### 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]

(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]))

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Rubi [A]  time = 0.119504, antiderivative size = 146, normalized size of antiderivative = 1., 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 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 )^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*}

Mathematica [A]  time = 0.37921, 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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Maple [B]  time = 0.132, size = 488, normalized size = 3.3 \begin{align*} 2\,{\frac{1}{{c}^{4}} \left ( -{\frac{ \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 \left ( x/2 \right ) \right ) ^{4}}{{c}^{2}}}-2\,{\frac{ \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 \left ( x/2 \right ) \right ) ^{3}}{{c}^{3}}}-4/3\,{\frac{ \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\,{c}^{4}b \right ) \left ( \tanh \left ( x/2 \right ) \right ) ^{2}}{{c}^{4}}}-2/3\,{\frac{ \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 ( x/2 \right ) }{{c}^{5}}}-1/15\,{\frac{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\,b{c}^{6}}{{c}^{6}}} \right ) \left ( \left ( \tanh \left ( x/2 \right ) \right ) ^{2}+2\,{\frac{\sqrt{{b}^{2}-{c}^{2}}\tanh \left ( x/2 \right ) }{c}}+2\,{\frac{\tanh \left ( x/2 \right ) b}{c}}+2\,{\frac{\sqrt{{b}^{2}-{c}^{2}}b}{{c}^{2}}}+2\,{\frac{{b}^{2}}{{c}^{2}}}-1 \right ) ^{-2} \left ( \tanh \left ( x/2 \right ) +{\frac{\sqrt{{b}^{2}-{c}^{2}}}{c}}+{\frac{b}{c}} \right ) ^{-1}} \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))^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*c^4*b)/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*b*c^6))/(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)^2/(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))^3,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [B]  time = 3.12206, size = 6947, normalized size = 47.58 \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))^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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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))**3,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))^3,x, algorithm="giac")

[Out]

Exception raised: TypeError