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

Optimal. Leaf size=78 \[ \frac {(2 a A-b B) x}{2 a^2}+\frac {B \cosh (x)}{2 a}+\frac {\left (2 a A b-a^2 B-b^2 B\right ) \log (a+b \cosh (x)-b \sinh (x))}{2 a^2 b}+\frac {B \sinh (x)}{2 a} \]

[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]
time = 0.03, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {3211} \begin {gather*} \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} \end {gather*}

Antiderivative was successfully verified.

[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*} \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}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.11, size = 86, normalized size = 1.10 \begin {gather*} \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} \end {gather*}

Antiderivative was successfully verified.

[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]
time = 1.10, size = 92, normalized size = 1.18

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	\(\frac {\left (2 A a b -B \,a^{2}-B \,b^{2}\right ) \ln \left (a \tanh \left (\frac {x}{2}\right )-b \tanh \left (\frac {x}{2}\right )+a +b \right )}{2 a^{2} b}+\frac {B \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{2 b}-\frac {B}{a \left (\tanh \left (\frac {x}{2}\right )-1\right )}+\frac {\left (-2 A a +B b \right ) \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{2 a^{2}}\)	\(92\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.27, size = 62, normalized size = 0.79 \begin {gather*} 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 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

________________________________________________________________________________________

Fricas [A]
time = 0.38, size = 56, normalized size = 0.72 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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] Leaf count of result is larger than twice the leaf count of optimal. 904 vs. \(2 (66) = 132\).
time = 2.68, size = 904, normalized size = 11.59 \begin {gather*} \begin {cases} \tilde {\infty } \left (A x + B \sinh {\left (x \right )}\right ) & \text {for}\: a = 0 \wedge b = 0 \\\frac {2 A x \tanh {\left (\frac {x}{2} \right )}}{2 b \tanh {\left (\frac {x}{2} \right )} - 2 b} - \frac {2 A x}{2 b \tanh {\left (\frac {x}{2} \right )} - 2 b} - \frac {2 A \log {\left (\tanh {\left (\frac {x}{2} \right )} + 1 \right )} \tanh {\left (\frac {x}{2} \right )}}{2 b \tanh {\left (\frac {x}{2} \right )} - 2 b} + \frac {2 A \log {\left (\tanh {\left (\frac {x}{2} \right )} + 1 \right )}}{2 b \tanh {\left (\frac {x}{2} \right )} - 2 b} - \frac {B x \tanh {\left (\frac {x}{2} \right )}}{2 b \tanh {\left (\frac {x}{2} \right )} - 2 b} + \frac {B x}{2 b \tanh {\left (\frac {x}{2} \right )} - 2 b} + \frac {2 B \log {\left (\tanh {\left (\frac {x}{2} \right )} + 1 \right )} \tanh {\left (\frac {x}{2} \right )}}{2 b \tanh {\left (\frac {x}{2} \right )} - 2 b} - \frac {2 B \log {\left (\tanh {\left (\frac {x}{2} \right )} + 1 \right )}}{2 b \tanh {\left (\frac {x}{2} \right )} - 2 b} - \frac {2 B}{2 b \tanh {\left (\frac {x}{2} \right )} - 2 b} & \text {for}\: a = b \\\frac {2 A}{- 2 b \sinh {\left (x \right )} + 2 b \cosh {\left (x \right )}} - \frac {B x \sinh {\left (x \right )}}{- 2 b \sinh {\left (x \right )} + 2 b \cosh {\left (x \right )}} + \frac {B x \cosh {\left (x \right )}}{- 2 b \sinh {\left (x \right )} + 2 b \cosh {\left (x \right )}} + \frac {B \cosh {\left (x \right )}}{- 2 b \sinh {\left (x \right )} + 2 b \cosh {\left (x \right )}} & \text {for}\: a = 0 \\\frac {A x + B \sinh {\left (x \right )}}{a} & \text {for}\: b = 0 \\\frac {2 A a b x \tanh {\left (\frac {x}{2} \right )}}{2 a^{2} b \tanh {\left (\frac {x}{2} \right )} - 2 a^{2} b} - \frac {2 A a b x}{2 a^{2} b \tanh {\left (\frac {x}{2} \right )} - 2 a^{2} b} - \frac {2 A a b \log {\left (\tanh {\left (\frac {x}{2} \right )} + 1 \right )} \tanh {\left (\frac {x}{2} \right )}}{2 a^{2} b \tanh {\left (\frac {x}{2} \right )} - 2 a^{2} b} + \frac {2 A a b \log {\left (\tanh {\left (\frac {x}{2} \right )} + 1 \right )}}{2 a^{2} b \tanh {\left (\frac {x}{2} \right )} - 2 a^{2} b} + \frac {2 A a b \log {\left (\frac {a}{a - b} + \frac {b}{a - b} + \tanh {\left (\frac {x}{2} \right )} \right )} \tanh {\left (\frac {x}{2} \right )}}{2 a^{2} b \tanh {\left (\frac {x}{2} \right )} - 2 a^{2} b} - \frac {2 A a b \log {\left (\frac {a}{a - b} + \frac {b}{a - b} + \tanh {\left (\frac {x}{2} \right )} \right )}}{2 a^{2} b \tanh {\left (\frac {x}{2} \right )} - 2 a^{2} b} + \frac {B a^{2} \log {\left (\tanh {\left (\frac {x}{2} \right )} + 1 \right )} \tanh {\left (\frac {x}{2} \right )}}{2 a^{2} b \tanh {\left (\frac {x}{2} \right )} - 2 a^{2} b} - \frac {B a^{2} \log {\left (\tanh {\left (\frac {x}{2} \right )} + 1 \right )}}{2 a^{2} b \tanh {\left (\frac {x}{2} \right )} - 2 a^{2} b} - \frac {B a^{2} \log {\left (\frac {a}{a - b} + \frac {b}{a - b} + \tanh {\left (\frac {x}{2} \right )} \right )} \tanh {\left (\frac {x}{2} \right )}}{2 a^{2} b \tanh {\left (\frac {x}{2} \right )} - 2 a^{2} b} + \frac {B a^{2} \log {\left (\frac {a}{a - b} + \frac {b}{a - b} + \tanh {\left (\frac {x}{2} \right )} \right )}}{2 a^{2} b \tanh {\left (\frac {x}{2} \right )} - 2 a^{2} b} - \frac {2 B a b}{2 a^{2} b \tanh {\left (\frac {x}{2} \right )} - 2 a^{2} b} - \frac {B b^{2} x \tanh {\left (\frac {x}{2} \right )}}{2 a^{2} b \tanh {\left (\frac {x}{2} \right )} - 2 a^{2} b} + \frac {B b^{2} x}{2 a^{2} b \tanh {\left (\frac {x}{2} \right )} - 2 a^{2} b} + \frac {B b^{2} \log {\left (\tanh {\left (\frac {x}{2} \right )} + 1 \right )} \tanh {\left (\frac {x}{2} \right )}}{2 a^{2} b \tanh {\left (\frac {x}{2} \right )} - 2 a^{2} b} - \frac {B b^{2} \log {\left (\tanh {\left (\frac {x}{2} \right )} + 1 \right )}}{2 a^{2} b \tanh {\left (\frac {x}{2} \right )} - 2 a^{2} b} - \frac {B b^{2} \log {\left (\frac {a}{a - b} + \frac {b}{a - b} + \tanh {\left (\frac {x}{2} \right )} \right )} \tanh {\left (\frac {x}{2} \right )}}{2 a^{2} b \tanh {\left (\frac {x}{2} \right )} - 2 a^{2} b} + \frac {B b^{2} \log {\left (\frac {a}{a - b} + \frac {b}{a - b} + \tanh {\left (\frac {x}{2} \right )} \right )}}{2 a^{2} b \tanh {\left (\frac {x}{2} \right )} - 2 a^{2} b} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

________________________________________________________________________________________

Giac [A]
time = 0.40, size = 48, normalized size = 0.62 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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]
time = 1.59, size = 47, normalized size = 0.60 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________