3.727 \(\int \frac{A+B \cosh (x)}{b \cosh (x)+c \sinh (x)} \, dx\)

Optimal. Leaf size=80 \[ \frac{A \tan ^{-1}\left (\frac{b \sinh (x)+c \cosh (x)}{\sqrt{b^2-c^2}}\right )}{\sqrt{b^2-c^2}}+\frac{b B x}{b^2-c^2}-\frac{B c \log (b \cosh (x)+c \sinh (x))}{b^2-c^2} \]

[Out]

(b*B*x)/(b^2 - c^2) + (A*ArcTan[(c*Cosh[x] + b*Sinh[x])/Sqrt[b^2 - c^2]])/Sqrt[b^2 - c^2] - (B*c*Log[b*Cosh[x]
+ c*Sinh[x]])/(b^2 - c^2)

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Rubi [A]  time = 0.0614946, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3138, 3074, 206} \[ \frac{A \tan ^{-1}\left (\frac{b \sinh (x)+c \cosh (x)}{\sqrt{b^2-c^2}}\right )}{\sqrt{b^2-c^2}}+\frac{b B x}{b^2-c^2}-\frac{B c \log (b \cosh (x)+c \sinh (x))}{b^2-c^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*B*x)/(b^2 - c^2) + (A*ArcTan[(c*Cosh[x] + b*Sinh[x])/Sqrt[b^2 - c^2]])/Sqrt[b^2 - c^2] - (B*c*Log[b*Cosh[x]
+ c*Sinh[x]])/(b^2 - c^2)

Rule 3138

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

Rule 3074

Int[(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> -Dist[d^(-1), Subst[Int
[1/(a^2 + b^2 - x^2), x], x, b*Cos[c + d*x] - a*Sin[c + d*x]], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2,
0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{A+B \cosh (x)}{b \cosh (x)+c \sinh (x)} \, dx &=\frac{b B x}{b^2-c^2}-\frac{B c \log (b \cosh (x)+c \sinh (x))}{b^2-c^2}+A \int \frac{1}{b \cosh (x)+c \sinh (x)} \, dx\\ &=\frac{b B x}{b^2-c^2}-\frac{B c \log (b \cosh (x)+c \sinh (x))}{b^2-c^2}+(i A) \operatorname{Subst}\left (\int \frac{1}{b^2-c^2-x^2} \, dx,x,-i c \cosh (x)-i b \sinh (x)\right )\\ &=\frac{b B x}{b^2-c^2}+\frac{A \tan ^{-1}\left (\frac{c \cosh (x)+b \sinh (x)}{\sqrt{b^2-c^2}}\right )}{\sqrt{b^2-c^2}}-\frac{B c \log (b \cosh (x)+c \sinh (x))}{b^2-c^2}\\ \end{align*}

Mathematica [A]  time = 0.170774, size = 78, normalized size = 0.98 \[ \frac{2 A \sqrt{b-c} \sqrt{b+c} \tan ^{-1}\left (\frac{b \tanh \left (\frac{x}{2}\right )+c}{\sqrt{b-c} \sqrt{b+c}}\right )-B c \log (b \cosh (x)+c \sinh (x))+b B x}{b^2-c^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*B*x + 2*A*Sqrt[b - c]*Sqrt[b + c]*ArcTan[(c + b*Tanh[x/2])/(Sqrt[b - c]*Sqrt[b + c])] - B*c*Log[b*Cosh[x] +
c*Sinh[x]])/(b^2 - c^2)

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Maple [B]  time = 0.055, size = 182, normalized size = 2.3 \begin{align*} 2\,{\frac{B\ln \left ( \tanh \left ( x/2 \right ) +1 \right ) }{2\,b-2\,c}}-{\frac{Bc}{ \left ( b-c \right ) \left ( b+c \right ) }\ln \left ( \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}b+2\,c\tanh \left ( x/2 \right ) +b \right ) }+2\,{\frac{A{b}^{2}}{ \left ( b-c \right ) \left ( b+c \right ) \sqrt{{b}^{2}-{c}^{2}}}\arctan \left ( 1/2\,{\frac{2\,\tanh \left ( x/2 \right ) b+2\,c}{\sqrt{{b}^{2}-{c}^{2}}}} \right ) }-2\,{\frac{A{c}^{2}}{ \left ( b-c \right ) \left ( b+c \right ) \sqrt{{b}^{2}-{c}^{2}}}\arctan \left ( 1/2\,{\frac{2\,\tanh \left ( x/2 \right ) b+2\,c}{\sqrt{{b}^{2}-{c}^{2}}}} \right ) }-2\,{\frac{B\ln \left ( \tanh \left ( x/2 \right ) -1 \right ) }{2\,b+2\,c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*B/(2*b-2*c)*ln(tanh(1/2*x)+1)-1/(b-c)/(b+c)*B*c*ln(tanh(1/2*x)^2*b+2*c*tanh(1/2*x)+b)+2/(b-c)/(b+c)/(b^2-c^2
)^(1/2)*arctan(1/2*(2*tanh(1/2*x)*b+2*c)/(b^2-c^2)^(1/2))*A*b^2-2/(b-c)/(b+c)/(b^2-c^2)^(1/2)*arctan(1/2*(2*ta
nh(1/2*x)*b+2*c)/(b^2-c^2)^(1/2))*A*c^2-2*B/(2*b+2*c)*ln(tanh(1/2*x)-1)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.46231, size = 647, normalized size = 8.09 \begin{align*} \left [-\frac{B c \log \left (\frac{2 \,{\left (b \cosh \left (x\right ) + c \sinh \left (x\right )\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) + \sqrt{-b^{2} + c^{2}} A \log \left (\frac{{\left (b + c\right )} \cosh \left (x\right )^{2} + 2 \,{\left (b + c\right )} \cosh \left (x\right ) \sinh \left (x\right ) +{\left (b + c\right )} \sinh \left (x\right )^{2} - 2 \, \sqrt{-b^{2} + c^{2}}{\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} - b + c}{{\left (b + c\right )} \cosh \left (x\right )^{2} + 2 \,{\left (b + c\right )} \cosh \left (x\right ) \sinh \left (x\right ) +{\left (b + c\right )} \sinh \left (x\right )^{2} + b - c}\right ) -{\left (B b + B c\right )} x}{b^{2} - c^{2}}, -\frac{B c \log \left (\frac{2 \,{\left (b \cosh \left (x\right ) + c \sinh \left (x\right )\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) + 2 \, \sqrt{b^{2} - c^{2}} A \arctan \left (\frac{\sqrt{b^{2} - c^{2}}}{{\left (b + c\right )} \cosh \left (x\right ) +{\left (b + c\right )} \sinh \left (x\right )}\right ) -{\left (B b + B c\right )} x}{b^{2} - c^{2}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-(B*c*log(2*(b*cosh(x) + c*sinh(x))/(cosh(x) - sinh(x))) + sqrt(-b^2 + c^2)*A*log(((b + c)*cosh(x)^2 + 2*(b +
c)*cosh(x)*sinh(x) + (b + c)*sinh(x)^2 - 2*sqrt(-b^2 + c^2)*(cosh(x) + sinh(x)) - b + c)/((b + c)*cosh(x)^2 +
2*(b + c)*cosh(x)*sinh(x) + (b + c)*sinh(x)^2 + b - c)) - (B*b + B*c)*x)/(b^2 - c^2), -(B*c*log(2*(b*cosh(x)
+ c*sinh(x))/(cosh(x) - sinh(x))) + 2*sqrt(b^2 - c^2)*A*arctan(sqrt(b^2 - c^2)/((b + c)*cosh(x) + (b + c)*sinh
(x))) - (B*b + B*c)*x)/(b^2 - c^2)]

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Sympy [A]  time = 46.9949, size = 408, normalized size = 5.1 \begin{align*} \begin{cases} \tilde{\infty } \left (A \log{\left (\tanh{\left (\frac{x}{2} \right )} \right )} + B x - 2 B \log{\left (\tanh{\left (\frac{x}{2} \right )} + 1 \right )} + B \log{\left (\tanh{\left (\frac{x}{2} \right )} \right )}\right ) & \text{for}\: b = 0 \wedge c = 0 \\\frac{A \log{\left (\tanh{\left (\frac{x}{2} \right )} \right )} + B x - 2 B \log{\left (\tanh{\left (\frac{x}{2} \right )} + 1 \right )} + B \log{\left (\tanh{\left (\frac{x}{2} \right )} \right )}}{c} & \text{for}\: b = 0 \\- \frac{2 A}{- 2 c \sinh{\left (x \right )} + 2 c \cosh{\left (x \right )}} + \frac{B x \sinh{\left (x \right )}}{- 2 c \sinh{\left (x \right )} + 2 c \cosh{\left (x \right )}} - \frac{B x \cosh{\left (x \right )}}{- 2 c \sinh{\left (x \right )} + 2 c \cosh{\left (x \right )}} - \frac{B \cosh{\left (x \right )}}{- 2 c \sinh{\left (x \right )} + 2 c \cosh{\left (x \right )}} & \text{for}\: b = - c \\- \frac{2 A}{2 c \sinh{\left (x \right )} + 2 c \cosh{\left (x \right )}} + \frac{B x \sinh{\left (x \right )}}{2 c \sinh{\left (x \right )} + 2 c \cosh{\left (x \right )}} + \frac{B x \cosh{\left (x \right )}}{2 c \sinh{\left (x \right )} + 2 c \cosh{\left (x \right )}} - \frac{B \cosh{\left (x \right )}}{2 c \sinh{\left (x \right )} + 2 c \cosh{\left (x \right )}} & \text{for}\: b = c \\- \frac{A \sqrt{- b^{2} + c^{2}} \log{\left (\tanh{\left (\frac{x}{2} \right )} + \frac{c}{b} - \frac{\sqrt{- b^{2} + c^{2}}}{b} \right )}}{b^{2} - c^{2}} + \frac{A \sqrt{- b^{2} + c^{2}} \log{\left (\tanh{\left (\frac{x}{2} \right )} + \frac{c}{b} + \frac{\sqrt{- b^{2} + c^{2}}}{b} \right )}}{b^{2} - c^{2}} + \frac{B b x}{b^{2} - c^{2}} - \frac{B c x}{b^{2} - c^{2}} + \frac{2 B c \log{\left (\tanh{\left (\frac{x}{2} \right )} + 1 \right )}}{b^{2} - c^{2}} - \frac{B c \log{\left (\tanh{\left (\frac{x}{2} \right )} + \frac{c}{b} - \frac{\sqrt{- b^{2} + c^{2}}}{b} \right )}}{b^{2} - c^{2}} - \frac{B c \log{\left (\tanh{\left (\frac{x}{2} \right )} + \frac{c}{b} + \frac{\sqrt{- b^{2} + c^{2}}}{b} \right )}}{b^{2} - c^{2}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((zoo*(A*log(tanh(x/2)) + B*x - 2*B*log(tanh(x/2) + 1) + B*log(tanh(x/2))), Eq(b, 0) & Eq(c, 0)), ((A
*log(tanh(x/2)) + B*x - 2*B*log(tanh(x/2) + 1) + B*log(tanh(x/2)))/c, Eq(b, 0)), (-2*A/(-2*c*sinh(x) + 2*c*cos
h(x)) + B*x*sinh(x)/(-2*c*sinh(x) + 2*c*cosh(x)) - B*x*cosh(x)/(-2*c*sinh(x) + 2*c*cosh(x)) - B*cosh(x)/(-2*c*
sinh(x) + 2*c*cosh(x)), Eq(b, -c)), (-2*A/(2*c*sinh(x) + 2*c*cosh(x)) + B*x*sinh(x)/(2*c*sinh(x) + 2*c*cosh(x)
) + B*x*cosh(x)/(2*c*sinh(x) + 2*c*cosh(x)) - B*cosh(x)/(2*c*sinh(x) + 2*c*cosh(x)), Eq(b, c)), (-A*sqrt(-b**2
+ c**2)*log(tanh(x/2) + c/b - sqrt(-b**2 + c**2)/b)/(b**2 - c**2) + A*sqrt(-b**2 + c**2)*log(tanh(x/2) + c/b
+ sqrt(-b**2 + c**2)/b)/(b**2 - c**2) + B*b*x/(b**2 - c**2) - B*c*x/(b**2 - c**2) + 2*B*c*log(tanh(x/2) + 1)/(
b**2 - c**2) - B*c*log(tanh(x/2) + c/b - sqrt(-b**2 + c**2)/b)/(b**2 - c**2) - B*c*log(tanh(x/2) + c/b + sqrt(
-b**2 + c**2)/b)/(b**2 - c**2), True))

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Giac [A]  time = 1.13014, size = 108, normalized size = 1.35 \begin{align*} -\frac{B c \log \left (b e^{\left (2 \, x\right )} + c e^{\left (2 \, x\right )} + b - c\right )}{b^{2} - c^{2}} + \frac{2 \, A \arctan \left (\frac{b e^{x} + c e^{x}}{\sqrt{b^{2} - c^{2}}}\right )}{\sqrt{b^{2} - c^{2}}} + \frac{B x}{b - c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-B*c*log(b*e^(2*x) + c*e^(2*x) + b - c)/(b^2 - c^2) + 2*A*arctan((b*e^x + c*e^x)/sqrt(b^2 - c^2))/sqrt(b^2 - c
^2) + B*x/(b - c)