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

Optimal. Leaf size=322 \[ \frac {2 i \sqrt {\frac {a+b \cosh (x)+c \sinh (x)}{a+\sqrt {b^2-c^2}}} F\left (\frac {1}{2} \left (i x-\tan ^{-1}(b,-i c)\right )|\frac {2 \sqrt {b^2-c^2}}{a+\sqrt {b^2-c^2}}\right )}{3 \left (a^2-b^2+c^2\right ) \sqrt {a+b \cosh (x)+c \sinh (x)}}-\frac {8 i a \sqrt {a+b \cosh (x)+c \sinh (x)} E\left (\frac {1}{2} \left (i x-\tan ^{-1}(b,-i c)\right )|\frac {2 \sqrt {b^2-c^2}}{a+\sqrt {b^2-c^2}}\right )}{3 \left (a^2-b^2+c^2\right )^2 \sqrt {\frac {a+b \cosh (x)+c \sinh (x)}{a+\sqrt {b^2-c^2}}}}-\frac {8 (a b \sinh (x)+a c \cosh (x))}{3 \left (a^2-b^2+c^2\right )^2 \sqrt {a+b \cosh (x)+c \sinh (x)}}-\frac {2 (b \sinh (x)+c \cosh (x))}{3 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^{3/2}} \]

[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)

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Rubi [A]  time = 0.34, antiderivative size = 322, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3129, 3156, 3149, 3119, 2653, 3127, 2661} \[ \frac {2 i \sqrt {\frac {a+b \cosh (x)+c \sinh (x)}{a+\sqrt {b^2-c^2}}} F\left (\frac {1}{2} \left (i x-\tan ^{-1}(b,-i c)\right )|\frac {2 \sqrt {b^2-c^2}}{a+\sqrt {b^2-c^2}}\right )}{3 \left (a^2-b^2+c^2\right ) \sqrt {a+b \cosh (x)+c \sinh (x)}}-\frac {8 i a \sqrt {a+b \cosh (x)+c \sinh (x)} E\left (\frac {1}{2} \left (i x-\tan ^{-1}(b,-i c)\right )|\frac {2 \sqrt {b^2-c^2}}{a+\sqrt {b^2-c^2}}\right )}{3 \left (a^2-b^2+c^2\right )^2 \sqrt {\frac {a+b \cosh (x)+c \sinh (x)}{a+\sqrt {b^2-c^2}}}}-\frac {8 (a b \sinh (x)+a c \cosh (x))}{3 \left (a^2-b^2+c^2\right )^2 \sqrt {a+b \cosh (x)+c \sinh (x)}}-\frac {2 (b \sinh (x)+c \cosh (x))}{3 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^{3/2}} \]

Antiderivative was successfully verified.

[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 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 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 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 3127

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]*Cos[d + e*x - ArcTan[b, c]])/(a + Sqrt[b^2 + c^2])], 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 3129

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 3149

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 3156

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*} \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 {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 F\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*}

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Mathematica [C]  time = 6.26, size = 2492, normalized size = 7.74 \[ \text {Result too large to show} \]

Warning: Unable to verify antiderivative.

[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)

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fricas [F]  time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b \cosh \relax (x) + c \sinh \relax (x) + a}}{b^{3} \cosh \relax (x)^{3} + c^{3} \sinh \relax (x)^{3} + 3 \, a b^{2} \cosh \relax (x)^{2} + 3 \, a^{2} b \cosh \relax (x) + a^{3} + 3 \, {\left (b c^{2} \cosh \relax (x) + a c^{2}\right )} \sinh \relax (x)^{2} + 3 \, {\left (b^{2} c \cosh \relax (x)^{2} + 2 \, a b c \cosh \relax (x) + a^{2} c\right )} \sinh \relax (x)}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(b*cosh(x) + c*sinh(x) + a)/(b^3*cosh(x)^3 + c^3*sinh(x)^3 + 3*a*b^2*cosh(x)^2 + 3*a^2*b*cosh(x)
+ a^3 + 3*(b*c^2*cosh(x) + a*c^2)*sinh(x)^2 + 3*(b^2*c*cosh(x)^2 + 2*a*b*c*cosh(x) + a^2*c)*sinh(x)), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b \cosh \relax (x) + c \sinh \relax (x) + a\right )}^{\frac {5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [B]  time = 2.96, size = 6075, normalized size = 18.87 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b \cosh \relax (x) + c \sinh \relax (x) + a\right )}^{\frac {5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{{\left (a+b\,\mathrm {cosh}\relax (x)+c\,\mathrm {sinh}\relax (x)\right )}^{5/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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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)+c*sinh(x))**(5/2),x)

[Out]

Timed out

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