∫ d+e cosh(x)____ a+b cosh(x)+c cosh2(x) dx [837]

3.9.37 \(\int \frac {d+e \cosh (x)}{a+b \cosh (x)+c \cosh ^2(x)} \, dx\) [837]

Optimal. Leaf size=246 \[ \frac {2 \left (e+\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) \tanh ^{-1}\left (\frac {\sqrt {b-2 c-\sqrt {b^2-4 a c}} \tanh \left (\frac {x}{2}\right )}{\sqrt {b+2 c-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b-2 c-\sqrt {b^2-4 a c}} \sqrt {b+2 c-\sqrt {b^2-4 a c}}}+\frac {2 \left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) \tanh ^{-1}\left (\frac {\sqrt {b-2 c+\sqrt {b^2-4 a c}} \tanh \left (\frac {x}{2}\right )}{\sqrt {b+2 c+\sqrt {b^2-4 a c}}}\right )}{\sqrt {b-2 c+\sqrt {b^2-4 a c}} \sqrt {b+2 c+\sqrt {b^2-4 a c}}} \]

[Out]

2*arctanh((b-2*c-(-4*a*c+b^2)^(1/2))^(1/2)*tanh(1/2*x)/(b+2*c-(-4*a*c+b^2)^(1/2))^(1/2))*(e+(-b*e+2*c*d)/(-4*a

*c+b^2)^(1/2))/(b-2*c-(-4*a*c+b^2)^(1/2))^(1/2)/(b+2*c-(-4*a*c+b^2)^(1/2))^(1/2)+2*arctanh((b-2*c+(-4*a*c+b^2)

^(1/2))^(1/2)*tanh(1/2*x)/(b+2*c+(-4*a*c+b^2)^(1/2))^(1/2))*(e+(b*e-2*c*d)/(-4*a*c+b^2)^(1/2))/(b-2*c+(-4*a*c+

b^2)^(1/2))^(1/2)/(b+2*c+(-4*a*c+b^2)^(1/2))^(1/2)

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Rubi [A]
time = 0.47, antiderivative size = 246, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3374, 2738, 214} \begin {gather*} \frac {2 \left (\frac {2 c d-b e}{\sqrt {b^2-4 a c}}+e\right ) \tanh ^{-1}\left (\frac {\tanh \left (\frac {x}{2}\right ) \sqrt {-\sqrt {b^2-4 a c}+b-2 c}}{\sqrt {-\sqrt {b^2-4 a c}+b+2 c}}\right )}{\sqrt {-\sqrt {b^2-4 a c}+b-2 c} \sqrt {-\sqrt {b^2-4 a c}+b+2 c}}+\frac {2 \left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) \tanh ^{-1}\left (\frac {\tanh \left (\frac {x}{2}\right ) \sqrt {\sqrt {b^2-4 a c}+b-2 c}}{\sqrt {\sqrt {b^2-4 a c}+b+2 c}}\right )}{\sqrt {\sqrt {b^2-4 a c}+b-2 c} \sqrt {\sqrt {b^2-4 a c}+b+2 c}} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*Cosh[x])/(a + b*Cosh[x] + c*Cosh[x]^2),x]

[Out]

(2*(e + (2*c*d - b*e)/Sqrt[b^2 - 4*a*c])*ArcTanh[(Sqrt[b - 2*c - Sqrt[b^2 - 4*a*c]]*Tanh[x/2])/Sqrt[b + 2*c -

Sqrt[b^2 - 4*a*c]]])/(Sqrt[b - 2*c - Sqrt[b^2 - 4*a*c]]*Sqrt[b + 2*c - Sqrt[b^2 - 4*a*c]]) + (2*(e - (2*c*d -

b*e)/Sqrt[b^2 - 4*a*c])*ArcTanh[(Sqrt[b - 2*c + Sqrt[b^2 - 4*a*c]]*Tanh[x/2])/Sqrt[b + 2*c + Sqrt[b^2 - 4*a*c]

]])/(Sqrt[b - 2*c + Sqrt[b^2 - 4*a*c]]*Sqrt[b + 2*c + Sqrt[b^2 - 4*a*c]])

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},

x] && NegQ[a/b]

Rule 2738

Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x

]}, Dist[2*(e/d), Subst[Int[1/(a + b + (a - b)*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}

, x] && NeQ[a^2 - b^2, 0]

Rule 3374

Int[(cos[(d_.) + (e_.)*(x_)]*(B_.) + (A_))/((a_.) + cos[(d_.) + (e_.)*(x_)]*(b_.) + cos[(d_.) + (e_.)*(x_)]^2*

(c_.)), x_Symbol] :> Module[{q = Rt[b^2 - 4*a*c, 2]}, Dist[B + (b*B - 2*A*c)/q, Int[1/(b + q + 2*c*Cos[d + e*x

]), x], x] + Dist[B - (b*B - 2*A*c)/q, Int[1/(b - q + 2*c*Cos[d + e*x]), x], x]] /; FreeQ[{a, b, c, d, e, A, B

}, x] && NeQ[b^2 - 4*a*c, 0]

Rubi steps

\begin {align*} \int \frac {d+e \cosh (x)}{a+b \cosh (x)+c \cosh ^2(x)} \, dx &=\left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) \int \frac {1}{b+\sqrt {b^2-4 a c}+2 c \cosh (x)} \, dx+\left (e+\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) \int \frac {1}{b-\sqrt {b^2-4 a c}+2 c \cosh (x)} \, dx\\ &=\left (2 \left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right )\right ) \text {Subst}\left (\int \frac {1}{b+2 c+\sqrt {b^2-4 a c}-\left (b-2 c+\sqrt {b^2-4 a c}\right ) x^2} \, dx,x,\tanh \left (\frac {x}{2}\right )\right )+\left (2 \left (e+\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right )\right ) \text {Subst}\left (\int \frac {1}{b+2 c-\sqrt {b^2-4 a c}-\left (b-2 c-\sqrt {b^2-4 a c}\right ) x^2} \, dx,x,\tanh \left (\frac {x}{2}\right )\right )\\ &=\frac {2 \left (e+\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) \tanh ^{-1}\left (\frac {\sqrt {b-2 c-\sqrt {b^2-4 a c}} \tanh \left (\frac {x}{2}\right )}{\sqrt {b+2 c-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b-2 c-\sqrt {b^2-4 a c}} \sqrt {b+2 c-\sqrt {b^2-4 a c}}}+\frac {2 \left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) \tanh ^{-1}\left (\frac {\sqrt {b-2 c+\sqrt {b^2-4 a c}} \tanh \left (\frac {x}{2}\right )}{\sqrt {b+2 c+\sqrt {b^2-4 a c}}}\right )}{\sqrt {b-2 c+\sqrt {b^2-4 a c}} \sqrt {b+2 c+\sqrt {b^2-4 a c}}}\\ \end {align*}

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Mathematica [A]
time = 0.31, size = 241, normalized size = 0.98 \begin {gather*} \frac {\sqrt {2} \left (-\frac {\left (-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e\right ) \text {ArcTan}\left (\frac {\left (b-2 c+\sqrt {b^2-4 a c}\right ) \tanh \left (\frac {x}{2}\right )}{\sqrt {-2 b^2+4 c (a+c)-2 b \sqrt {b^2-4 a c}}}\right )}{\sqrt {-b^2+2 c (a+c)-b \sqrt {b^2-4 a c}}}+\frac {\left (2 c d+\left (-b+\sqrt {b^2-4 a c}\right ) e\right ) \text {ArcTan}\left (\frac {\left (-b+2 c+\sqrt {b^2-4 a c}\right ) \tanh \left (\frac {x}{2}\right )}{\sqrt {-2 b^2+4 c (a+c)+2 b \sqrt {b^2-4 a c}}}\right )}{\sqrt {-b^2+2 c (a+c)+b \sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c}} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Integrate[(d + e*Cosh[x])/(a + b*Cosh[x] + c*Cosh[x]^2),x]

[Out]

(Sqrt[2]*(-(((-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e)*ArcTan[((b - 2*c + Sqrt[b^2 - 4*a*c])*Tanh[x/2])/Sqrt[-2*b^2

+ 4*c*(a + c) - 2*b*Sqrt[b^2 - 4*a*c]]])/Sqrt[-b^2 + 2*c*(a + c) - b*Sqrt[b^2 - 4*a*c]]) + ((2*c*d + (-b + Sq

rt[b^2 - 4*a*c])*e)*ArcTan[((-b + 2*c + Sqrt[b^2 - 4*a*c])*Tanh[x/2])/Sqrt[-2*b^2 + 4*c*(a + c) + 2*b*Sqrt[b^2

- 4*a*c]]])/Sqrt[-b^2 + 2*c*(a + c) + b*Sqrt[b^2 - 4*a*c]]))/Sqrt[b^2 - 4*a*c]

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Maple [A]
time = 6.79, size = 254, normalized size = 1.03


method	result	size

default	\(2 \left (a -b +c \right ) \left (\frac {\left (-d \sqrt {-4 a c +b^{2}}+e \sqrt {-4 a c +b^{2}}-2 e a +b d +b e -2 c d \right ) \arctan \left (\frac {\left (a -b +c \right ) \tanh \left (\frac {x}{2}\right )}{\sqrt {\left (\sqrt {-4 a c +b^{2}}-a +c \right ) \left (a -b +c \right )}}\right )}{2 \sqrt {-4 a c +b^{2}}\, \left (a -b +c \right ) \sqrt {\left (\sqrt {-4 a c +b^{2}}-a +c \right ) \left (a -b +c \right )}}+\frac {\left (-d \sqrt {-4 a c +b^{2}}+e \sqrt {-4 a c +b^{2}}+2 e a -b d -b e +2 c d \right ) \arctanh \left (\frac {\left (-a +b -c \right ) \tanh \left (\frac {x}{2}\right )}{\sqrt {\left (\sqrt {-4 a c +b^{2}}+a -c \right ) \left (a -b +c \right )}}\right )}{2 \sqrt {-4 a c +b^{2}}\, \left (a -b +c \right ) \sqrt {\left (\sqrt {-4 a c +b^{2}}+a -c \right ) \left (a -b +c \right )}}\right )\)	\(254\)
risch	\(\text {Expression too large to display}\)	\(8285\)

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((d+e*cosh(x))/(a+b*cosh(x)+c*cosh(x)^2),x,method=_RETURNVERBOSE)

[Out]

2*(a-b+c)*(1/2*(-d*(-4*a*c+b^2)^(1/2)+e*(-4*a*c+b^2)^(1/2)-2*e*a+b*d+b*e-2*c*d)/(-4*a*c+b^2)^(1/2)/(a-b+c)/(((

-4*a*c+b^2)^(1/2)-a+c)*(a-b+c))^(1/2)*arctan((a-b+c)*tanh(1/2*x)/(((-4*a*c+b^2)^(1/2)-a+c)*(a-b+c))^(1/2))+1/2

*(-d*(-4*a*c+b^2)^(1/2)+e*(-4*a*c+b^2)^(1/2)+2*e*a-b*d-b*e+2*c*d)/(-4*a*c+b^2)^(1/2)/(a-b+c)/(((-4*a*c+b^2)^(1

/2)+a-c)*(a-b+c))^(1/2)*arctanh((-a+b-c)*tanh(1/2*x)/(((-4*a*c+b^2)^(1/2)+a-c)*(a-b+c))^(1/2)))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((d+e*cosh(x))/(a+b*cosh(x)+c*cosh(x)^2),x, algorithm="maxima")

[Out]

integrate((cosh(x)*e + d)/(c*cosh(x)^2 + b*cosh(x) + a), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 15981 vs. \(2 (211) = 422\).
time = 3.12, size = 15981, normalized size = 64.96 \begin {gather*} \text {Too large to display} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((d+e*cosh(x))/(a+b*cosh(x)+c*cosh(x)^2),x, algorithm="fricas")

[Out]

1/2*sqrt(2)*sqrt(((b^2 - 2*a*c - 2*c^2)*d^2 - 2*(a*b - b*c)*d*cosh(1) + (2*a^2 - b^2 + 2*a*c)*cosh(1)^2 + (2*a

^2 - b^2 + 2*a*c)*sinh(1)^2 - 2*((a*b - b*c)*d - (2*a^2 - b^2 + 2*a*c)*cosh(1))*sinh(1) + (a^2*b^2 - b^4 - 4*a

*c^3 - (8*a^2 - b^2)*c^2 - 2*(2*a^3 - 3*a*b^2)*c)*sqrt((2*(2*a^2 + b^2 + 4*a*c + 2*c^2)*d^2 + (b^2*d^4 + b^2)*

cosh(1)^2 + (b^2*d^4 + b^2)*sinh(1)^2 - 4*((a*b + b*c)*d^3 + (a*b + b*c)*d)*cosh(1) + 2*(2*(a*b + b*c)*d^3 - 2

*(a*b + b*c)*d - (b^2*d^4 - b^2)*cosh(1))*sinh(1))/((a^4*b^2 - 2*a^2*b^4 + b^6 - 4*a*c^5 - (16*a^2 - b^2)*c^4

- 12*(2*a^3 - a*b^2)*c^3 - 2*(8*a^4 - 11*a^2*b^2 + b^4)*c^2 - 4*(a^5 - 3*a^3*b^2 + 2*a*b^4)*c)*cosh(1)^2 - 2*(

a^4*b^2 - 2*a^2*b^4 + b^6 - 4*a*c^5 - (16*a^2 - b^2)*c^4 - 12*(2*a^3 - a*b^2)*c^3 - 2*(8*a^4 - 11*a^2*b^2 + b^

4)*c^2 - 4*(a^5 - 3*a^3*b^2 + 2*a*b^4)*c)*cosh(1)*sinh(1) + (a^4*b^2 - 2*a^2*b^4 + b^6 - 4*a*c^5 - (16*a^2 - b

^2)*c^4 - 12*(2*a^3 - a*b^2)*c^3 - 2*(8*a^4 - 11*a^2*b^2 + b^4)*c^2 - 4*(a^5 - 3*a^3*b^2 + 2*a*b^4)*c)*sinh(1)

^2)))/(a^2*b^2 - b^4 - 4*a*c^3 - (8*a^2 - b^2)*c^2 - 2*(2*a^3 - 3*a*b^2)*c))*log(2*b^2*c*d^4 + 2*a*b^2*cosh(1)

^4 + 2*a*b^2*sinh(1)^4 - 2*(b^3 + 2*a*b*c + 2*b*c^2)*d^3*cosh(1) + 6*(a*b^2 + b^2*c)*d^2*cosh(1)^2 - 2*(2*a^2*

b + b^3 + 2*a*b*c)*d*cosh(1)^3 + 2*(4*a*b^2*cosh(1) - (2*a^2*b + b^3 + 2*a*b*c)*d)*sinh(1)^3 + 6*(2*a*b^2*cosh

(1)^2 + (a*b^2 + b^2*c)*d^2 - (2*a^2*b + b^3 + 2*a*b*c)*d*cosh(1))*sinh(1)^2 + sqrt(2)*((b^4 - 4*a*b^2*c)*d^3

- 3*(a*b^3 - 4*a*b*c^2 - (4*a^2*b - b^3)*c)*d^2*cosh(1) + (2*a^2*b^2 + b^4 - 8*a^3*c - 8*a*c^3 - 2*(8*a^2 - b^

2)*c^2)*d*cosh(1)^2 - (a*b^3 - 4*a*b*c^2 - (4*a^2*b - b^3)*c)*cosh(1)^3 - (a*b^3 - 4*a*b*c^2 - (4*a^2*b - b^3)

*c)*sinh(1)^3 + ((2*a^2*b^2 + b^4 - 8*a^3*c - 8*a*c^3 - 2*(8*a^2 - b^2)*c^2)*d - 3*(a*b^3 - 4*a*b*c^2 - (4*a^2

*b - b^3)*c)*cosh(1))*sinh(1)^2 - (3*(a*b^3 - 4*a*b*c^2 - (4*a^2*b - b^3)*c)*d^2 - 2*(2*a^2*b^2 + b^4 - 8*a^3*

c - 8*a*c^3 - 2*(8*a^2 - b^2)*c^2)*d*cosh(1) + 3*(a*b^3 - 4*a*b*c^2 - (4*a^2*b - b^3)*c)*cosh(1)^2)*sinh(1) -

((a^2*b^4 - b^6 + 8*a*c^5 + 2*(12*a^2 - b^2)*c^4 + 6*(4*a^3 - 3*a*b^2)*c^3 + (8*a^4 - 22*a^2*b^2 + 3*b^4)*c^2

- 2*(3*a^3*b^2 - 4*a*b^4)*c)*d - (a^3*b^3 - a*b^5 + 4*a*b*c^4 + (4*a^2*b - b^3)*c^3 - (4*a^3*b + 5*a*b^3)*c^2

- (4*a^4*b - 5*a^2*b^3 - b^5)*c)*cosh(1) - (a^3*b^3 - a*b^5 + 4*a*b*c^4 + (4*a^2*b - b^3)*c^3 - (4*a^3*b + 5*a

*b^3)*c^2 - (4*a^4*b - 5*a^2*b^3 - b^5)*c)*sinh(1))*sqrt((2*(2*a^2 + b^2 + 4*a*c + 2*c^2)*d^2 + (b^2*d^4 + b^2

)*cosh(1)^2 + (b^2*d^4 + b^2)*sinh(1)^2 - 4*((a*b + b*c)*d^3 + (a*b + b*c)*d)*cosh(1) + 2*(2*(a*b + b*c)*d^3 -

2*(a*b + b*c)*d - (b^2*d^4 - b^2)*cosh(1))*sinh(1))/((a^4*b^2 - 2*a^2*b^4 + b^6 - 4*a*c^5 - (16*a^2 - b^2)*c^

4 - 12*(2*a^3 - a*b^2)*c^3 - 2*(8*a^4 - 11*a^2*b^2 + b^4)*c^2 - 4*(a^5 - 3*a^3*b^2 + 2*a*b^4)*c)*cosh(1)^2 - 2

*(a^4*b^2 - 2*a^2*b^4 + b^6 - 4*a*c^5 - (16*a^2 - b^2)*c^4 - 12*(2*a^3 - a*b^2)*c^3 - 2*(8*a^4 - 11*a^2*b^2 +

b^4)*c^2 - 4*(a^5 - 3*a^3*b^2 + 2*a*b^4)*c)*cosh(1)*sinh(1) + (a^4*b^2 - 2*a^2*b^4 + b^6 - 4*a*c^5 - (16*a^2 -

b^2)*c^4 - 12*(2*a^3 - a*b^2)*c^3 - 2*(8*a^4 - 11*a^2*b^2 + b^4)*c^2 - 4*(a^5 - 3*a^3*b^2 + 2*a*b^4)*c)*sinh(

1)^2)))*sqrt(((b^2 - 2*a*c - 2*c^2)*d^2 - 2*(a*b - b*c)*d*cosh(1) + (2*a^2 - b^2 + 2*a*c)*cosh(1)^2 + (2*a^2 -

b^2 + 2*a*c)*sinh(1)^2 - 2*((a*b - b*c)*d - (2*a^2 - b^2 + 2*a*c)*cosh(1))*sinh(1) + (a^2*b^2 - b^4 - 4*a*c^3

- (8*a^2 - b^2)*c^2 - 2*(2*a^3 - 3*a*b^2)*c)*sqrt((2*(2*a^2 + b^2 + 4*a*c + 2*c^2)*d^2 + (b^2*d^4 + b^2)*cosh

(1)^2 + (b^2*d^4 + b^2)*sinh(1)^2 - 4*((a*b + b*c)*d^3 + (a*b + b*c)*d)*cosh(1) + 2*(2*(a*b + b*c)*d^3 - 2*(a*

b + b*c)*d - (b^2*d^4 - b^2)*cosh(1))*sinh(1))/((a^4*b^2 - 2*a^2*b^4 + b^6 - 4*a*c^5 - (16*a^2 - b^2)*c^4 - 12

*(2*a^3 - a*b^2)*c^3 - 2*(8*a^4 - 11*a^2*b^2 + b^4)*c^2 - 4*(a^5 - 3*a^3*b^2 + 2*a*b^4)*c)*cosh(1)^2 - 2*(a^4*

b^2 - 2*a^2*b^4 + b^6 - 4*a*c^5 - (16*a^2 - b^2)*c^4 - 12*(2*a^3 - a*b^2)*c^3 - 2*(8*a^4 - 11*a^2*b^2 + b^4)*c

^2 - 4*(a^5 - 3*a^3*b^2 + 2*a*b^4)*c)*cosh(1)*sinh(1) + (a^4*b^2 - 2*a^2*b^4 + b^6 - 4*a*c^5 - (16*a^2 - b^2)*

c^4 - 12*(2*a^3 - a*b^2)*c^3 - 2*(8*a^4 - 11*a^2*b^2 + b^4)*c^2 - 4*(a^5 - 3*a^3*b^2 + 2*a*b^4)*c)*sinh(1)^2))

)/(a^2*b^2 - b^4 - 4*a*c^3 - (8*a^2 - b^2)*c^2 - 2*(2*a^3 - 3*a*b^2)*c)) + 4*(b*c^2*d^4 + a*b*c*cosh(1)^4 + a*

b*c*sinh(1)^4 - (b^2*c + 2*a*c^2 + 2*c^3)*d^3*cosh(1) + 3*(a*b*c + b*c^2)*d^2*cosh(1)^2 - (2*a*c^2 + (2*a^2 +

b^2)*c)*d*cosh(1)^3 + (4*a*b*c*cosh(1) - (2*a*c^2 + (2*a^2 + b^2)*c)*d)*sinh(1)^3 + 3*(2*a*b*c*cosh(1)^2 + (a*

b*c + b*c^2)*d^2 - (2*a*c^2 + (2*a^2 + b^2)*c)*d*cosh(1))*sinh(1)^2 + (4*a*b*c*cosh(1)^3 - (b^2*c + 2*a*c^2 +

2*c^3)*d^3 + 6*(a*b*c + b*c^2)*d^2*cosh(1) - 3*(2*a*c^2 + (2*a^2 + b^2)*c)*d*cosh(1)^2)*sinh(1))*cosh(x) + 2*(

4*a*b^2*cosh(1)^3 - (b^3 + 2*a*b*c + 2*b*c^2)*d^3 + 6*(a*b^2 + b^2*c)*d^2*cosh(1) - 3*(2*a^2*b + b^3 + 2*a*b*c

)*d*cosh(1)^2)*sinh(1) + 4*(b*c^2*d^4 + a*b*c*cosh(1)^4 + a*b*c*sinh(1)^4 - (b^2*c + 2*a*c^2 + 2*c^3)*d^3*cosh

(1) + 3*(a*b*c + b*c^2)*d^2*cosh(1)^2 - (2*a*c^2 + (2*a^2 + b^2)*c)*d*cosh(1)^3 + (4*a*b*c*cosh(1) - (2*a*c^2

+ (2*a^2 + b^2)*c)*d)*sinh(1)^3 + 3*(2*a*b*c*co...

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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((d+e*cosh(x))/(a+b*cosh(x)+c*cosh(x)**2),x)

[Out]

Timed out

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Giac [A]
time = 1.52, size = 1, normalized size = 0.00 \begin {gather*} 0 \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((d+e*cosh(x))/(a+b*cosh(x)+c*cosh(x)^2),x, algorithm="giac")

[Out]

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Mupad [F(-1)]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \text {Hanged} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*cosh(x))/(a + b*cosh(x) + c*cosh(x)^2),x)

[Out]

\text{Hanged}

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