3.8.60 \(\int \frac {1}{(\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x))^4} \, dx\) [760]
Optimal. Leaf size=198 \[ \frac {c \cosh (x)+b \sinh (x)}{7 \sqrt {b^2-c^2} \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^4}+\frac {3 (c \cosh (x)+b \sinh (x))}{35 \left (b^2-c^2\right ) \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^3}+\frac {2 (c \cosh (x)+b \sinh (x))}{35 \left (b^2-c^2\right )^{3/2} \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^2}-\frac {2 \left (c+\sqrt {b^2-c^2} \sinh (x)\right )}{35 c \left (b^2-c^2\right )^{3/2} (c \cosh (x)+b \sinh (x))} \]
[Out]
1/7*(c*cosh(x)+b*sinh(x))/(b^2-c^2)^(1/2)/(b*cosh(x)+c*sinh(x)+(b^2-c^2)^(1/2))^4+3/35*(c*cosh(x)+b*sinh(x))/(
b^2-c^2)/(b*cosh(x)+c*sinh(x)+(b^2-c^2)^(1/2))^3+2/35*(c*cosh(x)+b*sinh(x))/(b^2-c^2)^(3/2)/(b*cosh(x)+c*sinh(
x)+(b^2-c^2)^(1/2))^2-2/35*(c+sinh(x)*(b^2-c^2)^(1/2))/c/(b^2-c^2)^(3/2)/(c*cosh(x)+b*sinh(x))
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Rubi [A]
time = 0.13, antiderivative size = 198, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {3195, 3193}
\begin {gather*} \frac {2 (b \sinh (x)+c \cosh (x))}{35 \left (b^2-c^2\right )^{3/2} \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^2}+\frac {3 (b \sinh (x)+c \cosh (x))}{35 \left (b^2-c^2\right ) \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^3}+\frac {b \sinh (x)+c \cosh (x)}{7 \sqrt {b^2-c^2} \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^4}-\frac {2 \left (\sqrt {b^2-c^2} \sinh (x)+c\right )}{35 c \left (b^2-c^2\right )^{3/2} (b \sinh (x)+c \cosh (x))} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x])^(-4),x]
[Out]
(c*Cosh[x] + b*Sinh[x])/(7*Sqrt[b^2 - c^2]*(Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x])^4) + (3*(c*Cosh[x] + b*Si
nh[x]))/(35*(b^2 - c^2)*(Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x])^3) + (2*(c*Cosh[x] + b*Sinh[x]))/(35*(b^2 -
c^2)^(3/2)*(Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x])^2) - (2*(c + Sqrt[b^2 - c^2]*Sinh[x]))/(35*c*(b^2 - c^2)^
(3/2)*(c*Cosh[x] + b*Sinh[x]))
Rule 3193
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 3195
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 )^4} \, dx &=\frac {c \cosh (x)+b \sinh (x)}{7 \sqrt {b^2-c^2} \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^4}+\frac {3 \int \frac {1}{\left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^3} \, dx}{7 \sqrt {b^2-c^2}}\\ &=\frac {c \cosh (x)+b \sinh (x)}{7 \sqrt {b^2-c^2} \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^4}+\frac {3 (c \cosh (x)+b \sinh (x))}{35 \left (b^2-c^2\right ) \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^3}+\frac {6 \int \frac {1}{\left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^2} \, dx}{35 \left (b^2-c^2\right )}\\ &=\frac {c \cosh (x)+b \sinh (x)}{7 \sqrt {b^2-c^2} \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^4}+\frac {3 (c \cosh (x)+b \sinh (x))}{35 \left (b^2-c^2\right ) \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^3}+\frac {2 (c \cosh (x)+b \sinh (x))}{35 \left (b^2-c^2\right )^{3/2} \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}{35 \left (b^2-c^2\right )^{3/2}}\\ &=\frac {c \cosh (x)+b \sinh (x)}{7 \sqrt {b^2-c^2} \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^4}+\frac {3 (c \cosh (x)+b \sinh (x))}{35 \left (b^2-c^2\right ) \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^3}+\frac {2 (c \cosh (x)+b \sinh (x))}{35 \left (b^2-c^2\right )^{3/2} \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 )}{35 c \left (b^2-c^2\right )^{3/2} (c \cosh (x)+b \sinh (x))}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(425\) vs. \(2(198)=396\).
time = 0.58, size = 425, normalized size = 2.15 \begin {gather*} -\frac {-832 b^4 c \sqrt {b^2-c^2}+1664 b^2 c^3 \sqrt {b^2-c^2}-832 c^5 \sqrt {b^2-c^2}+1190 b c \left (b^2-c^2\right )^2 \cosh (x)+448 c \sqrt {b^2-c^2} \left (-b^4+c^4\right ) \cosh (2 x)+112 b^5 c \cosh (3 x)+56 b^3 c^3 \cosh (3 x)-168 b c^5 \cosh (3 x)-28 b^5 c \cosh (5 x)+28 b c^5 \cosh (5 x)+6 b^5 c \cosh (7 x)+20 b^3 c^3 \cosh (7 x)+6 b c^5 \cosh (7 x)-35 b^6 \sinh (x)+1295 b^4 c^2 \sinh (x)-2485 b^2 c^4 \sinh (x)+1225 c^6 \sinh (x)-896 b^3 c^2 \sqrt {b^2-c^2} \sinh (2 x)+896 b c^4 \sqrt {b^2-c^2} \sinh (2 x)+21 b^6 \sinh (3 x)+189 b^4 c^2 \sinh (3 x)-161 b^2 c^4 \sinh (3 x)-49 c^6 \sinh (3 x)-7 b^6 \sinh (5 x)-35 b^4 c^2 \sinh (5 x)+35 b^2 c^4 \sinh (5 x)+7 c^6 \sinh (5 x)+b^6 \sinh (7 x)+15 b^4 c^2 \sinh (7 x)+15 b^2 c^4 \sinh (7 x)+c^6 \sinh (7 x)}{1120 (b-c) c (b+c) (c \cosh (x)+b \sinh (x))^7} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x])^(-4),x]
[Out]
-1/1120*(-832*b^4*c*Sqrt[b^2 - c^2] + 1664*b^2*c^3*Sqrt[b^2 - c^2] - 832*c^5*Sqrt[b^2 - c^2] + 1190*b*c*(b^2 -
c^2)^2*Cosh[x] + 448*c*Sqrt[b^2 - c^2]*(-b^4 + c^4)*Cosh[2*x] + 112*b^5*c*Cosh[3*x] + 56*b^3*c^3*Cosh[3*x] -
168*b*c^5*Cosh[3*x] - 28*b^5*c*Cosh[5*x] + 28*b*c^5*Cosh[5*x] + 6*b^5*c*Cosh[7*x] + 20*b^3*c^3*Cosh[7*x] + 6*b
*c^5*Cosh[7*x] - 35*b^6*Sinh[x] + 1295*b^4*c^2*Sinh[x] - 2485*b^2*c^4*Sinh[x] + 1225*c^6*Sinh[x] - 896*b^3*c^2
*Sqrt[b^2 - c^2]*Sinh[2*x] + 896*b*c^4*Sqrt[b^2 - c^2]*Sinh[2*x] + 21*b^6*Sinh[3*x] + 189*b^4*c^2*Sinh[3*x] -
161*b^2*c^4*Sinh[3*x] - 49*c^6*Sinh[3*x] - 7*b^6*Sinh[5*x] - 35*b^4*c^2*Sinh[5*x] + 35*b^2*c^4*Sinh[5*x] + 7*c
^6*Sinh[5*x] + b^6*Sinh[7*x] + 15*b^4*c^2*Sinh[7*x] + 15*b^2*c^4*Sinh[7*x] + c^6*Sinh[7*x])/((b - c)*c*(b + c)
*(c*Cosh[x] + b*Sinh[x])^7)
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(827\) vs.
\(2(176)=352\).
time = 2.66, size = 828, normalized size = 4.18
| | |
| method |
result |
size |
| | |
| risch |
\(-\frac {4 \left (35 \,{\mathrm e}^{3 x} b^{3}+105 \,{\mathrm e}^{3 x} b^{2} c +105 \,{\mathrm e}^{3 x} b \,c^{2}+35 \,{\mathrm e}^{3 x} c^{3}+21 \,{\mathrm e}^{2 x} \sqrt {b^{2}-c^{2}}\, b^{2}+42 \sqrt {b^{2}-c^{2}}\, b c \,{\mathrm e}^{2 x}+21 \,{\mathrm e}^{2 x} \sqrt {b^{2}-c^{2}}\, c^{2}+7 b^{3} {\mathrm e}^{x}+7 \,{\mathrm e}^{x} b^{2} c -7 \,{\mathrm e}^{x} b \,c^{2}-7 \,{\mathrm e}^{x} c^{3}+\sqrt {b^{2}-c^{2}}\, b^{2}-\sqrt {b^{2}-c^{2}}\, c^{2}\right )}{35 \left ({\mathrm e}^{x} b +c \,{\mathrm e}^{x}+\sqrt {b^{2}-c^{2}}\right )^{7}}\) |
\(184\) |
| default |
\(\frac {\frac {2 \left (8 b^{4}-8 b^{2} c^{2}+c^{4}+8 \sqrt {b^{2}-c^{2}}\, b^{3}-4 \sqrt {b^{2}-c^{2}}\, c^{2} b \right ) \left (\tanh ^{6}\left (\frac {x}{2}\right )\right )}{c^{2}}+\frac {6 \left (16 \sqrt {b^{2}-c^{2}}\, b^{4}-12 \sqrt {b^{2}-c^{2}}\, b^{2} c^{2}+\sqrt {b^{2}-c^{2}}\, c^{4}+16 b^{5}-20 b^{3} c^{2}+5 b \,c^{4}\right ) \left (\tanh ^{5}\left (\frac {x}{2}\right )\right )}{c^{3}}+\frac {4 \left (80 \sqrt {b^{2}-c^{2}}\, b^{5}-84 \sqrt {b^{2}-c^{2}}\, b^{3} c^{2}+17 \sqrt {b^{2}-c^{2}}\, b \,c^{4}+80 b^{6}-124 b^{4} c^{2}+49 b^{2} c^{4}-3 c^{6}\right ) \left (\tanh ^{4}\left (\frac {x}{2}\right )\right )}{c^{4}}+\frac {4 \left (160 b^{7}-288 b^{5} c^{2}+150 b^{3} c^{4}-20 c^{6} b +160 \sqrt {b^{2}-c^{2}}\, b^{6}-208 \sqrt {b^{2}-c^{2}}\, b^{4} c^{2}+66 \sqrt {b^{2}-c^{2}}\, b^{2} c^{4}-3 \sqrt {b^{2}-c^{2}}\, c^{6}\right ) \left (\tanh ^{3}\left (\frac {x}{2}\right )\right )}{c^{5}}+\frac {6 \left (640 b^{7} \sqrt {b^{2}-c^{2}}-992 \sqrt {b^{2}-c^{2}}\, b^{5} c^{2}+440 \sqrt {b^{2}-c^{2}}\, b^{3} c^{4}-50 \sqrt {b^{2}-c^{2}}\, b \,c^{6}+640 b^{8}-1312 b^{6} c^{2}+856 b^{4} c^{4}-186 c^{6} b^{2}+7 c^{8}\right ) \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )}{5 c^{6}}+\frac {2 \left (1280 b^{9}-2944 b^{7} c^{2}+2288 b^{5} c^{4}-676 b^{3} c^{6}+57 b \,c^{8}+1280 \sqrt {b^{2}-c^{2}}\, b^{8}-2304 \sqrt {b^{2}-c^{2}}\, b^{6} c^{2}+1296 \sqrt {b^{2}-c^{2}}\, b^{4} c^{4}-236 \sqrt {b^{2}-c^{2}}\, b^{2} c^{6}+7 \sqrt {b^{2}-c^{2}}\, c^{8}\right ) \tanh \left (\frac {x}{2}\right )}{5 c^{7}}+\frac {2 \left (\frac {512 \sqrt {b^{2}-c^{2}}\, b^{9}}{7}-\frac {5248 \sqrt {b^{2}-c^{2}}\, b^{7} c^{2}}{35}+\frac {512 \sqrt {b^{2}-c^{2}}\, b^{5} c^{4}}{5}-\frac {136 \sqrt {b^{2}-c^{2}}\, b^{3} c^{6}}{5}+\frac {12 \sqrt {b^{2}-c^{2}}\, b \,c^{8}}{5}+\frac {512 b^{10}}{7}-\frac {6528 b^{8} c^{2}}{35}+\frac {5888 b^{6} c^{4}}{35}-\frac {2248 b^{4} c^{6}}{35}+\frac {68 b^{2} c^{8}}{7}-\frac {12 c^{10}}{35}\right )}{c^{8}}}{c^{6} \left (\tanh ^{2}\left (\frac {x}{2}\right )+\frac {2 \sqrt {b^{2}-c^{2}}\, \tanh \left (\frac {x}{2}\right )}{c}+\frac {2 b \tanh \left (\frac {x}{2}\right )}{c}+\frac {2 \sqrt {b^{2}-c^{2}}\, b}{c^{2}}+\frac {2 b^{2}}{c^{2}}-1\right )^{3} \left (\tanh \left (\frac {x}{2}\right )+\frac {\sqrt {b^{2}-c^{2}}}{c}+\frac {b}{c}\right )}\) |
\(828\) |
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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))^4,x,method=_RETURNVERBOSE)
[Out]
2/c^6*((8*b^4-8*b^2*c^2+c^4+8*(b^2-c^2)^(1/2)*b^3-4*(b^2-c^2)^(1/2)*c^2*b)/c^2*tanh(1/2*x)^6+3*(16*(b^2-c^2)^(
1/2)*b^4-12*(b^2-c^2)^(1/2)*b^2*c^2+(b^2-c^2)^(1/2)*c^4+16*b^5-20*b^3*c^2+5*b*c^4)/c^3*tanh(1/2*x)^5+2*(80*(b^
2-c^2)^(1/2)*b^5-84*(b^2-c^2)^(1/2)*b^3*c^2+17*(b^2-c^2)^(1/2)*b*c^4+80*b^6-124*b^4*c^2+49*b^2*c^4-3*c^6)/c^4*
tanh(1/2*x)^4+2*(160*b^7-288*b^5*c^2+150*b^3*c^4-20*c^6*b+160*(b^2-c^2)^(1/2)*b^6-208*(b^2-c^2)^(1/2)*b^4*c^2+
66*(b^2-c^2)^(1/2)*b^2*c^4-3*(b^2-c^2)^(1/2)*c^6)/c^5*tanh(1/2*x)^3+3/5*(640*b^7*(b^2-c^2)^(1/2)-992*(b^2-c^2)
^(1/2)*b^5*c^2+440*(b^2-c^2)^(1/2)*b^3*c^4-50*(b^2-c^2)^(1/2)*b*c^6+640*b^8-1312*b^6*c^2+856*b^4*c^4-186*c^6*b
^2+7*c^8)/c^6*tanh(1/2*x)^2+1/5*(1280*b^9-2944*b^7*c^2+2288*b^5*c^4-676*b^3*c^6+57*b*c^8+1280*(b^2-c^2)^(1/2)*
b^8-2304*(b^2-c^2)^(1/2)*b^6*c^2+1296*(b^2-c^2)^(1/2)*b^4*c^4-236*(b^2-c^2)^(1/2)*b^2*c^6+7*(b^2-c^2)^(1/2)*c^
8)/c^7*tanh(1/2*x)+4/35*(640*(b^2-c^2)^(1/2)*b^9-1312*(b^2-c^2)^(1/2)*b^7*c^2+896*(b^2-c^2)^(1/2)*b^5*c^4-238*
(b^2-c^2)^(1/2)*b^3*c^6+21*(b^2-c^2)^(1/2)*b*c^8+640*b^10-1632*b^8*c^2+1472*b^6*c^4-562*b^4*c^6+85*b^2*c^8-3*c
^10)/c^8)/(tanh(1/2*x)^2+2/c*(b^2-c^2)^(1/2)*tanh(1/2*x)+2/c*b*tanh(1/2*x)+2/c^2*(b^2-c^2)^(1/2)*b+2*b^2/c^2-1
)^3/(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 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
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))^4,x, algorithm="maxima")
[Out]
Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is undefined.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 6590 vs.
\(2 (176) = 352\).
time = 0.45, size = 6590, normalized size = 33.28 \begin {gather*} \text {Too large to display} \end {gather*}
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))^4,x, algorithm="fricas")
[Out]
-4/35*(35*(b^5 + 5*b^4*c + 10*b^3*c^2 + 10*b^2*c^3 + 5*b*c^4 + c^5)*cosh(x)^10 + 350*(b^5 + 5*b^4*c + 10*b^3*c
^2 + 10*b^2*c^3 + 5*b*c^4 + c^5)*cosh(x)*sinh(x)^9 + 35*(b^5 + 5*b^4*c + 10*b^3*c^2 + 10*b^2*c^3 + 5*b*c^4 + c
^5)*sinh(x)^10 + 595*(b^5 + 3*b^4*c + 2*b^3*c^2 - 2*b^2*c^3 - 3*b*c^4 - c^5)*cosh(x)^8 + 35*(17*b^5 + 51*b^4*c
+ 34*b^3*c^2 - 34*b^2*c^3 - 51*b*c^4 - 17*c^5 + 45*(b^5 + 5*b^4*c + 10*b^3*c^2 + 10*b^2*c^3 + 5*b*c^4 + c^5)*
cosh(x)^2)*sinh(x)^8 + 280*(15*(b^5 + 5*b^4*c + 10*b^3*c^2 + 10*b^2*c^3 + 5*b*c^4 + c^5)*cosh(x)^3 + 17*(b^5 +
3*b^4*c + 2*b^3*c^2 - 2*b^2*c^3 - 3*b*c^4 - c^5)*cosh(x))*sinh(x)^7 + 630*(b^5 + b^4*c - 2*b^3*c^2 - 2*b^2*c^
3 + b*c^4 + c^5)*cosh(x)^6 + 70*(9*b^5 + 9*b^4*c - 18*b^3*c^2 - 18*b^2*c^3 + 9*b*c^4 + 9*c^5 + 105*(b^5 + 5*b^
4*c + 10*b^3*c^2 + 10*b^2*c^3 + 5*b*c^4 + c^5)*cosh(x)^4 + 238*(b^5 + 3*b^4*c + 2*b^3*c^2 - 2*b^2*c^3 - 3*b*c^
4 - c^5)*cosh(x)^2)*sinh(x)^6 + 140*(63*(b^5 + 5*b^4*c + 10*b^3*c^2 + 10*b^2*c^3 + 5*b*c^4 + c^5)*cosh(x)^5 +
238*(b^5 + 3*b^4*c + 2*b^3*c^2 - 2*b^2*c^3 - 3*b*c^4 - c^5)*cosh(x)^3 + 27*(b^5 + b^4*c - 2*b^3*c^2 - 2*b^2*c^
3 + b*c^4 + c^5)*cosh(x))*sinh(x)^5 - b^5 + 5*b^4*c - 10*b^3*c^2 + 10*b^2*c^3 - 5*b*c^4 + c^5 + 14*(b^5 - b^4*
c - 2*b^3*c^2 + 2*b^2*c^3 + b*c^4 - c^5)*cosh(x)^4 + 14*(525*(b^5 + 5*b^4*c + 10*b^3*c^2 + 10*b^2*c^3 + 5*b*c^
4 + c^5)*cosh(x)^6 + b^5 - b^4*c - 2*b^3*c^2 + 2*b^2*c^3 + b*c^4 - c^5 + 2975*(b^5 + 3*b^4*c + 2*b^3*c^2 - 2*b
^2*c^3 - 3*b*c^4 - c^5)*cosh(x)^4 + 675*(b^5 + b^4*c - 2*b^3*c^2 - 2*b^2*c^3 + b*c^4 + c^5)*cosh(x)^2)*sinh(x)
^4 + 56*(75*(b^5 + 5*b^4*c + 10*b^3*c^2 + 10*b^2*c^3 + 5*b*c^4 + c^5)*cosh(x)^7 + 595*(b^5 + 3*b^4*c + 2*b^3*c
^2 - 2*b^2*c^3 - 3*b*c^4 - c^5)*cosh(x)^5 + 225*(b^5 + b^4*c - 2*b^3*c^2 - 2*b^2*c^3 + b*c^4 + c^5)*cosh(x)^3
+ (b^5 - b^4*c - 2*b^3*c^2 + 2*b^2*c^3 + b*c^4 - c^5)*cosh(x))*sinh(x)^3 + 7*(b^5 - 3*b^4*c + 2*b^3*c^2 + 2*b^
2*c^3 - 3*b*c^4 + c^5)*cosh(x)^2 + 7*(225*(b^5 + 5*b^4*c + 10*b^3*c^2 + 10*b^2*c^3 + 5*b*c^4 + c^5)*cosh(x)^8
+ 2380*(b^5 + 3*b^4*c + 2*b^3*c^2 - 2*b^2*c^3 - 3*b*c^4 - c^5)*cosh(x)^6 + b^5 - 3*b^4*c + 2*b^3*c^2 + 2*b^2*c
^3 - 3*b*c^4 + c^5 + 1350*(b^5 + b^4*c - 2*b^3*c^2 - 2*b^2*c^3 + b*c^4 + c^5)*cosh(x)^4 + 12*(b^5 - b^4*c - 2*
b^3*c^2 + 2*b^2*c^3 + b*c^4 - c^5)*cosh(x)^2)*sinh(x)^2 + 14*(25*(b^5 + 5*b^4*c + 10*b^3*c^2 + 10*b^2*c^3 + 5*
b*c^4 + c^5)*cosh(x)^9 + 340*(b^5 + 3*b^4*c + 2*b^3*c^2 - 2*b^2*c^3 - 3*b*c^4 - c^5)*cosh(x)^7 + 270*(b^5 + b^
4*c - 2*b^3*c^2 - 2*b^2*c^3 + b*c^4 + c^5)*cosh(x)^5 + 4*(b^5 - b^4*c - 2*b^3*c^2 + 2*b^2*c^3 + b*c^4 - c^5)*c
osh(x)^3 + (b^5 - 3*b^4*c + 2*b^3*c^2 + 2*b^2*c^3 - 3*b*c^4 + c^5)*cosh(x))*sinh(x) - 32*(7*(b^4 + 4*b^3*c + 6
*b^2*c^2 + 4*b*c^3 + c^4)*cosh(x)^9 + 63*(b^4 + 4*b^3*c + 6*b^2*c^2 + 4*b*c^3 + c^4)*cosh(x)*sinh(x)^8 + 7*(b^
4 + 4*b^3*c + 6*b^2*c^2 + 4*b*c^3 + c^4)*sinh(x)^9 + 26*(b^4 + 2*b^3*c - 2*b*c^3 - c^4)*cosh(x)^7 + 2*(13*b^4
+ 26*b^3*c - 26*b*c^3 - 13*c^4 + 126*(b^4 + 4*b^3*c + 6*b^2*c^2 + 4*b*c^3 + c^4)*cosh(x)^2)*sinh(x)^7 + 14*(42
*(b^4 + 4*b^3*c + 6*b^2*c^2 + 4*b*c^3 + c^4)*cosh(x)^3 + 13*(b^4 + 2*b^3*c - 2*b*c^3 - c^4)*cosh(x))*sinh(x)^6
+ 7*(b^4 - 2*b^2*c^2 + c^4)*cosh(x)^5 + 7*(126*(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 + 78*(b^4 + 2*b^3*c - 2*b*c^3 - c^4)*cosh(x)^2)*sinh(x)^5 + 7*(126*(b^4 + 4*b^3*c + 6*b^2*c^2
+ 4*b*c^3 + c^4)*cosh(x)^5 + 130*(b^4 + 2*b^3*c - 2*b*c^3 - c^4)*cosh(x)^3 + 5*(b^4 - 2*b^2*c^2 + c^4)*cosh(x)
)*sinh(x)^4 + 14*(42*(b^4 + 4*b^3*c + 6*b^2*c^2 + 4*b*c^3 + c^4)*cosh(x)^6 + 65*(b^4 + 2*b^3*c - 2*b*c^3 - c^4
)*cosh(x)^4 + 5*(b^4 - 2*b^2*c^2 + c^4)*cosh(x)^2)*sinh(x)^3 + 14*(18*(b^4 + 4*b^3*c + 6*b^2*c^2 + 4*b*c^3 + c
^4)*cosh(x)^7 + 39*(b^4 + 2*b^3*c - 2*b*c^3 - c^4)*cosh(x)^5 + 5*(b^4 - 2*b^2*c^2 + c^4)*cosh(x)^3)*sinh(x)^2
+ 7*(9*(b^4 + 4*b^3*c + 6*b^2*c^2 + 4*b*c^3 + c^4)*cosh(x)^8 + 26*(b^4 + 2*b^3*c - 2*b*c^3 - c^4)*cosh(x)^6 +
5*(b^4 - 2*b^2*c^2 + c^4)*cosh(x)^4)*sinh(x))*sqrt(b^2 - c^2))/((b^9 + 9*b^8*c + 36*b^7*c^2 + 84*b^6*c^3 + 126
*b^5*c^4 + 126*b^4*c^5 + 84*b^3*c^6 + 36*b^2*c^7 + 9*b*c^8 + c^9)*cosh(x)^14 + 14*(b^9 + 9*b^8*c + 36*b^7*c^2
+ 84*b^6*c^3 + 126*b^5*c^4 + 126*b^4*c^5 + 84*b^3*c^6 + 36*b^2*c^7 + 9*b*c^8 + c^9)*cosh(x)*sinh(x)^13 + (b^9
+ 9*b^8*c + 36*b^7*c^2 + 84*b^6*c^3 + 126*b^5*c^4 + 126*b^4*c^5 + 84*b^3*c^6 + 36*b^2*c^7 + 9*b*c^8 + c^9)*sin
h(x)^14 - 7*(b^9 + 7*b^8*c + 20*b^7*c^2 + 28*b^6*c^3 + 14*b^5*c^4 - 14*b^4*c^5 - 28*b^3*c^6 - 20*b^2*c^7 - 7*b
*c^8 - c^9)*cosh(x)^12 - 7*(b^9 + 7*b^8*c + 20*b^7*c^2 + 28*b^6*c^3 + 14*b^5*c^4 - 14*b^4*c^5 - 28*b^3*c^6 - 2
0*b^2*c^7 - 7*b*c^8 - c^9 - 13*(b^9 + 9*b^8*c + 36*b^7*c^2 + 84*b^6*c^3 + 126*b^5*c^4 + 126*b^4*c^5 + 84*b^3*c
^6 + 36*b^2*c^7 + 9*b*c^8 + c^9)*cosh(x)^2)*sinh(x)^12 + 28*(13*(b^9 + 9*b^8*c + 36*b^7*c^2 + 84*b^6*c^3 + 126
*b^5*c^4 + 126*b^4*c^5 + 84*b^3*c^6 + 36*b^2*c^7 + 9*b*c^8 + c^9)*cosh(x)^3 - 3*(b^9 + 7*b^8*c + 20*b^7*c^2 +
28*b^6*c^3 + 14*b^5*c^4 - 14*b^4*c^5 - 28*b^3*c^6 - 20*b^2*c^7 - 7*b*c^8 - c^9)*cosh(x))*sinh(x)^11 + 21*(b^9
+ 5*b^8*c + 8*b^7*c^2 - 14*b^5*c^4 - 14*b^4*c^5...
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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))**4,x)
[Out]
Timed out
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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))^4,x, algorithm="giac")
[Out]
Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Unable to divide, perhaps due to rounding error%%%{%%%{1,[4,0]%%%}+%%%{4,[3,1]%%%}+%%%{6,[2,2]%%%}+%%%{4,[1
,3]%%%}+%%%
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{{\left (b\,\mathrm {cosh}\left (x\right )+\sqrt {b^2-c^2}+c\,\mathrm {sinh}\left (x\right )\right )}^4} \,d x \end {gather*}
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))^4,x)
[Out]
int(1/(b*cosh(x) + (b^2 - c^2)^(1/2) + c*sinh(x))^4, x)
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