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

Optimal. Leaf size=227 \[ -\frac {2 b \left (23 a^2+9 b^2\right ) \sinh (x)}{15 \left (a^2-b^2\right )^3 \sqrt {a+b \cosh (x)}}-\frac {16 a b \sinh (x)}{15 \left (a^2-b^2\right )^2 (a+b \cosh (x))^{3/2}}-\frac {2 b \sinh (x)}{5 \left (a^2-b^2\right ) (a+b \cosh (x))^{5/2}}+\frac {16 i a \sqrt {\frac {a+b \cosh (x)}{a+b}} F\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{15 \left (a^2-b^2\right )^2 \sqrt {a+b \cosh (x)}}-\frac {2 i \left (23 a^2+9 b^2\right ) \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{15 \left (a^2-b^2\right )^3 \sqrt {\frac {a+b \cosh (x)}{a+b}}} \]

[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)

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Rubi [A]  time = 0.31, antiderivative size = 227, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {2664, 2754, 2752, 2663, 2661, 2655, 2653} \[ -\frac {2 b \left (23 a^2+9 b^2\right ) \sinh (x)}{15 \left (a^2-b^2\right )^3 \sqrt {a+b \cosh (x)}}-\frac {16 a b \sinh (x)}{15 \left (a^2-b^2\right )^2 (a+b \cosh (x))^{3/2}}-\frac {2 b \sinh (x)}{5 \left (a^2-b^2\right ) (a+b \cosh (x))^{5/2}}+\frac {16 i a \sqrt {\frac {a+b \cosh (x)}{a+b}} F\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{15 \left (a^2-b^2\right )^2 \sqrt {a+b \cosh (x)}}-\frac {2 i \left (23 a^2+9 b^2\right ) \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{15 \left (a^2-b^2\right )^3 \sqrt {\frac {a+b \cosh (x)}{a+b}}} \]

Antiderivative was successfully verified.

[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 2653

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

Rule 2655

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*Sin[c + d*x])/(a + b)], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 -
b^2, 0] &&  !GtQ[a + b, 0]

Rule 2661

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

Rule 2663

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*Sin[c + d*x])/(a + b)], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a
^2 - b^2, 0] &&  !GtQ[a + b, 0]

Rule 2664

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] && Integer
Q[2*n]

Rule 2752

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 2754

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*} \int \frac {1}{(a+b \cosh (x))^{7/2}} \, dx &=-\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}} F\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*}

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Mathematica [A]  time = 0.72, size = 165, normalized size = 0.73 \[ \frac {2 \left (\frac {b \sinh (x) \left (34 a^4+b^2 \left (23 a^2+9 b^2\right ) \cosh ^2(x)+2 a b \left (27 a^2+5 b^2\right ) \cosh (x)-5 a^2 b^2+3 b^4\right )}{\left (b^2-a^2\right )^3}-\frac {i \left (\frac {a+b \cosh (x)}{a+b}\right )^{5/2} \left (\left (23 a^2+9 b^2\right ) E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )+8 a (b-a) F\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )\right )}{(a-b)^3}\right )}{15 (a+b \cosh (x))^{5/2}} \]

Antiderivative was successfully verified.

[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))

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fricas [F]  time = 1.01, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b \cosh \relax (x) + a}}{b^{4} \cosh \relax (x)^{4} + 4 \, a b^{3} \cosh \relax (x)^{3} + 6 \, a^{2} b^{2} \cosh \relax (x)^{2} + 4 \, a^{3} b \cosh \relax (x) + a^{4}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(b*cosh(x) + a)/(b^4*cosh(x)^4 + 4*a*b^3*cosh(x)^3 + 6*a^2*b^2*cosh(x)^2 + 4*a^3*b*cosh(x) + a^4)
, x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b \cosh \relax (x) + a\right )}^{\frac {7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*cosh(x) + a)^(-7/2), x)

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maple [B]  time = 1.12, size = 566, normalized size = 2.49 \[ \frac {\sqrt {\left (2 b \left (\cosh ^{2}\left (\frac {x}{2}\right )\right )+a -b \right ) \left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}\, \left (-\frac {\cosh \left (\frac {x}{2}\right ) \sqrt {2 b \left (\sinh ^{4}\left (\frac {x}{2}\right )\right )+\left (a +b \right ) \left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}}{10 b^{2} \left (a -b \right ) \left (a +b \right ) \left (\cosh ^{2}\left (\frac {x}{2}\right )+\frac {a -b}{2 b}\right )^{3}}-\frac {8 a \cosh \left (\frac {x}{2}\right ) \sqrt {2 b \left (\sinh ^{4}\left (\frac {x}{2}\right )\right )+\left (a +b \right ) \left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}}{15 b \left (a +b \right )^{2} \left (a -b \right )^{2} \left (\cosh ^{2}\left (\frac {x}{2}\right )+\frac {a -b}{2 b}\right )^{2}}-\frac {4 b \left (\sinh ^{2}\left (\frac {x}{2}\right )\right ) \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 b \left (\cosh ^{2}\left (\frac {x}{2}\right )\right )+a -b \right ) \left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}}+\frac {2 \left (15 a^{2}-8 a b +9 b^{2}\right ) \sqrt {\frac {2 b \left (\cosh ^{2}\left (\frac {x}{2}\right )\right )+a -b}{a -b}}\, \sqrt {-\left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}\, \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 b \left (\sinh ^{4}\left (\frac {x}{2}\right )\right )+\left (a +b \right ) \left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}}-\frac {8 b \left (23 a^{2}+9 b^{2}\right ) \left (-a +b \right ) \sqrt {\frac {2 b \left (\cosh ^{2}\left (\frac {x}{2}\right )\right )+a -b}{a -b}}\, \sqrt {-\left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}\, \left (\EllipticF \left (\cosh \left (\frac {x}{2}\right ) \sqrt {-\frac {2 b}{a -b}}, \frac {\sqrt {\frac {-2 a +2 b}{b}}}{2}\right )-\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 b \left (\sinh ^{4}\left (\frac {x}{2}\right )\right )+\left (a +b \right ) \left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}\, \left (2 a -2 b \right )}\right )}{\sinh \left (\frac {x}{2}\right ) \sqrt {2 b \left (\sinh ^{2}\left (\frac {x}{2}\right )\right )+a +b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((2*b*cosh(1/2*x)^2+a-b)*sinh(1/2*x)^2)^(1/2)*(-1/10/b^2/(a-b)/(a+b)*cosh(1/2*x)*(2*b*sinh(1/2*x)^4+(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*b*sinh(1/2*x)^4+(a+b)*s
inh(1/2*x)^2)^(1/2)/(cosh(1/2*x)^2+1/2*(a-b)/b)^2-4/15*b*sinh(1/2*x)^2/(a-b)^3/(a+b)^3*cosh(1/2*x)*(23*a^2+9*b
^2)/((2*b*cosh(1/2*x)^2+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*b*cosh(1/2*x)^2+a-b)/(a-b))^(1/2)*(-sinh(1/2*x)^2)^(1/2)/(2*b*sinh(1
/2*x)^4+(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*b*cosh(1/2*x)^2+a-b)/(a-b))^(1/2)*(-sinh(1/2*x)^2)
^(1/2)/(2*b*sinh(1/2*x)^4+(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*b*sin
h(1/2*x)^2+a+b)^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b \cosh \relax (x) + a\right )}^{\frac {7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*cosh(x) + a)^(-7/2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{{\left (a+b\,\mathrm {cosh}\relax (x)\right )}^{7/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b*cosh(x))^(7/2),x)

[Out]

int(1/(a + b*cosh(x))^(7/2), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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