Integrand size = 21, antiderivative size = 243 \[ \int \frac {\cosh ^{\frac {4}{3}}(a+b x)}{\sinh ^{\frac {4}{3}}(a+b x)} \, dx=\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}}{\sqrt {3}}\right )}{2 b}-\frac {\sqrt {3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}}{\sqrt {3}}\right )}{2 b}+\frac {\text {arctanh}\left (\frac {\sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}\right )}{b}-\frac {\log \left (1-\frac {\sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}+\frac {\sinh ^{\frac {2}{3}}(a+b x)}{\cosh ^{\frac {2}{3}}(a+b x)}\right )}{4 b}+\frac {\log \left (1+\frac {\sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}+\frac {\sinh ^{\frac {2}{3}}(a+b x)}{\cosh ^{\frac {2}{3}}(a+b x)}\right )}{4 b}-\frac {3 \sqrt [3]{\cosh (a+b x)}}{b \sqrt [3]{\sinh (a+b x)}} \]
arctanh(sinh(b*x+a)^(1/3)/cosh(b*x+a)^(1/3))/b-1/4*ln(1-sinh(b*x+a)^(1/3)/ cosh(b*x+a)^(1/3)+sinh(b*x+a)^(2/3)/cosh(b*x+a)^(2/3))/b+1/4*ln(1+sinh(b*x +a)^(1/3)/cosh(b*x+a)^(1/3)+sinh(b*x+a)^(2/3)/cosh(b*x+a)^(2/3))/b-3*cosh( b*x+a)^(1/3)/b/sinh(b*x+a)^(1/3)+1/2*arctan(1/3*(1-2*sinh(b*x+a)^(1/3)/cos h(b*x+a)^(1/3))*3^(1/2))*3^(1/2)/b-1/2*arctan(1/3*(1+2*sinh(b*x+a)^(1/3)/c osh(b*x+a)^(1/3))*3^(1/2))*3^(1/2)/b
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.02 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.23 \[ \int \frac {\cosh ^{\frac {4}{3}}(a+b x)}{\sinh ^{\frac {4}{3}}(a+b x)} \, dx=-\frac {3 \cosh ^2(a+b x)^{5/6} \operatorname {Hypergeometric2F1}\left (-\frac {1}{6},-\frac {1}{6},\frac {5}{6},-\sinh ^2(a+b x)\right )}{b \cosh ^{\frac {5}{3}}(a+b x) \sqrt [3]{\sinh (a+b x)}} \]
(-3*(Cosh[a + b*x]^2)^(5/6)*Hypergeometric2F1[-1/6, -1/6, 5/6, -Sinh[a + b *x]^2])/(b*Cosh[a + b*x]^(5/3)*Sinh[a + b*x]^(1/3))
Time = 0.45 (sec) , antiderivative size = 241, normalized size of antiderivative = 0.99, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.619, Rules used = {3042, 3047, 3042, 3054, 25, 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.
\(\displaystyle \int \frac {\cosh ^{\frac {4}{3}}(a+b x)}{\sinh ^{\frac {4}{3}}(a+b x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\cos (i a+i b x)^{4/3}}{(-i \sin (i a+i b x))^{4/3}}dx\) |
\(\Big \downarrow \) 3047 |
\(\displaystyle \int \frac {\sinh ^{\frac {2}{3}}(a+b x)}{\cosh ^{\frac {2}{3}}(a+b x)}dx-\frac {3 \sqrt [3]{\cosh (a+b x)}}{b \sqrt [3]{\sinh (a+b x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {3 \sqrt [3]{\cosh (a+b x)}}{b \sqrt [3]{\sinh (a+b x)}}+\int \frac {(-i \sin (i a+i b x))^{2/3}}{\cos (i a+i b x)^{2/3}}dx\) |
\(\Big \downarrow \) 3054 |
\(\displaystyle -\frac {3 \int -\frac {\sinh ^{\frac {4}{3}}(a+b x)}{\cosh ^{\frac {4}{3}}(a+b x) \left (1-\tanh ^2(a+b x)\right )}d\frac {\sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}}{b}-\frac {3 \sqrt [3]{\cosh (a+b x)}}{b \sqrt [3]{\sinh (a+b x)}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {3 \int \frac {\sinh ^{\frac {4}{3}}(a+b x)}{\cosh ^{\frac {4}{3}}(a+b x) \left (1-\tanh ^2(a+b x)\right )}d\frac {\sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}}{b}-\frac {3 \sqrt [3]{\cosh (a+b x)}}{b \sqrt [3]{\sinh (a+b x)}}\) |
\(\Big \downarrow \) 825 |
\(\displaystyle -\frac {3 \left (-\frac {1}{3} \int \frac {1}{1-\frac {\sinh ^{\frac {2}{3}}(a+b x)}{\cosh ^{\frac {2}{3}}(a+b x)}}d\frac {\sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}-\frac {1}{3} \int -\frac {\frac {\sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}+1}{2 \left (\frac {\sinh ^{\frac {2}{3}}(a+b x)}{\cosh ^{\frac {2}{3}}(a+b x)}-\frac {\sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}+1\right )}d\frac {\sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}-\frac {1}{3} \int -\frac {1-\frac {\sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}}{2 \left (\frac {\sinh ^{\frac {2}{3}}(a+b x)}{\cosh ^{\frac {2}{3}}(a+b x)}+\frac {\sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}+1\right )}d\frac {\sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}\right )}{b}-\frac {3 \sqrt [3]{\cosh (a+b x)}}{b \sqrt [3]{\sinh (a+b x)}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {3 \left (-\frac {1}{3} \int \frac {1}{1-\frac {\sinh ^{\frac {2}{3}}(a+b x)}{\cosh ^{\frac {2}{3}}(a+b x)}}d\frac {\sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}+\frac {1}{6} \int \frac {\frac {\sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}+1}{\frac {\sinh ^{\frac {2}{3}}(a+b x)}{\cosh ^{\frac {2}{3}}(a+b x)}-\frac {\sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}+1}d\frac {\sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}+\frac {1}{6} \int \frac {1-\frac {\sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}}{\frac {\sinh ^{\frac {2}{3}}(a+b x)}{\cosh ^{\frac {2}{3}}(a+b x)}+\frac {\sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}+1}d\frac {\sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}\right )}{b}-\frac {3 \sqrt [3]{\cosh (a+b x)}}{b \sqrt [3]{\sinh (a+b x)}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -\frac {3 \left (\frac {1}{6} \int \frac {\frac {\sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}+1}{\frac {\sinh ^{\frac {2}{3}}(a+b x)}{\cosh ^{\frac {2}{3}}(a+b x)}-\frac {\sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}+1}d\frac {\sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}+\frac {1}{6} \int \frac {1-\frac {\sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}}{\frac {\sinh ^{\frac {2}{3}}(a+b x)}{\cosh ^{\frac {2}{3}}(a+b x)}+\frac {\sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}+1}d\frac {\sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}-\frac {1}{3} \text {arctanh}\left (\frac {\sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}\right )\right )}{b}-\frac {3 \sqrt [3]{\cosh (a+b x)}}{b \sqrt [3]{\sinh (a+b x)}}\) |
\(\Big \downarrow \) 1142 |
\(\displaystyle -\frac {3 \left (\frac {1}{6} \left (\frac {3}{2} \int \frac {1}{\frac {\sinh ^{\frac {2}{3}}(a+b x)}{\cosh ^{\frac {2}{3}}(a+b x)}-\frac {\sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}+1}d\frac {\sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}+\frac {1}{2} \int -\frac {1-\frac {2 \sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}}{\frac {\sinh ^{\frac {2}{3}}(a+b x)}{\cosh ^{\frac {2}{3}}(a+b x)}-\frac {\sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}+1}d\frac {\sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}\right )+\frac {1}{6} \left (\frac {3}{2} \int \frac {1}{\frac {\sinh ^{\frac {2}{3}}(a+b x)}{\cosh ^{\frac {2}{3}}(a+b x)}+\frac {\sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}+1}d\frac {\sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}-\frac {1}{2} \int \frac {\frac {2 \sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}+1}{\frac {\sinh ^{\frac {2}{3}}(a+b x)}{\cosh ^{\frac {2}{3}}(a+b x)}+\frac {\sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}+1}d\frac {\sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}\right )-\frac {1}{3} \text {arctanh}\left (\frac {\sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}\right )\right )}{b}-\frac {3 \sqrt [3]{\cosh (a+b x)}}{b \sqrt [3]{\sinh (a+b x)}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {3 \left (\frac {1}{6} \left (\frac {3}{2} \int \frac {1}{\frac {\sinh ^{\frac {2}{3}}(a+b x)}{\cosh ^{\frac {2}{3}}(a+b x)}-\frac {\sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}+1}d\frac {\sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}-\frac {1}{2} \int \frac {1-\frac {2 \sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}}{\frac {\sinh ^{\frac {2}{3}}(a+b x)}{\cosh ^{\frac {2}{3}}(a+b x)}-\frac {\sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}+1}d\frac {\sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}\right )+\frac {1}{6} \left (\frac {3}{2} \int \frac {1}{\frac {\sinh ^{\frac {2}{3}}(a+b x)}{\cosh ^{\frac {2}{3}}(a+b x)}+\frac {\sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}+1}d\frac {\sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}-\frac {1}{2} \int \frac {\frac {2 \sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}+1}{\frac {\sinh ^{\frac {2}{3}}(a+b x)}{\cosh ^{\frac {2}{3}}(a+b x)}+\frac {\sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}+1}d\frac {\sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}\right )-\frac {1}{3} \text {arctanh}\left (\frac {\sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}\right )\right )}{b}-\frac {3 \sqrt [3]{\cosh (a+b x)}}{b \sqrt [3]{\sinh (a+b x)}}\) |
\(\Big \downarrow \) 1083 |
\(\displaystyle -\frac {3 \left (\frac {1}{6} \left (-3 \int \frac {1}{-\frac {\sinh ^{\frac {2}{3}}(a+b x)}{\cosh ^{\frac {2}{3}}(a+b x)}-3}d\left (\frac {2 \sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}-1\right )-\frac {1}{2} \int \frac {1-\frac {2 \sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}}{\frac {\sinh ^{\frac {2}{3}}(a+b x)}{\cosh ^{\frac {2}{3}}(a+b x)}-\frac {\sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}+1}d\frac {\sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}\right )+\frac {1}{6} \left (-3 \int \frac {1}{-\frac {\sinh ^{\frac {2}{3}}(a+b x)}{\cosh ^{\frac {2}{3}}(a+b x)}-3}d\left (\frac {2 \sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}+1\right )-\frac {1}{2} \int \frac {\frac {2 \sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}+1}{\frac {\sinh ^{\frac {2}{3}}(a+b x)}{\cosh ^{\frac {2}{3}}(a+b x)}+\frac {\sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}+1}d\frac {\sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}\right )-\frac {1}{3} \text {arctanh}\left (\frac {\sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}\right )\right )}{b}-\frac {3 \sqrt [3]{\cosh (a+b x)}}{b \sqrt [3]{\sinh (a+b x)}}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle -\frac {3 \left (\frac {1}{6} \left (\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}-1}{\sqrt {3}}\right )-\frac {1}{2} \int \frac {1-\frac {2 \sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}}{\frac {\sinh ^{\frac {2}{3}}(a+b x)}{\cosh ^{\frac {2}{3}}(a+b x)}-\frac {\sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}+1}d\frac {\sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}\right )+\frac {1}{6} \left (\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}+1}{\sqrt {3}}\right )-\frac {1}{2} \int \frac {\frac {2 \sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}+1}{\frac {\sinh ^{\frac {2}{3}}(a+b x)}{\cosh ^{\frac {2}{3}}(a+b x)}+\frac {\sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}+1}d\frac {\sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}\right )-\frac {1}{3} \text {arctanh}\left (\frac {\sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}\right )\right )}{b}-\frac {3 \sqrt [3]{\cosh (a+b x)}}{b \sqrt [3]{\sinh (a+b x)}}\) |
\(\Big \downarrow \) 1103 |
\(\displaystyle -\frac {3 \left (\frac {1}{6} \left (\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}-1}{\sqrt {3}}\right )+\frac {1}{2} \log \left (\frac {\sinh ^{\frac {2}{3}}(a+b x)}{\cosh ^{\frac {2}{3}}(a+b x)}-\frac {\sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}+1\right )\right )+\frac {1}{6} \left (\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}+1}{\sqrt {3}}\right )-\frac {1}{2} \log \left (\frac {\sinh ^{\frac {2}{3}}(a+b x)}{\cosh ^{\frac {2}{3}}(a+b x)}+\frac {\sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}+1\right )\right )-\frac {1}{3} \text {arctanh}\left (\frac {\sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}\right )\right )}{b}-\frac {3 \sqrt [3]{\cosh (a+b x)}}{b \sqrt [3]{\sinh (a+b x)}}\) |
(-3*(-1/3*ArcTanh[Sinh[a + b*x]^(1/3)/Cosh[a + b*x]^(1/3)] + (Sqrt[3]*ArcT an[(-1 + (2*Sinh[a + b*x]^(1/3))/Cosh[a + b*x]^(1/3))/Sqrt[3]] + Log[1 - S inh[a + b*x]^(1/3)/Cosh[a + b*x]^(1/3) + Sinh[a + b*x]^(2/3)/Cosh[a + b*x] ^(2/3)]/2)/6 + (Sqrt[3]*ArcTan[(1 + (2*Sinh[a + b*x]^(1/3))/Cosh[a + b*x]^ (1/3))/Sqrt[3]] - Log[1 + Sinh[a + b*x]^(1/3)/Cosh[a + b*x]^(1/3) + Sinh[a + b*x]^(2/3)/Cosh[a + b*x]^(2/3)]/2)/6))/b - (3*Cosh[a + b*x]^(1/3))/(b*S inh[a + b*x]^(1/3))
3.1.64.3.1 Defintions of rubi rules used
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[(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[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n _), x_Symbol] :> Simp[a*(a*Cos[e + f*x])^(m - 1)*((b*Sin[e + f*x])^(n + 1)/ (b*f*(n + 1))), x] + Simp[a^2*((m - 1)/(b^2*(n + 1))) Int[(a*Cos[e + f*x] )^(m - 2)*(b*Sin[e + f*x])^(n + 2), x], x] /; FreeQ[{a, b, e, f}, x] && GtQ [m, 1] && LtQ[n, -1] && (IntegersQ[2*m, 2*n] || EqQ[m + n, 0])
Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m _), x_Symbol] :> With[{k = Denominator[m]}, Simp[k*a*(b/f) Subst[Int[x^(k *(m + 1) - 1)/(a^2 + b^2*x^(2*k)), x], x, (a*Sin[e + f*x])^(1/k)/(b*Cos[e + f*x])^(1/k)], x]] /; FreeQ[{a, b, e, f}, x] && EqQ[m + n, 0] && GtQ[m, 0] && LtQ[m, 1]
\[\int \frac {\cosh \left (b x +a \right )^{\frac {4}{3}}}{\sinh \left (b x +a \right )^{\frac {4}{3}}}d x\]
Leaf count of result is larger than twice the leaf count of optimal. 1013 vs. \(2 (197) = 394\).
Time = 0.28 (sec) , antiderivative size = 1013, normalized size of antiderivative = 4.17 \[ \int \frac {\cosh ^{\frac {4}{3}}(a+b x)}{\sinh ^{\frac {4}{3}}(a+b x)} \, dx=\text {Too large to display} \]
1/4*(2*(sqrt(3)*cosh(b*x + a)^2 + 2*sqrt(3)*cosh(b*x + a)*sinh(b*x + a) + sqrt(3)*sinh(b*x + a)^2 - sqrt(3))*arctan(1/3*(sqrt(3)*cosh(b*x + a)^2 + 2 *sqrt(3)*cosh(b*x + a)*sinh(b*x + a) + sqrt(3)*sinh(b*x + a)^2 + 4*(sqrt(3 )*cosh(b*x + a) + sqrt(3)*sinh(b*x + a))*cosh(b*x + a)^(1/3)*sinh(b*x + a) ^(2/3) - sqrt(3))/(cosh(b*x + a)^2 + 2*cosh(b*x + a)*sinh(b*x + a) + sinh( b*x + a)^2 - 1)) + 2*(sqrt(3)*cosh(b*x + a)^2 + 2*sqrt(3)*cosh(b*x + a)*si nh(b*x + a) + sqrt(3)*sinh(b*x + a)^2 - sqrt(3))*arctan(-1/3*(sqrt(3)*cosh (b*x + a)^2 + 2*sqrt(3)*cosh(b*x + a)*sinh(b*x + a) + sqrt(3)*sinh(b*x + a )^2 - 4*(sqrt(3)*cosh(b*x + a) + sqrt(3)*sinh(b*x + a))*cosh(b*x + a)^(1/3 )*sinh(b*x + a)^(2/3) - sqrt(3))/(cosh(b*x + a)^2 + 2*cosh(b*x + a)*sinh(b *x + a) + sinh(b*x + a)^2 - 1)) + (cosh(b*x + a)^2 + 2*cosh(b*x + a)*sinh( b*x + a) + sinh(b*x + a)^2 - 1)*log((cosh(b*x + a)^2 + 2*(cosh(b*x + a) + sinh(b*x + a))*cosh(b*x + a)^(2/3)*sinh(b*x + a)^(1/3) + 2*(cosh(b*x + a) + sinh(b*x + a))*cosh(b*x + a)^(1/3)*sinh(b*x + a)^(2/3) + 2*cosh(b*x + a) *sinh(b*x + a) + sinh(b*x + a)^2 - 1)/(cosh(b*x + a)^2 + 2*cosh(b*x + a)*s inh(b*x + a) + sinh(b*x + a)^2 - 1)) - (cosh(b*x + a)^2 + 2*cosh(b*x + a)* sinh(b*x + a) + sinh(b*x + a)^2 - 1)*log((cosh(b*x + a)^2 + 2*(cosh(b*x + a) + sinh(b*x + a))*cosh(b*x + a)^(2/3)*sinh(b*x + a)^(1/3) - 2*(cosh(b*x + a) + sinh(b*x + a))*cosh(b*x + a)^(1/3)*sinh(b*x + a)^(2/3) + 2*cosh(b*x + a)*sinh(b*x + a) + sinh(b*x + a)^2 - 1)/(cosh(b*x + a)^2 + 2*cosh(b*...
\[ \int \frac {\cosh ^{\frac {4}{3}}(a+b x)}{\sinh ^{\frac {4}{3}}(a+b x)} \, dx=\int \frac {\cosh ^{\frac {4}{3}}{\left (a + b x \right )}}{\sinh ^{\frac {4}{3}}{\left (a + b x \right )}}\, dx \]
\[ \int \frac {\cosh ^{\frac {4}{3}}(a+b x)}{\sinh ^{\frac {4}{3}}(a+b x)} \, dx=\int { \frac {\cosh \left (b x + a\right )^{\frac {4}{3}}}{\sinh \left (b x + a\right )^{\frac {4}{3}}} \,d x } \]
\[ \int \frac {\cosh ^{\frac {4}{3}}(a+b x)}{\sinh ^{\frac {4}{3}}(a+b x)} \, dx=\int { \frac {\cosh \left (b x + a\right )^{\frac {4}{3}}}{\sinh \left (b x + a\right )^{\frac {4}{3}}} \,d x } \]
Timed out. \[ \int \frac {\cosh ^{\frac {4}{3}}(a+b x)}{\sinh ^{\frac {4}{3}}(a+b x)} \, dx=\int \frac {{\mathrm {cosh}\left (a+b\,x\right )}^{4/3}}{{\mathrm {sinh}\left (a+b\,x\right )}^{4/3}} \,d x \]