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

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

[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 = 0.06, 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 = {3131} \[ \frac {x (2 a A+b C)}{2 a^2}-\frac {1}{2} \left (\frac {b C}{a^2}+\frac {2 A}{a}-\frac {C}{b}\right ) \log (a+b \sinh (x)+b \cosh (x))-\frac {C \sinh (x)}{2 a}+\frac {C \cosh (x)}{2 a} \]

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 3131

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 = 0.24, size = 86, normalized size = 1.21 \[ \frac {x \left (a^2 C+2 a A b+b^2 C\right )+2 \left (a^2 C-2 a A b-b^2 C\right ) \log \left ((b-a) \sinh \left (\frac {x}{2}\right )+(a+b) \cosh \left (\frac {x}{2}\right )\right )-2 a b C \sinh (x)+2 a b C \cosh (x)}{4 a^2 b} \]

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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fricas [A]  time = 0.45, size = 107, normalized size = 1.51 \[ \frac {C a b + {\left (2 \, A a b + C b^{2}\right )} x \cosh \relax (x) + {\left (2 \, A a b + C b^{2}\right )} x \sinh \relax (x) + {\left ({\left (C a^{2} - 2 \, A a b - C b^{2}\right )} \cosh \relax (x) + {\left (C a^{2} - 2 \, A a b - C b^{2}\right )} \sinh \relax (x)\right )} \log \left (b \cosh \relax (x) + b \sinh \relax (x) + a\right )}{2 \, {\left (a^{2} b \cosh \relax (x) + a^{2} b \sinh \relax (x)\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+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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giac [A]  time = 0.14, size = 58, normalized size = 0.82 \[ \frac {C e^{\left (-x\right )}}{2 \, a} + \frac {{\left (2 \, A a + C b\right )} x}{2 \, a^{2}} + \frac {{\left (C a^{2} - 2 \, A a b - C b^{2}\right )} \log \left ({\left | b e^{x} + a \right |}\right )}{2 \, a^{2} b} \]

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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maple [B]  time = 0.19, size = 136, normalized size = 1.92 \[ -\frac {C \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{2 b}-\frac {\ln \left (a \tanh \left (\frac {x}{2}\right )-\tanh \left (\frac {x}{2}\right ) b -a -b \right ) A}{a}+\frac {\ln \left (a \tanh \left (\frac {x}{2}\right )-\tanh \left (\frac {x}{2}\right ) b -a -b \right ) C}{2 b}-\frac {b \ln \left (a \tanh \left (\frac {x}{2}\right )-\tanh \left (\frac {x}{2}\right ) b -a -b \right ) C}{2 a^{2}}+\frac {C}{a \left (\tanh \left (\frac {x}{2}\right )+1\right )}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right ) A}{a}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right ) b C}{2 a^{2}} \]

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]

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

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maxima [A]  time = 0.32, size = 58, normalized size = 0.82 \[ \frac {1}{2} \, C {\left (\frac {x}{b} + \frac {e^{\left (-x\right )}}{a} + \frac {{\left (a^{2} - b^{2}\right )} \log \left (a e^{\left (-x\right )} + b\right )}{a^{2} b}\right )} - \frac {A \log \left (a e^{\left (-x\right )} + b\right )}{a} \]

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

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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sympy [A]  time = 5.11, size = 753, normalized size = 10.61 \[ \text {result too large to display} \]

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/(2*b*sinh(x) + 2*b*cosh(x)) + C*x*sinh(x)/(2*b*s
inh(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*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*t
anh(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*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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