### 3.757 $$\int \frac{1}{\sqrt{b^2-c^2}+b \cosh (x)+c \sinh (x)} \, dx$$

Optimal. Leaf size=34 $-\frac{\sqrt{b^2-c^2} \sinh (x)+c}{c (b \sinh (x)+c \cosh (x))}$

[Out]

-((c + Sqrt[b^2 - c^2]*Sinh[x])/(c*(c*Cosh[x] + b*Sinh[x])))

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Rubi [A]  time = 0.0367554, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 24, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.042, Rules used = {3114} $-\frac{\sqrt{b^2-c^2} \sinh (x)+c}{c (b \sinh (x)+c \cosh (x))}$

Antiderivative was successfully veriﬁed.

[In]

Int[(Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x])^(-1),x]

[Out]

-((c + Sqrt[b^2 - c^2]*Sinh[x])/(c*(c*Cosh[x] + b*Sinh[x])))

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}{\sqrt{b^2-c^2}+b \cosh (x)+c \sinh (x)} \, dx &=-\frac{c+\sqrt{b^2-c^2} \sinh (x)}{c (c \cosh (x)+b \sinh (x))}\\ \end{align*}

Mathematica [A]  time = 0.0734254, size = 36, normalized size = 1.06 $\frac{-\sqrt{b^2-c^2} \sinh (x)-c}{c (b \sinh (x)+c \cosh (x))}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x])^(-1),x]

[Out]

(-c - Sqrt[b^2 - c^2]*Sinh[x])/(c*(c*Cosh[x] + b*Sinh[x]))

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Maple [C]  time = 0.254, size = 596, normalized size = 17.5 \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)),x)

[Out]

1/2*sum((2*b*_R^2+4*c*_R+2*b+abs(b^2-c^2)^(1/2)*(1-I+_R^2*(-1+I-I*signum(b^2-c^2)-signum(b^2-c^2))+I*signum(b^
2-c^2)+signum(b^2-c^2)))/(abs((b-c)*(b+c))*_R^3*signum((b-c)*(b+c))^2-abs((b-c)*(b+c))*_R*signum((b-c)*(b+c))^
2+abs((b-c)*(b+c))*_R^3+2*_R^3*b^2+6*_R^2*b*c-abs((b-c)*(b+c))*_R+2*b^2*_R+4*c^2*_R+2*c*b+abs((b-c)*(b+c))^(1/
2)*(-2*signum((b-c)*(b+c))*b*_R^3-3*signum((b-c)*(b+c))*c*_R^2-2*b*_R^3-3*c*_R^2+signum((b-c)*(b+c))*c+c))*ln(
tanh(1/2*x)-_R),_R=RootOf((abs((b-c)*(b+c))*signum((b-c)*(b+c))^2-2*abs((b-c)*(b+c))^(1/2)*signum((b-c)*(b+c))
*b+abs((b-c)*(b+c))-2*abs((b-c)*(b+c))^(1/2)*b+2*b^2)*_Z^4+(-4*abs((b-c)*(b+c))^(1/2)*signum((b-c)*(b+c))*c-4*
abs((b-c)*(b+c))^(1/2)*c+8*c*b)*_Z^3+(-2*abs((b-c)*(b+c))*signum((b-c)*(b+c))^2-2*abs((b-c)*(b+c))+4*b^2+8*c^2
)*_Z^2+(4*abs((b-c)*(b+c))^(1/2)*signum((b-c)*(b+c))*c+4*abs((b-c)*(b+c))^(1/2)*c+8*c*b)*_Z+abs((b-c)*(b+c))*s
ignum((b-c)*(b+c))^2+2*abs((b-c)*(b+c))^(1/2)*signum((b-c)*(b+c))*b+abs((b-c)*(b+c))+2*abs((b-c)*(b+c))^(1/2)*
b+2*b^2))

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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)),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [B]  time = 2.98676, size = 235, normalized size = 6.91 \begin{align*} -\frac{2 \,{\left ({\left (b + c\right )} \cosh \left (x\right ) +{\left (b + c\right )} \sinh \left (x\right ) - \sqrt{b^{2} - c^{2}}\right )}}{{\left (b^{2} + 2 \, b c + c^{2}\right )} \cosh \left (x\right )^{2} + 2 \,{\left (b^{2} + 2 \, b c + c^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right ) +{\left (b^{2} + 2 \, b c + c^{2}\right )} \sinh \left (x\right )^{2} - b^{2} + c^{2}} \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)),x, algorithm="fricas")

[Out]

-2*((b + c)*cosh(x) + (b + c)*sinh(x) - sqrt(b^2 - c^2))/((b^2 + 2*b*c + c^2)*cosh(x)^2 + 2*(b^2 + 2*b*c + c^2
)*cosh(x)*sinh(x) + (b^2 + 2*b*c + c^2)*sinh(x)^2 - b^2 + c^2)

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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)),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)),x, algorithm="giac")

[Out]

Exception raised: TypeError