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

Optimal. Leaf size=156 \[ -\frac{2 (b \sinh (x)+c \cosh (x))}{\left (a^2-b^2+c^2\right ) \sqrt{a+b \cosh (x)+c \sinh (x)}}-\frac{2 i \sqrt{a+b \cosh (x)+c \sinh (x)} E\left (\frac{1}{2} \left (i x-\tan ^{-1}(b,-i c)\right )|\frac{2 \sqrt{b^2-c^2}}{a+\sqrt{b^2-c^2}}\right )}{\left (a^2-b^2+c^2\right ) \sqrt{\frac{a+b \cosh (x)+c \sinh (x)}{a+\sqrt{b^2-c^2}}}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0979996, antiderivative size = 156, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {3128, 3119, 2653} \[ -\frac{2 (b \sinh (x)+c \cosh (x))}{\left (a^2-b^2+c^2\right ) \sqrt{a+b \cosh (x)+c \sinh (x)}}-\frac{2 i \sqrt{a+b \cosh (x)+c \sinh (x)} E\left (\frac{1}{2} \left (i x-\tan ^{-1}(b,-i c)\right )|\frac{2 \sqrt{b^2-c^2}}{a+\sqrt{b^2-c^2}}\right )}{\left (a^2-b^2+c^2\right ) \sqrt{\frac{a+b \cosh (x)+c \sinh (x)}{a+\sqrt{b^2-c^2}}}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Cosh[x] + c*Sinh[x])^(-3/2),x]

[Out]

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

Rule 3128

Int[(cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)])^(-3/2), x_Symbol] :> Simp[(2*(c*Cos
[d + e*x] - b*Sin[d + e*x]))/(e*(a^2 - b^2 - c^2)*Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]]), x] + Dist[1/(a^2
- b^2 - c^2), Int[Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]], x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[a^2 -
b^2 - c^2, 0]

Rule 3119

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

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]

Rubi steps

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

Mathematica [C]  time = 6.19654, size = 1522, normalized size = 9.76 \[ \text{result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*Cosh[x] + c*Sinh[x])^(-3/2),x]

[Out]

Sqrt[a + b*Cosh[x] + c*Sinh[x]]*((-2*(b^2 - c^2))/(b*c*(-a^2 + b^2 - c^2)) - (2*(a*c - b^2*Sinh[x] + c^2*Sinh[
x]))/(b*(-a^2 + b^2 - c^2)*(a + b*Cosh[x] + c*Sinh[x]))) + (2*a*AppellF1[1/2, 1/2, 1/2, 3/2, ((-I)*(a + Sqrt[1
- b^2/c^2]*c*Sinh[x + ArcTanh[b/c]]))/(Sqrt[1 - b^2/c^2]*(1 - (I*a)/(Sqrt[1 - b^2/c^2]*c))*c), ((-I)*(a + Sqr
t[1 - b^2/c^2]*c*Sinh[x + ArcTanh[b/c]]))/(Sqrt[1 - b^2/c^2]*(-1 - (I*a)/(Sqrt[1 - b^2/c^2]*c))*c)]*Sech[x + A
rcTanh[b/c]]*Sqrt[-1 + I*Sinh[x + ArcTanh[b/c]]]*Sqrt[(c*Sqrt[(-b^2 + c^2)/c^2] - I*c*Sqrt[(-b^2 + c^2)/c^2]*S
inh[x + ArcTanh[b/c]])/(I*a + c*Sqrt[(-b^2 + c^2)/c^2])]*Sqrt[(c*Sqrt[(-b^2 + c^2)/c^2] + I*c*Sqrt[(-b^2 + c^2
)/c^2]*Sinh[x + ArcTanh[b/c]])/((-I)*a + c*Sqrt[(-b^2 + c^2)/c^2])]*Sqrt[a + c*Sqrt[(-b^2 + c^2)/c^2]*Sinh[x +
ArcTanh[b/c]]])/(Sqrt[1 - b^2/c^2]*c*(a^2 - b^2 + c^2)*Sqrt[I*(I + Sinh[x + ArcTanh[b/c]])]) - (b^2*((c*Appel
lF1[-1/2, -1/2, -1/2, 1/2, (a + b*Sqrt[1 - c^2/b^2]*Cosh[x + ArcTanh[c/b]])/(b*Sqrt[1 - c^2/b^2]*(1 + a/(b*Sqr
t[1 - c^2/b^2]))), (a + b*Sqrt[1 - c^2/b^2]*Cosh[x + ArcTanh[c/b]])/(b*Sqrt[1 - c^2/b^2]*(-1 + a/(b*Sqrt[1 - c
^2/b^2])))]*Sinh[x + ArcTanh[c/b]])/(b*Sqrt[1 - c^2/b^2]*Sqrt[(b*Sqrt[(b^2 - c^2)/b^2] - b*Sqrt[(b^2 - c^2)/b^
2]*Cosh[x + ArcTanh[c/b]])/(a + b*Sqrt[(b^2 - c^2)/b^2])]*Sqrt[a + b*Sqrt[(b^2 - c^2)/b^2]*Cosh[x + ArcTanh[c/
b]]]*Sqrt[(b*Sqrt[(b^2 - c^2)/b^2] + b*Sqrt[(b^2 - c^2)/b^2]*Cosh[x + ArcTanh[c/b]])/(-a + b*Sqrt[(b^2 - c^2)/
b^2])]) - ((-2*b*(a + b*Sqrt[1 - c^2/b^2]*Cosh[x + ArcTanh[c/b]]))/(b^2 - c^2) + (c*Sinh[x + ArcTanh[c/b]])/(b
*Sqrt[1 - c^2/b^2]))/Sqrt[a + b*Sqrt[1 - c^2/b^2]*Cosh[x + ArcTanh[c/b]]]))/(c*(a^2 - b^2 + c^2)) + (c*((c*App
ellF1[-1/2, -1/2, -1/2, 1/2, (a + b*Sqrt[1 - c^2/b^2]*Cosh[x + ArcTanh[c/b]])/(b*Sqrt[1 - c^2/b^2]*(1 + a/(b*S
qrt[1 - c^2/b^2]))), (a + b*Sqrt[1 - c^2/b^2]*Cosh[x + ArcTanh[c/b]])/(b*Sqrt[1 - c^2/b^2]*(-1 + a/(b*Sqrt[1 -
c^2/b^2])))]*Sinh[x + ArcTanh[c/b]])/(b*Sqrt[1 - c^2/b^2]*Sqrt[(b*Sqrt[(b^2 - c^2)/b^2] - b*Sqrt[(b^2 - c^2)/
b^2]*Cosh[x + ArcTanh[c/b]])/(a + b*Sqrt[(b^2 - c^2)/b^2])]*Sqrt[a + b*Sqrt[(b^2 - c^2)/b^2]*Cosh[x + ArcTanh[
c/b]]]*Sqrt[(b*Sqrt[(b^2 - c^2)/b^2] + b*Sqrt[(b^2 - c^2)/b^2]*Cosh[x + ArcTanh[c/b]])/(-a + b*Sqrt[(b^2 - c^2
)/b^2])]) - ((-2*b*(a + b*Sqrt[1 - c^2/b^2]*Cosh[x + ArcTanh[c/b]]))/(b^2 - c^2) + (c*Sinh[x + ArcTanh[c/b]])/
(b*Sqrt[1 - c^2/b^2]))/Sqrt[a + b*Sqrt[1 - c^2/b^2]*Cosh[x + ArcTanh[c/b]]]))/(a^2 - b^2 + c^2)

________________________________________________________________________________________

Maple [B]  time = 0.958, size = 1430, normalized size = 9.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*(b^2-c^2)^(1/2)*arctanh((b^2-c^2)*cosh(x)/((a^2+b^2-c^2)*(b^2-c^2))^(1/2))*(-a^2/(b^2-c^2)^(1/2)*sinh(x
)+a^3/(b^2-c^2))^(1/2)*((a^2+b^2-c^2)*(b-c)*(b+c))^(1/2)*sinh(x)-a*((-b^2+c^2)/(b^2-c^2)^(1/2)*sinh(x)^3+a*sin
h(x)^2)^(1/2)*((a^2+b^2-c^2)*(b^2-c^2))^(1/2)*ln((cosh(x)*sinh(x)*(2*((a^2+b^2-c^2)*(b-c)*(b+c))^(1/2)*b^4-4*(
(a^2+b^2-c^2)*(b-c)*(b+c))^(1/2)*b^2*c^2+2*((a^2+b^2-c^2)*(b-c)*(b+c))^(1/2)*c^4)+cosh(x)*(-2*(b^2-c^2)^(1/2)*
((a^2+b^2-c^2)*(b-c)*(b+c))^(1/2)*a*b^2+2*(b^2-c^2)^(1/2)*((a^2+b^2-c^2)*(b-c)*(b+c))^(1/2)*a*c^2)+sinh(x)*(2*
b^6-6*b^4*c^2+6*b^2*c^4-2*c^6)+2*(-a^2/(b^2-c^2)^(1/2)*sinh(x)+a^3/(b^2-c^2))^(1/2)*((-b^2+c^2)/(b^2-c^2)^(1/2
)*sinh(x)^3+a*sinh(x)^2)^(1/2)*(b^2-c^2)^(3/2)*b^2-2*(-a^2/(b^2-c^2)^(1/2)*sinh(x)+a^3/(b^2-c^2))^(1/2)*((-b^2
+c^2)/(b^2-c^2)^(1/2)*sinh(x)^3+a*sinh(x)^2)^(1/2)*(b^2-c^2)^(3/2)*c^2+2*(b^2-c^2)^(3/2)*a^3-2*a^3*b^2*(b^2-c^
2)^(1/2)+2*a^3*c^2*(b^2-c^2)^(1/2)-2*(b^2-c^2)^(1/2)*a*b^4+4*(b^2-c^2)^(1/2)*a*b^2*c^2-2*(b^2-c^2)^(1/2)*a*c^4
)/(b^2*cosh(x)-c^2*cosh(x)+((a^2+b^2-c^2)*(b-c)*(b+c))^(1/2))/(b^2-c^2)^(3/2))+a*((-b^2+c^2)/(b^2-c^2)^(1/2)*s
inh(x)^3+a*sinh(x)^2)^(1/2)*((a^2+b^2-c^2)*(b^2-c^2))^(1/2)*ln((cosh(x)*sinh(x)*(2*((a^2+b^2-c^2)*(b-c)*(b+c))
^(1/2)*b^4-4*((a^2+b^2-c^2)*(b-c)*(b+c))^(1/2)*b^2*c^2+2*((a^2+b^2-c^2)*(b-c)*(b+c))^(1/2)*c^4)+cosh(x)*(-2*(b
^2-c^2)^(1/2)*((a^2+b^2-c^2)*(b-c)*(b+c))^(1/2)*a*b^2+2*(b^2-c^2)^(1/2)*((a^2+b^2-c^2)*(b-c)*(b+c))^(1/2)*a*c^
2)+sinh(x)*(-2*b^6+6*b^4*c^2-6*b^2*c^4+2*c^6)-2*(-a^2/(b^2-c^2)^(1/2)*sinh(x)+a^3/(b^2-c^2))^(1/2)*((-b^2+c^2)
/(b^2-c^2)^(1/2)*sinh(x)^3+a*sinh(x)^2)^(1/2)*(b^2-c^2)^(3/2)*b^2+2*(-a^2/(b^2-c^2)^(1/2)*sinh(x)+a^3/(b^2-c^2
))^(1/2)*((-b^2+c^2)/(b^2-c^2)^(1/2)*sinh(x)^3+a*sinh(x)^2)^(1/2)*(b^2-c^2)^(3/2)*c^2-2*(b^2-c^2)^(3/2)*a^3+2*
a^3*b^2*(b^2-c^2)^(1/2)-2*a^3*c^2*(b^2-c^2)^(1/2)+2*(b^2-c^2)^(1/2)*a*b^4-4*(b^2-c^2)^(1/2)*a*b^2*c^2+2*(b^2-c
^2)^(1/2)*a*c^4)/(-b^2*cosh(x)+c^2*cosh(x)+((a^2+b^2-c^2)*(b-c)*(b+c))^(1/2))/(b^2-c^2)^(3/2)))/((-b^2+c^2)/(b
^2-c^2)^(1/2)*sinh(x)+a)^(1/2)/((a^2+b^2-c^2)*(b^2-c^2))^(1/2)/(-a^2/(b^2-c^2)^(1/2)*sinh(x)+a^3/(b^2-c^2))^(1
/2)/((a^2+b^2-c^2)*(b-c)*(b+c))^(1/2)/sinh(x)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b \cosh \left (x\right ) + c \sinh \left (x\right ) + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*cosh(x) + c*sinh(x) + a)^(-3/2), x)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{b \cosh \left (x\right ) + c \sinh \left (x\right ) + a}}{b^{2} \cosh \left (x\right )^{2} + c^{2} \sinh \left (x\right )^{2} + 2 \, a b \cosh \left (x\right ) + a^{2} + 2 \,{\left (b c \cosh \left (x\right ) + a c\right )} \sinh \left (x\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a + b \cosh{\left (x \right )} + c \sinh{\left (x \right )}\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*cosh(x) + c*sinh(x))**(-3/2), x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b \cosh \left (x\right ) + c \sinh \left (x\right ) + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*cosh(x) + c*sinh(x) + a)^(-3/2), x)