∫ cosh(x)__∘ cosh (2x) dx

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

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Rubi [A] time = 0.02, 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 = {4356, 215} \[ \frac {\sinh ^{-1}\left (\sqrt {2} \sinh (x)\right )}{\sqrt {2}} \]

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

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 +

\begin {align*} \int \frac {\cosh (x)}{\sqrt {\cosh (2 x)}} \, dx &=\operatorname {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 \[ \frac {\sinh ^{-1}\left (\sqrt {2} \sinh (x)\right )}{\sqrt {2}} \]

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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cosh (x)}{\sqrt {\cosh (2 x)}} \, dx \]

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fricas [B] time = 0.97, size = 482, normalized size = 32.13 \[ \frac {1}{8} \, \sqrt {2} \log \left (-\frac {\cosh \relax (x)^{8} + 8 \, \cosh \relax (x) \sinh \relax (x)^{7} + \sinh \relax (x)^{8} + {\left (28 \, \cosh \relax (x)^{2} - 3\right )} \sinh \relax (x)^{6} - 3 \, \cosh \relax (x)^{6} + 2 \, {\left (28 \, \cosh \relax (x)^{3} - 9 \, \cosh \relax (x)\right )} \sinh \relax (x)^{5} + 5 \, {\left (14 \, \cosh \relax (x)^{4} - 9 \, \cosh \relax (x)^{2} + 1\right )} \sinh \relax (x)^{4} + 5 \, \cosh \relax (x)^{4} + 4 \, {\left (14 \, \cosh \relax (x)^{5} - 15 \, \cosh \relax (x)^{3} + 5 \, \cosh \relax (x)\right )} \sinh \relax (x)^{3} + {\left (28 \, \cosh \relax (x)^{6} - 45 \, \cosh \relax (x)^{4} + 30 \, \cosh \relax (x)^{2} - 4\right )} \sinh \relax (x)^{2} + \sqrt {2} {\left (\cosh \relax (x)^{6} + 6 \, \cosh \relax (x) \sinh \relax (x)^{5} + \sinh \relax (x)^{6} + 3 \, {\left (5 \, \cosh \relax (x)^{2} - 1\right )} \sinh \relax (x)^{4} - 3 \, \cosh \relax (x)^{4} + 4 \, {\left (5 \, \cosh \relax (x)^{3} - 3 \, \cosh \relax (x)\right )} \sinh \relax (x)^{3} + {\left (15 \, \cosh \relax (x)^{4} - 18 \, \cosh \relax (x)^{2} + 4\right )} \sinh \relax (x)^{2} + 4 \, \cosh \relax (x)^{2} + 2 \, {\left (3 \, \cosh \relax (x)^{5} - 6 \, \cosh \relax (x)^{3} + 4 \, \cosh \relax (x)\right )} \sinh \relax (x) - 4\right )} \sqrt {\frac {\cosh \relax (x)^{2} + \sinh \relax (x)^{2}}{\cosh \relax (x)^{2} - 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2}}} - 4 \, \cosh \relax (x)^{2} + 2 \, {\left (4 \, \cosh \relax (x)^{7} - 9 \, \cosh \relax (x)^{5} + 10 \, \cosh \relax (x)^{3} - 4 \, \cosh \relax (x)\right )} \sinh \relax (x) + 4}{\cosh \relax (x)^{6} + 6 \, \cosh \relax (x)^{5} \sinh \relax (x) + 15 \, \cosh \relax (x)^{4} \sinh \relax (x)^{2} + 20 \, \cosh \relax (x)^{3} \sinh \relax (x)^{3} + 15 \, \cosh \relax (x)^{2} \sinh \relax (x)^{4} + 6 \, \cosh \relax (x) \sinh \relax (x)^{5} + \sinh \relax (x)^{6}}\right ) + \frac {1}{8} \, \sqrt {2} \log \left (\frac {\cosh \relax (x)^{4} + 4 \, \cosh \relax (x) \sinh \relax (x)^{3} + \sinh \relax (x)^{4} + {\left (6 \, \cosh \relax (x)^{2} + 1\right )} \sinh \relax (x)^{2} + \sqrt {2} {\left (\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2} + 1\right )} \sqrt {\frac {\cosh \relax (x)^{2} + \sinh \relax (x)^{2}}{\cosh \relax (x)^{2} - 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2}}} + \cosh \relax (x)^{2} + 2 \, {\left (2 \, \cosh \relax (x)^{3} + \cosh \relax (x)\right )} \sinh \relax (x) + 1}{\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2}}\right ) \]

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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giac [B] time = 0.66, size = 58, normalized size = 3.87 \[ -\frac {1}{4} \, \sqrt {2} {\left (\log \left (\sqrt {e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )} + 1\right ) + \log \left (\sqrt {e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )}\right ) - \log \left (-\sqrt {e^{\left (4 \, x\right )} + 1} + e^{\left (2 \, x\right )} + 1\right )\right )} \]

-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)

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method	result	size

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

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

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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cosh \relax (x)}{\sqrt {\cosh \left (2 \, x\right )}}\,{d x} \]

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mupad [F] time = 0.00, size = -1, normalized size = -0.07 \[ \int \frac {\mathrm {cosh}\relax (x)}{\sqrt {\mathrm {cosh}\left (2\,x\right )}} \,d x \]

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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cosh {\relax (x )}}{\sqrt {\cosh {\left (2 x \right )}}}\, dx \]

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3.595 \(\int \frac {\cosh (x)}{\sqrt {\cosh (2 x)}} \, dx\)