\(\int \frac {1}{(a+b \cosh (x)+c \sinh (x))^{7/2}} \, dx\) [767]

Optimal result

Integrand size = 14, antiderivative size = 411 \[ \int \frac {1}{(a+b \cosh (x)+c \sinh (x))^{7/2}} \, dx=-\frac {2 (c \cosh (x)+b \sinh (x))}{5 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^{5/2}}-\frac {16 (a c \cosh (x)+a b \sinh (x))}{15 \left (a^2-b^2+c^2\right )^2 (a+b \cosh (x)+c \sinh (x))^{3/2}}-\frac {2 i \left (23 a^2+9 b^2-9 c^2\right ) E\left (\frac {1}{2} \left (i x-\tan ^{-1}(b,-i c)\right )|\frac {2 \sqrt {b^2-c^2}}{a+\sqrt {b^2-c^2}}\right ) \sqrt {a+b \cosh (x)+c \sinh (x)}}{15 \left (a^2-b^2+c^2\right )^3 \sqrt {\frac {a+b \cosh (x)+c \sinh (x)}{a+\sqrt {b^2-c^2}}}}+\frac {16 i a \operatorname {EllipticF}\left (\frac {1}{2} \left (i x-\tan ^{-1}(b,-i c)\right ),\frac {2 \sqrt {b^2-c^2}}{a+\sqrt {b^2-c^2}}\right ) \sqrt {\frac {a+b \cosh (x)+c \sinh (x)}{a+\sqrt {b^2-c^2}}}}{15 \left (a^2-b^2+c^2\right )^2 \sqrt {a+b \cosh (x)+c \sinh (x)}}-\frac {2 \left (c \left (23 a^2+9 b^2-9 c^2\right ) \cosh (x)+b \left (23 a^2+9 b^2-9 c^2\right ) \sinh (x)\right )}{15 \left (a^2-b^2+c^2\right )^3 \sqrt {a+b \cosh (x)+c \sinh (x)}} \]

[Out]

-2/5*(c*cosh(x)+b*sinh(x))/(a^2-b^2+c^2)/(a+b*cosh(x)+c*sinh(x))^(5/2)-16/15*(a*c*cosh(x)+a*b*sinh(x))/(a^2-b^
2+c^2)^2/(a+b*cosh(x)+c*sinh(x))^(3/2)-2/15*(c*(23*a^2+9*b^2-9*c^2)*cosh(x)+b*(23*a^2+9*b^2-9*c^2)*sinh(x))/(a
^2-b^2+c^2)^3/(a+b*cosh(x)+c*sinh(x))^(1/2)-2/15*I*(23*a^2+9*b^2-9*c^2)*(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*cosh(x)+c*sinh(x))^(1/2)/(a^2-b^2+c^2)^3/((a+b*cosh(x)+c*sinh(x))/(a+(b^2-c^2)^(1/2)
))^(1/2)+16/15*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))*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)^2/(a+b*cosh(x)+c*sinh(x))^(1/2)

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 411, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3208, 3235, 3228, 3198, 2732, 3206, 2740} \[ \int \frac {1}{(a+b \cosh (x)+c \sinh (x))^{7/2}} \, dx=\frac {16 i a \sqrt {\frac {a+b \cosh (x)+c \sinh (x)}{a+\sqrt {b^2-c^2}}} \operatorname {EllipticF}\left (\frac {1}{2} \left (i x-\tan ^{-1}(b,-i c)\right ),\frac {2 \sqrt {b^2-c^2}}{a+\sqrt {b^2-c^2}}\right )}{15 \left (a^2-b^2+c^2\right )^2 \sqrt {a+b \cosh (x)+c \sinh (x)}}-\frac {2 i \left (23 a^2+9 b^2-9 c^2\right ) \sqrt {a+b \cosh (x)+c \sinh (x)} E\left (\frac {1}{2} \left (i x-\tan ^{-1}(b,-i c)\right )|\frac {2 \sqrt {b^2-c^2}}{a+\sqrt {b^2-c^2}}\right )}{15 \left (a^2-b^2+c^2\right )^3 \sqrt {\frac {a+b \cosh (x)+c \sinh (x)}{a+\sqrt {b^2-c^2}}}}-\frac {2 \left (b \sinh (x) \left (23 a^2+9 b^2-9 c^2\right )+c \cosh (x) \left (23 a^2+9 b^2-9 c^2\right )\right )}{15 \left (a^2-b^2+c^2\right )^3 \sqrt {a+b \cosh (x)+c \sinh (x)}}-\frac {16 (a b \sinh (x)+a c \cosh (x))}{15 \left (a^2-b^2+c^2\right )^2 (a+b \cosh (x)+c \sinh (x))^{3/2}}-\frac {2 (b \sinh (x)+c \cosh (x))}{5 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^{5/2}} \]

[In]

Int[(a + b*Cosh[x] + c*Sinh[x])^(-7/2),x]

[Out]

(-2*(c*Cosh[x] + b*Sinh[x]))/(5*(a^2 - b^2 + c^2)*(a + b*Cosh[x] + c*Sinh[x])^(5/2)) - (16*(a*c*Cosh[x] + a*b*
Sinh[x]))/(15*(a^2 - b^2 + c^2)^2*(a + b*Cosh[x] + c*Sinh[x])^(3/2)) - (((2*I)/15)*(23*a^2 + 9*b^2 - 9*c^2)*El
lipticE[(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)^3*Sqrt[(a + b*Cosh[x] + c*Sinh[x])/(a + Sqrt[b^2 - c^2])]) + (((16*I)/15)*a*EllipticF[(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 + Sq
rt[b^2 - c^2])])/((a^2 - b^2 + c^2)^2*Sqrt[a + b*Cosh[x] + c*Sinh[x]]) - (2*(c*(23*a^2 + 9*b^2 - 9*c^2)*Cosh[x
] + b*(23*a^2 + 9*b^2 - 9*c^2)*Sinh[x]))/(15*(a^2 - b^2 + c^2)^3*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))}{5 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^{5/2}}-\frac {2 \int \frac {-\frac {5 a}{2}+\frac {3}{2} b \cosh (x)+\frac {3}{2} c \sinh (x)}{(a+b \cosh (x)+c \sinh (x))^{5/2}} \, dx}{5 \left (a^2-b^2+c^2\right )} \\ & = -\frac {2 (c \cosh (x)+b \sinh (x))}{5 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^{5/2}}-\frac {16 (a c \cosh (x)+a b \sinh (x))}{15 \left (a^2-b^2+c^2\right )^2 (a+b \cosh (x)+c \sinh (x))^{3/2}}+\frac {4 \int \frac {\frac {3}{4} \left (5 a^2+3 b^2-3 c^2\right )-2 a b \cosh (x)-2 a c \sinh (x)}{(a+b \cosh (x)+c \sinh (x))^{3/2}} \, dx}{15 \left (a^2-b^2+c^2\right )^2} \\ & = -\frac {2 (c \cosh (x)+b \sinh (x))}{5 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^{5/2}}-\frac {16 (a c \cosh (x)+a b \sinh (x))}{15 \left (a^2-b^2+c^2\right )^2 (a+b \cosh (x)+c \sinh (x))^{3/2}}-\frac {2 \left (c \left (23 a^2+9 b^2-9 c^2\right ) \cosh (x)+b \left (23 a^2+9 b^2-9 c^2\right ) \sinh (x)\right )}{15 \left (a^2-b^2+c^2\right )^3 \sqrt {a+b \cosh (x)+c \sinh (x)}}-\frac {8 \int \frac {-\frac {1}{8} a \left (15 a^2+17 b^2-17 c^2\right )-\frac {1}{8} b \left (23 a^2+9 b^2-9 c^2\right ) \cosh (x)-\frac {1}{8} c \left (23 a^2+9 b^2-9 c^2\right ) \sinh (x)}{\sqrt {a+b \cosh (x)+c \sinh (x)}} \, dx}{15 \left (a^2-b^2+c^2\right )^3} \\ & = -\frac {2 (c \cosh (x)+b \sinh (x))}{5 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^{5/2}}-\frac {16 (a c \cosh (x)+a b \sinh (x))}{15 \left (a^2-b^2+c^2\right )^2 (a+b \cosh (x)+c \sinh (x))^{3/2}}-\frac {2 \left (c \left (23 a^2+9 b^2-9 c^2\right ) \cosh (x)+b \left (23 a^2+9 b^2-9 c^2\right ) \sinh (x)\right )}{15 \left (a^2-b^2+c^2\right )^3 \sqrt {a+b \cosh (x)+c \sinh (x)}}+\frac {\left (23 a^2+9 b^2-9 c^2\right ) \int \sqrt {a+b \cosh (x)+c \sinh (x)} \, dx}{15 \left (a^2-b^2+c^2\right )^3}-\frac {(8 a) \int \frac {1}{\sqrt {a+b \cosh (x)+c \sinh (x)}} \, dx}{15 \left (a^2-b^2+c^2\right )^2} \\ & = -\frac {2 (c \cosh (x)+b \sinh (x))}{5 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^{5/2}}-\frac {16 (a c \cosh (x)+a b \sinh (x))}{15 \left (a^2-b^2+c^2\right )^2 (a+b \cosh (x)+c \sinh (x))^{3/2}}-\frac {2 \left (c \left (23 a^2+9 b^2-9 c^2\right ) \cosh (x)+b \left (23 a^2+9 b^2-9 c^2\right ) \sinh (x)\right )}{15 \left (a^2-b^2+c^2\right )^3 \sqrt {a+b \cosh (x)+c \sinh (x)}}+\frac {\left (\left (23 a^2+9 b^2-9 c^2\right ) \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}{15 \left (a^2-b^2+c^2\right )^3 \sqrt {\frac {a+b \cosh (x)+c \sinh (x)}{a+\sqrt {b^2-c^2}}}}-\frac {\left (8 a \sqrt {\frac {a+b \cosh (x)+c \sinh (x)}{a+\sqrt {b^2-c^2}}}\right ) \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}{15 \left (a^2-b^2+c^2\right )^2 \sqrt {a+b \cosh (x)+c \sinh (x)}} \\ & = -\frac {2 (c \cosh (x)+b \sinh (x))}{5 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^{5/2}}-\frac {16 (a c \cosh (x)+a b \sinh (x))}{15 \left (a^2-b^2+c^2\right )^2 (a+b \cosh (x)+c \sinh (x))^{3/2}}-\frac {2 i \left (23 a^2+9 b^2-9 c^2\right ) 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)}}{15 \left (a^2-b^2+c^2\right )^3 \sqrt {\frac {a+b \cosh (x)+c \sinh (x)}{a+\sqrt {b^2-c^2}}}}+\frac {16 i a \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}}}}{15 \left (a^2-b^2+c^2\right )^2 \sqrt {a+b \cosh (x)+c \sinh (x)}}-\frac {2 \left (c \left (23 a^2+9 b^2-9 c^2\right ) \cosh (x)+b \left (23 a^2+9 b^2-9 c^2\right ) \sinh (x)\right )}{15 \left (a^2-b^2+c^2\right )^3 \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 = 6.42 (sec) , antiderivative size = 4093, normalized size of antiderivative = 9.96 \[ \int \frac {1}{(a+b \cosh (x)+c \sinh (x))^{7/2}} \, dx=\text {Result too large to show} \]

[In]

Integrate[(a + b*Cosh[x] + c*Sinh[x])^(-7/2),x]

[Out]

Sqrt[a + b*Cosh[x] + c*Sinh[x]]*((-2*(23*a^2 + 9*b^2 - 9*c^2)*(b^2 - c^2))/(15*b*c*(-a^2 + b^2 - c^2)^3) - (2*
(a*c - b^2*Sinh[x] + c^2*Sinh[x]))/(5*b*(-a^2 + b^2 - c^2)*(a + b*Cosh[x] + c*Sinh[x])^3) - (2*(-5*a^2*c - 3*b
^2*c + 3*c^3 + 8*a*b^2*Sinh[x] - 8*a*c^2*Sinh[x]))/(15*b*(-a^2 + b^2 - c^2)^2*(a + b*Cosh[x] + c*Sinh[x])^2) +
(2*(-15*a^3*c - 17*a*b^2*c + 17*a*c^3 + 23*a^2*b^2*Sinh[x] + 9*b^4*Sinh[x] - 23*a^2*c^2*Sinh[x] - 18*b^2*c^2*
Sinh[x] + 9*c^4*Sinh[x]))/(15*b*(-a^2 + b^2 - c^2)^3*(a + b*Cosh[x] + c*Sinh[x]))) + (2*a^3*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*Sqr
t[(-b^2 + c^2)/c^2]*Sinh[x + ArcTanh[b/c]]])/(Sqrt[1 - b^2/c^2]*c*(a^2 - b^2 + c^2)^3*Sqrt[I*(I + Sinh[x + Arc
Tanh[b/c]])]) + (34*a*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 + ArcT
anh[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]]])/(15*Sqrt[1 - b^2/c^2]
*c*(a^2 - b^2 + c^2)^3*Sqrt[I*(I + Sinh[x + ArcTanh[b/c]])]) - (34*a*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)]*Sec
h[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]*S
inh[x + ArcTanh[b/c]]])/(15*Sqrt[1 - b^2/c^2]*(a^2 - b^2 + c^2)^3*Sqrt[I*(I + Sinh[x + ArcTanh[b/c]])]) - (23*
a^2*b^2*((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*Sq
rt[(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]*Co
sh[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]]]))/(15*c*(a^2 - b^2
+ c^2)^3) - (3*b^4*((c*AppellF1[-1/2, -1/2, -1/2, 1/2, (a + b*Sqrt[1 - c^2/b^2]*Cosh[x + ArcTanh[c/b]])/(b*Sqr
t[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]]]))/(5*c*(
a^2 - b^2 + c^2)^3) + (23*a^2*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]]]))/(15*(a^2 - b^2 + c^2)^3) + (6*b^2*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 + ArcTa
nh[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]*S
qrt[(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]]]))/(5*(a^2 - b^2 + c^2)^3) - (3*c^3*((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]]]))/(5*(a^2 - b^2 + c^2)^3)

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(57908\) vs. \(2(441)=882\).

Time = 8.22 (sec) , antiderivative size = 57909, normalized size of antiderivative = 140.90

method	result	size
default	\(\text {Expression too large to display}\)	\(57909\)

[In]

int(1/(a+b*cosh(x)+c*sinh(x))^(7/2),x,method=_RETURNVERBOSE)

[Out]

result too large to display

Fricas [C] (verification not implemented)

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

Time = 0.74 (sec) , antiderivative size = 13897, normalized size of antiderivative = 33.81 \[ \int \frac {1}{(a+b \cosh (x)+c \sinh (x))^{7/2}} \, dx=\text {Too large to display} \]

[In]

integrate(1/(a+b*cosh(x)+c*sinh(x))^(7/2),x, algorithm="fricas")

[Out]

Too large to include

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(a+b \cosh (x)+c \sinh (x))^{7/2}} \, dx=\text {Timed out} \]

[In]

integrate(1/(a+b*cosh(x)+c*sinh(x))**(7/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {1}{(a+b \cosh (x)+c \sinh (x))^{7/2}} \, dx=\int { \frac {1}{{\left (b \cosh \left (x\right ) + c \sinh \left (x\right ) + a\right )}^{\frac {7}{2}}} \,d x } \]

[In]

integrate(1/(a+b*cosh(x)+c*sinh(x))^(7/2),x, algorithm="maxima")

[Out]

integrate((b*cosh(x) + c*sinh(x) + a)^(-7/2), x)

Giac [F]

\[ \int \frac {1}{(a+b \cosh (x)+c \sinh (x))^{7/2}} \, dx=\int { \frac {1}{{\left (b \cosh \left (x\right ) + c \sinh \left (x\right ) + a\right )}^{\frac {7}{2}}} \,d x } \]

[In]

integrate(1/(a+b*cosh(x)+c*sinh(x))^(7/2),x, algorithm="giac")

[Out]

integrate((b*cosh(x) + c*sinh(x) + a)^(-7/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(a+b \cosh (x)+c \sinh (x))^{7/2}} \, dx=\int \frac {1}{{\left (a+b\,\mathrm {cosh}\left (x\right )+c\,\mathrm {sinh}\left (x\right )\right )}^{7/2}} \,d x \]

[In]

int(1/(a + b*cosh(x) + c*sinh(x))^(7/2),x)

[Out]

int(1/(a + b*cosh(x) + c*sinh(x))^(7/2), x)