3.742 \(\int \frac {1}{a+b \cosh (x)+c \sinh (x)} \, dx\)

Optimal. Leaf size=51 \[ -\frac {2 \tanh ^{-1}\left (\frac {c-(a-b) \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2+c^2}}\right )}{\sqrt {a^2-b^2+c^2}} \]

[Out]

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

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Rubi [A]  time = 0.07, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3124, 618, 206} \[ -\frac {2 \tanh ^{-1}\left (\frac {c-(a-b) \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2+c^2}}\right )}{\sqrt {a^2-b^2+c^2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Cosh[x] + c*Sinh[x])^(-1),x]

[Out]

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

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 3124

Int[(cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)])^(-1), x_Symbol] :> Module[{f = Free
Factors[Tan[(d + e*x)/2], x]}, Dist[(2*f)/e, Subst[Int[1/(a + b + 2*c*f*x + (a - b)*f^2*x^2), x], x, Tan[(d +
e*x)/2]/f], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[a^2 - b^2 - c^2, 0]

Rubi steps

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

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Mathematica [A]  time = 0.08, size = 54, normalized size = 1.06 \[ \frac {2 \tan ^{-1}\left (\frac {(b-a) \tanh \left (\frac {x}{2}\right )+c}{\sqrt {-a^2+b^2-c^2}}\right )}{\sqrt {-a^2+b^2-c^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Cosh[x] + c*Sinh[x])^(-1),x]

[Out]

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

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fricas [A]  time = 0.47, size = 248, normalized size = 4.86 \[ \left [\frac {\log \left (\frac {{\left (b^{2} + 2 \, b c + c^{2}\right )} \cosh \relax (x)^{2} + {\left (b^{2} + 2 \, b c + c^{2}\right )} \sinh \relax (x)^{2} + 2 \, a^{2} - b^{2} + c^{2} + 2 \, {\left (a b + a c\right )} \cosh \relax (x) + 2 \, {\left (a b + a c + {\left (b^{2} + 2 \, b c + c^{2}\right )} \cosh \relax (x)\right )} \sinh \relax (x) - 2 \, \sqrt {a^{2} - b^{2} + c^{2}} {\left ({\left (b + c\right )} \cosh \relax (x) + {\left (b + c\right )} \sinh \relax (x) + a\right )}}{{\left (b + c\right )} \cosh \relax (x)^{2} + {\left (b + c\right )} \sinh \relax (x)^{2} + 2 \, a \cosh \relax (x) + 2 \, {\left ({\left (b + c\right )} \cosh \relax (x) + a\right )} \sinh \relax (x) + b - c}\right )}{\sqrt {a^{2} - b^{2} + c^{2}}}, \frac {2 \, \sqrt {-a^{2} + b^{2} - c^{2}} \arctan \left (\frac {\sqrt {-a^{2} + b^{2} - c^{2}} {\left ({\left (b + c\right )} \cosh \relax (x) + {\left (b + c\right )} \sinh \relax (x) + a\right )}}{a^{2} - b^{2} + c^{2}}\right )}{a^{2} - b^{2} + c^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*cosh(x)+c*sinh(x)),x, algorithm="fricas")

[Out]

[log(((b^2 + 2*b*c + c^2)*cosh(x)^2 + (b^2 + 2*b*c + c^2)*sinh(x)^2 + 2*a^2 - b^2 + c^2 + 2*(a*b + a*c)*cosh(x
) + 2*(a*b + a*c + (b^2 + 2*b*c + c^2)*cosh(x))*sinh(x) - 2*sqrt(a^2 - b^2 + c^2)*((b + c)*cosh(x) + (b + c)*s
inh(x) + a))/((b + c)*cosh(x)^2 + (b + c)*sinh(x)^2 + 2*a*cosh(x) + 2*((b + c)*cosh(x) + a)*sinh(x) + b - c))/
sqrt(a^2 - b^2 + c^2), 2*sqrt(-a^2 + b^2 - c^2)*arctan(sqrt(-a^2 + b^2 - c^2)*((b + c)*cosh(x) + (b + c)*sinh(
x) + a)/(a^2 - b^2 + c^2))/(a^2 - b^2 + c^2)]

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giac [A]  time = 0.13, size = 46, normalized size = 0.90 \[ \frac {2 \, \arctan \left (\frac {b e^{x} + c e^{x} + a}{\sqrt {-a^{2} + b^{2} - c^{2}}}\right )}{\sqrt {-a^{2} + b^{2} - c^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*cosh(x)+c*sinh(x)),x, algorithm="giac")

[Out]

2*arctan((b*e^x + c*e^x + a)/sqrt(-a^2 + b^2 - c^2))/sqrt(-a^2 + b^2 - c^2)

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maple [A]  time = 0.19, size = 53, normalized size = 1.04 \[ -\frac {2 \arctan \left (\frac {2 \left (a -b \right ) \tanh \left (\frac {x}{2}\right )-2 c}{2 \sqrt {-a^{2}+b^{2}-c^{2}}}\right )}{\sqrt {-a^{2}+b^{2}-c^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*cosh(x)+c*sinh(x)),x)

[Out]

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

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*cosh(x)+c*sinh(x)),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(c^2-b^2+a^2>0)', see `assume?`
for more details)Is c^2-b^2+a^2 positive or negative?

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mupad [B]  time = 0.21, size = 78, normalized size = 1.53 \[ \frac {2\,\mathrm {atan}\left (\frac {a}{\sqrt {-a^2+b^2-c^2}}+\frac {b\,{\mathrm {e}}^x}{\sqrt {-a^2+b^2-c^2}}+\frac {c\,{\mathrm {e}}^x}{\sqrt {-a^2+b^2-c^2}}\right )}{\sqrt {-a^2+b^2-c^2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b*cosh(x) + c*sinh(x)),x)

[Out]

(2*atan(a/(b^2 - a^2 - c^2)^(1/2) + (b*exp(x))/(b^2 - a^2 - c^2)^(1/2) + (c*exp(x))/(b^2 - a^2 - c^2)^(1/2)))/
(b^2 - a^2 - c^2)^(1/2)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*cosh(x)+c*sinh(x)),x)

[Out]

Timed out

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