3.759 \(\int \frac {1}{(\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x))^3} \, dx\)
Optimal. Leaf size=146 \[ \frac {2 (b \sinh (x)+c \cosh (x))}{15 \left (b^2-c^2\right ) \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^2}+\frac {b \sinh (x)+c \cosh (x)}{5 \sqrt {b^2-c^2} \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^3}-\frac {2 \left (\sqrt {b^2-c^2} \sinh (x)+c\right )}{15 c \left (b^2-c^2\right ) (b \sinh (x)+c \cosh (x))} \]
[Out]
1/5*(c*cosh(x)+b*sinh(x))/(b^2-c^2)^(1/2)/(b*cosh(x)+c*sinh(x)+(b^2-c^2)^(1/2))^3+2/15*(c*cosh(x)+b*sinh(x))/(
b^2-c^2)/(b*cosh(x)+c*sinh(x)+(b^2-c^2)^(1/2))^2-2/15*(c+sinh(x)*(b^2-c^2)^(1/2))/c/(b^2-c^2)/(c*cosh(x)+b*sin
h(x))
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Rubi [A] time = 0.12, antiderivative size = 146, normalized size of antiderivative = 1.00,
number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used =
{3116, 3114} \[ \frac {2 (b \sinh (x)+c \cosh (x))}{15 \left (b^2-c^2\right ) \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^2}+\frac {b \sinh (x)+c \cosh (x)}{5 \sqrt {b^2-c^2} \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^3}-\frac {2 \left (\sqrt {b^2-c^2} \sinh (x)+c\right )}{15 c \left (b^2-c^2\right ) (b \sinh (x)+c \cosh (x))} \]
Antiderivative was successfully verified.
[In]
Int[(Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x])^(-3),x]
[Out]
(c*Cosh[x] + b*Sinh[x])/(5*Sqrt[b^2 - c^2]*(Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x])^3) + (2*(c*Cosh[x] + b*Si
nh[x]))/(15*(b^2 - c^2)*(Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x])^2) - (2*(c + Sqrt[b^2 - c^2]*Sinh[x]))/(15*c
*(b^2 - c^2)*(c*Cosh[x] + b*Sinh[x]))
Rule 3114
Int[(cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)])^(-1), x_Symbol] :> -Simp[(c - a*Sin
[d + e*x])/(c*e*(c*Cos[d + e*x] - b*Sin[d + e*x])), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[a^2 - b^2 - c^2, 0]
Rule 3116
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)/(a*e*(2*n + 1)), x] + Dist[(n + 1)/(a*(2*n +
1)), Int[(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[a^2 - b^2 -
c^2, 0] && LtQ[n, -1]
Rubi steps
\begin {align*} \int \frac {1}{\left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^3} \, dx &=\frac {c \cosh (x)+b \sinh (x)}{5 \sqrt {b^2-c^2} \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^3}+\frac {2 \int \frac {1}{\left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^2} \, dx}{5 \sqrt {b^2-c^2}}\\ &=\frac {c \cosh (x)+b \sinh (x)}{5 \sqrt {b^2-c^2} \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^3}+\frac {2 (c \cosh (x)+b \sinh (x))}{15 \left (b^2-c^2\right ) \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^2}+\frac {2 \int \frac {1}{\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)} \, dx}{15 \left (b^2-c^2\right )}\\ &=\frac {c \cosh (x)+b \sinh (x)}{5 \sqrt {b^2-c^2} \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^3}+\frac {2 (c \cosh (x)+b \sinh (x))}{15 \left (b^2-c^2\right ) \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^2}-\frac {2 \left (c+\sqrt {b^2-c^2} \sinh (x)\right )}{15 c \left (b^2-c^2\right ) (c \cosh (x)+b \sinh (x))}\\ \end {align*}
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Mathematica [A] time = 0.40, size = 184, normalized size = 1.26 \[ \frac {-12 \left (b^2-c^2\right ) \left (\sqrt {b^2-c^2} \sinh (x)+c\right )-\frac {2 \sqrt {b^2-c^2} \sinh (x) (b \sinh (x)+c \cosh (x))^4}{(b-c) (b+c)}-\frac {b \sqrt {b^2-c^2} (b \sinh (x)+c \cosh (x))^3}{(b-c) (b+c)}+\left (\sqrt {b^2-c^2} \sinh (x)-5 c\right ) (b \sinh (x)+c \cosh (x))^2+12 b \sqrt {b^2-c^2} (b \sinh (x)+c \cosh (x))}{15 c (b \sinh (x)+c \cosh (x))^5} \]
Antiderivative was successfully verified.
[In]
Integrate[(Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x])^(-3),x]
[Out]
(12*b*Sqrt[b^2 - c^2]*(c*Cosh[x] + b*Sinh[x]) - (b*Sqrt[b^2 - c^2]*(c*Cosh[x] + b*Sinh[x])^3)/((b - c)*(b + c)
) - (2*Sqrt[b^2 - c^2]*Sinh[x]*(c*Cosh[x] + b*Sinh[x])^4)/((b - c)*(b + c)) + (c*Cosh[x] + b*Sinh[x])^2*(-5*c
+ Sqrt[b^2 - c^2]*Sinh[x]) - 12*(b^2 - c^2)*(c + Sqrt[b^2 - c^2]*Sinh[x]))/(15*c*(c*Cosh[x] + b*Sinh[x])^5)
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fricas [B] time = 0.50, size = 3035, normalized size = 20.79 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(b*cosh(x)+c*sinh(x)+(b^2-c^2)^(1/2))^3,x, algorithm="fricas")
[Out]
-4/15*(10*(b^4 + 4*b^3*c + 6*b^2*c^2 + 4*b*c^3 + c^4)*cosh(x)^7 + 70*(b^4 + 4*b^3*c + 6*b^2*c^2 + 4*b*c^3 + c^
4)*cosh(x)*sinh(x)^6 + 10*(b^4 + 4*b^3*c + 6*b^2*c^2 + 4*b*c^3 + c^4)*sinh(x)^7 + 76*(b^4 + 2*b^3*c - 2*b*c^3
- c^4)*cosh(x)^5 + 2*(38*b^4 + 76*b^3*c - 76*b*c^3 - 38*c^4 + 105*(b^4 + 4*b^3*c + 6*b^2*c^2 + 4*b*c^3 + c^4)*
cosh(x)^2)*sinh(x)^5 + 10*(35*(b^4 + 4*b^3*c + 6*b^2*c^2 + 4*b*c^3 + c^4)*cosh(x)^3 + 38*(b^4 + 2*b^3*c - 2*b*
c^3 - c^4)*cosh(x))*sinh(x)^4 + 10*(b^4 - 2*b^2*c^2 + c^4)*cosh(x)^3 + 10*(35*(b^4 + 4*b^3*c + 6*b^2*c^2 + 4*b
*c^3 + c^4)*cosh(x)^4 + b^4 - 2*b^2*c^2 + c^4 + 76*(b^4 + 2*b^3*c - 2*b*c^3 - c^4)*cosh(x)^2)*sinh(x)^3 + 10*(
21*(b^4 + 4*b^3*c + 6*b^2*c^2 + 4*b*c^3 + c^4)*cosh(x)^5 + 76*(b^4 + 2*b^3*c - 2*b*c^3 - c^4)*cosh(x)^3 + 3*(b
^4 - 2*b^2*c^2 + c^4)*cosh(x))*sinh(x)^2 + 10*(7*(b^4 + 4*b^3*c + 6*b^2*c^2 + 4*b*c^3 + c^4)*cosh(x)^6 + 38*(b
^4 + 2*b^3*c - 2*b*c^3 - c^4)*cosh(x)^4 + 3*(b^4 - 2*b^2*c^2 + c^4)*cosh(x)^2)*sinh(x) - (45*(b^3 + 3*b^2*c +
3*b*c^2 + c^3)*cosh(x)^6 + 270*(b^3 + 3*b^2*c + 3*b*c^2 + c^3)*cosh(x)*sinh(x)^5 + 45*(b^3 + 3*b^2*c + 3*b*c^2
+ c^3)*sinh(x)^6 + 55*(b^3 + b^2*c - b*c^2 - c^3)*cosh(x)^4 + 5*(11*b^3 + 11*b^2*c - 11*b*c^2 - 11*c^3 + 135*
(b^3 + 3*b^2*c + 3*b*c^2 + c^3)*cosh(x)^2)*sinh(x)^4 + 20*(45*(b^3 + 3*b^2*c + 3*b*c^2 + c^3)*cosh(x)^3 + 11*(
b^3 + b^2*c - b*c^2 - c^3)*cosh(x))*sinh(x)^3 + b^3 - 3*b^2*c + 3*b*c^2 - c^3 - 5*(b^3 - b^2*c - b*c^2 + c^3)*
cosh(x)^2 + 5*(135*(b^3 + 3*b^2*c + 3*b*c^2 + c^3)*cosh(x)^4 - b^3 + b^2*c + b*c^2 - c^3 + 66*(b^3 + b^2*c - b
*c^2 - c^3)*cosh(x)^2)*sinh(x)^2 + 10*(27*(b^3 + 3*b^2*c + 3*b*c^2 + c^3)*cosh(x)^5 + 22*(b^3 + b^2*c - b*c^2
- c^3)*cosh(x)^3 - (b^3 - b^2*c - b*c^2 + c^3)*cosh(x))*sinh(x))*sqrt(b^2 - c^2))/((b^7 + 7*b^6*c + 21*b^5*c^2
+ 35*b^4*c^3 + 35*b^3*c^4 + 21*b^2*c^5 + 7*b*c^6 + c^7)*cosh(x)^10 + 10*(b^7 + 7*b^6*c + 21*b^5*c^2 + 35*b^4*
c^3 + 35*b^3*c^4 + 21*b^2*c^5 + 7*b*c^6 + c^7)*cosh(x)*sinh(x)^9 + (b^7 + 7*b^6*c + 21*b^5*c^2 + 35*b^4*c^3 +
35*b^3*c^4 + 21*b^2*c^5 + 7*b*c^6 + c^7)*sinh(x)^10 - 5*(b^7 + 5*b^6*c + 9*b^5*c^2 + 5*b^4*c^3 - 5*b^3*c^4 - 9
*b^2*c^5 - 5*b*c^6 - c^7)*cosh(x)^8 - 5*(b^7 + 5*b^6*c + 9*b^5*c^2 + 5*b^4*c^3 - 5*b^3*c^4 - 9*b^2*c^5 - 5*b*c
^6 - c^7 - 9*(b^7 + 7*b^6*c + 21*b^5*c^2 + 35*b^4*c^3 + 35*b^3*c^4 + 21*b^2*c^5 + 7*b*c^6 + c^7)*cosh(x)^2)*si
nh(x)^8 + 40*(3*(b^7 + 7*b^6*c + 21*b^5*c^2 + 35*b^4*c^3 + 35*b^3*c^4 + 21*b^2*c^5 + 7*b*c^6 + c^7)*cosh(x)^3
- (b^7 + 5*b^6*c + 9*b^5*c^2 + 5*b^4*c^3 - 5*b^3*c^4 - 9*b^2*c^5 - 5*b*c^6 - c^7)*cosh(x))*sinh(x)^7 - b^7 + 3
*b^6*c - b^5*c^2 - 5*b^4*c^3 + 5*b^3*c^4 + b^2*c^5 - 3*b*c^6 + c^7 + 10*(b^7 + 3*b^6*c + b^5*c^2 - 5*b^4*c^3 -
5*b^3*c^4 + b^2*c^5 + 3*b*c^6 + c^7)*cosh(x)^6 + 10*(b^7 + 3*b^6*c + b^5*c^2 - 5*b^4*c^3 - 5*b^3*c^4 + b^2*c^
5 + 3*b*c^6 + c^7 + 21*(b^7 + 7*b^6*c + 21*b^5*c^2 + 35*b^4*c^3 + 35*b^3*c^4 + 21*b^2*c^5 + 7*b*c^6 + c^7)*cos
h(x)^4 - 14*(b^7 + 5*b^6*c + 9*b^5*c^2 + 5*b^4*c^3 - 5*b^3*c^4 - 9*b^2*c^5 - 5*b*c^6 - c^7)*cosh(x)^2)*sinh(x)
^6 + 4*(63*(b^7 + 7*b^6*c + 21*b^5*c^2 + 35*b^4*c^3 + 35*b^3*c^4 + 21*b^2*c^5 + 7*b*c^6 + c^7)*cosh(x)^5 - 70*
(b^7 + 5*b^6*c + 9*b^5*c^2 + 5*b^4*c^3 - 5*b^3*c^4 - 9*b^2*c^5 - 5*b*c^6 - c^7)*cosh(x)^3 + 15*(b^7 + 3*b^6*c
+ b^5*c^2 - 5*b^4*c^3 - 5*b^3*c^4 + b^2*c^5 + 3*b*c^6 + c^7)*cosh(x))*sinh(x)^5 - 10*(b^7 + b^6*c - 3*b^5*c^2
- 3*b^4*c^3 + 3*b^3*c^4 + 3*b^2*c^5 - b*c^6 - c^7)*cosh(x)^4 - 10*(b^7 + b^6*c - 3*b^5*c^2 - 3*b^4*c^3 + 3*b^3
*c^4 + 3*b^2*c^5 - b*c^6 - c^7 - 21*(b^7 + 7*b^6*c + 21*b^5*c^2 + 35*b^4*c^3 + 35*b^3*c^4 + 21*b^2*c^5 + 7*b*c
^6 + c^7)*cosh(x)^6 + 35*(b^7 + 5*b^6*c + 9*b^5*c^2 + 5*b^4*c^3 - 5*b^3*c^4 - 9*b^2*c^5 - 5*b*c^6 - c^7)*cosh(
x)^4 - 15*(b^7 + 3*b^6*c + b^5*c^2 - 5*b^4*c^3 - 5*b^3*c^4 + b^2*c^5 + 3*b*c^6 + c^7)*cosh(x)^2)*sinh(x)^4 + 4
0*(3*(b^7 + 7*b^6*c + 21*b^5*c^2 + 35*b^4*c^3 + 35*b^3*c^4 + 21*b^2*c^5 + 7*b*c^6 + c^7)*cosh(x)^7 - 7*(b^7 +
5*b^6*c + 9*b^5*c^2 + 5*b^4*c^3 - 5*b^3*c^4 - 9*b^2*c^5 - 5*b*c^6 - c^7)*cosh(x)^5 + 5*(b^7 + 3*b^6*c + b^5*c^
2 - 5*b^4*c^3 - 5*b^3*c^4 + b^2*c^5 + 3*b*c^6 + c^7)*cosh(x)^3 - (b^7 + b^6*c - 3*b^5*c^2 - 3*b^4*c^3 + 3*b^3*
c^4 + 3*b^2*c^5 - b*c^6 - c^7)*cosh(x))*sinh(x)^3 + 5*(b^7 - b^6*c - 3*b^5*c^2 + 3*b^4*c^3 + 3*b^3*c^4 - 3*b^2
*c^5 - b*c^6 + c^7)*cosh(x)^2 + 5*(9*(b^7 + 7*b^6*c + 21*b^5*c^2 + 35*b^4*c^3 + 35*b^3*c^4 + 21*b^2*c^5 + 7*b*
c^6 + c^7)*cosh(x)^8 + b^7 - b^6*c - 3*b^5*c^2 + 3*b^4*c^3 + 3*b^3*c^4 - 3*b^2*c^5 - b*c^6 + c^7 - 28*(b^7 + 5
*b^6*c + 9*b^5*c^2 + 5*b^4*c^3 - 5*b^3*c^4 - 9*b^2*c^5 - 5*b*c^6 - c^7)*cosh(x)^6 + 30*(b^7 + 3*b^6*c + b^5*c^
2 - 5*b^4*c^3 - 5*b^3*c^4 + b^2*c^5 + 3*b*c^6 + c^7)*cosh(x)^4 - 12*(b^7 + b^6*c - 3*b^5*c^2 - 3*b^4*c^3 + 3*b
^3*c^4 + 3*b^2*c^5 - b*c^6 - c^7)*cosh(x)^2)*sinh(x)^2 + 10*((b^7 + 7*b^6*c + 21*b^5*c^2 + 35*b^4*c^3 + 35*b^3
*c^4 + 21*b^2*c^5 + 7*b*c^6 + c^7)*cosh(x)^9 - 4*(b^7 + 5*b^6*c + 9*b^5*c^2 + 5*b^4*c^3 - 5*b^3*c^4 - 9*b^2*c^
5 - 5*b*c^6 - c^7)*cosh(x)^7 + 6*(b^7 + 3*b^6*c + b^5*c^2 - 5*b^4*c^3 - 5*b^3*c^4 + b^2*c^5 + 3*b*c^6 + c^7)*c
osh(x)^5 - 4*(b^7 + b^6*c - 3*b^5*c^2 - 3*b^4*c^3 + 3*b^3*c^4 + 3*b^2*c^5 - b*c^6 - c^7)*cosh(x)^3 + (b^7 - b^
6*c - 3*b^5*c^2 + 3*b^4*c^3 + 3*b^3*c^4 - 3*b^2*c^5 - b*c^6 + c^7)*cosh(x))*sinh(x))
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(b*cosh(x)+c*sinh(x)+(b^2-c^2)^(1/2))^3,x, algorithm="giac")
[Out]
Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Unab
le to divide, perhaps due to rounding error%%%{%%%{1,[3,0]%%%}+%%%{3,[2,1]%%%}+%%%{3,[1,2]%%%}+%%%{1,[0,3]%%%}
,[6]%%%}+%%%{%%{[%%%{6,[2,0]%%%}+%%%{12,[1,1]%%%}+%%%{6,[0,2]%%%},0]:[1,0,%%%{-1,[2,0]%%%}+%%%{1,[0,2]%%%}]%%}
,[5]%%%}+%%%{%%%{15,[3,0]%%%}+%%%{15,[2,1]%%%}+%%%{-15,[1,2]%%%}+%%%{-15,[0,3]%%%},[4]%%%}+%%%{%%{[%%%{20,[2,0
]%%%}+%%%{-20,[0,2]%%%},0]:[1,0,%%%{-1,[2,0]%%%}+%%%{1,[0,2]%%%}]%%},[3]%%%}+%%%{%%%{15,[3,0]%%%}+%%%{-15,[2,1
]%%%}+%%%{-15,[1,2]%%%}+%%%{15,[0,3]%%%},[2]%%%}+%%%{%%{[%%%{6,[2,0]%%%}+%%%{-12,[1,1]%%%}+%%%{6,[0,2]%%%},0]:
[1,0,%%%{-1,[2,0]%%%}+%%%{1,[0,2]%%%}]%%},[1]%%%}+%%%{%%%{1,[3,0]%%%}+%%%{-3,[2,1]%%%}+%%%{3,[1,2]%%%}+%%%{-1,
[0,3]%%%},[0]%%%} / %%%{%%%{1,[6,0]%%%}+%%%{6,[5,1]%%%}+%%%{15,[4,2]%%%}+%%%{20,[3,3]%%%}+%%%{15,[2,4]%%%}+%%%
{6,[1,5]%%%}+%%%{1,[0,6]%%%},[6]%%%}+%%%{%%{[%%%{6,[5,0]%%%}+%%%{30,[4,1]%%%}+%%%{60,[3,2]%%%}+%%%{60,[2,3]%%%
}+%%%{30,[1,4]%%%}+%%%{6,[0,5]%%%},0]:[1,0,%%%{-1,[2,0]%%%}+%%%{1,[0,2]%%%}]%%},[5]%%%}+%%%{%%%{15,[6,0]%%%}+%
%%{60,[5,1]%%%}+%%%{75,[4,2]%%%}+%%%{-75,[2,4]%%%}+%%%{-60,[1,5]%%%}+%%%{-15,[0,6]%%%},[4]%%%}+%%%{%%{[%%%{20,
[5,0]%%%}+%%%{60,[4,1]%%%}+%%%{40,[3,2]%%%}+%%%{-40,[2,3]%%%}+%%%{-60,[1,4]%%%}+%%%{-20,[0,5]%%%},0]:[1,0,%%%{
-1,[2,0]%%%}+%%%{1,[0,2]%%%}]%%},[3]%%%}+%%%{%%%{15,[6,0]%%%}+%%%{30,[5,1]%%%}+%%%{-15,[4,2]%%%}+%%%{-60,[3,3]
%%%}+%%%{-15,[2,4]%%%}+%%%{30,[1,5]%%%}+%%%{15,[0,6]%%%},[2]%%%}+%%%{%%{[%%%{6,[5,0]%%%}+%%%{6,[4,1]%%%}+%%%{-
12,[3,2]%%%}+%%%{-12,[2,3]%%%}+%%%{6,[1,4]%%%}+%%%{6,[0,5]%%%},0]:[1,0,%%%{-1,[2,0]%%%}+%%%{1,[0,2]%%%}]%%},[1
]%%%}+%%%{%%%{1,[6,0]%%%}+%%%{-3,[4,2]%%%}+%%%{3,[2,4]%%%}+%%%{-1,[0,6]%%%},[0]%%%} Error: Bad Argument Value
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maple [B] time = 0.47, size = 488, normalized size = 3.34 \[ \frac {-\frac {2 \left (4 \sqrt {b^{2}-c^{2}}\, b^{2}-\sqrt {b^{2}-c^{2}}\, c^{2}+4 b^{3}-3 b \,c^{2}\right ) \left (\tanh ^{4}\left (\frac {x}{2}\right )\right )}{c^{2}}-\frac {4 \left (8 b^{4}-8 c^{2} b^{2}+c^{4}+8 \sqrt {b^{2}-c^{2}}\, b^{3}-4 \sqrt {b^{2}-c^{2}}\, c^{2} b \right ) \left (\tanh ^{3}\left (\frac {x}{2}\right )\right )}{c^{3}}-\frac {8 \left (24 \sqrt {b^{2}-c^{2}}\, b^{4}-20 \sqrt {b^{2}-c^{2}}\, b^{2} c^{2}+2 \sqrt {b^{2}-c^{2}}\, c^{4}+24 b^{5}-32 b^{3} c^{2}+9 b \,c^{4}\right ) \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )}{3 c^{4}}-\frac {4 \left (48 b^{6}-76 b^{4} c^{2}+31 b^{2} c^{4}-2 c^{6}+48 \sqrt {b^{2}-c^{2}}\, b^{5}-52 \sqrt {b^{2}-c^{2}}\, b^{3} c^{2}+11 \sqrt {b^{2}-c^{2}}\, b \,c^{4}\right ) \tanh \left (\frac {x}{2}\right )}{3 c^{5}}-\frac {2 \left (192 \sqrt {b^{2}-c^{2}}\, b^{6}-256 \sqrt {b^{2}-c^{2}}\, b^{4} c^{2}+96 \sqrt {b^{2}-c^{2}}\, b^{2} c^{4}-7 \sqrt {b^{2}-c^{2}}\, c^{6}+192 b^{7}-352 b^{5} c^{2}+200 b^{3} c^{4}-35 c^{6} b \right )}{15 c^{6}}}{c^{4} \left (\tanh ^{2}\left (\frac {x}{2}\right )+\frac {2 \sqrt {b^{2}-c^{2}}\, \tanh \left (\frac {x}{2}\right )}{c}+\frac {2 \tanh \left (\frac {x}{2}\right ) b}{c}+\frac {2 \sqrt {b^{2}-c^{2}}\, b}{c^{2}}+\frac {2 b^{2}}{c^{2}}-1\right )^{2} \left (\tanh \left (\frac {x}{2}\right )+\frac {\sqrt {b^{2}-c^{2}}}{c}+\frac {b}{c}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(b*cosh(x)+c*sinh(x)+(b^2-c^2)^(1/2))^3,x)
[Out]
2/c^4*(-(4*(b^2-c^2)^(1/2)*b^2-(b^2-c^2)^(1/2)*c^2+4*b^3-3*b*c^2)/c^2*tanh(1/2*x)^4-2*(8*b^4-8*c^2*b^2+c^4+8*(
b^2-c^2)^(1/2)*b^3-4*(b^2-c^2)^(1/2)*c^2*b)/c^3*tanh(1/2*x)^3-4/3*(24*(b^2-c^2)^(1/2)*b^4-20*(b^2-c^2)^(1/2)*b
^2*c^2+2*(b^2-c^2)^(1/2)*c^4+24*b^5-32*b^3*c^2+9*b*c^4)/c^4*tanh(1/2*x)^2-2/3*(48*b^6-76*b^4*c^2+31*b^2*c^4-2*
c^6+48*(b^2-c^2)^(1/2)*b^5-52*(b^2-c^2)^(1/2)*b^3*c^2+11*(b^2-c^2)^(1/2)*b*c^4)/c^5*tanh(1/2*x)-1/15/c^6*(192*
(b^2-c^2)^(1/2)*b^6-256*(b^2-c^2)^(1/2)*b^4*c^2+96*(b^2-c^2)^(1/2)*b^2*c^4-7*(b^2-c^2)^(1/2)*c^6+192*b^7-352*b
^5*c^2+200*b^3*c^4-35*c^6*b))/(tanh(1/2*x)^2+2/c*(b^2-c^2)^(1/2)*tanh(1/2*x)+2/c*tanh(1/2*x)*b+2/c^2*(b^2-c^2)
^(1/2)*b+2/c^2*b^2-1)^2/(tanh(1/2*x)+1/c*(b^2-c^2)^(1/2)+b/c)
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(b*cosh(x)+c*sinh(x)+(b^2-c^2)^(1/2))^3,x, algorithm="maxima")
[Out]
Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is undefined.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (b\,\mathrm {cosh}\relax (x)+\sqrt {b^2-c^2}+c\,\mathrm {sinh}\relax (x)\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(b*cosh(x) + (b^2 - c^2)^(1/2) + c*sinh(x))^3,x)
[Out]
int(1/(b*cosh(x) + (b^2 - c^2)^(1/2) + c*sinh(x))^3, 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/(b*cosh(x)+c*sinh(x)+(b**2-c**2)**(1/2))**3,x)
[Out]
Timed out
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