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

Optimal. Leaf size=86 \[ -\frac {\left (-\left (a^2 (B+C)\right )+2 a A b-b^2 (B-C)\right ) \log (a+b \sinh (x)+b \cosh (x))}{2 a^2 b}+\frac {x (2 a A-b (B-C))}{2 a^2}-\frac {(B-C) (\cosh (x)-\sinh (x))}{2 a} \]

[Out]

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

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Rubi [A]  time = 0.08, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {3130} \[ -\frac {\left (a^2 (-(B+C))+2 a A b-b^2 (B-C)\right ) \log (a+b \sinh (x)+b \cosh (x))}{2 a^2 b}+\frac {x (2 a A-b (B-C))}{2 a^2}-\frac {(B-C) (\cosh (x)-\sinh (x))}{2 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 3130

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

Rubi steps

\begin {align*} \int \frac {A+B \cosh (x)+C \sinh (x)}{a+b \cosh (x)+b \sinh (x)} \, dx &=\frac {(2 a A-b (B-C)) x}{2 a^2}-\frac {\left (2 a A b-b^2 (B-C)-a^2 (B+C)\right ) \log (a+b \cosh (x)+b \sinh (x))}{2 a^2 b}-\frac {(B-C) (\cosh (x)-\sinh (x))}{2 a}\\ \end {align*}

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Mathematica [A]  time = 0.31, size = 103, normalized size = 1.20 \[ \frac {\frac {2 \left (a^2 (B+C)-2 a A b+b^2 (B-C)\right ) \log \left ((b-a) \sinh \left (\frac {x}{2}\right )+(a+b) \cosh \left (\frac {x}{2}\right )\right )}{b}+x \left (\frac {a^2 (B+C)}{b}+2 a A+b (C-B)\right )+2 a (B-C) \sinh (x)-2 a (B-C) \cosh (x)}{4 a^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.43, size = 134, normalized size = 1.56 \[ -\frac {{\left (B - C\right )} a b - {\left (2 \, A a b - {\left (B - C\right )} b^{2}\right )} x \cosh \relax (x) - {\left (2 \, A a b - {\left (B - C\right )} b^{2}\right )} x \sinh \relax (x) - {\left ({\left ({\left (B + C\right )} a^{2} - 2 \, A a b + {\left (B - C\right )} b^{2}\right )} \cosh \relax (x) + {\left ({\left (B + C\right )} a^{2} - 2 \, A a b + {\left (B - C\right )} b^{2}\right )} \sinh \relax (x)\right )} \log \left (b \cosh \relax (x) + b \sinh \relax (x) + a\right )}{2 \, {\left (a^{2} b \cosh \relax (x) + a^{2} b \sinh \relax (x)\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*((B - C)*a*b - (2*A*a*b - (B - C)*b^2)*x*cosh(x) - (2*A*a*b - (B - C)*b^2)*x*sinh(x) - (((B + C)*a^2 - 2*
A*a*b + (B - C)*b^2)*cosh(x) + ((B + C)*a^2 - 2*A*a*b + (B - C)*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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giac [A]  time = 0.12, size = 79, normalized size = 0.92 \[ \frac {{\left (2 \, A a - B b + C b\right )} x}{2 \, a^{2}} - \frac {{\left (B a - C a\right )} e^{\left (-x\right )}}{2 \, a^{2}} + \frac {{\left (B a^{2} + C a^{2} - 2 \, A a b + B b^{2} - C b^{2}\right )} \log \left ({\left | b e^{x} + a \right |}\right )}{2 \, a^{2} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.19, size = 232, normalized size = 2.70 \[ -\frac {B \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{2 b}-\frac {C \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{2 b}-\frac {\ln \left (a \tanh \left (\frac {x}{2}\right )-\tanh \left (\frac {x}{2}\right ) b -a -b \right ) A}{a}+\frac {\ln \left (a \tanh \left (\frac {x}{2}\right )-\tanh \left (\frac {x}{2}\right ) b -a -b \right ) B}{2 b}+\frac {b \ln \left (a \tanh \left (\frac {x}{2}\right )-\tanh \left (\frac {x}{2}\right ) b -a -b \right ) B}{2 a^{2}}+\frac {\ln \left (a \tanh \left (\frac {x}{2}\right )-\tanh \left (\frac {x}{2}\right ) b -a -b \right ) C}{2 b}-\frac {b \ln \left (a \tanh \left (\frac {x}{2}\right )-\tanh \left (\frac {x}{2}\right ) b -a -b \right ) C}{2 a^{2}}-\frac {B}{a \left (\tanh \left (\frac {x}{2}\right )+1\right )}+\frac {C}{a \left (\tanh \left (\frac {x}{2}\right )+1\right )}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right ) A}{a}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right ) B b}{2 a^{2}}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right ) b C}{2 a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.32, size = 99, normalized size = 1.15 \[ \frac {1}{2} \, C {\left (\frac {x}{b} + \frac {e^{\left (-x\right )}}{a} + \frac {{\left (a^{2} - b^{2}\right )} \log \left (a e^{\left (-x\right )} + b\right )}{a^{2} b}\right )} + \frac {1}{2} \, B {\left (\frac {x}{b} - \frac {e^{\left (-x\right )}}{a} + \frac {{\left (a^{2} + b^{2}\right )} \log \left (a e^{\left (-x\right )} + b\right )}{a^{2} b}\right )} - \frac {A \log \left (a e^{\left (-x\right )} + b\right )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*C*(x/b + e^(-x)/a + (a^2 - b^2)*log(a*e^(-x) + b)/(a^2*b)) + 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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mupad [B]  time = 1.72, size = 75, normalized size = 0.87 \[ \frac {x\,\left (2\,A\,a-B\,b+C\,b\right )}{2\,a^2}-\frac {{\mathrm {e}}^{-x}\,\left (B-C\right )}{2\,a}+\frac {\ln \left (a+b\,{\mathrm {e}}^x\right )\,\left (B\,a^2+B\,b^2+C\,a^2-C\,b^2-2\,A\,a\,b\right )}{2\,a^2\,b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 5.94, size = 1321, normalized size = 15.36 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((zoo*(A*x + B*sinh(x) + C*cosh(x)), Eq(a, 0) & Eq(b, 0)), (-2*A/(2*b*sinh(x) + 2*b*cosh(x)) + B*x*si
nh(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*cos
h(x)) + C*x*sinh(x)/(2*b*sinh(x) + 2*b*cosh(x)) + C*x*cosh(x)/(2*b*sinh(x) + 2*b*cosh(x)) + C*cosh(x)/(2*b*sin
h(x) + 2*b*cosh(x)), Eq(a, 0)), ((A*x + B*sinh(x) + C*cosh(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(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) + C*x*tanh(x/2)/(2*b*tanh(x/2) + 2*b) + C*x/(2*b*tanh(x/2) + 2
*b) + 2*C/(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*tan
h(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*tan
h(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*ta
nh(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) + C*a**2*
x*tanh(x/2)/(2*a**2*b*tanh(x/2) + 2*a**2*b) + C*a**2*x/(2*a**2*b*tanh(x/2) + 2*a**2*b) - C*a**2*log(tanh(x/2)
+ 1)*tanh(x/2)/(2*a**2*b*tanh(x/2) + 2*a**2*b) - C*a**2*log(tanh(x/2) + 1)/(2*a**2*b*tanh(x/2) + 2*a**2*b) + C
*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) + C*a**2*log(-a/(a - b
) - b/(a - b) + tanh(x/2))/(2*a**2*b*tanh(x/2) + 2*a**2*b) + 2*C*a*b/(2*a**2*b*tanh(x/2) + 2*a**2*b) + C*b**2*
log(tanh(x/2) + 1)*tanh(x/2)/(2*a**2*b*tanh(x/2) + 2*a**2*b) + C*b**2*log(tanh(x/2) + 1)/(2*a**2*b*tanh(x/2) +
2*a**2*b) - C*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) - C*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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