3.6.95 \(\int \frac {\cosh (x)}{\sqrt {\cosh (2 x)}} \, dx\) [595]

Optimal. Leaf size=15 \[ \frac {\sinh ^{-1}\left (\sqrt {2} \sinh (x)\right )}{\sqrt {2}} \]

[Out]

1/2*arcsinh(sinh(x)*2^(1/2))*2^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {4441, 221} \begin {gather*} \frac {\sinh ^{-1}\left (\sqrt {2} \sinh (x)\right )}{\sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cosh[x]/Sqrt[Cosh[2*x]],x]

[Out]

ArcSinh[Sqrt[2]*Sinh[x]]/Sqrt[2]

Rule 221

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt[a])]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rule 4441

Int[(u_)*(F_)[(c_.)*((a_.) + (b_.)*(x_))], x_Symbol] :> With[{d = FreeFactors[Sin[c*(a + b*x)], x]}, Dist[d/(b
*c), Subst[Int[SubstFor[1, Sin[c*(a + b*x)]/d, u, x], x], x, Sin[c*(a + b*x)]/d], x] /; FunctionOfQ[Sin[c*(a +
b*x)]/d, u, x]] /; FreeQ[{a, b, c}, x] && (EqQ[F, Cos] || EqQ[F, cos])

Rubi steps

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

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Mathematica [A]
time = 0.01, size = 15, normalized size = 1.00 \begin {gather*} \frac {\sinh ^{-1}\left (\sqrt {2} \sinh (x)\right )}{\sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cosh[x]/Sqrt[Cosh[2*x]],x]

[Out]

ArcSinh[Sqrt[2]*Sinh[x]]/Sqrt[2]

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(62\) vs. \(2(12)=24\).
time = 0.17, size = 63, normalized size = 4.20

method	result	size
default	\(\frac {\sqrt {\left (2 \left (\cosh ^{2}\left (x \right )\right )-1\right ) \left (\sinh ^{2}\left (x \right )\right )}\, \ln \left (\sqrt {2}\, \left (\sinh ^{2}\left (x \right )\right )+\sqrt {2 \left (\sinh ^{4}\left (x \right )\right )+\sinh ^{2}\left (x \right )}+\frac {\sqrt {2}}{4}\right ) \sqrt {2}}{4 \sinh \left (x \right ) \sqrt {2 \left (\cosh ^{2}\left (x \right )\right )-1}}\)	\(63\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(x)/cosh(2*x)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/4*((2*cosh(x)^2-1)*sinh(x)^2)^(1/2)*ln(2^(1/2)*sinh(x)^2+(2*sinh(x)^4+sinh(x)^2)^(1/2)+1/4*2^(1/2))*2^(1/2)/
sinh(x)/(2*cosh(x)^2-1)^(1/2)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)/cosh(2*x)^(1/2),x, algorithm="maxima")

[Out]

integrate(cosh(x)/sqrt(cosh(2*x)), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 482 vs. \(2 (12) = 24\).
time = 0.88, size = 482, normalized size = 32.13 \begin {gather*} \frac {1}{8} \, \sqrt {2} \log \left (-\frac {\cosh \left (x\right )^{8} + 8 \, \cosh \left (x\right ) \sinh \left (x\right )^{7} + \sinh \left (x\right )^{8} + {\left (28 \, \cosh \left (x\right )^{2} - 3\right )} \sinh \left (x\right )^{6} - 3 \, \cosh \left (x\right )^{6} + 2 \, {\left (28 \, \cosh \left (x\right )^{3} - 9 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{5} + 5 \, {\left (14 \, \cosh \left (x\right )^{4} - 9 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{4} + 5 \, \cosh \left (x\right )^{4} + 4 \, {\left (14 \, \cosh \left (x\right )^{5} - 15 \, \cosh \left (x\right )^{3} + 5 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + {\left (28 \, \cosh \left (x\right )^{6} - 45 \, \cosh \left (x\right )^{4} + 30 \, \cosh \left (x\right )^{2} - 4\right )} \sinh \left (x\right )^{2} + \sqrt {2} {\left (\cosh \left (x\right )^{6} + 6 \, \cosh \left (x\right ) \sinh \left (x\right )^{5} + \sinh \left (x\right )^{6} + 3 \, {\left (5 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{4} - 3 \, \cosh \left (x\right )^{4} + 4 \, {\left (5 \, \cosh \left (x\right )^{3} - 3 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + {\left (15 \, \cosh \left (x\right )^{4} - 18 \, \cosh \left (x\right )^{2} + 4\right )} \sinh \left (x\right )^{2} + 4 \, \cosh \left (x\right )^{2} + 2 \, {\left (3 \, \cosh \left (x\right )^{5} - 6 \, \cosh \left (x\right )^{3} + 4 \, \cosh \left (x\right )\right )} \sinh \left (x\right ) - 4\right )} \sqrt {\frac {\cosh \left (x\right )^{2} + \sinh \left (x\right )^{2}}{\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}} - 4 \, \cosh \left (x\right )^{2} + 2 \, {\left (4 \, \cosh \left (x\right )^{7} - 9 \, \cosh \left (x\right )^{5} + 10 \, \cosh \left (x\right )^{3} - 4 \, \cosh \left (x\right )\right )} \sinh \left (x\right ) + 4}{\cosh \left (x\right )^{6} + 6 \, \cosh \left (x\right )^{5} \sinh \left (x\right ) + 15 \, \cosh \left (x\right )^{4} \sinh \left (x\right )^{2} + 20 \, \cosh \left (x\right )^{3} \sinh \left (x\right )^{3} + 15 \, \cosh \left (x\right )^{2} \sinh \left (x\right )^{4} + 6 \, \cosh \left (x\right ) \sinh \left (x\right )^{5} + \sinh \left (x\right )^{6}}\right ) + \frac {1}{8} \, \sqrt {2} \log \left (\frac {\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + {\left (6 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{2} + \sqrt {2} {\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 1\right )} \sqrt {\frac {\cosh \left (x\right )^{2} + \sinh \left (x\right )^{2}}{\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}} + \cosh \left (x\right )^{2} + 2 \, {\left (2 \, \cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1}{\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)/cosh(2*x)^(1/2),x, algorithm="fricas")

[Out]

1/8*sqrt(2)*log(-(cosh(x)^8 + 8*cosh(x)*sinh(x)^7 + sinh(x)^8 + (28*cosh(x)^2 - 3)*sinh(x)^6 - 3*cosh(x)^6 + 2
*(28*cosh(x)^3 - 9*cosh(x))*sinh(x)^5 + 5*(14*cosh(x)^4 - 9*cosh(x)^2 + 1)*sinh(x)^4 + 5*cosh(x)^4 + 4*(14*cos
h(x)^5 - 15*cosh(x)^3 + 5*cosh(x))*sinh(x)^3 + (28*cosh(x)^6 - 45*cosh(x)^4 + 30*cosh(x)^2 - 4)*sinh(x)^2 + sq
rt(2)*(cosh(x)^6 + 6*cosh(x)*sinh(x)^5 + sinh(x)^6 + 3*(5*cosh(x)^2 - 1)*sinh(x)^4 - 3*cosh(x)^4 + 4*(5*cosh(x
)^3 - 3*cosh(x))*sinh(x)^3 + (15*cosh(x)^4 - 18*cosh(x)^2 + 4)*sinh(x)^2 + 4*cosh(x)^2 + 2*(3*cosh(x)^5 - 6*co
sh(x)^3 + 4*cosh(x))*sinh(x) - 4)*sqrt((cosh(x)^2 + sinh(x)^2)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2)) -
4*cosh(x)^2 + 2*(4*cosh(x)^7 - 9*cosh(x)^5 + 10*cosh(x)^3 - 4*cosh(x))*sinh(x) + 4)/(cosh(x)^6 + 6*cosh(x)^5*s
inh(x) + 15*cosh(x)^4*sinh(x)^2 + 20*cosh(x)^3*sinh(x)^3 + 15*cosh(x)^2*sinh(x)^4 + 6*cosh(x)*sinh(x)^5 + sinh
(x)^6)) + 1/8*sqrt(2)*log((cosh(x)^4 + 4*cosh(x)*sinh(x)^3 + sinh(x)^4 + (6*cosh(x)^2 + 1)*sinh(x)^2 + sqrt(2)
*(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 + 1)*sqrt((cosh(x)^2 + sinh(x)^2)/(cosh(x)^2 - 2*cosh(x)*sinh(x) +
sinh(x)^2)) + cosh(x)^2 + 2*(2*cosh(x)^3 + cosh(x))*sinh(x) + 1)/(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\cosh {\left (x \right )}}{\sqrt {\cosh {\left (2 x \right )}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)/cosh(2*x)**(1/2),x)

[Out]

Integral(cosh(x)/sqrt(cosh(2*x)), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (12) = 24\).
time = 0.54, size = 58, normalized size = 3.87 \begin {gather*} -\frac {1}{4} \, \sqrt {2} {\left (\log \left (\sqrt {e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )} + 1\right ) + \log \left (\sqrt {e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )}\right ) - \log \left (-\sqrt {e^{\left (4 \, x\right )} + 1} + e^{\left (2 \, x\right )} + 1\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)/cosh(2*x)^(1/2),x, algorithm="giac")

[Out]

-1/4*sqrt(2)*(log(sqrt(e^(4*x) + 1) - e^(2*x) + 1) + log(sqrt(e^(4*x) + 1) - e^(2*x)) - log(-sqrt(e^(4*x) + 1)
+ e^(2*x) + 1))

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.07 \begin {gather*} \int \frac {\mathrm {cosh}\left (x\right )}{\sqrt {\mathrm {cosh}\left (2\,x\right )}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(x)/cosh(2*x)^(1/2),x)

[Out]

int(cosh(x)/cosh(2*x)^(1/2), x)

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