\(\int \frac {A+B \cosh (x)}{(a+b \cosh (x))^{5/2}} \, dx\) [120]

Optimal result

Integrand size = 17, antiderivative size = 231 \[ \int \frac {A+B \cosh (x)}{(a+b \cosh (x))^{5/2}} \, dx=-\frac {2 i \left (4 a A b-a^2 B-3 b^2 B\right ) \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{3 b \left (a^2-b^2\right )^2 \sqrt {\frac {a+b \cosh (x)}{a+b}}}+\frac {2 i (A b-a B) \sqrt {\frac {a+b \cosh (x)}{a+b}} \operatorname {EllipticF}\left (\frac {i x}{2},\frac {2 b}{a+b}\right )}{3 b \left (a^2-b^2\right ) \sqrt {a+b \cosh (x)}}-\frac {2 (A b-a B) \sinh (x)}{3 \left (a^2-b^2\right ) (a+b \cosh (x))^{3/2}}-\frac {2 \left (4 a A b-a^2 B-3 b^2 B\right ) \sinh (x)}{3 \left (a^2-b^2\right )^2 \sqrt {a+b \cosh (x)}} \]

[Out]

-2/3*(A*b-B*a)*sinh(x)/(a^2-b^2)/(a+b*cosh(x))^(3/2)-2/3*(4*A*a*b-B*a^2-3*B*b^2)*sinh(x)/(a^2-b^2)^2/(a+b*cosh
(x))^(1/2)-2/3*I*(4*A*a*b-B*a^2-3*B*b^2)*(cosh(1/2*x)^2)^(1/2)/cosh(1/2*x)*EllipticE(I*sinh(1/2*x),2^(1/2)*(b/
(a+b))^(1/2))*(a+b*cosh(x))^(1/2)/b/(a^2-b^2)^2/((a+b*cosh(x))/(a+b))^(1/2)+2/3*I*(A*b-B*a)*(cosh(1/2*x)^2)^(1
/2)/cosh(1/2*x)*EllipticF(I*sinh(1/2*x),2^(1/2)*(b/(a+b))^(1/2))*((a+b*cosh(x))/(a+b))^(1/2)/b/(a^2-b^2)/(a+b*
cosh(x))^(1/2)

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {2833, 2831, 2742, 2740, 2734, 2732} \[ \int \frac {A+B \cosh (x)}{(a+b \cosh (x))^{5/2}} \, dx=-\frac {2 \sinh (x) \left (a^2 (-B)+4 a A b-3 b^2 B\right )}{3 \left (a^2-b^2\right )^2 \sqrt {a+b \cosh (x)}}-\frac {2 \sinh (x) (A b-a B)}{3 \left (a^2-b^2\right ) (a+b \cosh (x))^{3/2}}+\frac {2 i (A b-a B) \sqrt {\frac {a+b \cosh (x)}{a+b}} \operatorname {EllipticF}\left (\frac {i x}{2},\frac {2 b}{a+b}\right )}{3 b \left (a^2-b^2\right ) \sqrt {a+b \cosh (x)}}-\frac {2 i \left (a^2 (-B)+4 a A b-3 b^2 B\right ) \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{3 b \left (a^2-b^2\right )^2 \sqrt {\frac {a+b \cosh (x)}{a+b}}} \]

[In]

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

[Out]

(((-2*I)/3)*(4*a*A*b - a^2*B - 3*b^2*B)*Sqrt[a + b*Cosh[x]]*EllipticE[(I/2)*x, (2*b)/(a + b)])/(b*(a^2 - b^2)^
2*Sqrt[(a + b*Cosh[x])/(a + b)]) + (((2*I)/3)*(A*b - a*B)*Sqrt[(a + b*Cosh[x])/(a + b)]*EllipticF[(I/2)*x, (2*
b)/(a + b)])/(b*(a^2 - b^2)*Sqrt[a + b*Cosh[x]]) - (2*(A*b - a*B)*Sinh[x])/(3*(a^2 - b^2)*(a + b*Cosh[x])^(3/2
)) - (2*(4*a*A*b - a^2*B - 3*b^2*B)*Sinh[x])/(3*(a^2 - b^2)^2*Sqrt[a + b*Cosh[x]])

Rule 2732

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[2*(Sqrt[a + b]/d)*EllipticE[(1/2)*(c - Pi/2
+ d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rule 2734

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[a + b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c +
d*x])/(a + b)], Int[Sqrt[a/(a + b) + (b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 -
b^2, 0] &&  !GtQ[a + b, 0]

Rule 2740

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*Sqrt[a + b]))*EllipticF[(1/2)*(c - P
i/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rule 2742

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[(a + b*Sin[c + d*x])/(a + b)]/Sqrt[a
+ b*Sin[c + d*x]], Int[1/Sqrt[a/(a + b) + (b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a
^2 - b^2, 0] &&  !GtQ[a + b, 0]

Rule 2831

Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])/Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[(b*c
- a*d)/b, Int[1/Sqrt[a + b*Sin[e + f*x]], x], x] + Dist[d/b, Int[Sqrt[a + b*Sin[e + f*x]], x], x] /; FreeQ[{a
, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]

Rule 2833

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 + 1)/(f*(m + 1)*(a^2 - b^2))), x] + Dist[1/((m + 1)*(a^2 - b
^2)), Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[(a*c - b*d)*(m + 1) - (b*c - a*d)*(m + 2)*Sin[e + f*x], x], x], x]
/; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && IntegerQ[2*m]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 (A b-a B) \sinh (x)}{3 \left (a^2-b^2\right ) (a+b \cosh (x))^{3/2}}-\frac {2 \int \frac {-\frac {3}{2} (a A-b B)+\frac {1}{2} (A b-a B) \cosh (x)}{(a+b \cosh (x))^{3/2}} \, dx}{3 \left (a^2-b^2\right )} \\ & = -\frac {2 (A b-a B) \sinh (x)}{3 \left (a^2-b^2\right ) (a+b \cosh (x))^{3/2}}-\frac {2 \left (4 a A b-a^2 B-3 b^2 B\right ) \sinh (x)}{3 \left (a^2-b^2\right )^2 \sqrt {a+b \cosh (x)}}+\frac {4 \int \frac {\frac {1}{4} \left (3 a^2 A+A b^2-4 a b B\right )+\frac {1}{4} \left (4 a A b-a^2 B-3 b^2 B\right ) \cosh (x)}{\sqrt {a+b \cosh (x)}} \, dx}{3 \left (a^2-b^2\right )^2} \\ & = -\frac {2 (A b-a B) \sinh (x)}{3 \left (a^2-b^2\right ) (a+b \cosh (x))^{3/2}}-\frac {2 \left (4 a A b-a^2 B-3 b^2 B\right ) \sinh (x)}{3 \left (a^2-b^2\right )^2 \sqrt {a+b \cosh (x)}}-\frac {(A b-a B) \int \frac {1}{\sqrt {a+b \cosh (x)}} \, dx}{3 b \left (a^2-b^2\right )}+\frac {\left (4 a A b-a^2 B-3 b^2 B\right ) \int \sqrt {a+b \cosh (x)} \, dx}{3 b \left (a^2-b^2\right )^2} \\ & = -\frac {2 (A b-a B) \sinh (x)}{3 \left (a^2-b^2\right ) (a+b \cosh (x))^{3/2}}-\frac {2 \left (4 a A b-a^2 B-3 b^2 B\right ) \sinh (x)}{3 \left (a^2-b^2\right )^2 \sqrt {a+b \cosh (x)}}+\frac {\left (\left (4 a A b-a^2 B-3 b^2 B\right ) \sqrt {a+b \cosh (x)}\right ) \int \sqrt {\frac {a}{a+b}+\frac {b \cosh (x)}{a+b}} \, dx}{3 b \left (a^2-b^2\right )^2 \sqrt {\frac {a+b \cosh (x)}{a+b}}}-\frac {\left ((A b-a B) \sqrt {\frac {a+b \cosh (x)}{a+b}}\right ) \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cosh (x)}{a+b}}} \, dx}{3 b \left (a^2-b^2\right ) \sqrt {a+b \cosh (x)}} \\ & = -\frac {2 i \left (4 a A b-a^2 B-3 b^2 B\right ) \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{3 b \left (a^2-b^2\right )^2 \sqrt {\frac {a+b \cosh (x)}{a+b}}}+\frac {2 i (A b-a B) \sqrt {\frac {a+b \cosh (x)}{a+b}} \operatorname {EllipticF}\left (\frac {i x}{2},\frac {2 b}{a+b}\right )}{3 b \left (a^2-b^2\right ) \sqrt {a+b \cosh (x)}}-\frac {2 (A b-a B) \sinh (x)}{3 \left (a^2-b^2\right ) (a+b \cosh (x))^{3/2}}-\frac {2 \left (4 a A b-a^2 B-3 b^2 B\right ) \sinh (x)}{3 \left (a^2-b^2\right )^2 \sqrt {a+b \cosh (x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.69 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.74 \[ \int \frac {A+B \cosh (x)}{(a+b \cosh (x))^{5/2}} \, dx=\frac {2 \left (\frac {i \left (\frac {a+b \cosh (x)}{a+b}\right )^{3/2} \left (\left (-4 a A b+a^2 B+3 b^2 B\right ) E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )-(a-b) (-A b+a B) \operatorname {EllipticF}\left (\frac {i x}{2},\frac {2 b}{a+b}\right )\right )}{(a-b)^2 b}+\frac {\left (-5 a^2 A b+A b^3+2 a^3 B+2 a b^2 B+b \left (-4 a A b+a^2 B+3 b^2 B\right ) \cosh (x)\right ) \sinh (x)}{\left (a^2-b^2\right )^2}\right )}{3 (a+b \cosh (x))^{3/2}} \]

[In]

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

[Out]

(2*((I*((a + b*Cosh[x])/(a + b))^(3/2)*((-4*a*A*b + a^2*B + 3*b^2*B)*EllipticE[(I/2)*x, (2*b)/(a + b)] - (a -
b)*(-(A*b) + a*B)*EllipticF[(I/2)*x, (2*b)/(a + b)]))/((a - b)^2*b) + ((-5*a^2*A*b + A*b^3 + 2*a^3*B + 2*a*b^2
*B + b*(-4*a*A*b + a^2*B + 3*b^2*B)*Cosh[x])*Sinh[x])/(a^2 - b^2)^2))/(3*(a + b*Cosh[x])^(3/2))

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(796\) vs. \(2(247)=494\).

Time = 4.98 (sec) , antiderivative size = 797, normalized size of antiderivative = 3.45

method	result	size
default	\(\frac {\sqrt {\left (2 \cosh \left (\frac {x}{2}\right )^{2} b +a -b \right ) \sinh \left (\frac {x}{2}\right )^{2}}\, \left (-\frac {2 B \sqrt {2 \sinh \left (\frac {x}{2}\right )^{4} b +\left (a +b \right ) \sinh \left (\frac {x}{2}\right )^{2}}\, \left (2 \cosh \left (\frac {x}{2}\right ) \sqrt {-\frac {2 b}{a -b}}\, \sinh \left (\frac {x}{2}\right )^{2} b -\sqrt {-\sinh \left (\frac {x}{2}\right )^{2}}\, \sqrt {\frac {2 b \sinh \left (\frac {x}{2}\right )^{2}}{a -b}+\frac {a +b}{a -b}}\, \operatorname {EllipticF}\left (\cosh \left (\frac {x}{2}\right ) \sqrt {-\frac {2 b}{a -b}}, \frac {\sqrt {\frac {-2 a +2 b}{b}}}{2}\right ) a -\sqrt {-\sinh \left (\frac {x}{2}\right )^{2}}\, \sqrt {\frac {2 b \sinh \left (\frac {x}{2}\right )^{2}}{a -b}+\frac {a +b}{a -b}}\, \operatorname {EllipticF}\left (\cosh \left (\frac {x}{2}\right ) \sqrt {-\frac {2 b}{a -b}}, \frac {\sqrt {\frac {-2 a +2 b}{b}}}{2}\right ) b +2 \sqrt {-\sinh \left (\frac {x}{2}\right )^{2}}\, \sqrt {\frac {2 b \sinh \left (\frac {x}{2}\right )^{2}}{a -b}+\frac {a +b}{a -b}}\, \operatorname {EllipticE}\left (\cosh \left (\frac {x}{2}\right ) \sqrt {-\frac {2 b}{a -b}}, \frac {\sqrt {\frac {-2 a +2 b}{b}}}{2}\right ) b \right )}{b \sinh \left (\frac {x}{2}\right )^{2} \left (2 \sinh \left (\frac {x}{2}\right )^{2} b +a +b \right ) \sqrt {-\frac {2 b}{a -b}}\, \left (a^{2}-b^{2}\right )}+\frac {2 \left (b A -B a \right ) \left (-\frac {\cosh \left (\frac {x}{2}\right ) \sqrt {2 \sinh \left (\frac {x}{2}\right )^{4} b +\left (a +b \right ) \sinh \left (\frac {x}{2}\right )^{2}}}{6 b \left (a -b \right ) \left (a +b \right ) \left (\cosh \left (\frac {x}{2}\right )^{2}+\frac {a -b}{2 b}\right )^{2}}-\frac {8 \sinh \left (\frac {x}{2}\right )^{2} b \cosh \left (\frac {x}{2}\right ) a}{3 \left (a -b \right )^{2} \left (a +b \right )^{2} \sqrt {\left (2 \cosh \left (\frac {x}{2}\right )^{2} b +a -b \right ) \sinh \left (\frac {x}{2}\right )^{2}}}+\frac {\left (3 a -b \right ) \sqrt {\frac {2 \cosh \left (\frac {x}{2}\right )^{2} b +a -b}{a -b}}\, \sqrt {-\sinh \left (\frac {x}{2}\right )^{2}}\, \operatorname {EllipticF}\left (\cosh \left (\frac {x}{2}\right ) \sqrt {-\frac {2 b}{a -b}}, \frac {\sqrt {\frac {-2 a +2 b}{b}}}{2}\right )}{\left (3 a^{3}+3 a^{2} b -3 a \,b^{2}-3 b^{3}\right ) \sqrt {-\frac {2 b}{a -b}}\, \sqrt {2 \sinh \left (\frac {x}{2}\right )^{4} b +\left (a +b \right ) \sinh \left (\frac {x}{2}\right )^{2}}}-\frac {16 a b \left (-a +b \right ) \sqrt {\frac {2 \cosh \left (\frac {x}{2}\right )^{2} b +a -b}{a -b}}\, \sqrt {-\sinh \left (\frac {x}{2}\right )^{2}}\, \left (\operatorname {EllipticF}\left (\cosh \left (\frac {x}{2}\right ) \sqrt {-\frac {2 b}{a -b}}, \frac {\sqrt {\frac {-2 a +2 b}{b}}}{2}\right )-\operatorname {EllipticE}\left (\cosh \left (\frac {x}{2}\right ) \sqrt {-\frac {2 b}{a -b}}, \frac {\sqrt {\frac {-2 a +2 b}{b}}}{2}\right )\right )}{3 \left (a +b \right )^{2} \left (a -b \right )^{2} \sqrt {-\frac {2 b}{a -b}}\, \sqrt {2 \sinh \left (\frac {x}{2}\right )^{4} b +\left (a +b \right ) \sinh \left (\frac {x}{2}\right )^{2}}\, \left (2 a -2 b \right )}\right )}{b}\right )}{\sinh \left (\frac {x}{2}\right ) \sqrt {2 \sinh \left (\frac {x}{2}\right )^{2} b +a +b}}\)	\(797\)
parts	\(\text {Expression too large to display}\)	\(1250\)

[In]

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

[Out]

((2*cosh(1/2*x)^2*b+a-b)*sinh(1/2*x)^2)^(1/2)*(-2*B/b/sinh(1/2*x)^2/(2*sinh(1/2*x)^2*b+a+b)/(-2*b/(a-b))^(1/2)
/(a^2-b^2)*(2*sinh(1/2*x)^4*b+(a+b)*sinh(1/2*x)^2)^(1/2)*(2*cosh(1/2*x)*(-2*b/(a-b))^(1/2)*sinh(1/2*x)^2*b-(-s
inh(1/2*x)^2)^(1/2)*(2*b/(a-b)*sinh(1/2*x)^2+(a+b)/(a-b))^(1/2)*EllipticF(cosh(1/2*x)*(-2*b/(a-b))^(1/2),1/2*(
(-2*a+2*b)/b)^(1/2))*a-(-sinh(1/2*x)^2)^(1/2)*(2*b/(a-b)*sinh(1/2*x)^2+(a+b)/(a-b))^(1/2)*EllipticF(cosh(1/2*x
)*(-2*b/(a-b))^(1/2),1/2*((-2*a+2*b)/b)^(1/2))*b+2*(-sinh(1/2*x)^2)^(1/2)*(2*b/(a-b)*sinh(1/2*x)^2+(a+b)/(a-b)
)^(1/2)*EllipticE(cosh(1/2*x)*(-2*b/(a-b))^(1/2),1/2*((-2*a+2*b)/b)^(1/2))*b)+2*(A*b-B*a)/b*(-1/6/b/(a-b)/(a+b
)*cosh(1/2*x)*(2*sinh(1/2*x)^4*b+(a+b)*sinh(1/2*x)^2)^(1/2)/(cosh(1/2*x)^2+1/2*(a-b)/b)^2-8/3*sinh(1/2*x)^2*b/
(a-b)^2/(a+b)^2*cosh(1/2*x)*a/((2*cosh(1/2*x)^2*b+a-b)*sinh(1/2*x)^2)^(1/2)+(3*a-b)/(3*a^3+3*a^2*b-3*a*b^2-3*b
^3)/(-2*b/(a-b))^(1/2)*((2*cosh(1/2*x)^2*b+a-b)/(a-b))^(1/2)*(-sinh(1/2*x)^2)^(1/2)/(2*sinh(1/2*x)^4*b+(a+b)*s
inh(1/2*x)^2)^(1/2)*EllipticF(cosh(1/2*x)*(-2*b/(a-b))^(1/2),1/2*((-2*a+2*b)/b)^(1/2))-16/3*a*b/(a+b)^2/(a-b)^
2*(-a+b)/(-2*b/(a-b))^(1/2)*((2*cosh(1/2*x)^2*b+a-b)/(a-b))^(1/2)*(-sinh(1/2*x)^2)^(1/2)/(2*sinh(1/2*x)^4*b+(a
+b)*sinh(1/2*x)^2)^(1/2)/(2*a-2*b)*(EllipticF(cosh(1/2*x)*(-2*b/(a-b))^(1/2),1/2*((-2*a+2*b)/b)^(1/2))-Ellipti
cE(cosh(1/2*x)*(-2*b/(a-b))^(1/2),1/2*((-2*a+2*b)/b)^(1/2)))))/sinh(1/2*x)/(2*sinh(1/2*x)^2*b+a+b)^(1/2)

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.14 (sec) , antiderivative size = 2153, normalized size of antiderivative = 9.32 \[ \int \frac {A+B \cosh (x)}{(a+b \cosh (x))^{5/2}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

2/9*((sqrt(2)*(2*B*a^3*b^2 + A*a^2*b^3 - 6*B*a*b^4 + 3*A*b^5)*cosh(x)^4 + sqrt(2)*(2*B*a^3*b^2 + A*a^2*b^3 - 6
*B*a*b^4 + 3*A*b^5)*sinh(x)^4 + 4*sqrt(2)*(2*B*a^4*b + A*a^3*b^2 - 6*B*a^2*b^3 + 3*A*a*b^4)*cosh(x)^3 + 4*(sqr
t(2)*(2*B*a^3*b^2 + A*a^2*b^3 - 6*B*a*b^4 + 3*A*b^5)*cosh(x) + sqrt(2)*(2*B*a^4*b + A*a^3*b^2 - 6*B*a^2*b^3 +
3*A*a*b^4))*sinh(x)^3 + 2*sqrt(2)*(4*B*a^5 + 2*A*a^4*b - 10*B*a^3*b^2 + 7*A*a^2*b^3 - 6*B*a*b^4 + 3*A*b^5)*cos
h(x)^2 + 2*(3*sqrt(2)*(2*B*a^3*b^2 + A*a^2*b^3 - 6*B*a*b^4 + 3*A*b^5)*cosh(x)^2 + 6*sqrt(2)*(2*B*a^4*b + A*a^3
*b^2 - 6*B*a^2*b^3 + 3*A*a*b^4)*cosh(x) + sqrt(2)*(4*B*a^5 + 2*A*a^4*b - 10*B*a^3*b^2 + 7*A*a^2*b^3 - 6*B*a*b^
4 + 3*A*b^5))*sinh(x)^2 + 4*sqrt(2)*(2*B*a^4*b + A*a^3*b^2 - 6*B*a^2*b^3 + 3*A*a*b^4)*cosh(x) + 4*(sqrt(2)*(2*
B*a^3*b^2 + A*a^2*b^3 - 6*B*a*b^4 + 3*A*b^5)*cosh(x)^3 + 3*sqrt(2)*(2*B*a^4*b + A*a^3*b^2 - 6*B*a^2*b^3 + 3*A*
a*b^4)*cosh(x)^2 + sqrt(2)*(4*B*a^5 + 2*A*a^4*b - 10*B*a^3*b^2 + 7*A*a^2*b^3 - 6*B*a*b^4 + 3*A*b^5)*cosh(x) +
sqrt(2)*(2*B*a^4*b + A*a^3*b^2 - 6*B*a^2*b^3 + 3*A*a*b^4))*sinh(x) + sqrt(2)*(2*B*a^3*b^2 + A*a^2*b^3 - 6*B*a*
b^4 + 3*A*b^5))*sqrt(b)*weierstrassPInverse(4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2)/b^3, 1/3*(3*b*cos
h(x) + 3*b*sinh(x) + 2*a)/b) + 3*(sqrt(2)*(B*a^2*b^3 - 4*A*a*b^4 + 3*B*b^5)*cosh(x)^4 + sqrt(2)*(B*a^2*b^3 - 4
*A*a*b^4 + 3*B*b^5)*sinh(x)^4 + 4*sqrt(2)*(B*a^3*b^2 - 4*A*a^2*b^3 + 3*B*a*b^4)*cosh(x)^3 + 4*(sqrt(2)*(B*a^2*
b^3 - 4*A*a*b^4 + 3*B*b^5)*cosh(x) + sqrt(2)*(B*a^3*b^2 - 4*A*a^2*b^3 + 3*B*a*b^4))*sinh(x)^3 + 2*sqrt(2)*(2*B
*a^4*b - 8*A*a^3*b^2 + 7*B*a^2*b^3 - 4*A*a*b^4 + 3*B*b^5)*cosh(x)^2 + 2*(3*sqrt(2)*(B*a^2*b^3 - 4*A*a*b^4 + 3*
B*b^5)*cosh(x)^2 + 6*sqrt(2)*(B*a^3*b^2 - 4*A*a^2*b^3 + 3*B*a*b^4)*cosh(x) + sqrt(2)*(2*B*a^4*b - 8*A*a^3*b^2
+ 7*B*a^2*b^3 - 4*A*a*b^4 + 3*B*b^5))*sinh(x)^2 + 4*sqrt(2)*(B*a^3*b^2 - 4*A*a^2*b^3 + 3*B*a*b^4)*cosh(x) + 4*
(sqrt(2)*(B*a^2*b^3 - 4*A*a*b^4 + 3*B*b^5)*cosh(x)^3 + 3*sqrt(2)*(B*a^3*b^2 - 4*A*a^2*b^3 + 3*B*a*b^4)*cosh(x)
^2 + sqrt(2)*(2*B*a^4*b - 8*A*a^3*b^2 + 7*B*a^2*b^3 - 4*A*a*b^4 + 3*B*b^5)*cosh(x) + sqrt(2)*(B*a^3*b^2 - 4*A*
a^2*b^3 + 3*B*a*b^4))*sinh(x) + sqrt(2)*(B*a^2*b^3 - 4*A*a*b^4 + 3*B*b^5))*sqrt(b)*weierstrassZeta(4/3*(4*a^2
- 3*b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2)/b^3, weierstrassPInverse(4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2
)/b^3, 1/3*(3*b*cosh(x) + 3*b*sinh(x) + 2*a)/b)) + 6*((B*a^2*b^3 - 4*A*a*b^4 + 3*B*b^5)*cosh(x)^4 + (B*a^2*b^3
- 4*A*a*b^4 + 3*B*b^5)*sinh(x)^4 + (4*B*a^3*b^2 - 13*A*a^2*b^3 + 8*B*a*b^4 + A*b^5)*cosh(x)^3 + (4*B*a^3*b^2
- 13*A*a^2*b^3 + 8*B*a*b^4 + A*b^5 + 4*(B*a^2*b^3 - 4*A*a*b^4 + 3*B*b^5)*cosh(x))*sinh(x)^3 + (2*B*a^4*b - 8*A
*a^3*b^2 + 7*B*a^2*b^3 - 4*A*a*b^4 + 3*B*b^5)*cosh(x)^2 + (2*B*a^4*b - 8*A*a^3*b^2 + 7*B*a^2*b^3 - 4*A*a*b^4 +
3*B*b^5 + 6*(B*a^2*b^3 - 4*A*a*b^4 + 3*B*b^5)*cosh(x)^2 + 3*(4*B*a^3*b^2 - 13*A*a^2*b^3 + 8*B*a*b^4 + A*b^5)*
cosh(x))*sinh(x)^2 - (3*A*a^2*b^3 - 4*B*a*b^4 + A*b^5)*cosh(x) - (3*A*a^2*b^3 - 4*B*a*b^4 + A*b^5 - 4*(B*a^2*b
^3 - 4*A*a*b^4 + 3*B*b^5)*cosh(x)^3 - 3*(4*B*a^3*b^2 - 13*A*a^2*b^3 + 8*B*a*b^4 + A*b^5)*cosh(x)^2 - 2*(2*B*a^
4*b - 8*A*a^3*b^2 + 7*B*a^2*b^3 - 4*A*a*b^4 + 3*B*b^5)*cosh(x))*sinh(x))*sqrt(b*cosh(x) + a))/(a^4*b^4 - 2*a^2
*b^6 + b^8 + (a^4*b^4 - 2*a^2*b^6 + b^8)*cosh(x)^4 + (a^4*b^4 - 2*a^2*b^6 + b^8)*sinh(x)^4 + 4*(a^5*b^3 - 2*a^
3*b^5 + a*b^7)*cosh(x)^3 + 4*(a^5*b^3 - 2*a^3*b^5 + a*b^7 + (a^4*b^4 - 2*a^2*b^6 + b^8)*cosh(x))*sinh(x)^3 + 2
*(2*a^6*b^2 - 3*a^4*b^4 + b^8)*cosh(x)^2 + 2*(2*a^6*b^2 - 3*a^4*b^4 + b^8 + 3*(a^4*b^4 - 2*a^2*b^6 + b^8)*cosh
(x)^2 + 6*(a^5*b^3 - 2*a^3*b^5 + a*b^7)*cosh(x))*sinh(x)^2 + 4*(a^5*b^3 - 2*a^3*b^5 + a*b^7)*cosh(x) + 4*(a^5*
b^3 - 2*a^3*b^5 + a*b^7 + (a^4*b^4 - 2*a^2*b^6 + b^8)*cosh(x)^3 + 3*(a^5*b^3 - 2*a^3*b^5 + a*b^7)*cosh(x)^2 +
(2*a^6*b^2 - 3*a^4*b^4 + b^8)*cosh(x))*sinh(x))

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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