\(\int \frac {1}{(a+b \cosh (x))^{7/2}} \, dx\) [85]

Optimal result

Integrand size = 10, antiderivative size = 227 \[ \int \frac {1}{(a+b \cosh (x))^{7/2}} \, dx=-\frac {2 i \left (23 a^2+9 b^2\right ) \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{15 \left (a^2-b^2\right )^3 \sqrt {\frac {a+b \cosh (x)}{a+b}}}+\frac {16 i a \sqrt {\frac {a+b \cosh (x)}{a+b}} \operatorname {EllipticF}\left (\frac {i x}{2},\frac {2 b}{a+b}\right )}{15 \left (a^2-b^2\right )^2 \sqrt {a+b \cosh (x)}}-\frac {2 b \sinh (x)}{5 \left (a^2-b^2\right ) (a+b \cosh (x))^{5/2}}-\frac {16 a b \sinh (x)}{15 \left (a^2-b^2\right )^2 (a+b \cosh (x))^{3/2}}-\frac {2 b \left (23 a^2+9 b^2\right ) \sinh (x)}{15 \left (a^2-b^2\right )^3 \sqrt {a+b \cosh (x)}} \]

[Out]

-2/5*b*sinh(x)/(a^2-b^2)/(a+b*cosh(x))^(5/2)-16/15*a*b*sinh(x)/(a^2-b^2)^2/(a+b*cosh(x))^(3/2)-2/15*b*(23*a^2+
9*b^2)*sinh(x)/(a^2-b^2)^3/(a+b*cosh(x))^(1/2)-2/15*I*(23*a^2+9*b^2)*(cosh(1/2*x)^2)^(1/2)/cosh(1/2*x)*Ellipti
cE(I*sinh(1/2*x),2^(1/2)*(b/(a+b))^(1/2))*(a+b*cosh(x))^(1/2)/(a^2-b^2)^3/((a+b*cosh(x))/(a+b))^(1/2)+16/15*I*
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)/(a^2-b^2)^2/(a+b*cosh(x))^(1/2)

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {2743, 2833, 2831, 2742, 2740, 2734, 2732} \[ \int \frac {1}{(a+b \cosh (x))^{7/2}} \, dx=-\frac {2 b \left (23 a^2+9 b^2\right ) \sinh (x)}{15 \left (a^2-b^2\right )^3 \sqrt {a+b \cosh (x)}}-\frac {16 a b \sinh (x)}{15 \left (a^2-b^2\right )^2 (a+b \cosh (x))^{3/2}}-\frac {2 b \sinh (x)}{5 \left (a^2-b^2\right ) (a+b \cosh (x))^{5/2}}+\frac {16 i a \sqrt {\frac {a+b \cosh (x)}{a+b}} \operatorname {EllipticF}\left (\frac {i x}{2},\frac {2 b}{a+b}\right )}{15 \left (a^2-b^2\right )^2 \sqrt {a+b \cosh (x)}}-\frac {2 i \left (23 a^2+9 b^2\right ) \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{15 \left (a^2-b^2\right )^3 \sqrt {\frac {a+b \cosh (x)}{a+b}}} \]

[In]

Int[(a + b*Cosh[x])^(-7/2),x]

[Out]

(((-2*I)/15)*(23*a^2 + 9*b^2)*Sqrt[a + b*Cosh[x]]*EllipticE[(I/2)*x, (2*b)/(a + b)])/((a^2 - b^2)^3*Sqrt[(a +
b*Cosh[x])/(a + b)]) + (((16*I)/15)*a*Sqrt[(a + b*Cosh[x])/(a + b)]*EllipticF[(I/2)*x, (2*b)/(a + b)])/((a^2 -
b^2)^2*Sqrt[a + b*Cosh[x]]) - (2*b*Sinh[x])/(5*(a^2 - b^2)*(a + b*Cosh[x])^(5/2)) - (16*a*b*Sinh[x])/(15*(a^2
- b^2)^2*(a + b*Cosh[x])^(3/2)) - (2*b*(23*a^2 + 9*b^2)*Sinh[x])/(15*(a^2 - b^2)^3*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 2743

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

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 b \sinh (x)}{5 \left (a^2-b^2\right ) (a+b \cosh (x))^{5/2}}-\frac {2 \int \frac {-\frac {5 a}{2}+\frac {3}{2} b \cosh (x)}{(a+b \cosh (x))^{5/2}} \, dx}{5 \left (a^2-b^2\right )} \\ & = -\frac {2 b \sinh (x)}{5 \left (a^2-b^2\right ) (a+b \cosh (x))^{5/2}}-\frac {16 a b \sinh (x)}{15 \left (a^2-b^2\right )^2 (a+b \cosh (x))^{3/2}}+\frac {4 \int \frac {\frac {3}{4} \left (5 a^2+3 b^2\right )-2 a b \cosh (x)}{(a+b \cosh (x))^{3/2}} \, dx}{15 \left (a^2-b^2\right )^2} \\ & = -\frac {2 b \sinh (x)}{5 \left (a^2-b^2\right ) (a+b \cosh (x))^{5/2}}-\frac {16 a b \sinh (x)}{15 \left (a^2-b^2\right )^2 (a+b \cosh (x))^{3/2}}-\frac {2 b \left (23 a^2+9 b^2\right ) \sinh (x)}{15 \left (a^2-b^2\right )^3 \sqrt {a+b \cosh (x)}}-\frac {8 \int \frac {-\frac {1}{8} a \left (15 a^2+17 b^2\right )-\frac {1}{8} b \left (23 a^2+9 b^2\right ) \cosh (x)}{\sqrt {a+b \cosh (x)}} \, dx}{15 \left (a^2-b^2\right )^3} \\ & = -\frac {2 b \sinh (x)}{5 \left (a^2-b^2\right ) (a+b \cosh (x))^{5/2}}-\frac {16 a b \sinh (x)}{15 \left (a^2-b^2\right )^2 (a+b \cosh (x))^{3/2}}-\frac {2 b \left (23 a^2+9 b^2\right ) \sinh (x)}{15 \left (a^2-b^2\right )^3 \sqrt {a+b \cosh (x)}}-\frac {(8 a) \int \frac {1}{\sqrt {a+b \cosh (x)}} \, dx}{15 \left (a^2-b^2\right )^2}+\frac {\left (23 a^2+9 b^2\right ) \int \sqrt {a+b \cosh (x)} \, dx}{15 \left (a^2-b^2\right )^3} \\ & = -\frac {2 b \sinh (x)}{5 \left (a^2-b^2\right ) (a+b \cosh (x))^{5/2}}-\frac {16 a b \sinh (x)}{15 \left (a^2-b^2\right )^2 (a+b \cosh (x))^{3/2}}-\frac {2 b \left (23 a^2+9 b^2\right ) \sinh (x)}{15 \left (a^2-b^2\right )^3 \sqrt {a+b \cosh (x)}}+\frac {\left (\left (23 a^2+9 b^2\right ) \sqrt {a+b \cosh (x)}\right ) \int \sqrt {\frac {a}{a+b}+\frac {b \cosh (x)}{a+b}} \, dx}{15 \left (a^2-b^2\right )^3 \sqrt {\frac {a+b \cosh (x)}{a+b}}}-\frac {\left (8 a \sqrt {\frac {a+b \cosh (x)}{a+b}}\right ) \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cosh (x)}{a+b}}} \, dx}{15 \left (a^2-b^2\right )^2 \sqrt {a+b \cosh (x)}} \\ & = -\frac {2 i \left (23 a^2+9 b^2\right ) \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{15 \left (a^2-b^2\right )^3 \sqrt {\frac {a+b \cosh (x)}{a+b}}}+\frac {16 i a \sqrt {\frac {a+b \cosh (x)}{a+b}} \operatorname {EllipticF}\left (\frac {i x}{2},\frac {2 b}{a+b}\right )}{15 \left (a^2-b^2\right )^2 \sqrt {a+b \cosh (x)}}-\frac {2 b \sinh (x)}{5 \left (a^2-b^2\right ) (a+b \cosh (x))^{5/2}}-\frac {16 a b \sinh (x)}{15 \left (a^2-b^2\right )^2 (a+b \cosh (x))^{3/2}}-\frac {2 b \left (23 a^2+9 b^2\right ) \sinh (x)}{15 \left (a^2-b^2\right )^3 \sqrt {a+b \cosh (x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.55 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.73 \[ \int \frac {1}{(a+b \cosh (x))^{7/2}} \, dx=\frac {2 \left (-\frac {i \left (\frac {a+b \cosh (x)}{a+b}\right )^{5/2} \left (\left (23 a^2+9 b^2\right ) E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )+8 a (-a+b) \operatorname {EllipticF}\left (\frac {i x}{2},\frac {2 b}{a+b}\right )\right )}{(a-b)^3}+\frac {b \left (34 a^4-5 a^2 b^2+3 b^4+2 a b \left (27 a^2+5 b^2\right ) \cosh (x)+b^2 \left (23 a^2+9 b^2\right ) \cosh ^2(x)\right ) \sinh (x)}{\left (-a^2+b^2\right )^3}\right )}{15 (a+b \cosh (x))^{5/2}} \]

[In]

Integrate[(a + b*Cosh[x])^(-7/2),x]

[Out]

(2*(((-I)*((a + b*Cosh[x])/(a + b))^(5/2)*((23*a^2 + 9*b^2)*EllipticE[(I/2)*x, (2*b)/(a + b)] + 8*a*(-a + b)*E
llipticF[(I/2)*x, (2*b)/(a + b)]))/(a - b)^3 + (b*(34*a^4 - 5*a^2*b^2 + 3*b^4 + 2*a*b*(27*a^2 + 5*b^2)*Cosh[x]
+ b^2*(23*a^2 + 9*b^2)*Cosh[x]^2)*Sinh[x])/(-a^2 + b^2)^3))/(15*(a + b*Cosh[x])^(5/2))

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(565\) vs. \(2(239)=478\).

Time = 2.96 (sec) , antiderivative size = 566, normalized size of antiderivative = 2.49

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 {\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}}}{10 b^{2} \left (a -b \right ) \left (a +b \right ) \left (\cosh \left (\frac {x}{2}\right )^{2}+\frac {a -b}{2 b}\right )^{3}}-\frac {8 a \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}}}{15 b \left (a +b \right )^{2} \left (a -b \right )^{2} \left (\cosh \left (\frac {x}{2}\right )^{2}+\frac {a -b}{2 b}\right )^{2}}-\frac {4 \sinh \left (\frac {x}{2}\right )^{2} b \cosh \left (\frac {x}{2}\right ) \left (23 a^{2}+9 b^{2}\right )}{15 \left (a -b \right )^{3} \left (a +b \right )^{3} \sqrt {\left (2 \cosh \left (\frac {x}{2}\right )^{2} b +a -b \right ) \sinh \left (\frac {x}{2}\right )^{2}}}+\frac {2 \left (15 a^{2}-8 a b +9 b^{2}\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 (15 a^{5}+15 a^{4} b -30 a^{3} b^{2}-30 a^{2} b^{3}+15 a \,b^{4}+15 b^{5}\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 {8 b \left (23 a^{2}+9 b^{2}\right ) \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 )}{15 \left (a +b \right )^{3} \left (a -b \right )^{3} \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 )}{\sinh \left (\frac {x}{2}\right ) \sqrt {2 \sinh \left (\frac {x}{2}\right )^{2} b +a +b}}\)	\(566\)

[In]

int(1/(a+b*cosh(x))^(7/2),x,method=_RETURNVERBOSE)

[Out]

((2*cosh(1/2*x)^2*b+a-b)*sinh(1/2*x)^2)^(1/2)*(-1/10/b^2/(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)^3-8/15*a/b/(a+b)^2/(a-b)^2*cosh(1/2*x)*(2*sinh(1/2*x)^4*b+(a+b)*s
inh(1/2*x)^2)^(1/2)/(cosh(1/2*x)^2+1/2*(a-b)/b)^2-4/15*sinh(1/2*x)^2*b/(a-b)^3/(a+b)^3*cosh(1/2*x)*(23*a^2+9*b
^2)/((2*cosh(1/2*x)^2*b+a-b)*sinh(1/2*x)^2)^(1/2)+2*(15*a^2-8*a*b+9*b^2)/(15*a^5+15*a^4*b-30*a^3*b^2-30*a^2*b^
3+15*a*b^4+15*b^5)/(-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)*EllipticF(cosh(1/2*x)*(-2*b/(a-b))^(1/2),1/2*((-2*a+2*b)/b)^(1/2))-8/15*b*(
23*a^2+9*b^2)/(a+b)^3/(a-b)^3*(-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))-EllipticE(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 = 0.15 (sec) , antiderivative size = 3315, normalized size of antiderivative = 14.60 \[ \int \frac {1}{(a+b \cosh (x))^{7/2}} \, dx=\text {Too large to display} \]

[In]

integrate(1/(a+b*cosh(x))^(7/2),x, algorithm="fricas")

[Out]

-2/45*((sqrt(2)*(a^3*b^3 - 33*a*b^5)*cosh(x)^6 + sqrt(2)*(a^3*b^3 - 33*a*b^5)*sinh(x)^6 + 6*sqrt(2)*(a^4*b^2 -
33*a^2*b^4)*cosh(x)^5 + 6*(sqrt(2)*(a^3*b^3 - 33*a*b^5)*cosh(x) + sqrt(2)*(a^4*b^2 - 33*a^2*b^4))*sinh(x)^5 +
3*sqrt(2)*(4*a^5*b - 131*a^3*b^3 - 33*a*b^5)*cosh(x)^4 + 3*(5*sqrt(2)*(a^3*b^3 - 33*a*b^5)*cosh(x)^2 + 10*sqr
t(2)*(a^4*b^2 - 33*a^2*b^4)*cosh(x) + sqrt(2)*(4*a^5*b - 131*a^3*b^3 - 33*a*b^5))*sinh(x)^4 + 4*sqrt(2)*(2*a^6
- 63*a^4*b^2 - 99*a^2*b^4)*cosh(x)^3 + 4*(5*sqrt(2)*(a^3*b^3 - 33*a*b^5)*cosh(x)^3 + 15*sqrt(2)*(a^4*b^2 - 33
*a^2*b^4)*cosh(x)^2 + 3*sqrt(2)*(4*a^5*b - 131*a^3*b^3 - 33*a*b^5)*cosh(x) + sqrt(2)*(2*a^6 - 63*a^4*b^2 - 99*
a^2*b^4))*sinh(x)^3 + 3*sqrt(2)*(4*a^5*b - 131*a^3*b^3 - 33*a*b^5)*cosh(x)^2 + 3*(5*sqrt(2)*(a^3*b^3 - 33*a*b^
5)*cosh(x)^4 + 20*sqrt(2)*(a^4*b^2 - 33*a^2*b^4)*cosh(x)^3 + 6*sqrt(2)*(4*a^5*b - 131*a^3*b^3 - 33*a*b^5)*cosh
(x)^2 + 4*sqrt(2)*(2*a^6 - 63*a^4*b^2 - 99*a^2*b^4)*cosh(x) + sqrt(2)*(4*a^5*b - 131*a^3*b^3 - 33*a*b^5))*sinh
(x)^2 + 6*sqrt(2)*(a^4*b^2 - 33*a^2*b^4)*cosh(x) + 6*(sqrt(2)*(a^3*b^3 - 33*a*b^5)*cosh(x)^5 + 5*sqrt(2)*(a^4*
b^2 - 33*a^2*b^4)*cosh(x)^4 + 2*sqrt(2)*(4*a^5*b - 131*a^3*b^3 - 33*a*b^5)*cosh(x)^3 + 2*sqrt(2)*(2*a^6 - 63*a
^4*b^2 - 99*a^2*b^4)*cosh(x)^2 + sqrt(2)*(4*a^5*b - 131*a^3*b^3 - 33*a*b^5)*cosh(x) + sqrt(2)*(a^4*b^2 - 33*a^
2*b^4))*sinh(x) + sqrt(2)*(a^3*b^3 - 33*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*cosh(x) + 3*b*sinh(x) + 2*a)/b) + 3*(sqrt(2)*(23*a^2*b^4 + 9*b^6)*cosh(x)^6 + sqr
t(2)*(23*a^2*b^4 + 9*b^6)*sinh(x)^6 + 6*sqrt(2)*(23*a^3*b^3 + 9*a*b^5)*cosh(x)^5 + 6*(sqrt(2)*(23*a^2*b^4 + 9*
b^6)*cosh(x) + sqrt(2)*(23*a^3*b^3 + 9*a*b^5))*sinh(x)^5 + 3*sqrt(2)*(92*a^4*b^2 + 59*a^2*b^4 + 9*b^6)*cosh(x)
^4 + 3*(5*sqrt(2)*(23*a^2*b^4 + 9*b^6)*cosh(x)^2 + 10*sqrt(2)*(23*a^3*b^3 + 9*a*b^5)*cosh(x) + sqrt(2)*(92*a^4
*b^2 + 59*a^2*b^4 + 9*b^6))*sinh(x)^4 + 4*sqrt(2)*(46*a^5*b + 87*a^3*b^3 + 27*a*b^5)*cosh(x)^3 + 4*(5*sqrt(2)*
(23*a^2*b^4 + 9*b^6)*cosh(x)^3 + 15*sqrt(2)*(23*a^3*b^3 + 9*a*b^5)*cosh(x)^2 + 3*sqrt(2)*(92*a^4*b^2 + 59*a^2*
b^4 + 9*b^6)*cosh(x) + sqrt(2)*(46*a^5*b + 87*a^3*b^3 + 27*a*b^5))*sinh(x)^3 + 3*sqrt(2)*(92*a^4*b^2 + 59*a^2*
b^4 + 9*b^6)*cosh(x)^2 + 3*(5*sqrt(2)*(23*a^2*b^4 + 9*b^6)*cosh(x)^4 + 20*sqrt(2)*(23*a^3*b^3 + 9*a*b^5)*cosh(
x)^3 + 6*sqrt(2)*(92*a^4*b^2 + 59*a^2*b^4 + 9*b^6)*cosh(x)^2 + 4*sqrt(2)*(46*a^5*b + 87*a^3*b^3 + 27*a*b^5)*co
sh(x) + sqrt(2)*(92*a^4*b^2 + 59*a^2*b^4 + 9*b^6))*sinh(x)^2 + 6*sqrt(2)*(23*a^3*b^3 + 9*a*b^5)*cosh(x) + 6*(s
qrt(2)*(23*a^2*b^4 + 9*b^6)*cosh(x)^5 + 5*sqrt(2)*(23*a^3*b^3 + 9*a*b^5)*cosh(x)^4 + 2*sqrt(2)*(92*a^4*b^2 + 5
9*a^2*b^4 + 9*b^6)*cosh(x)^3 + 2*sqrt(2)*(46*a^5*b + 87*a^3*b^3 + 27*a*b^5)*cosh(x)^2 + sqrt(2)*(92*a^4*b^2 +
59*a^2*b^4 + 9*b^6)*cosh(x) + sqrt(2)*(23*a^3*b^3 + 9*a*b^5))*sinh(x) + sqrt(2)*(23*a^2*b^4 + 9*b^6))*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*((23*a^2*b^4 + 9*b^6)*cosh(x)^
6 + (23*a^2*b^4 + 9*b^6)*sinh(x)^6 + (123*a^3*b^3 + 37*a*b^5)*cosh(x)^5 + (123*a^3*b^3 + 37*a*b^5 + 6*(23*a^2*
b^4 + 9*b^6)*cosh(x))*sinh(x)^5 + 2*(103*a^4*b^2 + 45*a^2*b^4 + 12*b^6)*cosh(x)^4 + (206*a^4*b^2 + 90*a^2*b^4
+ 24*b^6 + 15*(23*a^2*b^4 + 9*b^6)*cosh(x)^2 + 5*(123*a^3*b^3 + 37*a*b^5)*cosh(x))*sinh(x)^4 + 2*(46*a^5*b + 8
7*a^3*b^3 + 27*a*b^5)*cosh(x)^3 + 2*(46*a^5*b + 87*a^3*b^3 + 27*a*b^5 + 10*(23*a^2*b^4 + 9*b^6)*cosh(x)^3 + 5*
(123*a^3*b^3 + 37*a*b^5)*cosh(x)^2 + 4*(103*a^4*b^2 + 45*a^2*b^4 + 12*b^6)*cosh(x))*sinh(x)^3 + (70*a^4*b^2 +
87*a^2*b^4 + 3*b^6)*cosh(x)^2 + (70*a^4*b^2 + 87*a^2*b^4 + 3*b^6 + 15*(23*a^2*b^4 + 9*b^6)*cosh(x)^4 + 10*(123
*a^3*b^3 + 37*a*b^5)*cosh(x)^3 + 12*(103*a^4*b^2 + 45*a^2*b^4 + 12*b^6)*cosh(x)^2 + 6*(46*a^5*b + 87*a^3*b^3 +
27*a*b^5)*cosh(x))*sinh(x)^2 + (15*a^3*b^3 + 17*a*b^5)*cosh(x) + (15*a^3*b^3 + 17*a*b^5 + 6*(23*a^2*b^4 + 9*b
^6)*cosh(x)^5 + 5*(123*a^3*b^3 + 37*a*b^5)*cosh(x)^4 + 8*(103*a^4*b^2 + 45*a^2*b^4 + 12*b^6)*cosh(x)^3 + 6*(46
*a^5*b + 87*a^3*b^3 + 27*a*b^5)*cosh(x)^2 + 2*(70*a^4*b^2 + 87*a^2*b^4 + 3*b^6)*cosh(x))*sinh(x))*sqrt(b*cosh(
x) + a))/(a^6*b^4 - 3*a^4*b^6 + 3*a^2*b^8 - b^10 + (a^6*b^4 - 3*a^4*b^6 + 3*a^2*b^8 - b^10)*cosh(x)^6 + (a^6*b
^4 - 3*a^4*b^6 + 3*a^2*b^8 - b^10)*sinh(x)^6 + 6*(a^7*b^3 - 3*a^5*b^5 + 3*a^3*b^7 - a*b^9)*cosh(x)^5 + 6*(a^7*
b^3 - 3*a^5*b^5 + 3*a^3*b^7 - a*b^9 + (a^6*b^4 - 3*a^4*b^6 + 3*a^2*b^8 - b^10)*cosh(x))*sinh(x)^5 + 3*(4*a^8*b
^2 - 11*a^6*b^4 + 9*a^4*b^6 - a^2*b^8 - b^10)*cosh(x)^4 + 3*(4*a^8*b^2 - 11*a^6*b^4 + 9*a^4*b^6 - a^2*b^8 - b^
10 + 5*(a^6*b^4 - 3*a^4*b^6 + 3*a^2*b^8 - b^10)*cosh(x)^2 + 10*(a^7*b^3 - 3*a^5*b^5 + 3*a^3*b^7 - a*b^9)*cosh(
x))*sinh(x)^4 + 4*(2*a^9*b - 3*a^7*b^3 - 3*a^5*b^5 + 7*a^3*b^7 - 3*a*b^9)*cosh(x)^3 + 4*(2*a^9*b - 3*a^7*b^3 -
3*a^5*b^5 + 7*a^3*b^7 - 3*a*b^9 + 5*(a^6*b^4 - 3*a^4*b^6 + 3*a^2*b^8 - b^10)*cosh(x)^3 + 15*(a^7*b^3 - 3*a^5*
b^5 + 3*a^3*b^7 - a*b^9)*cosh(x)^2 + 3*(4*a^8*b^2 - 11*a^6*b^4 + 9*a^4*b^6 - a^2*b^8 - b^10)*cosh(x))*sinh(x)^
3 + 3*(4*a^8*b^2 - 11*a^6*b^4 + 9*a^4*b^6 - a^2*b^8 - b^10)*cosh(x)^2 + 3*(4*a^8*b^2 - 11*a^6*b^4 + 9*a^4*b^6
- a^2*b^8 - b^10 + 5*(a^6*b^4 - 3*a^4*b^6 + 3*a^2*b^8 - b^10)*cosh(x)^4 + 20*(a^7*b^3 - 3*a^5*b^5 + 3*a^3*b^7
- a*b^9)*cosh(x)^3 + 6*(4*a^8*b^2 - 11*a^6*b^4 + 9*a^4*b^6 - a^2*b^8 - b^10)*cosh(x)^2 + 4*(2*a^9*b - 3*a^7*b^
3 - 3*a^5*b^5 + 7*a^3*b^7 - 3*a*b^9)*cosh(x))*sinh(x)^2 + 6*(a^7*b^3 - 3*a^5*b^5 + 3*a^3*b^7 - a*b^9)*cosh(x)
+ 6*(a^7*b^3 - 3*a^5*b^5 + 3*a^3*b^7 - a*b^9 + (a^6*b^4 - 3*a^4*b^6 + 3*a^2*b^8 - b^10)*cosh(x)^5 + 5*(a^7*b^3
- 3*a^5*b^5 + 3*a^3*b^7 - a*b^9)*cosh(x)^4 + 2*(4*a^8*b^2 - 11*a^6*b^4 + 9*a^4*b^6 - a^2*b^8 - b^10)*cosh(x)^
3 + 2*(2*a^9*b - 3*a^7*b^3 - 3*a^5*b^5 + 7*a^3*b^7 - 3*a*b^9)*cosh(x)^2 + (4*a^8*b^2 - 11*a^6*b^4 + 9*a^4*b^6
- a^2*b^8 - b^10)*cosh(x))*sinh(x))

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(a+b \cosh (x))^{7/2}} \, dx=\text {Timed out} \]

[In]

integrate(1/(a+b*cosh(x))**(7/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {1}{(a+b \cosh (x))^{7/2}} \, dx=\int { \frac {1}{{\left (b \cosh \left (x\right ) + a\right )}^{\frac {7}{2}}} \,d x } \]

[In]

integrate(1/(a+b*cosh(x))^(7/2),x, algorithm="maxima")

[Out]

integrate((b*cosh(x) + a)^(-7/2), x)

Giac [F]

\[ \int \frac {1}{(a+b \cosh (x))^{7/2}} \, dx=\int { \frac {1}{{\left (b \cosh \left (x\right ) + a\right )}^{\frac {7}{2}}} \,d x } \]

[In]

integrate(1/(a+b*cosh(x))^(7/2),x, algorithm="giac")

[Out]

integrate((b*cosh(x) + a)^(-7/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(a+b \cosh (x))^{7/2}} \, dx=\int \frac {1}{{\left (a+b\,\mathrm {cosh}\left (x\right )\right )}^{7/2}} \,d x \]

[In]

int(1/(a + b*cosh(x))^(7/2),x)

[Out]

int(1/(a + b*cosh(x))^(7/2), x)