3.9.3 \(\int \frac {A+B \cosh (x)}{a+b \cosh (x)+b \sinh (x)} \, dx\) [803]

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

[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

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Rubi [A]
time = 0.03, 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 = {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*}

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Mathematica [A]
time = 0.13, size = 84, normalized size = 1.09 \begin {gather*} \frac {\left (2 a A+\frac {a^2 B}{b}-b B\right ) x-2 a B \cosh (x)+\frac {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 )}{b}+2 a B \sinh (x)}{4 a^2} \end {gather*}

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)

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Maple [A]
time = 1.08, size = 97, normalized size = 1.26

method	result	size
risch	\(-\frac {B \,{\mathrm e}^{-x}}{2 a}+\frac {x A}{a}-\frac {b x B}{2 a^{2}}-\frac {\ln \left ({\mathrm e}^{x}+\frac {a}{b}\right ) A}{a}+\frac {\ln \left ({\mathrm e}^{x}+\frac {a}{b}\right ) B}{2 b}+\frac {b \ln \left ({\mathrm e}^{x}+\frac {a}{b}\right ) B}{2 a^{2}}\)	\(72\)
default	\(-\frac {B}{a \left (\tanh \left (\frac {x}{2}\right )+1\right )}+\frac {\left (2 A a -B b \right ) \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{2 a^{2}}-\frac {B \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{2 b}-\frac {\left (2 A a b -B \,a^{2}-B \,b^{2}\right ) \ln \left (a \tanh \left (\frac {x}{2}\right )-b \tanh \left (\frac {x}{2}\right )-a -b \right )}{2 a^{2} b}\)	\(97\)

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]

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

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Maxima [A]
time = 0.26, size = 57, normalized size = 0.74 \begin {gather*} \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} \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]

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

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Fricas [A]
time = 0.40, size = 110, normalized size = 1.43 \begin {gather*} -\frac {B a b - {\left (2 \, A a b - B b^{2}\right )} x \cosh \left (x\right ) - {\left (2 \, A a b - B b^{2}\right )} x \sinh \left (x\right ) - {\left ({\left (B a^{2} - 2 \, A a b + B b^{2}\right )} \cosh \left (x\right ) + {\left (B a^{2} - 2 \, A a b + B b^{2}\right )} \sinh \left (x\right )\right )} \log \left (b \cosh \left (x\right ) + b \sinh \left (x\right ) + a\right )}{2 \, {\left (a^{2} b \cosh \left (x\right ) + a^{2} b \sinh \left (x\right )\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="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))

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 806 vs. \(2 (66) = 132\).
time = 2.86, size = 806, normalized size = 10.47 \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 \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 \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} x \tanh {\left (\frac {x}{2} \right )}}{2 a^{2} b \tanh {\left (\frac {x}{2} \right )} + 2 a^{2} b} + \frac {B a^{2} x}{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} \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*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*s
inh(x))/a, Eq(b, 0)), (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))

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Giac [A]
time = 0.41, size = 58, normalized size = 0.75 \begin {gather*} -\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} \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*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)

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Mupad [B]
time = 1.69, size = 57, normalized size = 0.74 \begin {gather*} \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} \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]

(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)

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