3.7.92 \(\int \frac {\cosh ^2(x)}{a \cosh (x)+b \sinh (x)} \, dx\) [692]

Optimal. Leaf size=74 \[ -\frac {b^2 \text {ArcTan}\left (\frac {b \cosh (x)+a \sinh (x)}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2}}-\frac {b \cosh (x)}{a^2-b^2}+\frac {a \sinh (x)}{a^2-b^2} \]

[Out]

-b^2*arctan((b*cosh(x)+a*sinh(x))/(a^2-b^2)^(1/2))/(a^2-b^2)^(3/2)-b*cosh(x)/(a^2-b^2)+a*sinh(x)/(a^2-b^2)

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Rubi [A]
time = 0.06, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3179, 2717, 3153, 212} \begin {gather*} -\frac {b^2 \text {ArcTan}\left (\frac {a \sinh (x)+b \cosh (x)}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2}}+\frac {a \sinh (x)}{a^2-b^2}-\frac {b \cosh (x)}{a^2-b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cosh[x]^2/(a*Cosh[x] + b*Sinh[x]),x]

[Out]

-((b^2*ArcTan[(b*Cosh[x] + a*Sinh[x])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(3/2)) - (b*Cosh[x])/(a^2 - b^2) + (a*Sinh
[x])/(a^2 - b^2)

Rule 212

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

Rule 2717

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3153

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 3179

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

Rubi steps

\begin {align*} \int \frac {\cosh ^2(x)}{a \cosh (x)+b \sinh (x)} \, dx &=-\frac {b \cosh (x)}{a^2-b^2}+\frac {a \int \cosh (x) \, dx}{a^2-b^2}-\frac {b^2 \int \frac {1}{a \cosh (x)+b \sinh (x)} \, dx}{a^2-b^2}\\ &=-\frac {b \cosh (x)}{a^2-b^2}+\frac {a \sinh (x)}{a^2-b^2}-\frac {\left (i b^2\right ) \text {Subst}\left (\int \frac {1}{a^2-b^2-x^2} \, dx,x,-i b \cosh (x)-i a \sinh (x)\right )}{a^2-b^2}\\ &=-\frac {b^2 \tan ^{-1}\left (\frac {b \cosh (x)+a \sinh (x)}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2}}-\frac {b \cosh (x)}{a^2-b^2}+\frac {a \sinh (x)}{a^2-b^2}\\ \end {align*}

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Mathematica [A]
time = 0.11, size = 80, normalized size = 1.08 \begin {gather*} -\frac {2 b^2 \text {ArcTan}\left (\frac {b+a \tanh \left (\frac {x}{2}\right )}{\sqrt {a-b} \sqrt {a+b}}\right )}{(a-b)^{3/2} (a+b)^{3/2}}+\frac {b \cosh (x)}{-a^2+b^2}+\frac {a \sinh (x)}{a^2-b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cosh[x]^2/(a*Cosh[x] + b*Sinh[x]),x]

[Out]

(-2*b^2*ArcTan[(b + a*Tanh[x/2])/(Sqrt[a - b]*Sqrt[a + b])])/((a - b)^(3/2)*(a + b)^(3/2)) + (b*Cosh[x])/(-a^2
+ b^2) + (a*Sinh[x])/(a^2 - b^2)

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Maple [A]
time = 1.08, size = 93, normalized size = 1.26

method	result	size
default	\(-\frac {2}{\left (2 b +2 a \right ) \left (\tanh \left (\frac {x}{2}\right )-1\right )}-\frac {2}{\left (2 a -2 b \right ) \left (\tanh \left (\frac {x}{2}\right )+1\right )}-\frac {2 b^{2} \arctan \left (\frac {2 a \tanh \left (\frac {x}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{\left (a -b \right ) \left (a +b \right ) \sqrt {a^{2}-b^{2}}}\)	\(93\)
risch	\(\frac {{\mathrm e}^{x}}{2 b +2 a}-\frac {{\mathrm e}^{-x}}{2 \left (a -b \right )}-\frac {b^{2} \ln \left ({\mathrm e}^{x}+\frac {a -b}{\sqrt {-a^{2}+b^{2}}}\right )}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right ) \left (a -b \right )}+\frac {b^{2} \ln \left ({\mathrm e}^{x}-\frac {a -b}{\sqrt {-a^{2}+b^{2}}}\right )}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right ) \left (a -b \right )}\)	\(122\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(x)^2/(a*cosh(x)+b*sinh(x)),x,method=_RETURNVERBOSE)

[Out]

-2/(2*b+2*a)/(tanh(1/2*x)-1)-2/(2*a-2*b)/(tanh(1/2*x)+1)-2*b^2/(a-b)/(a+b)/(a^2-b^2)^(1/2)*arctan(1/2*(2*a*tan
h(1/2*x)+2*b)/(a^2-b^2)^(1/2))

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)^2/(a*cosh(x)+b*sinh(x)),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*b^2-4*a^2>0)', see `assume?`
for more de

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 190 vs. \(2 (70) = 140\).
time = 0.37, size = 435, normalized size = 5.88 \begin {gather*} \left [-\frac {a^{3} + a^{2} b - a b^{2} - b^{3} - {\left (a^{3} - a^{2} b - a b^{2} + b^{3}\right )} \cosh \left (x\right )^{2} - 2 \, {\left (a^{3} - a^{2} b - a b^{2} + b^{3}\right )} \cosh \left (x\right ) \sinh \left (x\right ) - {\left (a^{3} - a^{2} b - a b^{2} + b^{3}\right )} \sinh \left (x\right )^{2} - 2 \, {\left (b^{2} \cosh \left (x\right ) + b^{2} \sinh \left (x\right )\right )} \sqrt {-a^{2} + b^{2}} \log \left (\frac {{\left (a + b\right )} \cosh \left (x\right )^{2} + 2 \, {\left (a + b\right )} \cosh \left (x\right ) \sinh \left (x\right ) + {\left (a + b\right )} \sinh \left (x\right )^{2} - 2 \, \sqrt {-a^{2} + b^{2}} {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} - a + b}{{\left (a + b\right )} \cosh \left (x\right )^{2} + 2 \, {\left (a + b\right )} \cosh \left (x\right ) \sinh \left (x\right ) + {\left (a + b\right )} \sinh \left (x\right )^{2} + a - b}\right )}{2 \, {\left ({\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \cosh \left (x\right ) + {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \sinh \left (x\right )\right )}}, -\frac {a^{3} + a^{2} b - a b^{2} - b^{3} - {\left (a^{3} - a^{2} b - a b^{2} + b^{3}\right )} \cosh \left (x\right )^{2} - 2 \, {\left (a^{3} - a^{2} b - a b^{2} + b^{3}\right )} \cosh \left (x\right ) \sinh \left (x\right ) - {\left (a^{3} - a^{2} b - a b^{2} + b^{3}\right )} \sinh \left (x\right )^{2} - 4 \, {\left (b^{2} \cosh \left (x\right ) + b^{2} \sinh \left (x\right )\right )} \sqrt {a^{2} - b^{2}} \arctan \left (\frac {\sqrt {a^{2} - b^{2}}}{{\left (a + b\right )} \cosh \left (x\right ) + {\left (a + b\right )} \sinh \left (x\right )}\right )}{2 \, {\left ({\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \cosh \left (x\right ) + {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \sinh \left (x\right )\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)^2/(a*cosh(x)+b*sinh(x)),x, algorithm="fricas")

[Out]

[-1/2*(a^3 + a^2*b - a*b^2 - b^3 - (a^3 - a^2*b - a*b^2 + b^3)*cosh(x)^2 - 2*(a^3 - a^2*b - a*b^2 + b^3)*cosh(
x)*sinh(x) - (a^3 - a^2*b - a*b^2 + b^3)*sinh(x)^2 - 2*(b^2*cosh(x) + b^2*sinh(x))*sqrt(-a^2 + b^2)*log(((a +
b)*cosh(x)^2 + 2*(a + b)*cosh(x)*sinh(x) + (a + b)*sinh(x)^2 - 2*sqrt(-a^2 + b^2)*(cosh(x) + sinh(x)) - a + b)
/((a + b)*cosh(x)^2 + 2*(a + b)*cosh(x)*sinh(x) + (a + b)*sinh(x)^2 + a - b)))/((a^4 - 2*a^2*b^2 + b^4)*cosh(x
) + (a^4 - 2*a^2*b^2 + b^4)*sinh(x)), -1/2*(a^3 + a^2*b - a*b^2 - b^3 - (a^3 - a^2*b - a*b^2 + b^3)*cosh(x)^2
- 2*(a^3 - a^2*b - a*b^2 + b^3)*cosh(x)*sinh(x) - (a^3 - a^2*b - a*b^2 + b^3)*sinh(x)^2 - 4*(b^2*cosh(x) + b^2
*sinh(x))*sqrt(a^2 - b^2)*arctan(sqrt(a^2 - b^2)/((a + b)*cosh(x) + (a + b)*sinh(x))))/((a^4 - 2*a^2*b^2 + b^4
)*cosh(x) + (a^4 - 2*a^2*b^2 + b^4)*sinh(x))]

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 774 vs. \(2 (58) = 116\).
time = 124.52, size = 774, normalized size = 10.46 \begin {gather*} \begin {cases} \tilde {\infty } \left (\frac {\log {\left (\tanh {\left (\frac {x}{2} \right )} \right )} \tanh ^{2}{\left (\frac {x}{2} \right )}}{\tanh ^{2}{\left (\frac {x}{2} \right )} - 1} - \frac {\log {\left (\tanh {\left (\frac {x}{2} \right )} \right )}}{\tanh ^{2}{\left (\frac {x}{2} \right )} - 1} - \frac {2}{\tanh ^{2}{\left (\frac {x}{2} \right )} - 1}\right ) & \text {for}\: a = 0 \wedge b = 0 \\\frac {\frac {\log {\left (\tanh {\left (\frac {x}{2} \right )} \right )} \tanh ^{2}{\left (\frac {x}{2} \right )}}{\tanh ^{2}{\left (\frac {x}{2} \right )} - 1} - \frac {\log {\left (\tanh {\left (\frac {x}{2} \right )} \right )}}{\tanh ^{2}{\left (\frac {x}{2} \right )} - 1} - \frac {2}{\tanh ^{2}{\left (\frac {x}{2} \right )} - 1}}{b} & \text {for}\: a = 0 \\\frac {2 \sinh ^{2}{\left (x \right )}}{- 3 b \sinh {\left (x \right )} + 3 b \cosh {\left (x \right )}} - \frac {2 \sinh {\left (x \right )} \cosh {\left (x \right )}}{- 3 b \sinh {\left (x \right )} + 3 b \cosh {\left (x \right )}} - \frac {\cosh ^{2}{\left (x \right )}}{- 3 b \sinh {\left (x \right )} + 3 b \cosh {\left (x \right )}} & \text {for}\: a = - b \\\frac {2 \sinh ^{2}{\left (x \right )}}{3 b \sinh {\left (x \right )} + 3 b \cosh {\left (x \right )}} + \frac {2 \sinh {\left (x \right )} \cosh {\left (x \right )}}{3 b \sinh {\left (x \right )} + 3 b \cosh {\left (x \right )}} - \frac {\cosh ^{2}{\left (x \right )}}{3 b \sinh {\left (x \right )} + 3 b \cosh {\left (x \right )}} & \text {for}\: a = b \\- \frac {2 a \sqrt {- a^{2} + b^{2}} \tanh {\left (\frac {x}{2} \right )}}{a^{2} \sqrt {- a^{2} + b^{2}} \tanh ^{2}{\left (\frac {x}{2} \right )} - a^{2} \sqrt {- a^{2} + b^{2}} - b^{2} \sqrt {- a^{2} + b^{2}} \tanh ^{2}{\left (\frac {x}{2} \right )} + b^{2} \sqrt {- a^{2} + b^{2}}} - \frac {b^{2} \log {\left (\tanh {\left (\frac {x}{2} \right )} + \frac {b}{a} - \frac {\sqrt {- a^{2} + b^{2}}}{a} \right )} \tanh ^{2}{\left (\frac {x}{2} \right )}}{a^{2} \sqrt {- a^{2} + b^{2}} \tanh ^{2}{\left (\frac {x}{2} \right )} - a^{2} \sqrt {- a^{2} + b^{2}} - b^{2} \sqrt {- a^{2} + b^{2}} \tanh ^{2}{\left (\frac {x}{2} \right )} + b^{2} \sqrt {- a^{2} + b^{2}}} + \frac {b^{2} \log {\left (\tanh {\left (\frac {x}{2} \right )} + \frac {b}{a} - \frac {\sqrt {- a^{2} + b^{2}}}{a} \right )}}{a^{2} \sqrt {- a^{2} + b^{2}} \tanh ^{2}{\left (\frac {x}{2} \right )} - a^{2} \sqrt {- a^{2} + b^{2}} - b^{2} \sqrt {- a^{2} + b^{2}} \tanh ^{2}{\left (\frac {x}{2} \right )} + b^{2} \sqrt {- a^{2} + b^{2}}} + \frac {b^{2} \log {\left (\tanh {\left (\frac {x}{2} \right )} + \frac {b}{a} + \frac {\sqrt {- a^{2} + b^{2}}}{a} \right )} \tanh ^{2}{\left (\frac {x}{2} \right )}}{a^{2} \sqrt {- a^{2} + b^{2}} \tanh ^{2}{\left (\frac {x}{2} \right )} - a^{2} \sqrt {- a^{2} + b^{2}} - b^{2} \sqrt {- a^{2} + b^{2}} \tanh ^{2}{\left (\frac {x}{2} \right )} + b^{2} \sqrt {- a^{2} + b^{2}}} - \frac {b^{2} \log {\left (\tanh {\left (\frac {x}{2} \right )} + \frac {b}{a} + \frac {\sqrt {- a^{2} + b^{2}}}{a} \right )}}{a^{2} \sqrt {- a^{2} + b^{2}} \tanh ^{2}{\left (\frac {x}{2} \right )} - a^{2} \sqrt {- a^{2} + b^{2}} - b^{2} \sqrt {- a^{2} + b^{2}} \tanh ^{2}{\left (\frac {x}{2} \right )} + b^{2} \sqrt {- a^{2} + b^{2}}} + \frac {2 b \sqrt {- a^{2} + b^{2}}}{a^{2} \sqrt {- a^{2} + b^{2}} \tanh ^{2}{\left (\frac {x}{2} \right )} - a^{2} \sqrt {- a^{2} + b^{2}} - b^{2} \sqrt {- a^{2} + b^{2}} \tanh ^{2}{\left (\frac {x}{2} \right )} + b^{2} \sqrt {- a^{2} + b^{2}}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)**2/(a*cosh(x)+b*sinh(x)),x)

[Out]

Piecewise((zoo*(log(tanh(x/2))*tanh(x/2)**2/(tanh(x/2)**2 - 1) - log(tanh(x/2))/(tanh(x/2)**2 - 1) - 2/(tanh(x
/2)**2 - 1)), Eq(a, 0) & Eq(b, 0)), ((log(tanh(x/2))*tanh(x/2)**2/(tanh(x/2)**2 - 1) - log(tanh(x/2))/(tanh(x/
2)**2 - 1) - 2/(tanh(x/2)**2 - 1))/b, Eq(a, 0)), (2*sinh(x)**2/(-3*b*sinh(x) + 3*b*cosh(x)) - 2*sinh(x)*cosh(x
)/(-3*b*sinh(x) + 3*b*cosh(x)) - cosh(x)**2/(-3*b*sinh(x) + 3*b*cosh(x)), Eq(a, -b)), (2*sinh(x)**2/(3*b*sinh(
x) + 3*b*cosh(x)) + 2*sinh(x)*cosh(x)/(3*b*sinh(x) + 3*b*cosh(x)) - cosh(x)**2/(3*b*sinh(x) + 3*b*cosh(x)), Eq
(a, b)), (-2*a*sqrt(-a**2 + b**2)*tanh(x/2)/(a**2*sqrt(-a**2 + b**2)*tanh(x/2)**2 - a**2*sqrt(-a**2 + b**2) -
b**2*sqrt(-a**2 + b**2)*tanh(x/2)**2 + b**2*sqrt(-a**2 + b**2)) - b**2*log(tanh(x/2) + b/a - sqrt(-a**2 + b**2
)/a)*tanh(x/2)**2/(a**2*sqrt(-a**2 + b**2)*tanh(x/2)**2 - a**2*sqrt(-a**2 + b**2) - b**2*sqrt(-a**2 + b**2)*ta
nh(x/2)**2 + b**2*sqrt(-a**2 + b**2)) + b**2*log(tanh(x/2) + b/a - sqrt(-a**2 + b**2)/a)/(a**2*sqrt(-a**2 + b*
*2)*tanh(x/2)**2 - a**2*sqrt(-a**2 + b**2) - b**2*sqrt(-a**2 + b**2)*tanh(x/2)**2 + b**2*sqrt(-a**2 + b**2)) +
b**2*log(tanh(x/2) + b/a + sqrt(-a**2 + b**2)/a)*tanh(x/2)**2/(a**2*sqrt(-a**2 + b**2)*tanh(x/2)**2 - a**2*sq
rt(-a**2 + b**2) - b**2*sqrt(-a**2 + b**2)*tanh(x/2)**2 + b**2*sqrt(-a**2 + b**2)) - b**2*log(tanh(x/2) + b/a
+ sqrt(-a**2 + b**2)/a)/(a**2*sqrt(-a**2 + b**2)*tanh(x/2)**2 - a**2*sqrt(-a**2 + b**2) - b**2*sqrt(-a**2 + b*
*2)*tanh(x/2)**2 + b**2*sqrt(-a**2 + b**2)) + 2*b*sqrt(-a**2 + b**2)/(a**2*sqrt(-a**2 + b**2)*tanh(x/2)**2 - a
**2*sqrt(-a**2 + b**2) - b**2*sqrt(-a**2 + b**2)*tanh(x/2)**2 + b**2*sqrt(-a**2 + b**2)), True))

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Giac [A]
time = 0.41, size = 61, normalized size = 0.82 \begin {gather*} -\frac {2 \, b^{2} \arctan \left (\frac {a e^{x} + b e^{x}}{\sqrt {a^{2} - b^{2}}}\right )}{{\left (a^{2} - b^{2}\right )}^{\frac {3}{2}}} - \frac {e^{\left (-x\right )}}{2 \, {\left (a - b\right )}} + \frac {e^{x}}{2 \, {\left (a + b\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)^2/(a*cosh(x)+b*sinh(x)),x, algorithm="giac")

[Out]

-2*b^2*arctan((a*e^x + b*e^x)/sqrt(a^2 - b^2))/(a^2 - b^2)^(3/2) - 1/2*e^(-x)/(a - b) + 1/2*e^x/(a + b)

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Mupad [B]
time = 1.72, size = 157, normalized size = 2.12 \begin {gather*} \frac {{\mathrm {e}}^x}{2\,a+2\,b}-\frac {{\mathrm {e}}^{-x}}{2\,a-2\,b}-\frac {b^2\,\ln \left (-\frac {2\,b^2}{{\left (a+b\right )}^{5/2}\,\sqrt {b-a}}-\frac {2\,b^2\,{\mathrm {e}}^x}{-a^3-a^2\,b+a\,b^2+b^3}\right )}{{\left (a+b\right )}^{3/2}\,{\left (b-a\right )}^{3/2}}+\frac {b^2\,\ln \left (\frac {2\,b^2}{{\left (a+b\right )}^{5/2}\,\sqrt {b-a}}-\frac {2\,b^2\,{\mathrm {e}}^x}{-a^3-a^2\,b+a\,b^2+b^3}\right )}{{\left (a+b\right )}^{3/2}\,{\left (b-a\right )}^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(x)^2/(a*cosh(x) + b*sinh(x)),x)

[Out]

exp(x)/(2*a + 2*b) - exp(-x)/(2*a - 2*b) - (b^2*log(- (2*b^2)/((a + b)^(5/2)*(b - a)^(1/2)) - (2*b^2*exp(x))/(
a*b^2 - a^2*b - a^3 + b^3)))/((a + b)^(3/2)*(b - a)^(3/2)) + (b^2*log((2*b^2)/((a + b)^(5/2)*(b - a)^(1/2)) -
(2*b^2*exp(x))/(a*b^2 - a^2*b - a^3 + b^3)))/((a + b)^(3/2)*(b - a)^(3/2))

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