Integrand size = 21, antiderivative size = 81 \[ \int \frac {\sinh ^{\frac {5}{2}}(a+b x)}{\cosh ^{\frac {5}{2}}(a+b x)} \, dx=-\frac {\arctan \left (\frac {\sqrt {\sinh (a+b x)}}{\sqrt {\cosh (a+b x)}}\right )}{b}+\frac {\text {arctanh}\left (\frac {\sqrt {\sinh (a+b x)}}{\sqrt {\cosh (a+b x)}}\right )}{b}-\frac {2 \sinh ^{\frac {3}{2}}(a+b x)}{3 b \cosh ^{\frac {3}{2}}(a+b x)} \]
-arctan(sinh(b*x+a)^(1/2)/cosh(b*x+a)^(1/2))/b+arctanh(sinh(b*x+a)^(1/2)/c osh(b*x+a)^(1/2))/b-2/3*sinh(b*x+a)^(3/2)/b/cosh(b*x+a)^(3/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.03 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.73 \[ \int \frac {\sinh ^{\frac {5}{2}}(a+b x)}{\cosh ^{\frac {5}{2}}(a+b x)} \, dx=\frac {2 \cosh ^2(a+b x)^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {7}{4},\frac {7}{4},\frac {11}{4},-\sinh ^2(a+b x)\right ) \sinh ^{\frac {7}{2}}(a+b x)}{7 b \cosh ^{\frac {3}{2}}(a+b x)} \]
(2*(Cosh[a + b*x]^2)^(3/4)*Hypergeometric2F1[7/4, 7/4, 11/4, -Sinh[a + b*x ]^2]*Sinh[a + b*x]^(7/2))/(7*b*Cosh[a + b*x]^(3/2))
Time = 0.34 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.06, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {3042, 3046, 3042, 3054, 25, 827, 216, 219}
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.
| \(\displaystyle \int \frac {\sinh ^{\frac {5}{2}}(a+b x)}{\cosh ^{\frac {5}{2}}(a+b x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(-i \sin (i a+i b x))^{5/2}}{\cos (i a+i b x)^{5/2}}dx\) |
| \(\Big \downarrow \) 3046 |
| \(\displaystyle \int \frac {\sqrt {\sinh (a+b x)}}{\sqrt {\cosh (a+b x)}}dx-\frac {2 \sinh ^{\frac {3}{2}}(a+b x)}{3 b \cosh ^{\frac {3}{2}}(a+b x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {2 \sinh ^{\frac {3}{2}}(a+b x)}{3 b \cosh ^{\frac {3}{2}}(a+b x)}+\int \frac {\sqrt {-i \sin (i a+i b x)}}{\sqrt {\cos (i a+i b x)}}dx\) |
| \(\Big \downarrow \) 3054 |
| \(\displaystyle -\frac {2 \int -\frac {\tanh (a+b x)}{1-\tanh ^2(a+b x)}d\frac {\sqrt {\sinh (a+b x)}}{\sqrt {\cosh (a+b x)}}}{b}-\frac {2 \sinh ^{\frac {3}{2}}(a+b x)}{3 b \cosh ^{\frac {3}{2}}(a+b x)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 \int \frac {\tanh (a+b x)}{1-\tanh ^2(a+b x)}d\frac {\sqrt {\sinh (a+b x)}}{\sqrt {\cosh (a+b x)}}}{b}-\frac {2 \sinh ^{\frac {3}{2}}(a+b x)}{3 b \cosh ^{\frac {3}{2}}(a+b x)}\) |
| \(\Big \downarrow \) 827 |
| \(\displaystyle -\frac {2 \left (\frac {1}{2} \int \frac {1}{\tanh (a+b x)+1}d\frac {\sqrt {\sinh (a+b x)}}{\sqrt {\cosh (a+b x)}}-\frac {1}{2} \int \frac {1}{1-\tanh (a+b x)}d\frac {\sqrt {\sinh (a+b x)}}{\sqrt {\cosh (a+b x)}}\right )}{b}-\frac {2 \sinh ^{\frac {3}{2}}(a+b x)}{3 b \cosh ^{\frac {3}{2}}(a+b x)}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle -\frac {2 \left (\frac {1}{2} \arctan \left (\frac {\sqrt {\sinh (a+b x)}}{\sqrt {\cosh (a+b x)}}\right )-\frac {1}{2} \int \frac {1}{1-\tanh (a+b x)}d\frac {\sqrt {\sinh (a+b x)}}{\sqrt {\cosh (a+b x)}}\right )}{b}-\frac {2 \sinh ^{\frac {3}{2}}(a+b x)}{3 b \cosh ^{\frac {3}{2}}(a+b x)}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {2 \left (\frac {1}{2} \arctan \left (\frac {\sqrt {\sinh (a+b x)}}{\sqrt {\cosh (a+b x)}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt {\sinh (a+b x)}}{\sqrt {\cosh (a+b x)}}\right )\right )}{b}-\frac {2 \sinh ^{\frac {3}{2}}(a+b x)}{3 b \cosh ^{\frac {3}{2}}(a+b x)}\) |
(-2*(ArcTan[Sqrt[Sinh[a + b*x]]/Sqrt[Cosh[a + b*x]]]/2 - ArcTanh[Sqrt[Sinh [a + b*x]]/Sqrt[Cosh[a + b*x]]]/2))/b - (2*Sinh[a + b*x]^(3/2))/(3*b*Cosh[ a + b*x]^(3/2))
3.1.50.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[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_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2*b) Int[1/(r + s*x^2), x], x] - Simp[s/(2*b) Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ [a/b, 0]
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_)]*(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]
\[\int \frac {\sinh \left (b x +a \right )^{\frac {5}{2}}}{\cosh \left (b x +a \right )^{\frac {5}{2}}}d x\]
Leaf count of result is larger than twice the leaf count of optimal. 591 vs. \(2 (67) = 134\).
Time = 0.27 (sec) , antiderivative size = 591, normalized size of antiderivative = 7.30 \[ \int \frac {\sinh ^{\frac {5}{2}}(a+b x)}{\cosh ^{\frac {5}{2}}(a+b x)} \, dx=-\frac {4 \, \cosh \left (b x + a\right )^{4} + 16 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + 4 \, \sinh \left (b x + a\right )^{4} + 8 \, {\left (3 \, \cosh \left (b x + a\right )^{2} + 1\right )} \sinh \left (b x + a\right )^{2} + 6 \, {\left (\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 )} \arctan \left (-\cosh \left (b x + a\right )^{2} + 2 \, {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} \sqrt {\cosh \left (b x + a\right )} \sqrt {\sinh \left (b x + a\right )} - 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) - \sinh \left (b x + a\right )^{2}\right ) + 8 \, \cosh \left (b x + a\right )^{2} + 3 \, {\left (\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 )} \log \left (-\cosh \left (b x + a\right )^{2} + 2 \, {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} \sqrt {\cosh \left (b x + a\right )} \sqrt {\sinh \left (b x + a\right )} - 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) - \sinh \left (b x + a\right )^{2}\right ) + 8 \, {\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 )} \sqrt {\cosh \left (b x + a\right )} \sqrt {\sinh \left (b x + a\right )} + 16 \, {\left (\cosh \left (b x + a\right )^{3} + \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 4}{6 \, {\left (b \cosh \left (b x + a\right )^{4} + 4 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + b \sinh \left (b x + a\right )^{4} + 2 \, b \cosh \left (b x + a\right )^{2} + 2 \, {\left (3 \, b \cosh \left (b x + a\right )^{2} + b\right )} \sinh \left (b x + a\right )^{2} + 4 \, {\left (b \cosh \left (b x + a\right )^{3} + b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + b\right )}} \]
-1/6*(4*cosh(b*x + a)^4 + 16*cosh(b*x + a)*sinh(b*x + a)^3 + 4*sinh(b*x + a)^4 + 8*(3*cosh(b*x + a)^2 + 1)*sinh(b*x + a)^2 + 6*(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)*arctan(-cosh(b*x + a)^2 + 2*(cosh(b*x + a) + sinh(b*x + a))*sqrt(cosh(b*x + a))*sqrt(sinh(b*x + a)) - 2*cosh(b*x + a)*sinh(b*x + a) - sinh(b*x + a)^2) + 8*cosh(b*x + a)^2 + 3*(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)*sin h(b*x + a)^2 + 2*cosh(b*x + a)^2 + 4*(cosh(b*x + a)^3 + cosh(b*x + a))*sin h(b*x + a) + 1)*log(-cosh(b*x + a)^2 + 2*(cosh(b*x + a) + sinh(b*x + a))*s qrt(cosh(b*x + a))*sqrt(sinh(b*x + a)) - 2*cosh(b*x + a)*sinh(b*x + a) - s inh(b*x + a)^2) + 8*(cosh(b*x + a)^3 + 3*cosh(b*x + a)*sinh(b*x + a)^2 + s inh(b*x + a)^3 + (3*cosh(b*x + a)^2 - 1)*sinh(b*x + a) - cosh(b*x + a))*sq rt(cosh(b*x + a))*sqrt(sinh(b*x + a)) + 16*(cosh(b*x + a)^3 + cosh(b*x + a ))*sinh(b*x + a) + 4)/(b*cosh(b*x + a)^4 + 4*b*cosh(b*x + a)*sinh(b*x + a) ^3 + b*sinh(b*x + a)^4 + 2*b*cosh(b*x + a)^2 + 2*(3*b*cosh(b*x + a)^2 + b) *sinh(b*x + a)^2 + 4*(b*cosh(b*x + a)^3 + b*cosh(b*x + a))*sinh(b*x + a) + b)
Timed out. \[ \int \frac {\sinh ^{\frac {5}{2}}(a+b x)}{\cosh ^{\frac {5}{2}}(a+b x)} \, dx=\text {Timed out} \]
\[ \int \frac {\sinh ^{\frac {5}{2}}(a+b x)}{\cosh ^{\frac {5}{2}}(a+b x)} \, dx=\int { \frac {\sinh \left (b x + a\right )^{\frac {5}{2}}}{\cosh \left (b x + a\right )^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {\sinh ^{\frac {5}{2}}(a+b x)}{\cosh ^{\frac {5}{2}}(a+b x)} \, dx=\int { \frac {\sinh \left (b x + a\right )^{\frac {5}{2}}}{\cosh \left (b x + a\right )^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {\sinh ^{\frac {5}{2}}(a+b x)}{\cosh ^{\frac {5}{2}}(a+b x)} \, dx=\int \frac {{\mathrm {sinh}\left (a+b\,x\right )}^{5/2}}{{\mathrm {cosh}\left (a+b\,x\right )}^{5/2}} \,d x \]