3.9.4 \(\int \frac {A+B \cosh (x)+C \sinh (x)}{a+b \cosh (x)+b \sinh (x)} \, dx\) [804]

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

[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 = 0.05, 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 = {3209} \begin {gather*} -\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} \end {gather*}

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 3209

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 = 0.21, size = 103, normalized size = 1.20 \begin {gather*} \frac {\left (2 a A+b (-B+C)+\frac {a^2 (B+C)}{b}\right ) x-2 a (B-C) \cosh (x)+\frac {2 \left (-2 a A b+b^2 (B-C)+a^2 (B+C)\right ) \log \left ((a+b) \cosh \left (\frac {x}{2}\right )+(-a+b) \sinh \left (\frac {x}{2}\right )\right )}{b}+2 a (B-C) \sinh (x)}{4 a^2} \end {gather*}

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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Maple [A]
time = 1.06, size = 121, normalized size = 1.41

method	result	size
default	\(\frac {\left (-B -C \right ) \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{2 b}-\frac {B -C}{a \left (\tanh \left (\frac {x}{2}\right )+1\right )}+\frac {\left (2 A a -B b +b C \right ) \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{2 a^{2}}-\frac {\left (2 A a b -B \,a^{2}-B \,b^{2}-a^{2} C +C \,b^{2}\right ) \ln \left (a \tanh \left (\frac {x}{2}\right )-b \tanh \left (\frac {x}{2}\right )-a -b \right )}{2 a^{2} b}\)	\(121\)
risch	\(-\frac {B \,{\mathrm e}^{-x}}{2 a}+\frac {C \,{\mathrm e}^{-x}}{2 a}+\frac {x A}{a}-\frac {b x B}{2 a^{2}}+\frac {b x C}{2 a^{2}}-\frac {\ln \left ({\mathrm e}^{x}+\frac {a}{b}\right ) A}{a}+\frac {\ln \left ({\mathrm e}^{x}+\frac {a}{b}\right ) B}{2 b}+\frac {b \ln \left ({\mathrm e}^{x}+\frac {a}{b}\right ) B}{2 a^{2}}+\frac {\ln \left ({\mathrm e}^{x}+\frac {a}{b}\right ) C}{2 b}-\frac {b \ln \left ({\mathrm e}^{x}+\frac {a}{b}\right ) C}{2 a^{2}}\)	\(121\)

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,method=_RETURNVERBOSE)

[Out]

1/2*(-B-C)/b*ln(tanh(1/2*x)-1)-(B-C)/a/(tanh(1/2*x)+1)+1/2*(2*A*a-B*b+C*b)/a^2*ln(tanh(1/2*x)+1)-1/2*(2*A*a*b-
B*a^2-B*b^2-C*a^2+C*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 = 99, normalized size = 1.15 \begin {gather*} \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} \end {gather*}

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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Fricas [A]
time = 0.41, size = 134, normalized size = 1.56 \begin {gather*} -\frac {{\left (B - C\right )} a b - {\left (2 \, A a b - {\left (B - C\right )} b^{2}\right )} x \cosh \left (x\right ) - {\left (2 \, A a b - {\left (B - C\right )} b^{2}\right )} x \sinh \left (x\right ) - {\left ({\left ({\left (B + C\right )} a^{2} - 2 \, A a b + {\left (B - C\right )} b^{2}\right )} \cosh \left (x\right ) + {\left ({\left (B + C\right )} a^{2} - 2 \, A a b + {\left (B - C\right )} 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)+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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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1321 vs. \(2 (70) = 140\).
time = 2.93, size = 1321, normalized size = 15.36 \begin {gather*} \text {Too large to display} \end {gather*}

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*log(tanh(x/2) + 1)*tanh(x/2)/(2*b*tan
h(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/(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)) + 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*sinh(x) + 2*b*cosh(x)), Eq(a, 0)), (
(A*x + B*sinh(x) + C*cosh(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*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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Giac [A]
time = 0.41, size = 79, normalized size = 0.92 \begin {gather*} \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} \end {gather*}

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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Mupad [B]
time = 1.72, size = 75, normalized size = 0.87 \begin {gather*} \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} \end {gather*}

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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