\(\int \frac {1}{(a+b \cosh (x)+c \sinh (x))^{5/2}} \, dx\) [766]

Optimal result

Integrand size = 14, antiderivative size = 322 \[ \int \frac {1}{(a+b \cosh (x)+c \sinh (x))^{5/2}} \, dx=-\frac {2 (c \cosh (x)+b \sinh (x))}{3 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^{3/2}}-\frac {8 (a c \cosh (x)+a b \sinh (x))}{3 \left (a^2-b^2+c^2\right )^2 \sqrt {a+b \cosh (x)+c \sinh (x)}}-\frac {8 i a E\left (\frac {1}{2} \left (i x-\tan ^{-1}(b,-i c)\right )|\frac {2 \sqrt {b^2-c^2}}{a+\sqrt {b^2-c^2}}\right ) \sqrt {a+b \cosh (x)+c \sinh (x)}}{3 \left (a^2-b^2+c^2\right )^2 \sqrt {\frac {a+b \cosh (x)+c \sinh (x)}{a+\sqrt {b^2-c^2}}}}+\frac {2 i \operatorname {EllipticF}\left (\frac {1}{2} \left (i x-\tan ^{-1}(b,-i c)\right ),\frac {2 \sqrt {b^2-c^2}}{a+\sqrt {b^2-c^2}}\right ) \sqrt {\frac {a+b \cosh (x)+c \sinh (x)}{a+\sqrt {b^2-c^2}}}}{3 \left (a^2-b^2+c^2\right ) \sqrt {a+b \cosh (x)+c \sinh (x)}} \]

[Out]

-2/3*(c*cosh(x)+b*sinh(x))/(a^2-b^2+c^2)/(a+b*cosh(x)+c*sinh(x))^(3/2)-8/3*(a*c*cosh(x)+a*b*sinh(x))/(a^2-b^2+
c^2)^2/(a+b*cosh(x)+c*sinh(x))^(1/2)-8/3*I*a*(cos(1/2*I*x-1/2*arctan(b,-I*c))^2)^(1/2)/cos(1/2*I*x-1/2*arctan(
b,-I*c))*EllipticE(sin(1/2*I*x-1/2*arctan(b,-I*c)),2^(1/2)*((b^2-c^2)^(1/2)/(a+(b^2-c^2)^(1/2)))^(1/2))*(a+b*c
osh(x)+c*sinh(x))^(1/2)/(a^2-b^2+c^2)^2/((a+b*cosh(x)+c*sinh(x))/(a+(b^2-c^2)^(1/2)))^(1/2)+2/3*I*(cos(1/2*I*x
-1/2*arctan(b,-I*c))^2)^(1/2)/cos(1/2*I*x-1/2*arctan(b,-I*c))*EllipticF(sin(1/2*I*x-1/2*arctan(b,-I*c)),2^(1/2
)*((b^2-c^2)^(1/2)/(a+(b^2-c^2)^(1/2)))^(1/2))*((a+b*cosh(x)+c*sinh(x))/(a+(b^2-c^2)^(1/2)))^(1/2)/(a^2-b^2+c^
2)/(a+b*cosh(x)+c*sinh(x))^(1/2)

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3208, 3235, 3228, 3198, 2732, 3206, 2740} \[ \int \frac {1}{(a+b \cosh (x)+c \sinh (x))^{5/2}} \, dx=\frac {2 i \sqrt {\frac {a+b \cosh (x)+c \sinh (x)}{a+\sqrt {b^2-c^2}}} \operatorname {EllipticF}\left (\frac {1}{2} \left (i x-\tan ^{-1}(b,-i c)\right ),\frac {2 \sqrt {b^2-c^2}}{a+\sqrt {b^2-c^2}}\right )}{3 \left (a^2-b^2+c^2\right ) \sqrt {a+b \cosh (x)+c \sinh (x)}}-\frac {8 i a \sqrt {a+b \cosh (x)+c \sinh (x)} E\left (\frac {1}{2} \left (i x-\tan ^{-1}(b,-i c)\right )|\frac {2 \sqrt {b^2-c^2}}{a+\sqrt {b^2-c^2}}\right )}{3 \left (a^2-b^2+c^2\right )^2 \sqrt {\frac {a+b \cosh (x)+c \sinh (x)}{a+\sqrt {b^2-c^2}}}}-\frac {8 (a b \sinh (x)+a c \cosh (x))}{3 \left (a^2-b^2+c^2\right )^2 \sqrt {a+b \cosh (x)+c \sinh (x)}}-\frac {2 (b \sinh (x)+c \cosh (x))}{3 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^{3/2}} \]

[In]

Int[(a + b*Cosh[x] + c*Sinh[x])^(-5/2),x]

[Out]

(-2*(c*Cosh[x] + b*Sinh[x]))/(3*(a^2 - b^2 + c^2)*(a + b*Cosh[x] + c*Sinh[x])^(3/2)) - (8*(a*c*Cosh[x] + a*b*S
inh[x]))/(3*(a^2 - b^2 + c^2)^2*Sqrt[a + b*Cosh[x] + c*Sinh[x]]) - (((8*I)/3)*a*EllipticE[(I*x - ArcTan[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)^2*Sqrt
[(a + b*Cosh[x] + c*Sinh[x])/(a + Sqrt[b^2 - c^2])]) + (((2*I)/3)*EllipticF[(I*x - ArcTan[b, (-I)*c])/2, (2*Sq
rt[b^2 - c^2])/(a + Sqrt[b^2 - c^2])]*Sqrt[(a + b*Cosh[x] + c*Sinh[x])/(a + Sqrt[b^2 - c^2])])/((a^2 - b^2 + c
^2)*Sqrt[a + b*Cosh[x] + c*Sinh[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 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 3198

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]/(a + Sqrt[b^2 + c^2]))*Cos[d + e*x - ArcTan[b, c]]], 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 3206

Int[1/Sqrt[cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)]], x_Symbol] :> Dist[Sqrt[(a +
b*Cos[d + e*x] + c*Sin[d + e*x])/(a + Sqrt[b^2 + c^2])]/Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]], Int[1/Sqrt[
a/(a + Sqrt[b^2 + c^2]) + (Sqrt[b^2 + c^2]/(a + Sqrt[b^2 + c^2]))*Cos[d + e*x - ArcTan[b, c]]], x], x] /; Free
Q[{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 3208

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

Rule 3228

Int[((A_.) + cos[(d_.) + (e_.)*(x_)]*(B_.) + (C_.)*sin[(d_.) + (e_.)*(x_)])/Sqrt[cos[(d_.) + (e_.)*(x_)]*(b_.)
+ (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)]], x_Symbol] :> Dist[B/b, Int[Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]]
, x], x] + Dist[(A*b - a*B)/b, Int[1/Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]], x], x] /; FreeQ[{a, b, c, d, e
, A, B, C}, x] && EqQ[B*c - b*C, 0] && NeQ[A*b - a*B, 0]

Rule 3235

Int[((a_.) + cos[(d_.) + (e_.)*(x_)]*(b_.) + (c_.)*sin[(d_.) + (e_.)*(x_)])^(n_)*((A_.) + cos[(d_.) + (e_.)*(x
_)]*(B_.) + (C_.)*sin[(d_.) + (e_.)*(x_)]), x_Symbol] :> Simp[(-(c*B - b*C - (a*C - c*A)*Cos[d + e*x] + (a*B -
b*A)*Sin[d + e*x]))*((a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n + 1)/(e*(n + 1)*(a^2 - b^2 - c^2))), x] + Dist[
1/((n + 1)*(a^2 - b^2 - c^2)), Int[(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n + 1)*Simp[(n + 1)*(a*A - b*B - c*C
) + (n + 2)*(a*B - b*A)*Cos[d + e*x] + (n + 2)*(a*C - c*A)*Sin[d + e*x], x], x], x] /; FreeQ[{a, b, c, d, e, A
, B, C}, x] && LtQ[n, -1] && NeQ[a^2 - b^2 - c^2, 0] && NeQ[n, -2]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 (c \cosh (x)+b \sinh (x))}{3 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^{3/2}}-\frac {2 \int \frac {-\frac {3 a}{2}+\frac {1}{2} b \cosh (x)+\frac {1}{2} c \sinh (x)}{(a+b \cosh (x)+c \sinh (x))^{3/2}} \, dx}{3 \left (a^2-b^2+c^2\right )} \\ & = -\frac {2 (c \cosh (x)+b \sinh (x))}{3 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^{3/2}}-\frac {8 (a c \cosh (x)+a b \sinh (x))}{3 \left (a^2-b^2+c^2\right )^2 \sqrt {a+b \cosh (x)+c \sinh (x)}}+\frac {4 \int \frac {\frac {1}{4} \left (3 a^2+b^2-c^2\right )+a b \cosh (x)+a c \sinh (x)}{\sqrt {a+b \cosh (x)+c \sinh (x)}} \, dx}{3 \left (a^2-b^2+c^2\right )^2} \\ & = -\frac {2 (c \cosh (x)+b \sinh (x))}{3 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^{3/2}}-\frac {8 (a c \cosh (x)+a b \sinh (x))}{3 \left (a^2-b^2+c^2\right )^2 \sqrt {a+b \cosh (x)+c \sinh (x)}}+\frac {(4 a) \int \sqrt {a+b \cosh (x)+c \sinh (x)} \, dx}{3 \left (a^2-b^2+c^2\right )^2}-\frac {\int \frac {1}{\sqrt {a+b \cosh (x)+c \sinh (x)}} \, dx}{3 \left (a^2-b^2+c^2\right )} \\ & = -\frac {2 (c \cosh (x)+b \sinh (x))}{3 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^{3/2}}-\frac {8 (a c \cosh (x)+a b \sinh (x))}{3 \left (a^2-b^2+c^2\right )^2 \sqrt {a+b \cosh (x)+c \sinh (x)}}+\frac {\left (4 a \sqrt {a+b \cosh (x)+c \sinh (x)}\right ) \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}{3 \left (a^2-b^2+c^2\right )^2 \sqrt {\frac {a+b \cosh (x)+c \sinh (x)}{a+\sqrt {b^2-c^2}}}}-\frac {\sqrt {\frac {a+b \cosh (x)+c \sinh (x)}{a+\sqrt {b^2-c^2}}} \int \frac {1}{\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}{3 \left (a^2-b^2+c^2\right ) \sqrt {a+b \cosh (x)+c \sinh (x)}} \\ & = -\frac {2 (c \cosh (x)+b \sinh (x))}{3 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^{3/2}}-\frac {8 (a c \cosh (x)+a b \sinh (x))}{3 \left (a^2-b^2+c^2\right )^2 \sqrt {a+b \cosh (x)+c \sinh (x)}}-\frac {8 i a 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)}}{3 \left (a^2-b^2+c^2\right )^2 \sqrt {\frac {a+b \cosh (x)+c \sinh (x)}{a+\sqrt {b^2-c^2}}}}+\frac {2 i \operatorname {EllipticF}\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 {\frac {a+b \cosh (x)+c \sinh (x)}{a+\sqrt {b^2-c^2}}}}{3 \left (a^2-b^2+c^2\right ) \sqrt {a+b \cosh (x)+c \sinh (x)}} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.

Time = 6.23 (sec) , antiderivative size = 2492, normalized size of antiderivative = 7.74 \[ \int \frac {1}{(a+b \cosh (x)+c \sinh (x))^{5/2}} \, dx=\text {Result too large to show} \]

[In]

Integrate[(a + b*Cosh[x] + c*Sinh[x])^(-5/2),x]

[Out]

Sqrt[a + b*Cosh[x] + c*Sinh[x]]*((8*a*(b^2 - c^2))/(3*b*c*(a^2 - b^2 + c^2)^2) - (2*(a*c - b^2*Sinh[x] + c^2*S
inh[x]))/(3*b*(-a^2 + b^2 - c^2)*(a + b*Cosh[x] + c*Sinh[x])^2) - (2*(-3*a^2*c - b^2*c + c^3 + 4*a*b^2*Sinh[x]
- 4*a*c^2*Sinh[x]))/(3*b*(-a^2 + b^2 - c^2)^2*(a + b*Cosh[x] + c*Sinh[x]))) + (2*a^2*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 + 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)]*Sech[x + ArcTanh[b/c]]*Sqrt[-1 + I*Sinh[x + ArcTanh[b/c]]]*Sqrt[(c*Sqrt[(-b^2 + c^2)/c^2] - I*c*Sq
rt[(-b^2 + c^2)/c^2]*Sinh[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)^2*Sqrt[I*(I + Sinh[x + ArcTanh[b
/c]])]) + (2*b^2*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 + 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)]*Sech[x + ArcTanh[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]*Sinh[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]]])/(3*Sqrt[1 - b^2/c^2]*c*(a^2 -
b^2 + c^2)^2*Sqrt[I*(I + Sinh[x + ArcTanh[b/c]])]) - (2*c*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 + 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)]*Sech[x + ArcTanh
[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]*Sinh[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 + ArcTa
nh[b/c]]])/(3*Sqrt[1 - b^2/c^2]*(a^2 - b^2 + c^2)^2*Sqrt[I*(I + Sinh[x + ArcTanh[b/c]])]) - (4*a*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]]]))/(3*c*(a^2 - b^2 + c^2)^2) + (4*a*c
*((c*AppellF1[-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*Sqrt[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]]]))/(3*(a^2 - b^2 + c^2)^2)

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(2159\) vs. \(2(356)=712\).

Time = 2.51 (sec) , antiderivative size = 2160, normalized size of antiderivative = 6.71

method	result	size
default	\(\text {Expression too large to display}\)	\(2160\)

[In]

int(1/(a+b*cosh(x)+c*sinh(x))^(5/2),x,method=_RETURNVERBOSE)

[Out]

2/((-b^2*sinh(x)+sinh(x)*c^2+a*(b^2-c^2)^(1/2))/(b^2-c^2)^(1/2))^(1/2)*a*(b^2-c^2)^(1/2)*(-1/2*cosh(x)/(a^2+b^
2-c^2)/(sinh(x)^2*(b^2-c^2)-a^2)+1/2/(a^2+b^2-c^2)/((a^2+b^2-c^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)))+((-b^2*sinh(x)+sinh(x)*c^2+a*(b^2-c^2)^(1/2))/(b^2-c^2)^(1/2)*sinh(x)^2)^(1/2
)*(1/(b^2-c^2)*a^2/(2*a^2+2*b^2-2*c^2)*(1/(sinh(x)*(b^2-c^2)^(1/2)-a)/a^2*(b^2-c^2)/(cosh(x)+((a^2+b^2-c^2)*(b
-c)*(b+c))^(1/2)/(b^2-c^2))*((-b^2*sinh(x)+sinh(x)*c^2+a*(b^2-c^2)^(1/2))/(b^2-c^2)^(1/2)*sinh(x)^2)^(1/2)-(b^
2*sinh(x)-sinh(x)*c^2-a*(b^2-c^2)^(1/2))/(b^2-c^2)^(1/2)*((a^2+b^2-c^2)*(b-c)*(b+c))^(1/2)/(sinh(x)*(b^2-c^2)^
(1/2)-a)/a^2/((-sinh(x)*(b^2-c^2)^(1/2)+a)*a^2/(b^2-c^2))^(1/2)*ln((-2*(sinh(x)*(b^2-c^2)^(1/2)-a)*a^2/(b^2-c^
2)+2*(b^2*sinh(x)-sinh(x)*c^2-a*(b^2-c^2)^(1/2))/(b^2-c^2)^(3/2)*((a^2+b^2-c^2)*(b-c)*(b+c))^(1/2)*(cosh(x)+((
a^2+b^2-c^2)*(b-c)*(b+c))^(1/2)/(b^2-c^2))+2*((-sinh(x)*(b^2-c^2)^(1/2)+a)*a^2/(b^2-c^2))^(1/2)*((-b^2*sinh(x)
+sinh(x)*c^2+a*(b^2-c^2)^(1/2))/(b^2-c^2)^(1/2)*sinh(x)^2)^(1/2))/(cosh(x)+((a^2+b^2-c^2)*(b-c)*(b+c))^(1/2)/(
b^2-c^2))))+(b^2-c^2)*a^2/(2*a^2+2*b^2-2*c^2)/(-b^2+c^2)^2*(1/(sinh(x)*(b^2-c^2)^(1/2)-a)/a^2*(b^2-c^2)/(cosh(
x)+((a^2+b^2-c^2)*(b-c)*(b+c))^(1/2)/(-b^2+c^2))*((-b^2*sinh(x)+sinh(x)*c^2+a*(b^2-c^2)^(1/2))/(b^2-c^2)^(1/2)
*sinh(x)^2)^(1/2)+(b^2*sinh(x)-sinh(x)*c^2-a*(b^2-c^2)^(1/2))/(b^2-c^2)^(1/2)*((a^2+b^2-c^2)*(b-c)*(b+c))^(1/2
)/(sinh(x)*(b^2-c^2)^(1/2)-a)/a^2/((-sinh(x)*(b^2-c^2)^(1/2)+a)*a^2/(b^2-c^2))^(1/2)*ln((-2*(sinh(x)*(b^2-c^2)
^(1/2)-a)*a^2/(b^2-c^2)-2*(b^2*sinh(x)-sinh(x)*c^2-a*(b^2-c^2)^(1/2))/(b^2-c^2)^(3/2)*((a^2+b^2-c^2)*(b-c)*(b+
c))^(1/2)*(cosh(x)+((a^2+b^2-c^2)*(b-c)*(b+c))^(1/2)/(-b^2+c^2))+2*((-sinh(x)*(b^2-c^2)^(1/2)+a)*a^2/(b^2-c^2)
)^(1/2)*((-b^2*sinh(x)+sinh(x)*c^2+a*(b^2-c^2)^(1/2))/(b^2-c^2)^(1/2)*sinh(x)^2)^(1/2))/(cosh(x)+((a^2+b^2-c^2
)*(b-c)*(b+c))^(1/2)/(-b^2+c^2))))-1/2*(a^2*b^2-c^2*a^2-((a^2+b^2-c^2)*(b-c)*(b+c))^(1/2)*((a^2+b^2-c^2)*(b^2-
c^2))^(1/2))/(a^2+b^2-c^2)/((a^2+b^2-c^2)*(b^2-c^2))^(1/2)/(b^2-c^2)/((-sinh(x)*(b^2-c^2)^(1/2)+a)*a^2/(b^2-c^
2))^(1/2)*ln((-2*(sinh(x)*(b^2-c^2)^(1/2)-a)*a^2/(b^2-c^2)+2*(b^2*sinh(x)-sinh(x)*c^2-a*(b^2-c^2)^(1/2))/(b^2-
c^2)^(3/2)*((a^2+b^2-c^2)*(b-c)*(b+c))^(1/2)*(cosh(x)+((a^2+b^2-c^2)*(b-c)*(b+c))^(1/2)/(b^2-c^2))+2*((-sinh(x
)*(b^2-c^2)^(1/2)+a)*a^2/(b^2-c^2))^(1/2)*((-b^2*sinh(x)+sinh(x)*c^2+a*(b^2-c^2)^(1/2))/(b^2-c^2)^(1/2)*sinh(x
)^2)^(1/2))/(cosh(x)+((a^2+b^2-c^2)*(b-c)*(b+c))^(1/2)/(b^2-c^2)))+1/2*(-a^2*b^2+c^2*a^2+((a^2+b^2-c^2)*(b-c)*
(b+c))^(1/2)*((a^2+b^2-c^2)*(b^2-c^2))^(1/2))/(a^2+b^2-c^2)/((a^2+b^2-c^2)*(b^2-c^2))^(1/2)/(-b^2+c^2)/((-sinh
(x)*(b^2-c^2)^(1/2)+a)*a^2/(b^2-c^2))^(1/2)*ln((-2*(sinh(x)*(b^2-c^2)^(1/2)-a)*a^2/(b^2-c^2)-2*(b^2*sinh(x)-si
nh(x)*c^2-a*(b^2-c^2)^(1/2))/(b^2-c^2)^(3/2)*((a^2+b^2-c^2)*(b-c)*(b+c))^(1/2)*(cosh(x)+((a^2+b^2-c^2)*(b-c)*(
b+c))^(1/2)/(-b^2+c^2))+2*((-sinh(x)*(b^2-c^2)^(1/2)+a)*a^2/(b^2-c^2))^(1/2)*((-b^2*sinh(x)+sinh(x)*c^2+a*(b^2
-c^2)^(1/2))/(b^2-c^2)^(1/2)*sinh(x)^2)^(1/2))/(cosh(x)+((a^2+b^2-c^2)*(b-c)*(b+c))^(1/2)/(-b^2+c^2))))/sinh(x
)/((-b^2*sinh(x)+sinh(x)*c^2+a*(b^2-c^2)^(1/2))/(b^2-c^2)^(1/2))^(1/2)

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.17 (sec) , antiderivative size = 3730, normalized size of antiderivative = 11.58 \[ \int \frac {1}{(a+b \cosh (x)+c \sinh (x))^{5/2}} \, dx=\text {Too large to display} \]

[In]

integrate(1/(a+b*cosh(x)+c*sinh(x))^(5/2),x, algorithm="fricas")

[Out]

2/9*((sqrt(2)*(a^2*b^2 + 3*b^4 + a^2*c^2 - 6*b*c^3 - 3*c^4 + 2*(a^2*b + 3*b^3)*c)*cosh(x)^4 + sqrt(2)*(a^2*b^2
+ 3*b^4 + a^2*c^2 - 6*b*c^3 - 3*c^4 + 2*(a^2*b + 3*b^3)*c)*sinh(x)^4 + 4*sqrt(2)*(a^3*b + 3*a*b^3 - 3*a*b*c^2
- 3*a*c^3 + (a^3 + 3*a*b^2)*c)*cosh(x)^3 + 4*(sqrt(2)*(a^2*b^2 + 3*b^4 + a^2*c^2 - 6*b*c^3 - 3*c^4 + 2*(a^2*b
+ 3*b^3)*c)*cosh(x) + sqrt(2)*(a^3*b + 3*a*b^3 - 3*a*b*c^2 - 3*a*c^3 + (a^3 + 3*a*b^2)*c))*sinh(x)^3 + 2*sqrt
(2)*(2*a^4 + 7*a^2*b^2 + 3*b^4 + 3*c^4 - (7*a^2 + 6*b^2)*c^2)*cosh(x)^2 + 2*(3*sqrt(2)*(a^2*b^2 + 3*b^4 + a^2*
c^2 - 6*b*c^3 - 3*c^4 + 2*(a^2*b + 3*b^3)*c)*cosh(x)^2 + 6*sqrt(2)*(a^3*b + 3*a*b^3 - 3*a*b*c^2 - 3*a*c^3 + (a
^3 + 3*a*b^2)*c)*cosh(x) + sqrt(2)*(2*a^4 + 7*a^2*b^2 + 3*b^4 + 3*c^4 - (7*a^2 + 6*b^2)*c^2))*sinh(x)^2 + 4*sq
rt(2)*(a^3*b + 3*a*b^3 - 3*a*b*c^2 + 3*a*c^3 - (a^3 + 3*a*b^2)*c)*cosh(x) + 4*(sqrt(2)*(a^2*b^2 + 3*b^4 + a^2*
c^2 - 6*b*c^3 - 3*c^4 + 2*(a^2*b + 3*b^3)*c)*cosh(x)^3 + 3*sqrt(2)*(a^3*b + 3*a*b^3 - 3*a*b*c^2 - 3*a*c^3 + (a
^3 + 3*a*b^2)*c)*cosh(x)^2 + sqrt(2)*(2*a^4 + 7*a^2*b^2 + 3*b^4 + 3*c^4 - (7*a^2 + 6*b^2)*c^2)*cosh(x) + sqrt(
2)*(a^3*b + 3*a*b^3 - 3*a*b*c^2 + 3*a*c^3 - (a^3 + 3*a*b^2)*c))*sinh(x) + sqrt(2)*(a^2*b^2 + 3*b^4 + a^2*c^2 +
6*b*c^3 - 3*c^4 - 2*(a^2*b + 3*b^3)*c))*sqrt(b + c)*weierstrassPInverse(4/3*(4*a^2 - 3*b^2 + 3*c^2)/(b^2 + 2*
b*c + c^2), -8/27*(8*a^3 - 9*a*b^2 + 9*a*c^2)/(b^3 + 3*b^2*c + 3*b*c^2 + c^3), 1/3*(3*(b + c)*cosh(x) + 3*(b +
c)*sinh(x) + 2*a)/(b + c)) - 12*(sqrt(2)*(a*b^3 + 3*a*b^2*c + 3*a*b*c^2 + a*c^3)*cosh(x)^4 + sqrt(2)*(a*b^3 +
3*a*b^2*c + 3*a*b*c^2 + a*c^3)*sinh(x)^4 + 4*sqrt(2)*(a^2*b^2 + 2*a^2*b*c + a^2*c^2)*cosh(x)^3 + 4*(sqrt(2)*(
a*b^3 + 3*a*b^2*c + 3*a*b*c^2 + a*c^3)*cosh(x) + sqrt(2)*(a^2*b^2 + 2*a^2*b*c + a^2*c^2))*sinh(x)^3 + 2*sqrt(2
)*(2*a^3*b + a*b^3 - a*b*c^2 - a*c^3 + (2*a^3 + a*b^2)*c)*cosh(x)^2 + 2*(3*sqrt(2)*(a*b^3 + 3*a*b^2*c + 3*a*b*
c^2 + a*c^3)*cosh(x)^2 + 6*sqrt(2)*(a^2*b^2 + 2*a^2*b*c + a^2*c^2)*cosh(x) + sqrt(2)*(2*a^3*b + a*b^3 - a*b*c^
2 - a*c^3 + (2*a^3 + a*b^2)*c))*sinh(x)^2 + 4*sqrt(2)*(a^2*b^2 - a^2*c^2)*cosh(x) + 4*(sqrt(2)*(a*b^3 + 3*a*b^
2*c + 3*a*b*c^2 + a*c^3)*cosh(x)^3 + 3*sqrt(2)*(a^2*b^2 + 2*a^2*b*c + a^2*c^2)*cosh(x)^2 + sqrt(2)*(2*a^3*b +
a*b^3 - a*b*c^2 - a*c^3 + (2*a^3 + a*b^2)*c)*cosh(x) + sqrt(2)*(a^2*b^2 - a^2*c^2))*sinh(x) + sqrt(2)*(a*b^3 -
a*b^2*c - a*b*c^2 + a*c^3))*sqrt(b + c)*weierstrassZeta(4/3*(4*a^2 - 3*b^2 + 3*c^2)/(b^2 + 2*b*c + c^2), -8/2
7*(8*a^3 - 9*a*b^2 + 9*a*c^2)/(b^3 + 3*b^2*c + 3*b*c^2 + c^3), weierstrassPInverse(4/3*(4*a^2 - 3*b^2 + 3*c^2)
/(b^2 + 2*b*c + c^2), -8/27*(8*a^3 - 9*a*b^2 + 9*a*c^2)/(b^3 + 3*b^2*c + 3*b*c^2 + c^3), 1/3*(3*(b + c)*cosh(x
) + 3*(b + c)*sinh(x) + 2*a)/(b + c))) - 6*(4*(a*b^3 + 3*a*b^2*c + 3*a*b*c^2 + a*c^3)*cosh(x)^4 + 4*(a*b^3 + 3
*a*b^2*c + 3*a*b*c^2 + a*c^3)*sinh(x)^4 + (13*a^2*b^2 - b^4 + 13*a^2*c^2 + 2*b*c^3 + c^4 + 2*(13*a^2*b - b^3)*
c)*cosh(x)^3 + (13*a^2*b^2 - b^4 + 13*a^2*c^2 + 2*b*c^3 + c^4 + 2*(13*a^2*b - b^3)*c + 16*(a*b^3 + 3*a*b^2*c +
3*a*b*c^2 + a*c^3)*cosh(x))*sinh(x)^3 + 4*(2*a^3*b + a*b^3 - a*b*c^2 - a*c^3 + (2*a^3 + a*b^2)*c)*cosh(x)^2 +
(8*a^3*b + 4*a*b^3 - 4*a*b*c^2 - 4*a*c^3 + 24*(a*b^3 + 3*a*b^2*c + 3*a*b*c^2 + a*c^3)*cosh(x)^2 + 4*(2*a^3 +
a*b^2)*c + 3*(13*a^2*b^2 - b^4 + 13*a^2*c^2 + 2*b*c^3 + c^4 + 2*(13*a^2*b - b^3)*c)*cosh(x))*sinh(x)^2 + (3*a^
2*b^2 + b^4 + c^4 - (3*a^2 + 2*b^2)*c^2)*cosh(x) + (3*a^2*b^2 + b^4 + c^4 + 16*(a*b^3 + 3*a*b^2*c + 3*a*b*c^2
+ a*c^3)*cosh(x)^3 - (3*a^2 + 2*b^2)*c^2 + 3*(13*a^2*b^2 - b^4 + 13*a^2*c^2 + 2*b*c^3 + c^4 + 2*(13*a^2*b - b^
3)*c)*cosh(x)^2 + 8*(2*a^3*b + a*b^3 - a*b*c^2 - a*c^3 + (2*a^3 + a*b^2)*c)*cosh(x))*sinh(x))*sqrt(b*cosh(x) +
c*sinh(x) + a))/(a^4*b^3 - 2*a^2*b^5 + b^7 - b*c^6 + c^7 + (2*a^2 - 3*b^2)*c^5 - (2*a^2*b - 3*b^3)*c^4 + (a^4
*b^3 - 2*a^2*b^5 + b^7 + 3*b*c^6 + c^7 + (2*a^2 + b^2)*c^5 + (6*a^2*b - 5*b^3)*c^4 + (a^4 + 4*a^2*b^2 - 5*b^4)
*c^3 + (3*a^4*b - 4*a^2*b^3 + b^5)*c^2 + 3*(a^4*b^2 - 2*a^2*b^4 + b^6)*c)*cosh(x)^4 + (a^4*b^3 - 2*a^2*b^5 + b
^7 + 3*b*c^6 + c^7 + (2*a^2 + b^2)*c^5 + (6*a^2*b - 5*b^3)*c^4 + (a^4 + 4*a^2*b^2 - 5*b^4)*c^3 + (3*a^4*b - 4*
a^2*b^3 + b^5)*c^2 + 3*(a^4*b^2 - 2*a^2*b^4 + b^6)*c)*sinh(x)^4 + (a^4 - 4*a^2*b^2 + 3*b^4)*c^3 + 4*(a^5*b^2 -
2*a^3*b^4 + a*b^6 + 2*a*b*c^5 + a*c^6 + (2*a^3 - a*b^2)*c^4 + 4*(a^3*b - a*b^3)*c^3 + (a^5 - a*b^4)*c^2 + 2*(
a^5*b - 2*a^3*b^3 + a*b^5)*c)*cosh(x)^3 + 4*(a^5*b^2 - 2*a^3*b^4 + a*b^6 + 2*a*b*c^5 + a*c^6 + (2*a^3 - a*b^2)
*c^4 + 4*(a^3*b - a*b^3)*c^3 + (a^5 - a*b^4)*c^2 + 2*(a^5*b - 2*a^3*b^3 + a*b^5)*c + (a^4*b^3 - 2*a^2*b^5 + b^
7 + 3*b*c^6 + c^7 + (2*a^2 + b^2)*c^5 + (6*a^2*b - 5*b^3)*c^4 + (a^4 + 4*a^2*b^2 - 5*b^4)*c^3 + (3*a^4*b - 4*a
^2*b^3 + b^5)*c^2 + 3*(a^4*b^2 - 2*a^2*b^4 + b^6)*c)*cosh(x))*sinh(x)^3 - (a^4*b - 4*a^2*b^3 + 3*b^5)*c^2 + 2*
(2*a^6*b - 3*a^4*b^3 + b^7 + 3*b^3*c^4 + 3*b^2*c^5 - b*c^6 - c^7 + 3*(a^4 - b^4)*c^3 + 3*(a^4*b - b^5)*c^2 + (
2*a^6 - 3*a^4*b^2 + b^6)*c)*cosh(x)^2 + 2*(2*a^6*b - 3*a^4*b^3 + b^7 + 3*b^3*c^4 + 3*b^2*c^5 - b*c^6 - c^7 + 3
*(a^4 - b^4)*c^3 + 3*(a^4*b - b^5)*c^2 + 3*(a^4*b^3 - 2*a^2*b^5 + b^7 + 3*b*c^6 + c^7 + (2*a^2 + b^2)*c^5 + (6
*a^2*b - 5*b^3)*c^4 + (a^4 + 4*a^2*b^2 - 5*b^4)*c^3 + (3*a^4*b - 4*a^2*b^3 + b^5)*c^2 + 3*(a^4*b^2 - 2*a^2*b^4
+ b^6)*c)*cosh(x)^2 + (2*a^6 - 3*a^4*b^2 + b^6)*c + 6*(a^5*b^2 - 2*a^3*b^4 + a*b^6 + 2*a*b*c^5 + a*c^6 + (2*a
^3 - a*b^2)*c^4 + 4*(a^3*b - a*b^3)*c^3 + (a^5 - a*b^4)*c^2 + 2*(a^5*b - 2*a^3*b^3 + a*b^5)*c)*cosh(x))*sinh(x
)^2 - (a^4*b^2 - 2*a^2*b^4 + b^6)*c + 4*(a^5*b^2 - 2*a^3*b^4 + a*b^6 - a*c^6 - (2*a^3 - 3*a*b^2)*c^4 - (a^5 -
4*a^3*b^2 + 3*a*b^4)*c^2)*cosh(x) + 4*(a^5*b^2 - 2*a^3*b^4 + a*b^6 - a*c^6 - (2*a^3 - 3*a*b^2)*c^4 + (a^4*b^3
- 2*a^2*b^5 + b^7 + 3*b*c^6 + c^7 + (2*a^2 + b^2)*c^5 + (6*a^2*b - 5*b^3)*c^4 + (a^4 + 4*a^2*b^2 - 5*b^4)*c^3
+ (3*a^4*b - 4*a^2*b^3 + b^5)*c^2 + 3*(a^4*b^2 - 2*a^2*b^4 + b^6)*c)*cosh(x)^3 - (a^5 - 4*a^3*b^2 + 3*a*b^4)*c
^2 + 3*(a^5*b^2 - 2*a^3*b^4 + a*b^6 + 2*a*b*c^5 + a*c^6 + (2*a^3 - a*b^2)*c^4 + 4*(a^3*b - a*b^3)*c^3 + (a^5 -
a*b^4)*c^2 + 2*(a^5*b - 2*a^3*b^3 + a*b^5)*c)*cosh(x)^2 + (2*a^6*b - 3*a^4*b^3 + b^7 + 3*b^3*c^4 + 3*b^2*c^5
- b*c^6 - c^7 + 3*(a^4 - b^4)*c^3 + 3*(a^4*b - b^5)*c^2 + (2*a^6 - 3*a^4*b^2 + b^6)*c)*cosh(x))*sinh(x))

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(a+b \cosh (x)+c \sinh (x))^{5/2}} \, dx=\text {Timed out} \]

[In]

integrate(1/(a+b*cosh(x)+c*sinh(x))**(5/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {1}{(a+b \cosh (x)+c \sinh (x))^{5/2}} \, dx=\int { \frac {1}{{\left (b \cosh \left (x\right ) + c \sinh \left (x\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate(1/(a+b*cosh(x)+c*sinh(x))^(5/2),x, algorithm="maxima")

[Out]

integrate((b*cosh(x) + c*sinh(x) + a)^(-5/2), x)

Giac [F]

\[ \int \frac {1}{(a+b \cosh (x)+c \sinh (x))^{5/2}} \, dx=\int { \frac {1}{{\left (b \cosh \left (x\right ) + c \sinh \left (x\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate(1/(a+b*cosh(x)+c*sinh(x))^(5/2),x, algorithm="giac")

[Out]

integrate((b*cosh(x) + c*sinh(x) + a)^(-5/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(a+b \cosh (x)+c \sinh (x))^{5/2}} \, dx=\int \frac {1}{{\left (a+b\,\mathrm {cosh}\left (x\right )+c\,\mathrm {sinh}\left (x\right )\right )}^{5/2}} \,d x \]

[In]

int(1/(a + b*cosh(x) + c*sinh(x))^(5/2),x)

[Out]

int(1/(a + b*cosh(x) + c*sinh(x))^(5/2), x)