3.803 \(\int \frac {A+B \cosh (x)}{a+b \cosh (x)+b \sinh (x)} \, dx\)

Optimal. Leaf size=77 \[ -\frac {\left (a^2 (-B)+2 a A b-b^2 B\right ) \log (a+b \sinh (x)+b \cosh (x))}{2 a^2 b}+\frac {x (2 a A-b B)}{2 a^2}+\frac {B \sinh (x)}{2 a}-\frac {B \cosh (x)}{2 a} \]

[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 = 0.05, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {3132} \[ -\frac {\left (a^2 (-B)+2 a A b-b^2 B\right ) \log (a+b \sinh (x)+b \cosh (x))}{2 a^2 b}+\frac {x (2 a A-b B)}{2 a^2}+\frac {B \sinh (x)}{2 a}-\frac {B \cosh (x)}{2 a} \]

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 3132

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 = 0.19, size = 84, normalized size = 1.09 \[ \frac {\frac {2 \left (a^2 B-2 a A b+b^2 B\right ) \log \left ((b-a) \sinh \left (\frac {x}{2}\right )+(a+b) \cosh \left (\frac {x}{2}\right )\right )}{b}+x \left (\frac {a^2 B}{b}+2 a A-b B\right )+2 a B \sinh (x)-2 a B \cosh (x)}{4 a^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.44, size = 110, normalized size = 1.43 \[ -\frac {B a b - {\left (2 \, A a b - B b^{2}\right )} x \cosh \relax (x) - {\left (2 \, A a b - B b^{2}\right )} x \sinh \relax (x) - {\left ({\left (B a^{2} - 2 \, A a b + B b^{2}\right )} \cosh \relax (x) + {\left (B a^{2} - 2 \, A a b + B b^{2}\right )} \sinh \relax (x)\right )} \log \left (b \cosh \relax (x) + b \sinh \relax (x) + a\right )}{2 \, {\left (a^{2} b \cosh \relax (x) + a^{2} b \sinh \relax (x)\right )}} \]

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

________________________________________________________________________________________

giac [A]  time = 0.13, size = 58, normalized size = 0.75 \[ -\frac {B e^{\left (-x\right )}}{2 \, a} + \frac {{\left (2 \, A a - B b\right )} x}{2 \, a^{2}} + \frac {{\left (B a^{2} - 2 \, A a b + B b^{2}\right )} \log \left ({\left | b e^{x} + a \right |}\right )}{2 \, a^{2} b} \]

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

________________________________________________________________________________________

maple [A]  time = 0.20, size = 137, normalized size = 1.78 \[ -\frac {B \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{2 b}-\frac {\ln \left (a \tanh \left (\frac {x}{2}\right )-\tanh \left (\frac {x}{2}\right ) b -a -b \right ) A}{a}+\frac {\ln \left (a \tanh \left (\frac {x}{2}\right )-\tanh \left (\frac {x}{2}\right ) b -a -b \right ) B}{2 b}+\frac {b \ln \left (a \tanh \left (\frac {x}{2}\right )-\tanh \left (\frac {x}{2}\right ) b -a -b \right ) B}{2 a^{2}}-\frac {B}{a \left (\tanh \left (\frac {x}{2}\right )+1\right )}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right ) A}{a}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right ) B b}{2 a^{2}} \]

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]

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

________________________________________________________________________________________

maxima [A]  time = 0.44, size = 57, normalized size = 0.74 \[ \frac {1}{2} \, B {\left (\frac {x}{b} - \frac {e^{\left (-x\right )}}{a} + \frac {{\left (a^{2} + b^{2}\right )} \log \left (a e^{\left (-x\right )} + b\right )}{a^{2} b}\right )} - \frac {A \log \left (a e^{\left (-x\right )} + b\right )}{a} \]

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]

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

________________________________________________________________________________________

mupad [B]  time = 1.69, size = 57, normalized size = 0.74 \[ \frac {x\,\left (2\,A\,a-B\,b\right )}{2\,a^2}-\frac {B\,{\mathrm {e}}^{-x}}{2\,a}+\frac {\ln \left (a+b\,{\mathrm {e}}^x\right )\,\left (B\,a^2-2\,A\,a\,b+B\,b^2\right )}{2\,a^2\,b} \]

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]

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

________________________________________________________________________________________

sympy [A]  time = 5.15, size = 806, normalized size = 10.47 \[ \text {result too large to display} \]

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/(2*b*sinh(x) + 2*b*cosh(x)) + B*x*sinh(x)/(2*b*s
inh(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*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(ta
nh(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*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*x*tanh(x/2)/(2*a**2*b*tanh(x/2) + 2*a**2*b) + B*a**2*x/(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*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))

________________________________________________________________________________________