3.9.2 \(\int \frac {A+C \sinh (x)}{a+b \cosh (x)+b \sinh (x)} \, dx\) [802]

Optimal. Leaf size=71 \[ \frac {(2 a A+b C) x}{2 a^2}+\frac {C \cosh (x)}{2 a}-\frac {1}{2} \left (\frac {2 A}{a}-\frac {C}{b}+\frac {b C}{a^2}\right ) \log (a+b \cosh (x)+b \sinh (x))-\frac {C \sinh (x)}{2 a} \]

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 71, 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 = {3210} \begin {gather*} \frac {x (2 a A+b C)}{2 a^2}-\frac {1}{2} \left (\frac {b C}{a^2}+\frac {2 A}{a}-\frac {C}{b}\right ) \log (a+b \sinh (x)+b \cosh (x))-\frac {C \sinh (x)}{2 a}+\frac {C \cosh (x)}{2 a} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 3210

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

Rubi steps

\begin {align*} \int \frac {A+C \sinh (x)}{a+b \cosh (x)+b \sinh (x)} \, dx &=\frac {(2 a A+b C) x}{2 a^2}+\frac {C \cosh (x)}{2 a}-\frac {1}{2} \left (\frac {2 A}{a}-\frac {C}{b}+\frac {b C}{a^2}\right ) \log (a+b \cosh (x)+b \sinh (x))-\frac {C \sinh (x)}{2 a}\\ \end {align*}

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Mathematica [A]
time = 0.15, size = 86, normalized size = 1.21 \begin {gather*} \frac {\left (2 a A b+a^2 C+b^2 C\right ) x+2 a b C \cosh (x)+2 \left (-2 a A b+a^2 C-b^2 C\right ) \log \left ((a+b) \cosh \left (\frac {x}{2}\right )+(-a+b) \sinh \left (\frac {x}{2}\right )\right )-2 a b C \sinh (x)}{4 a^2 b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 1.09, size = 94, normalized size = 1.32

method	result	size
risch	\(\frac {C \,{\mathrm e}^{-x}}{2 a}+\frac {x A}{a}+\frac {b x C}{2 a^{2}}-\frac {\ln \left ({\mathrm e}^{x}+\frac {a}{b}\right ) A}{a}+\frac {\ln \left ({\mathrm e}^{x}+\frac {a}{b}\right ) C}{2 b}-\frac {b \ln \left ({\mathrm e}^{x}+\frac {a}{b}\right ) C}{2 a^{2}}\)	\(72\)
default	\(-\frac {\left (2 A a b -a^{2} C +C \,b^{2}\right ) \ln \left (a \tanh \left (\frac {x}{2}\right )-b \tanh \left (\frac {x}{2}\right )-a -b \right )}{2 a^{2} b}+\frac {C}{a \left (\tanh \left (\frac {x}{2}\right )+1\right )}+\frac {\left (2 A a +b C \right ) \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{2 a^{2}}-\frac {C \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{2 b}\)	\(94\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(2*A*a*b-C*a^2+C*b^2)/a^2/b*ln(a*tanh(1/2*x)-b*tanh(1/2*x)-a-b)+C/a/(tanh(1/2*x)+1)+1/2*(2*A*a+C*b)/a^2*l
n(tanh(1/2*x)+1)-1/2*C/b*ln(tanh(1/2*x)-1)

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Maxima [A]
time = 0.28, size = 58, normalized size = 0.82 \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 {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+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)) - A*log(a*e^(-x) + b)/a

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Fricas [A]
time = 0.38, size = 107, normalized size = 1.51 \begin {gather*} \frac {C a b + {\left (2 \, A a b + C b^{2}\right )} x \cosh \left (x\right ) + {\left (2 \, A a b + C b^{2}\right )} x \sinh \left (x\right ) + {\left ({\left (C a^{2} - 2 \, A a b - C b^{2}\right )} \cosh \left (x\right ) + {\left (C a^{2} - 2 \, A a b - C 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+C*sinh(x))/(a+b*cosh(x)+b*sinh(x)),x, algorithm="fricas")

[Out]

1/2*(C*a*b + (2*A*a*b + C*b^2)*x*cosh(x) + (2*A*a*b + C*b^2)*x*sinh(x) + ((C*a^2 - 2*A*a*b - C*b^2)*cosh(x) +
(C*a^2 - 2*A*a*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. 753 vs. \(2 (66) = 132\).
time = 2.65, size = 753, normalized size = 10.61 \begin {gather*} \begin {cases} \tilde {\infty } \left (A x + C \cosh {\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 {C x \tanh {\left (\frac {x}{2} \right )}}{2 b \tanh {\left (\frac {x}{2} \right )} + 2 b} + \frac {C x}{2 b \tanh {\left (\frac {x}{2} \right )} + 2 b} + \frac {2 C}{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 {C x \sinh {\left (x \right )}}{2 b \sinh {\left (x \right )} + 2 b \cosh {\left (x \right )}} + \frac {C x \cosh {\left (x \right )}}{2 b \sinh {\left (x \right )} + 2 b \cosh {\left (x \right )}} + \frac {C \cosh {\left (x \right )}}{2 b \sinh {\left (x \right )} + 2 b \cosh {\left (x \right )}} & \text {for}\: a = 0 \\\frac {A x + C \cosh {\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 {C a^{2} x \tanh {\left (\frac {x}{2} \right )}}{2 a^{2} b \tanh {\left (\frac {x}{2} \right )} + 2 a^{2} b} + \frac {C a^{2} x}{2 a^{2} b \tanh {\left (\frac {x}{2} \right )} + 2 a^{2} b} - \frac {C 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 {C 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 {C 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 {C 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 C a b}{2 a^{2} b \tanh {\left (\frac {x}{2} \right )} + 2 a^{2} b} + \frac {C 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 {C 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 {C 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 {C 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+C*sinh(x))/(a+b*cosh(x)+b*sinh(x)),x)

[Out]

Piecewise((zoo*(A*x + C*cosh(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) + 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)) + 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
+ 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*lo
g(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) + 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*lo
g(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*l
og(-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 = 58, normalized size = 0.82 \begin {gather*} \frac {C e^{\left (-x\right )}}{2 \, a} + \frac {{\left (2 \, A a + C b\right )} x}{2 \, a^{2}} + \frac {{\left (C a^{2} - 2 \, A a b - 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+C*sinh(x))/(a+b*cosh(x)+b*sinh(x)),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 1.66, size = 57, normalized size = 0.80 \begin {gather*} \frac {C\,{\mathrm {e}}^{-x}}{2\,a}+\frac {x\,\left (2\,A\,a+C\,b\right )}{2\,a^2}-\frac {\ln \left (a+b\,{\mathrm {e}}^x\right )\,\left (-C\,a^2+2\,A\,a\,b+C\,b^2\right )}{2\,a^2\,b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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