3.2.42 \(\int \frac {1}{x \sqrt {a+a \cosh (c+d x)}} \, dx\) [142]

Optimal. Leaf size=21 \[ \text {Int}\left (\frac {1}{x \sqrt {a+a \cosh (c+d x)}},x\right ) \]

[Out]

Unintegrable(1/x/(a+a*cosh(d*x+c))^(1/2),x)

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Rubi [A]
time = 0.05, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {1}{x \sqrt {a+a \cosh (c+d x)}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[1/(x*Sqrt[a + a*Cosh[c + d*x]]),x]

[Out]

Defer[Int][1/(x*Sqrt[a + a*Cosh[c + d*x]]), x]

Rubi steps

\begin {align*} \int \frac {1}{x \sqrt {a+a \cosh (c+d x)}} \, dx &=\int \frac {1}{x \sqrt {a+a \cosh (c+d x)}} \, dx\\ \end {align*}

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Mathematica [A]
time = 1.96, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x \sqrt {a+a \cosh (c+d x)}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[1/(x*Sqrt[a + a*Cosh[c + d*x]]),x]

[Out]

Integrate[1/(x*Sqrt[a + a*Cosh[c + d*x]]), x]

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Maple [A]
time = 0.43, size = 0, normalized size = 0.00 \[\int \frac {1}{x \sqrt {a +a \cosh \left (d x +c \right )}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/(a+a*cosh(d*x+c))^(1/2),x)

[Out]

int(1/x/(a+a*cosh(d*x+c))^(1/2),x)

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Maxima [A]
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(1/x/(a+a*cosh(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(a*cosh(d*x + c) + a)*x), x)

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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(a+a*cosh(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(a*cosh(d*x + c) + a)/(a*x*cosh(d*x + c) + a*x), x)

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x \sqrt {a \left (\cosh {\left (c + d x \right )} + 1\right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(a+a*cosh(d*x+c))**(1/2),x)

[Out]

Integral(1/(x*sqrt(a*(cosh(c + d*x) + 1))), x)

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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(a+a*cosh(d*x+c))^(1/2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(a*cosh(d*x + c) + a)*x), x)

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \int \frac {1}{x\,\sqrt {a+a\,\mathrm {cosh}\left (c+d\,x\right )}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(a + a*cosh(c + d*x))^(1/2)),x)

[Out]

int(1/(x*(a + a*cosh(c + d*x))^(1/2)), x)

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