3.119 \(\int \frac {A+B \cosh (x)}{(a+b \cosh (x))^{3/2}} \, dx\)

Optimal. Leaf size=152 \[ -\frac {2 \sinh (x) (A b-a B)}{\left (a^2-b^2\right ) \sqrt {a+b \cosh (x)}}-\frac {2 i (A b-a B) \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{b \left (a^2-b^2\right ) \sqrt {\frac {a+b \cosh (x)}{a+b}}}-\frac {2 i B \sqrt {\frac {a+b \cosh (x)}{a+b}} F\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{b \sqrt {a+b \cosh (x)}} \]

[Out]

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

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Rubi [A]  time = 0.20, antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {2754, 2752, 2663, 2661, 2655, 2653} \[ -\frac {2 \sinh (x) (A b-a B)}{\left (a^2-b^2\right ) \sqrt {a+b \cosh (x)}}-\frac {2 i (A b-a B) \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{b \left (a^2-b^2\right ) \sqrt {\frac {a+b \cosh (x)}{a+b}}}-\frac {2 i B \sqrt {\frac {a+b \cosh (x)}{a+b}} F\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{b \sqrt {a+b \cosh (x)}} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*Cosh[x])/(a + b*Cosh[x])^(3/2),x]

[Out]

((-2*I)*(A*b - a*B)*Sqrt[a + b*Cosh[x]]*EllipticE[(I/2)*x, (2*b)/(a + b)])/(b*(a^2 - b^2)*Sqrt[(a + b*Cosh[x])
/(a + b)]) - ((2*I)*B*Sqrt[(a + b*Cosh[x])/(a + b)]*EllipticF[(I/2)*x, (2*b)/(a + b)])/(b*Sqrt[a + b*Cosh[x]])
- (2*(A*b - a*B)*Sinh[x])/((a^2 - b^2)*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 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 {A+B \cosh (x)}{(a+b \cosh (x))^{3/2}} \, dx &=-\frac {2 (A b-a B) \sinh (x)}{\left (a^2-b^2\right ) \sqrt {a+b \cosh (x)}}-\frac {2 \int \frac {\frac {1}{2} (-a A+b B)-\frac {1}{2} (A b-a B) \cosh (x)}{\sqrt {a+b \cosh (x)}} \, dx}{a^2-b^2}\\ &=-\frac {2 (A b-a B) \sinh (x)}{\left (a^2-b^2\right ) \sqrt {a+b \cosh (x)}}+\frac {B \int \frac {1}{\sqrt {a+b \cosh (x)}} \, dx}{b}+\frac {(A b-a B) \int \sqrt {a+b \cosh (x)} \, dx}{b \left (a^2-b^2\right )}\\ &=-\frac {2 (A b-a B) \sinh (x)}{\left (a^2-b^2\right ) \sqrt {a+b \cosh (x)}}+\frac {\left ((A b-a B) \sqrt {a+b \cosh (x)}\right ) \int \sqrt {\frac {a}{a+b}+\frac {b \cosh (x)}{a+b}} \, dx}{b \left (a^2-b^2\right ) \sqrt {\frac {a+b \cosh (x)}{a+b}}}+\frac {\left (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}{b \sqrt {a+b \cosh (x)}}\\ &=-\frac {2 i (A b-a B) \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{b \left (a^2-b^2\right ) \sqrt {\frac {a+b \cosh (x)}{a+b}}}-\frac {2 i B \sqrt {\frac {a+b \cosh (x)}{a+b}} F\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{b \sqrt {a+b \cosh (x)}}-\frac {2 (A b-a B) \sinh (x)}{\left (a^2-b^2\right ) \sqrt {a+b \cosh (x)}}\\ \end {align*}

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Mathematica [A]  time = 0.38, size = 133, normalized size = 0.88 \[ \frac {-2 i B \left (a^2-b^2\right ) \sqrt {\frac {a+b \cosh (x)}{a+b}} F\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )+2 b \sinh (x) (a B-A b)+2 i (a+b) (a B-A b) \sqrt {\frac {a+b \cosh (x)}{a+b}} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{b (a-b) (a+b) \sqrt {a+b \cosh (x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*Cosh[x])/(a + b*Cosh[x])^(3/2),x]

[Out]

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

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fricas [F]  time = 0.77, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (B \cosh \relax (x) + A\right )} \sqrt {b \cosh \relax (x) + a}}{b^{2} \cosh \relax (x)^{2} + 2 \, a b \cosh \relax (x) + a^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((B*cosh(x) + A)*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 \[ \int \frac {B \cosh \relax (x) + A}{{\left (b \cosh \relax (x) + a\right )}^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((B*cosh(x) + A)/(b*cosh(x) + a)^(3/2), x)

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maple [B]  time = 1.05, size = 483, normalized size = 3.18 \[ \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 {2 B \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 )}{b \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 {2 \left (A b -a B \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 )}\, \left (-2 \sqrt {-\frac {2 b}{a -b}}\, b \cosh \left (\frac {x}{2}\right ) \left (\sinh ^{2}\left (\frac {x}{2}\right )\right )+\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 ) \sqrt {\frac {2 b \left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}{a -b}+\frac {a +b}{a -b}}\, a +\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 ) \sqrt {\frac {2 b \left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}{a -b}+\frac {a +b}{a -b}}\, b -2 \sqrt {-\left (\sinh ^{2}\left (\frac {x}{2}\right )\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 ) \sqrt {\frac {2 b \left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}{a -b}+\frac {a +b}{a -b}}\, b \right )}{b \sinh \left (\frac {x}{2}\right )^{2} \left (2 b \left (\sinh ^{2}\left (\frac {x}{2}\right )\right )+a +b \right ) \sqrt {-\frac {2 b}{a -b}}\, \left (a^{2}-b^{2}\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((A+B*cosh(x))/(a+b*cosh(x))^(3/2),x)

[Out]

((2*b*cosh(1/2*x)^2+a-b)*sinh(1/2*x)^2)^(1/2)*(2*B/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)*EllipticF(cosh(1/2*x)*(-2*b/(a-b))^(1/2),
1/2*((-2*a+2*b)/b)^(1/2))+2*(A*b-B*a)/b/sinh(1/2*x)^2/(2*b*sinh(1/2*x)^2+a+b)/(-2*b/(a-b))^(1/2)/(a^2-b^2)*(2*
b*sinh(1/2*x)^4+(a+b)*sinh(1/2*x)^2)^(1/2)*(-2*(-2*b/(a-b))^(1/2)*b*cosh(1/2*x)*sinh(1/2*x)^2+(-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))*(2*b/(a-b)*sinh(1/2*x)^2+(a+b)/(a-b)
)^(1/2)*a+(-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))*(2*b/(a-b)
*sinh(1/2*x)^2+(a+b)/(a-b))^(1/2)*b-2*(-sinh(1/2*x)^2)^(1/2)*EllipticE(cosh(1/2*x)*(-2*b/(a-b))^(1/2),1/2*((-2
*a+2*b)/b)^(1/2))*(2*b/(a-b)*sinh(1/2*x)^2+(a+b)/(a-b))^(1/2)*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 \[ \int \frac {B \cosh \relax (x) + A}{{\left (b \cosh \relax (x) + a\right )}^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((B*cosh(x) + A)/(b*cosh(x) + a)^(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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