*exp(2/3*b*x+2/3*d)*(f+g)^(1/3)/(f-g)^(1/3))*3^(1/2))*3^(1/2)/b /(f-g)^(2/3)/(f+g)^(7/3)+8/9*exp(5/3*a-5/3*d)*ln((f-g)^(1/3)-exp(2/3*b*x+2 /3*d)*(f+g)^(1/3))/b/(f-g)^(2/3)/(f+g)^(7/3)-4/9*exp(5/3*a-5/3*d)*ln((f-g) ^(2/3)+exp(2/3*b*x+2/3*d)*(f-g)^(1/3)*(f+g)^(1/3)+exp(4/3*b*x+4/3*d)*(f+g) ^(2/3))/b/(f-g)^(2/3)/(f+g)^(7/3)
Time = 1.19 (sec) , antiderivative size = 343, normalized size of antiderivative = 1.01 \[ \int \frac {e^{\frac {5}{3} (a+b x)}}{(g \cosh (d+b x)+f \sinh (d+b x))^3} \, dx=\frac {12 e^{\frac {5 (a-d)}{3}} \left (\frac {e^{\frac {2}{3} (d+b x)}}{9 (f+g)^2 \left (\left (-1+e^{2 (d+b x)}\right ) f+\left (1+e^{2 (d+b x)}\right ) g\right )}+\frac {e^{\frac {2}{3} (d+b x)} (f-g)}{3 (f+g)^2 \left (-f+g+e^{2 (d+b x)} (f+g)\right )^2}-\frac {e^{\frac {8}{3} (d+b x)}}{2 (f+g) \left (-f+g+e^{2 (d+b x)} (f+g)\right )^2}-\frac {2 \arctan \left (\frac {1+\frac {2 e^{\frac {2}{3} (d+b x)} \sqrt [3]{f+g}}{\sqrt [3]{f-g}}}{\sqrt {3}}\right )}{9 \sqrt {3} (f-g)^{2/3} (f+g)^{7/3}}+\frac {2 \log \left (\sqrt [3]{f-g}-e^{\frac {2}{3} (d+b x)} \sqrt [3]{f+g}\right )}{27 (f-g)^{2/3} (f+g)^{7/3}}-\frac {\log \left ((f-g)^{2/3}+e^{\frac {2}{3} (d+b x)} \sqrt [3]{f-g} \sqrt [3]{f+g}+e^{\frac {4}{3} (d+b x)} (f+g)^{2/3}\right )}{27 (f-g)^{2/3} (f+g)^{7/3}}\right )}{b} \] Input:
Integrate[E^((5*(a + b*x))/3)/(g*Cosh[d + b*x] + f*Sinh[d + b*x])^3,x]
Output:
(12*E^((5*(a - d))/3)*(E^((2*(d + b*x))/3)/(9*(f + g)^2*((-1 + E^(2*(d + b *x)))*f + (1 + E^(2*(d + b*x)))*g)) + (E^((2*(d + b*x))/3)*(f - g))/(3*(f + g)^2*(-f + g + E^(2*(d + b*x))*(f + g))^2) - E^((8*(d + b*x))/3)/(2*(f + g)*(-f + g + E^(2*(d + b*x))*(f + g))^2) - (2*ArcTan[(1 + (2*E^((2*(d + b *x))/3)*(f + g)^(1/3))/(f - g)^(1/3))/Sqrt[3]])/(9*Sqrt[3]*(f - g)^(2/3)*( f + g)^(7/3)) + (2*Log[(f - g)^(1/3) - E^((2*(d + b*x))/3)*(f + g)^(1/3)]) /(27*(f - g)^(2/3)*(f + g)^(7/3)) - Log[(f - g)^(2/3) + E^((2*(d + b*x))/3 )*(f - g)^(1/3)*(f + g)^(1/3) + E^((4*(d + b*x))/3)*(f + g)^(2/3)]/(27*(f - g)^(2/3)*(f + g)^(7/3))))/b
Time = 0.54 (sec) , antiderivative size = 274, normalized size of antiderivative = 0.80, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.387, Rules used = {2720, 27, 807, 817, 817, 750, 16, 1142, 27, 1082, 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.
| \(\displaystyle \int \frac {e^{\frac {5}{3} (a+b x)}}{(f \sinh (b x+d)+g \cosh (b x+d))^3} \, dx\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle \frac {3 \int -\frac {8 e^{\frac {5 a}{3}+\frac {13 b x}{3}}}{\left (f-g-e^{2 b x} (f+g)\right )^3}de^{\frac {b x}{3}}}{b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {24 e^{5 a/3} \int \frac {e^{\frac {13 b x}{3}}}{\left (f-g-e^{2 b x} (f+g)\right )^3}de^{\frac {b x}{3}}}{b}\) |
| \(\Big \downarrow \) 807 |
| \(\displaystyle -\frac {12 e^{5 a/3} \int \frac {e^{2 b x}}{\left (f-g-e^{b x} (f+g)\right )^3}de^{\frac {2 b x}{3}}}{b}\) |
| \(\Big \downarrow \) 817 |
| \(\displaystyle -\frac {12 e^{5 a/3} \left (\frac {e^{\frac {4 b x}{3}}}{6 (f+g) \left (-e^{b x} (f+g)+f-g\right )^2}-\frac {2 \int \frac {e^{b x}}{\left (f-g-e^{b x} (f+g)\right )^2}de^{\frac {2 b x}{3}}}{3 (f+g)}\right )}{b}\) |
| \(\Big \downarrow \) 817 |
| \(\displaystyle -\frac {12 e^{5 a/3} \left (\frac {e^{\frac {4 b x}{3}}}{6 (f+g) \left (-e^{b x} (f+g)+f-g\right )^2}-\frac {2 \left (\frac {e^{\frac {2 b x}{3}}}{3 (f+g) \left (-e^{b x} (f+g)+f-g\right )}-\frac {\int \frac {1}{f-g-e^{b x} (f+g)}de^{\frac {2 b x}{3}}}{3 (f+g)}\right )}{3 (f+g)}\right )}{b}\) |
| \(\Big \downarrow \) 750 |
| \(\displaystyle -\frac {12 e^{5 a/3} \left (\frac {e^{\frac {4 b x}{3}}}{6 (f+g) \left (-e^{b x} (f+g)+f-g\right )^2}-\frac {2 \left (\frac {e^{\frac {2 b x}{3}}}{3 (f+g) \left (-e^{b x} (f+g)+f-g\right )}-\frac {\frac {\int \frac {1}{\sqrt [3]{f-g}-e^{\frac {2 b x}{3}} \sqrt [3]{f+g}}de^{\frac {2 b x}{3}}}{3 (f-g)^{2/3}}+\frac {\int \frac {2 \sqrt [3]{f-g}+e^{\frac {2 b x}{3}} \sqrt [3]{f+g}}{(f-g)^{2/3}+e^{\frac {2 b x}{3}} \sqrt [3]{f+g} \sqrt [3]{f-g}+e^{\frac {2 b x}{3}} (f+g)^{2/3}}de^{\frac {2 b x}{3}}}{3 (f-g)^{2/3}}}{3 (f+g)}\right )}{3 (f+g)}\right )}{b}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle -\frac {12 e^{5 a/3} \left (\frac {e^{\frac {4 b x}{3}}}{6 (f+g) \left (-e^{b x} (f+g)+f-g\right )^2}-\frac {2 \left (\frac {e^{\frac {2 b x}{3}}}{3 (f+g) \left (-e^{b x} (f+g)+f-g\right )}-\frac {\frac {\int \frac {2 \sqrt [3]{f-g}+e^{\frac {2 b x}{3}} \sqrt [3]{f+g}}{(f-g)^{2/3}+e^{\frac {2 b x}{3}} \sqrt [3]{f+g} \sqrt [3]{f-g}+e^{\frac {2 b x}{3}} (f+g)^{2/3}}de^{\frac {2 b x}{3}}}{3 (f-g)^{2/3}}-\frac {\log \left (\sqrt [3]{f-g}-e^{\frac {2 b x}{3}} \sqrt [3]{f+g}\right )}{3 (f-g)^{2/3} \sqrt [3]{f+g}}}{3 (f+g)}\right )}{3 (f+g)}\right )}{b}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle -\frac {12 e^{5 a/3} \left (\frac {e^{\frac {4 b x}{3}}}{6 (f+g) \left (-e^{b x} (f+g)+f-g\right )^2}-\frac {2 \left (\frac {e^{\frac {2 b x}{3}}}{3 (f+g) \left (-e^{b x} (f+g)+f-g\right )}-\frac {\frac {\frac {3}{2} \sqrt [3]{f-g} \int \frac {1}{(f-g)^{2/3}+e^{\frac {2 b x}{3}} \sqrt [3]{f+g} \sqrt [3]{f-g}+e^{\frac {2 b x}{3}} (f+g)^{2/3}}de^{\frac {2 b x}{3}}+\frac {\int \frac {\sqrt [3]{f+g} \left (\sqrt [3]{f-g}+2 e^{\frac {2 b x}{3}} \sqrt [3]{f+g}\right )}{(f-g)^{2/3}+e^{\frac {2 b x}{3}} \sqrt [3]{f+g} \sqrt [3]{f-g}+e^{\frac {2 b x}{3}} (f+g)^{2/3}}de^{\frac {2 b x}{3}}}{2 \sqrt [3]{f+g}}}{3 (f-g)^{2/3}}-\frac {\log \left (\sqrt [3]{f-g}-e^{\frac {2 b x}{3}} \sqrt [3]{f+g}\right )}{3 (f-g)^{2/3} \sqrt [3]{f+g}}}{3 (f+g)}\right )}{3 (f+g)}\right )}{b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {12 e^{5 a/3} \left (\frac {e^{\frac {4 b x}{3}}}{6 (f+g) \left (-e^{b x} (f+g)+f-g\right )^2}-\frac {2 \left (\frac {e^{\frac {2 b x}{3}}}{3 (f+g) \left (-e^{b x} (f+g)+f-g\right )}-\frac {\frac {\frac {3}{2} \sqrt [3]{f-g} \int \frac {1}{(f-g)^{2/3}+e^{\frac {2 b x}{3}} \sqrt [3]{f+g} \sqrt [3]{f-g}+e^{\frac {2 b x}{3}} (f+g)^{2/3}}de^{\frac {2 b x}{3}}+\frac {1}{2} \int \frac {\sqrt [3]{f-g}+2 e^{\frac {2 b x}{3}} \sqrt [3]{f+g}}{(f-g)^{2/3}+e^{\frac {2 b x}{3}} \sqrt [3]{f+g} \sqrt [3]{f-g}+e^{\frac {2 b x}{3}} (f+g)^{2/3}}de^{\frac {2 b x}{3}}}{3 (f-g)^{2/3}}-\frac {\log \left (\sqrt [3]{f-g}-e^{\frac {2 b x}{3}} \sqrt [3]{f+g}\right )}{3 (f-g)^{2/3} \sqrt [3]{f+g}}}{3 (f+g)}\right )}{3 (f+g)}\right )}{b}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle -\frac {12 e^{5 a/3} \left (\frac {e^{\frac {4 b x}{3}}}{6 (f+g) \left (-e^{b x} (f+g)+f-g\right )^2}-\frac {2 \left (\frac {e^{\frac {2 b x}{3}}}{3 (f+g) \left (-e^{b x} (f+g)+f-g\right )}-\frac {\frac {\frac {1}{2} \int \frac {\sqrt [3]{f-g}+2 e^{\frac {2 b x}{3}} \sqrt [3]{f+g}}{(f-g)^{2/3}+e^{\frac {2 b x}{3}} \sqrt [3]{f+g} \sqrt [3]{f-g}+e^{\frac {2 b x}{3}} (f+g)^{2/3}}de^{\frac {2 b x}{3}}-\frac {3 \int \frac {1}{-\frac {2 e^{\frac {2 b x}{3}} \sqrt [3]{f+g}}{\sqrt [3]{f-g}}-4}d\left (\frac {2 e^{\frac {2 b x}{3}} \sqrt [3]{f+g}}{\sqrt [3]{f-g}}+1\right )}{\sqrt [3]{f+g}}}{3 (f-g)^{2/3}}-\frac {\log \left (\sqrt [3]{f-g}-e^{\frac {2 b x}{3}} \sqrt [3]{f+g}\right )}{3 (f-g)^{2/3} \sqrt [3]{f+g}}}{3 (f+g)}\right )}{3 (f+g)}\right )}{b}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {12 e^{5 a/3} \left (\frac {e^{\frac {4 b x}{3}}}{6 (f+g) \left (-e^{b x} (f+g)+f-g\right )^2}-\frac {2 \left (\frac {e^{\frac {2 b x}{3}}}{3 (f+g) \left (-e^{b x} (f+g)+f-g\right )}-\frac {\frac {\frac {1}{2} \int \frac {\sqrt [3]{f-g}+2 e^{\frac {2 b x}{3}} \sqrt [3]{f+g}}{(f-g)^{2/3}+e^{\frac {2 b x}{3}} \sqrt [3]{f+g} \sqrt [3]{f-g}+e^{\frac {2 b x}{3}} (f+g)^{2/3}}de^{\frac {2 b x}{3}}+\frac {\sqrt {3} \arctan \left (\frac {\frac {2 e^{\frac {2 b x}{3}} \sqrt [3]{f+g}}{\sqrt [3]{f-g}}+1}{\sqrt {3}}\right )}{\sqrt [3]{f+g}}}{3 (f-g)^{2/3}}-\frac {\log \left (\sqrt [3]{f-g}-e^{\frac {2 b x}{3}} \sqrt [3]{f+g}\right )}{3 (f-g)^{2/3} \sqrt [3]{f+g}}}{3 (f+g)}\right )}{3 (f+g)}\right )}{b}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle -\frac {12 e^{5 a/3} \left (\frac {e^{\frac {4 b x}{3}}}{6 (f+g) \left (-e^{b x} (f+g)+f-g\right )^2}-\frac {2 \left (\frac {e^{\frac {2 b x}{3}}}{3 (f+g) \left (-e^{b x} (f+g)+f-g\right )}-\frac {\frac {\frac {\sqrt {3} \arctan \left (\frac {\frac {2 e^{\frac {2 b x}{3}} \sqrt [3]{f+g}}{\sqrt [3]{f-g}}+1}{\sqrt {3}}\right )}{\sqrt [3]{f+g}}+\frac {\log \left (e^{\frac {2 b x}{3}} \sqrt [3]{f+g} \sqrt [3]{f-g}+e^{\frac {2 b x}{3}} (f+g)^{2/3}+(f-g)^{2/3}\right )}{2 \sqrt [3]{f+g}}}{3 (f-g)^{2/3}}-\frac {\log \left (\sqrt [3]{f-g}-e^{\frac {2 b x}{3}} \sqrt [3]{f+g}\right )}{3 (f-g)^{2/3} \sqrt [3]{f+g}}}{3 (f+g)}\right )}{3 (f+g)}\right )}{b}\) |
Input:
Int[E^((5*(a + b*x))/3)/(g*Cosh[d + b*x] + f*Sinh[d + b*x])^3,x]
Output:
(-12*E^((5*a)/3)*(E^((4*b*x)/3)/(6*(f + g)*(f - g - E^(b*x)*(f + g))^2) - (2*(E^((2*b*x)/3)/(3*(f + g)*(f - g - E^(b*x)*(f + g))) - (-1/3*Log[(f - g )^(1/3) - E^((2*b*x)/3)*(f + g)^(1/3)]/((f - g)^(2/3)*(f + g)^(1/3)) + ((S qrt[3]*ArcTan[(1 + (2*E^((2*b*x)/3)*(f + g)^(1/3))/(f - g)^(1/3))/Sqrt[3]] )/(f + g)^(1/3) + Log[(f - g)^(2/3) + E^((2*b*x)/3)*(f - g)^(1/3)*(f + g)^ (1/3) + E^((2*b*x)/3)*(f + g)^(2/3)]/(2*(f + g)^(1/3)))/(3*(f - g)^(2/3))) /(3*(f + g))))/(3*(f + g))))/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_)^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] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{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 higher order function than in optimal. Order 9 vs. order 3.
Time = 2.23 (sec) , antiderivative size = 224, normalized size of antiderivative = 0.66
\[-\frac {2 \left (7 f \,{\mathrm e}^{2 b x +2 d}+7 \,{\mathrm e}^{2 b x +2 d} g -4 f +4 g \right ) {\mathrm e}^{\frac {5 a}{3}-d +\frac {2 b x}{3}}}{3 \left (f \,{\mathrm e}^{2 b x +2 d}+{\mathrm e}^{2 b x +2 d} g -f +g \right )^{2} \left (f +g \right )^{2} b}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (-512+\left (729 b^{3} f^{9}+3645 b^{3} f^{8} g +5832 b^{3} f^{7} g^{2}-10206 b^{3} f^{5} g^{4}-10206 b^{3} f^{4} g^{5}+5832 b^{3} f^{2} g^{7}+3645 b^{3} f \,g^{8}+729 b^{3} g^{9}\right ) \textit {\_Z}^{3}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{\frac {2 b x}{3}+\frac {2 d}{3}}+\left (-\frac {9}{8} b \,f^{3}-\frac {9}{8} b \,f^{2} g +\frac {9}{8} b f \,g^{2}+\frac {9}{8} b \,g^{3}\right ) \textit {\_R} \right )\right ) {\mathrm e}^{\frac {5 a}{3}-\frac {5 d}{3}}\]
Input:
int(exp(5/3*b*x+5/3*a)/(g*cosh(b*x+d)+f*sinh(b*x+d))^3,x)
Output:
-2/3/(f*exp(2*b*x+2*d)+exp(2*b*x+2*d)*g-f+g)^2/(f+g)^2/b*(7*f*exp(2*b*x+2* d)+7*exp(2*b*x+2*d)*g-4*f+4*g)*exp(5/3*a-d+2/3*b*x)+sum(_R*ln(exp(2/3*b*x+ 2/3*d)+(-9/8*b*f^3-9/8*b*f^2*g+9/8*b*f*g^2+9/8*b*g^3)*_R),_R=RootOf(-512+( 729*b^3*f^9+3645*b^3*f^8*g+5832*b^3*f^7*g^2-10206*b^3*f^5*g^4-10206*b^3*f^ 4*g^5+5832*b^3*f^2*g^7+3645*b^3*f*g^8+729*b^3*g^9)*_Z^3))*exp(5/3*a-5/3*d)
Leaf count of result is larger than twice the leaf count of optimal. 7534 vs. \(2 (263) = 526\).
Time = 0.44 (sec) , antiderivative size = 15668, normalized size of antiderivative = 45.95 \[ \int \frac {e^{\frac {5}{3} (a+b x)}}{(g \cosh (d+b x)+f \sinh (d+b x))^3} \, dx=\text {Too large to display} \] Input:
integrate(exp(5/3*b*x+5/3*a)/(g*cosh(b*x+d)+f*sinh(b*x+d))^3,x, algorithm= "fricas")
Output:
Too large to include
\[ \int \frac {e^{\frac {5}{3} (a+b x)}}{(g \cosh (d+b x)+f \sinh (d+b x))^3} \, dx=e^{\frac {5 a}{3}} \int \frac {e^{\frac {5 b x}{3}}}{f^{3} \sinh ^{3}{\left (b x + d \right )} + 3 f^{2} g \sinh ^{2}{\left (b x + d \right )} \cosh {\left (b x + d \right )} + 3 f g^{2} \sinh {\left (b x + d \right )} \cosh ^{2}{\left (b x + d \right )} + g^{3} \cosh ^{3}{\left (b x + d \right )}}\, dx \] Input:
integrate(exp(5/3*b*x+5/3*a)/(g*cosh(b*x+d)+f*sinh(b*x+d))**3,x)
Output:
exp(5*a/3)*Integral(exp(5*b*x/3)/(f**3*sinh(b*x + d)**3 + 3*f**2*g*sinh(b* x + d)**2*cosh(b*x + d) + 3*f*g**2*sinh(b*x + d)*cosh(b*x + d)**2 + g**3*c osh(b*x + d)**3), x)
Exception generated. \[ \int \frac {e^{\frac {5}{3} (a+b x)}}{(g \cosh (d+b x)+f \sinh (d+b x))^3} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(exp(5/3*b*x+5/3*a)/(g*cosh(b*x+d)+f*sinh(b*x+d))^3,x, algorithm= "maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(g-f>0)', see `assume?` for more details)Is
Time = 0.14 (sec) , antiderivative size = 487, normalized size of antiderivative = 1.43 \[ \int \frac {e^{\frac {5}{3} (a+b x)}}{(g \cosh (d+b x)+f \sinh (d+b x))^3} \, dx =\text {Too large to display} \] Input:
integrate(exp(5/3*b*x+5/3*a)/(g*cosh(b*x+d)+f*sinh(b*x+d))^3,x, algorithm= "giac")
Output:
-2/9*(12*(f^3*e^d + f^2*g*e^d - f*g^2*e^d - g^3*e^d)^(1/3)*arctan(1/3*sqrt (3)*(((f - g)/(f*e^(2*d) + g*e^(2*d)))^(1/3) + 2*e^(2/3*b*x))/((f - g)/(f* e^(2*d) + g*e^(2*d)))^(1/3))/(sqrt(3)*f^4*e^(5*d) + 2*sqrt(3)*f^3*g*e^(5*d ) - 2*sqrt(3)*f*g^3*e^(5*d) - sqrt(3)*g^4*e^(5*d)) + 2*(f^3*e^d + f^2*g*e^ d - f*g^2*e^d - g^3*e^d)^(1/3)*log(((f - g)/(f*e^(2*d) + g*e^(2*d)))^(1/3) *e^(2/3*b*x) + ((f - g)/(f*e^(2*d) + g*e^(2*d)))^(2/3) + e^(4/3*b*x))/(f^4 *e^(5*d) + 2*f^3*g*e^(5*d) - 2*f*g^3*e^(5*d) - g^4*e^(5*d)) - 4*((f - g)/( f*e^(2*d) + g*e^(2*d)))^(1/3)*log(abs(-((f - g)/(f*e^(2*d) + g*e^(2*d)))^( 1/3) + e^(2/3*b*x)))/(f^3*e^(4*d) + f^2*g*e^(4*d) - f*g^2*e^(4*d) - g^3*e^ (4*d)) + 3*(7*f*e^(8/3*b*x + 2*d) + 7*g*e^(8/3*b*x + 2*d) - 4*f*e^(2/3*b*x ) + 4*g*e^(2/3*b*x))/((f^2*e^(4*d) + 2*f*g*e^(4*d) + g^2*e^(4*d))*(f*e^(2* b*x + 2*d) + g*e^(2*b*x + 2*d) - f + g)^2))*e^(5/3*a + 3*d)/b
Time = 8.02 (sec) , antiderivative size = 623, normalized size of antiderivative = 1.83 \[ \int \frac {e^{\frac {5}{3} (a+b x)}}{(g \cosh (d+b x)+f \sinh (d+b x))^3} \, dx =\text {Too large to display} \] Input:
int(exp((5*a)/3 + (5*b*x)/3)/(g*cosh(d + b*x) + f*sinh(d + b*x))^3,x)
Output:
(8*log(g^3*exp(5*a - 5*d)^(4/3) - f^3*exp(5*a - 5*d)^(4/3) + f*g^2*exp(5*a - 5*d)^(4/3) - f^2*g*exp(5*a - 5*d)^(4/3) + exp((20*a)/3 - 6*d + (2*b*x)/ 3)*(f + g)^(7/3)*(f - g)^(2/3))*exp(5*a - 5*d)^(1/3))/(9*b*(f + g)^(7/3)*( f - g)^(2/3)) - (8*exp((5*a)/3 - d + (2*b*x)/3))/(3*b*(f + g)^2*(g - f + e xp(2*d + 2*b*x)*(f + g))) - (2*exp((5*a)/3 + d + (8*b*x)/3))/(b*(f + g)*(e xp(4*d + 4*b*x)*(f + g)^2 + (f - g)^2 - 2*exp(2*d + 2*b*x)*(f + g)*(f - g) )) + (8*exp(5*a - 5*d)^(1/3)*log(f^3*exp(5*a - 5*d)^(4/3) - g^3*exp(5*a - 5*d)^(4/3) - 3^(1/2)*f^3*exp(5*a - 5*d)^(4/3)*1i + 3^(1/2)*g^3*exp(5*a - 5 *d)^(4/3)*1i - f*g^2*exp(5*a - 5*d)^(4/3) + f^2*g*exp(5*a - 5*d)^(4/3) + 2 *exp((20*a)/3 - 6*d + (2*b*x)/3)*(f + g)^(7/3)*(f - g)^(2/3) + 3^(1/2)*f*g ^2*exp(5*a - 5*d)^(4/3)*1i - 3^(1/2)*f^2*g*exp(5*a - 5*d)^(4/3)*1i)*((3^(1 /2)*1i)/2 - 1/2))/(9*b*(f + g)^(7/3)*(f - g)^(2/3)) - (8*exp(5*a - 5*d)^(1 /3)*log(f^3*exp(5*a - 5*d)^(4/3) - g^3*exp(5*a - 5*d)^(4/3) + 3^(1/2)*f^3* exp(5*a - 5*d)^(4/3)*1i - 3^(1/2)*g^3*exp(5*a - 5*d)^(4/3)*1i - f*g^2*exp( 5*a - 5*d)^(4/3) + f^2*g*exp(5*a - 5*d)^(4/3) + 2*exp((20*a)/3 - 6*d + (2* b*x)/3)*(f + g)^(7/3)*(f - g)^(2/3) - 3^(1/2)*f*g^2*exp(5*a - 5*d)^(4/3)*1 i + 3^(1/2)*f^2*g*exp(5*a - 5*d)^(4/3)*1i)*((3^(1/2)*1i)/2 + 1/2))/(9*b*(f + g)^(7/3)*(f - g)^(2/3))
\[ \int \frac {e^{\frac {5}{3} (a+b x)}}{(g \cosh (d+b x)+f \sinh (d+b x))^3} \, dx=\int \frac {e^{\frac {5 b x}{3}+\frac {5 a}{3}}}{\cosh \left (b x +d \right )^{3} g^{3}+3 \cosh \left (b x +d \right )^{2} \sinh \left (b x +d \right ) f \,g^{2}+3 \cosh \left (b x +d \right ) \sinh \left (b x +d \right )^{2} f^{2} g +\sinh \left (b x +d \right )^{3} f^{3}}d x \] Input:
int(exp(5/3*b*x+5/3*a)/(g*cosh(b*x+d)+f*sinh(b*x+d))^3,x)
Output:
int(e**((5*a + 5*b*x)/3)/(cosh(b*x + d)**3*g**3 + 3*cosh(b*x + d)**2*sinh( b*x + d)*f*g**2 + 3*cosh(b*x + d)*sinh(b*x + d)**2*f**2*g + sinh(b*x + d)* *3*f**3),x)