3.760 \(\int \frac{1}{(\sqrt{b^2-c^2}+b \cosh (x)+c \sinh (x))^4} \, dx\)

Optimal. Leaf size=198 \[ \frac{2 (b \sinh (x)+c \cosh (x))}{35 \left (b^2-c^2\right )^{3/2} \left (\sqrt{b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^2}+\frac{3 (b \sinh (x)+c \cosh (x))}{35 \left (b^2-c^2\right ) \left (\sqrt{b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^3}+\frac{b \sinh (x)+c \cosh (x)}{7 \sqrt{b^2-c^2} \left (\sqrt{b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^4}-\frac{2 \left (\sqrt{b^2-c^2} \sinh (x)+c\right )}{35 c \left (b^2-c^2\right )^{3/2} (b \sinh (x)+c \cosh (x))} \]

[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]))

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Rubi [A]  time = 0.177855, antiderivative size = 198, normalized size of antiderivative = 1., 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 = {3116, 3114} \[ \frac{2 (b \sinh (x)+c \cosh (x))}{35 \left (b^2-c^2\right )^{3/2} \left (\sqrt{b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^2}+\frac{3 (b \sinh (x)+c \cosh (x))}{35 \left (b^2-c^2\right ) \left (\sqrt{b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^3}+\frac{b \sinh (x)+c \cosh (x)}{7 \sqrt{b^2-c^2} \left (\sqrt{b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^4}-\frac{2 \left (\sqrt{b^2-c^2} \sinh (x)+c\right )}{35 c \left (b^2-c^2\right )^{3/2} (b \sinh (x)+c \cosh (x))} \]

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 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]

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]

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*}

Mathematica [B]  time = 0.79379, size = 425, normalized size = 2.15 \[ -\frac{1295 b^4 c^2 \sinh (x)+189 b^4 c^2 \sinh (3 x)-35 b^4 c^2 \sinh (5 x)+15 b^4 c^2 \sinh (7 x)-896 b^3 c^2 \sqrt{b^2-c^2} \sinh (2 x)-2485 b^2 c^4 \sinh (x)-161 b^2 c^4 \sinh (3 x)+35 b^2 c^4 \sinh (5 x)+15 b^2 c^4 \sinh (7 x)+896 b c^4 \sqrt{b^2-c^2} \sinh (2 x)+56 b^3 c^3 \cosh (3 x)+20 b^3 c^3 \cosh (7 x)+1190 b c \left (b^2-c^2\right )^2 \cosh (x)+448 c \sqrt{b^2-c^2} \left (c^4-b^4\right ) \cosh (2 x)-832 b^4 c \sqrt{b^2-c^2}+1664 b^2 c^3 \sqrt{b^2-c^2}-832 c^5 \sqrt{b^2-c^2}+112 b^5 c \cosh (3 x)-28 b^5 c \cosh (5 x)+6 b^5 c \cosh (7 x)-35 b^6 \sinh (x)+21 b^6 \sinh (3 x)-7 b^6 \sinh (5 x)+b^6 \sinh (7 x)-168 b c^5 \cosh (3 x)+28 b c^5 \cosh (5 x)+6 b c^5 \cosh (7 x)+1225 c^6 \sinh (x)-49 c^6 \sinh (3 x)+7 c^6 \sinh (5 x)+c^6 \sinh (7 x)}{1120 c (b-c) (b+c) (b \sinh (x)+c \cosh (x))^7} \]

Antiderivative was successfully verified.

[In]

Integrate[(Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x])^(-4),x]

[Out]

-(-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*Co
sh[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)

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Maple [B]  time = 0.229, size = 828, normalized size = 4.2 \begin{align*} \text{result too large to display} \end{align*}

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)

[Out]

2/c^6*((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^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*c^4*b)/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*b*c^6+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*c^4*b^4-186*b^2*c
^6+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*b/c*tanh(1/2*x)+2*b/c^2*(b^2-c^2)^(1/2)+2/c^2*b^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., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

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

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Fricas [B]  time = 4.05519, size = 15194, normalized size = 76.74 \begin{align*} \text{result too large to display} \end{align*}

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 + 8*b^2*c^7 + 5*b*c^8 + c^9)*cosh(x)^10 + 7*(3*b^9 + 15*b^8*c
+ 24*b^7*c^2 - 42*b^5*c^4 - 42*b^4*c^5 + 24*b^2*c^7 + 15*b*c^8 + 3*c^9 + 143*(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)^4 - 66*(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)^2)*sinh(
x)^10 + 14*(143*(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)^5 - 110*(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)^3 + 15*(b^9 + 5*b^8*c + 8*b^7*c^2 - 14*b^5*c^4 - 14*b^4*c^5 + 8*b^2*
c^7 + 5*b*c^8 + c^9)*cosh(x))*sinh(x)^9 - b^9 + 5*b^8*c - 8*b^7*c^2 + 14*b^5*c^4 - 14*b^4*c^5 + 8*b^2*c^7 - 5*
b*c^8 + c^9 - 35*(b^9 + 3*b^8*c - 8*b^6*c^3 - 6*b^5*c^4 + 6*b^4*c^5 + 8*b^3*c^6 - 3*b*c^8 - c^9)*cosh(x)^8 - 7
*(5*b^9 + 15*b^8*c - 40*b^6*c^3 - 30*b^5*c^4 + 30*b^4*c^5 + 40*b^3*c^6 - 15*b*c^8 - 5*c^9 - 429*(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)^6 +
495*(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)^4 - 135*(b^9 + 5*b^8*c + 8*b^7*c^2 - 14*b^5*c^4 - 14*b^4*c^5 + 8*b^2*c^7 + 5*b*c^8 + c^9)*cosh(x)^
2)*sinh(x)^8 + 8*(429*(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)^7 - 693*(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 - 2
8*b^3*c^6 - 20*b^2*c^7 - 7*b*c^8 - c^9)*cosh(x)^5 + 315*(b^9 + 5*b^8*c + 8*b^7*c^2 - 14*b^5*c^4 - 14*b^4*c^5 +
8*b^2*c^7 + 5*b*c^8 + c^9)*cosh(x)^3 - 35*(b^9 + 3*b^8*c - 8*b^6*c^3 - 6*b^5*c^4 + 6*b^4*c^5 + 8*b^3*c^6 - 3*
b*c^8 - c^9)*cosh(x))*sinh(x)^7 + 35*(b^9 + b^8*c - 4*b^7*c^2 - 4*b^6*c^3 + 6*b^5*c^4 + 6*b^4*c^5 - 4*b^3*c^6
- 4*b^2*c^7 + b*c^8 + c^9)*cosh(x)^6 + 7*(5*b^9 + 5*b^8*c - 20*b^7*c^2 - 20*b^6*c^3 + 30*b^5*c^4 + 30*b^4*c^5
- 20*b^3*c^6 - 20*b^2*c^7 + 5*b*c^8 + 5*c^9 + 429*(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)^8 - 924*(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)^6 + 630*(b^9 + 5*b^8*c + 8*b^7*c^2
- 14*b^5*c^4 - 14*b^4*c^5 + 8*b^2*c^7 + 5*b*c^8 + c^9)*cosh(x)^4 - 140*(b^9 + 3*b^8*c - 8*b^6*c^3 - 6*b^5*c^4
+ 6*b^4*c^5 + 8*b^3*c^6 - 3*b*c^8 - c^9)*cosh(x)^2)*sinh(x)^6 + 14*(143*(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)^9 - 396*(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)^7 + 378*(b^
9 + 5*b^8*c + 8*b^7*c^2 - 14*b^5*c^4 - 14*b^4*c^5 + 8*b^2*c^7 + 5*b*c^8 + c^9)*cosh(x)^5 - 140*(b^9 + 3*b^8*c
- 8*b^6*c^3 - 6*b^5*c^4 + 6*b^4*c^5 + 8*b^3*c^6 - 3*b*c^8 - c^9)*cosh(x)^3 + 15*(b^9 + b^8*c - 4*b^7*c^2 - 4*b
^6*c^3 + 6*b^5*c^4 + 6*b^4*c^5 - 4*b^3*c^6 - 4*b^2*c^7 + b*c^8 + c^9)*cosh(x))*sinh(x)^5 - 21*(b^9 - b^8*c - 4
*b^7*c^2 + 4*b^6*c^3 + 6*b^5*c^4 - 6*b^4*c^5 - 4*b^3*c^6 + 4*b^2*c^7 + b*c^8 - c^9)*cosh(x)^4 + 7*(143*(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)^10 - 3*b^9 + 3*b^8*c + 12*b^7*c^2 - 12*b^6*c^3 - 18*b^5*c^4 + 18*b^4*c^5 + 12*b^3*c^6 - 12*b^2*c^7 - 3*b*c^
8 + 3*c^9 - 495*(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)^8 + 630*(b^9 + 5*b^8*c + 8*b^7*c^2 - 14*b^5*c^4 - 14*b^4*c^5 + 8*b^2*c^7 + 5*b*c^8 + c
^9)*cosh(x)^6 - 350*(b^9 + 3*b^8*c - 8*b^6*c^3 - 6*b^5*c^4 + 6*b^4*c^5 + 8*b^3*c^6 - 3*b*c^8 - c^9)*cosh(x)^4
+ 75*(b^9 + b^8*c - 4*b^7*c^2 - 4*b^6*c^3 + 6*b^5*c^4 + 6*b^4*c^5 - 4*b^3*c^6 - 4*b^2*c^7 + b*c^8 + c^9)*cosh(
x)^2)*sinh(x)^4 + 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 + 3
6*b^2*c^7 + 9*b*c^8 + c^9)*cosh(x)^11 - 55*(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)^9 + 90*(b^9 + 5*b^8*c + 8*b^7*c^2 - 14*b^5*c^4 - 14*b^4*c^5
+ 8*b^2*c^7 + 5*b*c^8 + c^9)*cosh(x)^7 - 70*(b^9 + 3*b^8*c - 8*b^6*c^3 - 6*b^5*c^4 + 6*b^4*c^5 + 8*b^3*c^6 -
3*b*c^8 - c^9)*cosh(x)^5 + 25*(b^9 + b^8*c - 4*b^7*c^2 - 4*b^6*c^3 + 6*b^5*c^4 + 6*b^4*c^5 - 4*b^3*c^6 - 4*b^2
*c^7 + b*c^8 + c^9)*cosh(x)^3 - 3*(b^9 - b^8*c - 4*b^7*c^2 + 4*b^6*c^3 + 6*b^5*c^4 - 6*b^4*c^5 - 4*b^3*c^6 + 4
*b^2*c^7 + b*c^8 - c^9)*cosh(x))*sinh(x)^3 + 7*(b^9 - 3*b^8*c + 8*b^6*c^3 - 6*b^5*c^4 - 6*b^4*c^5 + 8*b^3*c^6
- 3*b*c^8 + c^9)*cosh(x)^2 + 7*(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)^12 - 66*(b^9 + 7*b^8*c + 20*b^7*c^2 + 28*b^6*c^3 + 14*b^5*c^4 - 1
4*b^4*c^5 - 28*b^3*c^6 - 20*b^2*c^7 - 7*b*c^8 - c^9)*cosh(x)^10 + b^9 - 3*b^8*c + 8*b^6*c^3 - 6*b^5*c^4 - 6*b^
4*c^5 + 8*b^3*c^6 - 3*b*c^8 + c^9 + 135*(b^9 + 5*b^8*c + 8*b^7*c^2 - 14*b^5*c^4 - 14*b^4*c^5 + 8*b^2*c^7 + 5*b
*c^8 + c^9)*cosh(x)^8 - 140*(b^9 + 3*b^8*c - 8*b^6*c^3 - 6*b^5*c^4 + 6*b^4*c^5 + 8*b^3*c^6 - 3*b*c^8 - c^9)*co
sh(x)^6 + 75*(b^9 + b^8*c - 4*b^7*c^2 - 4*b^6*c^3 + 6*b^5*c^4 + 6*b^4*c^5 - 4*b^3*c^6 - 4*b^2*c^7 + b*c^8 + c^
9)*cosh(x)^4 - 18*(b^9 - b^8*c - 4*b^7*c^2 + 4*b^6*c^3 + 6*b^5*c^4 - 6*b^4*c^5 - 4*b^3*c^6 + 4*b^2*c^7 + b*c^8
- c^9)*cosh(x)^2)*sinh(x)^2 + 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)^13 - 6*(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)^11 + 15*(b^9 + 5*b^8*c + 8*b^7*c^2 - 14*b^5*c^4 -
14*b^4*c^5 + 8*b^2*c^7 + 5*b*c^8 + c^9)*cosh(x)^9 - 20*(b^9 + 3*b^8*c - 8*b^6*c^3 - 6*b^5*c^4 + 6*b^4*c^5 + 8*
b^3*c^6 - 3*b*c^8 - c^9)*cosh(x)^7 + 15*(b^9 + b^8*c - 4*b^7*c^2 - 4*b^6*c^3 + 6*b^5*c^4 + 6*b^4*c^5 - 4*b^3*c
^6 - 4*b^2*c^7 + b*c^8 + c^9)*cosh(x)^5 - 6*(b^9 - b^8*c - 4*b^7*c^2 + 4*b^6*c^3 + 6*b^5*c^4 - 6*b^4*c^5 - 4*b
^3*c^6 + 4*b^2*c^7 + b*c^8 - c^9)*cosh(x)^3 + (b^9 - 3*b^8*c + 8*b^6*c^3 - 6*b^5*c^4 - 6*b^4*c^5 + 8*b^3*c^6 -
3*b*c^8 + c^9)*cosh(x))*sinh(x))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

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