\(\int \frac {A+B \cosh (x)}{a+b \cosh (x)-b \sinh (x)} \, dx\) [806]

Optimal result

Integrand size = 20, antiderivative size = 78 \[ \int \frac {A+B \cosh (x)}{a+b \cosh (x)-b \sinh (x)} \, dx=\frac {(2 a A-b B) x}{2 a^2}+\frac {B \cosh (x)}{2 a}+\frac {\left (2 a A b-a^2 B-b^2 B\right ) \log (a+b \cosh (x)-b \sinh (x))}{2 a^2 b}+\frac {B \sinh (x)}{2 a} \]

[Out]

1/2*(2*A*a-B*b)*x/a^2+1/2*B*cosh(x)/a+1/2*(2*A*a*b-B*a^2-B*b^2)*ln(a+b*cosh(x)-b*sinh(x))/a^2/b+1/2*B*sinh(x)/
a

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {3211} \[ \int \frac {A+B \cosh (x)}{a+b \cosh (x)-b \sinh (x)} \, dx=\frac {\left (a^2 (-B)+2 a A b-b^2 B\right ) \log (a-b \sinh (x)+b \cosh (x))}{2 a^2 b}+\frac {x (2 a A-b B)}{2 a^2}+\frac {B \sinh (x)}{2 a}+\frac {B \cosh (x)}{2 a} \]

[In]

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

[Out]

((2*a*A - b*B)*x)/(2*a^2) + (B*Cosh[x])/(2*a) + ((2*a*A*b - a^2*B - b^2*B)*Log[a + b*Cosh[x] - b*Sinh[x]])/(2*
a^2*b) + (B*Sinh[x])/(2*a)

Rule 3211

Int[((A_.) + cos[(d_.) + (e_.)*(x_)]*(B_.))/(cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x
_)]), x_Symbol] :> Simp[(2*a*A - b*B)*(x/(2*a^2)), x] + (Simp[B*(Sin[d + e*x]/(2*a*e)), x] - Simp[b*B*(Cos[d +
e*x]/(2*a*c*e)), x] + Simp[(a^2*B - 2*a*b*A + b^2*B)*(Log[RemoveContent[a + b*Cos[d + e*x] + c*Sin[d + e*x],
x]]/(2*a^2*c*e)), x]) /; FreeQ[{a, b, c, d, e, A, B}, x] && EqQ[b^2 + c^2, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {(2 a A-b B) x}{2 a^2}+\frac {B \cosh (x)}{2 a}+\frac {\left (2 a A b-a^2 B-b^2 B\right ) \log (a+b \cosh (x)-b \sinh (x))}{2 a^2 b}+\frac {B \sinh (x)}{2 a} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.10 \[ \int \frac {A+B \cosh (x)}{a+b \cosh (x)-b \sinh (x)} \, dx=\frac {\left (2 a A b+a^2 B-b^2 B\right ) x+2 a b B \cosh (x)-2 \left (-2 a A b+a^2 B+b^2 B\right ) \log \left ((a+b) \cosh \left (\frac {x}{2}\right )+(a-b) \sinh \left (\frac {x}{2}\right )\right )+2 a b B \sinh (x)}{4 a^2 b} \]

[In]

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

[Out]

((2*a*A*b + a^2*B - b^2*B)*x + 2*a*b*B*Cosh[x] - 2*(-2*a*A*b + a^2*B + b^2*B)*Log[(a + b)*Cosh[x/2] + (a - b)*
Sinh[x/2]] + 2*a*b*B*Sinh[x])/(4*a^2*b)

Maple [A] (verified)

Time = 0.26 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.79

method	result	size
risch	\(\frac {B \,{\mathrm e}^{x}}{2 a}+\frac {B x}{2 b}+\frac {\ln \left ({\mathrm e}^{x}+\frac {b}{a}\right ) A}{a}-\frac {\ln \left ({\mathrm e}^{x}+\frac {b}{a}\right ) B}{2 b}-\frac {b \ln \left ({\mathrm e}^{x}+\frac {b}{a}\right ) B}{2 a^{2}}\)	\(62\)
default	\(\frac {B \ln \left (1+\tanh \left (\frac {x}{2}\right )\right )}{2 b}-\frac {B}{a \left (\tanh \left (\frac {x}{2}\right )-1\right )}+\frac {\left (-2 A a +B b \right ) \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{2 a^{2}}+\frac {\left (2 A a b -B \,a^{2}-B \,b^{2}\right ) \ln \left (a \tanh \left (\frac {x}{2}\right )-b \tanh \left (\frac {x}{2}\right )+a +b \right )}{2 a^{2} b}\)	\(92\)

[In]

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

[Out]

1/2*B/a*exp(x)+1/2*B*x/b+1/a*ln(exp(x)+1/a*b)*A-1/2/b*ln(exp(x)+1/a*b)*B-1/2/a^2*b*ln(exp(x)+1/a*b)*B

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.72 \[ \int \frac {A+B \cosh (x)}{a+b \cosh (x)-b \sinh (x)} \, dx=\frac {B a^{2} x + B a b \cosh \left (x\right ) + B a b \sinh \left (x\right ) - {\left (B a^{2} - 2 \, A a b + B b^{2}\right )} \log \left (a \cosh \left (x\right ) + a \sinh \left (x\right ) + b\right )}{2 \, a^{2} b} \]

[In]

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

[Out]

1/2*(B*a^2*x + B*a*b*cosh(x) + B*a*b*sinh(x) - (B*a^2 - 2*A*a*b + B*b^2)*log(a*cosh(x) + a*sinh(x) + b))/(a^2*
b)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 904 vs. \(2 (66) = 132\).

Time = 2.35 (sec) , antiderivative size = 904, normalized size of antiderivative = 11.59 \[ \int \frac {A+B \cosh (x)}{a+b \cosh (x)-b \sinh (x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Piecewise((zoo*(A*x + B*sinh(x)), Eq(a, 0) & Eq(b, 0)), (2*A*x*tanh(x/2)/(2*b*tanh(x/2) - 2*b) - 2*A*x/(2*b*ta
nh(x/2) - 2*b) - 2*A*log(tanh(x/2) + 1)*tanh(x/2)/(2*b*tanh(x/2) - 2*b) + 2*A*log(tanh(x/2) + 1)/(2*b*tanh(x/2
) - 2*b) - B*x*tanh(x/2)/(2*b*tanh(x/2) - 2*b) + B*x/(2*b*tanh(x/2) - 2*b) + 2*B*log(tanh(x/2) + 1)*tanh(x/2)/
(2*b*tanh(x/2) - 2*b) - 2*B*log(tanh(x/2) + 1)/(2*b*tanh(x/2) - 2*b) - 2*B/(2*b*tanh(x/2) - 2*b), Eq(a, b)), (
2*A/(-2*b*sinh(x) + 2*b*cosh(x)) - B*x*sinh(x)/(-2*b*sinh(x) + 2*b*cosh(x)) + B*x*cosh(x)/(-2*b*sinh(x) + 2*b*
cosh(x)) + B*cosh(x)/(-2*b*sinh(x) + 2*b*cosh(x)), Eq(a, 0)), ((A*x + B*sinh(x))/a, Eq(b, 0)), (2*A*a*b*x*tanh
(x/2)/(2*a**2*b*tanh(x/2) - 2*a**2*b) - 2*A*a*b*x/(2*a**2*b*tanh(x/2) - 2*a**2*b) - 2*A*a*b*log(tanh(x/2) + 1)
*tanh(x/2)/(2*a**2*b*tanh(x/2) - 2*a**2*b) + 2*A*a*b*log(tanh(x/2) + 1)/(2*a**2*b*tanh(x/2) - 2*a**2*b) + 2*A*
a*b*log(a/(a - b) + b/(a - b) + tanh(x/2))*tanh(x/2)/(2*a**2*b*tanh(x/2) - 2*a**2*b) - 2*A*a*b*log(a/(a - b) +
b/(a - b) + tanh(x/2))/(2*a**2*b*tanh(x/2) - 2*a**2*b) + B*a**2*log(tanh(x/2) + 1)*tanh(x/2)/(2*a**2*b*tanh(x
/2) - 2*a**2*b) - B*a**2*log(tanh(x/2) + 1)/(2*a**2*b*tanh(x/2) - 2*a**2*b) - B*a**2*log(a/(a - b) + b/(a - b)
+ tanh(x/2))*tanh(x/2)/(2*a**2*b*tanh(x/2) - 2*a**2*b) + B*a**2*log(a/(a - b) + b/(a - b) + tanh(x/2))/(2*a**
2*b*tanh(x/2) - 2*a**2*b) - 2*B*a*b/(2*a**2*b*tanh(x/2) - 2*a**2*b) - B*b**2*x*tanh(x/2)/(2*a**2*b*tanh(x/2) -
2*a**2*b) + B*b**2*x/(2*a**2*b*tanh(x/2) - 2*a**2*b) + B*b**2*log(tanh(x/2) + 1)*tanh(x/2)/(2*a**2*b*tanh(x/2
) - 2*a**2*b) - B*b**2*log(tanh(x/2) + 1)/(2*a**2*b*tanh(x/2) - 2*a**2*b) - B*b**2*log(a/(a - b) + b/(a - b) +
tanh(x/2))*tanh(x/2)/(2*a**2*b*tanh(x/2) - 2*a**2*b) + B*b**2*log(a/(a - b) + b/(a - b) + tanh(x/2))/(2*a**2*
b*tanh(x/2) - 2*a**2*b), True))

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.79 \[ \int \frac {A+B \cosh (x)}{a+b \cosh (x)-b \sinh (x)} \, dx=A {\left (\frac {x}{a} + \frac {\log \left (b e^{\left (-x\right )} + a\right )}{a}\right )} - \frac {1}{2} \, B {\left (\frac {b x}{a^{2}} - \frac {e^{x}}{a} + \frac {{\left (a^{2} + b^{2}\right )} \log \left (b e^{\left (-x\right )} + a\right )}{a^{2} b}\right )} \]

[In]

integrate((A+B*cosh(x))/(a+b*cosh(x)-b*sinh(x)),x, algorithm="maxima")

[Out]

A*(x/a + log(b*e^(-x) + a)/a) - 1/2*B*(b*x/a^2 - e^x/a + (a^2 + b^2)*log(b*e^(-x) + a)/(a^2*b))

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.62 \[ \int \frac {A+B \cosh (x)}{a+b \cosh (x)-b \sinh (x)} \, dx=\frac {B x}{2 \, b} + \frac {B e^{x}}{2 \, a} - \frac {{\left (B a^{2} - 2 \, A a b + B b^{2}\right )} \log \left ({\left | a e^{x} + b \right |}\right )}{2 \, a^{2} b} \]

[In]

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

[Out]

1/2*B*x/b + 1/2*B*e^x/a - 1/2*(B*a^2 - 2*A*a*b + B*b^2)*log(abs(a*e^x + b))/(a^2*b)

Mupad [B] (verification not implemented)

Time = 2.30 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.60 \[ \int \frac {A+B \cosh (x)}{a+b \cosh (x)-b \sinh (x)} \, dx=\frac {B\,{\mathrm {e}}^x}{2\,a}+\frac {B\,x}{2\,b}-\frac {\ln \left (b+a\,{\mathrm {e}}^x\right )\,\left (B\,a^2-2\,A\,a\,b+B\,b^2\right )}{2\,a^2\,b} \]

[In]

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

[Out]

(B*exp(x))/(2*a) + (B*x)/(2*b) - (log(b + a*exp(x))*(B*a^2 + B*b^2 - 2*A*a*b))/(2*a^2*b)