Integrand size = 21, antiderivative size = 243 \[ \int \frac {\sinh ^{\frac {4}{3}}(a+b x)}{\cosh ^{\frac {4}{3}}(a+b x)} \, dx=\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}}{\sqrt {3}}\right )}{2 b}-\frac {\sqrt {3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}}{\sqrt {3}}\right )}{2 b}+\frac {\text {arctanh}\left (\frac {\sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}\right )}{b}-\frac {\log \left (1+\frac {\cosh ^{\frac {2}{3}}(a+b x)}{\sinh ^{\frac {2}{3}}(a+b x)}-\frac {\sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}\right )}{4 b}+\frac {\log \left (1+\frac {\cosh ^{\frac {2}{3}}(a+b x)}{\sinh ^{\frac {2}{3}}(a+b x)}+\frac {\sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}\right )}{4 b}-\frac {3 \sqrt [3]{\sinh (a+b x)}}{b \sqrt [3]{\cosh (a+b x)}} \]
arctanh(cosh(b*x+a)^(1/3)/sinh(b*x+a)^(1/3))/b-1/4*ln(1+cosh(b*x+a)^(2/3)/ sinh(b*x+a)^(2/3)-cosh(b*x+a)^(1/3)/sinh(b*x+a)^(1/3))/b+1/4*ln(1+cosh(b*x +a)^(2/3)/sinh(b*x+a)^(2/3)+cosh(b*x+a)^(1/3)/sinh(b*x+a)^(1/3))/b-3*sinh( b*x+a)^(1/3)/b/cosh(b*x+a)^(1/3)+1/2*arctan(1/3*(1-2*cosh(b*x+a)^(1/3)/sin h(b*x+a)^(1/3))*3^(1/2))*3^(1/2)/b-1/2*arctan(1/3*(1+2*cosh(b*x+a)^(1/3)/s inh(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 = 0.04 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.24 \[ \int \frac {\sinh ^{\frac {4}{3}}(a+b x)}{\cosh ^{\frac {4}{3}}(a+b x)} \, dx=\frac {3 \sqrt [6]{\cosh ^2(a+b x)} \operatorname {Hypergeometric2F1}\left (\frac {7}{6},\frac {7}{6},\frac {13}{6},-\sinh ^2(a+b x)\right ) \sinh ^{\frac {7}{3}}(a+b x)}{7 b \sqrt [3]{\cosh (a+b x)}} \]
(3*(Cosh[a + b*x]^2)^(1/6)*Hypergeometric2F1[7/6, 7/6, 13/6, -Sinh[a + b*x ]^2]*Sinh[a + b*x]^(7/3))/(7*b*Cosh[a + b*x]^(1/3))
Time = 0.45 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {3042, 3046, 3042, 3055, 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.
| \(\displaystyle \int \frac {\sinh ^{\frac {4}{3}}(a+b x)}{\cosh ^{\frac {4}{3}}(a+b x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(-i \sin (i a+i b x))^{4/3}}{\cos (i a+i b x)^{4/3}}dx\) |
| \(\Big \downarrow \) 3046 |
| \(\displaystyle \int \frac {\cosh ^{\frac {2}{3}}(a+b x)}{\sinh ^{\frac {2}{3}}(a+b x)}dx-\frac {3 \sqrt [3]{\sinh (a+b x)}}{b \sqrt [3]{\cosh (a+b x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {3 \sqrt [3]{\sinh (a+b x)}}{b \sqrt [3]{\cosh (a+b x)}}+\int \frac {\cos (i a+i b x)^{2/3}}{(-i \sin (i a+i b x))^{2/3}}dx\) |
| \(\Big \downarrow \) 3055 |
| \(\displaystyle \frac {3 \int \frac {\cosh ^{\frac {4}{3}}(a+b x)}{\left (1-\coth ^2(a+b x)\right ) \sinh ^{\frac {4}{3}}(a+b x)}d\frac {\sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}}{b}-\frac {3 \sqrt [3]{\sinh (a+b x)}}{b \sqrt [3]{\cosh (a+b x)}}\) |
| \(\Big \downarrow \) 825 |
| \(\displaystyle \frac {3 \left (\frac {1}{3} \int \frac {1}{1-\frac {\cosh ^{\frac {2}{3}}(a+b x)}{\sinh ^{\frac {2}{3}}(a+b x)}}d\frac {\sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}+\frac {1}{3} \int -\frac {\frac {\sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}+1}{2 \left (\frac {\cosh ^{\frac {2}{3}}(a+b x)}{\sinh ^{\frac {2}{3}}(a+b x)}-\frac {\sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}+1\right )}d\frac {\sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}+\frac {1}{3} \int -\frac {1-\frac {\sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}}{2 \left (\frac {\cosh ^{\frac {2}{3}}(a+b x)}{\sinh ^{\frac {2}{3}}(a+b x)}+\frac {\sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}+1\right )}d\frac {\sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}\right )}{b}-\frac {3 \sqrt [3]{\sinh (a+b x)}}{b \sqrt [3]{\cosh (a+b x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3 \left (\frac {1}{3} \int \frac {1}{1-\frac {\cosh ^{\frac {2}{3}}(a+b x)}{\sinh ^{\frac {2}{3}}(a+b x)}}d\frac {\sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}-\frac {1}{6} \int \frac {\frac {\sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}+1}{\frac {\cosh ^{\frac {2}{3}}(a+b x)}{\sinh ^{\frac {2}{3}}(a+b x)}-\frac {\sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}+1}d\frac {\sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}-\frac {1}{6} \int \frac {1-\frac {\sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}}{\frac {\cosh ^{\frac {2}{3}}(a+b x)}{\sinh ^{\frac {2}{3}}(a+b x)}+\frac {\sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}+1}d\frac {\sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}\right )}{b}-\frac {3 \sqrt [3]{\sinh (a+b x)}}{b \sqrt [3]{\cosh (a+b x)}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {3 \left (-\frac {1}{6} \int \frac {\frac {\sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}+1}{\frac {\cosh ^{\frac {2}{3}}(a+b x)}{\sinh ^{\frac {2}{3}}(a+b x)}-\frac {\sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}+1}d\frac {\sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}-\frac {1}{6} \int \frac {1-\frac {\sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}}{\frac {\cosh ^{\frac {2}{3}}(a+b x)}{\sinh ^{\frac {2}{3}}(a+b x)}+\frac {\sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}+1}d\frac {\sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}+\frac {1}{3} \text {arctanh}\left (\frac {\sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}\right )\right )}{b}-\frac {3 \sqrt [3]{\sinh (a+b x)}}{b \sqrt [3]{\cosh (a+b x)}}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {3 \left (\frac {1}{6} \left (-\frac {3}{2} \int \frac {1}{\frac {\cosh ^{\frac {2}{3}}(a+b x)}{\sinh ^{\frac {2}{3}}(a+b x)}-\frac {\sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}+1}d\frac {\sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}-\frac {1}{2} \int -\frac {1-\frac {2 \sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}}{\frac {\cosh ^{\frac {2}{3}}(a+b x)}{\sinh ^{\frac {2}{3}}(a+b x)}-\frac {\sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}+1}d\frac {\sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}\right )+\frac {1}{6} \left (\frac {1}{2} \int \frac {\frac {2 \sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}+1}{\frac {\cosh ^{\frac {2}{3}}(a+b x)}{\sinh ^{\frac {2}{3}}(a+b x)}+\frac {\sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}+1}d\frac {\sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}-\frac {3}{2} \int \frac {1}{\frac {\cosh ^{\frac {2}{3}}(a+b x)}{\sinh ^{\frac {2}{3}}(a+b x)}+\frac {\sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}+1}d\frac {\sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}\right )+\frac {1}{3} \text {arctanh}\left (\frac {\sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}\right )\right )}{b}-\frac {3 \sqrt [3]{\sinh (a+b x)}}{b \sqrt [3]{\cosh (a+b x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {3 \left (\frac {1}{6} \left (\frac {1}{2} \int \frac {1-\frac {2 \sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}}{\frac {\cosh ^{\frac {2}{3}}(a+b x)}{\sinh ^{\frac {2}{3}}(a+b x)}-\frac {\sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}+1}d\frac {\sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}-\frac {3}{2} \int \frac {1}{\frac {\cosh ^{\frac {2}{3}}(a+b x)}{\sinh ^{\frac {2}{3}}(a+b x)}-\frac {\sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}+1}d\frac {\sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}\right )+\frac {1}{6} \left (\frac {1}{2} \int \frac {\frac {2 \sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}+1}{\frac {\cosh ^{\frac {2}{3}}(a+b x)}{\sinh ^{\frac {2}{3}}(a+b x)}+\frac {\sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}+1}d\frac {\sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}-\frac {3}{2} \int \frac {1}{\frac {\cosh ^{\frac {2}{3}}(a+b x)}{\sinh ^{\frac {2}{3}}(a+b x)}+\frac {\sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}+1}d\frac {\sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}\right )+\frac {1}{3} \text {arctanh}\left (\frac {\sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}\right )\right )}{b}-\frac {3 \sqrt [3]{\sinh (a+b x)}}{b \sqrt [3]{\cosh (a+b x)}}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {3 \left (\frac {1}{6} \left (3 \int \frac {1}{-\frac {\cosh ^{\frac {2}{3}}(a+b x)}{\sinh ^{\frac {2}{3}}(a+b x)}-3}d\left (\frac {2 \sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}-1\right )+\frac {1}{2} \int \frac {1-\frac {2 \sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}}{\frac {\cosh ^{\frac {2}{3}}(a+b x)}{\sinh ^{\frac {2}{3}}(a+b x)}-\frac {\sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}+1}d\frac {\sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}\right )+\frac {1}{6} \left (3 \int \frac {1}{-\frac {\cosh ^{\frac {2}{3}}(a+b x)}{\sinh ^{\frac {2}{3}}(a+b x)}-3}d\left (\frac {2 \sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}+1\right )+\frac {1}{2} \int \frac {\frac {2 \sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}+1}{\frac {\cosh ^{\frac {2}{3}}(a+b x)}{\sinh ^{\frac {2}{3}}(a+b x)}+\frac {\sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}+1}d\frac {\sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}\right )+\frac {1}{3} \text {arctanh}\left (\frac {\sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}\right )\right )}{b}-\frac {3 \sqrt [3]{\sinh (a+b x)}}{b \sqrt [3]{\cosh (a+b x)}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {3 \left (\frac {1}{6} \left (\frac {1}{2} \int \frac {1-\frac {2 \sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}}{\frac {\cosh ^{\frac {2}{3}}(a+b x)}{\sinh ^{\frac {2}{3}}(a+b x)}-\frac {\sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}+1}d\frac {\sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}-\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}-1}{\sqrt {3}}\right )\right )+\frac {1}{6} \left (\frac {1}{2} \int \frac {\frac {2 \sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}+1}{\frac {\cosh ^{\frac {2}{3}}(a+b x)}{\sinh ^{\frac {2}{3}}(a+b x)}+\frac {\sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}+1}d\frac {\sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}-\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}+1}{\sqrt {3}}\right )\right )+\frac {1}{3} \text {arctanh}\left (\frac {\sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}\right )\right )}{b}-\frac {3 \sqrt [3]{\sinh (a+b x)}}{b \sqrt [3]{\cosh (a+b x)}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {3 \left (\frac {1}{6} \left (-\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}-1}{\sqrt {3}}\right )-\frac {1}{2} \log \left (\frac {\cosh ^{\frac {2}{3}}(a+b x)}{\sinh ^{\frac {2}{3}}(a+b x)}-\frac {\sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}+1\right )\right )+\frac {1}{6} \left (\frac {1}{2} \log \left (\frac {\cosh ^{\frac {2}{3}}(a+b x)}{\sinh ^{\frac {2}{3}}(a+b x)}+\frac {\sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}+1\right )-\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}+1}{\sqrt {3}}\right )\right )+\frac {1}{3} \text {arctanh}\left (\frac {\sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}\right )\right )}{b}-\frac {3 \sqrt [3]{\sinh (a+b x)}}{b \sqrt [3]{\cosh (a+b x)}}\) |
(3*(ArcTanh[Cosh[a + b*x]^(1/3)/Sinh[a + b*x]^(1/3)]/3 + (-(Sqrt[3]*ArcTan [(-1 + (2*Cosh[a + b*x]^(1/3))/Sinh[a + b*x]^(1/3))/Sqrt[3]]) - Log[1 + Co sh[a + b*x]^(2/3)/Sinh[a + b*x]^(2/3) - Cosh[a + b*x]^(1/3)/Sinh[a + b*x]^ (1/3)]/2)/6 + (-(Sqrt[3]*ArcTan[(1 + (2*Cosh[a + b*x]^(1/3))/Sinh[a + b*x] ^(1/3))/Sqrt[3]]) + Log[1 + Cosh[a + b*x]^(2/3)/Sinh[a + b*x]^(2/3) + Cosh [a + b*x]^(1/3)/Sinh[a + b*x]^(1/3)]/2)/6))/b - (3*Sinh[a + b*x]^(1/3))/(b *Cosh[a + b*x]^(1/3))
3.1.59.3.1 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_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m _), x_Symbol] :> Simp[(-a)*(a*Sin[e + f*x])^(m - 1)*((b*Cos[e + f*x])^(n + 1)/(b*f*(n + 1))), x] + Simp[a^2*((m - 1)/(b^2*(n + 1))) Int[(a*Sin[e + f *x])^(m - 2)*(b*Cos[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_)]*(a_.))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n _), 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*Cos[e + f*x])^(1/k)/(b*Sin[ e + f*x])^(1/k)], x]] /; FreeQ[{a, b, e, f}, x] && EqQ[m + n, 0] && GtQ[m, 0] && LtQ[m, 1]
\[\int \frac {\sinh \left (b x +a \right )^{\frac {4}{3}}}{\cosh \left (b x +a \right )^{\frac {4}{3}}}d x\]
Leaf count of result is larger than twice the leaf count of optimal. 1003 vs. \(2 (197) = 394\).
Time = 0.27 (sec) , antiderivative size = 1003, normalized size of antiderivative = 4.13 \[ \int \frac {\sinh ^{\frac {4}{3}}(a+b x)}{\cosh ^{\frac {4}{3}}(a+b x)} \, dx=\text {Too large to display} \]
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)^(2/3)*sinh(b*x + a) ^(1/3) + 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)^(2/3 )*sinh(b*x + a)^(1/3) + 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)) + 2*(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) + 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)) - (cosh(b*x + a)^2 + 2*cosh(b...
\[ \int \frac {\sinh ^{\frac {4}{3}}(a+b x)}{\cosh ^{\frac {4}{3}}(a+b x)} \, dx=\int \frac {\sinh ^{\frac {4}{3}}{\left (a + b x \right )}}{\cosh ^{\frac {4}{3}}{\left (a + b x \right )}}\, dx \]
\[ \int \frac {\sinh ^{\frac {4}{3}}(a+b x)}{\cosh ^{\frac {4}{3}}(a+b x)} \, dx=\int { \frac {\sinh \left (b x + a\right )^{\frac {4}{3}}}{\cosh \left (b x + a\right )^{\frac {4}{3}}} \,d x } \]
\[ \int \frac {\sinh ^{\frac {4}{3}}(a+b x)}{\cosh ^{\frac {4}{3}}(a+b x)} \, dx=\int { \frac {\sinh \left (b x + a\right )^{\frac {4}{3}}}{\cosh \left (b x + a\right )^{\frac {4}{3}}} \,d x } \]
Timed out. \[ \int \frac {\sinh ^{\frac {4}{3}}(a+b x)}{\cosh ^{\frac {4}{3}}(a+b x)} \, dx=\int \frac {{\mathrm {sinh}\left (a+b\,x\right )}^{4/3}}{{\mathrm {cosh}\left (a+b\,x\right )}^{4/3}} \,d x \]