\(\int \frac {1}{x \sqrt {a+a \cos (c+d x)}} \, dx\) [174]

Optimal result

Integrand size = 18, antiderivative size = 18 \[ \int \frac {1}{x \sqrt {a+a \cos (c+d x)}} \, dx=\text {Int}\left (\frac {1}{x \sqrt {a+a \cos (c+d x)}},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.09 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {1}{x \sqrt {a+a \cos (c+d x)}} \, dx=\int \frac {1}{x \sqrt {a+a \cos (c+d x)}} \, dx \]

[In]

Int[1/(x*Sqrt[a + a*Cos[c + d*x]]),x]

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{x \sqrt {a+a \cos (c+d x)}} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 1.09 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {1}{x \sqrt {a+a \cos (c+d x)}} \, dx=\int \frac {1}{x \sqrt {a+a \cos (c+d x)}} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.17 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89

\[\int \frac {1}{x \sqrt {a +\cos \left (d x +c \right ) a}}d x\]

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.67 \[ \int \frac {1}{x \sqrt {a+a \cos (c+d x)}} \, dx=\int { \frac {1}{\sqrt {a \cos \left (d x + c\right ) + a} x} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 1.86 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94 \[ \int \frac {1}{x \sqrt {a+a \cos (c+d x)}} \, dx=\int \frac {1}{x \sqrt {a \left (\cos {\left (c + d x \right )} + 1\right )}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.58 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \sqrt {a+a \cos (c+d x)}} \, dx=\int { \frac {1}{\sqrt {a \cos \left (d x + c\right ) + a} x} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.56 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \sqrt {a+a \cos (c+d x)}} \, dx=\int { \frac {1}{\sqrt {a \cos \left (d x + c\right ) + a} x} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 14.36 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \sqrt {a+a \cos (c+d x)}} \, dx=\int \frac {1}{x\,\sqrt {a+a\,\cos \left (c+d\,x\right )}} \,d x \]

[In]

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

[Out]

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