Integrand size = 18, antiderivative size = 98 \[ \int (a-a \cosh (x))^{5/2} (A+B \cosh (x)) \, dx=-\frac {64 a^3 (7 A-5 B) \sinh (x)}{105 \sqrt {a-a \cosh (x)}}-\frac {16}{105} a^2 (7 A-5 B) \sqrt {a-a \cosh (x)} \sinh (x)-\frac {2}{35} a (7 A-5 B) (a-a \cosh (x))^{3/2} \sinh (x)+\frac {2}{7} B (a-a \cosh (x))^{5/2} \sinh (x) \] Output:
-64/105*a^3*(7*A-5*B)*sinh(x)/(a-a*cosh(x))^(1/2)-16/105*a^2*(7*A-5*B)*(a- a*cosh(x))^(1/2)*sinh(x)-2/35*a*(7*A-5*B)*(a-a*cosh(x))^(3/2)*sinh(x)+2/7* B*(a-a*cosh(x))^(5/2)*sinh(x)
Time = 0.24 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.62 \[ \int (a-a \cosh (x))^{5/2} (A+B \cosh (x)) \, dx=\frac {1}{210} a^2 \sqrt {a-a \cosh (x)} (1246 A-1040 B+(-392 A+505 B) \cosh (x)+6 (7 A-20 B) \cosh (2 x)+15 B \cosh (3 x)) \coth \left (\frac {x}{2}\right ) \] Input:
Integrate[(a - a*Cosh[x])^(5/2)*(A + B*Cosh[x]),x]
Output:
(a^2*Sqrt[a - a*Cosh[x]]*(1246*A - 1040*B + (-392*A + 505*B)*Cosh[x] + 6*( 7*A - 20*B)*Cosh[2*x] + 15*B*Cosh[3*x])*Coth[x/2])/210
Time = 0.47 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.95, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {3042, 3230, 3042, 3126, 3042, 3126, 3042, 3125}
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.
| \(\displaystyle \int (a-a \cosh (x))^{5/2} (A+B \cosh (x)) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (a-a \sin \left (\frac {\pi }{2}+i x\right )\right )^{5/2} \left (A+B \sin \left (\frac {\pi }{2}+i x\right )\right )dx\) |
| \(\Big \downarrow \) 3230 |
| \(\displaystyle \frac {1}{7} (7 A-5 B) \int (a-a \cosh (x))^{5/2}dx+\frac {2}{7} B \sinh (x) (a-a \cosh (x))^{5/2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{7} B \sinh (x) (a-a \cosh (x))^{5/2}+\frac {1}{7} (7 A-5 B) \int \left (a-a \sin \left (i x+\frac {\pi }{2}\right )\right )^{5/2}dx\) |
| \(\Big \downarrow \) 3126 |
| \(\displaystyle \frac {1}{7} (7 A-5 B) \left (\frac {8}{5} a \int (a-a \cosh (x))^{3/2}dx-\frac {2}{5} a \sinh (x) (a-a \cosh (x))^{3/2}\right )+\frac {2}{7} B \sinh (x) (a-a \cosh (x))^{5/2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{7} B \sinh (x) (a-a \cosh (x))^{5/2}+\frac {1}{7} (7 A-5 B) \left (-\frac {2}{5} a \sinh (x) (a-a \cosh (x))^{3/2}+\frac {8}{5} a \int \left (a-a \sin \left (i x+\frac {\pi }{2}\right )\right )^{3/2}dx\right )\) |
| \(\Big \downarrow \) 3126 |
| \(\displaystyle \frac {1}{7} (7 A-5 B) \left (\frac {8}{5} a \left (\frac {4}{3} a \int \sqrt {a-a \cosh (x)}dx-\frac {2}{3} a \sinh (x) \sqrt {a-a \cosh (x)}\right )-\frac {2}{5} a \sinh (x) (a-a \cosh (x))^{3/2}\right )+\frac {2}{7} B \sinh (x) (a-a \cosh (x))^{5/2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{7} B \sinh (x) (a-a \cosh (x))^{5/2}+\frac {1}{7} (7 A-5 B) \left (-\frac {2}{5} a \sinh (x) (a-a \cosh (x))^{3/2}+\frac {8}{5} a \left (-\frac {2}{3} a \sinh (x) \sqrt {a-a \cosh (x)}+\frac {4}{3} a \int \sqrt {a-a \sin \left (i x+\frac {\pi }{2}\right )}dx\right )\right )\) |
| \(\Big \downarrow \) 3125 |
| \(\displaystyle \frac {1}{7} (7 A-5 B) \left (\frac {8}{5} a \left (-\frac {8 a^2 \sinh (x)}{3 \sqrt {a-a \cosh (x)}}-\frac {2}{3} a \sinh (x) \sqrt {a-a \cosh (x)}\right )-\frac {2}{5} a \sinh (x) (a-a \cosh (x))^{3/2}\right )+\frac {2}{7} B \sinh (x) (a-a \cosh (x))^{5/2}\) |
Input:
Int[(a - a*Cosh[x])^(5/2)*(A + B*Cosh[x]),x]
Output:
(2*B*(a - a*Cosh[x])^(5/2)*Sinh[x])/7 + ((7*A - 5*B)*((-2*a*(a - a*Cosh[x] )^(3/2)*Sinh[x])/5 + (8*a*((-8*a^2*Sinh[x])/(3*Sqrt[a - a*Cosh[x]]) - (2*a *Sqrt[a - a*Cosh[x]]*Sinh[x])/3))/5))/7
Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[-2*b*(Cos [c + d*x]/(d*Sqrt[a + b*Sin[c + d*x]])), x] /; FreeQ[{a, b, c, d}, x] && Eq Q[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 - 1)/(d*n)), x] + Simp[a*((2*n - 1)/n) Int[(a + b*Sin[c + d*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[ a^2 - b^2, 0] && IGtQ[n - 1/2, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-d)*Cos[e + f*x]*((a + b*Sin[e + f*x])^m/( f*(m + 1))), x] + Simp[(a*d*m + b*c*(m + 1))/(b*(m + 1)) Int[(a + b*Sin[e + f*x])^m, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && !LtQ[m, -2^(-1)]
Time = 1.71 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.70
| method | result | size |
| default | \(-\frac {16 \sinh \left (\frac {x}{2}\right ) a^{3} \cosh \left (\frac {x}{2}\right ) \left (30 B \sinh \left (\frac {x}{2}\right )^{6}+\left (21 A -15 B \right ) \sinh \left (\frac {x}{2}\right )^{4}+\left (-28 A +20 B \right ) \sinh \left (\frac {x}{2}\right )^{2}+56 A -40 B \right )}{105 \sqrt {-2 \sinh \left (\frac {x}{2}\right )^{2} a}}\) | \(69\) |
| parts | \(-\frac {16 A \sinh \left (\frac {x}{2}\right ) a^{3} \cosh \left (\frac {x}{2}\right ) \left (3 \sinh \left (\frac {x}{2}\right )^{4}-4 \sinh \left (\frac {x}{2}\right )^{2}+8\right )}{15 \sqrt {-2 \sinh \left (\frac {x}{2}\right )^{2} a}}-\frac {16 B \sinh \left (\frac {x}{2}\right ) a^{3} \cosh \left (\frac {x}{2}\right ) \left (6 \sinh \left (\frac {x}{2}\right )^{6}-3 \sinh \left (\frac {x}{2}\right )^{4}+4 \sinh \left (\frac {x}{2}\right )^{2}-8\right )}{21 \sqrt {-2 \sinh \left (\frac {x}{2}\right )^{2} a}}\) | \(96\) |
Input:
int((a-cosh(x)*a)^(5/2)*(A+B*cosh(x)),x,method=_RETURNVERBOSE)
Output:
-16/105*sinh(1/2*x)*a^3*cosh(1/2*x)*(30*B*sinh(1/2*x)^6+(21*A-15*B)*sinh(1 /2*x)^4+(-28*A+20*B)*sinh(1/2*x)^2+56*A-40*B)/(-2*sinh(1/2*x)^2*a)^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 564 vs. \(2 (82) = 164\).
Time = 0.10 (sec) , antiderivative size = 564, normalized size of antiderivative = 5.76 \[ \int (a-a \cosh (x))^{5/2} (A+B \cosh (x)) \, dx =\text {Too large to display} \] Input:
integrate((a-a*cosh(x))^(5/2)*(A+B*cosh(x)),x, algorithm="fricas")
Output:
1/420*sqrt(1/2)*(15*B*a^2*cosh(x)^7 + 15*B*a^2*sinh(x)^7 + 21*(2*A - 5*B)* a^2*cosh(x)^6 - 35*(10*A - 11*B)*a^2*cosh(x)^5 + 525*(4*A - 3*B)*a^2*cosh( x)^4 + 21*(5*B*a^2*cosh(x) + (2*A - 5*B)*a^2)*sinh(x)^6 + 525*(4*A - 3*B)* a^2*cosh(x)^3 + 7*(45*B*a^2*cosh(x)^2 + 18*(2*A - 5*B)*a^2*cosh(x) - 5*(10 *A - 11*B)*a^2)*sinh(x)^5 - 35*(10*A - 11*B)*a^2*cosh(x)^2 + 35*(15*B*a^2* cosh(x)^3 + 9*(2*A - 5*B)*a^2*cosh(x)^2 - 5*(10*A - 11*B)*a^2*cosh(x) + 15 *(4*A - 3*B)*a^2)*sinh(x)^4 + 21*(2*A - 5*B)*a^2*cosh(x) + 35*(15*B*a^2*co sh(x)^4 + 12*(2*A - 5*B)*a^2*cosh(x)^3 - 10*(10*A - 11*B)*a^2*cosh(x)^2 + 60*(4*A - 3*B)*a^2*cosh(x) + 15*(4*A - 3*B)*a^2)*sinh(x)^3 + 15*B*a^2 + 35 *(9*B*a^2*cosh(x)^5 + 9*(2*A - 5*B)*a^2*cosh(x)^4 - 10*(10*A - 11*B)*a^2*c osh(x)^3 + 90*(4*A - 3*B)*a^2*cosh(x)^2 + 45*(4*A - 3*B)*a^2*cosh(x) - (10 *A - 11*B)*a^2)*sinh(x)^2 + 7*(15*B*a^2*cosh(x)^6 + 18*(2*A - 5*B)*a^2*cos h(x)^5 - 25*(10*A - 11*B)*a^2*cosh(x)^4 + 300*(4*A - 3*B)*a^2*cosh(x)^3 + 225*(4*A - 3*B)*a^2*cosh(x)^2 - 10*(10*A - 11*B)*a^2*cosh(x) + 3*(2*A - 5* B)*a^2)*sinh(x))*sqrt(-a/(cosh(x) + sinh(x)))/(cosh(x)^3 + 3*cosh(x)^2*sin h(x) + 3*cosh(x)*sinh(x)^2 + sinh(x)^3)
Timed out. \[ \int (a-a \cosh (x))^{5/2} (A+B \cosh (x)) \, dx=\text {Timed out} \] Input:
integrate((a-a*cosh(x))**(5/2)*(A+B*cosh(x)),x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 288 vs. \(2 (82) = 164\).
Time = 0.14 (sec) , antiderivative size = 288, normalized size of antiderivative = 2.94 \[ \int (a-a \cosh (x))^{5/2} (A+B \cosh (x)) \, dx=\frac {1}{60} \, {\left (\frac {25 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (-x\right )}}{\left (-e^{\left (-x\right )}\right )^{\frac {5}{2}}} - \frac {150 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (-2 \, x\right )}}{\left (-e^{\left (-x\right )}\right )^{\frac {5}{2}}} - \frac {150 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (-3 \, x\right )}}{\left (-e^{\left (-x\right )}\right )^{\frac {5}{2}}} + \frac {25 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (-4 \, x\right )}}{\left (-e^{\left (-x\right )}\right )^{\frac {5}{2}}} - \frac {3 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (-5 \, x\right )}}{\left (-e^{\left (-x\right )}\right )^{\frac {5}{2}}} - \frac {3 \, \sqrt {2} a^{\frac {5}{2}}}{\left (-e^{\left (-x\right )}\right )^{\frac {5}{2}}}\right )} A + \frac {1}{168} \, B {\left (\frac {{\left (21 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (-x\right )} - 70 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (-2 \, x\right )} + 210 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (-3 \, x\right )} + 105 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (-4 \, x\right )} - 7 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (-5 \, x\right )} - 3 \, \sqrt {2} a^{\frac {5}{2}}\right )} e^{x}}{\left (-e^{\left (-x\right )}\right )^{\frac {5}{2}}} - \frac {7 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (-x\right )} - 105 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (-2 \, x\right )} - 210 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (-3 \, x\right )} + 70 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (-4 \, x\right )} - 21 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (-5 \, x\right )} + 3 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (-6 \, x\right )}}{\left (-e^{\left (-x\right )}\right )^{\frac {5}{2}}}\right )} \] Input:
integrate((a-a*cosh(x))^(5/2)*(A+B*cosh(x)),x, algorithm="maxima")
Output:
1/60*(25*sqrt(2)*a^(5/2)*e^(-x)/(-e^(-x))^(5/2) - 150*sqrt(2)*a^(5/2)*e^(- 2*x)/(-e^(-x))^(5/2) - 150*sqrt(2)*a^(5/2)*e^(-3*x)/(-e^(-x))^(5/2) + 25*s qrt(2)*a^(5/2)*e^(-4*x)/(-e^(-x))^(5/2) - 3*sqrt(2)*a^(5/2)*e^(-5*x)/(-e^( -x))^(5/2) - 3*sqrt(2)*a^(5/2)/(-e^(-x))^(5/2))*A + 1/168*B*((21*sqrt(2)*a ^(5/2)*e^(-x) - 70*sqrt(2)*a^(5/2)*e^(-2*x) + 210*sqrt(2)*a^(5/2)*e^(-3*x) + 105*sqrt(2)*a^(5/2)*e^(-4*x) - 7*sqrt(2)*a^(5/2)*e^(-5*x) - 3*sqrt(2)*a ^(5/2))*e^x/(-e^(-x))^(5/2) - (7*sqrt(2)*a^(5/2)*e^(-x) - 105*sqrt(2)*a^(5 /2)*e^(-2*x) - 210*sqrt(2)*a^(5/2)*e^(-3*x) + 70*sqrt(2)*a^(5/2)*e^(-4*x) - 21*sqrt(2)*a^(5/2)*e^(-5*x) + 3*sqrt(2)*a^(5/2)*e^(-6*x))/(-e^(-x))^(5/2 ))
Leaf count of result is larger than twice the leaf count of optimal. 295 vs. \(2 (82) = 164\).
Time = 0.13 (sec) , antiderivative size = 295, normalized size of antiderivative = 3.01 \[ \int (a-a \cosh (x))^{5/2} (A+B \cosh (x)) \, dx=\frac {1}{840} \, \sqrt {2} {\left (\frac {{\left (2100 \, A a^{6} e^{\left (3 \, x\right )} \mathrm {sgn}\left (-e^{x} + 1\right ) - 1575 \, B a^{6} e^{\left (3 \, x\right )} \mathrm {sgn}\left (-e^{x} + 1\right ) - 350 \, A a^{6} e^{\left (2 \, x\right )} \mathrm {sgn}\left (-e^{x} + 1\right ) + 385 \, B a^{6} e^{\left (2 \, x\right )} \mathrm {sgn}\left (-e^{x} + 1\right ) + 42 \, A a^{6} e^{x} \mathrm {sgn}\left (-e^{x} + 1\right ) - 105 \, B a^{6} e^{x} \mathrm {sgn}\left (-e^{x} + 1\right ) + 15 \, B a^{6} \mathrm {sgn}\left (-e^{x} + 1\right )\right )} e^{\left (-3 \, x\right )}}{\sqrt {-a e^{x}} a^{3}} - \frac {15 \, \sqrt {-a e^{x}} B a^{9} e^{\left (3 \, x\right )} \mathrm {sgn}\left (-e^{x} + 1\right ) + 42 \, \sqrt {-a e^{x}} A a^{9} e^{\left (2 \, x\right )} \mathrm {sgn}\left (-e^{x} + 1\right ) - 105 \, \sqrt {-a e^{x}} B a^{9} e^{\left (2 \, x\right )} \mathrm {sgn}\left (-e^{x} + 1\right ) - 350 \, \sqrt {-a e^{x}} A a^{9} e^{x} \mathrm {sgn}\left (-e^{x} + 1\right ) + 385 \, \sqrt {-a e^{x}} B a^{9} e^{x} \mathrm {sgn}\left (-e^{x} + 1\right ) + 2100 \, \sqrt {-a e^{x}} A a^{9} \mathrm {sgn}\left (-e^{x} + 1\right ) - 1575 \, \sqrt {-a e^{x}} B a^{9} \mathrm {sgn}\left (-e^{x} + 1\right )}{a^{7}}\right )} \] Input:
integrate((a-a*cosh(x))^(5/2)*(A+B*cosh(x)),x, algorithm="giac")
Output:
1/840*sqrt(2)*((2100*A*a^6*e^(3*x)*sgn(-e^x + 1) - 1575*B*a^6*e^(3*x)*sgn( -e^x + 1) - 350*A*a^6*e^(2*x)*sgn(-e^x + 1) + 385*B*a^6*e^(2*x)*sgn(-e^x + 1) + 42*A*a^6*e^x*sgn(-e^x + 1) - 105*B*a^6*e^x*sgn(-e^x + 1) + 15*B*a^6* sgn(-e^x + 1))*e^(-3*x)/(sqrt(-a*e^x)*a^3) - (15*sqrt(-a*e^x)*B*a^9*e^(3*x )*sgn(-e^x + 1) + 42*sqrt(-a*e^x)*A*a^9*e^(2*x)*sgn(-e^x + 1) - 105*sqrt(- a*e^x)*B*a^9*e^(2*x)*sgn(-e^x + 1) - 350*sqrt(-a*e^x)*A*a^9*e^x*sgn(-e^x + 1) + 385*sqrt(-a*e^x)*B*a^9*e^x*sgn(-e^x + 1) + 2100*sqrt(-a*e^x)*A*a^9*s gn(-e^x + 1) - 1575*sqrt(-a*e^x)*B*a^9*sgn(-e^x + 1))/a^7)
Timed out. \[ \int (a-a \cosh (x))^{5/2} (A+B \cosh (x)) \, dx=\int \left (A+B\,\mathrm {cosh}\left (x\right )\right )\,{\left (a-a\,\mathrm {cosh}\left (x\right )\right )}^{5/2} \,d x \] Input:
int((A + B*cosh(x))*(a - a*cosh(x))^(5/2),x)
Output:
int((A + B*cosh(x))*(a - a*cosh(x))^(5/2), x)
\[ \int (a-a \cosh (x))^{5/2} (A+B \cosh (x)) \, dx=\sqrt {a}\, a^{2} \left (\left (\int \sqrt {-\cosh \left (x \right )+1}d x \right ) a -2 \left (\int \sqrt {-\cosh \left (x \right )+1}\, \cosh \left (x \right )d x \right ) a +\left (\int \sqrt {-\cosh \left (x \right )+1}\, \cosh \left (x \right )d x \right ) b +\left (\int \sqrt {-\cosh \left (x \right )+1}\, \cosh \left (x \right )^{3}d x \right ) b +\left (\int \sqrt {-\cosh \left (x \right )+1}\, \cosh \left (x \right )^{2}d x \right ) a -2 \left (\int \sqrt {-\cosh \left (x \right )+1}\, \cosh \left (x \right )^{2}d x \right ) b \right ) \] Input:
int((a-a*cosh(x))^(5/2)*(A+B*cosh(x)),x)
Output:
sqrt(a)*a**2*(int(sqrt( - cosh(x) + 1),x)*a - 2*int(sqrt( - cosh(x) + 1)*c osh(x),x)*a + int(sqrt( - cosh(x) + 1)*cosh(x),x)*b + int(sqrt( - cosh(x) + 1)*cosh(x)**3,x)*b + int(sqrt( - cosh(x) + 1)*cosh(x)**2,x)*a - 2*int(sq rt( - cosh(x) + 1)*cosh(x)**2,x)*b)