Integrand size = 14, antiderivative size = 107 \[ \int \frac {1}{(a+a \cosh (c+d x))^{5/2}} \, dx=\frac {3 \arctan \left (\frac {\sqrt {a} \sinh (c+d x)}{\sqrt {2} \sqrt {a+a \cosh (c+d x)}}\right )}{16 \sqrt {2} a^{5/2} d}+\frac {\sinh (c+d x)}{4 d (a+a \cosh (c+d x))^{5/2}}+\frac {3 \sinh (c+d x)}{16 a d (a+a \cosh (c+d x))^{3/2}} \]
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 = 0.19 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.85 \[ \int \frac {1}{(a+a \cosh (c+d x))^{5/2}} \, dx=\frac {\cosh ^5\left (\frac {1}{2} (c+d x)\right ) \left (32 \text {csch}^4(c+d x) \sinh ^5\left (\frac {1}{2} (c+d x)\right )+3 \left (\arctan \left (\sinh \left (\frac {1}{2} (c+d x)\right )\right )+\text {sech}\left (\frac {1}{2} (c+d x)\right ) \tanh \left (\frac {1}{2} (c+d x)\right )\right )\right )}{4 d (a (1+\cosh (c+d x)))^{5/2}} \]
(Cosh[(c + d*x)/2]^5*(32*Csch[c + d*x]^4*Sinh[(c + d*x)/2]^5 + 3*(ArcTan[S inh[(c + d*x)/2]] + Sech[(c + d*x)/2]*Tanh[(c + d*x)/2])))/(4*d*(a*(1 + Co sh[c + d*x]))^(5/2))
Time = 0.38 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.05, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, 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.
\(\displaystyle \int \frac {1}{(a \cosh (c+d x)+a)^{5/2}} \, dx\) |
\(\Big \downarrow \) 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 |
\(\displaystyle \frac {3 \int \frac {1}{(\cosh (c+d x) a+a)^{3/2}}dx}{8 a}+\frac {\sinh (c+d x)}{4 d (a \cosh (c+d x)+a)^{5/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sinh (c+d x)}{4 d (a \cosh (c+d x)+a)^{5/2}}+\frac {3 \int \frac {1}{\left (\sin \left (i c+i d x+\frac {\pi }{2}\right ) a+a\right )^{3/2}}dx}{8 a}\) |
\(\Big \downarrow \) 3129 |
\(\displaystyle \frac {3 \left (\frac {\int \frac {1}{\sqrt {\cosh (c+d x) a+a}}dx}{4 a}+\frac {\sinh (c+d x)}{2 d (a \cosh (c+d x)+a)^{3/2}}\right )}{8 a}+\frac {\sinh (c+d x)}{4 d (a \cosh (c+d x)+a)^{5/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sinh (c+d x)}{4 d (a \cosh (c+d x)+a)^{5/2}}+\frac {3 \left (\frac {\sinh (c+d x)}{2 d (a \cosh (c+d x)+a)^{3/2}}+\frac {\int \frac {1}{\sqrt {\sin \left (i c+i d x+\frac {\pi }{2}\right ) a+a}}dx}{4 a}\right )}{8 a}\) |
\(\Big \downarrow \) 3128 |
\(\displaystyle \frac {\sinh (c+d x)}{4 d (a \cosh (c+d x)+a)^{5/2}}+\frac {3 \left (\frac {\sinh (c+d x)}{2 d (a \cosh (c+d x)+a)^{3/2}}+\frac {i \int \frac {1}{\frac {a^2 \sinh ^2(c+d x)}{\cosh (c+d x) a+a}+2 a}d\left (-\frac {i a \sinh (c+d x)}{\sqrt {\cosh (c+d x) a+a}}\right )}{2 a d}\right )}{8 a}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {3 \left (\frac {\arctan \left (\frac {\sqrt {a} \sinh (c+d x)}{\sqrt {2} \sqrt {a \cosh (c+d x)+a}}\right )}{2 \sqrt {2} a^{3/2} d}+\frac {\sinh (c+d x)}{2 d (a \cosh (c+d x)+a)^{3/2}}\right )}{8 a}+\frac {\sinh (c+d x)}{4 d (a \cosh (c+d x)+a)^{5/2}}\) |
Sinh[c + d*x]/(4*d*(a + a*Cosh[c + d*x])^(5/2)) + (3*(ArcTan[(Sqrt[a]*Sinh [c + d*x])/(Sqrt[2]*Sqrt[a + a*Cosh[c + d*x]])]/(2*Sqrt[2]*a^(3/2)*d) + Si nh[c + d*x]/(2*d*(a + a*Cosh[c + d*x])^(3/2))))/(8*a)
3.1.47.3.1 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]
Leaf count of result is larger than twice the leaf count of optimal. \(177\) vs. \(2(88)=176\).
Time = 0.14 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.66
method | result | size |
default | \(-\frac {\sqrt {\sinh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a}\, \left (3 \ln \left (\frac {2 \sqrt {-a}\, \sqrt {\sinh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a}-2 a}{\cosh \left (\frac {d x}{2}+\frac {c}{2}\right )}\right ) a \cosh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-3 \sqrt {-a}\, \sqrt {\sinh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a}\, \cosh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-2 \sqrt {-a}\, \sqrt {\sinh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a}\right ) \sqrt {2}}{32 a^{3} \cosh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} \sqrt {-a}\, \sinh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {a \cosh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, d}\) | \(178\) |
-1/32*(sinh(1/2*d*x+1/2*c)^2*a)^(1/2)*(3*ln(2*((-a)^(1/2)*(sinh(1/2*d*x+1/ 2*c)^2*a)^(1/2)-a)/cosh(1/2*d*x+1/2*c))*a*cosh(1/2*d*x+1/2*c)^4-3*(-a)^(1/ 2)*(sinh(1/2*d*x+1/2*c)^2*a)^(1/2)*cosh(1/2*d*x+1/2*c)^2-2*(-a)^(1/2)*(sin h(1/2*d*x+1/2*c)^2*a)^(1/2))/a^3/cosh(1/2*d*x+1/2*c)^3/(-a)^(1/2)/sinh(1/2 *d*x+1/2*c)*2^(1/2)/(a*cosh(1/2*d*x+1/2*c)^2)^(1/2)/d
Leaf count of result is larger than twice the leaf count of optimal. 522 vs. \(2 (88) = 176\).
Time = 0.27 (sec) , antiderivative size = 522, normalized size of antiderivative = 4.88 \[ \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} \arctan \left (\frac {\sqrt {2} \sqrt {\frac {1}{2}} \sqrt {\frac {a}{\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )}}}{\sqrt {a}}\right ) - 2 \, \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 )}}}{16 \, {\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 )}} \]
-1/16*(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)*arctan(sqrt(2)*sqrt(1/2)*sqrt(a/(cosh(d*x + c) + sinh(d*x + c)))/sqrt (a)) - 2*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) + sinh(d*x + c))))/(a^3*d*cosh(d*x + c)^4 + a^3*d*sinh(d*x + c)^4 + 4*a^3*d*cosh(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))
\[ \int \frac {1}{(a+a \cosh (c+d x))^{5/2}} \, dx=\int \frac {1}{\left (a \cosh {\left (c + d x \right )} + a\right )^{\frac {5}{2}}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 250 vs. \(2 (88) = 176\).
Time = 0.33 (sec) , antiderivative size = 250, normalized size of antiderivative = 2.34 \[ \int \frac {1}{(a+a \cosh (c+d x))^{5/2}} \, dx=\frac {1}{80} \, \sqrt {2} {\left (\frac {15 \, e^{\left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right )} + 70 \, e^{\left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right )} + 128 \, e^{\left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right )} - 70 \, e^{\left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right )} - 15 \, e^{\left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}}{{\left (a^{\frac {5}{2}} e^{\left (5 \, d x + 5 \, c\right )} + 5 \, a^{\frac {5}{2}} e^{\left (4 \, d x + 4 \, c\right )} + 10 \, a^{\frac {5}{2}} e^{\left (3 \, d x + 3 \, c\right )} + 10 \, a^{\frac {5}{2}} e^{\left (2 \, d x + 2 \, c\right )} + 5 \, a^{\frac {5}{2}} e^{\left (d x + c\right )} + a^{\frac {5}{2}}\right )} d} + \frac {15 \, \arctan \left (e^{\left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}\right )}{a^{\frac {5}{2}} d}\right )} - \frac {8 \, \sqrt {2} e^{\left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right )}}{5 \, {\left (a^{\frac {5}{2}} d e^{\left (5 \, d x + 5 \, c\right )} + 5 \, a^{\frac {5}{2}} d e^{\left (4 \, d x + 4 \, c\right )} + 10 \, a^{\frac {5}{2}} d e^{\left (3 \, d x + 3 \, c\right )} + 10 \, a^{\frac {5}{2}} d e^{\left (2 \, d x + 2 \, c\right )} + 5 \, a^{\frac {5}{2}} d e^{\left (d x + c\right )} + a^{\frac {5}{2}} d\right )}} \]
1/80*sqrt(2)*((15*e^(9/2*d*x + 9/2*c) + 70*e^(7/2*d*x + 7/2*c) + 128*e^(5/ 2*d*x + 5/2*c) - 70*e^(3/2*d*x + 3/2*c) - 15*e^(1/2*d*x + 1/2*c))/((a^(5/2 )*e^(5*d*x + 5*c) + 5*a^(5/2)*e^(4*d*x + 4*c) + 10*a^(5/2)*e^(3*d*x + 3*c) + 10*a^(5/2)*e^(2*d*x + 2*c) + 5*a^(5/2)*e^(d*x + c) + a^(5/2))*d) + 15*a rctan(e^(1/2*d*x + 1/2*c))/(a^(5/2)*d)) - 8/5*sqrt(2)*e^(5/2*d*x + 5/2*c)/ (a^(5/2)*d*e^(5*d*x + 5*c) + 5*a^(5/2)*d*e^(4*d*x + 4*c) + 10*a^(5/2)*d*e^ (3*d*x + 3*c) + 10*a^(5/2)*d*e^(2*d*x + 2*c) + 5*a^(5/2)*d*e^(d*x + c) + a ^(5/2)*d)
Exception generated. \[ \int \frac {1}{(a+a \cosh (c+d x))^{5/2}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro unding error%%%{%%{[%%%{%%{[147456,0]:[1,0,-2]%%},[0]%%%},0]:[1,0,%%%{-1,[ 1]%%%}]%%
Timed out. \[ \int \frac {1}{(a+a \cosh (c+d x))^{5/2}} \, dx=\int \frac {1}{{\left (a+a\,\mathrm {cosh}\left (c+d\,x\right )\right )}^{5/2}} \,d x \]