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-sinh(x)*(b^2-c^2)^(1/2))/c/(c*cosh(x)+b*sinh(x))

________________________________________________________________________________________

Rubi [A]  time = 0.04, antiderivative size = 34, normalized size of antiderivative = 1.00, 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 = 0.08, 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]))

________________________________________________________________________________________

fricas [B]  time = 0.40, size = 88, normalized size = 2.59 \[ -\frac {2 \, {\left ({\left (b + c\right )} \cosh \relax (x) + {\left (b + c\right )} \sinh \relax (x) - \sqrt {b^{2} - c^{2}}\right )}}{{\left (b^{2} + 2 \, b c + c^{2}\right )} \cosh \relax (x)^{2} + 2 \, {\left (b^{2} + 2 \, b c + c^{2}\right )} \cosh \relax (x) \sinh \relax (x) + {\left (b^{2} + 2 \, b c + c^{2}\right )} \sinh \relax (x)^{2} - b^{2} + c^{2}} \]

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)

________________________________________________________________________________________

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

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 >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Unab
le to divide, perhaps due to rounding error%%%{%%%{1,[1,0]%%%}+%%%{1,[0,1]%%%},[2]%%%}+%%%{%%{[2,0]:[1,0,%%%{-
1,[2,0]%%%}+%%%{1,[0,2]%%%}]%%},[1]%%%}+%%%{%%%{1,[1,0]%%%}+%%%{-1,[0,1]%%%},[0]%%%} / %%%{%%%{1,[2,0]%%%}+%%%
{2,[1,1]%%%}+%%%{1,[0,2]%%%},[2]%%%}+%%%{%%{[%%%{2,[1,0]%%%}+%%%{2,[0,1]%%%},0]:[1,0,%%%{-1,[2,0]%%%}+%%%{1,[0
,2]%%%}]%%},[1]%%%}+%%%{%%%{1,[2,0]%%%}+%%%{-1,[0,2]%%%},[0]%%%} Error: Bad Argument Value

________________________________________________________________________________________

maple [C]  time = 0.62, size = 596, normalized size = 17.53 \[ \frac {\left (\munderset {\textit {\_R} =\RootOf \left (\left ({| \left (b -c \right ) \left (b +c \right )|} \mathrm {signum}\left (\left (b -c \right ) \left (b +c \right )\right )^{2}-2 \sqrt {{| \left (b -c \right ) \left (b +c \right )|}}\, \mathrm {signum}\left (\left (b -c \right ) \left (b +c \right )\right ) b +{| \left (b -c \right ) \left (b +c \right )|}-2 \sqrt {{| \left (b -c \right ) \left (b +c \right )|}}\, b +2 b^{2}\right ) \textit {\_Z}^{4}+\left (-4 \sqrt {{| \left (b -c \right ) \left (b +c \right )|}}\, \mathrm {signum}\left (\left (b -c \right ) \left (b +c \right )\right ) c -4 \sqrt {{| \left (b -c \right ) \left (b +c \right )|}}\, c +8 c b \right ) \textit {\_Z}^{3}+\left (-2 {| \left (b -c \right ) \left (b +c \right )|} \mathrm {signum}\left (\left (b -c \right ) \left (b +c \right )\right )^{2}-2 {| \left (b -c \right ) \left (b +c \right )|}+4 b^{2}+8 c^{2}\right ) \textit {\_Z}^{2}+\left (4 \sqrt {{| \left (b -c \right ) \left (b +c \right )|}}\, \mathrm {signum}\left (\left (b -c \right ) \left (b +c \right )\right ) c +4 \sqrt {{| \left (b -c \right ) \left (b +c \right )|}}\, c +8 c b \right ) \textit {\_Z} +{| \left (b -c \right ) \left (b +c \right )|} \mathrm {signum}\left (\left (b -c \right ) \left (b +c \right )\right )^{2}+2 \sqrt {{| \left (b -c \right ) \left (b +c \right )|}}\, \mathrm {signum}\left (\left (b -c \right ) \left (b +c \right )\right ) b +{| \left (b -c \right ) \left (b +c \right )|}+2 \sqrt {{| \left (b -c \right ) \left (b +c \right )|}}\, b +2 b^{2}\right )}{\sum }\frac {\left (2 b \,\textit {\_R}^{2}+4 \textit {\_R} c +2 b +\sqrt {{| b^{2}-c^{2}|}}\, \left (1-i+\textit {\_R}^{2} \left (-1+i-i \mathrm {signum}\left (b^{2}-c^{2}\right )-\mathrm {signum}\left (b^{2}-c^{2}\right )\right )+i \mathrm {signum}\left (b^{2}-c^{2}\right )+\mathrm {signum}\left (b^{2}-c^{2}\right )\right )\right ) \ln \left (\tanh \left (\frac {x}{2}\right )-\textit {\_R} \right )}{{| \left (b -c \right ) \left (b +c \right )|} \textit {\_R}^{3} \mathrm {signum}\left (\left (b -c \right ) \left (b +c \right )\right )^{2}-{| \left (b -c \right ) \left (b +c \right )|} \textit {\_R} \mathrm {signum}\left (\left (b -c \right ) \left (b +c \right )\right )^{2}+{| \left (b -c \right ) \left (b +c \right )|} \textit {\_R}^{3}+2 b^{2} \textit {\_R}^{3}+6 \textit {\_R}^{2} b c -{| \left (b -c \right ) \left (b +c \right )|} \textit {\_R} +2 b^{2} \textit {\_R} +4 c^{2} \textit {\_R} +2 c b +\sqrt {{| \left (b -c \right ) \left (b +c \right )|}}\, \left (-2 \,\mathrm {signum}\left (\left (b -c \right ) \left (b +c \right )\right ) b \,\textit {\_R}^{3}-3 \,\mathrm {signum}\left (\left (b -c \right ) \left (b +c \right )\right ) c \,\textit {\_R}^{2}-2 b \,\textit {\_R}^{3}-3 c \,\textit {\_R}^{2}+\mathrm {signum}\left (\left (b -c \right ) \left (b +c \right )\right ) c +c \right )}\right )}{2} \]

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*_R*c+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*b^2*_R^3+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))

________________________________________________________________________________________

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

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 >> ECL says: THROW: The catch RAT-ERR is undefined.

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {1}{b\,\mathrm {cosh}\relax (x)+\sqrt {b^2-c^2}+c\,\mathrm {sinh}\relax (x)} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*cosh(x) + (b^2 - c^2)^(1/2) + c*sinh(x)),x)

[Out]

int(1/(b*cosh(x) + (b^2 - c^2)^(1/2) + c*sinh(x)), x)

________________________________________________________________________________________

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/(b*cosh(x)+c*sinh(x)+(b**2-c**2)**(1/2)),x)

[Out]

Timed out

________________________________________________________________________________________