Integrand size = 21, antiderivative size = 128 \[ \int \frac {\sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {1+\frac {2 \cosh ^{\frac {2}{3}}(a+b x)}{\sinh ^{\frac {2}{3}}(a+b x)}}{\sqrt {3}}\right )}{2 b}-\frac {\log \left (1-\frac {\cosh ^{\frac {2}{3}}(a+b x)}{\sinh ^{\frac {2}{3}}(a+b x)}\right )}{2 b}+\frac {\log \left (1+\frac {\cosh ^{\frac {4}{3}}(a+b x)}{\sinh ^{\frac {4}{3}}(a+b x)}+\frac {\cosh ^{\frac {2}{3}}(a+b x)}{\sinh ^{\frac {2}{3}}(a+b x)}\right )}{4 b} \]
-1/2*ln(1-cosh(b*x+a)^(2/3)/sinh(b*x+a)^(2/3))/b+1/4*ln(1+cosh(b*x+a)^(4/3 )/sinh(b*x+a)^(4/3)+cosh(b*x+a)^(2/3)/sinh(b*x+a)^(2/3))/b-1/2*arctan(1/3* (1+2*cosh(b*x+a)^(2/3)/sinh(b*x+a)^(2/3))*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 = 59, normalized size of antiderivative = 0.46 \[ \int \frac {\sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}} \, dx=\frac {3 \sqrt [3]{\cosh ^2(a+b x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{3},\frac {4}{3},-\sinh ^2(a+b x)\right ) \sinh ^{\frac {2}{3}}(a+b x)}{2 b \cosh ^{\frac {2}{3}}(a+b x)} \]
(3*(Cosh[a + b*x]^2)^(1/3)*Hypergeometric2F1[1/3, 1/3, 4/3, -Sinh[a + b*x] ^2]*Sinh[a + b*x]^(2/3))/(2*b*Cosh[a + b*x]^(2/3))
Time = 0.30 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.85, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {3042, 3055, 807, 821, 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.
| \(\displaystyle \int \frac {\sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sqrt [3]{\cos (i a+i b x)}}{\sqrt [3]{-i \sin (i a+i b x)}}dx\) |
| \(\Big \downarrow \) 3055 |
| \(\displaystyle \frac {3 \int \frac {\coth (a+b x)}{1-\coth ^2(a+b x)}d\frac {\sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}}{b}\) |
| \(\Big \downarrow \) 807 |
| \(\displaystyle \frac {3 \int \frac {\cosh ^{\frac {2}{3}}(a+b x)}{(1-\coth (a+b x)) \sinh ^{\frac {2}{3}}(a+b x)}d\frac {\cosh ^{\frac {2}{3}}(a+b x)}{\sinh ^{\frac {2}{3}}(a+b x)}}{2 b}\) |
| \(\Big \downarrow \) 821 |
| \(\displaystyle \frac {3 \left (\frac {1}{3} \int \frac {1}{1-\frac {\cosh ^{\frac {2}{3}}(a+b x)}{\sinh ^{\frac {2}{3}}(a+b x)}}d\frac {\cosh ^{\frac {2}{3}}(a+b x)}{\sinh ^{\frac {2}{3}}(a+b x)}-\frac {1}{3} \int \frac {1-\frac {\cosh ^{\frac {2}{3}}(a+b x)}{\sinh ^{\frac {2}{3}}(a+b x)}}{\frac {2 \cosh ^{\frac {2}{3}}(a+b x)}{\sinh ^{\frac {2}{3}}(a+b x)}+1}d\frac {\cosh ^{\frac {2}{3}}(a+b x)}{\sinh ^{\frac {2}{3}}(a+b x)}\right )}{2 b}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {3 \left (-\frac {1}{3} \int \frac {1-\frac {\cosh ^{\frac {2}{3}}(a+b x)}{\sinh ^{\frac {2}{3}}(a+b x)}}{\frac {2 \cosh ^{\frac {2}{3}}(a+b x)}{\sinh ^{\frac {2}{3}}(a+b x)}+1}d\frac {\cosh ^{\frac {2}{3}}(a+b x)}{\sinh ^{\frac {2}{3}}(a+b x)}-\frac {1}{3} \log \left (1-\frac {\cosh ^{\frac {2}{3}}(a+b x)}{\sinh ^{\frac {2}{3}}(a+b x)}\right )\right )}{2 b}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {3 \left (\frac {1}{3} \left (\frac {1}{2} \int 1d\frac {\cosh ^{\frac {2}{3}}(a+b x)}{\sinh ^{\frac {2}{3}}(a+b x)}-\frac {3}{2} \int \frac {1}{\frac {2 \cosh ^{\frac {2}{3}}(a+b x)}{\sinh ^{\frac {2}{3}}(a+b x)}+1}d\frac {\cosh ^{\frac {2}{3}}(a+b x)}{\sinh ^{\frac {2}{3}}(a+b x)}\right )-\frac {1}{3} \log \left (1-\frac {\cosh ^{\frac {2}{3}}(a+b x)}{\sinh ^{\frac {2}{3}}(a+b x)}\right )\right )}{2 b}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {3 \left (\frac {1}{3} \left (\frac {1}{2} \int 1d\frac {\cosh ^{\frac {2}{3}}(a+b x)}{\sinh ^{\frac {2}{3}}(a+b x)}+3 \int \frac {1}{-\frac {2 \cosh ^{\frac {2}{3}}(a+b x)}{\sinh ^{\frac {2}{3}}(a+b x)}-4}d\left (\frac {2 \cosh ^{\frac {2}{3}}(a+b x)}{\sinh ^{\frac {2}{3}}(a+b x)}+1\right )\right )-\frac {1}{3} \log \left (1-\frac {\cosh ^{\frac {2}{3}}(a+b x)}{\sinh ^{\frac {2}{3}}(a+b x)}\right )\right )}{2 b}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {3 \left (\frac {1}{3} \left (\frac {1}{2} \int 1d\frac {\cosh ^{\frac {2}{3}}(a+b x)}{\sinh ^{\frac {2}{3}}(a+b x)}-\sqrt {3} \arctan \left (\frac {\frac {2 \cosh ^{\frac {2}{3}}(a+b x)}{\sinh ^{\frac {2}{3}}(a+b x)}+1}{\sqrt {3}}\right )\right )-\frac {1}{3} \log \left (1-\frac {\cosh ^{\frac {2}{3}}(a+b x)}{\sinh ^{\frac {2}{3}}(a+b x)}\right )\right )}{2 b}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {3 \left (\frac {1}{3} \left (\frac {1}{2} \log \left (\frac {2 \cosh ^{\frac {2}{3}}(a+b x)}{\sinh ^{\frac {2}{3}}(a+b x)}+1\right )-\sqrt {3} \arctan \left (\frac {\frac {2 \cosh ^{\frac {2}{3}}(a+b x)}{\sinh ^{\frac {2}{3}}(a+b x)}+1}{\sqrt {3}}\right )\right )-\frac {1}{3} \log \left (1-\frac {\cosh ^{\frac {2}{3}}(a+b x)}{\sinh ^{\frac {2}{3}}(a+b x)}\right )\right )}{2 b}\) |
(3*(-1/3*Log[1 - Cosh[a + b*x]^(2/3)/Sinh[a + b*x]^(2/3)] + (-(Sqrt[3]*Arc Tan[(1 + (2*Cosh[a + b*x]^(2/3))/Sinh[a + b*x]^(2/3))/Sqrt[3]]) + Log[1 + (2*Cosh[a + b*x]^(2/3))/Sinh[a + b*x]^(2/3)]/2)/3))/(2*b)
3.1.62.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, 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[(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[(x_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> Simp[-(3*Rt[a, 3]*Rt[b, 3])^(- 1) Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Simp[1/(3*Rt[a, 3]*Rt[b, 3]) Int[(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[((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] :> 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]
\[\int \frac {\cosh \left (b x +a \right )^{\frac {1}{3}}}{\sinh \left (b x +a \right )^{\frac {1}{3}}}d x\]
Leaf count of result is larger than twice the leaf count of optimal. 578 vs. \(2 (103) = 206\).
Time = 0.27 (sec) , antiderivative size = 578, normalized size of antiderivative = 4.52 \[ \int \frac {\sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}} \, dx=-\frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} \cosh \left (b x + a\right )^{2} + 2 \, \sqrt {3} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sqrt {3} \sinh \left (b x + a\right )^{2} + 4 \, {\left (\sqrt {3} \cosh \left (b x + a\right ) + \sqrt {3} \sinh \left (b x + a\right )\right )} \cosh \left (b x + a\right )^{\frac {2}{3}} \sinh \left (b x + a\right )^{\frac {1}{3}} - \sqrt {3}}{3 \, {\left (\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} - 1\right )}}\right ) - \log \left (\frac {\cosh \left (b x + a\right )^{4} + 4 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + \sinh \left (b x + a\right )^{4} + 2 \, {\left (3 \, \cosh \left (b x + a\right )^{2} - 1\right )} \sinh \left (b x + a\right )^{2} - 2 \, \cosh \left (b x + a\right )^{2} + 2 \, {\left (\cosh \left (b x + a\right )^{3} + 3 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + \sinh \left (b x + a\right )^{3} + {\left (3 \, \cosh \left (b x + a\right )^{2} - 1\right )} \sinh \left (b x + a\right ) - \cosh \left (b x + a\right )\right )} \cosh \left (b x + a\right )^{\frac {2}{3}} \sinh \left (b x + a\right )^{\frac {1}{3}} + 2 \, {\left (\cosh \left (b x + a\right )^{3} + 3 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + \sinh \left (b x + a\right )^{3} + {\left (3 \, \cosh \left (b x + a\right )^{2} + 1\right )} \sinh \left (b x + a\right ) + \cosh \left (b x + a\right )\right )} \cosh \left (b x + a\right )^{\frac {1}{3}} \sinh \left (b x + a\right )^{\frac {2}{3}} + 4 \, {\left (\cosh \left (b x + a\right )^{3} - \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 1}{\cosh \left (b x + a\right )^{4} + 4 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + \sinh \left (b x + a\right )^{4} + 2 \, {\left (3 \, \cosh \left (b x + a\right )^{2} - 1\right )} \sinh \left (b x + a\right )^{2} - 2 \, \cosh \left (b x + a\right )^{2} + 4 \, {\left (\cosh \left (b x + a\right )^{3} - \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 1}\right ) + 2 \, \log \left (-\frac {\cosh \left (b x + a\right )^{2} - 2 \, {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} \cosh \left (b x + a\right )^{\frac {2}{3}} \sinh \left (b x + a\right )^{\frac {1}{3}} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} - 1}{\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} - 1}\right )}{4 \, b} \]
-1/4*(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) + s qrt(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)) - log((cosh(b*x + a)^4 + 4*cosh(b*x + a)*sinh(b*x + a)^3 + sinh(b*x + a)^4 + 2*(3*cosh(b*x + a)^2 - 1)*sinh(b*x + a)^2 - 2*cosh(b*x + a)^2 + 2*(cosh(b *x + a)^3 + 3*cosh(b*x + a)*sinh(b*x + a)^2 + sinh(b*x + a)^3 + (3*cosh(b* x + a)^2 - 1)*sinh(b*x + a) - cosh(b*x + a))*cosh(b*x + a)^(2/3)*sinh(b*x + a)^(1/3) + 2*(cosh(b*x + a)^3 + 3*cosh(b*x + a)*sinh(b*x + a)^2 + sinh(b *x + a)^3 + (3*cosh(b*x + a)^2 + 1)*sinh(b*x + a) + cosh(b*x + a))*cosh(b* x + a)^(1/3)*sinh(b*x + a)^(2/3) + 4*(cosh(b*x + a)^3 - cosh(b*x + a))*sin h(b*x + a) + 1)/(cosh(b*x + a)^4 + 4*cosh(b*x + a)*sinh(b*x + a)^3 + sinh( b*x + a)^4 + 2*(3*cosh(b*x + a)^2 - 1)*sinh(b*x + a)^2 - 2*cosh(b*x + a)^2 + 4*(cosh(b*x + a)^3 - cosh(b*x + a))*sinh(b*x + a) + 1)) + 2*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)))/b
\[ \int \frac {\sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}} \, dx=\int \frac {\sqrt [3]{\cosh {\left (a + b x \right )}}}{\sqrt [3]{\sinh {\left (a + b x \right )}}}\, dx \]
\[ \int \frac {\sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}} \, dx=\int { \frac {\cosh \left (b x + a\right )^{\frac {1}{3}}}{\sinh \left (b x + a\right )^{\frac {1}{3}}} \,d x } \]
\[ \int \frac {\sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}} \, dx=\int { \frac {\cosh \left (b x + a\right )^{\frac {1}{3}}}{\sinh \left (b x + a\right )^{\frac {1}{3}}} \,d x } \]
Timed out. \[ \int \frac {\sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}} \, dx=\int \frac {{\mathrm {cosh}\left (a+b\,x\right )}^{1/3}}{{\mathrm {sinh}\left (a+b\,x\right )}^{1/3}} \,d x \]