\(\int (a+a \cosh (x))^{5/2} (A+B \cosh (x)) \, dx\) [87]

Optimal result

Integrand size = 17, antiderivative size = 94 \[ \int (a+a \cosh (x))^{5/2} (A+B \cosh (x)) \, dx=\frac {64 a^3 (7 A+5 B) \sinh (x)}{105 \sqrt {a+a \cosh (x)}}+\frac {16}{105} a^2 (7 A+5 B) \sqrt {a+a \cosh (x)} \sinh (x)+\frac {2}{35} a (7 A+5 B) (a+a \cosh (x))^{3/2} \sinh (x)+\frac {2}{7} B (a+a \cosh (x))^{5/2} \sinh (x) \]

[Out]

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)+64/105*a^3*(7*A+5*B)*sinh(x)/(a
+a*cosh(x))^(1/2)+16/105*a^2*(7*A+5*B)*sinh(x)*(a+a*cosh(x))^(1/2)

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {2830, 2726, 2725} \[ \int (a+a \cosh (x))^{5/2} (A+B \cosh (x)) \, dx=\frac {64 a^3 (7 A+5 B) \sinh (x)}{105 \sqrt {a \cosh (x)+a}}+\frac {16}{105} a^2 (7 A+5 B) \sinh (x) \sqrt {a \cosh (x)+a}+\frac {2}{35} a (7 A+5 B) \sinh (x) (a \cosh (x)+a)^{3/2}+\frac {2}{7} B \sinh (x) (a \cosh (x)+a)^{5/2} \]

[In]

Int[(a + a*Cosh[x])^(5/2)*(A + B*Cosh[x]),x]

[Out]

(64*a^3*(7*A + 5*B)*Sinh[x])/(105*Sqrt[a + a*Cosh[x]]) + (16*a^2*(7*A + 5*B)*Sqrt[a + a*Cosh[x]]*Sinh[x])/105
+ (2*a*(7*A + 5*B)*(a + a*Cosh[x])^(3/2)*Sinh[x])/35 + (2*B*(a + a*Cosh[x])^(5/2)*Sinh[x])/7

Rule 2725

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] && EqQ[a^2 - b^2, 0]

Rule 2726

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] + Dist[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]

Rule 2830

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] + Dist[(a*d*m + b*c*(m + 1))/(b*(m + 1)), Int[(a + b*S
in[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)]

Rubi steps \begin{align*} \text {integral}& = \frac {2}{7} B (a+a \cosh (x))^{5/2} \sinh (x)+\frac {1}{7} (7 A+5 B) \int (a+a \cosh (x))^{5/2} \, dx \\ & = \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)+\frac {1}{35} (8 a (7 A+5 B)) \int (a+a \cosh (x))^{3/2} \, dx \\ & = \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)+\frac {1}{105} \left (32 a^2 (7 A+5 B)\right ) \int \sqrt {a+a \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) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.64 \[ \int (a+a \cosh (x))^{5/2} (A+B \cosh (x)) \, dx=\frac {1}{210} a^2 \sqrt {a (1+\cosh (x))} (1246 A+1040 B+(392 A+505 B) \cosh (x)+6 (7 A+20 B) \cosh (2 x)+15 B \cosh (3 x)) \tanh \left (\frac {x}{2}\right ) \]

[In]

Integrate[(a + a*Cosh[x])^(5/2)*(A + B*Cosh[x]),x]

[Out]

(a^2*Sqrt[a*(1 + 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])*Tanh[x/2])/210

Maple [A] (verified)

Time = 0.68 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.76

method	result	size
default	\(\frac {8 \cosh \left (\frac {x}{2}\right ) a^{3} \sinh \left (\frac {x}{2}\right ) \left (30 B \sinh \left (\frac {x}{2}\right )^{6}+\left (21 A +105 B \right ) \sinh \left (\frac {x}{2}\right )^{4}+\left (70 A +140 B \right ) \sinh \left (\frac {x}{2}\right )^{2}+105 A +105 B \right ) \sqrt {2}}{105 \sqrt {a \cosh \left (\frac {x}{2}\right )^{2}}}\)	\(71\)
parts	\(\frac {8 A \,a^{3} \cosh \left (\frac {x}{2}\right ) \sinh \left (\frac {x}{2}\right ) \left (3 \cosh \left (\frac {x}{2}\right )^{4}+4 \cosh \left (\frac {x}{2}\right )^{2}+8\right ) \sqrt {2}}{15 \sqrt {a \cosh \left (\frac {x}{2}\right )^{2}}}+\frac {8 B \cosh \left (\frac {x}{2}\right ) a^{3} \sinh \left (\frac {x}{2}\right ) \left (6 \cosh \left (\frac {x}{2}\right )^{6}+3 \cosh \left (\frac {x}{2}\right )^{4}+4 \cosh \left (\frac {x}{2}\right )^{2}+8\right ) \sqrt {2}}{21 \sqrt {a \cosh \left (\frac {x}{2}\right )^{2}}}\)	\(100\)

[In]

int((a+a*cosh(x))^(5/2)*(A+B*cosh(x)),x,method=_RETURNVERBOSE)

[Out]

8/105*cosh(1/2*x)*a^3*sinh(1/2*x)*(30*B*sinh(1/2*x)^6+(21*A+105*B)*sinh(1/2*x)^4+(70*A+140*B)*sinh(1/2*x)^2+10
5*A+105*B)*2^(1/2)/(a*cosh(1/2*x)^2)^(1/2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 563 vs. \(2 (78) = 156\).

Time = 0.26 (sec) , antiderivative size = 563, normalized size of antiderivative = 5.99 \[ \int (a+a \cosh (x))^{5/2} (A+B \cosh (x)) \, dx=\frac {\sqrt {\frac {1}{2}} {\left (15 \, B a^{2} \cosh \left (x\right )^{7} + 15 \, B a^{2} \sinh \left (x\right )^{7} + 21 \, {\left (2 \, A + 5 \, B\right )} a^{2} \cosh \left (x\right )^{6} + 35 \, {\left (10 \, A + 11 \, B\right )} a^{2} \cosh \left (x\right )^{5} + 525 \, {\left (4 \, A + 3 \, B\right )} a^{2} \cosh \left (x\right )^{4} + 21 \, {\left (5 \, B a^{2} \cosh \left (x\right ) + {\left (2 \, A + 5 \, B\right )} a^{2}\right )} \sinh \left (x\right )^{6} - 525 \, {\left (4 \, A + 3 \, B\right )} a^{2} \cosh \left (x\right )^{3} + 7 \, {\left (45 \, B a^{2} \cosh \left (x\right )^{2} + 18 \, {\left (2 \, A + 5 \, B\right )} a^{2} \cosh \left (x\right ) + 5 \, {\left (10 \, A + 11 \, B\right )} a^{2}\right )} \sinh \left (x\right )^{5} - 35 \, {\left (10 \, A + 11 \, B\right )} a^{2} \cosh \left (x\right )^{2} + 35 \, {\left (15 \, B a^{2} \cosh \left (x\right )^{3} + 9 \, {\left (2 \, A + 5 \, B\right )} a^{2} \cosh \left (x\right )^{2} + 5 \, {\left (10 \, A + 11 \, B\right )} a^{2} \cosh \left (x\right ) + 15 \, {\left (4 \, A + 3 \, B\right )} a^{2}\right )} \sinh \left (x\right )^{4} - 21 \, {\left (2 \, A + 5 \, B\right )} a^{2} \cosh \left (x\right ) + 35 \, {\left (15 \, B a^{2} \cosh \left (x\right )^{4} + 12 \, {\left (2 \, A + 5 \, B\right )} a^{2} \cosh \left (x\right )^{3} + 10 \, {\left (10 \, A + 11 \, B\right )} a^{2} \cosh \left (x\right )^{2} + 60 \, {\left (4 \, A + 3 \, B\right )} a^{2} \cosh \left (x\right ) - 15 \, {\left (4 \, A + 3 \, B\right )} a^{2}\right )} \sinh \left (x\right )^{3} - 15 \, B a^{2} + 35 \, {\left (9 \, B a^{2} \cosh \left (x\right )^{5} + 9 \, {\left (2 \, A + 5 \, B\right )} a^{2} \cosh \left (x\right )^{4} + 10 \, {\left (10 \, A + 11 \, B\right )} a^{2} \cosh \left (x\right )^{3} + 90 \, {\left (4 \, A + 3 \, B\right )} a^{2} \cosh \left (x\right )^{2} - 45 \, {\left (4 \, A + 3 \, B\right )} a^{2} \cosh \left (x\right ) - {\left (10 \, A + 11 \, B\right )} a^{2}\right )} \sinh \left (x\right )^{2} + 7 \, {\left (15 \, B a^{2} \cosh \left (x\right )^{6} + 18 \, {\left (2 \, A + 5 \, B\right )} a^{2} \cosh \left (x\right )^{5} + 25 \, {\left (10 \, A + 11 \, B\right )} a^{2} \cosh \left (x\right )^{4} + 300 \, {\left (4 \, A + 3 \, B\right )} a^{2} \cosh \left (x\right )^{3} - 225 \, {\left (4 \, A + 3 \, B\right )} a^{2} \cosh \left (x\right )^{2} - 10 \, {\left (10 \, A + 11 \, B\right )} a^{2} \cosh \left (x\right ) - 3 \, {\left (2 \, A + 5 \, B\right )} a^{2}\right )} \sinh \left (x\right )\right )} \sqrt {\frac {a}{\cosh \left (x\right ) + \sinh \left (x\right )}}}{420 \, {\left (\cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right )^{2} \sinh \left (x\right ) + 3 \, \cosh \left (x\right ) \sinh \left (x\right )^{2} + \sinh \left (x\right )^{3}\right )}} \]

[In]

integrate((a+a*cosh(x))^(5/2)*(A+B*cosh(x)),x, algorithm="fricas")

[Out]

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*cosh(x)^4 + 12*(2*A + 5*B)*a^2*cos
h(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*cosh(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*cosh(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*sinh(x) + 3*cosh(x)*sinh(x)^2 + sinh(x)^3)

Sympy [F(-1)]

Timed out. \[ \int (a+a \cosh (x))^{5/2} (A+B \cosh (x)) \, dx=\text {Timed out} \]

[In]

integrate((a+a*cosh(x))**(5/2)*(A+B*cosh(x)),x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 237 vs. \(2 (78) = 156\).

Time = 0.30 (sec) , antiderivative size = 237, normalized size of antiderivative = 2.52 \[ \int (a+a \cosh (x))^{5/2} (A+B \cosh (x)) \, dx=\frac {1}{60} \, {\left (3 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (\frac {5}{2} \, x\right )} + 25 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (\frac {3}{2} \, x\right )} + 150 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (\frac {1}{2} \, x\right )} - 150 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (-\frac {1}{2} \, x\right )} - 25 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (-\frac {3}{2} \, x\right )} - 3 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (-\frac {5}{2} \, x\right )}\right )} A + \frac {1}{168} \, {\left ({\left (3 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (-x\right )} + 21 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (-2 \, x\right )} + 70 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (-3 \, x\right )} + 210 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (-4 \, x\right )} - 105 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (-5 \, x\right )} - 7 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (-6 \, x\right )}\right )} e^{\left (\frac {9}{2} \, x\right )} + {\left (7 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (-x\right )} + 105 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (-2 \, x\right )} - 210 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (-3 \, x\right )} - 70 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (-4 \, x\right )} - 21 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (-5 \, x\right )} - 3 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (-6 \, x\right )}\right )} e^{\left (\frac {5}{2} \, x\right )}\right )} B \]

[In]

integrate((a+a*cosh(x))^(5/2)*(A+B*cosh(x)),x, algorithm="maxima")

[Out]

1/60*(3*sqrt(2)*a^(5/2)*e^(5/2*x) + 25*sqrt(2)*a^(5/2)*e^(3/2*x) + 150*sqrt(2)*a^(5/2)*e^(1/2*x) - 150*sqrt(2)
*a^(5/2)*e^(-1/2*x) - 25*sqrt(2)*a^(5/2)*e^(-3/2*x) - 3*sqrt(2)*a^(5/2)*e^(-5/2*x))*A + 1/168*((3*sqrt(2)*a^(5
/2)*e^(-x) + 21*sqrt(2)*a^(5/2)*e^(-2*x) + 70*sqrt(2)*a^(5/2)*e^(-3*x) + 210*sqrt(2)*a^(5/2)*e^(-4*x) - 105*sq
rt(2)*a^(5/2)*e^(-5*x) - 7*sqrt(2)*a^(5/2)*e^(-6*x))*e^(9/2*x) + (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*sqr
t(2)*a^(5/2)*e^(-6*x))*e^(5/2*x))*B

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.63 \[ \int (a+a \cosh (x))^{5/2} (A+B \cosh (x)) \, dx=-\frac {1}{840} \, \sqrt {2} {\left (\frac {{\left (2100 \, A a^{6} e^{\left (3 \, x\right )} + 1575 \, B a^{6} e^{\left (3 \, x\right )} + 350 \, A a^{6} e^{\left (2 \, x\right )} + 385 \, B a^{6} e^{\left (2 \, x\right )} + 42 \, A a^{6} e^{x} + 105 \, B a^{6} e^{x} + 15 \, B a^{6}\right )} e^{\left (-\frac {7}{2} \, x\right )}}{a^{\frac {7}{2}}} - \frac {15 \, B a^{\frac {19}{2}} e^{\left (\frac {7}{2} \, x\right )} + 42 \, A a^{\frac {19}{2}} e^{\left (\frac {5}{2} \, x\right )} + 105 \, B a^{\frac {19}{2}} e^{\left (\frac {5}{2} \, x\right )} + 350 \, A a^{\frac {19}{2}} e^{\left (\frac {3}{2} \, x\right )} + 385 \, B a^{\frac {19}{2}} e^{\left (\frac {3}{2} \, x\right )} + 2100 \, A a^{\frac {19}{2}} e^{\left (\frac {1}{2} \, x\right )} + 1575 \, B a^{\frac {19}{2}} e^{\left (\frac {1}{2} \, x\right )}}{a^{7}}\right )} \]

[In]

integrate((a+a*cosh(x))^(5/2)*(A+B*cosh(x)),x, algorithm="giac")

[Out]

-1/840*sqrt(2)*((2100*A*a^6*e^(3*x) + 1575*B*a^6*e^(3*x) + 350*A*a^6*e^(2*x) + 385*B*a^6*e^(2*x) + 42*A*a^6*e^
x + 105*B*a^6*e^x + 15*B*a^6)*e^(-7/2*x)/a^(7/2) - (15*B*a^(19/2)*e^(7/2*x) + 42*A*a^(19/2)*e^(5/2*x) + 105*B*
a^(19/2)*e^(5/2*x) + 350*A*a^(19/2)*e^(3/2*x) + 385*B*a^(19/2)*e^(3/2*x) + 2100*A*a^(19/2)*e^(1/2*x) + 1575*B*
a^(19/2)*e^(1/2*x))/a^7)

Mupad [F(-1)]

Timed out. \[ \int (a+a \cosh (x))^{5/2} (A+B \cosh (x)) \, dx=\int \left (A+B\,\mathrm {cosh}\left (x\right )\right )\,{\left (a+a\,\mathrm {cosh}\left (x\right )\right )}^{5/2} \,d x \]

[In]

int((A + B*cosh(x))*(a + a*cosh(x))^(5/2),x)

[Out]

int((A + B*cosh(x))*(a + a*cosh(x))^(5/2), x)