Integrand size = 17, antiderivative size = 93 \[ \int \frac {A+B \cosh (x)}{(a+a \cosh (x))^{5/2}} \, dx=\frac {(3 A+5 B) \arctan \left (\frac {\sqrt {a} \sinh (x)}{\sqrt {2} \sqrt {a+a \cosh (x)}}\right )}{16 \sqrt {2} a^{5/2}}+\frac {(A-B) \sinh (x)}{4 (a+a \cosh (x))^{5/2}}+\frac {(3 A+5 B) \sinh (x)}{16 a (a+a \cosh (x))^{3/2}} \]
1/4*(A-B)*sinh(x)/(a+a*cosh(x))^(5/2)+1/16*(3*A+5*B)*sinh(x)/a/(a+a*cosh(x ))^(3/2)+1/32*(3*A+5*B)*arctan(1/2*sinh(x)*a^(1/2)*2^(1/2)/(a+a*cosh(x))^( 1/2))/a^(5/2)*2^(1/2)
Time = 0.11 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.61 \[ \int \frac {A+B \cosh (x)}{(a+a \cosh (x))^{5/2}} \, dx=\frac {4 (3 A+5 B) \arctan \left (\sinh \left (\frac {x}{2}\right )\right ) \cosh ^5\left (\frac {x}{2}\right )+(7 A+B+(3 A+5 B) \cosh (x)) \sinh (x)}{16 (a (1+\cosh (x)))^{5/2}} \]
(4*(3*A + 5*B)*ArcTan[Sinh[x/2]]*Cosh[x/2]^5 + (7*A + B + (3*A + 5*B)*Cosh [x])*Sinh[x])/(16*(a*(1 + Cosh[x]))^(5/2))
Time = 0.39 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.98, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {3042, 3229, 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.
| \(\displaystyle \int \frac {A+B \cosh (x)}{(a \cosh (x)+a)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+B \sin \left (\frac {\pi }{2}+i x\right )}{\left (a+a \sin \left (\frac {\pi }{2}+i x\right )\right )^{5/2}}dx\) |
| \(\Big \downarrow \) 3229 |
| \(\displaystyle \frac {(3 A+5 B) \int \frac {1}{(\cosh (x) a+a)^{3/2}}dx}{8 a}+\frac {(A-B) \sinh (x)}{4 (a \cosh (x)+a)^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(A-B) \sinh (x)}{4 (a \cosh (x)+a)^{5/2}}+\frac {(3 A+5 B) \int \frac {1}{\left (\sin \left (i x+\frac {\pi }{2}\right ) a+a\right )^{3/2}}dx}{8 a}\) |
| \(\Big \downarrow \) 3129 |
| \(\displaystyle \frac {(3 A+5 B) \left (\frac {\int \frac {1}{\sqrt {\cosh (x) a+a}}dx}{4 a}+\frac {\sinh (x)}{2 (a \cosh (x)+a)^{3/2}}\right )}{8 a}+\frac {(A-B) \sinh (x)}{4 (a \cosh (x)+a)^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(A-B) \sinh (x)}{4 (a \cosh (x)+a)^{5/2}}+\frac {(3 A+5 B) \left (\frac {\sinh (x)}{2 (a \cosh (x)+a)^{3/2}}+\frac {\int \frac {1}{\sqrt {\sin \left (i x+\frac {\pi }{2}\right ) a+a}}dx}{4 a}\right )}{8 a}\) |
| \(\Big \downarrow \) 3128 |
| \(\displaystyle \frac {(A-B) \sinh (x)}{4 (a \cosh (x)+a)^{5/2}}+\frac {(3 A+5 B) \left (\frac {\sinh (x)}{2 (a \cosh (x)+a)^{3/2}}+\frac {i \int \frac {1}{\frac {a^2 \sinh ^2(x)}{\cosh (x) a+a}+2 a}d\left (-\frac {i a \sinh (x)}{\sqrt {\cosh (x) a+a}}\right )}{2 a}\right )}{8 a}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {(3 A+5 B) \left (\frac {\arctan \left (\frac {\sqrt {a} \sinh (x)}{\sqrt {2} \sqrt {a \cosh (x)+a}}\right )}{2 \sqrt {2} a^{3/2}}+\frac {\sinh (x)}{2 (a \cosh (x)+a)^{3/2}}\right )}{8 a}+\frac {(A-B) \sinh (x)}{4 (a \cosh (x)+a)^{5/2}}\) |
((A - B)*Sinh[x])/(4*(a + a*Cosh[x])^(5/2)) + ((3*A + 5*B)*(ArcTan[(Sqrt[a ]*Sinh[x])/(Sqrt[2]*Sqrt[a + a*Cosh[x]])]/(2*Sqrt[2]*a^(3/2)) + Sinh[x]/(2 *(a + a*Cosh[x])^(3/2))))/(8*a)
3.2.3.3.1 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]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*c - a*d)*Cos[e + f*x]*((a + b*Sin[e + f* x])^m/(a*f*(2*m + 1))), x] + Simp[(a*d*m + b*c*(m + 1))/(a*b*(2*m + 1)) I nt[(a + b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && LtQ[m, -2^(-1)]
Leaf count of result is larger than twice the leaf count of optimal. \(208\) vs. \(2(74)=148\).
Time = 0.51 (sec) , antiderivative size = 209, normalized size of antiderivative = 2.25
| method | result | size |
| default | \(-\frac {\sqrt {\sinh \left (\frac {x}{2}\right )^{2} a}\, \left (3 A \ln \left (\frac {2 \sqrt {\sinh \left (\frac {x}{2}\right )^{2} a}\, \sqrt {-a}-2 a}{\cosh \left (\frac {x}{2}\right )}\right ) \cosh \left (\frac {x}{2}\right )^{4} a +5 B \ln \left (\frac {2 \sqrt {\sinh \left (\frac {x}{2}\right )^{2} a}\, \sqrt {-a}-2 a}{\cosh \left (\frac {x}{2}\right )}\right ) \cosh \left (\frac {x}{2}\right )^{4} a -3 A \sqrt {\sinh \left (\frac {x}{2}\right )^{2} a}\, \cosh \left (\frac {x}{2}\right )^{2} \sqrt {-a}-5 B \sqrt {\sinh \left (\frac {x}{2}\right )^{2} a}\, \sqrt {-a}\, \cosh \left (\frac {x}{2}\right )^{2}-2 A \sqrt {-a}\, \sqrt {\sinh \left (\frac {x}{2}\right )^{2} a}+2 B \sqrt {\sinh \left (\frac {x}{2}\right )^{2} a}\, \sqrt {-a}\right ) \sqrt {2}}{32 \cosh \left (\frac {x}{2}\right )^{3} a^{3} \sqrt {-a}\, \sinh \left (\frac {x}{2}\right ) \sqrt {a \cosh \left (\frac {x}{2}\right )^{2}}}\) | \(209\) |
| parts | \(-\frac {A \sqrt {\sinh \left (\frac {x}{2}\right )^{2} a}\, \left (3 \ln \left (\frac {2 \sqrt {\sinh \left (\frac {x}{2}\right )^{2} a}\, \sqrt {-a}-2 a}{\cosh \left (\frac {x}{2}\right )}\right ) a \cosh \left (\frac {x}{2}\right )^{4}-3 \sqrt {\sinh \left (\frac {x}{2}\right )^{2} a}\, \cosh \left (\frac {x}{2}\right )^{2} \sqrt {-a}-2 \sqrt {\sinh \left (\frac {x}{2}\right )^{2} a}\, \sqrt {-a}\right ) \sqrt {2}}{32 a^{3} \cosh \left (\frac {x}{2}\right )^{3} \sqrt {-a}\, \sinh \left (\frac {x}{2}\right ) \sqrt {a \cosh \left (\frac {x}{2}\right )^{2}}}-\frac {B \sqrt {\sinh \left (\frac {x}{2}\right )^{2} a}\, \left (5 \ln \left (\frac {2 \sqrt {\sinh \left (\frac {x}{2}\right )^{2} a}\, \sqrt {-a}-2 a}{\cosh \left (\frac {x}{2}\right )}\right ) a \cosh \left (\frac {x}{2}\right )^{4}-5 \sqrt {\sinh \left (\frac {x}{2}\right )^{2} a}\, \cosh \left (\frac {x}{2}\right )^{2} \sqrt {-a}+2 \sqrt {\sinh \left (\frac {x}{2}\right )^{2} a}\, \sqrt {-a}\right ) \sqrt {2}}{32 \cosh \left (\frac {x}{2}\right )^{3} a^{3} \sqrt {-a}\, \sinh \left (\frac {x}{2}\right ) \sqrt {a \cosh \left (\frac {x}{2}\right )^{2}}}\) | \(252\) |
-1/32*(sinh(1/2*x)^2*a)^(1/2)*(3*A*ln(2/cosh(1/2*x)*((sinh(1/2*x)^2*a)^(1/ 2)*(-a)^(1/2)-a))*cosh(1/2*x)^4*a+5*B*ln(2/cosh(1/2*x)*((sinh(1/2*x)^2*a)^ (1/2)*(-a)^(1/2)-a))*cosh(1/2*x)^4*a-3*A*(sinh(1/2*x)^2*a)^(1/2)*cosh(1/2* x)^2*(-a)^(1/2)-5*B*(sinh(1/2*x)^2*a)^(1/2)*(-a)^(1/2)*cosh(1/2*x)^2-2*A*( -a)^(1/2)*(sinh(1/2*x)^2*a)^(1/2)+2*B*(sinh(1/2*x)^2*a)^(1/2)*(-a)^(1/2))/ cosh(1/2*x)^3/a^3/(-a)^(1/2)/sinh(1/2*x)*2^(1/2)/(a*cosh(1/2*x)^2)^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 509 vs. \(2 (74) = 148\).
Time = 0.28 (sec) , antiderivative size = 509, normalized size of antiderivative = 5.47 \[ \int \frac {A+B \cosh (x)}{(a+a \cosh (x))^{5/2}} \, dx=-\frac {\sqrt {2} {\left ({\left (3 \, A + 5 \, B\right )} \cosh \left (x\right )^{4} + {\left (3 \, A + 5 \, B\right )} \sinh \left (x\right )^{4} + 4 \, {\left (3 \, A + 5 \, B\right )} \cosh \left (x\right )^{3} + 4 \, {\left ({\left (3 \, A + 5 \, B\right )} \cosh \left (x\right ) + 3 \, A + 5 \, B\right )} \sinh \left (x\right )^{3} + 6 \, {\left (3 \, A + 5 \, B\right )} \cosh \left (x\right )^{2} + 6 \, {\left ({\left (3 \, A + 5 \, B\right )} \cosh \left (x\right )^{2} + 2 \, {\left (3 \, A + 5 \, B\right )} \cosh \left (x\right ) + 3 \, A + 5 \, B\right )} \sinh \left (x\right )^{2} + 4 \, {\left (3 \, A + 5 \, B\right )} \cosh \left (x\right ) + 4 \, {\left ({\left (3 \, A + 5 \, B\right )} \cosh \left (x\right )^{3} + 3 \, {\left (3 \, A + 5 \, B\right )} \cosh \left (x\right )^{2} + 3 \, {\left (3 \, A + 5 \, B\right )} \cosh \left (x\right ) + 3 \, A + 5 \, B\right )} \sinh \left (x\right ) + 3 \, A + 5 \, B\right )} \sqrt {a} \arctan \left (\frac {\sqrt {2} \sqrt {\frac {1}{2}} \sqrt {\frac {a}{\cosh \left (x\right ) + \sinh \left (x\right )}}}{\sqrt {a}}\right ) - 2 \, \sqrt {\frac {1}{2}} {\left ({\left (3 \, A + 5 \, B\right )} \cosh \left (x\right )^{4} + {\left (3 \, A + 5 \, B\right )} \sinh \left (x\right )^{4} + {\left (11 \, A - 3 \, B\right )} \cosh \left (x\right )^{3} + {\left (4 \, {\left (3 \, A + 5 \, B\right )} \cosh \left (x\right ) + 11 \, A - 3 \, B\right )} \sinh \left (x\right )^{3} - {\left (11 \, A - 3 \, B\right )} \cosh \left (x\right )^{2} + {\left (6 \, {\left (3 \, A + 5 \, B\right )} \cosh \left (x\right )^{2} + 3 \, {\left (11 \, A - 3 \, B\right )} \cosh \left (x\right ) - 11 \, A + 3 \, B\right )} \sinh \left (x\right )^{2} - {\left (3 \, A + 5 \, B\right )} \cosh \left (x\right ) + {\left (4 \, {\left (3 \, A + 5 \, B\right )} \cosh \left (x\right )^{3} + 3 \, {\left (11 \, A - 3 \, B\right )} \cosh \left (x\right )^{2} - 2 \, {\left (11 \, A - 3 \, B\right )} \cosh \left (x\right ) - 3 \, A - 5 \, B\right )} \sinh \left (x\right )\right )} \sqrt {\frac {a}{\cosh \left (x\right ) + \sinh \left (x\right )}}}{16 \, {\left (a^{3} \cosh \left (x\right )^{4} + a^{3} \sinh \left (x\right )^{4} + 4 \, a^{3} \cosh \left (x\right )^{3} + 6 \, a^{3} \cosh \left (x\right )^{2} + 4 \, a^{3} \cosh \left (x\right ) + 4 \, {\left (a^{3} \cosh \left (x\right ) + a^{3}\right )} \sinh \left (x\right )^{3} + a^{3} + 6 \, {\left (a^{3} \cosh \left (x\right )^{2} + 2 \, a^{3} \cosh \left (x\right ) + a^{3}\right )} \sinh \left (x\right )^{2} + 4 \, {\left (a^{3} \cosh \left (x\right )^{3} + 3 \, a^{3} \cosh \left (x\right )^{2} + 3 \, a^{3} \cosh \left (x\right ) + a^{3}\right )} \sinh \left (x\right )\right )}} \]
-1/16*(sqrt(2)*((3*A + 5*B)*cosh(x)^4 + (3*A + 5*B)*sinh(x)^4 + 4*(3*A + 5 *B)*cosh(x)^3 + 4*((3*A + 5*B)*cosh(x) + 3*A + 5*B)*sinh(x)^3 + 6*(3*A + 5 *B)*cosh(x)^2 + 6*((3*A + 5*B)*cosh(x)^2 + 2*(3*A + 5*B)*cosh(x) + 3*A + 5 *B)*sinh(x)^2 + 4*(3*A + 5*B)*cosh(x) + 4*((3*A + 5*B)*cosh(x)^3 + 3*(3*A + 5*B)*cosh(x)^2 + 3*(3*A + 5*B)*cosh(x) + 3*A + 5*B)*sinh(x) + 3*A + 5*B) *sqrt(a)*arctan(sqrt(2)*sqrt(1/2)*sqrt(a/(cosh(x) + sinh(x)))/sqrt(a)) - 2 *sqrt(1/2)*((3*A + 5*B)*cosh(x)^4 + (3*A + 5*B)*sinh(x)^4 + (11*A - 3*B)*c osh(x)^3 + (4*(3*A + 5*B)*cosh(x) + 11*A - 3*B)*sinh(x)^3 - (11*A - 3*B)*c osh(x)^2 + (6*(3*A + 5*B)*cosh(x)^2 + 3*(11*A - 3*B)*cosh(x) - 11*A + 3*B) *sinh(x)^2 - (3*A + 5*B)*cosh(x) + (4*(3*A + 5*B)*cosh(x)^3 + 3*(11*A - 3* B)*cosh(x)^2 - 2*(11*A - 3*B)*cosh(x) - 3*A - 5*B)*sinh(x))*sqrt(a/(cosh(x ) + sinh(x))))/(a^3*cosh(x)^4 + a^3*sinh(x)^4 + 4*a^3*cosh(x)^3 + 6*a^3*co sh(x)^2 + 4*a^3*cosh(x) + 4*(a^3*cosh(x) + a^3)*sinh(x)^3 + a^3 + 6*(a^3*c osh(x)^2 + 2*a^3*cosh(x) + a^3)*sinh(x)^2 + 4*(a^3*cosh(x)^3 + 3*a^3*cosh( x)^2 + 3*a^3*cosh(x) + a^3)*sinh(x))
\[ \int \frac {A+B \cosh (x)}{(a+a \cosh (x))^{5/2}} \, dx=\int \frac {A + B \cosh {\left (x \right )}}{\left (a \left (\cosh {\left (x \right )} + 1\right )\right )^{\frac {5}{2}}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 427 vs. \(2 (74) = 148\).
Time = 0.39 (sec) , antiderivative size = 427, normalized size of antiderivative = 4.59 \[ \int \frac {A+B \cosh (x)}{(a+a \cosh (x))^{5/2}} \, dx=\frac {1}{80} \, {\left (\sqrt {2} {\left (\frac {15 \, e^{\left (\frac {9}{2} \, x\right )} + 70 \, e^{\left (\frac {7}{2} \, x\right )} + 128 \, e^{\left (\frac {5}{2} \, x\right )} - 70 \, e^{\left (\frac {3}{2} \, x\right )} - 15 \, e^{\left (\frac {1}{2} \, x\right )}}{a^{\frac {5}{2}} e^{\left (5 \, x\right )} + 5 \, a^{\frac {5}{2}} e^{\left (4 \, x\right )} + 10 \, a^{\frac {5}{2}} e^{\left (3 \, x\right )} + 10 \, a^{\frac {5}{2}} e^{\left (2 \, x\right )} + 5 \, a^{\frac {5}{2}} e^{x} + a^{\frac {5}{2}}} + \frac {15 \, \arctan \left (e^{\left (\frac {1}{2} \, x\right )}\right )}{a^{\frac {5}{2}}}\right )} - \frac {128 \, \sqrt {2} e^{\left (\frac {5}{2} \, x\right )}}{a^{\frac {5}{2}} e^{\left (5 \, x\right )} + 5 \, a^{\frac {5}{2}} e^{\left (4 \, x\right )} + 10 \, a^{\frac {5}{2}} e^{\left (3 \, x\right )} + 10 \, a^{\frac {5}{2}} e^{\left (2 \, x\right )} + 5 \, a^{\frac {5}{2}} e^{x} + a^{\frac {5}{2}}}\right )} A + \frac {1}{672} \, {\left (\sqrt {2} {\left (\frac {105 \, e^{\left (\frac {9}{2} \, x\right )} + 490 \, e^{\left (\frac {7}{2} \, x\right )} + 896 \, e^{\left (\frac {5}{2} \, x\right )} + 790 \, e^{\left (\frac {3}{2} \, x\right )} - 105 \, e^{\left (\frac {1}{2} \, x\right )}}{a^{\frac {5}{2}} e^{\left (5 \, x\right )} + 5 \, a^{\frac {5}{2}} e^{\left (4 \, x\right )} + 10 \, a^{\frac {5}{2}} e^{\left (3 \, x\right )} + 10 \, a^{\frac {5}{2}} e^{\left (2 \, x\right )} + 5 \, a^{\frac {5}{2}} e^{x} + a^{\frac {5}{2}}} + \frac {105 \, \arctan \left (e^{\left (\frac {1}{2} \, x\right )}\right )}{a^{\frac {5}{2}}}\right )} + 7 \, \sqrt {2} {\left (\frac {15 \, e^{\left (\frac {9}{2} \, x\right )} + 70 \, e^{\left (\frac {7}{2} \, x\right )} - 128 \, e^{\left (\frac {5}{2} \, x\right )} - 70 \, e^{\left (\frac {3}{2} \, x\right )} - 15 \, e^{\left (\frac {1}{2} \, x\right )}}{a^{\frac {5}{2}} e^{\left (5 \, x\right )} + 5 \, a^{\frac {5}{2}} e^{\left (4 \, x\right )} + 10 \, a^{\frac {5}{2}} e^{\left (3 \, x\right )} + 10 \, a^{\frac {5}{2}} e^{\left (2 \, x\right )} + 5 \, a^{\frac {5}{2}} e^{x} + a^{\frac {5}{2}}} + \frac {15 \, \arctan \left (e^{\left (\frac {1}{2} \, x\right )}\right )}{a^{\frac {5}{2}}}\right )} - \frac {128 \, {\left (7 \, \sqrt {2} \sqrt {a} e^{\left (\frac {7}{2} \, x\right )} + 3 \, \sqrt {2} \sqrt {a} e^{\left (\frac {3}{2} \, x\right )}\right )}}{a^{3} e^{\left (5 \, x\right )} + 5 \, a^{3} e^{\left (4 \, x\right )} + 10 \, a^{3} e^{\left (3 \, x\right )} + 10 \, a^{3} e^{\left (2 \, x\right )} + 5 \, a^{3} e^{x} + a^{3}}\right )} B \]
1/80*(sqrt(2)*((15*e^(9/2*x) + 70*e^(7/2*x) + 128*e^(5/2*x) - 70*e^(3/2*x) - 15*e^(1/2*x))/(a^(5/2)*e^(5*x) + 5*a^(5/2)*e^(4*x) + 10*a^(5/2)*e^(3*x) + 10*a^(5/2)*e^(2*x) + 5*a^(5/2)*e^x + a^(5/2)) + 15*arctan(e^(1/2*x))/a^ (5/2)) - 128*sqrt(2)*e^(5/2*x)/(a^(5/2)*e^(5*x) + 5*a^(5/2)*e^(4*x) + 10*a ^(5/2)*e^(3*x) + 10*a^(5/2)*e^(2*x) + 5*a^(5/2)*e^x + a^(5/2)))*A + 1/672* (sqrt(2)*((105*e^(9/2*x) + 490*e^(7/2*x) + 896*e^(5/2*x) + 790*e^(3/2*x) - 105*e^(1/2*x))/(a^(5/2)*e^(5*x) + 5*a^(5/2)*e^(4*x) + 10*a^(5/2)*e^(3*x) + 10*a^(5/2)*e^(2*x) + 5*a^(5/2)*e^x + a^(5/2)) + 105*arctan(e^(1/2*x))/a^ (5/2)) + 7*sqrt(2)*((15*e^(9/2*x) + 70*e^(7/2*x) - 128*e^(5/2*x) - 70*e^(3 /2*x) - 15*e^(1/2*x))/(a^(5/2)*e^(5*x) + 5*a^(5/2)*e^(4*x) + 10*a^(5/2)*e^ (3*x) + 10*a^(5/2)*e^(2*x) + 5*a^(5/2)*e^x + a^(5/2)) + 15*arctan(e^(1/2*x ))/a^(5/2)) - 128*(7*sqrt(2)*sqrt(a)*e^(7/2*x) + 3*sqrt(2)*sqrt(a)*e^(3/2* x))/(a^3*e^(5*x) + 5*a^3*e^(4*x) + 10*a^3*e^(3*x) + 10*a^3*e^(2*x) + 5*a^3 *e^x + a^3))*B
Time = 0.27 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.27 \[ \int \frac {A+B \cosh (x)}{(a+a \cosh (x))^{5/2}} \, dx=\frac {\sqrt {2} {\left (3 \, A + 5 \, B\right )} \arctan \left (e^{\left (\frac {1}{2} \, x\right )}\right )}{16 \, a^{\frac {5}{2}}} + \frac {\sqrt {2} {\left (3 \, A a^{\frac {7}{2}} e^{\left (\frac {7}{2} \, x\right )} + 5 \, B a^{\frac {7}{2}} e^{\left (\frac {7}{2} \, x\right )} + 11 \, A a^{\frac {7}{2}} e^{\left (\frac {5}{2} \, x\right )} - 3 \, B a^{\frac {7}{2}} e^{\left (\frac {5}{2} \, x\right )} - 11 \, A a^{\frac {7}{2}} e^{\left (\frac {3}{2} \, x\right )} + 3 \, B a^{\frac {7}{2}} e^{\left (\frac {3}{2} \, x\right )} - 3 \, A a^{\frac {7}{2}} e^{\left (\frac {1}{2} \, x\right )} - 5 \, B a^{\frac {7}{2}} e^{\left (\frac {1}{2} \, x\right )}\right )}}{16 \, {\left (a e^{x} + a\right )}^{4} a^{2}} \]
1/16*sqrt(2)*(3*A + 5*B)*arctan(e^(1/2*x))/a^(5/2) + 1/16*sqrt(2)*(3*A*a^( 7/2)*e^(7/2*x) + 5*B*a^(7/2)*e^(7/2*x) + 11*A*a^(7/2)*e^(5/2*x) - 3*B*a^(7 /2)*e^(5/2*x) - 11*A*a^(7/2)*e^(3/2*x) + 3*B*a^(7/2)*e^(3/2*x) - 3*A*a^(7/ 2)*e^(1/2*x) - 5*B*a^(7/2)*e^(1/2*x))/((a*e^x + a)^4*a^2)
Timed out. \[ \int \frac {A+B \cosh (x)}{(a+a \cosh (x))^{5/2}} \, dx=\int \frac {A+B\,\mathrm {cosh}\left (x\right )}{{\left (a+a\,\mathrm {cosh}\left (x\right )\right )}^{5/2}} \,d x \]