Integrand size = 18, antiderivative size = 71 \[ \int (a-a \cosh (x))^{3/2} (A+B \cosh (x)) \, dx=-\frac {8 a^2 (5 A-3 B) \sinh (x)}{15 \sqrt {a-a \cosh (x)}}-\frac {2}{15} a (5 A-3 B) \sqrt {a-a \cosh (x)} \sinh (x)+\frac {2}{5} B (a-a \cosh (x))^{3/2} \sinh (x) \] Output:
-8/15*a^2*(5*A-3*B)*sinh(x)/(a-a*cosh(x))^(1/2)-2/15*a*(5*A-3*B)*(a-a*cosh (x))^(1/2)*sinh(x)+2/5*B*(a-a*cosh(x))^(3/2)*sinh(x)
Time = 0.19 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.66 \[ \int (a-a \cosh (x))^{3/2} (A+B \cosh (x)) \, dx=-\frac {1}{15} a \sqrt {a-a \cosh (x)} (-50 A+39 B+2 (5 A-9 B) \cosh (x)+3 B \cosh (2 x)) \coth \left (\frac {x}{2}\right ) \] Input:
Integrate[(a - a*Cosh[x])^(3/2)*(A + B*Cosh[x]),x]
Output:
-1/15*(a*Sqrt[a - a*Cosh[x]]*(-50*A + 39*B + 2*(5*A - 9*B)*Cosh[x] + 3*B*C osh[2*x])*Coth[x/2])
Time = 0.37 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.97, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3042, 3230, 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.
| \(\displaystyle \int (a-a \cosh (x))^{3/2} (A+B \cosh (x)) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (a-a \sin \left (\frac {\pi }{2}+i x\right )\right )^{3/2} \left (A+B \sin \left (\frac {\pi }{2}+i x\right )\right )dx\) |
| \(\Big \downarrow \) 3230 |
| \(\displaystyle \frac {1}{5} (5 A-3 B) \int (a-a \cosh (x))^{3/2}dx+\frac {2}{5} B \sinh (x) (a-a \cosh (x))^{3/2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{5} B \sinh (x) (a-a \cosh (x))^{3/2}+\frac {1}{5} (5 A-3 B) \int \left (a-a \sin \left (i x+\frac {\pi }{2}\right )\right )^{3/2}dx\) |
| \(\Big \downarrow \) 3126 |
| \(\displaystyle \frac {1}{5} (5 A-3 B) \left (\frac {4}{3} a \int \sqrt {a-a \cosh (x)}dx-\frac {2}{3} a \sinh (x) \sqrt {a-a \cosh (x)}\right )+\frac {2}{5} B \sinh (x) (a-a \cosh (x))^{3/2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{5} B \sinh (x) (a-a \cosh (x))^{3/2}+\frac {1}{5} (5 A-3 B) \left (-\frac {2}{3} a \sinh (x) \sqrt {a-a \cosh (x)}+\frac {4}{3} a \int \sqrt {a-a \sin \left (i x+\frac {\pi }{2}\right )}dx\right )\) |
| \(\Big \downarrow \) 3125 |
| \(\displaystyle \frac {1}{5} (5 A-3 B) \left (-\frac {8 a^2 \sinh (x)}{3 \sqrt {a-a \cosh (x)}}-\frac {2}{3} a \sinh (x) \sqrt {a-a \cosh (x)}\right )+\frac {2}{5} B \sinh (x) (a-a \cosh (x))^{3/2}\) |
Input:
Int[(a - a*Cosh[x])^(3/2)*(A + B*Cosh[x]),x]
Output:
(2*B*(a - a*Cosh[x])^(3/2)*Sinh[x])/5 + ((5*A - 3*B)*((-8*a^2*Sinh[x])/(3* Sqrt[a - a*Cosh[x]]) - (2*a*Sqrt[a - a*Cosh[x]]*Sinh[x])/3))/5
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 = 0.90 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.77
| method | result | size |
| default | \(\frac {8 \sinh \left (\frac {x}{2}\right ) a^{2} \cosh \left (\frac {x}{2}\right ) \left (6 B \sinh \left (\frac {x}{2}\right )^{4}+\left (5 A -3 B \right ) \sinh \left (\frac {x}{2}\right )^{2}-10 A +6 B \right )}{15 \sqrt {-2 \sinh \left (\frac {x}{2}\right )^{2} a}}\) | \(55\) |
| parts | \(\frac {8 A \sinh \left (\frac {x}{2}\right ) a^{2} \cosh \left (\frac {x}{2}\right ) \left (\cosh \left (\frac {x}{2}\right )^{2}-3\right )}{3 \sqrt {-2 \sinh \left (\frac {x}{2}\right )^{2} a}}+\frac {8 B \sinh \left (\frac {x}{2}\right ) a^{2} \cosh \left (\frac {x}{2}\right ) \left (2 \cosh \left (\frac {x}{2}\right )^{4}-5 \cosh \left (\frac {x}{2}\right )^{2}+5\right )}{5 \sqrt {-2 \sinh \left (\frac {x}{2}\right )^{2} a}}\) | \(78\) |
Input:
int((a-cosh(x)*a)^(3/2)*(A+B*cosh(x)),x,method=_RETURNVERBOSE)
Output:
8/15*sinh(1/2*x)*a^2*cosh(1/2*x)*(6*B*sinh(1/2*x)^4+(5*A-3*B)*sinh(1/2*x)^ 2-10*A+6*B)/(-2*sinh(1/2*x)^2*a)^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 279 vs. \(2 (59) = 118\).
Time = 0.10 (sec) , antiderivative size = 279, normalized size of antiderivative = 3.93 \[ \int (a-a \cosh (x))^{3/2} (A+B \cosh (x)) \, dx=-\frac {\sqrt {\frac {1}{2}} {\left (3 \, B a \cosh \left (x\right )^{5} + 3 \, B a \sinh \left (x\right )^{5} + 5 \, {\left (2 \, A - 3 \, B\right )} a \cosh \left (x\right )^{4} - 30 \, {\left (3 \, A - 2 \, B\right )} a \cosh \left (x\right )^{3} + 5 \, {\left (3 \, B a \cosh \left (x\right ) + {\left (2 \, A - 3 \, B\right )} a\right )} \sinh \left (x\right )^{4} - 30 \, {\left (3 \, A - 2 \, B\right )} a \cosh \left (x\right )^{2} + 10 \, {\left (3 \, B a \cosh \left (x\right )^{2} + 2 \, {\left (2 \, A - 3 \, B\right )} a \cosh \left (x\right ) - 3 \, {\left (3 \, A - 2 \, B\right )} a\right )} \sinh \left (x\right )^{3} + 5 \, {\left (2 \, A - 3 \, B\right )} a \cosh \left (x\right ) + 30 \, {\left (B a \cosh \left (x\right )^{3} + {\left (2 \, A - 3 \, B\right )} a \cosh \left (x\right )^{2} - 3 \, {\left (3 \, A - 2 \, B\right )} a \cosh \left (x\right ) - {\left (3 \, A - 2 \, B\right )} a\right )} \sinh \left (x\right )^{2} + 3 \, B a + 5 \, {\left (3 \, B a \cosh \left (x\right )^{4} + 4 \, {\left (2 \, A - 3 \, B\right )} a \cosh \left (x\right )^{3} - 18 \, {\left (3 \, A - 2 \, B\right )} a \cosh \left (x\right )^{2} - 12 \, {\left (3 \, A - 2 \, B\right )} a \cosh \left (x\right ) + {\left (2 \, A - 3 \, B\right )} a\right )} \sinh \left (x\right )\right )} \sqrt {-\frac {a}{\cosh \left (x\right ) + \sinh \left (x\right )}}}{30 \, {\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}\right )}} \] Input:
integrate((a-a*cosh(x))^(3/2)*(A+B*cosh(x)),x, algorithm="fricas")
Output:
-1/30*sqrt(1/2)*(3*B*a*cosh(x)^5 + 3*B*a*sinh(x)^5 + 5*(2*A - 3*B)*a*cosh( x)^4 - 30*(3*A - 2*B)*a*cosh(x)^3 + 5*(3*B*a*cosh(x) + (2*A - 3*B)*a)*sinh (x)^4 - 30*(3*A - 2*B)*a*cosh(x)^2 + 10*(3*B*a*cosh(x)^2 + 2*(2*A - 3*B)*a *cosh(x) - 3*(3*A - 2*B)*a)*sinh(x)^3 + 5*(2*A - 3*B)*a*cosh(x) + 30*(B*a* cosh(x)^3 + (2*A - 3*B)*a*cosh(x)^2 - 3*(3*A - 2*B)*a*cosh(x) - (3*A - 2*B )*a)*sinh(x)^2 + 3*B*a + 5*(3*B*a*cosh(x)^4 + 4*(2*A - 3*B)*a*cosh(x)^3 - 18*(3*A - 2*B)*a*cosh(x)^2 - 12*(3*A - 2*B)*a*cosh(x) + (2*A - 3*B)*a)*sin h(x))*sqrt(-a/(cosh(x) + sinh(x)))/(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x )^2)
\[ \int (a-a \cosh (x))^{3/2} (A+B \cosh (x)) \, dx=\int \left (- a \left (\cosh {\left (x \right )} - 1\right )\right )^{\frac {3}{2}} \left (A + B \cosh {\left (x \right )}\right )\, dx \] Input:
integrate((a-a*cosh(x))**(3/2)*(A+B*cosh(x)),x)
Output:
Integral((-a*(cosh(x) - 1))**(3/2)*(A + B*cosh(x)), x)
Leaf count of result is larger than twice the leaf count of optimal. 199 vs. \(2 (59) = 118\).
Time = 0.14 (sec) , antiderivative size = 199, normalized size of antiderivative = 2.80 \[ \int (a-a \cosh (x))^{3/2} (A+B \cosh (x)) \, dx=\frac {1}{6} \, {\left (\frac {9 \, \sqrt {2} a^{\frac {3}{2}} e^{\left (-x\right )}}{\left (-e^{\left (-x\right )}\right )^{\frac {3}{2}}} + \frac {9 \, \sqrt {2} a^{\frac {3}{2}} e^{\left (-2 \, x\right )}}{\left (-e^{\left (-x\right )}\right )^{\frac {3}{2}}} - \frac {\sqrt {2} a^{\frac {3}{2}} e^{\left (-3 \, x\right )}}{\left (-e^{\left (-x\right )}\right )^{\frac {3}{2}}} - \frac {\sqrt {2} a^{\frac {3}{2}}}{\left (-e^{\left (-x\right )}\right )^{\frac {3}{2}}}\right )} A + \frac {1}{20} \, B {\left (\frac {{\left (5 \, \sqrt {2} a^{\frac {3}{2}} e^{\left (-x\right )} - 15 \, \sqrt {2} a^{\frac {3}{2}} e^{\left (-2 \, x\right )} - 5 \, \sqrt {2} a^{\frac {3}{2}} e^{\left (-3 \, x\right )} - \sqrt {2} a^{\frac {3}{2}}\right )} e^{x}}{\left (-e^{\left (-x\right )}\right )^{\frac {3}{2}}} - \frac {5 \, \sqrt {2} a^{\frac {3}{2}} e^{\left (-x\right )} + 15 \, \sqrt {2} a^{\frac {3}{2}} e^{\left (-2 \, x\right )} - 5 \, \sqrt {2} a^{\frac {3}{2}} e^{\left (-3 \, x\right )} + \sqrt {2} a^{\frac {3}{2}} e^{\left (-4 \, x\right )}}{\left (-e^{\left (-x\right )}\right )^{\frac {3}{2}}}\right )} \] Input:
integrate((a-a*cosh(x))^(3/2)*(A+B*cosh(x)),x, algorithm="maxima")
Output:
1/6*(9*sqrt(2)*a^(3/2)*e^(-x)/(-e^(-x))^(3/2) + 9*sqrt(2)*a^(3/2)*e^(-2*x) /(-e^(-x))^(3/2) - sqrt(2)*a^(3/2)*e^(-3*x)/(-e^(-x))^(3/2) - sqrt(2)*a^(3 /2)/(-e^(-x))^(3/2))*A + 1/20*B*((5*sqrt(2)*a^(3/2)*e^(-x) - 15*sqrt(2)*a^ (3/2)*e^(-2*x) - 5*sqrt(2)*a^(3/2)*e^(-3*x) - sqrt(2)*a^(3/2))*e^x/(-e^(-x ))^(3/2) - (5*sqrt(2)*a^(3/2)*e^(-x) + 15*sqrt(2)*a^(3/2)*e^(-2*x) - 5*sqr t(2)*a^(3/2)*e^(-3*x) + sqrt(2)*a^(3/2)*e^(-4*x))/(-e^(-x))^(3/2))
Leaf count of result is larger than twice the leaf count of optimal. 212 vs. \(2 (59) = 118\).
Time = 0.12 (sec) , antiderivative size = 212, normalized size of antiderivative = 2.99 \[ \int (a-a \cosh (x))^{3/2} (A+B \cosh (x)) \, dx=\frac {1}{60} \, \sqrt {2} {\left (\frac {{\left (90 \, A a^{4} e^{\left (2 \, x\right )} \mathrm {sgn}\left (-e^{x} + 1\right ) - 60 \, B a^{4} e^{\left (2 \, x\right )} \mathrm {sgn}\left (-e^{x} + 1\right ) - 10 \, A a^{4} e^{x} \mathrm {sgn}\left (-e^{x} + 1\right ) + 15 \, B a^{4} e^{x} \mathrm {sgn}\left (-e^{x} + 1\right ) - 3 \, B a^{4} \mathrm {sgn}\left (-e^{x} + 1\right )\right )} e^{\left (-2 \, x\right )}}{\sqrt {-a e^{x}} a^{2}} + \frac {3 \, \sqrt {-a e^{x}} B a^{6} e^{\left (2 \, x\right )} \mathrm {sgn}\left (-e^{x} + 1\right ) + 10 \, \sqrt {-a e^{x}} A a^{6} e^{x} \mathrm {sgn}\left (-e^{x} + 1\right ) - 15 \, \sqrt {-a e^{x}} B a^{6} e^{x} \mathrm {sgn}\left (-e^{x} + 1\right ) - 90 \, \sqrt {-a e^{x}} A a^{6} \mathrm {sgn}\left (-e^{x} + 1\right ) + 60 \, \sqrt {-a e^{x}} B a^{6} \mathrm {sgn}\left (-e^{x} + 1\right )}{a^{5}}\right )} \] Input:
integrate((a-a*cosh(x))^(3/2)*(A+B*cosh(x)),x, algorithm="giac")
Output:
1/60*sqrt(2)*((90*A*a^4*e^(2*x)*sgn(-e^x + 1) - 60*B*a^4*e^(2*x)*sgn(-e^x + 1) - 10*A*a^4*e^x*sgn(-e^x + 1) + 15*B*a^4*e^x*sgn(-e^x + 1) - 3*B*a^4*s gn(-e^x + 1))*e^(-2*x)/(sqrt(-a*e^x)*a^2) + (3*sqrt(-a*e^x)*B*a^6*e^(2*x)* sgn(-e^x + 1) + 10*sqrt(-a*e^x)*A*a^6*e^x*sgn(-e^x + 1) - 15*sqrt(-a*e^x)* B*a^6*e^x*sgn(-e^x + 1) - 90*sqrt(-a*e^x)*A*a^6*sgn(-e^x + 1) + 60*sqrt(-a *e^x)*B*a^6*sgn(-e^x + 1))/a^5)
Timed out. \[ \int (a-a \cosh (x))^{3/2} (A+B \cosh (x)) \, dx=\int \left (A+B\,\mathrm {cosh}\left (x\right )\right )\,{\left (a-a\,\mathrm {cosh}\left (x\right )\right )}^{3/2} \,d x \] Input:
int((A + B*cosh(x))*(a - a*cosh(x))^(3/2),x)
Output:
int((A + B*cosh(x))*(a - a*cosh(x))^(3/2), x)
\[ \int (a-a \cosh (x))^{3/2} (A+B \cosh (x)) \, dx=\sqrt {a}\, a \left (\left (\int \sqrt {-\cosh \left (x \right )+1}d x \right ) a -\left (\int \sqrt {-\cosh \left (x \right )+1}\, \cosh \left (x \right )d x \right ) a +\left (\int \sqrt {-\cosh \left (x \right )+1}\, \cosh \left (x \right )d x \right ) b -\left (\int \sqrt {-\cosh \left (x \right )+1}\, \cosh \left (x \right )^{2}d x \right ) b \right ) \] Input:
int((a-a*cosh(x))^(3/2)*(A+B*cosh(x)),x)
Output:
sqrt(a)*a*(int(sqrt( - cosh(x) + 1),x)*a - int(sqrt( - cosh(x) + 1)*cosh(x ),x)*a + int(sqrt( - cosh(x) + 1)*cosh(x),x)*b - int(sqrt( - cosh(x) + 1)* cosh(x)**2,x)*b)