Integrand size = 15, antiderivative size = 110 \[ \int \frac {1}{(a-a \cosh (c+d x))^{5/2}} \, dx=-\frac {3 \arctan \left (\frac {\sqrt {a} \sinh (c+d x)}{\sqrt {2} \sqrt {a-a \cosh (c+d x)}}\right )}{16 \sqrt {2} a^{5/2} d}-\frac {\sinh (c+d x)}{4 d (a-a \cosh (c+d x))^{5/2}}-\frac {3 \sinh (c+d x)}{16 a d (a-a \cosh (c+d x))^{3/2}} \]
-1/4*sinh(d*x+c)/d/(a-a*cosh(d*x+c))^(5/2)-3/16*sinh(d*x+c)/a/d/(a-a*cosh( d*x+c))^(3/2)-3/32*arctan(1/2*sinh(d*x+c)*a^(1/2)*2^(1/2)/(a-a*cosh(d*x+c) )^(1/2))/a^(5/2)/d*2^(1/2)
Time = 0.28 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.17 \[ \int \frac {1}{(a-a \cosh (c+d x))^{5/2}} \, dx=\frac {\left (6 \text {csch}^2\left (\frac {1}{4} (c+d x)\right )-\text {csch}^4\left (\frac {1}{4} (c+d x)\right )+24 \left (-\log \left (\cosh \left (\frac {1}{4} (c+d x)\right )\right )+\log \left (\sinh \left (\frac {1}{4} (c+d x)\right )\right )\right )+6 \text {sech}^2\left (\frac {1}{4} (c+d x)\right )+\text {sech}^4\left (\frac {1}{4} (c+d x)\right )\right ) \sinh ^5\left (\frac {1}{2} (c+d x)\right )}{32 a^2 d (-1+\cosh (c+d x))^2 \sqrt {a-a \cosh (c+d x)}} \]
((6*Csch[(c + d*x)/4]^2 - Csch[(c + d*x)/4]^4 + 24*(-Log[Cosh[(c + d*x)/4] ] + Log[Sinh[(c + d*x)/4]]) + 6*Sech[(c + d*x)/4]^2 + Sech[(c + d*x)/4]^4) *Sinh[(c + d*x)/2]^5)/(32*a^2*d*(-1 + Cosh[c + d*x])^2*Sqrt[a - a*Cosh[c + d*x]])
Time = 0.39 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.05, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {3042, 3129, 3042, 3129, 3042, 3128, 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.
| \(\displaystyle \int \frac {1}{(a-a \cosh (c+d x))^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\left (a-a \sin \left (i c+i d x+\frac {\pi }{2}\right )\right )^{5/2}}dx\) |
| \(\Big \downarrow \) 3129 |
| \(\displaystyle \frac {3 \int \frac {1}{(a-a \cosh (c+d x))^{3/2}}dx}{8 a}-\frac {\sinh (c+d x)}{4 d (a-a \cosh (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\sinh (c+d x)}{4 d (a-a \cosh (c+d x))^{5/2}}+\frac {3 \int \frac {1}{\left (a-a \sin \left (i c+i d x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{8 a}\) |
| \(\Big \downarrow \) 3129 |
| \(\displaystyle \frac {3 \left (\frac {\int \frac {1}{\sqrt {a-a \cosh (c+d x)}}dx}{4 a}-\frac {\sinh (c+d x)}{2 d (a-a \cosh (c+d x))^{3/2}}\right )}{8 a}-\frac {\sinh (c+d x)}{4 d (a-a \cosh (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\sinh (c+d x)}{4 d (a-a \cosh (c+d x))^{5/2}}+\frac {3 \left (-\frac {\sinh (c+d x)}{2 d (a-a \cosh (c+d x))^{3/2}}+\frac {\int \frac {1}{\sqrt {a-a \sin \left (i c+i d x+\frac {\pi }{2}\right )}}dx}{4 a}\right )}{8 a}\) |
| \(\Big \downarrow \) 3128 |
| \(\displaystyle -\frac {\sinh (c+d x)}{4 d (a-a \cosh (c+d x))^{5/2}}+\frac {3 \left (-\frac {\sinh (c+d x)}{2 d (a-a \cosh (c+d x))^{3/2}}+\frac {i \int \frac {1}{\frac {a^2 \sinh ^2(c+d x)}{a-a \cosh (c+d x)}+2 a}d\frac {i a \sinh (c+d x)}{\sqrt {a-a \cosh (c+d x)}}}{2 a d}\right )}{8 a}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {3 \left (-\frac {\arctan \left (\frac {\sqrt {a} \sinh (c+d x)}{\sqrt {2} \sqrt {a-a \cosh (c+d x)}}\right )}{2 \sqrt {2} a^{3/2} d}-\frac {\sinh (c+d x)}{2 d (a-a \cosh (c+d x))^{3/2}}\right )}{8 a}-\frac {\sinh (c+d x)}{4 d (a-a \cosh (c+d x))^{5/2}}\) |
-1/4*Sinh[c + d*x]/(d*(a - a*Cosh[c + d*x])^(5/2)) + (3*(-1/2*ArcTan[(Sqrt [a]*Sinh[c + d*x])/(Sqrt[2]*Sqrt[a - a*Cosh[c + d*x]])]/(Sqrt[2]*a^(3/2)*d ) - Sinh[c + d*x]/(2*d*(a - a*Cosh[c + d*x])^(3/2))))/(8*a)
3.1.53.3.1 Defintions of rubi rules used
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[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[-2/d Subst[Int[1/(2*a - x^2), x], x, b*(Cos[c + d*x]/Sqrt[a + b*Sin[c + d*x]])], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*Cos[c + d*x]*((a + b*Sin[c + d*x])^n/(a*d*(2*n + 1))), x] + Simp[(n + 1)/(a*(2*n + 1)) Int[(a + b*Sin[c + d*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0] && LtQ[n, -1] && IntegerQ[2*n]
Time = 0.13 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.25
| method | result | size |
| default | \(\frac {6 \cosh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sinh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-4 \cosh \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (3 \ln \left (\cosh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-3 \ln \left (\cosh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )\right ) \sinh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{32 a^{2} \left (\cosh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \left (\cosh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \sinh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \sinh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a}\, d}\) | \(137\) |
1/32/a^2*(6*cosh(1/2*d*x+1/2*c)*sinh(1/2*d*x+1/2*c)^2-4*cosh(1/2*d*x+1/2*c )+(3*ln(cosh(1/2*d*x+1/2*c)-1)-3*ln(cosh(1/2*d*x+1/2*c)+1))*sinh(1/2*d*x+1 /2*c)^4)/(cosh(1/2*d*x+1/2*c)+1)/(cosh(1/2*d*x+1/2*c)-1)/sinh(1/2*d*x+1/2* c)/(-2*sinh(1/2*d*x+1/2*c)^2*a)^(1/2)/d
Leaf count of result is larger than twice the leaf count of optimal. 580 vs. \(2 (91) = 182\).
Time = 0.25 (sec) , antiderivative size = 580, normalized size of antiderivative = 5.27 \[ \int \frac {1}{(a-a \cosh (c+d x))^{5/2}} \, dx=-\frac {3 \, \sqrt {2} {\left (\cosh \left (d x + c\right )^{4} + 4 \, {\left (\cosh \left (d x + c\right ) - 1\right )} \sinh \left (d x + c\right )^{3} + \sinh \left (d x + c\right )^{4} - 4 \, \cosh \left (d x + c\right )^{3} + 6 \, {\left (\cosh \left (d x + c\right )^{2} - 2 \, \cosh \left (d x + c\right ) + 1\right )} \sinh \left (d x + c\right )^{2} + 6 \, \cosh \left (d x + c\right )^{2} + 4 \, {\left (\cosh \left (d x + c\right )^{3} - 3 \, \cosh \left (d x + c\right )^{2} + 3 \, \cosh \left (d x + c\right ) - 1\right )} \sinh \left (d x + c\right ) - 4 \, \cosh \left (d x + c\right ) + 1\right )} \sqrt {-a} \log \left (-\frac {2 \, \sqrt {2} \sqrt {\frac {1}{2}} \sqrt {-a} \sqrt {-\frac {a}{\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )}} {\left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right )} + a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right ) + a}{\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) - 1}\right ) + 4 \, \sqrt {\frac {1}{2}} {\left (3 \, \cosh \left (d x + c\right )^{4} + {\left (12 \, \cosh \left (d x + c\right ) - 11\right )} \sinh \left (d x + c\right )^{3} + 3 \, \sinh \left (d x + c\right )^{4} - 11 \, \cosh \left (d x + c\right )^{3} + {\left (18 \, \cosh \left (d x + c\right )^{2} - 33 \, \cosh \left (d x + c\right ) - 11\right )} \sinh \left (d x + c\right )^{2} - 11 \, \cosh \left (d x + c\right )^{2} + {\left (12 \, \cosh \left (d x + c\right )^{3} - 33 \, \cosh \left (d x + c\right )^{2} - 22 \, \cosh \left (d x + c\right ) + 3\right )} \sinh \left (d x + c\right ) + 3 \, \cosh \left (d x + c\right )\right )} \sqrt {-\frac {a}{\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )}}}{32 \, {\left (a^{3} d \cosh \left (d x + c\right )^{4} + a^{3} d \sinh \left (d x + c\right )^{4} - 4 \, a^{3} d \cosh \left (d x + c\right )^{3} + 6 \, a^{3} d \cosh \left (d x + c\right )^{2} - 4 \, a^{3} d \cosh \left (d x + c\right ) + a^{3} d + 4 \, {\left (a^{3} d \cosh \left (d x + c\right ) - a^{3} d\right )} \sinh \left (d x + c\right )^{3} + 6 \, {\left (a^{3} d \cosh \left (d x + c\right )^{2} - 2 \, a^{3} d \cosh \left (d x + c\right ) + a^{3} d\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left (a^{3} d \cosh \left (d x + c\right )^{3} - 3 \, a^{3} d \cosh \left (d x + c\right )^{2} + 3 \, a^{3} d \cosh \left (d x + c\right ) - a^{3} d\right )} \sinh \left (d x + c\right )\right )}} \]
-1/32*(3*sqrt(2)*(cosh(d*x + c)^4 + 4*(cosh(d*x + c) - 1)*sinh(d*x + c)^3 + sinh(d*x + c)^4 - 4*cosh(d*x + c)^3 + 6*(cosh(d*x + c)^2 - 2*cosh(d*x + c) + 1)*sinh(d*x + c)^2 + 6*cosh(d*x + c)^2 + 4*(cosh(d*x + c)^3 - 3*cosh( d*x + c)^2 + 3*cosh(d*x + c) - 1)*sinh(d*x + c) - 4*cosh(d*x + c) + 1)*sqr t(-a)*log(-(2*sqrt(2)*sqrt(1/2)*sqrt(-a)*sqrt(-a/(cosh(d*x + c) + sinh(d*x + c)))*(cosh(d*x + c) + sinh(d*x + c)) + a*cosh(d*x + c) + a*sinh(d*x + c ) + a)/(cosh(d*x + c) + sinh(d*x + c) - 1)) + 4*sqrt(1/2)*(3*cosh(d*x + c) ^4 + (12*cosh(d*x + c) - 11)*sinh(d*x + c)^3 + 3*sinh(d*x + c)^4 - 11*cosh (d*x + c)^3 + (18*cosh(d*x + c)^2 - 33*cosh(d*x + c) - 11)*sinh(d*x + c)^2 - 11*cosh(d*x + c)^2 + (12*cosh(d*x + c)^3 - 33*cosh(d*x + c)^2 - 22*cosh (d*x + c) + 3)*sinh(d*x + c) + 3*cosh(d*x + c))*sqrt(-a/(cosh(d*x + c) + s inh(d*x + c))))/(a^3*d*cosh(d*x + c)^4 + a^3*d*sinh(d*x + c)^4 - 4*a^3*d*c osh(d*x + c)^3 + 6*a^3*d*cosh(d*x + c)^2 - 4*a^3*d*cosh(d*x + c) + a^3*d + 4*(a^3*d*cosh(d*x + c) - a^3*d)*sinh(d*x + c)^3 + 6*(a^3*d*cosh(d*x + c)^ 2 - 2*a^3*d*cosh(d*x + c) + a^3*d)*sinh(d*x + c)^2 + 4*(a^3*d*cosh(d*x + c )^3 - 3*a^3*d*cosh(d*x + c)^2 + 3*a^3*d*cosh(d*x + c) - a^3*d)*sinh(d*x + c))
\[ \int \frac {1}{(a-a \cosh (c+d x))^{5/2}} \, dx=\int \frac {1}{\left (- a \cosh {\left (c + d x \right )} + a\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {1}{(a-a \cosh (c+d x))^{5/2}} \, dx=\int { \frac {1}{{\left (-a \cosh \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
Time = 0.28 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.61 \[ \int \frac {1}{(a-a \cosh (c+d x))^{5/2}} \, dx=-\frac {3 \, \sqrt {2} \arctan \left (\frac {\sqrt {-a e^{\left (d x + c\right )}}}{\sqrt {a}}\right )}{16 \, a^{\frac {5}{2}} d \mathrm {sgn}\left (-e^{\left (d x + c\right )} + 1\right )} + \frac {3 \, \sqrt {2} \sqrt {-a e^{\left (d x + c\right )}} a^{3} e^{\left (3 \, d x + 3 \, c\right )} - 11 \, \sqrt {2} \sqrt {-a e^{\left (d x + c\right )}} a^{3} e^{\left (2 \, d x + 2 \, c\right )} - 11 \, \sqrt {2} \sqrt {-a e^{\left (d x + c\right )}} a^{3} e^{\left (d x + c\right )} + 3 \, \sqrt {2} \sqrt {-a e^{\left (d x + c\right )}} a^{3}}{16 \, {\left (a e^{\left (d x + c\right )} - a\right )}^{4} a^{2} d \mathrm {sgn}\left (-e^{\left (d x + c\right )} + 1\right )} \]
-3/16*sqrt(2)*arctan(sqrt(-a*e^(d*x + c))/sqrt(a))/(a^(5/2)*d*sgn(-e^(d*x + c) + 1)) + 1/16*(3*sqrt(2)*sqrt(-a*e^(d*x + c))*a^3*e^(3*d*x + 3*c) - 11 *sqrt(2)*sqrt(-a*e^(d*x + c))*a^3*e^(2*d*x + 2*c) - 11*sqrt(2)*sqrt(-a*e^( d*x + c))*a^3*e^(d*x + c) + 3*sqrt(2)*sqrt(-a*e^(d*x + c))*a^3)/((a*e^(d*x + c) - a)^4*a^2*d*sgn(-e^(d*x + c) + 1))
Timed out. \[ \int \frac {1}{(a-a \cosh (c+d x))^{5/2}} \, dx=\int \frac {1}{{\left (a-a\,\mathrm {cosh}\left (c+d\,x\right )\right )}^{5/2}} \,d x \]