3.143 \(\int \frac {1}{x^2 \sqrt {a+a \cosh (c+d x)}} \, dx\)

Optimal. Leaf size=21 \[ \text {Int}\left (\frac {1}{x^2 \sqrt {a \cosh (c+d x)+a}},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.07, 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 = {} \[ \int \frac {1}{x^2 \sqrt {a+a \cosh (c+d x)}} \, dx \]

Verification is Not applicable to the result.

[In]

Int[1/(x^2*Sqrt[a + a*Cosh[c + d*x]]),x]

[Out]

Defer[Int][1/(x^2*Sqrt[a + a*Cosh[c + d*x]]), x]

Rubi steps

\begin {align*} \int \frac {1}{x^2 \sqrt {a+a \cosh (c+d x)}} \, dx &=\int \frac {1}{x^2 \sqrt {a+a \cosh (c+d x)}} \, dx\\ \end {align*}

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Mathematica [A]  time = 1.76, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^2 \sqrt {a+a \cosh (c+d x)}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[1/(x^2*Sqrt[a + a*Cosh[c + d*x]]),x]

[Out]

Integrate[1/(x^2*Sqrt[a + a*Cosh[c + d*x]]), x]

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fricas [A]  time = 0.63, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {a \cosh \left (d x + c\right ) + a}}{a x^{2} \cosh \left (d x + c\right ) + a x^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(a*cosh(d*x + c) + a)/(a*x^2*cosh(d*x + c) + a*x^2), x)

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a \cosh \left (d x + c\right ) + a} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.10, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{2} \sqrt {a +a \cosh \left (d x +c \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a \cosh \left (d x + c\right ) + a} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [A]  time = 0.00, size = -1, normalized size = -0.05 \[ \int \frac {1}{x^2\,\sqrt {a+a\,\mathrm {cosh}\left (c+d\,x\right )}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{2} \sqrt {a \left (\cosh {\left (c + d x \right )} + 1\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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