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

Optimal result

Integrand size = 21, antiderivative size = 246 \[ \int \frac {d+e \cosh (x)}{a+b \cosh (x)+c \cosh ^2(x)} \, dx=\frac {2 \left (e+\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) \text {arctanh}\left (\frac {\sqrt {b-2 c-\sqrt {b^2-4 a c}} \tanh \left (\frac {x}{2}\right )}{\sqrt {b+2 c-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b-2 c-\sqrt {b^2-4 a c}} \sqrt {b+2 c-\sqrt {b^2-4 a c}}}+\frac {2 \left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) \text {arctanh}\left (\frac {\sqrt {b-2 c+\sqrt {b^2-4 a c}} \tanh \left (\frac {x}{2}\right )}{\sqrt {b+2 c+\sqrt {b^2-4 a c}}}\right )}{\sqrt {b-2 c+\sqrt {b^2-4 a c}} \sqrt {b+2 c+\sqrt {b^2-4 a c}}} \]

[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)

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3374, 2738, 214} \[ \int \frac {d+e \cosh (x)}{a+b \cosh (x)+c \cosh ^2(x)} \, dx=\frac {2 \left (\frac {2 c d-b e}{\sqrt {b^2-4 a c}}+e\right ) \text {arctanh}\left (\frac {\tanh \left (\frac {x}{2}\right ) \sqrt {-\sqrt {b^2-4 a c}+b-2 c}}{\sqrt {-\sqrt {b^2-4 a c}+b+2 c}}\right )}{\sqrt {-\sqrt {b^2-4 a c}+b-2 c} \sqrt {-\sqrt {b^2-4 a c}+b+2 c}}+\frac {2 \left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) \text {arctanh}\left (\frac {\tanh \left (\frac {x}{2}\right ) \sqrt {\sqrt {b^2-4 a c}+b-2 c}}{\sqrt {\sqrt {b^2-4 a c}+b+2 c}}\right )}{\sqrt {\sqrt {b^2-4 a c}+b-2 c} \sqrt {\sqrt {b^2-4 a c}+b+2 c}} \]

[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*} \text {integral}& = \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 ) \text {arctanh}\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 ) \text {arctanh}\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*}

Mathematica [A] (verified)

Time = 0.77 (sec) , antiderivative size = 241, normalized size of antiderivative = 0.98 \[ \int \frac {d+e \cosh (x)}{a+b \cosh (x)+c \cosh ^2(x)} \, dx=\frac {\sqrt {2} \left (-\frac {\left (-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e\right ) \arctan \left (\frac {\left (b-2 c+\sqrt {b^2-4 a c}\right ) \tanh \left (\frac {x}{2}\right )}{\sqrt {-2 b^2+4 c (a+c)-2 b \sqrt {b^2-4 a c}}}\right )}{\sqrt {-b^2+2 c (a+c)-b \sqrt {b^2-4 a c}}}+\frac {\left (2 c d+\left (-b+\sqrt {b^2-4 a c}\right ) e\right ) \arctan \left (\frac {\left (-b+2 c+\sqrt {b^2-4 a c}\right ) \tanh \left (\frac {x}{2}\right )}{\sqrt {-2 b^2+4 c (a+c)+2 b \sqrt {b^2-4 a c}}}\right )}{\sqrt {-b^2+2 c (a+c)+b \sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c}} \]

[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]

Maple [A] (verified)

Time = 3.68 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.03

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

[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*a*e+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*a*e-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)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 6997 vs. \(2 (206) = 412\).

Time = 3.06 (sec) , antiderivative size = 6997, normalized size of antiderivative = 28.44 \[ \int \frac {d+e \cosh (x)}{a+b \cosh (x)+c \cosh ^2(x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [F(-1)]

Timed out. \[ \int \frac {d+e \cosh (x)}{a+b \cosh (x)+c \cosh ^2(x)} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 1.56 (sec) , antiderivative size = 1, normalized size of antiderivative = 0.00 \[ \int \frac {d+e \cosh (x)}{a+b \cosh (x)+c \cosh ^2(x)} \, dx=0 \]

[In]

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

[Out]

0

Mupad [F(-1)]

Timed out. \[ \int \frac {d+e \cosh (x)}{a+b \cosh (x)+c \cosh ^2(x)} \, dx=\text {Hanged} \]

[In]

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

[Out]

\text{Hanged}