3.83 \(\int \frac {1}{(a+b \cosh (x))^{3/2}} \, dx\)

Optimal. Leaf size=84 \[ -\frac {2 b \sinh (x)}{\left (a^2-b^2\right ) \sqrt {a+b \cosh (x)}}-\frac {2 i \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{\left (a^2-b^2\right ) \sqrt {\frac {a+b \cosh (x)}{a+b}}} \]

[Out]

-2*b*sinh(x)/(a^2-b^2)/(a+b*cosh(x))^(1/2)-2*I*(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)/(a^2-b^2)/((a+b*cosh(x))/(a+b))^(1/2)

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Rubi [A]  time = 0.06, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {2664, 21, 2655, 2653} \[ -\frac {2 b \sinh (x)}{\left (a^2-b^2\right ) \sqrt {a+b \cosh (x)}}-\frac {2 i \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{\left (a^2-b^2\right ) \sqrt {\frac {a+b \cosh (x)}{a+b}}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Cosh[x])^(-3/2),x]

[Out]

((-2*I)*Sqrt[a + b*Cosh[x]]*EllipticE[(I/2)*x, (2*b)/(a + b)])/((a^2 - b^2)*Sqrt[(a + b*Cosh[x])/(a + b)]) - (
2*b*Sinh[x])/((a^2 - b^2)*Sqrt[a + b*Cosh[x]])

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*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 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]

Rubi steps

\begin {align*} \int \frac {1}{(a+b \cosh (x))^{3/2}} \, dx &=-\frac {2 b \sinh (x)}{\left (a^2-b^2\right ) \sqrt {a+b \cosh (x)}}-\frac {2 \int \frac {-\frac {a}{2}-\frac {1}{2} b \cosh (x)}{\sqrt {a+b \cosh (x)}} \, dx}{a^2-b^2}\\ &=-\frac {2 b \sinh (x)}{\left (a^2-b^2\right ) \sqrt {a+b \cosh (x)}}+\frac {\int \sqrt {a+b \cosh (x)} \, dx}{a^2-b^2}\\ &=-\frac {2 b \sinh (x)}{\left (a^2-b^2\right ) \sqrt {a+b \cosh (x)}}+\frac {\sqrt {a+b \cosh (x)} \int \sqrt {\frac {a}{a+b}+\frac {b \cosh (x)}{a+b}} \, dx}{\left (a^2-b^2\right ) \sqrt {\frac {a+b \cosh (x)}{a+b}}}\\ &=-\frac {2 i \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{\left (a^2-b^2\right ) \sqrt {\frac {a+b \cosh (x)}{a+b}}}-\frac {2 b \sinh (x)}{\left (a^2-b^2\right ) \sqrt {a+b \cosh (x)}}\\ \end {align*}

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Mathematica [A]  time = 0.13, size = 68, normalized size = 0.81 \[ -\frac {2 \left (b \sinh (x)+i (a+b) \sqrt {\frac {a+b \cosh (x)}{a+b}} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )\right )}{(a-b) (a+b) \sqrt {a+b \cosh (x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Cosh[x])^(-3/2),x]

[Out]

(-2*(I*(a + b)*Sqrt[(a + b*Cosh[x])/(a + b)]*EllipticE[(I/2)*x, (2*b)/(a + b)] + b*Sinh[x]))/((a - b)*(a + b)*
Sqrt[a + b*Cosh[x]])

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fricas [F]  time = 0.88, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b \cosh \relax (x) + a}}{b^{2} \cosh \relax (x)^{2} + 2 \, a b \cosh \relax (x) + a^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*cosh(x))^(3/2),x, algorithm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b \cosh \relax (x) + a\right )}^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*cosh(x))^(3/2),x, algorithm="giac")

[Out]

integrate((b*cosh(x) + a)^(-3/2), x)

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maple [B]  time = 0.50, size = 296, normalized size = 3.52 \[ \frac {-4 \sqrt {-\frac {2 b}{a -b}}\, b \cosh \left (\frac {x}{2}\right ) \left (\sinh ^{2}\left (\frac {x}{2}\right )\right )+2 \EllipticF \left (\cosh \left (\frac {x}{2}\right ) \sqrt {-\frac {2 b}{a -b}}, \frac {\sqrt {-\frac {2 \left (a -b \right )}{b}}}{2}\right ) \sqrt {\frac {2 b \left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}{a -b}+\frac {a +b}{a -b}}\, \sqrt {-\left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}\, a +2 \EllipticF \left (\cosh \left (\frac {x}{2}\right ) \sqrt {-\frac {2 b}{a -b}}, \frac {\sqrt {-\frac {2 \left (a -b \right )}{b}}}{2}\right ) \sqrt {\frac {2 b \left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}{a -b}+\frac {a +b}{a -b}}\, \sqrt {-\left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}\, b -4 \EllipticE \left (\cosh \left (\frac {x}{2}\right ) \sqrt {-\frac {2 b}{a -b}}, \frac {\sqrt {-\frac {2 \left (a -b \right )}{b}}}{2}\right ) \sqrt {\frac {2 b \left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}{a -b}+\frac {a +b}{a -b}}\, \sqrt {-\left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}\, b}{\sqrt {-\frac {2 b}{a -b}}\, \left (a -b \right ) \left (a +b \right ) \sinh \left (\frac {x}{2}\right ) \sqrt {2 b \left (\sinh ^{2}\left (\frac {x}{2}\right )\right )+a +b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*cosh(x))^(3/2),x)

[Out]

2*(-2*(-2*b/(a-b))^(1/2)*b*cosh(1/2*x)*sinh(1/2*x)^2+EllipticF(cosh(1/2*x)*(-2*b/(a-b))^(1/2),1/2*(-2*(a-b)/b)
^(1/2))*(2*b/(a-b)*sinh(1/2*x)^2+(a+b)/(a-b))^(1/2)*(-sinh(1/2*x)^2)^(1/2)*a+EllipticF(cosh(1/2*x)*(-2*b/(a-b)
)^(1/2),1/2*(-2*(a-b)/b)^(1/2))*(2*b/(a-b)*sinh(1/2*x)^2+(a+b)/(a-b))^(1/2)*(-sinh(1/2*x)^2)^(1/2)*b-2*Ellipti
cE(cosh(1/2*x)*(-2*b/(a-b))^(1/2),1/2*(-2*(a-b)/b)^(1/2))*(2*b/(a-b)*sinh(1/2*x)^2+(a+b)/(a-b))^(1/2)*(-sinh(1
/2*x)^2)^(1/2)*b)/(-2*b/(a-b))^(1/2)/(a-b)/(a+b)/sinh(1/2*x)/(2*b*sinh(1/2*x)^2+a+b)^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b \cosh \relax (x) + a\right )}^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*cosh(x))^(3/2),x, algorithm="maxima")

[Out]

integrate((b*cosh(x) + a)^(-3/2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (a+b\,\mathrm {cosh}\relax (x)\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b*cosh(x))^(3/2),x)

[Out]

int(1/(a + b*cosh(x))^(3/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a + b \cosh {\relax (x )}\right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*cosh(x))**(3/2),x)

[Out]

Integral((a + b*cosh(x))**(-3/2), x)

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