\(\int \frac {b+c+\sinh (x)}{a-b \cosh (x)} \, dx\) [569]

Optimal result

Integrand size = 15, antiderivative size = 59 \[ \int \frac {b+c+\sinh (x)}{a-b \cosh (x)} \, dx=\frac {2 (b+c) \text {arctanh}\left (\frac {\sqrt {a+b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a-b}}\right )}{\sqrt {a-b} \sqrt {a+b}}-\frac {\log (a-b \cosh (x))}{b} \]

[Out]

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

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4486, 2738, 214, 2747, 31} \[ \int \frac {b+c+\sinh (x)}{a-b \cosh (x)} \, dx=\frac {2 (b+c) \text {arctanh}\left (\frac {\sqrt {a+b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a-b}}\right )}{\sqrt {a-b} \sqrt {a+b}}-\frac {\log (a-b \cosh (x))}{b} \]

[In]

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

[Out]

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

Rule 31

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

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 2747

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^m*(b^2 - x^2)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && Integer
Q[(p - 1)/2] && NeQ[a^2 - b^2, 0]

Rule 4486

Int[u_, x_Symbol] :> With[{v = ExpandTrig[u, x]}, Int[v, x] /; SumQ[v]] /;  !InertTrigFreeQ[u]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {-b-c}{-a+b \cosh (x)}+\frac {\sinh (x)}{a-b \cosh (x)}\right ) \, dx \\ & = (-b-c) \int \frac {1}{-a+b \cosh (x)} \, dx+\int \frac {\sinh (x)}{a-b \cosh (x)} \, dx \\ & = -\frac {\text {Subst}\left (\int \frac {1}{a+x} \, dx,x,-b \cosh (x)\right )}{b}-(2 (b+c)) \text {Subst}\left (\int \frac {1}{-a+b-(-a-b) x^2} \, dx,x,\tanh \left (\frac {x}{2}\right )\right ) \\ & = \frac {2 (b+c) \text {arctanh}\left (\frac {\sqrt {a+b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a-b}}\right )}{\sqrt {a-b} \sqrt {a+b}}-\frac {\log (a-b \cosh (x))}{b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.95 \[ \int \frac {b+c+\sinh (x)}{a-b \cosh (x)} \, dx=-\frac {2 (b+c) \arctan \left (\frac {(a+b) \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2+b^2}}\right )}{\sqrt {-a^2+b^2}}-\frac {\log (a-b \cosh (x))}{b} \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(108\) vs. \(2(49)=98\).

Time = 0.41 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.85

method	result	size
default	\(\frac {\frac {2 \left (-a -b \right ) \ln \left (\tanh \left (\frac {x}{2}\right )^{2} a +\tanh \left (\frac {x}{2}\right )^{2} b -a +b \right )}{2 a +2 b}-\frac {2 \left (-b^{2}-c b \right ) \operatorname {arctanh}\left (\frac {\left (a +b \right ) \tanh \left (\frac {x}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}}{b}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{b}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{b}\)	\(109\)
risch	\(-\frac {x}{b}-\frac {2 x \,a^{2} b}{-a^{2} b^{2}+b^{4}}+\frac {2 x \,b^{3}}{-a^{2} b^{2}+b^{4}}-\frac {\ln \left ({\mathrm e}^{x}+\frac {-a \,b^{2}-c a b +\sqrt {a^{2} b^{4}+2 a^{2} b^{3} c +a^{2} b^{2} c^{2}-b^{6}-2 b^{5} c -b^{4} c^{2}}}{b^{2} \left (b +c \right )}\right ) a^{2}}{\left (a^{2}-b^{2}\right ) b}+\frac {b \ln \left ({\mathrm e}^{x}+\frac {-a \,b^{2}-c a b +\sqrt {a^{2} b^{4}+2 a^{2} b^{3} c +a^{2} b^{2} c^{2}-b^{6}-2 b^{5} c -b^{4} c^{2}}}{b^{2} \left (b +c \right )}\right )}{a^{2}-b^{2}}+\frac {\ln \left ({\mathrm e}^{x}+\frac {-a \,b^{2}-c a b +\sqrt {a^{2} b^{4}+2 a^{2} b^{3} c +a^{2} b^{2} c^{2}-b^{6}-2 b^{5} c -b^{4} c^{2}}}{b^{2} \left (b +c \right )}\right ) \sqrt {a^{2} b^{4}+2 a^{2} b^{3} c +a^{2} b^{2} c^{2}-b^{6}-2 b^{5} c -b^{4} c^{2}}}{\left (a^{2}-b^{2}\right ) b}-\frac {\ln \left ({\mathrm e}^{x}-\frac {a \,b^{2}+c a b +\sqrt {a^{2} b^{4}+2 a^{2} b^{3} c +a^{2} b^{2} c^{2}-b^{6}-2 b^{5} c -b^{4} c^{2}}}{b^{2} \left (b +c \right )}\right ) a^{2}}{\left (a^{2}-b^{2}\right ) b}+\frac {b \ln \left ({\mathrm e}^{x}-\frac {a \,b^{2}+c a b +\sqrt {a^{2} b^{4}+2 a^{2} b^{3} c +a^{2} b^{2} c^{2}-b^{6}-2 b^{5} c -b^{4} c^{2}}}{b^{2} \left (b +c \right )}\right )}{a^{2}-b^{2}}-\frac {\ln \left ({\mathrm e}^{x}-\frac {a \,b^{2}+c a b +\sqrt {a^{2} b^{4}+2 a^{2} b^{3} c +a^{2} b^{2} c^{2}-b^{6}-2 b^{5} c -b^{4} c^{2}}}{b^{2} \left (b +c \right )}\right ) \sqrt {a^{2} b^{4}+2 a^{2} b^{3} c +a^{2} b^{2} c^{2}-b^{6}-2 b^{5} c -b^{4} c^{2}}}{\left (a^{2}-b^{2}\right ) b}\)	\(675\)

[In]

int((b+c+sinh(x))/(a-b*cosh(x)),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 114 vs. \(2 (49) = 98\).

Time = 0.28 (sec) , antiderivative size = 299, normalized size of antiderivative = 5.07 \[ \int \frac {b+c+\sinh (x)}{a-b \cosh (x)} \, dx=\left [\frac {\sqrt {a^{2} - b^{2}} {\left (b^{2} + b c\right )} \log \left (\frac {b^{2} \cosh \left (x\right )^{2} + b^{2} \sinh \left (x\right )^{2} - 2 \, a b \cosh \left (x\right ) + 2 \, a^{2} - b^{2} + 2 \, {\left (b^{2} \cosh \left (x\right ) - a b\right )} \sinh \left (x\right ) + 2 \, \sqrt {a^{2} - b^{2}} {\left (b \cosh \left (x\right ) + b \sinh \left (x\right ) - a\right )}}{b \cosh \left (x\right )^{2} + b \sinh \left (x\right )^{2} - 2 \, a \cosh \left (x\right ) + 2 \, {\left (b \cosh \left (x\right ) - a\right )} \sinh \left (x\right ) + b}\right ) + {\left (a^{2} - b^{2}\right )} x - {\left (a^{2} - b^{2}\right )} \log \left (\frac {2 \, {\left (b \cosh \left (x\right ) - a\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right )}{a^{2} b - b^{3}}, \frac {2 \, \sqrt {-a^{2} + b^{2}} {\left (b^{2} + b c\right )} \arctan \left (-\frac {\sqrt {-a^{2} + b^{2}} {\left (b \cosh \left (x\right ) + b \sinh \left (x\right ) - a\right )}}{a^{2} - b^{2}}\right ) + {\left (a^{2} - b^{2}\right )} x - {\left (a^{2} - b^{2}\right )} \log \left (\frac {2 \, {\left (b \cosh \left (x\right ) - a\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right )}{a^{2} b - b^{3}}\right ] \]

[In]

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

[Out]

[(sqrt(a^2 - b^2)*(b^2 + b*c)*log((b^2*cosh(x)^2 + b^2*sinh(x)^2 - 2*a*b*cosh(x) + 2*a^2 - b^2 + 2*(b^2*cosh(x
) - a*b)*sinh(x) + 2*sqrt(a^2 - b^2)*(b*cosh(x) + b*sinh(x) - a))/(b*cosh(x)^2 + b*sinh(x)^2 - 2*a*cosh(x) + 2
*(b*cosh(x) - a)*sinh(x) + b)) + (a^2 - b^2)*x - (a^2 - b^2)*log(2*(b*cosh(x) - a)/(cosh(x) - sinh(x))))/(a^2*
b - b^3), (2*sqrt(-a^2 + b^2)*(b^2 + b*c)*arctan(-sqrt(-a^2 + b^2)*(b*cosh(x) + b*sinh(x) - a)/(a^2 - b^2)) +
(a^2 - b^2)*x - (a^2 - b^2)*log(2*(b*cosh(x) - a)/(cosh(x) - sinh(x))))/(a^2*b - b^3)]

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 840 vs. \(2 (49) = 98\).

Time = 15.82 (sec) , antiderivative size = 840, normalized size of antiderivative = 14.24 \[ \int \frac {b+c+\sinh (x)}{a-b \cosh (x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Piecewise((zoo*(2*c*atan(tanh(x/2)) + x - 2*log(tanh(x/2) + 1) + log(tanh(x/2)**2 + 1)), Eq(a, 0) & Eq(b, 0)),
(-tanh(x/2) - c*tanh(x/2)/b - x/b + 2*log(tanh(x/2) + 1)/b, Eq(a, -b)), (1/tanh(x/2) + c/(b*tanh(x/2)) - x/b
+ 2*log(tanh(x/2) + 1)/b - 2*log(tanh(x/2))/b, Eq(a, b)), ((c*x + cosh(x))/a, Eq(b, 0)), (-a*x*sqrt(a/(a + b)
- b/(a + b))/(a*b*sqrt(a/(a + b) - b/(a + b)) + b**2*sqrt(a/(a + b) - b/(a + b))) - a*sqrt(a/(a + b) - b/(a +
b))*log(-sqrt(a/(a + b) - b/(a + b)) + tanh(x/2))/(a*b*sqrt(a/(a + b) - b/(a + b)) + b**2*sqrt(a/(a + b) - b/(
a + b))) - a*sqrt(a/(a + b) - b/(a + b))*log(sqrt(a/(a + b) - b/(a + b)) + tanh(x/2))/(a*b*sqrt(a/(a + b) - b/
(a + b)) + b**2*sqrt(a/(a + b) - b/(a + b))) + 2*a*sqrt(a/(a + b) - b/(a + b))*log(tanh(x/2) + 1)/(a*b*sqrt(a/
(a + b) - b/(a + b)) + b**2*sqrt(a/(a + b) - b/(a + b))) - b**2*log(-sqrt(a/(a + b) - b/(a + b)) + tanh(x/2))/
(a*b*sqrt(a/(a + b) - b/(a + b)) + b**2*sqrt(a/(a + b) - b/(a + b))) + b**2*log(sqrt(a/(a + b) - b/(a + b)) +
tanh(x/2))/(a*b*sqrt(a/(a + b) - b/(a + b)) + b**2*sqrt(a/(a + b) - b/(a + b))) - b*c*log(-sqrt(a/(a + b) - b/
(a + b)) + tanh(x/2))/(a*b*sqrt(a/(a + b) - b/(a + b)) + b**2*sqrt(a/(a + b) - b/(a + b))) + b*c*log(sqrt(a/(a
+ b) - b/(a + b)) + tanh(x/2))/(a*b*sqrt(a/(a + b) - b/(a + b)) + b**2*sqrt(a/(a + b) - b/(a + b))) - b*x*sqr
t(a/(a + b) - b/(a + b))/(a*b*sqrt(a/(a + b) - b/(a + b)) + b**2*sqrt(a/(a + b) - b/(a + b))) - b*sqrt(a/(a +
b) - b/(a + b))*log(-sqrt(a/(a + b) - b/(a + b)) + tanh(x/2))/(a*b*sqrt(a/(a + b) - b/(a + b)) + b**2*sqrt(a/(
a + b) - b/(a + b))) - b*sqrt(a/(a + b) - b/(a + b))*log(sqrt(a/(a + b) - b/(a + b)) + tanh(x/2))/(a*b*sqrt(a/
(a + b) - b/(a + b)) + b**2*sqrt(a/(a + b) - b/(a + b))) + 2*b*sqrt(a/(a + b) - b/(a + b))*log(tanh(x/2) + 1)/
(a*b*sqrt(a/(a + b) - b/(a + b)) + b**2*sqrt(a/(a + b) - b/(a + b))), True))

Maxima [F(-2)]

Exception generated. \[ \int \frac {b+c+\sinh (x)}{a-b \cosh (x)} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((b+c+sinh(x))/(a-b*cosh(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(4*a^2-4*b^2>0)', see `assume?`
for more de

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.05 \[ \int \frac {b+c+\sinh (x)}{a-b \cosh (x)} \, dx=-\frac {2 \, {\left (b + c\right )} \arctan \left (\frac {b e^{x} - a}{\sqrt {-a^{2} + b^{2}}}\right )}{\sqrt {-a^{2} + b^{2}}} + \frac {x}{b} - \frac {\log \left (b e^{\left (2 \, x\right )} - 2 \, a e^{x} + b\right )}{b} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 2.49 (sec) , antiderivative size = 199, normalized size of antiderivative = 3.37 \[ \int \frac {b+c+\sinh (x)}{a-b \cosh (x)} \, dx=\frac {x}{b}-\frac {\ln \left (b\,\sqrt {\left (a+b\right )\,\left (a-b\right )}+a^2\,{\mathrm {e}}^x-b^2\,{\mathrm {e}}^x-a\,{\mathrm {e}}^x\,\sqrt {\left (a+b\right )\,\left (a-b\right )}\right )\,\left (b^2\,\sqrt {\left (a+b\right )\,\left (a-b\right )}+a^2-b^2+b\,c\,\sqrt {\left (a+b\right )\,\left (a-b\right )}\right )}{a^2\,b-b^3}+\frac {\ln \left (b\,\sqrt {\left (a+b\right )\,\left (a-b\right )}-a^2\,{\mathrm {e}}^x+b^2\,{\mathrm {e}}^x-a\,{\mathrm {e}}^x\,\sqrt {\left (a+b\right )\,\left (a-b\right )}\right )\,\left (b^2\,\sqrt {\left (a+b\right )\,\left (a-b\right )}-a^2+b^2+b\,c\,\sqrt {\left (a+b\right )\,\left (a-b\right )}\right )}{a^2\,b-b^3} \]

[In]

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

[Out]

x/b - (log(b*((a + b)*(a - b))^(1/2) + a^2*exp(x) - b^2*exp(x) - a*exp(x)*((a + b)*(a - b))^(1/2))*(b^2*((a +
b)*(a - b))^(1/2) + a^2 - b^2 + b*c*((a + b)*(a - b))^(1/2)))/(a^2*b - b^3) + (log(b*((a + b)*(a - b))^(1/2) -
a^2*exp(x) + b^2*exp(x) - a*exp(x)*((a + b)*(a - b))^(1/2))*(b^2*((a + b)*(a - b))^(1/2) - a^2 + b^2 + b*c*((
a + b)*(a - b))^(1/2)))/(a^2*b - b^3)